Research ArticleOn Darboux-Type Differential Inclusions with Uncertainty

Nayyar Mehmood 12 Ahmed Al-Rawashdeh 2 and Akbar Azam 3

1Department of Mathematics amp Statistics International Islamic University H-10 Islamabad Pakistan2Department of Mathematical Sciences College of Science UAE University 15551 Al Ain UAE3Department of Mathematics COMSATS Institute of Information Technology Islamabad Pakistan

Correspondence should be addressed to Ahmed Al-Rawashdeh aalrawashdehuaeuacae

Received 25 January 2019 Revised 12 April 2019 Accepted 20 May 2019 Published 16 July 2019

Academic Editor Michele Scarpiniti

Copyright copy 2019 Nayyar Mehmood 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

In this article we prove the existence results for solutions of the Darboux-type problems in fuzzy partial differential inclusionswith local conditions of integral types We present two problems involving open and closed level sets of a given fuzzy mappingIn the first case fuzzy differential inclusion has been transformed into an equivalent Darboux-type problem for partial differentialequations and then using the Tychonoff fixed point theorem we prove the existence result for this crisp case For the second casewe use Nadlerrsquos fixed point theorem and selection theorem of Kuratowski-Ryll-Nardzewski to find the solution of given differentialinclusions problem We furnish an example to validate our results

1 Introduction

The uncertainties that occur in modelling of the physicalproblems may originate some types of ambiguities Zadehin [1] introduced the fuzzy sets to deal with ambiguitiesFor modelling the physical problems in a better way theformulation of these problems by virtue of fuzzy differen-tial equations plays a significant role The notion of fuzzyderivatives was familiarized in [2] Afterwards this idea wasexplored by using Zadehrsquos extension principle (ZEP) in [3]The generalized Hukuhara derivative for fuzzy functions wasintroduced and studied in [4] For detailed overview of theoryof fuzzy differential equations we refer the readers to consultthe monographs [5 6]

The ambiguous and complex physical models can bedemonstrated in a convenient way by means of fuzzy dif-ferential equations (FDEs) The basic idea of fuzzy partialdifferential equations (FPDEs) has been originated in [7]Many of the physical problems were designed solved andanalyzed with fuzzy differential equations Many authors puttheir efforts to study and extend this theory see for example[8ndash12] The importance of both the ODEs and the PDEsin fuzzy sense is evident Up till now the theory of FDEshas been studied in various ways but the analysis of fuzzy

partial differential equations still needs to be explored Mostof the results are not concerned for fuzzy partial differentialinclusions In [13] the authors discuss the existence of solu-tion of a fuzzy differential inclusion problem constructed vialevel sets of fuzzy mappings In [14 15] the authors extendedthis technique for system of fuzzy differential inclusions Wegeneralize and extend this study to discuss the existenceof solutions of fuzzy partial differential inclusions (FPDIs)In this article we investigate two models for Darboux-typepartial fuzzy differential inclusions with local conditions ofintegral type We furnish an example to justify our mainresults

2 Preliminaries

LetR119896 be 119896-dimensional Euclidean space for 119860 sube R119896 119860 and119860119900 denote the closure and interior of119860 respectively Let119870(119883)be the family of all nonempty convex and compact subsetsof a given linear normed space 119883 The Hausdorff metric on119870(R119896) is defined as

ℎ119863 (119860 119861) = maxsup119887isin119861

119889 (119887 119860) sup119886isin119860

119889 (119886 119861) (1)

HindawiComplexityVolume 2019 Article ID 2161230 10 pageshttpsdoiorg10115520192161230

2 Complexity

(119870(R119896) ℎ119863) is a complete and separable metric space ByΩ119886119887119888 we mean the open subset (0 119886) times (0 119887) times (0 119888) of R3denote 119878120600120598 by an open ball with center at 120600 and radius 120598 gt0 For a given nonempty set 119883 let 2119883 be the family of allnonempty subsets of 119883Definition 1 (see [14]) A fuzzy subset 120596 of R119896 is a function120596 R119896 997888rarr [0 1] For 120574 isin [0 1] denote the 120574 open level and120574 closed level sets of 120596 by (120596)120574 and [120596]120574 respectively definedas follows

(120596)120574 = 120600 isin R119896 120596 (120600) gt 120574 (2)

and

[120596]120574 = 120600 isin R119896 120596 (120600) ge 120574 (3)

Definition 2 (see [14]) A fuzzy subset 120596 of R119896 is called(i) fuzzy normal set if there exists an 119905 isin R119896 such that120596(119905) = 1(ii) fuzzy convex fuzzy set if for 119904 119905 isin R119896 and 120582 isin [0 1]

120596 (120582119904 + (1 minus 120582) 119905) ge min 120596 (119904) 120596 (119905) (4)

(iii) fuzzy upper semicontinuous (usc) if for any 120574 isin(0 1] [120596]120574 is closed set in the usual topology on R119896

For a fuzzy subset 120596 of R119896 the support of 120596 is denotedand defined as

[120596]0 = (119904 isin R119896 120596 (119904) ge 0) (5)

Definition 3 (see [13]) (i)Let (119884 1205881) and (119885 1205882) be twometricspaces and let 120595 119884 997888rarr 2119885 be a set-valued mapping Then120595 is said to be upper semicontinuous (119906119904119888) on 119884 if for each120599 isin 119884 and any open set 119881 sub 119885 with 120595(120599) sub 119881 there exists anopen neighborhood 119880120599 of 120599 such that 120595(119880120599) sub 119881

(ii) 120595 is called lower semicontinuous (119897119904119888) on 119884 if foreach 120599 isin 119884 and any open set 119881 sub 119885 with 120595(120599) cap 119881 = 120601 thereexists an open neighborhood 119880120599 of 120599 such that 120595(120599) cap 119881 = 120601for all 120599 isin 119880120599Definition 4 (see [14]) Let 119864119896 denote the set of all fuzzy setsof R119896 that are fuzzy normal fuzzy convex and fuzzy upper119906119904119888 and have compact support

It is well known that for each 120596 isin 119864119896 and 120574 isin [0 1]the closed level sets [120596]120574 are compact and convex in the usualtopology on R119896 [14]

For any nonempty subset 119881 of R the fuzzy map 119881 times119881 997888rarr 119864119896 can be presented by a real valued function 119865 119881 times119881timesR119896 997888rarr [0 1] where 119904 119905 isin 119881 120600 isin R119896 we denote 119865(119904 119905 120600)by (119904119905)(120600)Definition 5 (see [14]) A fuzzy mapping Ω 997888rarr 119864119896 whereΩ sub Ω119886119887119888 times R119896 is called lower open if (120600120599120596)(120603) is 119897119904119888 ateach (120600 120599 120596) isin Ω

There is extraordinary useful weakening of compactnessthat is satisfied by many topological spaces that arise ingeometry and analysis called paracompactness Paracom-pactness is weaker than compactness and is often adequatefor many purposes For example R119896 with usual topology isnot compact but a paracomapct space Before defining theparacompact spaces we first recall some notions used in thedefinition of a paracompact space

Definition 6 (see [16]) An open covering of a topologicalspace 119883 is the collection of open sets so that their unioncovers (contains) 119883An open covering 119880119894 of119883 refines opencovering 119881119895 of 119883 if each 119880119894 is contained in some 119881119895Definition 7 (see [16]) An open covering 119880119894 of 119883 is locallyfinite if every 119909 isin 119883 admits a neighborhood 119873 such that 119873 cap119880119894 is empty for all but finitely many 119894

For example the covering of R by open intervals (119899 minus1 119899 + 1) for 119899 isin 119885 is locally finite whereas the covering ofthe interval (minus1 1) by intervals (minus1119899 1119899) for 119899 ge 1 barelyfails to be locally finite as there is a problem at the originnow we define the paracompact space

Definition 8 (see [16]) A topological space 119883 is called para-compact if every open covering has a locally finite refinement

Definition 9 (see [16]) A Hausdorff space is a topologicalspace in which every two distinct points can be separated bydisjoint open sets

Definition 10 (see [17]) Let119883 and119884 be two topological spacesand let 119883 997888rarr 2119884 be a set valued mapping is said to haveopen lower sections if

minus1 (120600) = 119909 isin 119883 120600 isin (119909) (6)

is open in 119883 for any 120600 isin 119884Definition 11 (see [14]) For a given complete 120590-finitemeasurespace (MA 120583) and a separable Banach space (119883 sdot ) acompact set-valued operator 120595 M 997888rarr 119870(119883) is called anintegrably bounded if and only if there exits a 120601 isin 1198711(M)such that for all 120603 isin 120595(119904) 120603 le 120601(119904) ae

For 1 le 119901 le infin define the set

119878119901120595 = 119892 isin 119871119901 (Ω119883) 119892 (119904) isin 120595 (119904) 119886119890 (7)

119878119901120595 being the set of selections of 120595 and the closed subsetof 119871119901(Ω119883) 119878119901120595 is nonempty if and only if 120595 is integrablybounded [18] A fuzzy mapping Ω 997888rarr 119864119896 is integrablybounded if []120574 is integrably bounded for any 120574 isin [0 1]Definition 12 (see [14]) A fuzzy mapping Ω 997888rarr 119864119896 iscalled 120572-level uniformly continuous if for a given 120598 gt 0 thereexists 120575 gt 0 such that for any (119909 120600 120580) isin Ω and for 120572 120573 isin [0 1]such that |120573 minus 120572| lt 120575 implies

ℎ119863 ([ (119909 120600 120580)]120572 [ (119909 120600 120580)]120573) lt 120598 (8)

Complexity 3

3 Main Results

In this section first we recall some important results that willbe useful in the sequel then we state and prove our mainresults

Proposition 13 (see [17]) Let 119884 and 119885 be the paracompactHausdorff topological space and a topological vector spacerespectively Let 120595 119884 997888rarr 2119885 be a convex valuedmapping If120595possesses an open lower sections then there exists a continuousselection 120593 of 120595 that is 120593 119884 997888rarr 119885 is continuous and120593(120599) isin 120595(120599) for all 120599 isin 119884Lemma 14 (see [18]) Consider an open subset Ω sub R timesR119896 timesR119898 and let 119879 Ω 997888rarr 119870(R119896times119877119898) be an upper semicontinuousoperator Then there exists an open interval 119869 ofR and 119886119872 gt0 such that

(i) 119869 times 119878(120580120600120599)119886119872 sub Ω for (120580 120600 120599) isin Ω(ii) 119879(120580 120600 120599) le 119872 on 119869 times 119878(120580120600120599)119886119872

Lemma 15 ( see [18]) Let 119883 be a Banach space with borelmeasure space (119883A 120583) Let 119878 119879 119869 997888rarr 119870(119883) be twomeasurable compact operators Then if ] 119869 997888rarr 119883 is ameasurable selection of 119878 then there exists a measurableselection 120596 119869 997888rarr 119883 of 119879 such that

120596 (120580) minus ] (120580) le ℎ119863 (119878 (120580) 119879 (120580)) (9)

for all 120580 isin 119869 where ℎ119863 is Pompeiu Hausdorff metric on 119870(119883)Let us propose our first Darboux type differential inclu-

sion problem which involves open level sets of a fuzzymapping defined on an open subset of given space We statethe first problem as follows

Problem 16 Let (R119896 sdot infin) be a Banach space and let Ωbe an open subset of Ω119886119887119888 times R119896 Let 120596 Ω119886119887119888 997888rarr R119896

be a continuous function and let Ω 997888rarr 119864119896 be a fuzzymapping with 120574 R119896 997888rarr [0 1) an 119906119904119888 function Considerthe following FPDI

(120580120600120599120596(120580120600120599))( 1205973120596120597120580120597120600120597120599) gt 120574 (120596) (10)

for (120580 120600 120599 120596) isin Ω with

120596 (120580 120600 0) + int119888

01198961 (120580 120600) 120596 (120580 120600 120599) 119889120599 = 1205771 (120580 120600)

120596 (120580 0 120599) + int11988701198962 (120580 120599) 120596 (120580 120600 120599) 119889120600 = 1205772 (120580 120599)

120596 (0 120600 120599) + int119886

01198963 (120600 120599) 120596 (120580 120600 120599) 119889120580 = 1205773 (120600 120599)

(C1)

The problem is equivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin ((120580120600120599120596(120580120600120599)))120574(120596) (11)

with (C1)

In the next theorem we discuss the conditions underwhich the above problem can be transformed into a nonfuzzyproblem of partial differential equation known as Darbouxproblem

Theorem 17 Let Ω 997888rarr 119864119896 be a bounded convex andlower open fuzzy surjection If there exists an 119906119904119888 function120574 R119896 997888rarr [0 1) such that ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599)) is nonemptyfor each (120580 120600 120599 120596) isin Ω then there exists a continuous ℎ Ω 997888rarrR119896 such that ℎ(120580 120600 120599 120596(120580 120600 120599)) isin ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599))satisfying

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 = ℎ (120580 120600 120599 120596 (120580 120600 120599))

for all (120580 120600 120599 120596) isin Ω(12)

Proof Define a set-valued map 119865 Ω 997888rarr 2R119896 by119865 (120580 120600 120599 120596 (120580 120600 120599)) = ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599)) (13)

for each (120580 120600 120599 120596) isin Ω where ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599)) is120574(120596(120580 120600 120599)) open level set of the fuzzymapping By assump-tions 119865(120580 120600 120599 120596(120580 120600 120599)) is nonempty for each (120580 120600 120599 120596) isin Ωconsider for 120603 119911 isin 119865(120580 120600 120599 120596(120580 120600 120599)) and 120579 isin [0 1]

(120580120600120599120596(120580120600120599)) (120579120603 + (1 minus 120579) 119911)ge min (120580120600120599120596(120580120600120599)) (120603) (120580120600120599120596(120580120600120599)) (119911)gt 120574 (120596 (120580 120600 120599))

(14)

by convexity of Hence 120579120603 + (1 minus 120579)119911 isin 119865(120580 120600 120599 120596(120580 120600 120599))thus 119865(120580 120600 120599 120596(120580 120600 120599)) is convex on Ω

Clearly 119865 has open lower sections since for any 119911 isin R119896

119865minus1 (119911) = (120580 120600 120599 120596) isin Ω 119911 isin 119865 (120580 120600 120599 120596 (120580 120600 120599)) = (120580 120600 120599 120596) isin Ω ((120580120600120599120596(120580120600120599))) (119911)

gt 120574 (120596 (120580 120600 120599)) (15)

We show that

(119865minus1 (119911))119888 = (120580 120600 120599 120596) isin Ω ((120580120600120599120596(120580120600120599))) (119911)le 120574 (120596 (120580 120600 120599)) (16)

is closed Let (120580119899 120600119899 120599119899 120596119899) be a sequence in (119865minus1(119911))119888 suchthat (120580119899 120600119899 120599119899 120596119899) 997888rarr (120580 120600 120599 120596) As is lower open and 120574 is119906119904119888 therefore

(120580119899120600119899 120599119899120596119899) (119911) le 120574 (120596 (120580119899 120600119899 120599119899)) (17)

which implies

(120580120600120599120596) (119911) le lim119899997888rarrinfin

inf (120580119899120600119899 120599119899120596119899) (119911)le lim119899997888rarrinfin

sup 120574 (120596 (120580119899 120600119899 120599119899))le 120574 (120596 (120580 120600 120599))

(18)

4 Complexity

Thus (120580 120600 120599 120596) isin (119865minus1(119911))119888 which shows that 119865 hasopen lower sections Thus by Proposition 13 there exists acontinuous ℎ isin 119865(120580 120600 120599 120596(120580 120600 120599)) for each (120580 120600 120599 120596) isin ΩAs is a surjection and 119865(120580 120600 120599 120596(120580 120600 120599)) is bounded we getthe problem

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 = ℎ (120580 120600 120599 120596 (120580 120600 120599))

for all (120580 120600 120599 120596) isin Ω(P1)

with local conditions (C1)Now we use the following Tychonoff Theorem to prove

the existence of solution of problem (P1) with local condi-tions (C1)Theorem 18 (see [19] [Tychonoff ]) Let119883 be a complete locallyconvex vector space Let 119862 be a closed and convex subset of 119883and let 119879 119883 997888rarr 119883 be continuous compact operator such that119879(119862) sub 119862 Then 119879 admits a fixed point

LetR119896 be a 119896-dimensional Euclidean space and we denote

119862(R119896R)= 120577 120577 is continuous and 120577 R119896 997888rarr R (19)

The topology on 119862(R2R) is induced by the families ofseminorms

120572120582119896119896 (120595) = sup 1003816100381610038161003816120595 (120600 120599)1003816100381610038161003816 119890minus120582119896(|120600|+|120599|) 120600 120599 isin Ω (20)

where Ω is the bounded region in R2 119896 isin N and 120582119896 ge 0119862(R2R) is complete and locally convex linear space [20]Weuse the similar technique used in [21]

Define the topology on 119862(R3R) induced by the familiesof seminorms

120573120582119896119896

(120595)= max 120572120582119896

119896(120595 (120600 120599)) 120572120582119896119896 (120595 (120599 120580)) 120572120582119896119896 (120595 (120580 120600)) (21)

then with this topology 119862(R3R) is complete and locallyconvex linear space

Problem (P1) with (C1) is equivalent to the fixed pointproblem 119879V = V where 119879 119862(R3R) 997888rarr 119862(R3R) is givenby

(119879]) (120580 120600 120599)= 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0) minus 1205773 (0 120599)

minus 1205773 (120600 0) + 1205773 (0 0) minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int1205990

int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int1205800int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(22)

Condition 1 119891(120580 120600 120599 120596) isin 119862(R4R) and1003816100381610038161003816119891 (120580 120600 120599 120596)1003816100381610038161003816

le max 120595 (120580 120600 |120596|) 120595 (120580 120599 |120596|) 120595 (120600 120599 |120596|) (23)

120595 isin 119862(R2 times R+R+) and120595 is subadditive in |120596| for all 120600 120599 isinΩ

Condition 2 1205771 1205772 1198961 1198962 1198963 isin 119862(R2R)Theorem 19 If Conditions 1 and 2 hold then problem (P1)with (C1) has a solution in 119883 = 119862(R3R)Proof Clearly 119879 is continuous and compact in the topologyof 119883 We need to use Tychonoff Theorem to find the fixedpoint of 119879 It is sufficient to prove 119879(119862) sub 119862 for a closed set119862 Consider

(119879]) (120580 120600 120599)= 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) + 1205772 (120580 0) minus 1205773 (0 120599)

minus 1205773 (120600 0) + 1205773 (0 0) minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(24)

This implies

|(119879]) (120580 120600 120599)|le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816

+ 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816+ 10038161003816100381610038161003816100381610038161003816int

119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int1205990

119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(25)

The subadditivity of 120595 implies the following inequality asgiven in [21]

120595 (120600 120599 |120596|) le 120595 (120600 120599 1) [1 + |120596|] (26)

Complexity 5

Using the above inequality we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816int119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int119886

01198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1)

sdot [1 + ] (120580 119904 119903)] 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816 (27)

Let 119872 = (int1198870|1198961(120580 120600)|2119889120599)12 and 119870 = (int119887

0|](120580 120600 119904)|2119889119904)12

By applying Schwartzrsquos inequality and using above values of119872 and 119870 we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

times 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0|] (120580 119903 119904)|2 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

(28)

Using the definition of given seminorm we obtain

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int1198860

1198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int1206000

int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 12120582119896

100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120580100381610038161003816100381610038161003816100381610038161003816

12

120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|)

(29)

6 Complexity

Let

119861 (120580 120600 120599)

= max

10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816int120599

0int1206000

int1198860

1198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(30)

Using the above inequality we have

|(119879) ] (120580 120600 120599)| le 4119861 (120580 120600 120599)+ 1

120582119896119861 (120580 120600 120599) 120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|) (31)

and equivalently

120573120582119896119896 (119879]) le 41205730119896 (119861) + 1

1205821198961205730119896 (119861) 120572120582119896119896 (]) (32)

Choose 120582119896 = 51205730119896(119861) and set 119862 = 120596 isin 119862(R3R) 120573120582119896119896

(120596) le51205730119896(119861) a closed and bounded subset of 119883 we have for any120596 isin 119862

120573120582119896119896 (119879120596) le 51205730119896 (119861) (33)

that is

119879 (119862) sub 119862 (34)

Hence by the above Tychonoff theorem there exists ] isin 119862such that 119879] = ] that is ] is the required solution

Now we state our second Darboux type fuzzy differentialinclusion involving closed level sets of a fuzzy mappingdefined on an open subset of given space This problem isstated as follows

Problem 20 Consider the following partial fuzzy differentialinclusion

(120580120600120599120596(120580120600120599))(1205973120596 (120580 120600 120599)120597120580120597120600120597120599 ) ge 120574 (120596) (35)

for (120580 120600 120599 120596) isin Ω with local conditions (C1) which areequivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin [(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) (36)

with (C1)The next theorem describes the conditions under which

the solution of above problem exists

Theorem 21 Let Ω119886119887119888 997888rarr 119864119896 be a fuzzy integrablybounded and 120574- level uniformly continuous mapping Let 120574

R119896 997888rarr [0 1] be uniformly continuous If for every (120580 120600 120599 120596)(120580 120600 120599 ]) isin Ω119886119887119888 there exists 120575 isin (0 119886119887119888) satisfyingℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) le 120575

(119886119887119888)2 120596 minus ] (37)

Then Problem 20 possesses a solution

Proof Define 119865 Ω119886119887119888 997888rarr 2R119896 by 119865(120580 120600 120599 120596) = [(120580 120600 120599 120596)]120574(120596(120580120600120599)) We show that 119865(120580 120600 120599 120596) is 119906119904119888 For agiven (∘120580 ∘119910 ∘119911 ∘119906) isin Ω119886119887119888 we can write the neighborhood of119865(∘120580 ∘119910 ∘119911 ∘119906) as

119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903 = 120603 isin R119896 120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (38)

For (120580 120600 120599 120596) isin Ω119886119887119888 and ] isin 119865(120580 120600 120599 120596) we have

120588 (] 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) le ℎ119863 (119865 (120580 120600 120599 120596) 119865 ( ∘119905 ∘119910 ∘119911 ∘119906))= ℎ119863 ([ (120580 120600 120599 120596)]120574(120596) [ ( ∘119905 ∘119910 ∘119911 ∘119906)]

120574(∘119906))

le ℎ119863 ([ ( ∘119905 ∘119910 ∘119911 ∘119906)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(∘119906))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))le ℎ119863 (119865 ( ∘119905 ∘119910 ∘119911 ∘119906) 119865 (120580 120600 120599 120596))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))

(39)

Since [ (120580 120600 120599 120596)]120574(120596) is 120574-level uniformly continuous and 120574is uniformly continuous using the above inequality we canfind a small enough neighborhood 119880 of ( ∘119905 ∘119910 ∘119911 ∘119906) in Ω119886119887119888such that for all (120580 120600 120599 ]) isin 119880 and 120603 isin 119865(120580 120600 120599 ])

120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (40)

thus

119865 (119880) sub 119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903(41)

which means that 119865(120580 120600 120599 120596) is 119906119904119888 So by Lemma 14 thereexists a real constant 120579 gt 0 such that

max(120580120600120599120596)isinΩ

10038171003817100381710038171003817119865 (120580 120600 120599 120596)10038171003817100381710038171003817 le 120579 (42)

Complexity 7

Let 119883 = 120596 isin 119862(119868R119896) 120596 minus 1205960 le 120579 for all (120580 120600 120599) isinΩ119886119887119888 120596(1205800 1206000 1205990) = 1205960 with a metric 120588119883 119883 times 119883 997888rarr R cup+infin defined by

120588119883 = sup(120580120600120599)isin119868

120596 (120580 120600 120599) minus ] (120580 120600 120599) (43)

Then (119883 120588119883) is a complete generalized metric space [14]Define 119879 119883 997888rarr 2119883 by

119879 (120596) = 119906 (120580 120600 120599) 119906 (120580 120600 120599) isin 1199060 (120580 120600 120599) + 1199061 (120580 120600 120599)

+ int120580

0int120600

0int120599

0[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119904120580)) 119889119903119889119904119889120591 119886119890 119868

(44)

where

1199060 (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0) (45)

and

1199061 (120580 120600 120599) = int119888

01198961 (120580 120600) 120596 (120580 120600 119904) 119889119904

+ int1205990

int119887

01198962 (120580 120599) 120596 (120580 119904 120599) 119889119904119889120599

minus int1205990

int120600

0int119886

01198963 (120600 120599) 120596 (119904 120600 120599) 119889119904119889120600119889120599

(46)

Here int1205800int1206000

int1205990[ (119903 119904 120580 120596(119903 119904 120580))]120574(120596(119903119904120580))119889119903119889119904119889120591 is multival-

ued triple integral of Aumann [22] defined by

int1205800int120600

0int120599

0119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 = int120580

0int1206000

int1205990

[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119903119904120580)) 119889119903119889119904119889120591

= 1199060 + 1199061 + int1205800int120600

0int120599

0119891 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 | 119891 Ω 997888rarr R

119896 is measurable selection for [ (119903 119904 120580 120596 (119903 119904 120580))]120574 (47)

for each 120574 isin (0 1]Clearly 119879(120596) = 120593 for all 120596 isin 119883 Since the operator

119865(120580 120600 120599 120596) = [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) is compact 119906119904119888so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23] 119865(120580 120600 120599 120596) has measurableselection 119891(120580 120600 120599 120596) isin [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) for all(120580 120600 120599 120596) isin Ω times R119896 and 119891(120580 120600 120599 120596) is Lebesgue integrable[4]

Let 120596(120580 120600 120599) = 1205960 + int1205800int1206000

int1205990119891(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 isin

119879(120596) where 1205960 = 1199060 + 1199061 thus 119879(120596) = 120593Next we show that 119879(120596) is closed for all 120596 isin 119883

Consider a sequence 120596119899 in 119879(120596) such that 120596119899 997888rarr ∘119906 isin 119883Since

120596119899 isin 1205960 + int120580

0int1206000

int1205990

119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 (48)

and int1205800int1206000

int1205990119865(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 is closed [22] so we

have ∘119906 isin 119879(120596)

We show that 119879 is multivalued contraction let 1205962 isin119879(1205961) which implies that there exists 119891(120580 120600 120599 120596) isin119865(120580 120600 120599 120596) such that

1205962 (120580 120600 120599) = 1205960 + int1205800int120600

0int120599

0119891 (119904 120580 1205961 (119904 120580)) 119889119903119889119904119889120591 (49)

By virtue of Lemma 15 there exists a measurable selection119891(120580 120600 120599 120596) isin 119865(120580 120600 120599 120596) such that

1003817100381710038171003817119891 (120580 120600 120599 120596) minus 119891 (119904 120580 1205961 (119904 120580))1003817100381710038171003817 le ℎ119863 (119865 (120580 120600 120599 120596) 119865 (120580 120600 120599 120596)) = ℎ119863 ([ (120580 120600 120599 120596)]120574(1205961(120580120600120599)) [ (120580 120600 120599 120596)]120574(1205962(120580120600120599))) le ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 120596))

(50)

Let

1205963 (120580 120600 120599) = 1205960+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 (51)

and consider

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817int

120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 minus int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205961 (119903 119904 120580)) 119889119903119889119904119889120591100381710038171003817100381710038171003817100381710038171003817

le int1205800int120600

0int120599

0

1003817100381710038171003817119891 (120580 120600 120599 120596)

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

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Complexity 3

3 Main Results

In this section first we recall some important results that willbe useful in the sequel then we state and prove our mainresults

Proposition 13 (see [17]) Let 119884 and 119885 be the paracompactHausdorff topological space and a topological vector spacerespectively Let 120595 119884 997888rarr 2119885 be a convex valuedmapping If120595possesses an open lower sections then there exists a continuousselection 120593 of 120595 that is 120593 119884 997888rarr 119885 is continuous and120593(120599) isin 120595(120599) for all 120599 isin 119884Lemma 14 (see [18]) Consider an open subset Ω sub R timesR119896 timesR119898 and let 119879 Ω 997888rarr 119870(R119896times119877119898) be an upper semicontinuousoperator Then there exists an open interval 119869 ofR and 119886119872 gt0 such that

(i) 119869 times 119878(120580120600120599)119886119872 sub Ω for (120580 120600 120599) isin Ω(ii) 119879(120580 120600 120599) le 119872 on 119869 times 119878(120580120600120599)119886119872

Lemma 15 ( see [18]) Let 119883 be a Banach space with borelmeasure space (119883A 120583) Let 119878 119879 119869 997888rarr 119870(119883) be twomeasurable compact operators Then if ] 119869 997888rarr 119883 is ameasurable selection of 119878 then there exists a measurableselection 120596 119869 997888rarr 119883 of 119879 such that

120596 (120580) minus ] (120580) le ℎ119863 (119878 (120580) 119879 (120580)) (9)

for all 120580 isin 119869 where ℎ119863 is Pompeiu Hausdorff metric on 119870(119883)Let us propose our first Darboux type differential inclu-

sion problem which involves open level sets of a fuzzymapping defined on an open subset of given space We statethe first problem as follows

Problem 16 Let (R119896 sdot infin) be a Banach space and let Ωbe an open subset of Ω119886119887119888 times R119896 Let 120596 Ω119886119887119888 997888rarr R119896

be a continuous function and let Ω 997888rarr 119864119896 be a fuzzymapping with 120574 R119896 997888rarr [0 1) an 119906119904119888 function Considerthe following FPDI

(120580120600120599120596(120580120600120599))( 1205973120596120597120580120597120600120597120599) gt 120574 (120596) (10)

for (120580 120600 120599 120596) isin Ω with

120596 (120580 120600 0) + int119888

01198961 (120580 120600) 120596 (120580 120600 120599) 119889120599 = 1205771 (120580 120600)

120596 (120580 0 120599) + int11988701198962 (120580 120599) 120596 (120580 120600 120599) 119889120600 = 1205772 (120580 120599)

120596 (0 120600 120599) + int119886

01198963 (120600 120599) 120596 (120580 120600 120599) 119889120580 = 1205773 (120600 120599)

(C1)

The problem is equivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin ((120580120600120599120596(120580120600120599)))120574(120596) (11)

with (C1)

In the next theorem we discuss the conditions underwhich the above problem can be transformed into a nonfuzzyproblem of partial differential equation known as Darbouxproblem

Theorem 17 Let Ω 997888rarr 119864119896 be a bounded convex andlower open fuzzy surjection If there exists an 119906119904119888 function120574 R119896 997888rarr [0 1) such that ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599)) is nonemptyfor each (120580 120600 120599 120596) isin Ω then there exists a continuous ℎ Ω 997888rarrR119896 such that ℎ(120580 120600 120599 120596(120580 120600 120599)) isin ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599))satisfying

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 = ℎ (120580 120600 120599 120596 (120580 120600 120599))

for all (120580 120600 120599 120596) isin Ω(12)

Proof Define a set-valued map 119865 Ω 997888rarr 2R119896 by119865 (120580 120600 120599 120596 (120580 120600 120599)) = ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599)) (13)

for each (120580 120600 120599 120596) isin Ω where ((120580120600120599120596(120580120600120599)))120574(120596(120580120600120599)) is120574(120596(120580 120600 120599)) open level set of the fuzzymapping By assump-tions 119865(120580 120600 120599 120596(120580 120600 120599)) is nonempty for each (120580 120600 120599 120596) isin Ωconsider for 120603 119911 isin 119865(120580 120600 120599 120596(120580 120600 120599)) and 120579 isin [0 1]

(120580120600120599120596(120580120600120599)) (120579120603 + (1 minus 120579) 119911)ge min (120580120600120599120596(120580120600120599)) (120603) (120580120600120599120596(120580120600120599)) (119911)gt 120574 (120596 (120580 120600 120599))

(14)

by convexity of Hence 120579120603 + (1 minus 120579)119911 isin 119865(120580 120600 120599 120596(120580 120600 120599))thus 119865(120580 120600 120599 120596(120580 120600 120599)) is convex on Ω

Clearly 119865 has open lower sections since for any 119911 isin R119896

119865minus1 (119911) = (120580 120600 120599 120596) isin Ω 119911 isin 119865 (120580 120600 120599 120596 (120580 120600 120599)) = (120580 120600 120599 120596) isin Ω ((120580120600120599120596(120580120600120599))) (119911)

gt 120574 (120596 (120580 120600 120599)) (15)

We show that

(119865minus1 (119911))119888 = (120580 120600 120599 120596) isin Ω ((120580120600120599120596(120580120600120599))) (119911)le 120574 (120596 (120580 120600 120599)) (16)

is closed Let (120580119899 120600119899 120599119899 120596119899) be a sequence in (119865minus1(119911))119888 suchthat (120580119899 120600119899 120599119899 120596119899) 997888rarr (120580 120600 120599 120596) As is lower open and 120574 is119906119904119888 therefore

(120580119899120600119899 120599119899120596119899) (119911) le 120574 (120596 (120580119899 120600119899 120599119899)) (17)

which implies

(120580120600120599120596) (119911) le lim119899997888rarrinfin

inf (120580119899120600119899 120599119899120596119899) (119911)le lim119899997888rarrinfin

sup 120574 (120596 (120580119899 120600119899 120599119899))le 120574 (120596 (120580 120600 120599))

(18)

4 Complexity

Thus (120580 120600 120599 120596) isin (119865minus1(119911))119888 which shows that 119865 hasopen lower sections Thus by Proposition 13 there exists acontinuous ℎ isin 119865(120580 120600 120599 120596(120580 120600 120599)) for each (120580 120600 120599 120596) isin ΩAs is a surjection and 119865(120580 120600 120599 120596(120580 120600 120599)) is bounded we getthe problem

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 = ℎ (120580 120600 120599 120596 (120580 120600 120599))

for all (120580 120600 120599 120596) isin Ω(P1)

with local conditions (C1)Now we use the following Tychonoff Theorem to prove

the existence of solution of problem (P1) with local condi-tions (C1)Theorem 18 (see [19] [Tychonoff ]) Let119883 be a complete locallyconvex vector space Let 119862 be a closed and convex subset of 119883and let 119879 119883 997888rarr 119883 be continuous compact operator such that119879(119862) sub 119862 Then 119879 admits a fixed point

LetR119896 be a 119896-dimensional Euclidean space and we denote

119862(R119896R)= 120577 120577 is continuous and 120577 R119896 997888rarr R (19)

The topology on 119862(R2R) is induced by the families ofseminorms

120572120582119896119896 (120595) = sup 1003816100381610038161003816120595 (120600 120599)1003816100381610038161003816 119890minus120582119896(|120600|+|120599|) 120600 120599 isin Ω (20)

where Ω is the bounded region in R2 119896 isin N and 120582119896 ge 0119862(R2R) is complete and locally convex linear space [20]Weuse the similar technique used in [21]

Define the topology on 119862(R3R) induced by the familiesof seminorms

120573120582119896119896

(120595)= max 120572120582119896

119896(120595 (120600 120599)) 120572120582119896119896 (120595 (120599 120580)) 120572120582119896119896 (120595 (120580 120600)) (21)

then with this topology 119862(R3R) is complete and locallyconvex linear space

Problem (P1) with (C1) is equivalent to the fixed pointproblem 119879V = V where 119879 119862(R3R) 997888rarr 119862(R3R) is givenby

(119879]) (120580 120600 120599)= 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0) minus 1205773 (0 120599)

minus 1205773 (120600 0) + 1205773 (0 0) minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int1205990

int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int1205800int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(22)

Condition 1 119891(120580 120600 120599 120596) isin 119862(R4R) and1003816100381610038161003816119891 (120580 120600 120599 120596)1003816100381610038161003816

le max 120595 (120580 120600 |120596|) 120595 (120580 120599 |120596|) 120595 (120600 120599 |120596|) (23)

120595 isin 119862(R2 times R+R+) and120595 is subadditive in |120596| for all 120600 120599 isinΩ

Condition 2 1205771 1205772 1198961 1198962 1198963 isin 119862(R2R)Theorem 19 If Conditions 1 and 2 hold then problem (P1)with (C1) has a solution in 119883 = 119862(R3R)Proof Clearly 119879 is continuous and compact in the topologyof 119883 We need to use Tychonoff Theorem to find the fixedpoint of 119879 It is sufficient to prove 119879(119862) sub 119862 for a closed set119862 Consider

(119879]) (120580 120600 120599)= 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) + 1205772 (120580 0) minus 1205773 (0 120599)

minus 1205773 (120600 0) + 1205773 (0 0) minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(24)

This implies

|(119879]) (120580 120600 120599)|le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816

+ 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816+ 10038161003816100381610038161003816100381610038161003816int

119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int1205990

119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(25)

The subadditivity of 120595 implies the following inequality asgiven in [21]

120595 (120600 120599 |120596|) le 120595 (120600 120599 1) [1 + |120596|] (26)

Complexity 5

Using the above inequality we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816int119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int119886

01198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1)

sdot [1 + ] (120580 119904 119903)] 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816 (27)

Let 119872 = (int1198870|1198961(120580 120600)|2119889120599)12 and 119870 = (int119887

0|](120580 120600 119904)|2119889119904)12

By applying Schwartzrsquos inequality and using above values of119872 and 119870 we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

times 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0|] (120580 119903 119904)|2 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

(28)

Using the definition of given seminorm we obtain

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int1198860

1198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int1206000

int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 12120582119896

100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120580100381610038161003816100381610038161003816100381610038161003816

12

120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|)

(29)

6 Complexity

Let

119861 (120580 120600 120599)

= max

10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816int120599

0int1206000

int1198860

1198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(30)

Using the above inequality we have

|(119879) ] (120580 120600 120599)| le 4119861 (120580 120600 120599)+ 1

120582119896119861 (120580 120600 120599) 120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|) (31)

and equivalently

120573120582119896119896 (119879]) le 41205730119896 (119861) + 1

1205821198961205730119896 (119861) 120572120582119896119896 (]) (32)

Choose 120582119896 = 51205730119896(119861) and set 119862 = 120596 isin 119862(R3R) 120573120582119896119896

(120596) le51205730119896(119861) a closed and bounded subset of 119883 we have for any120596 isin 119862

120573120582119896119896 (119879120596) le 51205730119896 (119861) (33)

that is

119879 (119862) sub 119862 (34)

Hence by the above Tychonoff theorem there exists ] isin 119862such that 119879] = ] that is ] is the required solution

Now we state our second Darboux type fuzzy differentialinclusion involving closed level sets of a fuzzy mappingdefined on an open subset of given space This problem isstated as follows

Problem 20 Consider the following partial fuzzy differentialinclusion

(120580120600120599120596(120580120600120599))(1205973120596 (120580 120600 120599)120597120580120597120600120597120599 ) ge 120574 (120596) (35)

for (120580 120600 120599 120596) isin Ω with local conditions (C1) which areequivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin [(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) (36)

with (C1)The next theorem describes the conditions under which

the solution of above problem exists

Theorem 21 Let Ω119886119887119888 997888rarr 119864119896 be a fuzzy integrablybounded and 120574- level uniformly continuous mapping Let 120574

R119896 997888rarr [0 1] be uniformly continuous If for every (120580 120600 120599 120596)(120580 120600 120599 ]) isin Ω119886119887119888 there exists 120575 isin (0 119886119887119888) satisfyingℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) le 120575

(119886119887119888)2 120596 minus ] (37)

Then Problem 20 possesses a solution

Proof Define 119865 Ω119886119887119888 997888rarr 2R119896 by 119865(120580 120600 120599 120596) = [(120580 120600 120599 120596)]120574(120596(120580120600120599)) We show that 119865(120580 120600 120599 120596) is 119906119904119888 For agiven (∘120580 ∘119910 ∘119911 ∘119906) isin Ω119886119887119888 we can write the neighborhood of119865(∘120580 ∘119910 ∘119911 ∘119906) as

119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903 = 120603 isin R119896 120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (38)

For (120580 120600 120599 120596) isin Ω119886119887119888 and ] isin 119865(120580 120600 120599 120596) we have

120588 (] 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) le ℎ119863 (119865 (120580 120600 120599 120596) 119865 ( ∘119905 ∘119910 ∘119911 ∘119906))= ℎ119863 ([ (120580 120600 120599 120596)]120574(120596) [ ( ∘119905 ∘119910 ∘119911 ∘119906)]

120574(∘119906))

le ℎ119863 ([ ( ∘119905 ∘119910 ∘119911 ∘119906)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(∘119906))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))le ℎ119863 (119865 ( ∘119905 ∘119910 ∘119911 ∘119906) 119865 (120580 120600 120599 120596))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))

(39)

Since [ (120580 120600 120599 120596)]120574(120596) is 120574-level uniformly continuous and 120574is uniformly continuous using the above inequality we canfind a small enough neighborhood 119880 of ( ∘119905 ∘119910 ∘119911 ∘119906) in Ω119886119887119888such that for all (120580 120600 120599 ]) isin 119880 and 120603 isin 119865(120580 120600 120599 ])

120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (40)

thus

119865 (119880) sub 119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903(41)

which means that 119865(120580 120600 120599 120596) is 119906119904119888 So by Lemma 14 thereexists a real constant 120579 gt 0 such that

max(120580120600120599120596)isinΩ

10038171003817100381710038171003817119865 (120580 120600 120599 120596)10038171003817100381710038171003817 le 120579 (42)

Complexity 7

Let 119883 = 120596 isin 119862(119868R119896) 120596 minus 1205960 le 120579 for all (120580 120600 120599) isinΩ119886119887119888 120596(1205800 1206000 1205990) = 1205960 with a metric 120588119883 119883 times 119883 997888rarr R cup+infin defined by

120588119883 = sup(120580120600120599)isin119868

120596 (120580 120600 120599) minus ] (120580 120600 120599) (43)

Then (119883 120588119883) is a complete generalized metric space [14]Define 119879 119883 997888rarr 2119883 by

119879 (120596) = 119906 (120580 120600 120599) 119906 (120580 120600 120599) isin 1199060 (120580 120600 120599) + 1199061 (120580 120600 120599)

+ int120580

0int120600

0int120599

0[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119904120580)) 119889119903119889119904119889120591 119886119890 119868

(44)

where

1199060 (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0) (45)

and

1199061 (120580 120600 120599) = int119888

01198961 (120580 120600) 120596 (120580 120600 119904) 119889119904

+ int1205990

int119887

01198962 (120580 120599) 120596 (120580 119904 120599) 119889119904119889120599

minus int1205990

int120600

0int119886

01198963 (120600 120599) 120596 (119904 120600 120599) 119889119904119889120600119889120599

(46)

Here int1205800int1206000

int1205990[ (119903 119904 120580 120596(119903 119904 120580))]120574(120596(119903119904120580))119889119903119889119904119889120591 is multival-

ued triple integral of Aumann [22] defined by

int1205800int120600

0int120599

0119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 = int120580

0int1206000

int1205990

[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119903119904120580)) 119889119903119889119904119889120591

= 1199060 + 1199061 + int1205800int120600

0int120599

0119891 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 | 119891 Ω 997888rarr R

119896 is measurable selection for [ (119903 119904 120580 120596 (119903 119904 120580))]120574 (47)

for each 120574 isin (0 1]Clearly 119879(120596) = 120593 for all 120596 isin 119883 Since the operator

119865(120580 120600 120599 120596) = [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) is compact 119906119904119888so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23] 119865(120580 120600 120599 120596) has measurableselection 119891(120580 120600 120599 120596) isin [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) for all(120580 120600 120599 120596) isin Ω times R119896 and 119891(120580 120600 120599 120596) is Lebesgue integrable[4]

Let 120596(120580 120600 120599) = 1205960 + int1205800int1206000

int1205990119891(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 isin

119879(120596) where 1205960 = 1199060 + 1199061 thus 119879(120596) = 120593Next we show that 119879(120596) is closed for all 120596 isin 119883

Consider a sequence 120596119899 in 119879(120596) such that 120596119899 997888rarr ∘119906 isin 119883Since

120596119899 isin 1205960 + int120580

0int1206000

int1205990

119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 (48)

and int1205800int1206000

int1205990119865(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 is closed [22] so we

have ∘119906 isin 119879(120596)

We show that 119879 is multivalued contraction let 1205962 isin119879(1205961) which implies that there exists 119891(120580 120600 120599 120596) isin119865(120580 120600 120599 120596) such that

1205962 (120580 120600 120599) = 1205960 + int1205800int120600

0int120599

0119891 (119904 120580 1205961 (119904 120580)) 119889119903119889119904119889120591 (49)

By virtue of Lemma 15 there exists a measurable selection119891(120580 120600 120599 120596) isin 119865(120580 120600 120599 120596) such that

1003817100381710038171003817119891 (120580 120600 120599 120596) minus 119891 (119904 120580 1205961 (119904 120580))1003817100381710038171003817 le ℎ119863 (119865 (120580 120600 120599 120596) 119865 (120580 120600 120599 120596)) = ℎ119863 ([ (120580 120600 120599 120596)]120574(1205961(120580120600120599)) [ (120580 120600 120599 120596)]120574(1205962(120580120600120599))) le ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 120596))

(50)

Let

1205963 (120580 120600 120599) = 1205960+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 (51)

and consider

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817int

120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 minus int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205961 (119903 119904 120580)) 119889119903119889119904119889120591100381710038171003817100381710038171003817100381710038171003817

le int1205800int120600

0int120599

0

1003817100381710038171003817119891 (120580 120600 120599 120596)

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

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4 Complexity

Thus (120580 120600 120599 120596) isin (119865minus1(119911))119888 which shows that 119865 hasopen lower sections Thus by Proposition 13 there exists acontinuous ℎ isin 119865(120580 120600 120599 120596(120580 120600 120599)) for each (120580 120600 120599 120596) isin ΩAs is a surjection and 119865(120580 120600 120599 120596(120580 120600 120599)) is bounded we getthe problem

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 = ℎ (120580 120600 120599 120596 (120580 120600 120599))

for all (120580 120600 120599 120596) isin Ω(P1)

with local conditions (C1)Now we use the following Tychonoff Theorem to prove

the existence of solution of problem (P1) with local condi-tions (C1)Theorem 18 (see [19] [Tychonoff ]) Let119883 be a complete locallyconvex vector space Let 119862 be a closed and convex subset of 119883and let 119879 119883 997888rarr 119883 be continuous compact operator such that119879(119862) sub 119862 Then 119879 admits a fixed point

LetR119896 be a 119896-dimensional Euclidean space and we denote

119862(R119896R)= 120577 120577 is continuous and 120577 R119896 997888rarr R (19)

The topology on 119862(R2R) is induced by the families ofseminorms

120572120582119896119896 (120595) = sup 1003816100381610038161003816120595 (120600 120599)1003816100381610038161003816 119890minus120582119896(|120600|+|120599|) 120600 120599 isin Ω (20)

where Ω is the bounded region in R2 119896 isin N and 120582119896 ge 0119862(R2R) is complete and locally convex linear space [20]Weuse the similar technique used in [21]

Define the topology on 119862(R3R) induced by the familiesof seminorms

120573120582119896119896

(120595)= max 120572120582119896

119896(120595 (120600 120599)) 120572120582119896119896 (120595 (120599 120580)) 120572120582119896119896 (120595 (120580 120600)) (21)

then with this topology 119862(R3R) is complete and locallyconvex linear space

Problem (P1) with (C1) is equivalent to the fixed pointproblem 119879V = V where 119879 119862(R3R) 997888rarr 119862(R3R) is givenby

(119879]) (120580 120600 120599)= 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0) minus 1205773 (0 120599)

minus 1205773 (120600 0) + 1205773 (0 0) minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int1205990

int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int1205800int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(22)

Condition 1 119891(120580 120600 120599 120596) isin 119862(R4R) and1003816100381610038161003816119891 (120580 120600 120599 120596)1003816100381610038161003816

le max 120595 (120580 120600 |120596|) 120595 (120580 120599 |120596|) 120595 (120600 120599 |120596|) (23)

120595 isin 119862(R2 times R+R+) and120595 is subadditive in |120596| for all 120600 120599 isinΩ

Condition 2 1205771 1205772 1198961 1198962 1198963 isin 119862(R2R)Theorem 19 If Conditions 1 and 2 hold then problem (P1)with (C1) has a solution in 119883 = 119862(R3R)Proof Clearly 119879 is continuous and compact in the topologyof 119883 We need to use Tychonoff Theorem to find the fixedpoint of 119879 It is sufficient to prove 119879(119862) sub 119862 for a closed set119862 Consider

(119879]) (120580 120600 120599)= 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) + 1205772 (120580 0) minus 1205773 (0 120599)

minus 1205773 (120600 0) + 1205773 (0 0) minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(24)

This implies

|(119879]) (120580 120600 120599)|le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816

+ 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816+ 10038161003816100381610038161003816100381610038161003816int

119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int1205990

119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(25)

The subadditivity of 120595 implies the following inequality asgiven in [21]

120595 (120600 120599 |120596|) le 120595 (120600 120599 1) [1 + |120596|] (26)

Complexity 5

Using the above inequality we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816int119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int119886

01198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1)

sdot [1 + ] (120580 119904 119903)] 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816 (27)

Let 119872 = (int1198870|1198961(120580 120600)|2119889120599)12 and 119870 = (int119887

0|](120580 120600 119904)|2119889119904)12

By applying Schwartzrsquos inequality and using above values of119872 and 119870 we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

times 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0|] (120580 119903 119904)|2 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

(28)

Using the definition of given seminorm we obtain

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int1198860

1198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int1206000

int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 12120582119896

100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120580100381610038161003816100381610038161003816100381610038161003816

12

120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|)

(29)

6 Complexity

Let

119861 (120580 120600 120599)

= max

10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816int120599

0int1206000

int1198860

1198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(30)

Using the above inequality we have

|(119879) ] (120580 120600 120599)| le 4119861 (120580 120600 120599)+ 1

120582119896119861 (120580 120600 120599) 120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|) (31)

and equivalently

120573120582119896119896 (119879]) le 41205730119896 (119861) + 1

1205821198961205730119896 (119861) 120572120582119896119896 (]) (32)

Choose 120582119896 = 51205730119896(119861) and set 119862 = 120596 isin 119862(R3R) 120573120582119896119896

(120596) le51205730119896(119861) a closed and bounded subset of 119883 we have for any120596 isin 119862

120573120582119896119896 (119879120596) le 51205730119896 (119861) (33)

that is

119879 (119862) sub 119862 (34)

Hence by the above Tychonoff theorem there exists ] isin 119862such that 119879] = ] that is ] is the required solution

Now we state our second Darboux type fuzzy differentialinclusion involving closed level sets of a fuzzy mappingdefined on an open subset of given space This problem isstated as follows

Problem 20 Consider the following partial fuzzy differentialinclusion

(120580120600120599120596(120580120600120599))(1205973120596 (120580 120600 120599)120597120580120597120600120597120599 ) ge 120574 (120596) (35)

for (120580 120600 120599 120596) isin Ω with local conditions (C1) which areequivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin [(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) (36)

with (C1)The next theorem describes the conditions under which

the solution of above problem exists

Theorem 21 Let Ω119886119887119888 997888rarr 119864119896 be a fuzzy integrablybounded and 120574- level uniformly continuous mapping Let 120574

R119896 997888rarr [0 1] be uniformly continuous If for every (120580 120600 120599 120596)(120580 120600 120599 ]) isin Ω119886119887119888 there exists 120575 isin (0 119886119887119888) satisfyingℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) le 120575

(119886119887119888)2 120596 minus ] (37)

Then Problem 20 possesses a solution

Proof Define 119865 Ω119886119887119888 997888rarr 2R119896 by 119865(120580 120600 120599 120596) = [(120580 120600 120599 120596)]120574(120596(120580120600120599)) We show that 119865(120580 120600 120599 120596) is 119906119904119888 For agiven (∘120580 ∘119910 ∘119911 ∘119906) isin Ω119886119887119888 we can write the neighborhood of119865(∘120580 ∘119910 ∘119911 ∘119906) as

119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903 = 120603 isin R119896 120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (38)

For (120580 120600 120599 120596) isin Ω119886119887119888 and ] isin 119865(120580 120600 120599 120596) we have

120588 (] 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) le ℎ119863 (119865 (120580 120600 120599 120596) 119865 ( ∘119905 ∘119910 ∘119911 ∘119906))= ℎ119863 ([ (120580 120600 120599 120596)]120574(120596) [ ( ∘119905 ∘119910 ∘119911 ∘119906)]

120574(∘119906))

le ℎ119863 ([ ( ∘119905 ∘119910 ∘119911 ∘119906)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(∘119906))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))le ℎ119863 (119865 ( ∘119905 ∘119910 ∘119911 ∘119906) 119865 (120580 120600 120599 120596))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))

(39)

Since [ (120580 120600 120599 120596)]120574(120596) is 120574-level uniformly continuous and 120574is uniformly continuous using the above inequality we canfind a small enough neighborhood 119880 of ( ∘119905 ∘119910 ∘119911 ∘119906) in Ω119886119887119888such that for all (120580 120600 120599 ]) isin 119880 and 120603 isin 119865(120580 120600 120599 ])

120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (40)

thus

119865 (119880) sub 119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903(41)

which means that 119865(120580 120600 120599 120596) is 119906119904119888 So by Lemma 14 thereexists a real constant 120579 gt 0 such that

max(120580120600120599120596)isinΩ

10038171003817100381710038171003817119865 (120580 120600 120599 120596)10038171003817100381710038171003817 le 120579 (42)

Complexity 7

Let 119883 = 120596 isin 119862(119868R119896) 120596 minus 1205960 le 120579 for all (120580 120600 120599) isinΩ119886119887119888 120596(1205800 1206000 1205990) = 1205960 with a metric 120588119883 119883 times 119883 997888rarr R cup+infin defined by

120588119883 = sup(120580120600120599)isin119868

120596 (120580 120600 120599) minus ] (120580 120600 120599) (43)

Then (119883 120588119883) is a complete generalized metric space [14]Define 119879 119883 997888rarr 2119883 by

119879 (120596) = 119906 (120580 120600 120599) 119906 (120580 120600 120599) isin 1199060 (120580 120600 120599) + 1199061 (120580 120600 120599)

+ int120580

0int120600

0int120599

0[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119904120580)) 119889119903119889119904119889120591 119886119890 119868

(44)

where

1199060 (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0) (45)

and

1199061 (120580 120600 120599) = int119888

01198961 (120580 120600) 120596 (120580 120600 119904) 119889119904

+ int1205990

int119887

01198962 (120580 120599) 120596 (120580 119904 120599) 119889119904119889120599

minus int1205990

int120600

0int119886

01198963 (120600 120599) 120596 (119904 120600 120599) 119889119904119889120600119889120599

(46)

Here int1205800int1206000

int1205990[ (119903 119904 120580 120596(119903 119904 120580))]120574(120596(119903119904120580))119889119903119889119904119889120591 is multival-

ued triple integral of Aumann [22] defined by

int1205800int120600

0int120599

0119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 = int120580

0int1206000

int1205990

[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119903119904120580)) 119889119903119889119904119889120591

= 1199060 + 1199061 + int1205800int120600

0int120599

0119891 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 | 119891 Ω 997888rarr R

119896 is measurable selection for [ (119903 119904 120580 120596 (119903 119904 120580))]120574 (47)

for each 120574 isin (0 1]Clearly 119879(120596) = 120593 for all 120596 isin 119883 Since the operator

119865(120580 120600 120599 120596) = [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) is compact 119906119904119888so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23] 119865(120580 120600 120599 120596) has measurableselection 119891(120580 120600 120599 120596) isin [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) for all(120580 120600 120599 120596) isin Ω times R119896 and 119891(120580 120600 120599 120596) is Lebesgue integrable[4]

Let 120596(120580 120600 120599) = 1205960 + int1205800int1206000

int1205990119891(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 isin

119879(120596) where 1205960 = 1199060 + 1199061 thus 119879(120596) = 120593Next we show that 119879(120596) is closed for all 120596 isin 119883

Consider a sequence 120596119899 in 119879(120596) such that 120596119899 997888rarr ∘119906 isin 119883Since

120596119899 isin 1205960 + int120580

0int1206000

int1205990

119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 (48)

and int1205800int1206000

int1205990119865(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 is closed [22] so we

have ∘119906 isin 119879(120596)

We show that 119879 is multivalued contraction let 1205962 isin119879(1205961) which implies that there exists 119891(120580 120600 120599 120596) isin119865(120580 120600 120599 120596) such that

1205962 (120580 120600 120599) = 1205960 + int1205800int120600

0int120599

0119891 (119904 120580 1205961 (119904 120580)) 119889119903119889119904119889120591 (49)

By virtue of Lemma 15 there exists a measurable selection119891(120580 120600 120599 120596) isin 119865(120580 120600 120599 120596) such that

1003817100381710038171003817119891 (120580 120600 120599 120596) minus 119891 (119904 120580 1205961 (119904 120580))1003817100381710038171003817 le ℎ119863 (119865 (120580 120600 120599 120596) 119865 (120580 120600 120599 120596)) = ℎ119863 ([ (120580 120600 120599 120596)]120574(1205961(120580120600120599)) [ (120580 120600 120599 120596)]120574(1205962(120580120600120599))) le ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 120596))

(50)

Let

1205963 (120580 120600 120599) = 1205960+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 (51)

and consider

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817int

120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 minus int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205961 (119903 119904 120580)) 119889119903119889119904119889120591100381710038171003817100381710038171003817100381710038171003817

le int1205800int120600

0int120599

0

1003817100381710038171003817119891 (120580 120600 120599 120596)

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

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Submit your manuscripts atwwwhindawicom

Complexity 5

Using the above inequality we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 +

10038161003816100381610038161003816100381610038161003816int119888

01198961 (120580 120600) ] (120580 120600 119904) 11988911990410038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int119886

01198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1)

sdot [1 + ] (120580 119904 119903)] 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816 (27)

Let 119872 = (int1198870|1198961(120580 120600)|2119889120599)12 and 119870 = (int119887

0|](120580 120600 119904)|2119889119904)12

By applying Schwartzrsquos inequality and using above values of119872 and 119870 we get

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816int120599

0int120600

0int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

times 100381610038161003816100381610038161003816100381610038161003816int120580

0int120600

0int120599

0|] (120580 119903 119904)|2 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

12

(28)

Using the definition of given seminorm we obtain

|(119879]) (120580 120600 120599)| le 10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816+ 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816+ 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int119887

01198962 (120580 120599) ] (120580

119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120599

0int120600

0int1198860

1198963 (120600 120599)

sdot ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816+ 100381610038161003816100381610038161003816100381610038161003816int

120580

0int1206000

int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

+ 12120582119896

100381610038161003816100381610038161003816100381610038161003816int120580

0int1206000

int120599

0max 1003816100381610038161003816120595 (120580 120600 1)10038161003816100381610038162 1003816100381610038161003816120595 (120580 120599 1)10038161003816100381610038162 1003816100381610038161003816120595 (120600 120599 1)10038161003816100381610038162 119889119903119889119904119889120580100381610038161003816100381610038161003816100381610038161003816

12

120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|)

(29)

6 Complexity

Let

119861 (120580 120600 120599)

= max

10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816int120599

0int1206000

int1198860

1198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(30)

Using the above inequality we have

|(119879) ] (120580 120600 120599)| le 4119861 (120580 120600 120599)+ 1

120582119896119861 (120580 120600 120599) 120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|) (31)

and equivalently

120573120582119896119896 (119879]) le 41205730119896 (119861) + 1

1205821198961205730119896 (119861) 120572120582119896119896 (]) (32)

Choose 120582119896 = 51205730119896(119861) and set 119862 = 120596 isin 119862(R3R) 120573120582119896119896

(120596) le51205730119896(119861) a closed and bounded subset of 119883 we have for any120596 isin 119862

120573120582119896119896 (119879120596) le 51205730119896 (119861) (33)

that is

119879 (119862) sub 119862 (34)

Hence by the above Tychonoff theorem there exists ] isin 119862such that 119879] = ] that is ] is the required solution

Now we state our second Darboux type fuzzy differentialinclusion involving closed level sets of a fuzzy mappingdefined on an open subset of given space This problem isstated as follows

Problem 20 Consider the following partial fuzzy differentialinclusion

(120580120600120599120596(120580120600120599))(1205973120596 (120580 120600 120599)120597120580120597120600120597120599 ) ge 120574 (120596) (35)

for (120580 120600 120599 120596) isin Ω with local conditions (C1) which areequivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin [(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) (36)

with (C1)The next theorem describes the conditions under which

the solution of above problem exists

Theorem 21 Let Ω119886119887119888 997888rarr 119864119896 be a fuzzy integrablybounded and 120574- level uniformly continuous mapping Let 120574

R119896 997888rarr [0 1] be uniformly continuous If for every (120580 120600 120599 120596)(120580 120600 120599 ]) isin Ω119886119887119888 there exists 120575 isin (0 119886119887119888) satisfyingℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) le 120575

(119886119887119888)2 120596 minus ] (37)

Then Problem 20 possesses a solution

Proof Define 119865 Ω119886119887119888 997888rarr 2R119896 by 119865(120580 120600 120599 120596) = [(120580 120600 120599 120596)]120574(120596(120580120600120599)) We show that 119865(120580 120600 120599 120596) is 119906119904119888 For agiven (∘120580 ∘119910 ∘119911 ∘119906) isin Ω119886119887119888 we can write the neighborhood of119865(∘120580 ∘119910 ∘119911 ∘119906) as

119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903 = 120603 isin R119896 120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (38)

For (120580 120600 120599 120596) isin Ω119886119887119888 and ] isin 119865(120580 120600 120599 120596) we have

120588 (] 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) le ℎ119863 (119865 (120580 120600 120599 120596) 119865 ( ∘119905 ∘119910 ∘119911 ∘119906))= ℎ119863 ([ (120580 120600 120599 120596)]120574(120596) [ ( ∘119905 ∘119910 ∘119911 ∘119906)]

120574(∘119906))

le ℎ119863 ([ ( ∘119905 ∘119910 ∘119911 ∘119906)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(∘119906))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))le ℎ119863 (119865 ( ∘119905 ∘119910 ∘119911 ∘119906) 119865 (120580 120600 120599 120596))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))

(39)

Since [ (120580 120600 120599 120596)]120574(120596) is 120574-level uniformly continuous and 120574is uniformly continuous using the above inequality we canfind a small enough neighborhood 119880 of ( ∘119905 ∘119910 ∘119911 ∘119906) in Ω119886119887119888such that for all (120580 120600 120599 ]) isin 119880 and 120603 isin 119865(120580 120600 120599 ])

120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (40)

thus

119865 (119880) sub 119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903(41)

which means that 119865(120580 120600 120599 120596) is 119906119904119888 So by Lemma 14 thereexists a real constant 120579 gt 0 such that

max(120580120600120599120596)isinΩ

10038171003817100381710038171003817119865 (120580 120600 120599 120596)10038171003817100381710038171003817 le 120579 (42)

Complexity 7

Let 119883 = 120596 isin 119862(119868R119896) 120596 minus 1205960 le 120579 for all (120580 120600 120599) isinΩ119886119887119888 120596(1205800 1206000 1205990) = 1205960 with a metric 120588119883 119883 times 119883 997888rarr R cup+infin defined by

120588119883 = sup(120580120600120599)isin119868

120596 (120580 120600 120599) minus ] (120580 120600 120599) (43)

Then (119883 120588119883) is a complete generalized metric space [14]Define 119879 119883 997888rarr 2119883 by

119879 (120596) = 119906 (120580 120600 120599) 119906 (120580 120600 120599) isin 1199060 (120580 120600 120599) + 1199061 (120580 120600 120599)

+ int120580

0int120600

0int120599

0[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119904120580)) 119889119903119889119904119889120591 119886119890 119868

(44)

where

1199060 (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0) (45)

and

1199061 (120580 120600 120599) = int119888

01198961 (120580 120600) 120596 (120580 120600 119904) 119889119904

+ int1205990

int119887

01198962 (120580 120599) 120596 (120580 119904 120599) 119889119904119889120599

minus int1205990

int120600

0int119886

01198963 (120600 120599) 120596 (119904 120600 120599) 119889119904119889120600119889120599

(46)

Here int1205800int1206000

int1205990[ (119903 119904 120580 120596(119903 119904 120580))]120574(120596(119903119904120580))119889119903119889119904119889120591 is multival-

ued triple integral of Aumann [22] defined by

int1205800int120600

0int120599

0119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 = int120580

0int1206000

int1205990

[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119903119904120580)) 119889119903119889119904119889120591

= 1199060 + 1199061 + int1205800int120600

0int120599

0119891 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 | 119891 Ω 997888rarr R

119896 is measurable selection for [ (119903 119904 120580 120596 (119903 119904 120580))]120574 (47)

for each 120574 isin (0 1]Clearly 119879(120596) = 120593 for all 120596 isin 119883 Since the operator

119865(120580 120600 120599 120596) = [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) is compact 119906119904119888so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23] 119865(120580 120600 120599 120596) has measurableselection 119891(120580 120600 120599 120596) isin [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) for all(120580 120600 120599 120596) isin Ω times R119896 and 119891(120580 120600 120599 120596) is Lebesgue integrable[4]

Let 120596(120580 120600 120599) = 1205960 + int1205800int1206000

int1205990119891(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 isin

119879(120596) where 1205960 = 1199060 + 1199061 thus 119879(120596) = 120593Next we show that 119879(120596) is closed for all 120596 isin 119883

Consider a sequence 120596119899 in 119879(120596) such that 120596119899 997888rarr ∘119906 isin 119883Since

120596119899 isin 1205960 + int120580

0int1206000

int1205990

119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 (48)

and int1205800int1206000

int1205990119865(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 is closed [22] so we

have ∘119906 isin 119879(120596)

We show that 119879 is multivalued contraction let 1205962 isin119879(1205961) which implies that there exists 119891(120580 120600 120599 120596) isin119865(120580 120600 120599 120596) such that

1205962 (120580 120600 120599) = 1205960 + int1205800int120600

0int120599

0119891 (119904 120580 1205961 (119904 120580)) 119889119903119889119904119889120591 (49)

By virtue of Lemma 15 there exists a measurable selection119891(120580 120600 120599 120596) isin 119865(120580 120600 120599 120596) such that

1003817100381710038171003817119891 (120580 120600 120599 120596) minus 119891 (119904 120580 1205961 (119904 120580))1003817100381710038171003817 le ℎ119863 (119865 (120580 120600 120599 120596) 119865 (120580 120600 120599 120596)) = ℎ119863 ([ (120580 120600 120599 120596)]120574(1205961(120580120600120599)) [ (120580 120600 120599 120596)]120574(1205962(120580120600120599))) le ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 120596))

(50)

Let

1205963 (120580 120600 120599) = 1205960+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 (51)

and consider

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817int

120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 minus int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205961 (119903 119904 120580)) 119889119903119889119904119889120591100381710038171003817100381710038171003817100381710038171003817

le int1205800int120600

0int120599

0

1003817100381710038171003817119891 (120580 120600 120599 120596)

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

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Submit your manuscripts atwwwhindawicom

6 Complexity

Let

119861 (120580 120600 120599)

= max

10038161003816100381610038161205771 (120580 120600)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 120599)1003816100381610038161003816 + 10038161003816100381610038161205772 (120580 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 120599)1003816100381610038161003816 + 10038161003816100381610038161205773 (120600 0)1003816100381610038161003816 + 10038161003816100381610038161205773 (0 0)1003816100381610038161003816 + 119872119870100381610038161003816100381610038161003816100381610038161003816int120599

0int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816int120599

0int1206000

int1198860

1198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599100381610038161003816100381610038161003816100381610038161003816 100381610038161003816100381610038161003816100381610038161003816int

120580

0int120600

0int1205990max 120595 (120580 120600 1) 120595 (120580 120599 1) 120595 (120600 120599 1) 119889119903119889119904119889120591100381610038161003816100381610038161003816100381610038161003816

(30)

Using the above inequality we have

|(119879) ] (120580 120600 120599)| le 4119861 (120580 120600 120599)+ 1

120582119896119861 (120580 120600 120599) 120573120582119896119896 (]) 119890120582119896(|120600|+|120599|+|120580|) (31)

and equivalently

120573120582119896119896 (119879]) le 41205730119896 (119861) + 1

1205821198961205730119896 (119861) 120572120582119896119896 (]) (32)

Choose 120582119896 = 51205730119896(119861) and set 119862 = 120596 isin 119862(R3R) 120573120582119896119896

(120596) le51205730119896(119861) a closed and bounded subset of 119883 we have for any120596 isin 119862

120573120582119896119896 (119879120596) le 51205730119896 (119861) (33)

that is

119879 (119862) sub 119862 (34)

Hence by the above Tychonoff theorem there exists ] isin 119862such that 119879] = ] that is ] is the required solution

Now we state our second Darboux type fuzzy differentialinclusion involving closed level sets of a fuzzy mappingdefined on an open subset of given space This problem isstated as follows

Problem 20 Consider the following partial fuzzy differentialinclusion

(120580120600120599120596(120580120600120599))(1205973120596 (120580 120600 120599)120597120580120597120600120597120599 ) ge 120574 (120596) (35)

for (120580 120600 120599 120596) isin Ω with local conditions (C1) which areequivalent to

1205973120596 (120580 120600 120599)120597120580120597120600120597120599 isin [(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) (36)

with (C1)The next theorem describes the conditions under which

the solution of above problem exists

Theorem 21 Let Ω119886119887119888 997888rarr 119864119896 be a fuzzy integrablybounded and 120574- level uniformly continuous mapping Let 120574

R119896 997888rarr [0 1] be uniformly continuous If for every (120580 120600 120599 120596)(120580 120600 120599 ]) isin Ω119886119887119888 there exists 120575 isin (0 119886119887119888) satisfyingℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) le 120575

(119886119887119888)2 120596 minus ] (37)

Then Problem 20 possesses a solution

Proof Define 119865 Ω119886119887119888 997888rarr 2R119896 by 119865(120580 120600 120599 120596) = [(120580 120600 120599 120596)]120574(120596(120580120600120599)) We show that 119865(120580 120600 120599 120596) is 119906119904119888 For agiven (∘120580 ∘119910 ∘119911 ∘119906) isin Ω119886119887119888 we can write the neighborhood of119865(∘120580 ∘119910 ∘119911 ∘119906) as

119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903 = 120603 isin R119896 120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (38)

For (120580 120600 120599 120596) isin Ω119886119887119888 and ] isin 119865(120580 120600 120599 120596) we have

120588 (] 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) le ℎ119863 (119865 (120580 120600 120599 120596) 119865 ( ∘119905 ∘119910 ∘119911 ∘119906))= ℎ119863 ([ (120580 120600 120599 120596)]120574(120596) [ ( ∘119905 ∘119910 ∘119911 ∘119906)]

120574(∘119906))

le ℎ119863 ([ ( ∘119905 ∘119910 ∘119911 ∘119906)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(∘119906))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))le ℎ119863 (119865 ( ∘119905 ∘119910 ∘119911 ∘119906) 119865 (120580 120600 120599 120596))

+ ℎ119863 ([ (120580 120600 120599 120596)]120574(∘119906)

[ (120580 120600 120599 120596)]120574(120596))

(39)

Since [ (120580 120600 120599 120596)]120574(120596) is 120574-level uniformly continuous and 120574is uniformly continuous using the above inequality we canfind a small enough neighborhood 119880 of ( ∘119905 ∘119910 ∘119911 ∘119906) in Ω119886119887119888such that for all (120580 120600 120599 ]) isin 119880 and 120603 isin 119865(120580 120600 120599 ])

120588 (120603 119865 ( ∘119905 ∘119910 ∘119911 ∘119906)) lt 119903 (40)

thus

119865 (119880) sub 119878119865(∘119905 ∘119910 ∘119911 ∘119906)119903(41)

which means that 119865(120580 120600 120599 120596) is 119906119904119888 So by Lemma 14 thereexists a real constant 120579 gt 0 such that

max(120580120600120599120596)isinΩ

10038171003817100381710038171003817119865 (120580 120600 120599 120596)10038171003817100381710038171003817 le 120579 (42)

Complexity 7

Let 119883 = 120596 isin 119862(119868R119896) 120596 minus 1205960 le 120579 for all (120580 120600 120599) isinΩ119886119887119888 120596(1205800 1206000 1205990) = 1205960 with a metric 120588119883 119883 times 119883 997888rarr R cup+infin defined by

120588119883 = sup(120580120600120599)isin119868

120596 (120580 120600 120599) minus ] (120580 120600 120599) (43)

Then (119883 120588119883) is a complete generalized metric space [14]Define 119879 119883 997888rarr 2119883 by

119879 (120596) = 119906 (120580 120600 120599) 119906 (120580 120600 120599) isin 1199060 (120580 120600 120599) + 1199061 (120580 120600 120599)

+ int120580

0int120600

0int120599

0[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119904120580)) 119889119903119889119904119889120591 119886119890 119868

(44)

where

1199060 (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0) (45)

and

1199061 (120580 120600 120599) = int119888

01198961 (120580 120600) 120596 (120580 120600 119904) 119889119904

+ int1205990

int119887

01198962 (120580 120599) 120596 (120580 119904 120599) 119889119904119889120599

minus int1205990

int120600

0int119886

01198963 (120600 120599) 120596 (119904 120600 120599) 119889119904119889120600119889120599

(46)

Here int1205800int1206000

int1205990[ (119903 119904 120580 120596(119903 119904 120580))]120574(120596(119903119904120580))119889119903119889119904119889120591 is multival-

ued triple integral of Aumann [22] defined by

int1205800int120600

0int120599

0119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 = int120580

0int1206000

int1205990

[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119903119904120580)) 119889119903119889119904119889120591

= 1199060 + 1199061 + int1205800int120600

0int120599

0119891 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 | 119891 Ω 997888rarr R

119896 is measurable selection for [ (119903 119904 120580 120596 (119903 119904 120580))]120574 (47)

for each 120574 isin (0 1]Clearly 119879(120596) = 120593 for all 120596 isin 119883 Since the operator

119865(120580 120600 120599 120596) = [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) is compact 119906119904119888so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23] 119865(120580 120600 120599 120596) has measurableselection 119891(120580 120600 120599 120596) isin [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) for all(120580 120600 120599 120596) isin Ω times R119896 and 119891(120580 120600 120599 120596) is Lebesgue integrable[4]

Let 120596(120580 120600 120599) = 1205960 + int1205800int1206000

int1205990119891(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 isin

119879(120596) where 1205960 = 1199060 + 1199061 thus 119879(120596) = 120593Next we show that 119879(120596) is closed for all 120596 isin 119883

Consider a sequence 120596119899 in 119879(120596) such that 120596119899 997888rarr ∘119906 isin 119883Since

120596119899 isin 1205960 + int120580

0int1206000

int1205990

119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 (48)

and int1205800int1206000

int1205990119865(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 is closed [22] so we

have ∘119906 isin 119879(120596)

We show that 119879 is multivalued contraction let 1205962 isin119879(1205961) which implies that there exists 119891(120580 120600 120599 120596) isin119865(120580 120600 120599 120596) such that

1205962 (120580 120600 120599) = 1205960 + int1205800int120600

0int120599

0119891 (119904 120580 1205961 (119904 120580)) 119889119903119889119904119889120591 (49)

By virtue of Lemma 15 there exists a measurable selection119891(120580 120600 120599 120596) isin 119865(120580 120600 120599 120596) such that

1003817100381710038171003817119891 (120580 120600 120599 120596) minus 119891 (119904 120580 1205961 (119904 120580))1003817100381710038171003817 le ℎ119863 (119865 (120580 120600 120599 120596) 119865 (120580 120600 120599 120596)) = ℎ119863 ([ (120580 120600 120599 120596)]120574(1205961(120580120600120599)) [ (120580 120600 120599 120596)]120574(1205962(120580120600120599))) le ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 120596))

(50)

Let

1205963 (120580 120600 120599) = 1205960+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 (51)

and consider

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817int

120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 minus int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205961 (119903 119904 120580)) 119889119903119889119904119889120591100381710038171003817100381710038171003817100381710038171003817

le int1205800int120600

0int120599

0

1003817100381710038171003817119891 (120580 120600 120599 120596)

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

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Complexity 7

Let 119883 = 120596 isin 119862(119868R119896) 120596 minus 1205960 le 120579 for all (120580 120600 120599) isinΩ119886119887119888 120596(1205800 1206000 1205990) = 1205960 with a metric 120588119883 119883 times 119883 997888rarr R cup+infin defined by

120588119883 = sup(120580120600120599)isin119868

120596 (120580 120600 120599) minus ] (120580 120600 120599) (43)

Then (119883 120588119883) is a complete generalized metric space [14]Define 119879 119883 997888rarr 2119883 by

119879 (120596) = 119906 (120580 120600 120599) 119906 (120580 120600 120599) isin 1199060 (120580 120600 120599) + 1199061 (120580 120600 120599)

+ int120580

0int120600

0int120599

0[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119904120580)) 119889119903119889119904119889120591 119886119890 119868

(44)

where

1199060 (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0) (45)

and

1199061 (120580 120600 120599) = int119888

01198961 (120580 120600) 120596 (120580 120600 119904) 119889119904

+ int1205990

int119887

01198962 (120580 120599) 120596 (120580 119904 120599) 119889119904119889120599

minus int1205990

int120600

0int119886

01198963 (120600 120599) 120596 (119904 120600 120599) 119889119904119889120600119889120599

(46)

Here int1205800int1206000

int1205990[ (119903 119904 120580 120596(119903 119904 120580))]120574(120596(119903119904120580))119889119903119889119904119889120591 is multival-

ued triple integral of Aumann [22] defined by

int1205800int120600

0int120599

0119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 = int120580

0int1206000

int1205990

[ (119903 119904 120580 120596 (119903 119904 120580))]120574(120596(119903119904120580)) 119889119903119889119904119889120591

= 1199060 + 1199061 + int1205800int120600

0int120599

0119891 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 | 119891 Ω 997888rarr R

119896 is measurable selection for [ (119903 119904 120580 120596 (119903 119904 120580))]120574 (47)

for each 120574 isin (0 1]Clearly 119879(120596) = 120593 for all 120596 isin 119883 Since the operator

119865(120580 120600 120599 120596) = [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) is compact 119906119904119888so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23] 119865(120580 120600 120599 120596) has measurableselection 119891(120580 120600 120599 120596) isin [ (120580 120600 120599 120596)]120574(120596(120580120600120599)) for all(120580 120600 120599 120596) isin Ω times R119896 and 119891(120580 120600 120599 120596) is Lebesgue integrable[4]

Let 120596(120580 120600 120599) = 1205960 + int1205800int1206000

int1205990119891(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 isin

119879(120596) where 1205960 = 1199060 + 1199061 thus 119879(120596) = 120593Next we show that 119879(120596) is closed for all 120596 isin 119883

Consider a sequence 120596119899 in 119879(120596) such that 120596119899 997888rarr ∘119906 isin 119883Since

120596119899 isin 1205960 + int120580

0int1206000

int1205990

119865 (119903 119904 120580 120596 (119903 119904 120580)) 119889119903119889119904119889120591 (48)

and int1205800int1206000

int1205990119865(119903 119904 120580 120596(119903 119904 120580))119889119903119889119904119889120591 is closed [22] so we

have ∘119906 isin 119879(120596)

We show that 119879 is multivalued contraction let 1205962 isin119879(1205961) which implies that there exists 119891(120580 120600 120599 120596) isin119865(120580 120600 120599 120596) such that

1205962 (120580 120600 120599) = 1205960 + int1205800int120600

0int120599

0119891 (119904 120580 1205961 (119904 120580)) 119889119903119889119904119889120591 (49)

By virtue of Lemma 15 there exists a measurable selection119891(120580 120600 120599 120596) isin 119865(120580 120600 120599 120596) such that

1003817100381710038171003817119891 (120580 120600 120599 120596) minus 119891 (119904 120580 1205961 (119904 120580))1003817100381710038171003817 le ℎ119863 (119865 (120580 120600 120599 120596) 119865 (120580 120600 120599 120596)) = ℎ119863 ([ (120580 120600 120599 120596)]120574(1205961(120580120600120599)) [ (120580 120600 120599 120596)]120574(1205962(120580120600120599))) le ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 120596))

(50)

Let

1205963 (120580 120600 120599) = 1205960+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 (51)

and consider

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817= 100381710038171003817100381710038171003817100381710038171003817int

120580

0int120600

0int120599

0119891 (119903 119904 120580 1205962 (119903 119904 120580)) 119889119903119889119904119889120591 minus int120580

0int120600

0int120599

0119891 (119903 119904 120580 1205961 (119903 119904 120580)) 119889119903119889119904119889120591100381710038171003817100381710038171003817100381710038171003817

le int1205800int120600

0int120599

0

1003817100381710038171003817119891 (120580 120600 120599 120596)

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

minus 119891 (119903 119904 120580 1205961 (119903 119904 120580))1003817100381710038171003817 119889119903119889119904119889120591le int120580

0int120600

0int120599

0ℎ119863 ( (120580 120600 120599 120596) (120580 120600 120599 ])) 119889119903119889119904119889120591

le int1205800int120600

0int120599

0

120575(119886119887119888)2

10038171003817100381710038171205961 (120580 120600 120599)minus 1205962 (120580 120600 120599)1003817100381710038171003817 119889119903119889119904119889120591le 120575

(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) int120580

0int120600

0int120599

0119889119903119889119904119889120591

le 120575(119886119887119888)2 120588 (1205961 (120580 120600 120599) 1205962 (120580 120600

120599)) (120599 minus 1205990) (120600 minus 1206000)(52)

which gives

10038171003817100381710038171205963 (120580 120600 120599) minus 1205962 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (53)

similarly we will get

10038171003817100381710038171205962 (120580 120600 120599) minus 1205963 (120580 120600 120599)1003817100381710038171003817le 120575

119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (54)

Thus we have

ℎ119863 (119879 (1205961) 119879 (1205962)) le 120575119886119887119888120588 (1205961 (120580 120600 120599) 1205962 (120580 120600 120599)) (55)

for all 1205961(120580 120600 120599) 1205962(120580 120600 120599) isin 119883Since 120575119886119887119888 isin (0 1) so by Nadlerrsquos Theorem [24] there

exist a fixed point ] isin 119879] Thus

] (120580 120600 120599) = 1205771 (120580 120600) + 1205772 (120580 120599) + 1205773 (120600 120599) minus 1205772 (120580 0)minus 1205773 (0 120599) minus 1205773 (120600 0) + 1205773 (0 0)minus int119888

01198961 (120580 120600) ] (120580 120600 119904) 119889119904

minus int1205990

int119887

01198962 (120580 120599) ] (120580 119904 120599) 119889119904119889120599

+ int120599

0int1206000

int119886

01198963 (120600 120599) ] (119904 120600 120599) 119889119904119889120600119889120599

+ int120580

0int120600

0int120599

0119891 (119903 119904 120580 ] (119903 119904 120580)) 119889119903119889119904119889120591

(56)

In the next example we use Theorems 17 and 21 to findthe solution of given fuzzy differential inclusion

Example 22 Consider the hyperbolic PDI

1205973120596120597120580120597120600120597120599 isin [(120580120600120599120596)]120572(120596) (57)

with boundary conditions

120596 (120580 120600 0) = sin (120580 + 120600) 120596 (120580 0 120599) = sin (120580 + 120599)120596 (0 120600 120599) = sin (120600 + 120599)

(58)

where [0 1]2 times R timesR 997888rarr 1198641 defined by

(120580120600120599120596) = cos (120587 minus 120580 minus 120600 minus 120599) (119890minus2(120580) 2119890minus2(120580))119878

(59)

where (119890minus2(120600+120603) 2119890minus2(120600+120603))119878 is a symmetric triangular fuzzynumber with compact support [119890minus2(120600+120603) 2119890minus2(120600+120603)] For any120572 isin [0 1] the 120572-level set is

[(120580120600120599120596)]120572 = cos (120587 minus 120580 minus 120600 minus 120599)sdot [(1 + 05120572) 119890minus2(120580) (2 minus 05120572) 119890minus2(120580)] (60)

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 9

Clearly is convex and bounded Using Theorem 17there exists a continuous selection 119891(120580 120600 120599 120596) isin[(120580120600120599120596(120580120600120599))]120574(120596(120580120600120599)) for each (120580 120600 120599 120596) isin Ω suchthat

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (61)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (62)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (63)

Now for 120572 isin [0 1] considerℎ119863 ([(120580120600120599120596)]120572 [(120580120600120599])]120572)

= 10038161003816100381610038161003816cos (120587 minus 120580 minus 120600 minus 120599) 119890minus212058010038161003816100381610038161003816sdot ℎ119863 ([1 + 05120572 2 minus 05120572] [1 + 05120572 2 minus 05120572])= 0 le 10038161003816100381610038161003816119890minus2(120580)10038161003816100381610038161003816 120596 minus ]

(64)

Using Theorem 21 there exists a continuous selection119891(120580 120600 120599 120596) isin [(120580120600120599])]120572 for each (120580 120600 120599 120596) isin Ω such that

minus cos (120580 + 120600 + 120599) = 119891 (120580 120600 120599 120596) (65)

which implies

minus cos (120580 + 120600 + 120599) = 1205973120596120597120580120597120600120597120599 (66)

Hence the solution of 1205973120596120597120580120597120600120597120599 + cos(120580 + 120600 + 120599) = 0 is120596 (120580 120600 120599) = sin (120580 + 120600 + 120599) (67)

Data Availability

No data were used to support this study

Conflicts of Interest

The authors have declared that they have no conflicts ofinterest

Acknowledgments

The second author is grateful to the Department of ResearchAffairs at UAEU for Grant UPAR (11) 2016 Fund No 31S249(COS)Thefirst authorwould like to thank theDepartment ofMathematical ScienceUAEU andDepartment ofMathemat-ics and Statistics International Islamic University IslamabadPakistan for their support during his tenure of Post Doc

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8no 3 pp 338ndash353 1965

[2] S C Sheldon and L A Zadeh ldquoOn fuzzy mapping and controlrdquoIEEE Transactions on Systems Man and Cybernetics no 1 pp30ndash34 1972

[3] D Dubois and H Prade ldquoTowards fuzzy differential calculuspart 3 differentiationrdquo Fuzzy Sets and Systems vol 8 no 3 pp225ndash233 1982

[4] S Seikkala ldquoOn the fuzzy initial value problemrdquo Fuzzy Sets andSystems vol 24 no 3 pp 319ndash330 1987

[5] V Lakshmikantham and R N Mohapatra Theory of FuzzyDifferential Equations and Inclusions CRC Press 2004

[6] L T Gomes L C de Barros and B Bede Fuzzy DifferentialEquations in Various Approaches Springer Berlin Germany2015

[7] J J Buckley and T Feuring ldquoIntroduction to fuzzy partialdifferential equationsrdquo Fuzzy Sets and Systems vol 105 no 2pp 241ndash248 1999

[8] A Arara and M Benchohra ldquoFuzzy solutions for boundaryvalue problems with integral boundary conditionsrdquo ActaMath-ematica Universitatis Comenianae vol 75 no 1 pp 119ndash1262006

[9] A Arara M Benchohra S K Ntouyas and A OuahabldquoFuzzy solutions for hyperbolic partial differential equationsrdquoInternational Journal of Applied Mathematical Sciences vol 2no 2 pp 181ndash195 2005

[10] Y-Y Chen Y-T Chang and B-S Chen ldquoFuzzy solutions topartial differential equations adaptive approachrdquo IEEE Trans-actions on Fuzzy Systems vol 17 no 1 pp 116ndash127 2009

[11] H V Long N T K Son N T M Ha and L H Son ldquoTheexistence and uniqueness of fuzzy solutions for hyperbolicpartial differential equationsrdquo Fuzzy Optimization and DecisionMaking vol 13 no 4 pp 435ndash462 2014

[12] M Nikravesh and L A Zadeh Fuzzy Partial DifferentialEquations And Relational Equations Reservoir Characterizationand Modeling Springer Science amp Business Media 2004

[13] Y G Zhu and L Rao ldquoDifferential inclusions for fuzzy mapsrdquoFuzzy Sets and Systems vol 112 no 2 pp 257ndash261 2000

[14] C Min Z-B Liu L-H Zhang and N-J Huang ldquoOn a systemof fuzzy differential inclusionsrdquo Filomat vol 29 no 6 pp 1231ndash1244 2015

[15] N Mehmood and A Azam ldquoExistence results for fuzzy partialdifferential inclusionsrdquo Journal of Function Spaces vol 2016Article ID 6759294 8 pages 2016

[16] J R Munkres and R James Munkres Topology Prentice HallIncorporated 2000

[17] N C Yannelis and N D Prabhakar ldquoExistence of maximalelements and equilibria in linear topological spacesrdquo Journal ofMathematical Economics vol 12 no 3 pp 233ndash245 1983

[18] J P Aubin and A Cellina Differential Inclusions Set-ValuedMaps And Viability Theory Springer Science amp Business Media2012

[19] A Tychonoff ldquoEin fixpunktsatzrdquo Mathematische Annalen vol111 no 1 pp 767ndash776 1935

[20] M A NaimarkNormed Rings Noordhoff Groningen Nether-lands 1959

[21] A K Aziz and J P Maloney ldquoAn application of Tychonoff rsquosfixed point theorem to hyperbolic partial differential equationsrdquoMathematische Annalen vol 162 no 1 pp 77ndash82 1965

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Complexity

[22] R J Aumann ldquoIntegrals of set-valued functionsrdquo Journal ofMathematical Analysis and Applications vol 12 no 1 pp 1ndash121965

[23] K Kuratowski and C Z Ryll-Nardzewski ldquoA general theoremon selectorsrdquoBulletin LrsquoAcademie Polonaise des Science Serie desSciences Mathematiques Astronomiques et Physiques vol 13 no6 pp 397ndash403 1965

[24] J Nadler ldquoMulti-valued contraction mappingsrdquo Pacific Journalof Mathematics vol 30 no 2 pp 475ndash488 1969

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom