Research Article Generalized ( ,,) -Cocoercive Operators...
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Hindawi Publishing Corporation Journal of Mathematics Volume 2013, Article ID 738491, 10 pages http://dx.doi.org/10.1155/2013/738491 Research Article Generalized (⋅, ⋅, ⋅)--Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions Shamshad Husain, 1 Sanjeev Gupta, 1 and Vishnu Narayan Mishra 2 1 Department of Applied Mathematics, Faculty of Engineering & Technology, Aligarh Muslim University, Aligarh 202002, India 2 Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat 395 007, India Correspondence should be addressed to Sanjeev Gupta; [email protected] Received 18 March 2013; Revised 4 May 2013; Accepted 8 May 2013 Academic Editor: Kaleem R. Kazmi Copyright © 2013 Shamshad Husain et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate a new class of cocoercive operators named generalized (⋅, ⋅, ⋅)--cocoercive operators in Hilbert spaces. We prove that generalized (⋅, ⋅, ⋅)--cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with (⋅, ⋅)-cocoercive operators to the generalized (⋅, ⋅, ⋅)--cocoercive operators. Some examples are given to justify the definition of generalized (⋅, ⋅, ⋅)--cocoercive operators. Further, we consider a generalized set-valued variational- like inclusion problem involving generalized (⋅, ⋅, ⋅)--cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature. 1. Introduction Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years. One of the most interesting and important problems in the theory of variational inclusions include variational, quasi- variational, and variational-like inequalities as special cases. For applications of variational inclusions, see, for example, [1]. Various kinds of iterative methods have been studied to solve the variational inclusions. Among these methods, the resolvent operator technique for the study of variational inclusions has been widely used by many authors. For details, we refer to [2–16]. Recently Fang and Huang, Kazmi and Khan, and Lan et al. investigated several resolvent operators for generalized operators such as -monotone [5], -accretive [6], (, )- proximal point [11], (, )-accretive [12], (, )-monotone [7], (, )-accretive [13], and mappings. Very recently, Zou and Huang [16] introduced and studied (⋅, ⋅)-accretive oper- ators, Kazmi et al. [8–10] introduced and studied generalized (⋅, ⋅)-accretive operators, (⋅, ⋅)--proximal point map- ping, Xu and Wang [15] introduced and studied ((⋅, ⋅), )- monotone operators, and Ahmad et al. [2] introduced and studied (⋅, ⋅)-cocoercive operators. Motivated by the recent work going in this direction, we consider a class of cocoercive operators called generalized (⋅, ⋅, ⋅)--cocoercive, a natural generalization of monotone (accretive) operators in Hilbert (Banach) spaces. For details, we refer to [2, 5–7, 13–16]. We prove that generalized (⋅, ⋅, ⋅)- -cocoercive operator is single-valued and Lipschitz continu- ous and extends the concept of resolvent operators associated with (⋅, ⋅)-cocoercive operators to the generalized (⋅, ⋅, ⋅)- -cocoercive operators. Further, we consider the general- ized set-valued variational-like inclusion problem involving generalized (⋅, ⋅, ⋅)--cocoercive operator in Hilbert spaces. Using new a resolvent operator technique, we prove the existence of solutions and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Fur- thermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can
Transcript of Research Article Generalized ( ,,) -Cocoercive Operators...
Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 738491 10 pageshttpdxdoiorg1011552013738491
Research ArticleGeneralized119867(sdot sdot sdot)-120578-Cocoercive Operators andGeneralized Set-Valued Variational-Like Inclusions
Shamshad Husain1 Sanjeev Gupta1 and Vishnu Narayan Mishra2
1 Department of Applied Mathematics Faculty of Engineering amp Technology Aligarh Muslim University Aligarh 202002 India2Department of Applied Mathematics amp Humanities Sardar Vallabhbhai National Institute of TechnologyIchchhanath Mahadev Road Surat 395 007 India
Correspondence should be addressed to Sanjeev Gupta guptasanmpgmailcom
Received 18 March 2013 Revised 4 May 2013 Accepted 8 May 2013
Academic Editor Kaleem R Kazmi
Copyright copy 2013 Shamshad Husain et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We investigate a new class of cocoercive operators named generalized119867(sdot sdot sdot)-120578-cocoercive operators in Hilbert spaces We provethat generalized 119867(sdot sdot sdot)-120578-cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolventoperators associated with119867(sdot sdot)-cocoercive operators to the generalized119867(sdot sdot sdot)-120578-cocoercive operators Some examples are givento justify the definition of generalized 119867(sdot sdot sdot)-120578-cocoercive operators Further we consider a generalized set-valued variational-like inclusion problem involving generalized 119867(sdot sdot sdot)-120578-cocoercive operator In terms of the new resolvent operator techniquewe give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusionsFurthermore we discuss the convergence criteria of iterative algorithm under some suitable conditions Our results can be viewedas a generalization of some known results in the literature
1 Introduction
Variational inclusions as the generalization of variationalinequalities have been widely studied in recent years Oneof the most interesting and important problems in thetheory of variational inclusions include variational quasi-variational and variational-like inequalities as special casesFor applications of variational inclusions see for example[1] Various kinds of iterative methods have been studiedto solve the variational inclusions Among these methodsthe resolvent operator technique for the study of variationalinclusions has been widely used bymany authors For detailswe refer to [2ndash16]
Recently Fang and Huang Kazmi and Khan and Lanet al investigated several resolvent operators for generalizedoperators such as 119867-monotone [5] 119867-accretive [6] (119875 120578)-proximal point [11] (119875 120578)-accretive [12] (119867 120578)-monotone[7] (119860 120578)-accretive [13] and mappings Very recently ZouandHuang [16] introduced and studied119867(sdot sdot)-accretive oper-ators Kazmi et al [8ndash10] introduced and studied generalized
119867(sdot sdot)-accretive operators 119867(sdot sdot)-120578-proximal point map-ping Xu and Wang [15] introduced and studied (119867(sdot sdot) 120578)-monotone operators and Ahmad et al [2] introduced andstudied119867(sdot sdot)-cocoercive operators
Motivated by the recent work going in this direction weconsider a class of cocoercive operators called generalized119867(sdot sdot sdot)-120578-cocoercive a natural generalization of monotone(accretive) operators in Hilbert (Banach) spaces For detailswe refer to [2 5ndash7 13ndash16]We prove that generalized119867(sdot sdot sdot)-120578-cocoercive operator is single-valued and Lipschitz continu-ous and extends the concept of resolvent operators associatedwith 119867(sdot sdot)-cocoercive operators to the generalized 119867(sdot sdot sdot)-120578-cocoercive operators Further we consider the general-ized set-valued variational-like inclusion problem involvinggeneralized119867(sdot sdot sdot)-120578-cocoercive operator in Hilbert spacesUsing new a resolvent operator technique we prove theexistence of solutions and suggest an iterative algorithm forthe generalized set-valued variational-like inclusions Fur-thermore we discuss the convergence criteria of the iterativealgorithm under some suitable conditions Our results can
2 Journal of Mathematics
be viewed as an extension and generalization of some knownresults [2 15 16] For illustration of Definitions 2 and 7 andexamples 23 and 32 are given respectively
2 Preliminaries
Throughout this paper we suppose that 119883 is a real Hilbertspace endowedwith a norm sdot and an inner product ⟨sdot sdot⟩ 2119883(resp CB(119883)) is the family of all the nonempty (resp closedand bounded) subsets of119883 andD(sdot sdot) is theHausdorffmetricon CB(119883) defined by
D (119875 119876) = max sup119909isin119875
119889 (119909 119876) sup119910isin119876
119889 (119875 119910)
forall119875 119876 isin CB (119883)
(1)
where 119889(119909 119876) = inf119910isin119876
119909 minus 119910 and 119889(119875 119910) = inf119909isin119875
119909 minus 119910In the sequel let us recall some concepts
Definition 1 (see [4 17]) Let119875 119883 rarr 119883 and 120578 119883times119883 rarr 119883
be two mappings Then 119875 is said to be
(i) 120578-monotone if
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 0 forall119909 119910 isin 119883 (2)
(ii) 1205751-120578-stronglymonotone if there exists a constant 120575
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205751
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2 forall119909 119910 isin 119883 (3)
(iii) 1205831-120578-cocoercive if there exists a constant 120583
1gt 0 such
that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205831
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817
2
forall119909 119910 isin 119883
(4)
(iv) 1205741-120578-relaxed cocoercive if there exists a constant 120574
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge (minus1205741)1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817
2
forall119909 119910 isin 119883
(5)
(v) 120582119875-Lipschitz continuous if there exists a constant
120582119875gt 0 such that
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817 le 120582
119875
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (6)
(vi) 120572-expansive if there exists a constant 120572 gt 0 such that1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817 ge 1205721003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (7)
if 120572 = 1 then it is expansive(vii) 120578 is said to be 120591-Lipschitz continuous if there exists a
constant 120591 gt 0 such that1003817100381710038171003817120578 (119909 119910)
1003817100381710038171003817 le 1205911003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (8)
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119883 then definitions(i) to (iv) reduce to the Definitions of monotonicity strongmonotonicity [18] cocoercivity [19] and relaxed cocoerciverespectively
Definition 2 Let 119875119876 119877 119883 rarr 119883 120578 119883 times 119883 rarr 119883 and119867 119883 times 119883 times 119883 rarr 119883 be the single-valued mappings Then
(i) 119867(119875 sdot sdot) is said to be 120583-120578-cocoercive with respect to 119875if there exists a constant 120583 gt 0 such that
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge 1205831003817100381710038171003817119875119909 minus 119875119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(9)
(ii) 119867(sdot 119876 sdot) is said to be 120574-120578-relaxed cocoercive withrespect to 119876 if there exists a constant 120574 gt 0 such that
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge (minus120574)1003817100381710038171003817119876119909 minus 119876119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(10)
(iii) 119867(sdot sdot 119877) is said to be 120575-120578-strongly monotone withrespect to 119877 if there exists a constant 120575 gt 0 such that
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 1205751003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(11)
(iv) 119867(119875 sdot sdot) is said to be 1199031-Lipschitz continuous with
respect to 119875 if there exists a constant 1199031gt 0 such that
1003817100381710038171003817119867 (119875119909 sdot sdot) minus 119867 (119875119910 sdot sdot)1003817100381710038171003817 le 1199031
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (12)
(v) 119867(sdot 119876 sdot) is said to be 1199032-Lipschitz continuous with
respect to 119876 if there exists a constant 1199032gt 0 such that
1003817100381710038171003817119867 (sdot 119876119909 sdot) minus 119867 (sdot 119876119910 sdot)1003817100381710038171003817 le 1199032
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (13)
(vi) 119867(sdot sdot 119877) is said to be 1199033-Lipschitz continuous with
respect to 119877 if there exists a constant 1199033gt 0 such that
1003817100381710038171003817119867 (sdot sdot 119877119909) minus 119867 (sdot sdot 119877119910)1003817100381710038171003817 le 1199033
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (14)
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119867 and 119867(sdot sdot sdot) =
119867(sdot sdot) then Definitions (i)-(ii) are reduced to the definitionof cocoercivity and relaxed cocoercive [2] respectively and(iii) reduces to strong accretivity [16]
Example 3 Let 119883 = R2 with usual inner product Let 120578
R2 timesR2 rarr R2 119875 119876 119877 R2 rarr R2 be defined by
120578 (119909 119910) = (1198981199091+ 1198991199092
minus1198981199101minus 1198991199102
) 119875119909 = (1198981199091minus 1198981199092
minus1198981199091+ 1198991199092
)
119876119910 = (minus1198981199101+ 1198981199102
minus1198991199102
) 119877119911 = (1198991199111
1198981199112
)
(15)
Journal of Mathematics 3
for all scalers 119899119898 isin Rwith 119899 gt 119898 and for all119909 = (1199091 1199092) 119910 =
(1199101 1199102) 119911 = (119911
1 1199112) isin R2
Suppose that119867(119875119909119876119910 119877119911) = 119875119909 +119876119910+119877119911 Then119867(119875
119876 119877) is (12)-120578-cocoercive with respect to 119875 (12)-120578-relaxedcocoercive with respect to 119876 1198982-120578-strongly monotone withrespect to 119877 and 120578(119909 119910) is 119899-Lipschitz continuous
Indeed let for any 119906 isin 119883
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩
= ⟨119875119909 minus 119875119910 120578 (119909 119910)⟩
= ⟨ (1198981199091minus1198981199092 minus119898119909
1+ 1198991199092)minus(119898119910
1minus 1198981199102 minus1198981199101+ 1198991199102)
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= ⟨ (119898 (1199091minus 1199101)minus119898 (119909
2minus 1199102) minus119898 (119909
1minus 1199101)+119899 (119909
2minus 1199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= 1198982(1199091minus 1199101)2minus 119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ 1198992(1199092minus 1199102)2
1003817100381710038171003817119875119909 minus 1198751199101003817100381710038171003817
2
= ⟨119875119909 minus 119875119910 119875119909 minus 119875119910⟩
= ⟨ (1198981199091minus 1198981199092 minus119898119909
1+1198991199092) minus (119898119910
1minus 1199102 minus1198981199101+1198991199102)
(1198981199091minus 1198981199092 minus119898119909
1+1198991199092) minus (119898119910
1minus 1199102 minus1198981199101+1198991199102)⟩
= 21198982(1199091minus 1199101)2minus 2119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ (1198982+ 1198992) (1199092minus 1199102)2
le 21198982(1199091minus 1199101)2minus 2119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ 21198992(1199092minus 1199102)2
= 2 ⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge1
2
1003817100381710038171003817119875119909 minus 1198751199101003817100381710038171003817
2
(16)
that is 119867(119875119876 119877) is (12)-120578-cocoercive with respect to 119875Consider
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩
= ⟨119876119909 minus 119876119910 120578 (119909 119910)⟩
= ⟨ (minus1198981199091+ 1198981199092 minus1198991199092) minus (minus119898119910
1+ 1198981199102 minus1198991199102)
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= ⟨ (minus119898 (1199091minus 1199101) + 119898 (119909
2minus 1199102) minus119899 (119909
2minus 1199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= minus1198982(1199091minus 1199101)2+ 1198982(1199091minus 1199101) (1199092minus 1199102) minus 1198992(1199092minus 1199102)2
1003817100381710038171003817119876119909 minus 1198761199101003817100381710038171003817
2
= ⟨119876119909 minus 119876119910119876119909 minus 119876119910⟩
= ⟨ (minus1198981199091+ 1198981199092 minus1198991199092)minus (minus119898119910
1+ 1198981199102 minus1198991199102)
(minus1198981199091+ 1198981199092 minus1198991199092) minus (minus119898119910
1minus 1198981199102 minus1198991199102)⟩
= 1198982(1199091minus 1199101)2minus 21198982(1199091minus 1199101) (1199092minus 1199102)
+ (1198982+ 1198992) (1199092minus 1199102)2
le 21198982(1199091minus 1199101)2minus 21198982(1199091minus 1199101) (1199092minus 1199102) + 2119899
2(1199092minus 1199102)2
= (minus2) ⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge minus1
2
1003817100381710038171003817119876119909 minus 1198761199101003817100381710038171003817
2
(17)
that is 119867(119875119876 119877) is (12)-120578-relaxed cocoercive with respectto 119876 In addition
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩
= ⟨119877119909 minus 119877119910 120578 (119909 119910)⟩
= ⟨ (1198991199091 1198981199092) minus (119899119910
2 1198981199102) (119898 (119909
1minus 1199101) 119899 (119909
2minus 1199102))⟩
= 119898119899(1199091minus 1199101)2+ 119898119899(119909
2minus 1199102)2
ge 11989821003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 11989821003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(18)
that is119867(119875119876 119877) is1198982-120578-strongly monotone with respect to119877 Moreover
1003817100381710038171003817120578 (119909 119910)1003817100381710038171003817 =
1003817100381710038171003817119898 (1199091minus 1199101) 119899 (119909
2minus 1199102)1003817100381710038171003817
= radic1198982(1199091minus 1199101)2+ 1198992(119909
2minus 1199102)2
le radic1198992(1199091minus 1199101)2+ 1198992(119909
2minus 1199102)2
le 1198991003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(19)
that is 120578(119909 119910) is 119899-Lipschitz continuous
Definition 4 A set-valued mapping 119872 119883 rarr 2119883 is said to
be 119898-120578-relaxed monotone if there exists a constant 119898 gt 0
such that
⟨119906 minus V 120578 (119909 119910)⟩ ge (minus119898)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 forall119909 119910 isin 119883
119906 isin 119872 (119909) V isin 119872(119910)
(20)
Definition 5 A set-valued mapping 119879 119883 rarr CB(119883) is saidto beD-Lipschitz continuous if there exists a constant 119897 gt 0
such that
D (119879119909 119879119910) le 1198971003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (21)
4 Journal of Mathematics
Definition 6 Let 119878 119879 119883 rarr CB(119883) be the set-valuedmappings A mapping119873 119883 times 119883 rarr 119883 is said to be
(i) 1205981-Lipschitz continuous in the first argument with
respect to 119878 if there exists a constant 1205981gt 0 such that
1003817100381710038171003817119873 (1199081 sdot) minus 119873 (119908
2 sdot)
1003817100381710038171003817 le 1205981
10038171003817100381710038171199081 minus 1199082
1003817100381710038171003817 forall1199061 1199062isin 119883
1199081isin 119878 (119906
1) 119908
2isin 119878 (119906
2)
(22)
(ii) 1205982-Lipschitz continuous in the second argument with
respect to 119879 if there exists a constant 1205982gt 0 such that
1003817100381710038171003817119873 (sdot V1) minus 119873 (sdot V
2)1003817100381710038171003817 le 120598
2
1003817100381710038171003817V1 minus V2
1003817100381710038171003817 forall1199061 1199062isin 119883
V1isin 119879 (119906
1) V
2isin 119879 (119906
2)
(23)
3 Generalized 119867(sdot sdot sdot)-120578-CocoerciveOperators
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
1 1199102) isin R2 we have
⟨119872119909 minus119872119910 120578 (119909 119910)⟩
= ⟨ ((minus1198991199091minus 1198981199092) minus (minus119899119910
1minus 1198981199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
=⟨ (minus119899 (1199091minus1199101) minus119898 (119909
2minus1199102)) (119898 (119909
1minus1199101) 119899 (119909
2minus1199102))⟩
= minus119898119899 (1199091minus 1199101)2+ (1199092minus 1199102)2
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Graph (119872) = (119886 119887) isin 119883 times 119883 119887 isin 119872 (119886) (26)
Proof Suppose on the contrary that there exists (1199090 1199060) notin
Graph119872 such that
⟨1199060minus V 120578 (119909
0 119910)⟩ ge 0 forall (119910 V) isin Graph (119872) (27)
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
1 1199061) isin Graph (119872) such that
119867(1198751199090 1198761199090 1198771199090) + 120582119906
0= 119867 (119875119909
1 1198761199091 1198771199091) + 120582119906
1isin 119883
(28)
Now 1205821199060minus1205821199061= 119867(119875119909
1 1198761199091 1198771199091) minus119867(119875119909
0 1198761199090 1198771199090) isin 119883
⟨1205821199060minus 1205821199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
It follows that
minus119867 (119875119906119876119906 119877119906) + 119909 isin 120582119872119906
minus119867 (119875V 119876V 119877V) + 119909 isin 120582119872V(32)
Since119872 is119898-120578-relaxed monotone we have
minus 119898119906 minus V 2
le1
120582⟨ minus 119867 (119875119906119876119906 119877119906)
+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
minus 120582119898119906 minus V 2
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is defined by
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906) forall119906 isin 119883 (35)
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V forall119906 V isin 119883
(36)
Proof Let 119906 V isin 119883 be any given points It follows from (35)that
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906)
R119867(sdotsdotsdot)-120578120582119872
(V) = (119867 (119875119876 119877) + 120582119872)minus1
(V)
1
120582(119906 minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(119906)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(119906))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(119906)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(119906))
1
120582(V minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(V)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(V))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(V)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(V))
(37)
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
119879(119906) such that
0 isin 119873 (119908 V) + 119872 (119901 (119906)) (43)
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
119901 (119906) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ] (46)
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Mathematics
be viewed as an extension and generalization of some knownresults [2 15 16] For illustration of Definitions 2 and 7 andexamples 23 and 32 are given respectively
2 Preliminaries
Throughout this paper we suppose that 119883 is a real Hilbertspace endowedwith a norm sdot and an inner product ⟨sdot sdot⟩ 2119883(resp CB(119883)) is the family of all the nonempty (resp closedand bounded) subsets of119883 andD(sdot sdot) is theHausdorffmetricon CB(119883) defined by
D (119875 119876) = max sup119909isin119875
119889 (119909 119876) sup119910isin119876
119889 (119875 119910)
forall119875 119876 isin CB (119883)
(1)
where 119889(119909 119876) = inf119910isin119876
119909 minus 119910 and 119889(119875 119910) = inf119909isin119875
119909 minus 119910In the sequel let us recall some concepts
Definition 1 (see [4 17]) Let119875 119883 rarr 119883 and 120578 119883times119883 rarr 119883
be two mappings Then 119875 is said to be
(i) 120578-monotone if
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 0 forall119909 119910 isin 119883 (2)
(ii) 1205751-120578-stronglymonotone if there exists a constant 120575
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205751
1003817100381710038171003817119909 minus 1199101003817100381710038171003817
2 forall119909 119910 isin 119883 (3)
(iii) 1205831-120578-cocoercive if there exists a constant 120583
1gt 0 such
that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge 1205831
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817
2
forall119909 119910 isin 119883
(4)
(iv) 1205741-120578-relaxed cocoercive if there exists a constant 120574
1gt
0 such that
⟨119875 (119909) minus 119875 (119910) 120578 (119909 119910)⟩ ge (minus1205741)1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817
2
forall119909 119910 isin 119883
(5)
(v) 120582119875-Lipschitz continuous if there exists a constant
120582119875gt 0 such that
1003817100381710038171003817119875 (119909) minus 119875 (119910)1003817100381710038171003817 le 120582
119875
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (6)
(vi) 120572-expansive if there exists a constant 120572 gt 0 such that1003817100381710038171003817119875 (119909) minus 119875 (119910)
1003817100381710038171003817 ge 1205721003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (7)
if 120572 = 1 then it is expansive(vii) 120578 is said to be 120591-Lipschitz continuous if there exists a
constant 120591 gt 0 such that1003817100381710038171003817120578 (119909 119910)
1003817100381710038171003817 le 1205911003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (8)
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119883 then definitions(i) to (iv) reduce to the Definitions of monotonicity strongmonotonicity [18] cocoercivity [19] and relaxed cocoerciverespectively
Definition 2 Let 119875119876 119877 119883 rarr 119883 120578 119883 times 119883 rarr 119883 and119867 119883 times 119883 times 119883 rarr 119883 be the single-valued mappings Then
(i) 119867(119875 sdot sdot) is said to be 120583-120578-cocoercive with respect to 119875if there exists a constant 120583 gt 0 such that
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge 1205831003817100381710038171003817119875119909 minus 119875119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(9)
(ii) 119867(sdot 119876 sdot) is said to be 120574-120578-relaxed cocoercive withrespect to 119876 if there exists a constant 120574 gt 0 such that
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge (minus120574)1003817100381710038171003817119876119909 minus 119876119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(10)
(iii) 119867(sdot sdot 119877) is said to be 120575-120578-strongly monotone withrespect to 119877 if there exists a constant 120575 gt 0 such that
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 1205751003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
forall119909 119910 119906 isin 119883
(11)
(iv) 119867(119875 sdot sdot) is said to be 1199031-Lipschitz continuous with
respect to 119875 if there exists a constant 1199031gt 0 such that
1003817100381710038171003817119867 (119875119909 sdot sdot) minus 119867 (119875119910 sdot sdot)1003817100381710038171003817 le 1199031
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (12)
(v) 119867(sdot 119876 sdot) is said to be 1199032-Lipschitz continuous with
respect to 119876 if there exists a constant 1199032gt 0 such that
1003817100381710038171003817119867 (sdot 119876119909 sdot) minus 119867 (sdot 119876119910 sdot)1003817100381710038171003817 le 1199032
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (13)
(vi) 119867(sdot sdot 119877) is said to be 1199033-Lipschitz continuous with
respect to 119877 if there exists a constant 1199033gt 0 such that
1003817100381710038171003817119867 (sdot sdot 119877119909) minus 119867 (sdot sdot 119877119910)1003817100381710038171003817 le 1199033
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119883 (14)
If 120578(119909 119910) = 119909 minus 119910 for all 119909 119910 isin 119867 and 119867(sdot sdot sdot) =
119867(sdot sdot) then Definitions (i)-(ii) are reduced to the definitionof cocoercivity and relaxed cocoercive [2] respectively and(iii) reduces to strong accretivity [16]
Example 3 Let 119883 = R2 with usual inner product Let 120578
R2 timesR2 rarr R2 119875 119876 119877 R2 rarr R2 be defined by
120578 (119909 119910) = (1198981199091+ 1198991199092
minus1198981199101minus 1198991199102
) 119875119909 = (1198981199091minus 1198981199092
minus1198981199091+ 1198991199092
)
119876119910 = (minus1198981199101+ 1198981199102
minus1198991199102
) 119877119911 = (1198991199111
1198981199112
)
(15)
Journal of Mathematics 3
for all scalers 119899119898 isin Rwith 119899 gt 119898 and for all119909 = (1199091 1199092) 119910 =
(1199101 1199102) 119911 = (119911
1 1199112) isin R2
Suppose that119867(119875119909119876119910 119877119911) = 119875119909 +119876119910+119877119911 Then119867(119875
119876 119877) is (12)-120578-cocoercive with respect to 119875 (12)-120578-relaxedcocoercive with respect to 119876 1198982-120578-strongly monotone withrespect to 119877 and 120578(119909 119910) is 119899-Lipschitz continuous
Indeed let for any 119906 isin 119883
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩
= ⟨119875119909 minus 119875119910 120578 (119909 119910)⟩
= ⟨ (1198981199091minus1198981199092 minus119898119909
1+ 1198991199092)minus(119898119910
1minus 1198981199102 minus1198981199101+ 1198991199102)
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= ⟨ (119898 (1199091minus 1199101)minus119898 (119909
2minus 1199102) minus119898 (119909
1minus 1199101)+119899 (119909
2minus 1199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= 1198982(1199091minus 1199101)2minus 119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ 1198992(1199092minus 1199102)2
1003817100381710038171003817119875119909 minus 1198751199101003817100381710038171003817
2
= ⟨119875119909 minus 119875119910 119875119909 minus 119875119910⟩
= ⟨ (1198981199091minus 1198981199092 minus119898119909
1+1198991199092) minus (119898119910
1minus 1199102 minus1198981199101+1198991199102)
(1198981199091minus 1198981199092 minus119898119909
1+1198991199092) minus (119898119910
1minus 1199102 minus1198981199101+1198991199102)⟩
= 21198982(1199091minus 1199101)2minus 2119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ (1198982+ 1198992) (1199092minus 1199102)2
le 21198982(1199091minus 1199101)2minus 2119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ 21198992(1199092minus 1199102)2
= 2 ⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge1
2
1003817100381710038171003817119875119909 minus 1198751199101003817100381710038171003817
2
(16)
that is 119867(119875119876 119877) is (12)-120578-cocoercive with respect to 119875Consider
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩
= ⟨119876119909 minus 119876119910 120578 (119909 119910)⟩
= ⟨ (minus1198981199091+ 1198981199092 minus1198991199092) minus (minus119898119910
1+ 1198981199102 minus1198991199102)
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= ⟨ (minus119898 (1199091minus 1199101) + 119898 (119909
2minus 1199102) minus119899 (119909
2minus 1199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= minus1198982(1199091minus 1199101)2+ 1198982(1199091minus 1199101) (1199092minus 1199102) minus 1198992(1199092minus 1199102)2
1003817100381710038171003817119876119909 minus 1198761199101003817100381710038171003817
2
= ⟨119876119909 minus 119876119910119876119909 minus 119876119910⟩
= ⟨ (minus1198981199091+ 1198981199092 minus1198991199092)minus (minus119898119910
1+ 1198981199102 minus1198991199102)
(minus1198981199091+ 1198981199092 minus1198991199092) minus (minus119898119910
1minus 1198981199102 minus1198991199102)⟩
= 1198982(1199091minus 1199101)2minus 21198982(1199091minus 1199101) (1199092minus 1199102)
+ (1198982+ 1198992) (1199092minus 1199102)2
le 21198982(1199091minus 1199101)2minus 21198982(1199091minus 1199101) (1199092minus 1199102) + 2119899
2(1199092minus 1199102)2
= (minus2) ⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge minus1
2
1003817100381710038171003817119876119909 minus 1198761199101003817100381710038171003817
2
(17)
that is 119867(119875119876 119877) is (12)-120578-relaxed cocoercive with respectto 119876 In addition
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩
= ⟨119877119909 minus 119877119910 120578 (119909 119910)⟩
= ⟨ (1198991199091 1198981199092) minus (119899119910
2 1198981199102) (119898 (119909
1minus 1199101) 119899 (119909
2minus 1199102))⟩
= 119898119899(1199091minus 1199101)2+ 119898119899(119909
2minus 1199102)2
ge 11989821003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 11989821003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(18)
that is119867(119875119876 119877) is1198982-120578-strongly monotone with respect to119877 Moreover
1003817100381710038171003817120578 (119909 119910)1003817100381710038171003817 =
1003817100381710038171003817119898 (1199091minus 1199101) 119899 (119909
2minus 1199102)1003817100381710038171003817
= radic1198982(1199091minus 1199101)2+ 1198992(119909
2minus 1199102)2
le radic1198992(1199091minus 1199101)2+ 1198992(119909
2minus 1199102)2
le 1198991003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(19)
that is 120578(119909 119910) is 119899-Lipschitz continuous
Definition 4 A set-valued mapping 119872 119883 rarr 2119883 is said to
be 119898-120578-relaxed monotone if there exists a constant 119898 gt 0
such that
⟨119906 minus V 120578 (119909 119910)⟩ ge (minus119898)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 forall119909 119910 isin 119883
119906 isin 119872 (119909) V isin 119872(119910)
(20)
Definition 5 A set-valued mapping 119879 119883 rarr CB(119883) is saidto beD-Lipschitz continuous if there exists a constant 119897 gt 0
such that
D (119879119909 119879119910) le 1198971003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (21)
4 Journal of Mathematics
Definition 6 Let 119878 119879 119883 rarr CB(119883) be the set-valuedmappings A mapping119873 119883 times 119883 rarr 119883 is said to be
(i) 1205981-Lipschitz continuous in the first argument with
respect to 119878 if there exists a constant 1205981gt 0 such that
1003817100381710038171003817119873 (1199081 sdot) minus 119873 (119908
2 sdot)
1003817100381710038171003817 le 1205981
10038171003817100381710038171199081 minus 1199082
1003817100381710038171003817 forall1199061 1199062isin 119883
1199081isin 119878 (119906
1) 119908
2isin 119878 (119906
2)
(22)
(ii) 1205982-Lipschitz continuous in the second argument with
respect to 119879 if there exists a constant 1205982gt 0 such that
1003817100381710038171003817119873 (sdot V1) minus 119873 (sdot V
2)1003817100381710038171003817 le 120598
2
1003817100381710038171003817V1 minus V2
1003817100381710038171003817 forall1199061 1199062isin 119883
V1isin 119879 (119906
1) V
2isin 119879 (119906
2)
(23)
3 Generalized 119867(sdot sdot sdot)-120578-CocoerciveOperators
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
1 1199102) isin R2 we have
⟨119872119909 minus119872119910 120578 (119909 119910)⟩
= ⟨ ((minus1198991199091minus 1198981199092) minus (minus119899119910
1minus 1198981199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
=⟨ (minus119899 (1199091minus1199101) minus119898 (119909
2minus1199102)) (119898 (119909
1minus1199101) 119899 (119909
2minus1199102))⟩
= minus119898119899 (1199091minus 1199101)2+ (1199092minus 1199102)2
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Graph (119872) = (119886 119887) isin 119883 times 119883 119887 isin 119872 (119886) (26)
Proof Suppose on the contrary that there exists (1199090 1199060) notin
Graph119872 such that
⟨1199060minus V 120578 (119909
0 119910)⟩ ge 0 forall (119910 V) isin Graph (119872) (27)
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
1 1199061) isin Graph (119872) such that
119867(1198751199090 1198761199090 1198771199090) + 120582119906
0= 119867 (119875119909
1 1198761199091 1198771199091) + 120582119906
1isin 119883
(28)
Now 1205821199060minus1205821199061= 119867(119875119909
1 1198761199091 1198771199091) minus119867(119875119909
0 1198761199090 1198771199090) isin 119883
⟨1205821199060minus 1205821199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
It follows that
minus119867 (119875119906119876119906 119877119906) + 119909 isin 120582119872119906
minus119867 (119875V 119876V 119877V) + 119909 isin 120582119872V(32)
Since119872 is119898-120578-relaxed monotone we have
minus 119898119906 minus V 2
le1
120582⟨ minus 119867 (119875119906119876119906 119877119906)
+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
minus 120582119898119906 minus V 2
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is defined by
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906) forall119906 isin 119883 (35)
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V forall119906 V isin 119883
(36)
Proof Let 119906 V isin 119883 be any given points It follows from (35)that
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906)
R119867(sdotsdotsdot)-120578120582119872
(V) = (119867 (119875119876 119877) + 120582119872)minus1
(V)
1
120582(119906 minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(119906)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(119906))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(119906)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(119906))
1
120582(V minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(V)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(V))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(V)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(V))
(37)
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
119879(119906) such that
0 isin 119873 (119908 V) + 119872 (119901 (119906)) (43)
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
119901 (119906) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ] (46)
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
for all scalers 119899119898 isin Rwith 119899 gt 119898 and for all119909 = (1199091 1199092) 119910 =
(1199101 1199102) 119911 = (119911
1 1199112) isin R2
Suppose that119867(119875119909119876119910 119877119911) = 119875119909 +119876119910+119877119911 Then119867(119875
119876 119877) is (12)-120578-cocoercive with respect to 119875 (12)-120578-relaxedcocoercive with respect to 119876 1198982-120578-strongly monotone withrespect to 119877 and 120578(119909 119910) is 119899-Lipschitz continuous
Indeed let for any 119906 isin 119883
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩
= ⟨119875119909 minus 119875119910 120578 (119909 119910)⟩
= ⟨ (1198981199091minus1198981199092 minus119898119909
1+ 1198991199092)minus(119898119910
1minus 1198981199102 minus1198981199101+ 1198991199102)
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= ⟨ (119898 (1199091minus 1199101)minus119898 (119909
2minus 1199102) minus119898 (119909
1minus 1199101)+119899 (119909
2minus 1199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= 1198982(1199091minus 1199101)2minus 119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ 1198992(1199092minus 1199102)2
1003817100381710038171003817119875119909 minus 1198751199101003817100381710038171003817
2
= ⟨119875119909 minus 119875119910 119875119909 minus 119875119910⟩
= ⟨ (1198981199091minus 1198981199092 minus119898119909
1+1198991199092) minus (119898119910
1minus 1199102 minus1198981199101+1198991199102)
(1198981199091minus 1198981199092 minus119898119909
1+1198991199092) minus (119898119910
1minus 1199102 minus1198981199101+1198991199102)⟩
= 21198982(1199091minus 1199101)2minus 2119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ (1198982+ 1198992) (1199092minus 1199102)2
le 21198982(1199091minus 1199101)2minus 2119898 (119898 + 119899) (1199091 minus 119910
1) (1199092minus 1199102)
+ 21198992(1199092minus 1199102)2
= 2 ⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩
⟨119867 (119875119909 119906 119906) minus 119867 (119875119910 119906 119906) 120578 (119909 119910)⟩ ge1
2
1003817100381710038171003817119875119909 minus 1198751199101003817100381710038171003817
2
(16)
that is 119867(119875119876 119877) is (12)-120578-cocoercive with respect to 119875Consider
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩
= ⟨119876119909 minus 119876119910 120578 (119909 119910)⟩
= ⟨ (minus1198981199091+ 1198981199092 minus1198991199092) minus (minus119898119910
1+ 1198981199102 minus1198991199102)
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= ⟨ (minus119898 (1199091minus 1199101) + 119898 (119909
2minus 1199102) minus119899 (119909
2minus 1199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
= minus1198982(1199091minus 1199101)2+ 1198982(1199091minus 1199101) (1199092minus 1199102) minus 1198992(1199092minus 1199102)2
1003817100381710038171003817119876119909 minus 1198761199101003817100381710038171003817
2
= ⟨119876119909 minus 119876119910119876119909 minus 119876119910⟩
= ⟨ (minus1198981199091+ 1198981199092 minus1198991199092)minus (minus119898119910
1+ 1198981199102 minus1198991199102)
(minus1198981199091+ 1198981199092 minus1198991199092) minus (minus119898119910
1minus 1198981199102 minus1198991199102)⟩
= 1198982(1199091minus 1199101)2minus 21198982(1199091minus 1199101) (1199092minus 1199102)
+ (1198982+ 1198992) (1199092minus 1199102)2
le 21198982(1199091minus 1199101)2minus 21198982(1199091minus 1199101) (1199092minus 1199102) + 2119899
2(1199092minus 1199102)2
= (minus2) ⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩
⟨119867 (119906 119876119909 119906) minus 119867 (119906 119876119910 119906) 120578 (119909 119910)⟩ ge minus1
2
1003817100381710038171003817119876119909 minus 1198761199101003817100381710038171003817
2
(17)
that is 119867(119875119876 119877) is (12)-120578-relaxed cocoercive with respectto 119876 In addition
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩
= ⟨119877119909 minus 119877119910 120578 (119909 119910)⟩
= ⟨ (1198991199091 1198981199092) minus (119899119910
2 1198981199102) (119898 (119909
1minus 1199101) 119899 (119909
2minus 1199102))⟩
= 119898119899(1199091minus 1199101)2+ 119898119899(119909
2minus 1199102)2
ge 11989821003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119867 (119906 119906 119877119909) minus 119867 (119906 119906 119877119910) 120578 (119909 119910)⟩ ge 11989821003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(18)
that is119867(119875119876 119877) is1198982-120578-strongly monotone with respect to119877 Moreover
1003817100381710038171003817120578 (119909 119910)1003817100381710038171003817 =
1003817100381710038171003817119898 (1199091minus 1199101) 119899 (119909
2minus 1199102)1003817100381710038171003817
= radic1198982(1199091minus 1199101)2+ 1198992(119909
2minus 1199102)2
le radic1198992(1199091minus 1199101)2+ 1198992(119909
2minus 1199102)2
le 1198991003817100381710038171003817119909 minus 119910
1003817100381710038171003817
(19)
that is 120578(119909 119910) is 119899-Lipschitz continuous
Definition 4 A set-valued mapping 119872 119883 rarr 2119883 is said to
be 119898-120578-relaxed monotone if there exists a constant 119898 gt 0
such that
⟨119906 minus V 120578 (119909 119910)⟩ ge (minus119898)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 forall119909 119910 isin 119883
119906 isin 119872 (119909) V isin 119872(119910)
(20)
Definition 5 A set-valued mapping 119879 119883 rarr CB(119883) is saidto beD-Lipschitz continuous if there exists a constant 119897 gt 0
such that
D (119879119909 119879119910) le 1198971003817100381710038171003817119909 minus 119910
1003817100381710038171003817 forall119909 119910 isin 119883 (21)
4 Journal of Mathematics
Definition 6 Let 119878 119879 119883 rarr CB(119883) be the set-valuedmappings A mapping119873 119883 times 119883 rarr 119883 is said to be
(i) 1205981-Lipschitz continuous in the first argument with
respect to 119878 if there exists a constant 1205981gt 0 such that
1003817100381710038171003817119873 (1199081 sdot) minus 119873 (119908
2 sdot)
1003817100381710038171003817 le 1205981
10038171003817100381710038171199081 minus 1199082
1003817100381710038171003817 forall1199061 1199062isin 119883
1199081isin 119878 (119906
1) 119908
2isin 119878 (119906
2)
(22)
(ii) 1205982-Lipschitz continuous in the second argument with
respect to 119879 if there exists a constant 1205982gt 0 such that
1003817100381710038171003817119873 (sdot V1) minus 119873 (sdot V
2)1003817100381710038171003817 le 120598
2
1003817100381710038171003817V1 minus V2
1003817100381710038171003817 forall1199061 1199062isin 119883
V1isin 119879 (119906
1) V
2isin 119879 (119906
2)
(23)
3 Generalized 119867(sdot sdot sdot)-120578-CocoerciveOperators
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
1 1199102) isin R2 we have
⟨119872119909 minus119872119910 120578 (119909 119910)⟩
= ⟨ ((minus1198991199091minus 1198981199092) minus (minus119899119910
1minus 1198981199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
=⟨ (minus119899 (1199091minus1199101) minus119898 (119909
2minus1199102)) (119898 (119909
1minus1199101) 119899 (119909
2minus1199102))⟩
= minus119898119899 (1199091minus 1199101)2+ (1199092minus 1199102)2
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Graph (119872) = (119886 119887) isin 119883 times 119883 119887 isin 119872 (119886) (26)
Proof Suppose on the contrary that there exists (1199090 1199060) notin
Graph119872 such that
⟨1199060minus V 120578 (119909
0 119910)⟩ ge 0 forall (119910 V) isin Graph (119872) (27)
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
1 1199061) isin Graph (119872) such that
119867(1198751199090 1198761199090 1198771199090) + 120582119906
0= 119867 (119875119909
1 1198761199091 1198771199091) + 120582119906
1isin 119883
(28)
Now 1205821199060minus1205821199061= 119867(119875119909
1 1198761199091 1198771199091) minus119867(119875119909
0 1198761199090 1198771199090) isin 119883
⟨1205821199060minus 1205821199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
It follows that
minus119867 (119875119906119876119906 119877119906) + 119909 isin 120582119872119906
minus119867 (119875V 119876V 119877V) + 119909 isin 120582119872V(32)
Since119872 is119898-120578-relaxed monotone we have
minus 119898119906 minus V 2
le1
120582⟨ minus 119867 (119875119906119876119906 119877119906)
+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
minus 120582119898119906 minus V 2
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is defined by
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906) forall119906 isin 119883 (35)
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V forall119906 V isin 119883
(36)
Proof Let 119906 V isin 119883 be any given points It follows from (35)that
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906)
R119867(sdotsdotsdot)-120578120582119872
(V) = (119867 (119875119876 119877) + 120582119872)minus1
(V)
1
120582(119906 minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(119906)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(119906))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(119906)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(119906))
1
120582(V minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(V)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(V))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(V)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(V))
(37)
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
119879(119906) such that
0 isin 119873 (119908 V) + 119872 (119901 (119906)) (43)
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
119901 (119906) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ] (46)
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Mathematics
Definition 6 Let 119878 119879 119883 rarr CB(119883) be the set-valuedmappings A mapping119873 119883 times 119883 rarr 119883 is said to be
(i) 1205981-Lipschitz continuous in the first argument with
respect to 119878 if there exists a constant 1205981gt 0 such that
1003817100381710038171003817119873 (1199081 sdot) minus 119873 (119908
2 sdot)
1003817100381710038171003817 le 1205981
10038171003817100381710038171199081 minus 1199082
1003817100381710038171003817 forall1199061 1199062isin 119883
1199081isin 119878 (119906
1) 119908
2isin 119878 (119906
2)
(22)
(ii) 1205982-Lipschitz continuous in the second argument with
respect to 119879 if there exists a constant 1205982gt 0 such that
1003817100381710038171003817119873 (sdot V1) minus 119873 (sdot V
2)1003817100381710038171003817 le 120598
2
1003817100381710038171003817V1 minus V2
1003817100381710038171003817 forall1199061 1199062isin 119883
V1isin 119879 (119906
1) V
2isin 119879 (119906
2)
(23)
3 Generalized 119867(sdot sdot sdot)-120578-CocoerciveOperators
Definition 7 Let 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883and 120578 119883 times 119883 rarr 119883 be the single-valued mappings Let119867(119875119876 119877) be 120583-120578-cocoercive with respect to 119875 120574-120578-relaxedcocoercive with respect to 119876 and 120575-120578-strongly monotonewith respect to 119877 Then the set-valued mapping 119872 119883 rarr
2119883 is said to be a generalized 119867(sdot sdot sdot)-120578-cocoercive withrespect to the mappings 119875119876 and 119877 if
(i) 119872 is119898-120578-relaxed monotone(ii) (119867(119875 119876 119877) + 120582119872)(119883) = 119883 for all 120582 gt 0
Example 8 Let 119883119867 119875 119876 119877 and 120578 be the same as inExample 3 and let 119872 R2 rarr R2 be defined by 119872(119909) =
(minus1198991199091 minus119898119909
2) for all 119909 = (119909
1 1199092) isin R2
We claim that 119872 is 1198992-120578-relaxed monotone mapping
Indeed for any 119909 = (1199091 1199092) 119910 = (119910
1 1199102) isin R2 we have
⟨119872119909 minus119872119910 120578 (119909 119910)⟩
= ⟨ ((minus1198991199091minus 1198981199092) minus (minus119899119910
1minus 1198981199102))
(119898 (1199091minus 1199101) 119899 (119909
2minus 1199102))⟩
=⟨ (minus119899 (1199091minus1199101) minus119898 (119909
2minus1199102)) (119898 (119909
1minus1199101) 119899 (119909
2minus1199102))⟩
= minus119898119899 (1199091minus 1199101)2+ (1199092minus 1199102)2
ge minus 1198992(1199091minus 1199101)2+ (1199092minus 1199102)2
= minus11989921003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
⟨119872119909 minus119872119910 120578 (119909 119910)⟩ ge (minus1198992)1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2
(24)
Furthermore 119872 is also a generalized 119867(sdot sdot sdot)-120578-cocoerciveoperator since (119867(119875119876 119877) + 120582119872)(R2) = R2 for any 120582 gt 0
Remark 9 If 119867(119875119876 119877) = 119867(119875119876) 119875 is 120572-strongly mono-tone and119876 is120573-relaxedmonotone then generalized119867(sdot sdot sdot)-120578-cocoercive operator reduces to 119867(sdot sdot)-120578-monotone opera-tor introduced and studied by Xu and Wang [15]
Proposition 10 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 119898
then the following inequality
⟨119906 minus V 120578 (119909 119910)⟩ ge 0 (25)
holds for all (119910 V) isin Graph (119872) and implies 119906 isin 119872119909 where
Graph (119872) = (119886 119887) isin 119883 times 119883 119887 isin 119872 (119886) (26)
Proof Suppose on the contrary that there exists (1199090 1199060) notin
Graph119872 such that
⟨1199060minus V 120578 (119909
0 119910)⟩ ge 0 forall (119910 V) isin Graph (119872) (27)
Since119872 is a generalized119867(sdot sdot sdot)-120578-cocoercive we know that(119867(119875 119876 119877) + 120582119872)(119883) = 119883 holds for all 120582 gt 0 and so thereexists (119909
1 1199061) isin Graph (119872) such that
119867(1198751199090 1198761199090 1198771199090) + 120582119906
0= 119867 (119875119909
1 1198761199091 1198771199091) + 120582119906
1isin 119883
(28)
Now 1205821199060minus1205821199061= 119867(119875119909
1 1198761199091 1198771199091) minus119867(119875119909
0 1198761199090 1198771199090) isin 119883
⟨1205821199060minus 1205821199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(29)
Setting (119910 V) = (1199091 1199061) in (27) and then from the resultant
(28) and119898-120578-relaxed monotonicity of119872 we obtain
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le 120582 ⟨1199060minus 1199061 120578 (1199090 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
= minus ⟨119867 (1198751199090 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199090 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199090 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199090) 120578 (119909
0 1199091)⟩
minus ⟨119867 (1198751199091 1198761199091 1198771199090) minus 119867 (119875119909
1 1198761199091 1198771199091) 120578 (119909
0 1199091)⟩
(30)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
It follows that
minus119867 (119875119906119876119906 119877119906) + 119909 isin 120582119872119906
minus119867 (119875V 119876V 119877V) + 119909 isin 120582119872V(32)
Since119872 is119898-120578-relaxed monotone we have
minus 119898119906 minus V 2
le1
120582⟨ minus 119867 (119875119906119876119906 119877119906)
+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
minus 120582119898119906 minus V 2
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is defined by
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906) forall119906 isin 119883 (35)
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V forall119906 V isin 119883
(36)
Proof Let 119906 V isin 119883 be any given points It follows from (35)that
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906)
R119867(sdotsdotsdot)-120578120582119872
(V) = (119867 (119875119876 119877) + 120582119872)minus1
(V)
1
120582(119906 minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(119906)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(119906))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(119906)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(119906))
1
120582(V minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(V)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(V))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(V)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(V))
(37)
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
119879(119906) such that
0 isin 119873 (119908 V) + 119872 (119901 (119906)) (43)
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
119901 (119906) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ] (46)
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 5
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (30) becomes
minus 11989810038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2
le minus 12058310038171003817100381710038171198751199090 minus 119875119909
1
1003817100381710038171003817
2+ 120574
10038171003817100381710038171198761199090 minus 1198761199091
1003817100381710038171003817
2minus 120575
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
le minus (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199090 minus 1199091
1003817100381710038171003817
2
= minus11990310038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0
le minus (119903 minus 119898)10038171003817100381710038171199090 minus 119909
1
1003817100381710038171003817
2le 0 where 119903 = 120583120572
2minus 1205741205732+ 120575
(31)
which gives 1199090= 1199091since 119903 gt 119898 By (27) we have 119906
0= 1199061 a
contradiction This completes the proof
Theorem 11 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 with 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then (119867(119875 119876 119877) + 120582119872)minus1 is single-valued
Proof For any given 119909 isin 119883 let 119906 V isin (119867(119875119876 119877)+120582119872)minus1(119909)
It follows that
minus119867 (119875119906119876119906 119877119906) + 119909 isin 120582119872119906
minus119867 (119875V 119876V 119877V) + 119909 isin 120582119872V(32)
Since119872 is119898-120578-relaxed monotone we have
minus 119898119906 minus V 2
le1
120582⟨ minus 119867 (119875119906119876119906 119877119906)
+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
minus 120582119898119906 minus V 2
le ⟨ minus 119867 (119875119906119876119906 119877119906)+119909 minus (minus119867 (119875V 119876V 119877V) + 119909) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩
= minus ⟨119867 (119875119906 119876119906 119877119906) minus 119867 (119875V 119876119906 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876119906 119877119906) minus 119867 (119875V 119876V 119877119906) 120578 (119906 V)⟩
minus ⟨119867 (119875V 119876V 119877119906) minus 119867 (119875V 119876V 119877V) 120578 (119906 V)⟩ (33)
Since 119867(119875119876 119877) is 120583-120578-cocoercive with respect to 119875 120574-120578-relaxed cocoercive with respect to119876 120575-120578-strongly monotone
with respect to 119877 and 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous thus (33) becomes
minus 120582119898119906 minus V 2
le minus 120583119875119906 minus 119875V 2 + 120574119876119906 minus 119876V 2 minus 120575119906 minus V 2
le minus (1205831205722minus 1205741205732+ 120575) 119906 minus V 2
= minus119903119906 minus V 2 le 0
le minus (119903 minus 120582119898) 119906 minus V 2 le 0 where 119903 = 1205831205722minus 1205741205732+ 120575
(34)
since 119903 gt 120582119898 Hence it follows that 119906 minus V le 0 This impliesthat 119906 = V and so (119867(119875 119876 119877) + 120582119872)
minus1 is single-valued
Definition 12 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized119867(sdot sdot sdot)-120578-cocoercive operatorwith respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous and 120583 gt 120574 120572 gt 120573 and 119903 = 120583120572
2minus 1205741205732+ 120575 gt 120582119898
then the resolvent operator 119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is defined by
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906) forall119906 isin 119883 (35)
Now we prove that the resolvent operator defined by (35)is Lipschitz continuous
Theorem 13 Let set-valued mapping 119872 119883 rarr 2119883 be a
generalized 119867(sdot sdot sdot)-120578-cocoercive operator with respect to themappings 119875119876 and 119877 If 119875 is 120572-expansive 119876 is 120573-Lipschitzcontinuous 120578 is 120591-Lipschitz continuous and 120583 gt 120574 120572 gt 120573
with 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 then the resolvent operator
119877119867(sdotsdotsdot)-120578120582119872
119883 rarr 119883 is (120591(119903 minus 120582119898))-Lipschitz continuous thatis
100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V forall119906 V isin 119883
(36)
Proof Let 119906 V isin 119883 be any given points It follows from (35)that
119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867 (119875119876 119877) + 120582119872)minus1
(119906)
R119867(sdotsdotsdot)-120578120582119872
(V) = (119867 (119875119876 119877) + 120582119872)minus1
(V)
1
120582(119906 minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(119906)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(119906))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(119906)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(119906))
1
120582(V minus 119867(119875 (119877
119867(sdotsdotsdot)-120578120582119872
(V)) 119876 (119877119867(sdotsdotsdot)-120578120582119872
(V))
119877 (119877119867(sdotsdotsdot)-120578120582119872
(V)))) isin 119872(119877119867(sdotsdotsdot)-120578120582119872
(V))
(37)
Let 1199111= 119877119867(sdotsdotsdot)-120578120582119872
(119906) and 1199112= 119877119867(sdotsdotsdot)-120578120582119872
(V)
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
119879(119906) such that
0 isin 119873 (119908 V) + 119872 (119901 (119906)) (43)
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
119901 (119906) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ] (46)
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Mathematics
Since119872 is119898-120578-relaxed monotone we have
⟨1
120582(119906 minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
minus (V minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
⟨1
120582(119906 minus V minus 119867 (119875 (119911
1) 119876 (119911
1) 119877 (119911
1))
+119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2)) 120578 (119911
1 1199112) ⟩
ge minus 11989810038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2
(38)
which implies
⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))minus119867 (119875 (119911
2) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
(39)
Further we have120591 119906 minus V
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
ge 119906 minus V1003817100381710038171003817120578 (1199111 1199112)
1003817100381710038171003817
ge ⟨119906 minus V 120578 (1199111 1199112)⟩
ge ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= ⟨119867 (119875 (1199111) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
1) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1)) 120578 (119911
1 1199112)⟩
+ ⟨119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
1))
minus 119867 (119875 (1199112) 119876 (119911
2) 119877 (119911
2))
120578 (1199111 1199112)⟩ minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
ge 1205831003817100381710038171003817119875 (1199111) minus 119875 (119911
2)1003817100381710038171003817
2minus 120574
1003817100381710038171003817119876 (1199111) minus 119876 (119911
2)1003817100381710038171003817
2
+ 1205751003817100381710038171003817119877 (1199111) minus 119877 (119911
2)1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (1205831205722minus 1205741205732+ 120575)
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2minus 120582119898
10038171003817100381710038171199111 minus 1199112
1003817100381710038171003817
2
= (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 where 119903 = 120583120572
2minus 1205741205732+ 120575
(40)
and hence
120591 119906 minus V10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817 ge (119903 minus 120582119898)10038171003817100381710038171199111 minus 119911
2
1003817100381710038171003817
2 (41)
that is100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
(119906) minus 119877119867(sdotsdotsdot)-120578120582119872
(V)100381710038171003817100381710038171003817le
120591
119903 minus 120582119898119906 minus V
forall119906 V isin 119883
(42)
This completes the proof
4 An Application of 119867(sdot sdot sdot)-120578-CocoerciveOperators for Solving GeneralizedVariational Inclusions
In this section we will show that under suitable assumptionsthe generalized 119867(sdot sdot sdot)-120578-cocoercive operator can also playimportant roles for solving the variational inclusion problemin Hilbert spaces
Let119873 119883times119883 rarr 119883 120578 119883times119883 rarr 119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883 be the single-valuedmap-pings and 119878 119879 119883 rarr CB(119883)119872 119883 rarr 2
119883 be the set-valued mappings such that119872 is generalized119867(sdot sdot sdot)-120578-coco-ercive with respect to119875119876 and119877 and range (119901) cap dom119872 = 0Then we consider the problem to find 119906 isin 119883119908 isin 119878(119906) V isin
119879(119906) such that
0 isin 119873 (119908 V) + 119872 (119901 (119906)) (43)
Theproblem (43) is called generalized set-valued variatio-nal-like inclusion problem The problem of type (43) wasintroduced and studied by Chidume et al [3] by applying120578-proximal mapping If 119879 = 0 and 120578(119906 V) = 119906 minus V for all119906 V isin 119883 and 119873(sdot sdot) = 119878(sdot) where 119878 119883 rarr CB(119883) is a set-valued mapping Then problem (43) reduces to the problemof finding 119906 isin 119883119908 isin 119878(119906) such that
0 isin 119908 +119872(119901 (119906)) (44)
The problem of type (44) was studied by Ahmad et al [2] byapplying119867(sdot sdot)-cocoercive operators
If 119878 119879 = 0119873(sdot sdot) = 0 and 120578(119906 V) = 119906 minus V for all 119906 V isin 119883then problem (43) reduces to the problem of finding 119906 isin 119883
such that
0 isin 119872(119901 (119906)) (45)
The problem of type (45) was studied by Verma [14] inthe setting of Banach spaces when 119872 is 119860-maximal-relaxedaccretive
Lemma 14 The (119906 119908 V) where 119906 isin 119883119908 isin 119878(119906) V isin 119879(119906) isa solution of the problem (43) if and only if (119906 119908 V) satisfiesthe following relation
119901 (119906) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ] (46)
where 119877119867(sdotsdotsdot)-120578120582119872
(119906) = (119867(119875119876 119877) + 120582119872)minus1(119906) and 120582 gt 0 is a
constant
Proof By using the definitions of resolvent operators119877119867(sdotsdotsdot)-120578120582119872
the conclusion follows directly
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 7
Using Lemma 14 and using the technique of Chidume etal [3] and Nadler [20] we develop an iterative algorithm forfinding the approximate solution of problem (43) as follows
Algorithm 15 Let 119873 119883 times 119883 rarr 119883 120578 119883 times 119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 119901 119883 rarr 119883 and119878 119879 119883 rarr CB(119883) be such that for each119906 isin 119883119865(119906) sube 119901(119883)where 119865 119883 rarr 2
119883 is the set-valued mappings defined by
119865 (119906)
= ⋃
119908isin119878(119906)
⋃
Visin119879(119906)
(119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (119906)) 119876 (119901 (119906)) 119877 (119901 (119906)))
minus120582119873 (119908 V) ) )
(47)
where 119872 119883 rarr 2119883 is a set-valued mapping such that 119872
is generalized119867(sdot sdot sdot)-120578-cocoercive with respect to the map-pings 119875119876 and 119877
For given 1199060isin 119883 take 119908
0isin 119878(1199060) and V
0isin 119879(119906
0) Let
1199090= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199060)) 119876 (119901 (119906
0)) 119877 (119901 (119906
0)))
minus120582119873 (1199080 V0)) isin 119865 (119906
0) sube 119901 (119883)
(48)
Hence there exists 1199061isin 119883 such that 119909
0= 119901(119906
1) Since 119908
0isin
119878(1199060) isin CB(119883) and V
0isin 119879(119906
0) isin CB(119883) then by Nadlerrsquos
result [20] there exist 1199081isin 119878(1199061) and V
1isin 119879(119906
1) such that
10038171003817100381710038171199080 minus 1199081
1003817100381710038171003817 le (1 + 1minus1)D (119878 (119906
0) 119878 (119906
1))
1003817100381710038171003817V0 minus V1
1003817100381710038171003817 le (1 + 1minus1)D (119879 (119906
0) 119879 (119906
1))
(49)
Let
1199091= 119877119867(sdotsdotsdot)-120578120582119872
(119867 (119875 (119901 (1199061)) 119876 (119901 (119906
1)) 119877 (119901 (119906
1)))
minus120582119873 (1199081 V1)) isin 119865 (119906
1) sube 119901 (119883)
(50)
Hence there exists 1199062isin 119883 such that 119909
1= 119901(119906
2) By induc-
tion we can define iterative sequences 119906119899 119901(119906
119899) 119908119899 and
V119899 as follows
119901 (119906119899+1
) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)]
(51)
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(52)
V119899isin 119879 (119906
119899)
1003817100381710038171003817V119899 minus V119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119879 (119906
119899) 119879 (119906
119899+1))
(53)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If 119873(119908 V) = 119908 for all 119908 V isin 119883 then Algorithm 15reduces to the following algorithm for solving the problem(44)
Algorithm 16 For any 1199060isin 119883 and 119908
0isin 119878(1199060) compute the
sequence 119906119899 and 119908
119899 by the following
119901 (119906119899)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))minus120582119908
119899]
119908119899isin 119878 (119906
119899)
1003817100381710038171003817119908119899 minus 119908119899+1
1003817100381710038171003817
le (1 +1
119899 + 1)D (119878 (119906
119899) 119878 (119906
119899+1))
(54)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
If119873(119908 V) = 0 for all119908 V isin 119883 thenAlgorithm 15 reducesto the following algorithm for solving the problem (45)
Algorithm 17 For any 1199060isin 119883 compute the sequence 119906
119899 by
the following
119901 (119906119899) = 119877119867(sdotsdotsdot)-120578120582119872
[119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))]
(55)
for all 119899 = 0 1 2 and 120582 gt 0 is a constant
Now we prove the existence of a solution of problem (43)and the convergence of Algorithm 15
Theorem 18 Let 119883 be a real Hilbert space Let 120578119873 119883 times
119883 rarr 119883 119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901
119883 rarr 119883 be single-valued mappings and 119878 119879 119883 rarr CB(119883)
and 119872 119883 rarr 2119883 be the set-valued mappings such that 119872
is generalized119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 119879 are D-Lipschitz continuous with constants 1198971and
1198972 respectively
(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(iii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877(v) 119873 is 120598
1-Lipschitz continuous in the first argument with
respect to 119878 and 1205982-Lipschitz continuous in the second
argument with respect to 119879
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) lt 120585 (119903 minus 120582119898) (56)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Mathematics
Then generalized set-valued variational inclusion problem(43) has a solution (119906 119908 V) isin 119883 and the iterative sequences119906119899 119901(119906
119899) 119908
119899 and V
119899 generated by Algorithm 15 con-
verge strongly to 119906 119901(119906) 119908 and V respectively
Proof Since 119878 119879 are D-Lipschitz continuous with constants1198971and 1198972 respectively it follows from (52) and (53) that
1003817100381710038171003817119908119899+1 minus 119908119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119878 (119906
119899+1) 119878 (119906
119899))
le (1 +1
119899 + 1) 1198971
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
1003817100381710038171003817V119899+1 minus V119899
1003817100381710038171003817 le (1 +1
119899 + 1)D (119879 (119906
119899+1) 119879 (119906
119899))
le (1 +1
119899 + 1) 1198972
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
(57)
for 119899 = 0 1 2 It follows from (51) andTheorem 13 that
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
=100381710038171003817100381710038171003817119877119867(sdotsdotsdot)-120578120582119872
119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus120582119873 (119908119899 V119899)
minus [119877119867(sdotsdotsdot)-120578120582119872
119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))
minus120582119873 (119908119899minus1
V119899minus1
) ]100381710038171003817100381710038171003817
+120591
119903 minus 120582119898
1003817100381710038171003817119867 (119875 (119901 (119906119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867(119875(119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
+120591120582
119903 minus 120582119898
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
(58)
Since 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect
to 119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877 and 119901 is 120582119901-Lipschitz
continuous we have1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
=1003817100381710038171003817119867 (119875 (119901 (119906
119899)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899)) 119877 (119901 (119906
119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899)))
1003817100381710038171003817
+1003817100381710038171003817119867 (119875 (119901 (119906
119899minus1)) 119876 (119901 (119906
119899minus1)) 119877 (119901 (119906
119899)))
minus119867 (119875 (119901 (119906119899minus1
)) 119876 (119901 (119906119899minus1
)) 119877 (119901 (119906119899minus1
)))1003817100381710038171003817
le (1199031+ 1199032+ 1199033) 120582119901
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
(59)
Since 119873 is 1205981-Lipschitz continuous in the first argument
with respect to 119878 and 1205982- Lipschitz continuous in the second
argument with respect to 119879 and 119878 119879 areD-Lipschitz contin-uous with constants 119897
1and 1198972 respectively we have
1003817100381710038171003817119873 (119908119899 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le1003817100381710038171003817119873 (119908
119899 V119899) minus 119873 (119908
119899minus1 V119899)1003817100381710038171003817
+1003817100381710038171003817119873 (119908
119899minus1 V119899) minus 119873 (119908
119899minus1 V119899minus1
)1003817100381710038171003817
le 1205981
1003817100381710038171003817119908119899 minus 119908119899minus1
1003817100381710038171003817 + 1205982
1003817100381710038171003817V119899 minus V119899minus1
1003817100381710038171003817
le 1205981(1 +
1
119899)D (119878 (119906
119899) 119878 (119906
119899minus1))
+ 1205982(1 +
1
119899)D (119879 (119906
119899) 119879 (119906
119899minus1))
le 1205981(1 +
1
119899) 1198971
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
+ 1205982(1 +
1
119899) 1198972
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817
= (12059811198971+ 12059821198972) (1 +
1
119899)1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(60)
Using (59) (60) in (58) we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
le
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
119903 minus 120582119898
times1003817100381710038171003817119906119899 minus 119906
119899minus1
1003817100381710038171003817
(61)
Using the 120585-strong monotonicity of 119901 we have
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817
ge ⟨119901 (119906119899+1
) minus 119901 (119906119899) 119906119899+1
minus 119906119899⟩
ge 1205851003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817
(62)
which implies that
1003817100381710038171003817119906119899+1 minus 119906119899
1003817100381710038171003817 le1
120585
1003817100381710038171003817119901 (119906119899+1
) minus 119901 (119906119899)1003817100381710038171003817 (63)
Combining (61) and (63) we have1003817100381710038171003817119906119899+1 minus 119906
119899
1003817100381710038171003817 le 120579119899
1003817100381710038171003817119906119899 minus 119906119899minus1
1003817100381710038171003817 (64)
where
120579119899=
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972) (1 + (1119899))
120585 (119903 minus 120582119898) (65)
Let
120579 =
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582 (120598
11198971+ 12059821198972)
120585 (119903 minus 120582119898) (66)
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 9
From (56) it is easy to see that 120579 lt 1 Therefore (64)implies that 119906
119899 is a Cauchy sequence in 119883 Since 119883 is a
Hilbert space there exists 119906 isin 119883 such that 119906119899rarr 119906 as 119899 rarr
infin From (57) 119908119899 and V
119899 are also Cauchy sequences in119883
thus there exist 119908 V isin 119883 such that 119908119899
rarr 119908 and V119899
rarr V
as 119899 rarr infin By the continuity of 119901 119877119867(sdotsdotsdot)-120578120582119872
119867 120578119873 119875 119876 119877and (51) of Algorithm 15 we have
119901 (119906)
= 119877119867(sdotsdotsdot)-120578120582119872
[119867(119875(119901 (119906)) 119876(119901 (119906)) 119877(119901 (119906)))minus120582119873 (119908 V)]
(67)
Now we prove that 119908 isin 119878(119906) In fact since 119908119899isin 119878(119906
119899)
we have
119889 (119908 119878 (119906)) le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 119889 (119908119899 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 +D (119878 (119906119899) 119878 (119906))
le1003817100381710038171003817119908 minus 119908
119899
1003817100381710038171003817 + 1198971
1003817100381710038171003817119906119899 minus 1199061003817100381710038171003817 997888rarr 0 as 119899 997888rarr infin
(68)
which implies that 119889(119908 119878(119906)) = 0 Since 119878(119906) isin CB(119883) itfollows that119908 isin 119878(119906) Similarly it is easy to see that V isin 119879(119906)By Lemma 14 (119906 119908 V) is the solution of problem (43) Thiscompletes the proof
Based on Lemma 14 and Algorithm 16 Theorem 18reduced to the following result for solving problem (44)
Theorem19 Let119883 be a realHilbert space Let 120578 119883times119883 rarr 119883119867 119883 times 119883 times 119883 rarr 119883 119875119876 119877 119883 rarr 119883 and 119901 119883 rarr 119883
be single-valued mappings and let 119878 119883 rarr CB(119883) and119872 119883 rarr 2
119883 be the set-valued mappings such that 119872 isgeneralized 119867(119875119876 119877)-120578-cocoercive operator with respect tothe mappings 119875119876 and 119877 and range (119901) cap dom119872 = 0 andfor each 119906 isin 119883 let 119865(119906) sube 119901(119883) where 119865 is defined by (47)Assume that
(i) 119878 isD-Lipschitz continuous with constants 119897(ii) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is
120573-Lipschitz continuous(iii) 119901 is 120582
119901-Lipschitz continuous and 120585-strongly monotone
(iv) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 + 120582119897 lt 120585 (119903 minus 120582119898) (69)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(44) has a solution (119906 119908) isin 119883 and the iterative sequences 119906
119899
119901(119906119899) and 119908
119899 generated by Algorithm 16 converge strongly
to 119906 119901(119906) and 119908 respectively
Based on Lemma 14 and Algorithm 17 Theorem 18reduced to the following result for solving problem (45)
Theorem 20 Let 119883 be a real Hilbert space Let 120578 119883 times 119883 rarr
119883119867 119883times119883times119883 rarr 119883119875119876 119877 119883 rarr 119883 and119901 119883 rarr 119883 besingle-valuedmappings and let119872 119883 rarr 2
119883 be the set-valuedmappings such that 119872 is generalized 119867(119875119876 119877)-120578-cocoerciveoperator with respect to the mappings 119875119876 and 119877 and range(119901) cap dom119872 = 0 and for each 119906 isin 119883 let 119865(119906) sube 119901(119883)where 119865 is defined by (47) Assume that
(i) 120578 is 120591-Lipschitz continuous 119875 is 120572-expansive and 119876 is120573-Lipschitz continuous
(ii) 119901 is 120582119901-Lipschitz continuous and 120585-strongly monotone
(iii) 119867(119875119876 119877) is 1199031-Lipschitz continuous with respect to
119875 1199032-Lipschitz continuous with respect to 119876 and 119903
3-
Lipschitz continuous with respect to 119877
In addition
(1199031+ 1199032+ 1199033) 120582119901120591 lt 120585 (119903 minus 120582119898) (70)
where 119903 = 1205831205722minus 1205741205732+ 120575 gt 120582119898 and 120583 gt 120574 120572 gt 120573 120575 gt 0
Then generalized set-valued variational inclusion problem(45) has a solution 119906 isin 119883 and the iterative sequence 119906
119899 and
119901(119906119899) generated by Algorithm 17 converge strongly to 119906 and
119901(119906) respectively
Acknowledgments
The authors are grateful to the editor and referees for valuablecomments and suggestions
References
[1] J-P Aubin and A Cellina Differential Inclusions SpringerBerlin Germany 1984
[2] R Ahmad M Dilshad M-M Wong and J-C Yao ldquo119867(sdot sdot)-cocoercive operator and an application for solving generalizedvariational inclusionsrdquo Abstract and Applied Analysis vol 2011Article ID 261534 12 pages 2011
[3] C E Chidume K R Kazmi and H Zegeye ldquoIterative approx-imation of a solution of a general variational-like inclusionin Banach spacesrdquo International Journal of Mathematics andMathematical Sciences vol 2004 no 21ndash24 pp 1159ndash1168 2004
[4] X P Ding and J-C Yao ldquoExistence and algorithm of solutionsfor mixed quasi-variational-like inclusions in Banach spacesrdquoComputers amp Mathematics with Applications vol 49 no 5-6pp 857ndash869 2005
[5] Y-P Fang and N-J Huang ldquo119867-monotone operator and resol-vent operator technique for variational inclusionsrdquo AppliedMathematics and Computation vol 145 no 2-3 pp 795ndash8032003
[6] Y-P Fang and N-J Huang ldquo119867-accretive operators and resol-vent operator technique for solving variational inclusions inBanach spacesrdquo Applied Mathematics Letters vol 17 no 6 pp647ndash653 2004
[7] Y-P Fang N-J Huang and H B Thompson ldquoA new systemof variational inclusions with (119867 120578)-monotone operators inHilbert spacesrdquo Computers amp Mathematics with Applicationsvol 49 no 2-3 pp 365ndash374 2005
[8] K R Kazmi N Ahmad and M Shahzad ldquoConvergence andstability of an iterative algorithm for a system of generalized
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Mathematics
implicit variational-like inclusions in Banach spacesrdquo AppliedMathematics and Computation vol 218 no 18 pp 9208ndash92192012
[9] K R Kazmi M I Bhat and N Ahmad ldquoAn iterative algorithmbased on 119872-proximal mappings for a system of generalizedimplicit variational inclusions in Banach spacesrdquo Journal ofComputational and Applied Mathematics vol 233 no 2 pp361ndash371 2009
[10] K R Kazmi F A Khan and M Shahzad ldquoA system ofgeneralized variational inclusions involving generalized119867(sdot sdot)-accretive mapping in real 119902-uniformly smooth Banach spacesrdquoApplied Mathematics and Computation vol 217 no 23 pp9679ndash9688 2011
[11] K R Kazmi and F A Khan ldquoIterative approximation of asolution of multi-valued variational-like inclusion in Banachspaces a 119875-120578-proximal-point mapping approachrdquo Journal ofMathematical Analysis and Applications vol 325 no 1 pp 665ndash674 2007
[12] K R Kazmi and F A Khan ldquoSensitivity analysis for parametricgeneralized implicit quasi-variational-like inclusions involving119875-120578-accretive mappingsrdquo Journal of Mathematical Analysis andApplications vol 337 no 2 pp 1198ndash1210 2008
[13] H-Y Lan Y J Cho and R U Verma ldquoNonlinear relaxedcocoercive variational inclusions involving (119860 120578)-accretivemappings in Banach spacesrdquo Computers amp Mathematics withApplications vol 51 no 9-10 pp 1529ndash1538 2006
[14] RUVerma ldquoThe generalized relaxed proximal point algorithminvolving119860-maximal-relaxed accretive mappings with applica-tions to Banach spacesrdquoMathematical and ComputerModellingvol 50 no 7-8 pp 1026ndash1032 2009
[15] Z Xu and Z Wang ldquoA generalized mixed variational inclusioninvolving (119867(sdot sdot) 120578)-monotone operators in Banach spacesrdquoJournal of Mathematics Research vol 2 no 3 pp 47ndash56 2010
[16] Y-Z Zou and N-J Huang ldquo119867(sdot sdot)-accretive operator with anapplication for solving variational inclusions in Banach spacesrdquoAppliedMathematics and Computation vol 204 no 2 pp 809ndash816 2008
[17] Q H Ansari and J C Yao ldquoIterative schemes for solving mixedvariational-like inequalitiesrdquo Journal of Optimization Theoryand Applications vol 108 no 3 pp 527ndash541 2001
[18] S Karamardian ldquoThe nonlinear complementarity problemwith applications I IIrdquo Journal of Optimization Theory andApplications vol 4 pp 167ndash181 1969
[19] P Tseng ldquoFurther applications of a splitting algorithm todecomposition in variational inequalities and convex program-mingrdquoMathematical Programming vol 48 no 2 pp 249ndash2631990
[20] S B Nadler ldquoMultivalued contractionmappingrdquo Pacific Journalof Mathematics vol 30 no 3 pp 457ndash488 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
