Research Article A Generalization of a Gregu Fixed Point...

Research ArticleA Generalization of a Greguš Fixed Point Theorem inMetric Spaces

Marwan A Kutbi1 A Amini-Harandi23 and N Hussain1

1 Department of Mathematics King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Department of Pure Mathematics University of Shahrekord Shahrekord 88186-34141 Iran3 School of Mathematics Institute for Research in Fundamental Sciences (IPM) Tehran 19395-5746 Iran

Correspondence should be addressed to N Hussain nhusainkauedusa

Received 30 January 2014 Accepted 22 February 2014 Published 25 March 2014

Academic Editor Giuseppe Marino

Copyright copy 2014 Marwan A Kutbi 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 first introduce a new class of contractive mappings in the setting of metric spaces and then we present certain Gregus type fixedpoint theorems for such mappings As an application we derive certain Gregus type common fixed theorems Our results extendGregus fixed point theorem inmetric spaces and generalize and unify some related results in the literature An example is also givento support our main result

1 Introduction and Preliminaries

Let 119883 be a Banach space and let 119862 be a closed convex subsetof119883 In 1980 Gregus [1] proved the following result

Theorem 1 Let 119879 119862 rarr 119862 be a mapping satisfying theinequality

1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 119886

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 119887 119909 minus 119879119909 + 119888

1003817100381710038171003817119910 minus 1198791199101003817100381710038171003817 (1)

for all 119909 119910 isin 119862 where 0 lt 119886 lt 1 119887 119888 ge 0 and 119886 + 119887 + 119888 = 1Then 119879 has a unique fixed point

Fisher and Sessa [2] Jungck [3] and Hussain et al [4]obtained common fixed point generalizations of Theorem 1In recent years many theorems which are closely related toGregusrsquos Theorem have appeared (see [1ndash21]) Very recentlyMoradi and Farajzadeh [21] extended Gregus fixed pointtheorem in complete metric spaces

Theorem 2 ([21 Theorem 24]) Let (119883 119889) be a completemetric space and let 119879 119883 rarr 119883 be a mapping such thatfor all 119909 119910 isin 119883

119889 (119879119909 119879119910) le 119886119889 (119909 119910) + 119887119889 (119909 119879119909)

+ 119888119889 (119910 119879119910) + 119890119889 (119910 119879119909) + 119891119889 (119909T119910) (2)

where 0 lt 119886 lt 1 119887 119888 119890 119891 ge 0 119887 + 119888 gt 0 119890 + 119891 gt 0 and119886 + 119887 + 119888 + 119890 + 119891 = 1 Then 119879 has a unique fixed point

Let 119868 and 119879 be self-maps of 119883 A point 119909 isin 119883 is acoincidence point (resp common fixed point) of 119868 and 119879 if119868119909 = 119879119909 (resp 119909 = I119909 = 119879119909) The pair 119868 119879 is called (1)commuting if119879119868119909 = 119868119879119909 for all119909 isin 119883 (2) weakly commuting[2] if for all 119909 isin 119883 119889(119868119879119909 119879119868119909) le 119889(119868119909 119879119909) (3) compatible[3] if lim

119899119889(119879119868119909

119899 119868119879119909119899) = 0 whenever 119909

119899 is a sequence

such that lim119899119879119909119899= lim119899119868119909119899= 119905 for some 119905 in119883 (4) weakly

compatible if they commute at their coincidence points thatis if 119868119879119909 = 119879119868119909 whenever 119868119909 = 119879119909 Clearly commutingmaps are weakly commuting and weakly commuting mapsare compatible References [2 3] give examples which showthat neither implication is reversible

The purpose of this paper is to define and to investigate aclass of new generalized contractive mappings (not necessar-ily continuous) on metric spaces We will prove certain fixedpoint and common fixed results which are generalizations ofthe above mentioned theorems

2 Fixed Point Results

We denote by R+the set of all nonnegative real numbers

and by U the set of all functions 119906 R5+

rarr R+satisfying

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 580297 5 pageshttpdxdoiorg1011552014580297

2 Journal of Applied Mathematics

the following conditions

(C1) 119906 is continuous

(C2) 119906(1199051 1199052 1199053 1199054 1199055) is nondecreasing in 119905

1 1199052 1199053 and 119905

5

(C3) 119904 lt 119905 rArr 119906(119904 119904 119905 0 119904 + 119905) lt 119905 for each 119904 119905 gt 0

(C4) 119906(119905 119905 119905 0 119906(2119905 119905 119905 119905 3119905)) lt 119905 for each 119905 gt 0

(C5) 119906(119905 0 0 119905 119905) lt 119905 for each 119905 gt 0

Example 3 If 119906(1199051 1199052 1199053 1199054 1199055) = 119886119905

1+ 1198871199052+ 1198871199053+ 1198901199054+ 1198901199055for

119905119894isin R+ where 0 lt 119886 lt 1 119887 119890 gt 0 and 119886 + 2119887 + 2119890 = 1 then

119906 isin U

Example 4 If 119906(1199051 1199052 1199053 1199054 1199055) = 119896max119905

1 1199052 1199053 (1199054+ 1199055)2

for 119905119894isin R+ where 119896 isin [0 1) then 119906 isin U

Example 5 Let 119906 isin UThen it is easy to see that119908 isin Uwhere

119908 (1199051 1199052 1199053 1199054 1199055)

= 119906 (1199051 1199052 1199053 1199054 1199055) + 119871min 119905

1 1199052 1199053 1199054 1199055

(3)

for each 119871 ge 0

Now we are ready to state our main result

Theorem 6 Let (119883 119889) be a complete metric space and let 119879

119883 rarr 119883 be a mapping satisfying

119889 (119879119909 119879119910)

le 119906 (119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910T119910) 119889 (119910 119879119909) 119889 (119909 119879119910))

(4)

for each 119909 119910 isin 119883 where 119906 isin U Then 119879 has a unique fixedpoint

Proof We first show that 120572 = inf119909isin119883

119889(119909 119879119909) = 0 If119889(119909 119879119909) = 0 for some 119909 isin 119883 then 119909 is a fixed point of 119879and we are done So we may assume that 119889(119909 119879119909) gt 0 foreach 119909 isin 119883 From (4) (C

2) and (C

3) we have

119889 (119879119911 1198792

119911)

le 119906 (119889 (119911 119879119911) 119889 (119911 119879119911) 119889 (119879119911 1198792

119911) 0 119889 (119911 1198792

119911))

le 119906 (119889 (119911 119879119911) 119889 (119911 119879119911) 119889 (119879119911 1198792

119911) 0

119889 (119911 119879119911) + 119889 (119879119911 1198792

119911))

(5)

and so

119889 (119879119911 1198792

119911) le 119889 (119911 119879119911) for each 119911 isin 119883 (6)

Now let 119909119899 be a sequence such that

120572 = inf119909isin119883

119889 (119909 119879119909) = lim119899rarrinfin

119889 (119909119899 119879119909119899) (7)

From (4) (6) and (C2) we get

119889 (119879119909119899 1198793

119909119899)

le 119906 (119889 (119909119899 1198792

119909119899) 119889 (119909

119899 119879119909119899) 119889 (119879

2

119909119899 1198793

119909119899)

119889 (1198792

119909119899 119879119909119899) 119889 (119909

119899 1198793

119909119899))

le 119906 (119889 (119909119899 119879119909119899) + 119889 (119879119909

119899 1198792

119909119899) 119889 (119909

119899 119879119909119899)

119889 (1198792

119909119899 1198793

119909119899) 119889 (119879

2

119909119899 119879119909119899) 119889 (119909

119899 119879119909119899)

+ 119889 (119879119909119899 1198792

119909119899) +119889 (119879

2

119909119899 1198793

119909119899))

le 119906 (2119889 (119909119899 119879119909119899) 119889 (119909

119899 119879119909119899) 119889 (119909

119899 119879119909119899)

119889 (119909119899 119879119909119899) 3119889 (119909

119899 119879119909119899))

(8)

for each 119899 isin N From (4) (6) (8) and (C2) we have

119889 (1198792

119909119899 1198793

119909119899)

le 119906 (119889 (119879119909119899 1198792

119909119899) 119889 (119879119909

119899 1198792

119909119899)

119889 (1198792

119909119899 1198793

119909119899) 0 119889 (119879119909

119899 1198793

119909119899))

le 119906 (119889 (119909119899 119879119909119899) 119889 (119909

119899 119879119909119899) 119889 (119879

2

119909119899 1198793

119909119899) 0

119906 (2119889 (119909119899 119879119909119899) 119889 (119909

119899 119879119909119899) 119889 (119909

119899 119879119909119899)

119889 (119909119899 119879119909119899) 3119889 (119909

119899 119879119909119899)))

(9)

for each 119899 isin N From (6) and (7) we get

120572 le 119889 (1198792

119909119899 1198793

119909119899) le 119889 (119909

119899 119879119909119899) (10)

and so by (7)

lim119899rarrinfin

119889 (1198792

119909119899 1198793

119909119899) = 120572 (11)

From (7) (9) (11) and (C1) we obtain

120572 le 119906 (120572 120572 120572 0 119906 (2120572 120572 120572 120572 3120572)) (12)

Hence by (C4)

120572 = inf119909isin119883

119889 (119909 119879119909) = 0 (13)

Now let

119862119899= 119909 isin 119883 119889 (119909 119879119909) le

1

119899 for each 119899 isin N (14)

Notice that by (13) 119862119899

= 0 for each 119899 isin N We show that

lim119899rarrinfin

diam (119862119899) = 0 (15)

On the contrary assume that there are sequences 119909119899 and

119910119899 with 119909

119899 119910119899isin 119862119899satisfying

120588 = lim119899rarrinfin

119889 (119909119899 119910119899) gt 0 (16)

Journal of Applied Mathematics 3

From (4) (14) and (C2) we have

119889 (119909119899 119910119899) le 119889 (119909

119899 119879119909119899) + 119889 (119879119909

119899 119879119910119899) + 119889 (119910

119899 119879119910119899)

le2

119899+ 119906 (119889 (119909

119899 119910119899) 119889 (119909

119899 119879119909119899) 119889 (119910

119899 119879119910119899)

119889 (119910119899 119879119909119899) 119889 (119909

119899 119879119910119899))

le2

119899+ 119906 (119889 (119909

119899 119910119899) 119889 (119909

119899 119879119909119899) 119889 (119910

119899 119879119910119899)

119889 (119909119899 119910119899) + 119889 (119909

119899 119879119909119899)

119889 (119909119899 119910119899) + 119889 (119910

119899 119879119910119899))

(17)

for each 119899 isin N From (14) (16) (17) and (C1) we get

120588 le 119906 (120588 0 0 120588 120588) (18)

which contradicts (C5) Thus (15) holds Hence 119862

119899

is a decreasing sequence of closed nonempty sets withdiam(119862

119899) rarr 0 and so by Cantorrsquos intersection theorem

infin

⋂

119899=1

119862119899= 119909 for some 119909 isin 119883 (19)

We show that 119909 is a fixed point of119879 Since 119909 isin 119862119899 there exists

119906119899isin 119862119899such that 119889(119909 119906

119899) lt 1119899 for all 119899 isin N Now for each

119899 isin N we have

119889 (119909 119879119909) le 119889 (119909 119906119899) + 119889 (119906

119899 119879119906119899) + 119889 (119879119906

119899 119879119909)

le2

119899+ 119906 (119889 (119906

119899 119909) 119889 (119906

119899 119879119906119899) 119889 (119909 119879119909)

119889 (119909 119879119906119899) 119889 (119906

119899 119879119909))

le2

119899+ 119906 (119889 (119906

119899 119909) 119889 (119906

119899 119879119906119899) 119889 (119909 119879119909)

119889 (119909 119906119899) + 119889 (119906

119899 119879119906119899) 119889 (119906

119899 119909)

+119889 (119909 119879119909))

(20)

Since 119906 is continuous from (20)

119889 (119909 119879119909) le 119906 (0 0 119889 (119909 119879119909) 0 119889 (119909 119879119909)) (21)

and hence by (C3) 119889(119909 119879119909) = 0 that is 119879119909 = 119909 To prove

the uniqueness note that if 119910 is a fixed point of 119879 then 119910 isin

⋂infin

119899=1119862119899sube ⋂infin

119899=1119862119899= 119909 and hence 119909 = 119910

Remark 7 Note that to prove Theorem 2 we may assumethat 119887 = 119888 and 119891 = 119890 (see the proof of Theorem 24 in[21]) Thus by Example 3 Theorem 6 is a generalization ofthe above mentionedTheorem 2 of Moradi and Farajzadeh

If we take 119906 as in Example 4 from Theorem 6 we get themain result of Ciric [10]

The following corollary improves Theorem 24 in [6]

Corollary 8 Let (119883 119889) be a complete metric space and let 119879


119889 (119879119909 119879119910)

le 119896max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119910 119879119909) + 119889 (119909 119879119910)

2

+ 119871min 119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119910 119879119909) 119889 (119909 119879119910)

(22)

for each 119909 119910 isin 119883 where 119896 isin [0 1) and 119871 ge 0 Then 119879 has aunique fixed point

As an easy consequence of the axiom of choice [13 page5] AC5 for every function 119891 119883 rarr 119883 there is a function 119892

such that 119863(119892) = 119877(119891) and for every 119909 isin 119863(119892) 119891(119892119909) = 119909]we obtain the following lemma (see also [22])

Lemma 9 Let 119883 be a nonempty set and let 119892 119883 rarr 119883 be amapping Then there exists a subset 119864 sube 119883 such that 119892(119864) =

119892(119883) and 119892 119864 rarr 119883 is one-to-one

As an application of Theorem 6 we now establish acommon fixed point result

Theorem 10 Let (119883 119889) be a metric space and let 119892119879 119883 rarr

119883 be mappings satisfying

119889 (119879119909 119879119910) le 119906 (119889 (119892119909 119892119910) 119889 (119892119909 119879119909)

119889 (119892119910 119879119910) 119889 (119892119910 119879119909) 119889 (119892119909 119879119910))

(23)

for each 119909 119910 isin 119883 where 119906 isin U Suppose that 119879119883 sube 119892119883 and119892119883 is complete subspace of 119883 Then 119892 and 119879 have a uniquecoincidence point Further if 119892 and 119879 are weakly compatiblethen they have a unique common fixed point

Proof ByLemma 9 there exists119864 sube 119883 such that119892(119864) = 119892(119883)

and 119892 119864 rarr 119883 is one-to-one We define a mapping 119866

119892(119864) rarr 119892(119864) by

119866 (119892119909) = 119879 (119909) (24)

for all 119892119909 isin 119892(119864) As 119892 is one-to-one on 119892(119864) and 119879(119883) sube

119892(119883) 119866 is well defined Thus it follows from (23) and (24)that

119889 (119866 (119892119909) 119866 (119892119910)) = 119889 (119879119909 119879119910)

le 119906 (119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910))

(25)

for all 119892119909 119892119910 isin 119892(119883) = 119892(119864) Thus the function 119866 119892(119864) rarr

119892(119864) satisfies all conditions of Theorem 6 so 119866 has a uniquefixed point 119911 isin 119892(119883) As 119911 isin 119892(119883) there exists 119908 isin 119883 such


that 119911 = 119892(119908) Thus 119879(119908) = 119866(119892119908) = 119866(119911) = 119911 = 119892(119908)

which implies that 119892 and 119879 have a unique coincidence pointFurther if 119892 and 119879 are weakly compatible then they have aunique common fixed point

If we take 119906 as in Example 5 then from Theorem 10 weobtain the following result which extendsmany related resultsin the literature (see [16 17])



119889 (119879119909 119879119910)

le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

+ 119871min 119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910)

(26)

for each 119909 119910 isin 119883 where 119896 isin [0 1) and 119871 ge 0 Suppose that119879119883 sube 119892119883 and 119892119883 is a complete subspace of 119883 Then 119892 and 119879

have a unique coincidence point Further if 119892 and119879 are weaklycompatible then they have a unique common fixed point

Corollary 12 Let (119883 119889) be a metric space and let 119892119879 119883 rarr


119889 (119879119909 119879119910) le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

(27)

for each 119909 119910 isin 119883 where 119896 isin [0 1) Suppose that119879119883 sube 119892119883 and119892119883 is a complete subspace of 119883 Then 119892 and 119879 have a uniquecoincidence point Further if 119892 and 119879 are weakly compatiblethen they have a unique common fixed point

As a linear continuous operator defined on a closed subsetof a normed space is closed operator we obtain the followingnew common fixed point results as corollaries toTheorem 11

Corollary 13 (see Fisher and Sessa [2]) Let 119879 and 119892 be twoweakly commuting mappings on a closed convex subset 119862 of aBanach space119883 into itself satisfying the inequality1003817100381710038171003817119879119909 minus 119879119910

1003817100381710038171003817 le 119886max 1003817100381710038171003817119879119909 minus 1198921199091003817100381710038171003817

1003817100381710038171003817119879119910 minus 1198921199101003817100381710038171003817

+ 119871min 1003817100381710038171003817119892119909 minus 119892119910

1003817100381710038171003817 1003817100381710038171003817119879119909 minus 119892119909

1003817100381710038171003817 1003817100381710038171003817119879119910 minus 119892119910

1003817100381710038171003817

1003817100381710038171003817119892119910 minus 1198791199091003817100381710038171003817

1003817100381710038171003817119892119909 minus 1198791199101003817100381710038171003817

(28)

for all 119909 119910 isin 119862 where 119886 isin (0 1) and 119871 ge 0 If 119892 is linear andnonexpansive on 119862 and 119879(119862) sube 119892(119862) then 119879 and 119892 have aunique common fixed point in 119862

Corollary 14 (see Jungck [3]) Let 119879 and 119892 be compatible self-maps of a closed convex subset 119862 of a Banach space119883 Supposethat 119892 is continuous and linear and that 119879(119862) sub 119892(119862) If 119879 and119892 satisfy inequality (28) then 119879 and 119892 have a unique commonfixed point in 119862

Now we illustrate our main result by the followingexample

Example 15 Let119883 = 1 2 3 4 and let119889(1 2) = 54119889(1 3) =1 119889(1 4) = 74 and 119889(2 3) = 119889(2 4) = 119889(3 4) = 2 Then(119883 119889) is a complete metric space Let119879 119883 rarr 119883 be given by1198791 = 11198792 = 41198793 = 4 and1198794 = 1Then it is straightforwardto show that

119889 (119879119909 119879119910) le7

8max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119909 119879119910) + 119889 (119910 119879119909)

2

(29)

for each 119909 119910 isin 119883 Then by Corollary 8 119879 has a unique fixedpoint (119909 = 1 is the unique fixed point of 119879)

Now we show that 119879 does not satisfy the conditionof Theorem 2 of Moradi and Farajzadeh On the contraryassume that there exist nonnegative numbers 0 lt 119886 lt 1119887 119888 119890 119891 ge 0 119887 + 119888 gt 0 119890 + 119891 gt 0 such that

119889 (119879119909 119879119910) le 119886119889 (119909 119910) + 119887119889 (119909 119879119909) + 119888119889 (119910 119879119910)

+ 119890119889 (119910 119879119909) + 119891119889 (119909 119879119910)

(30)

for all 119909 119910 isin 119883 Let (1199091 1199101) = (1 2) and (119909

2 1199102) = (2 1)

Then from (30) we have

7

4le

5

4119886 + 2119888 +

5

4119890 +

7

4119891

7

4le

5

4119886 + 2119887 +

7

4119890 +

5

4119891

(31)

which yield

7 le 5119886 + 4119887 + 4119888 + 6119890 + 6119891 le 6 (119886 + 119887 + 119888 + 119890 + 119891) le 6

(32)

a contradictionThus we cannot invoke the above mentionedtheorem of Moradi and Farajzadeh (Theorem 2) to show theexistence of a fixed point of 119879

Remark 16 The technique of proof of Theorem 6 is inline with the proof of Theorem 24 in [23] Thereforethe reader interested in fixed point results for generalizedcontractionsnonexpansive mappings in the general setup ofuniformly convex metric spaces is referred to [23]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah Therefore Mar-wan A Kutbi and N Hussain acknowledge with thanks DSRKAU for financial support A Amini-Harandi was partiallysupported by the Center of Excellence for MathematicsUniversity of Shahrekord Iran and by a Grant from IPM (no92470412)

References

[1] M Gregus ldquoA fixed point theorem in Banach spacerdquo Bollettinodella Unione Matematica Italiana A vol 5 pp 193ndash198 1980

[2] B Fisher and S Sessa ldquoOn a fixed point theorem of GregusrdquoInternational Journal of Mathematics and Mathematical Sci-ences vol 9 pp 23ndash28 1986

[3] G Jungck ldquoOn a fixed point theorem of Fisher and Sessardquo Inter-national Journal of Mathematics andMathematical Sciences vol13 pp 497ndash500 1990

[4] N Hussain B E Rhoades and G Jungck ldquoCommon fixedpoint and invariant approximation results for Gregus type I-contractionsrdquoNumerical Functional Analysis and Optimizationvol 28 no 9-10 pp 1139ndash1151 2007

[5] A Aliouche ldquoCommon fixed point theorems of Gregss type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[6] V Berinde ldquoSome remarks on a fixed point theorem for Ciric-type almost contractionsrdquo Carpathian Journal of Mathematicsvol 25 no 2 pp 157ndash162 2009

[7] L B Ciric ldquoOn some discontinuous fixed point mappings inconvex metric spacesrdquo Czechoslovak Mathematical Journal vol43 no 118 pp 319ndash326 1993

[8] L B Ciric ldquoOn a generalization of a Gregus fixed pointtheoremrdquo Czechoslovak Mathematical Journal vol 50 no 125pp 449ndash458 2000

[9] L B Ciric ldquoOn a common fixed point theorem of a Gregustyperdquo Publications de lrsquoInstitutMathematique vol 49 no 63 pp174ndash178 1991

[10] L B Ciric ldquoGeneralized contractions and fixed-point theo-remsrdquo Publications de lrsquoInstitutMathematique vol 12 pp 19ndash261971

[11] M L Diviccaro B Fisher and S Sessa ldquoA common fixed pointtheorem of Gregus typerdquo Publicationes Mathematicae Debrecenvol 34 pp 83ndash89 1987

[12] B Fisher ldquoCommon fixed points on a Banach spacerdquo ChungYuan vol 11 pp 19ndash26 1982

[13] R Herman and E R Jean Equivalents of the Axiom of ChoiceNorth-Holland Amsterdam The Netherlands 1970

[14] N J Huang and Y J Cho ldquoCommon fixed point theorems ofGregus type in convex metric spacesrdquo Japanese Mathematicsvol 48 pp 83ndash89 1998

[15] N Hussain ldquoCommon fixed points in best approximation forBanach operator pairs with Ciric type I-contractionsrdquo Journalof Mathematical Analysis and Applications vol 338 no 2 pp1351ndash1363 2008

[16] Y J Cho and N Hussain ldquoWeak contractions common fixedpoints and invariant approximationsrdquo Journal of Inequalitiesand Applications vol 2009 Article ID 390634 10 pages 2009

[17] G Jungck and N Hussain ldquoCompatible maps and invariantapproximationsrdquo Journal of Mathematical Analysis and Appli-cations vol 325 no 2 pp 1003ndash1012 2007

[18] L Bing-you ldquoFixed point theorem of nonexpansive mappingsin convex metric spacesrdquo Applied Mathematics and Mechanicsvol 10 no 2 pp 183ndash188 1989

[19] RNMukherjea andVVerma ldquoA note on a fixed point theoremof Gregusrdquo Japanese Journal of Mathematics vol 33 pp 745ndash749 1988

[20] P P Murthy Y J Cho and B Fisher ldquoCommon fixed points ofGregus type mappingsrdquo Glasnik Matematicki no 50 pp 335ndash341 1995

[21] S Moradi and A Farajzadeh ldquoOn Olalerursquos open problem onGregus fixed point theoremrdquo Journal of Global Optimizationvol 56 no 4 pp 1689ndash1697 2013

[22] R H Haghi S Rezapour and N Shahzad ldquoSome fixed pointgeneralizations are not real generalizationsrdquoNonlinear AnalysisTheory Methods and Applications vol 74 no 5 pp 1799ndash18032011

[23] H Fukhar-ud-din A R Khan and Z Akhtar ldquoFixed pointresults for a generalized nonexpansivemap in uniformly convexmetric spacesrdquo Nonlinear Analysis Theory Methods and Appli-cations vol 75 pp 4747ndash4760 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


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Journal of


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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the following conditions

(C1) 119906 is continuous

(C2) 119906(1199051 1199052 1199053 1199054 1199055) is nondecreasing in 119905

1 1199052 1199053 and 119905

5

(C3) 119904 lt 119905 rArr 119906(119904 119904 119905 0 119904 + 119905) lt 119905 for each 119904 119905 gt 0

(C4) 119906(119905 119905 119905 0 119906(2119905 119905 119905 119905 3119905)) lt 119905 for each 119905 gt 0

(C5) 119906(119905 0 0 119905 119905) lt 119905 for each 119905 gt 0

Example 3 If 119906(1199051 1199052 1199053 1199054 1199055) = 119886119905

1+ 1198871199052+ 1198871199053+ 1198901199054+ 1198901199055for

119905119894isin R+ where 0 lt 119886 lt 1 119887 119890 gt 0 and 119886 + 2119887 + 2119890 = 1 then

119906 isin U

Example 4 If 119906(1199051 1199052 1199053 1199054 1199055) = 119896max119905

1 1199052 1199053 (1199054+ 1199055)2

for 119905119894isin R+ where 119896 isin [0 1) then 119906 isin U

Example 5 Let 119906 isin UThen it is easy to see that119908 isin Uwhere

119908 (1199051 1199052 1199053 1199054 1199055)

= 119906 (1199051 1199052 1199053 1199054 1199055) + 119871min 119905

1 1199052 1199053 1199054 1199055

(3)

for each 119871 ge 0

Now we are ready to state our main result

Theorem 6 Let (119883 119889) be a complete metric space and let 119879


119889 (119879119909 119879119910)

le 119906 (119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910T119910) 119889 (119910 119879119909) 119889 (119909 119879119910))

(4)

for each 119909 119910 isin 119883 where 119906 isin U Then 119879 has a unique fixedpoint

Proof We first show that 120572 = inf119909isin119883

119889(119909 119879119909) = 0 If119889(119909 119879119909) = 0 for some 119909 isin 119883 then 119909 is a fixed point of 119879and we are done So we may assume that 119889(119909 119879119909) gt 0 foreach 119909 isin 119883 From (4) (C

2) and (C

3) we have

119889 (119879119911 1198792

119911)

le 119906 (119889 (119911 119879119911) 119889 (119911 119879119911) 119889 (119879119911 1198792

119911) 0 119889 (119911 1198792

119911))

le 119906 (119889 (119911 119879119911) 119889 (119911 119879119911) 119889 (119879119911 1198792

119911) 0

119889 (119911 119879119911) + 119889 (119879119911 1198792

119911))

(5)

and so

119889 (119879119911 1198792

119911) le 119889 (119911 119879119911) for each 119911 isin 119883 (6)

Now let 119909119899 be a sequence such that

120572 = inf119909isin119883

119889 (119909 119879119909) = lim119899rarrinfin

119889 (119909119899 119879119909119899) (7)

From (4) (6) and (C2) we get

119889 (119879119909119899 1198793

119909119899)

le 119906 (119889 (119909119899 1198792

119909119899) 119889 (119909

119899 119879119909119899) 119889 (119879

2

119909119899 1198793

119909119899)

119889 (1198792

119909119899 119879119909119899) 119889 (119909

119899 1198793

119909119899))

le 119906 (119889 (119909119899 119879119909119899) + 119889 (119879119909

119899 1198792

119909119899) 119889 (119909

119899 119879119909119899)

119889 (1198792

119909119899 1198793

119909119899) 119889 (119879

2

119909119899 119879119909119899) 119889 (119909

119899 119879119909119899)

+ 119889 (119879119909119899 1198792

119909119899) +119889 (119879

2

119909119899 1198793

119909119899))

le 119906 (2119889 (119909119899 119879119909119899) 119889 (119909

119899 119879119909119899) 119889 (119909

119899 119879119909119899)

119889 (119909119899 119879119909119899) 3119889 (119909

119899 119879119909119899))

(8)

for each 119899 isin N From (4) (6) (8) and (C2) we have

119889 (1198792

119909119899 1198793

119909119899)

le 119906 (119889 (119879119909119899 1198792

119909119899) 119889 (119879119909

119899 1198792

119909119899)

119889 (1198792

119909119899 1198793

119909119899) 0 119889 (119879119909

119899 1198793

119909119899))

le 119906 (119889 (119909119899 119879119909119899) 119889 (119909

119899 119879119909119899) 119889 (119879

2

119909119899 1198793

119909119899) 0

119906 (2119889 (119909119899 119879119909119899) 119889 (119909

119899 119879119909119899) 119889 (119909

119899 119879119909119899)

119889 (119909119899 119879119909119899) 3119889 (119909

119899 119879119909119899)))

(9)

for each 119899 isin N From (6) and (7) we get

120572 le 119889 (1198792

119909119899 1198793

119909119899) le 119889 (119909

119899 119879119909119899) (10)

and so by (7)

lim119899rarrinfin

119889 (1198792

119909119899 1198793

119909119899) = 120572 (11)

From (7) (9) (11) and (C1) we obtain

120572 le 119906 (120572 120572 120572 0 119906 (2120572 120572 120572 120572 3120572)) (12)

Hence by (C4)

120572 = inf119909isin119883

119889 (119909 119879119909) = 0 (13)

Now let

119862119899= 119909 isin 119883 119889 (119909 119879119909) le

1

119899 for each 119899 isin N (14)

Notice that by (13) 119862119899

= 0 for each 119899 isin N We show that

lim119899rarrinfin

diam (119862119899) = 0 (15)

On the contrary assume that there are sequences 119909119899 and

119910119899 with 119909

119899 119910119899isin 119862119899satisfying

120588 = lim119899rarrinfin

119889 (119909119899 119910119899) gt 0 (16)



119889 (119909119899 119910119899) le 119889 (119909

119899 119879119909119899) + 119889 (119879119909

119899 119879119910119899) + 119889 (119910

119899 119879119910119899)

le2

119899+ 119906 (119889 (119909

119899 119910119899) 119889 (119909

119899 119879119909119899) 119889 (119910

119899 119879119910119899)

119889 (119910119899 119879119909119899) 119889 (119909

119899 119879119910119899))

le2

119899+ 119906 (119889 (119909

119899 119910119899) 119889 (119909

119899 119879119909119899) 119889 (119910

119899 119879119910119899)

119889 (119909119899 119910119899) + 119889 (119909

119899 119879119909119899)

119889 (119909119899 119910119899) + 119889 (119910

119899 119879119910119899))

(17)


120588 le 119906 (120588 0 0 120588 120588) (18)


119899



infin

⋂

119899=1

119862119899= 119909 for some 119909 isin 119883 (19)


119906119899isin 119862119899such that 119889(119909 119906



119889 (119909 119879119909) le 119889 (119909 119906119899) + 119889 (119906

119899 119879119906119899) + 119889 (119879119906

119899 119879119909)

le2

119899+ 119906 (119889 (119906

119899 119909) 119889 (119906

119899 119879119906119899) 119889 (119909 119879119909)

119889 (119909 119879119906119899) 119889 (119906

119899 119879119909))

le2

119899+ 119906 (119889 (119906

119899 119909) 119889 (119906

119899 119879119906119899) 119889 (119909 119879119909)

119889 (119909 119906119899) + 119889 (119906

119899 119879119906119899) 119889 (119906

119899 119909)

+119889 (119909 119879119909))

(20)


119889 (119909 119879119909) le 119906 (0 0 119889 (119909 119879119909) 0 119889 (119909 119879119909)) (21)



⋂infin

119899=1119862119899sube ⋂infin

119899=1119862119899= 119909 and hence 119909 = 119910






119889 (119879119909 119879119910)

le 119896max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119910 119879119909) + 119889 (119909 119879119910)

2

+ 119871min 119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119910 119879119909) 119889 (119909 119879119910)

(22)









119889 (119879119909 119879119910) le 119906 (119889 (119892119909 119892119910) 119889 (119892119909 119879119909)

119889 (119892119910 119879119910) 119889 (119892119910 119879119909) 119889 (119892119909 119879119910))

(23)




119892(119864) rarr 119892(119864) by

119866 (119892119909) = 119879 (119909) (24)



119889 (119866 (119892119909) 119866 (119892119910)) = 119889 (119879119909 119879119910)

le 119906 (119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910))

(25)




that 119911 = 119892(119908) Thus 119879(119908) = 119866(119892119908) = 119866(119911) = 119911 = 119892(119908)





119889 (119879119909 119879119910)

le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

+ 119871min 119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910)

(26)





119889 (119879119909 119879119910) le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

(27)




1003817100381710038171003817 le 119886max 1003817100381710038171003817119879119909 minus 1198921199091003817100381710038171003817

1003817100381710038171003817119879119910 minus 1198921199101003817100381710038171003817

+ 119871min 1003817100381710038171003817119892119909 minus 119892119910

1003817100381710038171003817 1003817100381710038171003817119879119909 minus 119892119909

1003817100381710038171003817 1003817100381710038171003817119879119910 minus 119892119910

1003817100381710038171003817

1003817100381710038171003817119892119910 minus 1198791199091003817100381710038171003817

1003817100381710038171003817119892119909 minus 1198791199101003817100381710038171003817

(28)





119889 (119879119909 119879119910) le7

8max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119909 119879119910) + 119889 (119910 119879119909)

2

(29)



119889 (119879119909 119879119910) le 119886119889 (119909 119910) + 119887119889 (119909 119879119909) + 119888119889 (119910 119879119910)

+ 119890119889 (119910 119879119909) + 119891119889 (119909 119879119910)

(30)


2 1199102) = (2 1)


7

4le

5

4119886 + 2119888 +

5

4119890 +

7

4119891

7

4le

5

4119886 + 2119887 +

7

4119890 +

5

4119891

(31)

which yield

7 le 5119886 + 4119887 + 4119888 + 6119890 + 6119891 le 6 (119886 + 119887 + 119888 + 119890 + 119891) le 6

(32)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889 (119909119899 119910119899) le 119889 (119909

119899 119879119909119899) + 119889 (119879119909

119899 119879119910119899) + 119889 (119910

119899 119879119910119899)

le2

119899+ 119906 (119889 (119909

119899 119910119899) 119889 (119909

119899 119879119909119899) 119889 (119910

119899 119879119910119899)

119889 (119910119899 119879119909119899) 119889 (119909

119899 119879119910119899))

le2

119899+ 119906 (119889 (119909

119899 119910119899) 119889 (119909

119899 119879119909119899) 119889 (119910

119899 119879119910119899)

119889 (119909119899 119910119899) + 119889 (119909

119899 119879119909119899)

119889 (119909119899 119910119899) + 119889 (119910

119899 119879119910119899))

(17)


120588 le 119906 (120588 0 0 120588 120588) (18)


119899



infin

⋂

119899=1

119862119899= 119909 for some 119909 isin 119883 (19)


119906119899isin 119862119899such that 119889(119909 119906



119889 (119909 119879119909) le 119889 (119909 119906119899) + 119889 (119906

119899 119879119906119899) + 119889 (119879119906

119899 119879119909)

le2

119899+ 119906 (119889 (119906

119899 119909) 119889 (119906

119899 119879119906119899) 119889 (119909 119879119909)

119889 (119909 119879119906119899) 119889 (119906

119899 119879119909))

le2

119899+ 119906 (119889 (119906

119899 119909) 119889 (119906

119899 119879119906119899) 119889 (119909 119879119909)

119889 (119909 119906119899) + 119889 (119906

119899 119879119906119899) 119889 (119906

119899 119909)

+119889 (119909 119879119909))

(20)


119889 (119909 119879119909) le 119906 (0 0 119889 (119909 119879119909) 0 119889 (119909 119879119909)) (21)



⋂infin

119899=1119862119899sube ⋂infin

119899=1119862119899= 119909 and hence 119909 = 119910






119889 (119879119909 119879119910)

le 119896max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119910 119879119909) + 119889 (119909 119879119910)

2

+ 119871min 119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119910 119879119909) 119889 (119909 119879119910)

(22)









119889 (119879119909 119879119910) le 119906 (119889 (119892119909 119892119910) 119889 (119892119909 119879119909)

119889 (119892119910 119879119910) 119889 (119892119910 119879119909) 119889 (119892119909 119879119910))

(23)




119892(119864) rarr 119892(119864) by

119866 (119892119909) = 119879 (119909) (24)



119889 (119866 (119892119909) 119866 (119892119910)) = 119889 (119879119909 119879119910)

le 119906 (119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910))

(25)




that 119911 = 119892(119908) Thus 119879(119908) = 119866(119892119908) = 119866(119911) = 119911 = 119892(119908)





119889 (119879119909 119879119910)

le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

+ 119871min 119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910)

(26)





119889 (119879119909 119879119910) le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

(27)




1003817100381710038171003817 le 119886max 1003817100381710038171003817119879119909 minus 1198921199091003817100381710038171003817

1003817100381710038171003817119879119910 minus 1198921199101003817100381710038171003817

+ 119871min 1003817100381710038171003817119892119909 minus 119892119910

1003817100381710038171003817 1003817100381710038171003817119879119909 minus 119892119909

1003817100381710038171003817 1003817100381710038171003817119879119910 minus 119892119910

1003817100381710038171003817

1003817100381710038171003817119892119910 minus 1198791199091003817100381710038171003817

1003817100381710038171003817119892119909 minus 1198791199101003817100381710038171003817

(28)





119889 (119879119909 119879119910) le7

8max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119909 119879119910) + 119889 (119910 119879119909)

2

(29)



119889 (119879119909 119879119910) le 119886119889 (119909 119910) + 119887119889 (119909 119879119909) + 119888119889 (119910 119879119910)

+ 119890119889 (119910 119879119909) + 119891119889 (119909 119879119910)

(30)


2 1199102) = (2 1)


7

4le

5

4119886 + 2119888 +

5

4119890 +

7

4119891

7

4le

5

4119886 + 2119887 +

7

4119890 +

5

4119891

(31)

which yield

7 le 5119886 + 4119887 + 4119888 + 6119890 + 6119891 le 6 (119886 + 119887 + 119888 + 119890 + 119891) le 6

(32)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










that 119911 = 119892(119908) Thus 119879(119908) = 119866(119892119908) = 119866(119911) = 119911 = 119892(119908)





119889 (119879119909 119879119910)

le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

+ 119871min 119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) 119889 (119892119909 119879119910)

(26)





119889 (119879119909 119879119910) le 119896max119889 (119892119909 119892119910) 119889 (119892119909 119879119909) 119889 (119892119910 119879119910)

119889 (119892119910 119879119909) + 119889 (119892119909 119879119910)

2

(27)




1003817100381710038171003817 le 119886max 1003817100381710038171003817119879119909 minus 1198921199091003817100381710038171003817

1003817100381710038171003817119879119910 minus 1198921199101003817100381710038171003817

+ 119871min 1003817100381710038171003817119892119909 minus 119892119910

1003817100381710038171003817 1003817100381710038171003817119879119909 minus 119892119909

1003817100381710038171003817 1003817100381710038171003817119879119910 minus 119892119910

1003817100381710038171003817

1003817100381710038171003817119892119910 minus 1198791199091003817100381710038171003817

1003817100381710038171003817119892119909 minus 1198791199101003817100381710038171003817

(28)





119889 (119879119909 119879119910) le7

8max119889 (119909 119910) 119889 (119909 119879119909) 119889 (119910 119879119910)

119889 (119909 119879119910) + 119889 (119910 119879119909)

2

(29)



119889 (119879119909 119879119910) le 119886119889 (119909 119910) + 119887119889 (119909 119879119909) + 119888119889 (119910 119879119910)

+ 119890119889 (119910 119879119909) + 119891119889 (119909 119879119910)

(30)


2 1199102) = (2 1)


7

4le

5

4119886 + 2119888 +

5

4119890 +

7

4119891

7

4le

5

4119886 + 2119887 +

7

4119890 +

5

4119891

(31)

which yield

7 le 5119886 + 4119887 + 4119888 + 6119890 + 6119891 le 6 (119886 + 119887 + 119888 + 119890 + 119891) le 6

(32)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Generalization of a Gregu Fixed Point...

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