Research Article The Partial Averaging of Fuzzy ...inclusions with the fuzzy right-hand side. eir...

Research ArticleThe Partial Averaging of Fuzzy Differential Inclusions onFinite Interval

Andrej V Plotnikov1 and Tatyana A Komleva2

1 Department of Applied Mathematics Odessa State Academy of Civil Engineering and Architecture Didrihsona Street 4Odessa 65029 Ukraine

2Department of Mathematics Odessa State Academy of Civil Engineering and Architecture Didrihsona Street 4Odessa 65029 Ukraine

Correspondence should be addressed to Andrej V Plotnikov a-plotnikovukrnet

Received 12 February 2014 Accepted 23 April 2014 Published 4 May 2014

Academic Editor Nikolai N Leonenko

Copyright copy 2014 A V Plotnikov and T A Komleva This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The substantiation of a possibility of application of partial averaging method on finite interval for differential inclusions with thefuzzy right-hand side with a small parameter is considered

1 Introduction

In 1990 Aubin [1] and Baidosov [2 3] introduced differentialinclusions with the fuzzy right-hand side Their approach isbased on usual differential inclusions In 1995 Hullermeier[4ndash6] introduced the concept of 119877-solution similar to how ithas been done in [7] Further in [8ndash20] various propertiesof solutions of fuzzy differential inclusions and their applica-tions at modeling of various natural-science processes wereconsidered

The averaging methods combined with the asymptoticrepresentations (in Poincare sense) began to be applied as thebasic constructive tool for solving the complicated problemsof analytical dynamics described by the differential equationsAfter the systematic researches done by N M Krylov NN Bogoliubov Yu A Mitropolsky and so forth in 1930sthe averaging method gradually became one of the classicalmethods in analyzing nonlinear oscillations

In works [21 22] the possibility of application of schemesof full and partial averaging for differential inclusions withthe fuzzy right-hand side containing a small parameter wasproved By proving these theorems the scheme offered byPlotnikov et al for a substantiation of schemes of an averageof usual differential inclusions [23ndash28] was used In this workthe possibility of application of partial averaging method for

fuzzy differential inclusions without passage to reviewing ofseparate solutions is proved that is all estimations are spentfor 119877-solution corresponding fuzzy systems

2 Preliminaries

Let comp(119877119899)(conv(119877119899)) be a family of all nonempty (convex)compact subsets from the space119877119899 with theHausdorffmetric

ℎ (119860 119861) = min119903ge0

119878119903(119860) sup 119861 119878

119903(119861) sup 119860 (1)

where 119860 119861 isin comp(119877119899) and 119878119903(119860) is 119903-neighborhood of set

119860Let 119864119899 be a family of all 119906 119877119899 rarr [0 1] such that 119906

satisfies the following conditions

(1) 119906 is normal that is there exists an 1199090isin 119877119899 such that

119906(1199090) = 1

(2) 119906 is fuzzy convex that is

119906 (120582119909 + (1 minus 120582) 119910) ge min 119906 (119909) 119906 (119910) (2)

for any 119909 119910 isin 119877119899 and 0 le 120582 le 1

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2014 Article ID 307941 5 pageshttpdxdoiorg1011552014307941

brought to you by COREView metadata citation and similar papers at coreacuk

provided by Crossref

2 International Journal of Differential Equations

(3) 119906 is upper semicontinuous that is for any1199090isin 119877119899 and

120576 gt 0 exists 120575(1199090 120576) gt 0 such that 119906(119909) lt 119906(119909

0) + 120576

whenever 119909 minus 1199090 lt 120575(119909

0 120576) 119909 isin 119877119899

(4) the closure of the set 119909 isin 119877119899 119906(119909) gt 0 is compact

If 119906 isin 119864119899 then 119906 is called a fuzzy number and 119864119899 is saidto be a fuzzy number space

Definition 1 The set 119909 isin 119877119899 119906(119909) ge 120572 is called the 120572-level[119906]120572 of a fuzzy number 119906 isin 119864119899 for 0 lt 120572 le 1 The closure of

the set 119909 isin 119877119899 119906(119909) gt 0 is called the 0-level [119906]0 of a fuzzynumber 119906 isin 119864119899

It is clear that the set [119906]120572 isin conv(119877119899) for all 0 le 120572 le 1

Theorem 2 (see [29] (stacking theorem)) If 119906 isin 119864119899 then

(1) [119906]120572 isin conv(119877119899) for all 120572 isin [0 1](2) [119906]1205722 sub [119906]1205721 for all 0 le 120572

1le 1205722le 1

(3) if 120572119896 is a nondecreasing sequence converging to 120572 gt 0

then [119906]120572 = ⋂119896ge1[119906]120572119896

Conversely if 119860120572 120572 isin [0 1] is the family of subsets of 119877119899

satisfying conditions (1)ndash(3) then there exists 119906 isin 119864119899 such that[119906]120572

= 119860120572for 0 lt 120572 le 1 and [119906]0 = ⋃

0lt120572le1119860120572sub 1198600

Let 120579 be the fuzzy number defined by 120579(119909) = 0 if 119909 = 0and 120579(0) = 1

Define119863 119864119899 times 119864119899 rarr [0infin) by the relation

119863(119906 V) = sup0le120572le1

ℎ ([119906]120572

[V]120572) (3)

Then119863 is a metric in 119864119899 Further we know that [30]

(i) (119864119899 119863) is a complete metric space(ii) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for all 119906 V 119908 isin 119864119899(iii) 119863(120582119906 120582V) = |120582|119863(119906 V) for all 119906 V isin 119864119899 and 120582 isin 119877

3 Fuzzy Differential Inclusion 119877-Solution

Consider the fuzzy differential inclusion

isin 119865 (119905 119909) 119909 (0) isin 1198830 (4)

where 119909 isin 119877119899 119905 isin [0 119879] sub 119877+119865 [0 119879]times119877119899 rarr 119864119899119883

0isin 119864119899

We interpret (4) as a family of differential inclusions (see[7 9 10])

120572isin [119865 (119905 119909

120572)]120572

119909120572(0) isin [119883

0]120572

120572 isin [0 1] (5)

An 120572-solution 119909120572(sdot) of (4) is understood to be an

absolutely continuous function 119909120572 [0 119879] rarr 119877

119899 whichsatisfies (5) almost everywhere Let119883

120572denote the 120572-solution

set of (5) and let 119883120572(119905) = 119909

120572(119905) 119909

120572(sdot) isin 119883

120572 Clearly a

family of subsets 119883119905= 119883120572(119905) 120572 isin [0 1] cannot satisfy the

conditions of Theorem 2 (see [5 6 9])Therefore we will consider an 119877-solution of fuzzy differ-

ential inclusion (4)

Definition 3 The upper semicontinuous fuzzy mapping 119883 [0 119879] rarr 119864

119899 which satisfies the system

sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + int

119905+120590

119905

[119865 (119904 119909)]120572d119904)

= 119900 (120590) 119883 (0) = 1198830

(6)

is called the 119877-solution119883(sdot) of differential inclusion (4) wherelim120590rarr0

+

(119900(120590)120590) = 0

Theorem 4 Suppose that the following conditions hold

(1) fuzzy mapping 119865(sdot 119909) is measurable for all 119909 isin 119877119899(2) there exists 120582 gt 0 such that for all 1199091015840 11990910158401015840 isin 119877119899

119863(119865 (119905 1199091015840

) 119865 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817 (7)

for almost every 119905 isin [0 119879]

(3) there exists 120574 gt 0 such that119863(119865(119905 119909) 0) le 120574 for almostevery 119905 isin [0 119879] and every 119909 isin 119877119899

(4) for all 120573 isin [0 1] 1199091015840 11990910158401015840 isin 119877119899 and almost every 119905 isin[0 119879]

120573119865 (119905 1199091015840

) + (1 minus 120573) 119865 (119905 11990910158401015840

) sub F (t 1205731199091015840 + (1 minus 120573) 11990910158401015840) (8)

Then there exists a unique 119877-solution 119883(sdot) of fuzzy system (4)defined on the interval [0 120591] sube [0 119879]

Proof Let 119878119903(1198830) = 119883 isin 119864

119899

119863(1198831198830) le 119903 and 120591 =

min119879 119903120574By [5 6] it follows that a family of subsets 119883

119905= 119883120572(119905)

120572 isin [0 1] satisfy the conditions of Theorem 2 that is 119883119905isin

119864119899 for every 119905 isin [0 120591]Divide the interval [0 120591] into partial intervals by the

points 119905119875119896= 1198961205912

minus119901 119896 = 0 119875 119875 = 2119901 119901 isin 119873 We useEuler algorithm let the mapping119883119875(sdot) be given by

[119883119875

(119905)]120572

= ⋃

119909isin[119883(119905119875

119896)]120572

119909 + int

119905

119905119875

119896

[119865 (119904 119909)]120572d119904 (9)

where 119905 isin [119905119875119896 119905119875

119896+1] 119896 isin 0 119875119883(0) = 119883

0 120572 isin [0 1]

By [7 28] it follows that the sequence [119883119875(sdot)]120572infin119901=1

isequicontinuous and fundamental and its limit is a unique 119877-solution [119883(sdot)]120572 of differential inclusion (5) and [119883(119905)]120572 =119883120572(119905) for every 119905 isin [0 120591] and 120572 isin [0 1] This concludes the

proof

Also we consider the differential inclusion

119910 isin 119866 (119905 119910) 119910 (0) isin 1198840 (10)

where 119910 isin 119877119899 119905 isin [0 119879] sub 119877+ 119865 [0 119879]times119877119899 rarr 119864119899 119884

0isin 119864119899

International Journal of Differential Equations 3

Lemma 5 Let 119865(119905 119909) and 119866(119905 119910) satisfy conditions (1)ndash(4) ofTheorem 4 and there exist 120578 gt 0 and 120583 gt 0 such that

119863(int

1199052

1199051

119865 (119904 119909) 119889119904 int

1199052

1199051

119866 (119904 119909) 119889119904) lt 120578 (1199052minus 1199051)

119863 (1198830 1198840) lt 120583

(11)

for every 119909 isin 119877119899 and 1199052gt 1199051 1199051 1199052isin [0 119879]

Then 119863(119883(119905) 119884(119905)) le 120583119890120582119905 + (120578120582)(119890120582119905 minus 1) for every 119905 isin[0 119879]

Proof Divide the interval [0 119879] into partial intervals by thepoints 119905119898

119896= 119896Δ Δ = (119879119898) 119896 = 0 119898 119898 isin 119873 By

Definition 3 we have

119863 (119883 (119905) 119884 (119905))

le sup120572isin[01]

ℎ( ⋃

119909isin[119883(119905119896)]120572

119909 + int

119905

119905119896

[119865 (119904 119909)]120572d119904

⋃

119910isin[119884(119905119896)]120572

119910 + int

119905

119905119896

[119866 (119904 119910)]120572d119904)

+ 119900 (Δ)


ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ 119900 (Δ)


ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

[119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ([119884 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ 119900 (Δ)


ℎ(int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ ([119883 (119905119896)]120572

[119884 (119905119896)]120572

)


ℎ(int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904) + 119900 (Δ)

le int

119905

119905119896

120582119863 (119883 (119905119896) 119884 (119905

119896)) d119904

+ 119863 (119883 (119905119896) 119884 (119905

119896)) + 120578 (119905 minus 119905

119896) + 119900 (Δ)

le ((119905 minus 119905119896) 120582 + 1)119863 (119883 (119905

119896) 119884 (119905

119896))

+ 120578 (119905 minus 119905119896) + 119900 (Δ) le 120583119890

120582119905

+120578

120582(119890120582119905

minus 1)

(12)

for every 119905 isin [0 119879] This concludes the proof

Remark 6 If 1198830= 1198840 then 119863(119883(119905) 119884(119905)) le (120578120582)(119890120582119905 minus 1)

for every 119905 isin [0 119879]

4 The Method of Partial Averaging

Now consider the fuzzy differential inclusion with a smallparameter

isin 120576119865 (119905 119909) 119909 (0) isin 1198830 (13)

where119883 isin 119877119899 119905 isin 119877+ 119865 119877

+times119877119899

rarr 1198641198991198830isin 119864119899 and 120576 gt 0

is a small parameterIn this work we associate the following partial averaged

fuzzy differential inclusion with the inclusion (10)

119910 isin 120576119866 (119905 119910) 119910 (0) isin 1198830 (14)

where 119866 119877+times 119877119899

rarr 119864119899 such that

lim119879rarrinfin

119863(1

119879int

119879

0

119865 (119905 119909) d119905 1119879int

119879

0

119866 (119905 119909) d119905) = 0 (15)

Theorem 7 Let in domain 119876 = (119905 119909) 119905 ge 0 119909 isin 119863 isin119888119900119899V(119877119899) the following conditions hold

(1) mappings 119865(sdot 119909) 119866(sdot 119909) are measurable on 119877+

(2) mappings 119865(119905 sdot) 119866(119905 sdot) satisfy a Lipschitz condition

119863(119865 (119905 1199091015840

) 119865 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817

119863 (119866 (119905 1199091015840

) 119866 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817

(16)

with a Lipschitz constant 120582 gt 0


(3) there exists 120574 gt 0 such that

119863(119865 (119905 119909) 0) le 120574 119863 (119866 (119905 119909) 0) le 120574 (17)

for almost every 119905 isin [0 119879] and every 119909 isin 119877119899


120573119865 (119905 1199091015840

) + (1 minus 120573) 119865 (119905 11990910158401015840

)

sub 119865 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

120573 119866 (119905 1199091015840

) + (1 minus 120573)119866 (119905 11990910158401015840

)

sub 119866 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

(18)

(5) limit (15) exists uniformly with respect to x in thedomain G

(6) for any 1198830([1198830]0

sub 1198631015840

sub 119863) 120576 isin (0 ]] and119905 gt 0 the 119877-solution of the inclusion (10) togetherwith a 120588-neighborhood belongs to the domain G thatis [119883(119905)]0 + 119878

120588(0) sub 119863 for every 119905 gt 0

Then for any 120578 isin (0 120588] and L gt 0 there exists 1205760(120578 119871) gt 0

such that for all 120576 isin (0 1205760] and 119905 isin [0 119871120576minus1] the following

inequality holds

119863(119883 (119905) 119884 (119905)) lt 120578 (19)

where 119883(sdot) 119884(sdot) are the 119877-solutions of initial and partialaveraged inclusions

Proof Divide the interval [0 119871120576minus1] on the partial intervals bythe points 119905

119896= (119896119871120576119898) 119896 isin 0 1 119898 minus 1 We denote

fuzzy mappings119883119898(sdot) and 119884119898(sdot) such that

[119883119898

(119905)]120572

= ⋃

119909isin[119883(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904

[119883119898

(0)]120572

= [1198830]120572

(20)

[119884119898

(119905)]120572

= ⋃

119910isin[119884(119905119896)]120572

119910 + 120576int

119905

119905119896

[119866 (119904 119910)]120572d119904

[119884119898

(0)]120572

= [1198830]120572

(21)

for every 120572 isin [0 1] 119905 isin [119905119896 119905119896+1] 119896 isin 0 1 119898 minus 1

Then

119863(119883119898

(119905119896) 119883 (119905

119896))


ℎ( ⋃

119909isin[119883119898(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904

⋃

119909isin[119883(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904)

+ 119900 (119905119896minus 119905119896minus1)

le (1 + 120576 (119905119896minus 119905119896minus1) 120582)119863 (119883

119898

(119905119896minus1) 119883 (119905

119896minus1))

+ 119900 (119905119896minus 119905119896minus1) le119900 (119905119896minus 119905119896minus1)

119905119896minus 119905119896minus1

(119890120582119871

minus 1)

(22)

Also we take

119863(119884119898

(119905119896) 119884 (119905

119896)) le119900 (119905119896minus 119905119896minus1)

119905119896minus 119905119896minus1

(119890120582119871

minus 1) (23)

As for 119905 isin [119905119896 119905119896+1]

119863(119883119898

(119905) 119883119898

(119905119896))


ℎ

times ( ⋃

119909isin[119883119898(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904 [119883119898 (119905

119896)]120572

)

le 120576120574 (119905 minus 119905119896) le120574119871

119898

(24)

119863(119884119898

(119905) 119884119898

(119905119896)) le 120576120574 (119905 minus 119905

119896) le120574119871

119898 (25)

Using estimates (22)ndash(25) for any 120578 gt 0 there exists 1198980

such that for119898 gt 1198980 we have

119863(119883119898

(119905) 119883 (119905)) le120578

4

119863 (119884119898

(119905) 119884 (119905)) le120578

4

(26)

Taking into account Lemma 5 for any ] gt 0 there exists1205760gt 0 such that for all 120576 isin (0 120576

0] the following inequality

holds

119863(119883119898

(119905119896+1) 119884 (119905

119896+1)) le

]120582(119890120582119871

minus 1) (27)

By combining (26) and (27) and choosing 119898 ge

max1198980 8120574119871120578 and ] lt (1205781205824(119890120582 119871 minus 1)) we obtain (19) The

theorem is proved


5 Conclusion

If 119865(sdot 119909) is continuous on [0 119879] then instead of (5) it ispossible to consider the following more simple equation

sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + 120590[119865 (119905 119909)]120572

)

= 119900 (120590) 119883 (0) = 1198830

(28)

and similarly we can prove all the results received earlier

Conflict of Interests

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

References

[1] J-P Aubin ldquoFuzzy differential inclusionsrdquo Problems of Controland Information Theory vol 19 no 1 pp 55ndash67 1990

[2] V A Baidosov ldquoDifferential inclusions with fuzzy right-handsiderdquo Soviet Mathematics vol 40 no 3 pp 567ndash569 1990

[3] VA Baidosov ldquoFuzzy differential inclusionsrdquo Journal of AppliedMathematics and Mechanics vol 54 no 1 pp 8ndash13 1990

[4] E Hullermeier ldquoTowards modelling of fuzzy functionsrdquo in Pro-ceedings of the 3rd European Congress on Intelligent Techniquesand Soft Computing (EUFIT rsquo95) pp 150ndash154 1995

[5] E Hullermeier ldquoAn approach to modelling and simulation ofuncertain dynamical systemrdquo International Journal of Uncer-tainty Fuzziness and Knowledge-Based Systems vol 7 pp 117ndash137 1997

[6] E Hullermeier ldquoA fuzzy simulation methodrdquo in Proceedings ofthe 1st International ICSC Symposium on Intelligent IndustrialAutomation (IIA rsquo96) and Soft Computing (SOCO rsquo96) ReadingUK March 1996

[7] A I Panasyuk and V I Panasyuk ldquoAn equation generated bya differential inclusionrdquo Mathematical Notes of the Academy ofSciences of the USSR vol 27 no 3 pp 213ndash218 1980

[8] S Abbasbandy T A Viranloo O Lopez-Pouso and J JNieto ldquoNumerical methods forfuzzy differential inclusionsrdquoComputers amp Mathematics with Applications vol 48 no 10-11pp 1633ndash1641 2004

[9] R P Agarwal D OrsquoRegan andV Lakshmikantham ldquoA stackingtheorem approach for fuzzy differential equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 55 no 3 pp299ndash312 2003

[10] R P Agarwal D OrsquoRegan and V Lakshmikantham ldquoMaximalsolutions and existence theory for fuzzy differential and integralequationsrdquo Journal of Applied Analysis vol 11 no 2 pp 171ndash1862005

[11] P L Antonelli and V Krivan ldquoFuzzy differential inclusions assubstitutes for stochastic differential equations in populationbiologyrdquo Open Systems amp Information Dynamics vol 1 no 2pp 217ndash232 1992

[12] M Chen D Li and X Xue ldquoPeriodic problems of first orderuncertain dynamical systemsrdquo Fuzzy Sets and Systems vol 162no 1 pp 67ndash78 2011

[13] M Chen and C Han ldquoPeriodic behavior of semi-linear uncer-tain dynamical systemsrdquo Fuzzy Sets and Systems vol 230 no 1pp 82ndash91 2013

[14] G Colombo and V Krivan ldquoFuzzy differential inclusions andnonprobabilistic likelihoodrdquo Dynamic Systems and Applica-tions vol 1 pp 419ndash440 1992

[15] M Guo X Xue and R Li ldquoImpulsive functional differentialinclusions and fuzzy population modelsrdquo Fuzzy Sets and Sys-tems vol 138 no 3 pp 601ndash615 2003

[16] V Laksmikantham ldquoSet differential equations versus fuzzydifferential equationsrdquo Applied Mathematics and Computationvol 164 no 2 pp 277ndash297 2005

[17] V Lakshmikantham T Granna Bhaskar and J VasundharaDevi Theory of Set Differential Equations in Metric SpacesCambridge Scientific Publishers Cambridge UK 2006

[18] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions Taylor amp Francis LondonUK 2003

[19] D Li M Chen and X Xue ldquoTwo-point boundary valueproblems of uncertain dynamical systemsrdquo Fuzzy Sets andSystems vol 179 no 1 pp 50ndash61 2011

[20] K K Majumdar and D D Majumder ldquoFuzzy differentialinclusions in atmospheric and medical cyberneticsrdquo IEEETransactions on Systems Man and Cybernetics B Cyberneticsvol 34 no 2 pp 877ndash887 2004

[21] A V Plotnikov T A Komleva and L I Plotnikova ldquoThe partialaveraging of differential inclusions with fuzzy right-hand siderdquoJournal of AdvancedResearch inDynamical andControl Systemsvol 2 no 2 pp 26ndash34 2010

[22] A V Plotnikov T A Komleva and L I Plotnikova ldquoOn theaveraging of differential inclusions with fuzzy right-hand sidewhen the average of the right-hand side is absentrdquo IranianJournal of Optimization vol 2 no 3 pp 506ndash517 2010

[23] S Klymchuk A Plotnikov and N Skripnik ldquoOverview of VAPlotnikovrsquos research on averaging of differential inclusionsrdquoPhysica D Nonlinear Phenomena vol 241 no 22 pp 1932ndash19472012

[24] N A Perestyuk V A Plotnikov A M Samoılenko andN V Skripnik Differential Equations with Impulse EffectsMultivalued Right-Hand Sides with Discontinuities vol 40 ofde Gruyter Studies in Mathematics Walter de Gruyter BerlinGermany 2011

[25] V A Plotnikov ldquoAveraging of differential inclusionsrdquoUkrainianMathematical Journal vol 31 no 5 pp 454ndash457 1979

[26] V A Plotnikov ldquoPartial averaging of differential inclusionsrdquoMathematical Notes of the Academy of Sciences of the USSR vol27 no 6 pp 456ndash459 1980

[27] V A Plotnikov The Averaging Method in Control ProblemsLybid Kiev Ukraine 1992

[28] V A Plotnikov A V Plotnikov and A N Vityuk DifferentialEquations with a Multivalued Right-Hand Side AsymptoticMethods AstroPrint Odessa Ukraine 1999

[29] C V Negoita and D A Ralescu Application of Fuzzy Sets toSystems Analysis JohnWiley amp Sons New York NY USA 1975

[30] M L Puri and D A Ralescu ldquoFuzzy random variablesrdquo Journalof Mathematical Analysis and Applications vol 114 no 2 pp409ndash422 1986

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(3) 119906 is upper semicontinuous that is for any1199090isin 119877119899 and

120576 gt 0 exists 120575(1199090 120576) gt 0 such that 119906(119909) lt 119906(119909

0) + 120576

whenever 119909 minus 1199090 lt 120575(119909

0 120576) 119909 isin 119877119899

(4) the closure of the set 119909 isin 119877119899 119906(119909) gt 0 is compact

If 119906 isin 119864119899 then 119906 is called a fuzzy number and 119864119899 is saidto be a fuzzy number space

Definition 1 The set 119909 isin 119877119899 119906(119909) ge 120572 is called the 120572-level[119906]120572 of a fuzzy number 119906 isin 119864119899 for 0 lt 120572 le 1 The closure of

the set 119909 isin 119877119899 119906(119909) gt 0 is called the 0-level [119906]0 of a fuzzynumber 119906 isin 119864119899

It is clear that the set [119906]120572 isin conv(119877119899) for all 0 le 120572 le 1

Theorem 2 (see [29] (stacking theorem)) If 119906 isin 119864119899 then

(1) [119906]120572 isin conv(119877119899) for all 120572 isin [0 1](2) [119906]1205722 sub [119906]1205721 for all 0 le 120572

1le 1205722le 1

(3) if 120572119896 is a nondecreasing sequence converging to 120572 gt 0

then [119906]120572 = ⋂119896ge1[119906]120572119896

Conversely if 119860120572 120572 isin [0 1] is the family of subsets of 119877119899

satisfying conditions (1)ndash(3) then there exists 119906 isin 119864119899 such that[119906]120572

= 119860120572for 0 lt 120572 le 1 and [119906]0 = ⋃

0lt120572le1119860120572sub 1198600

Let 120579 be the fuzzy number defined by 120579(119909) = 0 if 119909 = 0and 120579(0) = 1

Define119863 119864119899 times 119864119899 rarr [0infin) by the relation

119863(119906 V) = sup0le120572le1

ℎ ([119906]120572

[V]120572) (3)

Then119863 is a metric in 119864119899 Further we know that [30]

(i) (119864119899 119863) is a complete metric space(ii) 119863(119906 + 119908 V + 119908) = 119863(119906 V) for all 119906 V 119908 isin 119864119899(iii) 119863(120582119906 120582V) = |120582|119863(119906 V) for all 119906 V isin 119864119899 and 120582 isin 119877

3 Fuzzy Differential Inclusion 119877-Solution

Consider the fuzzy differential inclusion

isin 119865 (119905 119909) 119909 (0) isin 1198830 (4)

where 119909 isin 119877119899 119905 isin [0 119879] sub 119877+119865 [0 119879]times119877119899 rarr 119864119899119883

0isin 119864119899

We interpret (4) as a family of differential inclusions (see[7 9 10])

120572isin [119865 (119905 119909

120572)]120572

119909120572(0) isin [119883

0]120572

120572 isin [0 1] (5)

An 120572-solution 119909120572(sdot) of (4) is understood to be an

absolutely continuous function 119909120572 [0 119879] rarr 119877

119899 whichsatisfies (5) almost everywhere Let119883

120572denote the 120572-solution

set of (5) and let 119883120572(119905) = 119909

120572(119905) 119909

120572(sdot) isin 119883

120572 Clearly a

family of subsets 119883119905= 119883120572(119905) 120572 isin [0 1] cannot satisfy the

conditions of Theorem 2 (see [5 6 9])Therefore we will consider an 119877-solution of fuzzy differ-

ential inclusion (4)

Definition 3 The upper semicontinuous fuzzy mapping 119883 [0 119879] rarr 119864

119899 which satisfies the system

sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + int

119905+120590

119905

[119865 (119904 119909)]120572d119904)

= 119900 (120590) 119883 (0) = 1198830

(6)

is called the 119877-solution119883(sdot) of differential inclusion (4) wherelim120590rarr0

+

(119900(120590)120590) = 0

Theorem 4 Suppose that the following conditions hold

(1) fuzzy mapping 119865(sdot 119909) is measurable for all 119909 isin 119877119899(2) there exists 120582 gt 0 such that for all 1199091015840 11990910158401015840 isin 119877119899

119863(119865 (119905 1199091015840

) 119865 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817 (7)

for almost every 119905 isin [0 119879]

(3) there exists 120574 gt 0 such that119863(119865(119905 119909) 0) le 120574 for almostevery 119905 isin [0 119879] and every 119909 isin 119877119899


120573119865 (119905 1199091015840

) + (1 minus 120573) 119865 (119905 11990910158401015840

) sub F (t 1205731199091015840 + (1 minus 120573) 11990910158401015840) (8)

Then there exists a unique 119877-solution 119883(sdot) of fuzzy system (4)defined on the interval [0 120591] sube [0 119879]

Proof Let 119878119903(1198830) = 119883 isin 119864

119899

119863(1198831198830) le 119903 and 120591 =

min119879 119903120574By [5 6] it follows that a family of subsets 119883

119905= 119883120572(119905)

120572 isin [0 1] satisfy the conditions of Theorem 2 that is 119883119905isin

119864119899 for every 119905 isin [0 120591]Divide the interval [0 120591] into partial intervals by the

points 119905119875119896= 1198961205912

minus119901 119896 = 0 119875 119875 = 2119901 119901 isin 119873 We useEuler algorithm let the mapping119883119875(sdot) be given by

[119883119875

(119905)]120572

= ⋃

119909isin[119883(119905119875

119896)]120572

119909 + int

119905

119905119875

119896

[119865 (119904 119909)]120572d119904 (9)

where 119905 isin [119905119875119896 119905119875

119896+1] 119896 isin 0 119875119883(0) = 119883

0 120572 isin [0 1]

By [7 28] it follows that the sequence [119883119875(sdot)]120572infin119901=1

isequicontinuous and fundamental and its limit is a unique 119877-solution [119883(sdot)]120572 of differential inclusion (5) and [119883(119905)]120572 =119883120572(119905) for every 119905 isin [0 120591] and 120572 isin [0 1] This concludes the

proof

Also we consider the differential inclusion

119910 isin 119866 (119905 119910) 119910 (0) isin 1198840 (10)

where 119910 isin 119877119899 119905 isin [0 119879] sub 119877+ 119865 [0 119879]times119877119899 rarr 119864119899 119884

0isin 119864119899



119863(int

1199052

1199051

119865 (119904 119909) 119889119904 int

1199052

1199051

119866 (119904 119909) 119889119904) lt 120578 (1199052minus 1199051)

119863 (1198830 1198840) lt 120583

(11)




119896= 119896Δ Δ = (119879119898) 119896 = 0 119898 119898 isin 119873 By


119863 (119883 (119905) 119884 (119905))


ℎ( ⋃

119909isin[119883(119905119896)]120572

119909 + int

119905

119905119896

[119865 (119904 119909)]120572d119904

⋃

119910isin[119884(119905119896)]120572

119910 + int

119905

119905119896

[119866 (119904 119910)]120572d119904)

+ 119900 (Δ)


ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ 119900 (Δ)


ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

[119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ([119884 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ 119900 (Δ)


ℎ(int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ ([119883 (119905119896)]120572

[119884 (119905119896)]120572

)


ℎ(int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904) + 119900 (Δ)

le int

119905

119905119896

120582119863 (119883 (119905119896) 119884 (119905

119896)) d119904

+ 119863 (119883 (119905119896) 119884 (119905

119896)) + 120578 (119905 minus 119905

119896) + 119900 (Δ)

le ((119905 minus 119905119896) 120582 + 1)119863 (119883 (119905

119896) 119884 (119905

119896))

+ 120578 (119905 minus 119905119896) + 119900 (Δ) le 120583119890

120582119905

+120578

120582(119890120582119905

minus 1)

(12)



for every 119905 isin [0 119879]



isin 120576119865 (119905 119909) 119909 (0) isin 1198830 (13)

where119883 isin 119877119899 119905 isin 119877+ 119865 119877

+times119877119899

rarr 1198641198991198830isin 119864119899 and 120576 gt 0



119910 isin 120576119866 (119905 119910) 119910 (0) isin 1198830 (14)

where 119866 119877+times 119877119899


lim119879rarrinfin

119863(1

119879int

119879

0

119865 (119905 119909) d119905 1119879int

119879

0

119866 (119905 119909) d119905) = 0 (15)




119863(119865 (119905 1199091015840

) 119865 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817

119863 (119866 (119905 1199091015840

) 119866 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817

(16)




119863(119865 (119905 119909) 0) le 120574 119863 (119866 (119905 119909) 0) le 120574 (17)



120573119865 (119905 1199091015840

) + (1 minus 120573) 119865 (119905 11990910158401015840

)

sub 119865 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

120573 119866 (119905 1199091015840

) + (1 minus 120573)119866 (119905 11990910158401015840

)

sub 119866 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

(18)


(6) for any 1198830([1198830]0

sub 1198631015840


120588(0) sub 119863 for every 119905 gt 0



inequality holds

119863(119883 (119905) 119884 (119905)) lt 120578 (19)





[119883119898

(119905)]120572

= ⋃

119909isin[119883(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904

[119883119898

(0)]120572

= [1198830]120572

(20)

[119884119898

(119905)]120572

= ⋃

119910isin[119884(119905119896)]120572

119910 + 120576int

119905

119905119896

[119866 (119904 119910)]120572d119904

[119884119898

(0)]120572

= [1198830]120572

(21)


Then

119863(119883119898

(119905119896) 119883 (119905

119896))


ℎ( ⋃

119909isin[119883119898(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904

⋃

119909isin[119883(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904)

+ 119900 (119905119896minus 119905119896minus1)

le (1 + 120576 (119905119896minus 119905119896minus1) 120582)119863 (119883

119898

(119905119896minus1) 119883 (119905

119896minus1))


119905119896minus 119905119896minus1

(119890120582119871

minus 1)

(22)

Also we take

119863(119884119898

(119905119896) 119884 (119905

119896)) le119900 (119905119896minus 119905119896minus1)

119905119896minus 119905119896minus1

(119890120582119871

minus 1) (23)

As for 119905 isin [119905119896 119905119896+1]

119863(119883119898

(119905) 119883119898

(119905119896))


ℎ

times ( ⋃

119909isin[119883119898(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904 [119883119898 (119905

119896)]120572

)

le 120576120574 (119905 minus 119905119896) le120574119871

119898

(24)

119863(119884119898

(119905) 119884119898

(119905119896)) le 120576120574 (119905 minus 119905

119896) le120574119871

119898 (25)



119863(119883119898

(119905) 119883 (119905)) le120578

4

119863 (119884119898

(119905) 119884 (119905)) le120578

4

(26)



holds

119863(119883119898

(119905119896+1) 119884 (119905

119896+1)) le

]120582(119890120582119871

minus 1) (27)



theorem is proved


5 Conclusion


sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + 120590[119865 (119905 119909)]120572

)

= 119900 (120590) 119883 (0) = 1198830

(28)




References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119863(int

1199052

1199051

119865 (119904 119909) 119889119904 int

1199052

1199051

119866 (119904 119909) 119889119904) lt 120578 (1199052minus 1199051)

119863 (1198830 1198840) lt 120583

(11)




119896= 119896Δ Δ = (119879119898) 119896 = 0 119898 119898 isin 119873 By


119863 (119883 (119905) 119884 (119905))


ℎ( ⋃

119909isin[119883(119905119896)]120572

119909 + int

119905

119905119896

[119865 (119904 119909)]120572d119904

⋃

119910isin[119884(119905119896)]120572

119910 + int

119905

119905119896

[119866 (119904 119910)]120572d119904)

+ 119900 (Δ)


ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ 119900 (Δ)


ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

[119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ([119883 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ([119884 (119905119896)]120572

+ int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904

[119884 (119905119896)]120572

+ int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ 119900 (Δ)


ℎ(int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

int

119905

119905119896

[119865 (119904 [119884 (119905119896)]120572

)]120572

d119904)

+ sup120572isin[01]

ℎ ([119883 (119905119896)]120572

[119884 (119905119896)]120572

)


ℎ(int

119905

119905119896

[119865 (119904 [119883 (119905119896)]120572

)]120572

d119904

int

119905

119905119896

[119866 (119904 [119884 (119905119896)]120572

)]120572

d119904) + 119900 (Δ)

le int

119905

119905119896

120582119863 (119883 (119905119896) 119884 (119905

119896)) d119904

+ 119863 (119883 (119905119896) 119884 (119905

119896)) + 120578 (119905 minus 119905

119896) + 119900 (Δ)

le ((119905 minus 119905119896) 120582 + 1)119863 (119883 (119905

119896) 119884 (119905

119896))

+ 120578 (119905 minus 119905119896) + 119900 (Δ) le 120583119890

120582119905

+120578

120582(119890120582119905

minus 1)

(12)



for every 119905 isin [0 119879]



isin 120576119865 (119905 119909) 119909 (0) isin 1198830 (13)

where119883 isin 119877119899 119905 isin 119877+ 119865 119877

+times119877119899

rarr 1198641198991198830isin 119864119899 and 120576 gt 0



119910 isin 120576119866 (119905 119910) 119910 (0) isin 1198830 (14)

where 119866 119877+times 119877119899


lim119879rarrinfin

119863(1

119879int

119879

0

119865 (119905 119909) d119905 1119879int

119879

0

119866 (119905 119909) d119905) = 0 (15)




119863(119865 (119905 1199091015840

) 119865 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817

119863 (119866 (119905 1199091015840

) 119866 (119905 11990910158401015840

)) le 120582100381710038171003817100381710038171199091015840

minus 1199091015840101584010038171003817100381710038171003817

(16)




119863(119865 (119905 119909) 0) le 120574 119863 (119866 (119905 119909) 0) le 120574 (17)



120573119865 (119905 1199091015840

) + (1 minus 120573) 119865 (119905 11990910158401015840

)

sub 119865 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

120573 119866 (119905 1199091015840

) + (1 minus 120573)119866 (119905 11990910158401015840

)

sub 119866 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

(18)


(6) for any 1198830([1198830]0

sub 1198631015840


120588(0) sub 119863 for every 119905 gt 0



inequality holds

119863(119883 (119905) 119884 (119905)) lt 120578 (19)





[119883119898

(119905)]120572

= ⋃

119909isin[119883(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904

[119883119898

(0)]120572

= [1198830]120572

(20)

[119884119898

(119905)]120572

= ⋃

119910isin[119884(119905119896)]120572

119910 + 120576int

119905

119905119896

[119866 (119904 119910)]120572d119904

[119884119898

(0)]120572

= [1198830]120572

(21)


Then

119863(119883119898

(119905119896) 119883 (119905

119896))


ℎ( ⋃

119909isin[119883119898(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904

⋃

119909isin[119883(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904)

+ 119900 (119905119896minus 119905119896minus1)

le (1 + 120576 (119905119896minus 119905119896minus1) 120582)119863 (119883

119898

(119905119896minus1) 119883 (119905

119896minus1))


119905119896minus 119905119896minus1

(119890120582119871

minus 1)

(22)

Also we take

119863(119884119898

(119905119896) 119884 (119905

119896)) le119900 (119905119896minus 119905119896minus1)

119905119896minus 119905119896minus1

(119890120582119871

minus 1) (23)

As for 119905 isin [119905119896 119905119896+1]

119863(119883119898

(119905) 119883119898

(119905119896))


ℎ

times ( ⋃

119909isin[119883119898(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904 [119883119898 (119905

119896)]120572

)

le 120576120574 (119905 minus 119905119896) le120574119871

119898

(24)

119863(119884119898

(119905) 119884119898

(119905119896)) le 120576120574 (119905 minus 119905

119896) le120574119871

119898 (25)



119863(119883119898

(119905) 119883 (119905)) le120578

4

119863 (119884119898

(119905) 119884 (119905)) le120578

4

(26)



holds

119863(119883119898

(119905119896+1) 119884 (119905

119896+1)) le

]120582(119890120582119871

minus 1) (27)



theorem is proved


5 Conclusion


sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + 120590[119865 (119905 119909)]120572

)

= 119900 (120590) 119883 (0) = 1198830

(28)




References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119863(119865 (119905 119909) 0) le 120574 119863 (119866 (119905 119909) 0) le 120574 (17)



120573119865 (119905 1199091015840

) + (1 minus 120573) 119865 (119905 11990910158401015840

)

sub 119865 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

120573 119866 (119905 1199091015840

) + (1 minus 120573)119866 (119905 11990910158401015840

)

sub 119866 (119905 1205731199091015840

+ (1 minus 120573) 11990910158401015840

)

(18)


(6) for any 1198830([1198830]0

sub 1198631015840


120588(0) sub 119863 for every 119905 gt 0



inequality holds

119863(119883 (119905) 119884 (119905)) lt 120578 (19)





[119883119898

(119905)]120572

= ⋃

119909isin[119883(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904

[119883119898

(0)]120572

= [1198830]120572

(20)

[119884119898

(119905)]120572

= ⋃

119910isin[119884(119905119896)]120572

119910 + 120576int

119905

119905119896

[119866 (119904 119910)]120572d119904

[119884119898

(0)]120572

= [1198830]120572

(21)


Then

119863(119883119898

(119905119896) 119883 (119905

119896))


ℎ( ⋃

119909isin[119883119898(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904

⋃

119909isin[119883(119905119896minus1)]120572

119909 + 120576int

119905119896

119905119896minus1

[119865 (119904 119909)]120572d119904)

+ 119900 (119905119896minus 119905119896minus1)

le (1 + 120576 (119905119896minus 119905119896minus1) 120582)119863 (119883

119898

(119905119896minus1) 119883 (119905

119896minus1))


119905119896minus 119905119896minus1

(119890120582119871

minus 1)

(22)

Also we take

119863(119884119898

(119905119896) 119884 (119905

119896)) le119900 (119905119896minus 119905119896minus1)

119905119896minus 119905119896minus1

(119890120582119871

minus 1) (23)

As for 119905 isin [119905119896 119905119896+1]

119863(119883119898

(119905) 119883119898

(119905119896))


ℎ

times ( ⋃

119909isin[119883119898(119905119896)]120572

119909 + 120576int

119905

119905119896

[119865 (119904 119909)]120572d119904 [119883119898 (119905

119896)]120572

)

le 120576120574 (119905 minus 119905119896) le120574119871

119898

(24)

119863(119884119898

(119905) 119884119898

(119905119896)) le 120576120574 (119905 minus 119905

119896) le120574119871

119898 (25)



119863(119883119898

(119905) 119883 (119905)) le120578

4

119863 (119884119898

(119905) 119884 (119905)) le120578

4

(26)



holds

119863(119883119898

(119905119896+1) 119884 (119905

119896+1)) le

]120582(119890120582119871

minus 1) (27)



theorem is proved


5 Conclusion


sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + 120590[119865 (119905 119909)]120572

)

= 119900 (120590) 119883 (0) = 1198830

(28)




References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusion


sup120572isin[01]

ℎ([119883 (119905 + 120590)]120572

⋃

119909isin[119883(119905)]120572

119909 + 120590[119865 (119905 119909)]120572

)

= 119900 (120590) 119883 (0) = 1198830

(28)




References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article The Partial Averaging of Fuzzy ...inclusions with the fuzzy right-hand side. eir...

Documents

Transcript of Research Article The Partial Averaging of Fuzzy ...inclusions with the fuzzy right-hand side. eir...