# On Darboux-Type Differential Inclusions with Uncertainty

Authors: Nayyar Mehmood; Ahmed Al-Rawashdeh; Akbar Azam
Journal: Complexity
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

The uncertainties that occur in modelling of the physical problems may originate some types of ambiguities. Zadeh, in [1], introduced the fuzzy sets to deal with ambiguities. For modelling the physical problems in a better way, the formulation of these problems by virtue of fuzzy differential equations plays a significant role. The notion of fuzzy derivatives was familiarized in [2]. Afterwards, this idea was explored by using Zadeh’s extension principle (ZEP), in [3]. The generalized Hukuhara derivative for fuzzy functions was introduced and studied in [4]. For detailed overview of theory of fuzzy differential equations, we refer the readers to consult the monographs [5, 6].The ambiguous and complex physical models can be demonstrated in a convenient way by means of fuzzy differential equations (FDEs). The basic idea of fuzzy partial differential equations (FPDEs) has been originated in [7]. Many of the physical problems were designed, solved, and analyzed with fuzzy differential equations. Many authors put their efforts to study and extend this theory; see, for example [8–12]. The importance of both the ODEs and the PDEs in fuzzy sense is evident. Up till now the theory of FDEs has been studied in various ways but the analysis of fuzzy partial differential equations still needs to be explored. Most of the results are not concerned for fuzzy partial differential inclusions. In [13], the authors discuss the existence of solution of a fuzzy differential inclusion problem constructed via level sets of fuzzy mappings. In [14, 15], the authors extended this technique for system of fuzzy differential inclusions. We generalize and extend this study to discuss the existence of solutions of fuzzy partial differential inclusions (FPDIs). In this article we investigate two models for Darboux-type partial fuzzy differential inclusions with local conditions of integral type. We furnish an example to justify our main results.


## 2. Preliminaries

LetRk be k-dimensional Euclidean space; for A⊆Rk, A¯ and Ao denote the closure and interior of A, respectively. Let KX be the family of all nonempty, convex, and compact subsets of a given linear normed space X. The Hausdorff metric on KRk is defined as (1)hDA,B=max⁡supb∈Bdb,A,supa∈Ada,B.KRk,hD is a complete and separable metric space. By Ωa,b,c we mean the open subset 0,a×0,b×0,c of R3; denote Sϵϰ by an open ball with center at ϰ and radius ϵ>0. For a given nonempty set X, let 2X be the family of all nonempty subsets of X.Definition 1 (see [14]).
A fuzzy subsetω of Rk is a function ω:Rk→0,1. For γ∈0,1, denote the γ open level and γ closed level sets of ω by ωγ and ωγ, respectively, defined as follows: (2)ωγ=ϰ∈Rk:ωϰ>γ,and (3)ωγ=ϰ∈Rk:ωϰ≥γ.Definition 2 (see [14]).
A fuzzy subsetω of Rk is called
i fuzzy normal set if there exists an t∈Rk such that ωt=1;
i i fuzzy convex fuzzy set, if, for s,t∈Rk and λ∈0,1, (4)ωλs+1-λt≥min⁡ωs,ωt;
i i i fuzzy upper semicontinuous (u.s.c), if, for any γ∈(0,1], ωγ is closed set in the usual topology on Rk.For a fuzzy subsetω of Rk, the support of ω is denoted and defined as (5)ω0=s∈Rk:ωs≥0¯.Definition 3 (see [13]).
i Let Y,ρ1 and Z,ρ2 be two metric spaces and let ψ:Y→2Z be a set-valued mapping. Then ψ is said to be upper semicontinuous u.s.c on Y, if, for each ϑ∈Y and any open set V⊂Z, with ψϑ⊂V, there exists an open neighborhood Uϑ of ϑ such that ψUϑ⊂V.
(ii)  ψ is called lower semicontinuous l.s.c on Y if, for each ϑ∈Y and any open set V⊂Z with ψϑ∩V≠ϕ, there exists an open neighborhood Uϑ of ϑ such that ψϑ∩V≠ϕ for all ϑ∈Uϑ.Definition 4 (see [14]).
LetEk denote the set of all fuzzy sets of Rk that are fuzzy normal, fuzzy convex, and fuzzy upper u.s.c and have compact support.It is well known that, for eachω∈Ek and γ∈0,1, the closed level sets ωγ are compact and convex in the usual topology on Rk [14].For any nonempty subsetV of R, the fuzzy map ϝ:V×V→Ek can be presented by a real valued function F~:V×V×Rk→0,1, where, s,t∈V,ϰ∈Rk, we denote F~s,t,ϰ by ϝs,tϰ.Definition 5 (see [14]).
A fuzzy mappingϝ:Ω→Ek, where Ω⊂Ωa,b,c×Rk, is called lower open if ϝϰ,ϑ,ωϖ is l.s.c at each ϰ,ϑ,ω∈Ω.There is extraordinary useful weakening of compactness that is satisfied by many topological spaces that arise in geometry and analysis, called paracompactness. Paracompactness is weaker than compactness and is often adequate for many purposes. For example,Rk with usual topology is not compact but a paracomapct space. Before defining the paracompact spaces, we first recall some notions used in the definition of a paracompact space.Definition 6 (see [16]).
An open covering of a topological spaceX is the collection of open sets, so that their union covers (contains) X. An open covering Ui of X refines open covering Vj of X, if each Ui is contained in some Vj.Definition 7 (see [16]).
An open coveringUi of X is locally finite if every x∈X admits a neighborhood N such that N∩Ui is empty for all but finitely many i.For example, the covering ofR by open intervals n-1,n+1 for n∈Z is locally finite, whereas the covering of the interval -1,1 by intervals -1/n,1/n for n≥1 barely fails to be locally finite, as there is a problem at the origin; now we define the paracompact space.Definition 8 (see [16]).
A topological spaceX is called paracompact if every open covering has a locally finite refinement.Definition 9 (see [16]).
A Hausdorff space is a topological space in which every two distinct points can be separated by disjoint open sets.Definition 10 (see [17]).
LetX and Y be two topological spaces and let ϝ:X→2Y be a set valued mapping. ϝ is said to have open lower sections if (6)ϝ-1ϰ=x∈X:ϰ∈ϝxis open in X for any ϰ∈Y.Definition 11 (see [14]).
For a given completeσ-finite measure space M,A,μ and a separable Banach space X,· a compact set-valued operator ψ:M→KX is called an integrably bounded if and only if there exits a ϕ∈L1M such that for all ϖ∈ψs, ϖ≤ϕs a.e.For1≤p≤∞, define the set (7)Sψp=g∈LpΩ,X:gs∈ψsa.e.,Sψp being the set of selections of ψ and the closed subset of LpΩ,X. Sψp is nonempty if and only if ψ is integrably bounded [18]. A fuzzy mapping ϝ:Ω→Ek is integrably bounded if ϝγ is integrably bounded for any γ∈0,1.Definition 12 (see [14]).
A fuzzy mappingϝ:Ω→Ek is called α-level uniformly continuous, if for a given ϵ>0 there exists δ>0 such that for any x,ϰ,ι∈Ω and for α,β∈0,1 such that β-α<δ implies (8)hDϝx,ϰ,ια,ϝx,ϰ,ιβ<ϵ.


## 3. Main Results

In this section, first we recall some important results that will be useful in the sequel; then we state and prove our main results.Proposition 13 (see [17]).
LetY and Z be the paracompact Hausdorff topological space and a topological vector space, respectively. Let ψ:Y→2Z be a convex valued mapping. If ψ possesses an open lower sections, then there exists a continuous selection φ of ψ; that is, φ:Y→Z is continuous and φϑ∈ψϑ for all ϑ∈Y.Lemma 14 (see [18]).
Consider an open subsetΩ⊂R×Rk×Rm and let T:Ω→KRk×Rm be an upper semicontinuous operator. Then there exists an open interval J of R, and a,M>0, such that(i)
J×SaMι,ϰ,ϑ⊂Ω for ι,ϰ,ϑ∈Ω.(ii)
Tι,ϰ,ϑ≤M on J×SaMι,ϰ,ϑ.Lemma 15 ( see [18]).
LetX be a Banach space with borel measure space X,A,μ. Let S,T:J→KX be two measurable compact operators. Then if ν:J→X is a measurable selection of S, then there exists a measurable selection ω:J→X of T such that (9)ωι-νι≤hDSι,Tιfor all ι∈J, where hD is Pompeiu Hausdorff metric on KX.Let us propose our first Darboux type differential inclusion problem, which involves open level sets of a fuzzy mapping defined on an open subset of given space. We state the first problem as follows.Problem 16.
LetRk,·∞ be a Banach space and let Ω be an open subset of Ωa,b,c×Rk. Let ω:Ωa,b,c→Rk be a continuous function and let ϝ:Ω→Ek be a fuzzy mapping with γ:Rk→[0,1) an u.s.c function. Consider the following FPDI: (10)ϝι,ϰ,ϑ,ωι,ϰ,ϑ∂3ω∂ι∂ϰ∂ϑ>γω,for ι,ϰ,ϑ,ω∈Ω, withC1ωι,ϰ,0+∫0ck1ι,ϰωι,ϰ,ϑdϑ=ζ1ι,ϰ,ωι,0,ϑ+∫0bk2ι,ϑωι,ϰ,ϑdϰ=ζ2ι,ϑ,ω0,ϰ,ϑ+∫0ak3ϰ,ϑωι,ϰ,ϑdι=ζ3ϰ,ϑ.The problem is equivalent to(11)∂3ωι,ϰ,ϑ∂ι∂ϰ∂ϑ∈ϝι,ϰ,ϑ,ωι,ϰ,ϑγωwith C1.In the next theorem we discuss the conditions, under which the above problem can be transformed into a nonfuzzy problem of partial differential equation, known as Darboux problem.Theorem 17.
Letϝ:Ω→Ek be a bounded convex and lower open fuzzy surjection. If there exists an u.s.c function γ:Rk→[0,1) such that ϝ(ι,ϰ,ϑ,ω(ι,ϰ,ϑ))γωι,ϰ,ϑ is nonempty for each (ι,ϰ,ϑ,ω)∈Ω then there exists a continuous h~:Ω→Rk such that h~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ))∈ϝ(ι,ϰ,ϑ,ω(ι,ϰ,ϑ))γωι,ϰ,ϑ satisfying (12)∂3ωι,ϰ,ϑ∂ι∂ϰ∂ϑ=h~ι,ϰ,ϑ,ωι,ϰ,ϑfor  allι,ϰ,ϑ,ω∈Ω.Proof.
Define a set-valued mapF~:Ω→2Rk by (13)F~ι,ϰ,ϑ,ωι,ϰ,ϑ=ϝι,ϰ,ϑ,ωι,ϰ,ϑγωι,ϰ,ϑ,for each (ι,ϰ,ϑ,ω)∈Ω, where ϝ(ι,ϰ,ϑ,ω(ι,ϰ,ϑ))γωι,ϰ,ϑ is γωι,ϰ,ϑ open level set of the fuzzy mapping ϝ. By assumptions F~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ)) is nonempty for each ι,ϰ,ϑ,ω∈Ω; consider for ϖ,z∈F~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ)) and θ∈0,1(14)ϝι,ϰ,ϑ,ωι,ϰ,ϑθϖ+1-θz≥min⁡ϝι,ϰ,ϑ,ωι,ϰ,ϑϖ,ϝι,ϰ,ϑ,ωι,ϰ,ϑz>γωι,ϰ,ϑby convexity of ϝ. Hence θϖ+1-θz∈F~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ)); thus F~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ)) is convex on Ω.
ClearlyF~ has open lower sections, since for any z∈Rk(15)F~-1z=ι,ϰ,ϑ,ω∈Ω:z∈F~ι,ϰ,ϑ,ωι,ϰ,ϑ,=ι,ϰ,ϑ,ω∈Ω:ϝι,ϰ,ϑ,ωι,ϰ,ϑz>γωι,ϰ,ϑ.We show that (16)F~-1zc=ι,ϰ,ϑ,ω∈Ω:ϝι,ϰ,ϑ,ωι,ϰ,ϑz≤γωι,ϰ,ϑ,is closed. Let ιn,ϰn,ϑn,ωn be a sequence in F~-1zc such that ιn,ϰn,ϑn,ωn→(ι,ϰ,ϑ,ω). As ϝ is lower open and γ is u.s.c therefore (17)ϝιn,ϰn,ϑn,ωnz≤γωιn,ϰn,ϑn,which implies (18)ϝι,ϰ,ϑ,ωz≤limn→∞inf⁡ϝιn,ϰn,ϑn,ωnz≤limn→∞sup⁡γωιn,ϰn,ϑn≤γωι,ϰ,ϑ.Thus (ι,ϰ,ϑ,ω)∈F~-1zc which shows that F~ has open lower sections. Thus, by Proposition 13, there exists a continuous h~∈F~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ)) for each (ι,ϰ,ϑ,ω)∈Ω. As ϝ is a surjection and F~(ι,ϰ,ϑ,ω(ι,ϰ,ϑ)) is bounded we get the problemP1∂3ωι,ϰ,ϑ∂ι∂ϰ∂ϑ=h~ι,ϰ,ϑ,ωι,ϰ,ϑfor  allι,ϰ,ϑ,ω∈Ωwith local conditions C1.Now we use the following Tychonoff Theorem to prove the existence of solution of problemP1 with local conditions C1.Theorem 18 (see [19] [Tychonoff]).
LetX be a complete locally convex vector space. Let C be a closed and convex subset of X and let T:X→X be continuous compact operator such that TC⊂C. Then T admits a fixed point.LetRk be a k-dimensional Euclidean space and we denote (19)CRk,R=ζ:ζis  continuous  andζ:Rk→R.The topology on CR2,R is induced by the families of seminorms(20)αkλkψ=sup⁡ψϰ,ϑe-λkϰ+ϑ:ϰ,ϑ∈Ωwhere Ω is the bounded region in R2, k∈N and λk≥0. CR2,R is complete and locally convex linear space [20]. We use the similar technique used in [21].Define the topology onCR3,R induced by the families of seminorms(21)βkλkψ=max⁡αkλkψϰ,ϑ,αkλkψϑ,ι,αkλkψι,ϰ;then with this topology, CR3,R is complete and locally convex linear space.ProblemP1 with C1 is equivalent to the fixed point problem Tv=v, where T:CR3,R→CR3,R is given by(22)Tνι,ϰ,ϑ=ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ-ζ2ι,0-ζ30,ϑ-ζ3ϰ,0+ζ30,0-∫0ck1ι,ϰνι,ϰ,sds-∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑfr,s,ι,νr,s,ιdrdsdτ.Condition 1.
f ι , ϰ , ϑ , ω ∈ C R 4 , R and (23)fι,ϰ,ϑ,ω≤max⁡ψι,ϰ,ω,ψι,ϑ,ω,ψϰ,ϑ,ω,ψ∈CR2×R+,R+andψ is subadditive in ω, for all ϰ,ϑ∈Ω.Condition 2.
ζ 1 , ζ 2 , k 1, k2,k3∈CR2,R.Theorem 19.
If Conditions1 and 2 hold then problem P1 with C1 has a solution in X=CR3,R.Proof.
ClearlyT is continuous and compact in the topology of X. We need to use Tychonoff Theorem to find the fixed point of T. It is sufficient to prove TC⊂C for a closed set C. Consider(24)Tνι,ϰ,ϑ=ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ+ζ2ι,0-ζ30,ϑ-ζ3ϰ,0+ζ30,0-∫0ck1ι,ϰνι,ϰ,sds-∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑfr,s,ι,νr,s,ιdrdsdτ.This implies(25)Tνι,ϰ,ϑ≤ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ+ζ2ι,0+ζ30,ϑ+ζ3ϰ,0+ζ30,0+∫0ck1ι,ϰνι,ϰ,sds+∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑfr,s,ι,νr,s,ιdrdsdτ.The subadditivity of ψ implies the following inequality, as given in [21]: (26)ψϰ,ϑ,ω≤ψϰ,ϑ,11+ωUsing the above inequality, we get(27)Tνι,ϰ,ϑ≤ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ+ζ2ι,0+ζ30,ϑ+ζ3ϰ,0+ζ30,0+∫0ck1ι,ϰνι,ϰ,sds+∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑmax⁡ψι,ϰ,1,ψι,ϑ,1,ψϰ,ϑ,11+νι,s,rdrdsdτ.Let M=∫0bk1ι,ϰ2dϑ1/2 and K=∫0bνι,ϰ,s2ds1/2. By applying Schwartz’s inequality and using above values of M and K, we get(28)Tνι,ϰ,ϑ≤ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ+ζ2ι,0+ζ30,ϑ+ζ3ϰ,0+ζ30,0+MK+∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑmax⁡ψι,ϰ,1,ψι,ϑ,1,ψϰ,ϑ,1drdsdτ+∫0ι∫0ϰ∫0ϑmax⁡ψι,ϰ,12,ψι,ϑ,12,ψϰ,ϑ,12drdsdτ1/2×∫0ι∫0ϰ∫0ϑνι,r,s2drdsdτ1/2.Using the definition of given seminorm, we obtain(29)Tνι,ϰ,ϑ≤ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ+ζ2ι,0+ζ30,ϑ+ζ3ϰ,0+ζ30,0+MK+∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑmax⁡ψι,ϰ,1,ψι,ϑ,1,ψϰ,ϑ,1drdsdτ+12λk∫0ι∫0ϰ∫0ϑmax⁡ψι,ϰ,12,ψι,ϑ,12,ψϰ,ϑ,12drdsdι1/2βkλkνeλkϰ+ϑ+ι.Let (30)Bι,ϰ,ϑ=max⁡ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ+ζ2ι,0+ζ30,ϑ+ζ3ϰ,0+ζ30,0+MK∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ,∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ,∫0ι∫0ϰ∫0ϑmax⁡ψι,ϰ,1,ψι,ϑ,1,ψϰ,ϑ,1drdsdτUsing the above inequality, we have (31)Tνι,ϰ,ϑ≤4Bι,ϰ,ϑ+1λkBι,ϰ,ϑβkλkνeλkϰ+ϑ+ι,and equivalently (32)βkλkTν≤4βk0B+1λkβk0BαkλkνChoose λk=5βk0B and set C=ω∈CR3,R:βkλkω≤5βk0B, a closed and bounded subset of X; we have for any ω∈C(33)βkλkTω≤5βk0B;that is (34)TC⊂C.Hence, by the aboveTychonoff theorem, there exists ν∈C such that Tν=ν; that is, ν is the required solution.Now we state our second Darboux type fuzzy differential inclusion, involving closed level sets of a fuzzy mapping defined on an open subset of given space. This problem is stated as follows.Problem 20.
Consider the following partial fuzzy differential inclusion:(35)ϝι,ϰ,ϑ,ωι,ϰ,ϑ∂3ωι,ϰ,ϑ∂ι∂ϰ∂ϑ≥γω,for (ι,ϰ,ϑ,ω)∈Ω, with local conditions C1, which are equivalent to (36)∂3ωι,ϰ,ϑ∂ι∂ϰ∂ϑ∈ϝι,ϰ,ϑ,ωι,ϰ,ϑγωι,ϰ,ϑwith C1.The next theorem describes the conditions under which the solution of above problem exists.Theorem 21.
Letϝ:Ωa,b,c→Ek be a fuzzy integrably bounded and γ- level uniformly continuous mapping. Let γ:Rk→0,1 be uniformly continuous. If, for every (ι,ϰ,ϑ,ω), ι,ϰ,ϑ,ν∈Ωa,b,c, there exists δ∈0,abc satisfying (37)hDϝι,ϰ,ϑ,ω,ϝι,ϰ,ϑ,ν≤δabc2ω-ν.Then Problem 20 possesses a solution.Proof.
DefineF~:Ωa,b,c→2Rk by F~(ι,ϰ,ϑ,ω)=ϝ(ι,ϰ,ϑ,ω)γωι,ϰ,ϑ. We show that F~(ι,ϰ,ϑ,ω) is u.s.c. For a given ι°,y°,z°,u°∈Ωa,b,c, we can write the neighborhood of F~ι°,y°,z°,u° as (38)SrF~t°,y°,z°,u°=ϖ∈Rk:ρϖ,F~t°,y°,z°,u°<r.For (ι,ϰ,ϑ,ω)∈Ωa,b,c and ν∈F~(ι,ϰ,ϑ,ω), we have (39)ρν,F~t°,y°,z°,u°≤hDF~ι,ϰ,ϑ,ω,F~t°,y°,z°,u°=hDϝι,ϰ,ϑ,ωγω,ϝt°,y°,z°,u°γu°≤hDϝt°,y°,z°,u°γu°,ϝι,ϰ,ϑ,ωγu°+hDϝι,ϰ,ϑ,ωγu°,ϝι,ϰ,ϑ,ωγω≤hDF~t°,y°,z°,u°,F~ι,ϰ,ϑ,ω+hDϝι,ϰ,ϑ,ωγu°,ϝι,ϰ,ϑ,ωγωSince ϝ(ι,ϰ,ϑ,ω)γω is γ-level uniformly continuous and γ is uniformly continuous, using the above inequality, we can find a small enough neighborhood U of t°,y°,z°,u° in Ωa,b,c, such that for all ι,ϰ,ϑ,ν∈U and ϖ∈F~ι,ϰ,ϑ,ν(40)ρϖ,F~t°,y°,z°,u°<r;thus (41)F~U⊂SrF~t°,y°,z°,u°which means that F~(ι,ϰ,ϑ,ω) is u.s.c. So by Lemma 14 there exists a real constant θ>0 such that (42)maxι,ϰ,ϑ,ω∈ΩF~ι,ϰ,ϑ,ω≤θ.Let X=ω∈CI,Rk:ω-ω0≤θ,forallι,ϰ,ϑ∈Ωa,b,c,ωι0,ϰ0,ϑ0=ω0, with a metric ρX:X×X→R∪+∞ defined by (43)ρX=supι,ϰ,ϑ∈Iωι,ϰ,ϑ-νι,ϰ,ϑ.Then X,ρX is a complete generalized metric space [14]. Define T:X→2X by (44)Tω=u¯ι,ϰ,ϑ:u¯ι,ϰ,ϑ∈u¯0ι,ϰ,ϑ+u¯1ι,ϰ,ϑ+∫0ι∫0ϰ∫0ϑϝr,s,ι,ωr,s,ιγωs,ιdrdsdτa.eI,where (45)u¯0ι,ϰ,ϑ=ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ-ζ2ι,0-ζ30,ϑ-ζ3ϰ,0+ζ30,0and (46)u¯1ι,ϰ,ϑ=∫0ck1ι,ϰωι,ϰ,sds+∫0ϑ∫0bk2ι,ϑωι,s,ϑdsdϑ-∫0ϑ∫0ϰ∫0ak3ϰ,ϑωs,ϰ,ϑdsdϰdϑHere ∫0ι∫0ϰ∫0ϑϝr,s,ι,ωr,s,ιγωr,s,ιdrdsdτ is multivalued triple integral of Aumann [22], defined by (47)∫0ι∫0ϰ∫0ϑF~r,s,ι,ωr,s,ιdrdsdτ=∫0ι∫0ϰ∫0ϑϝr,s,ι,ωr,s,ιγωr,s,ιdrdsdτ=u¯0+u¯1+∫0ι∫0ϰ∫0ϑfr,s,ι,ωr,s,ιdrdsdτ∣f:Ω→Rkis  measurable  selection  forϝr,s,ι,ωr,s,ιγ,for each γ∈(0,1].
ClearlyTω≠φ for all ω∈X. Since the operator F~(ι,ϰ,ϑ,ω)=ϝ(ι,ϰ,ϑ,ω)γωι,ϰ,ϑ is compact u.s.c so by well-known selection theorem of Kuratowski-Ryll-Nardzewski [23], F~(ι,ϰ,ϑ,ω) has measurable selection f(ι,ϰ,ϑ,ω)∈ϝ(ι,ϰ,ϑ,ω)γωι,ϰ,ϑ for all (ι,ϰ,ϑ,ω)∈Ω×Rk, and f(ι,ϰ,ϑ,ω) is Lebesgue integrable [4].
Letωι,ϰ,ϑ=ω0+∫0ι∫0ϰ∫0ϑfr,s,ι,ωr,s,ιdrdsdτ∈Tω, where ω0=u¯0+u¯1; thus Tω≠φ.
Next we show thatTω is closed for all ω∈X. Consider a sequence ωn in Tω such that ωn→u°∈X. Since (48)ωn∈ω0+∫0ι∫0ϰ∫0ϑF~r,s,ι,ωr,s,ιdrdsdτand ∫0ι∫0ϰ∫0ϑF~r,s,ι,ωr,s,ιdrdsdτ is closed [22], so we have u°∈Tω.
We show thatT is multivalued contraction; let ω2∈Tω1, which implies that there exists f(ι,ϰ,ϑ,ω)∈F~(ι,ϰ,ϑ,ω) such that (49)ω2ι,ϰ,ϑ=ω0+∫0ι∫0ϰ∫0ϑfs,ι,ω1s,ιdrdsdτBy virtue of Lemma 15, there exists a measurable selection f(ι,ϰ,ϑ,ω)∈F~(ι,ϰ,ϑ,ω) such that (50)fι,ϰ,ϑ,ω-fs,ι,ω1s,ι≤hDF~ι,ϰ,ϑ,ω,F~ι,ϰ,ϑ,ω=hDϝι,ϰ,ϑ,ωγω1ι,ϰ,ϑ,ϝι,ϰ,ϑ,ωγω2ι,ϰ,ϑ≤hDϝι,ϰ,ϑ,ω,ϝι,ϰ,ϑ,ω.Let (51)ω3ι,ϰ,ϑ=ω0+∫0ι∫0ϰ∫0ϑfr,s,ι,ω2r,s,ιdrdsdτ,and consider (52)ω3ι,ϰ,ϑ-ω2ι,ϰ,ϑ=∫0ι∫0ϰ∫0ϑfr,s,ι,ω2r,s,ιdrdsdτ-∫0ι∫0ϰ∫0ϑfr,s,ι,ω1r,s,ιdrdsdτ≤∫0ι∫0ϰ∫0ϑfι,ϰ,ϑ,ω-fr,s,ι,ω1r,s,ιdrdsdτ≤∫0ι∫0ϰ∫0ϑhDϝι,ϰ,ϑ,ω,ϝι,ϰ,ϑ,νdrdsdτ≤∫0ι∫0ϰ∫0ϑδabc2ω1ι,ϰ,ϑ-ω2ι,ϰ,ϑdrdsdτ≤δabc2ρω1ι,ϰ,ϑ,ω2ι,ϰ,ϑ∫0ι∫0ϰ∫0ϑdrdsdτ≤δabc2ρω1ι,ϰ,ϑ,ω2ι,ϰ,ϑϑ-ϑ0ϰ-ϰ0which gives (53)ω3ι,ϰ,ϑ-ω2ι,ϰ,ϑ≤δabcρω1ι,ϰ,ϑ,ω2ι,ϰ,ϑ;similarly we will get (54)ω2ι,ϰ,ϑ-ω3ι,ϰ,ϑ≤δabcρω1ι,ϰ,ϑ,ω2ι,ϰ,ϑThus we have (55)hDTω1,Tω2≤δabcρω1ι,ϰ,ϑ,ω2ι,ϰ,ϑfor all ω1ι,ϰ,ϑ,ω2ι,ϰ,ϑ∈X.
Sinceδ/abc∈0,1 so by Nadler’s Theorem [24], there exist a fixed point ν∈Tν. Thus (56)νι,ϰ,ϑ=ζ1ι,ϰ+ζ2ι,ϑ+ζ3ϰ,ϑ-ζ2ι,0-ζ30,ϑ-ζ3ϰ,0+ζ30,0-∫0ck1ι,ϰνι,ϰ,sds-∫0ϑ∫0bk2ι,ϑνι,s,ϑdsdϑ+∫0ϑ∫0ϰ∫0ak3ϰ,ϑνs,ϰ,ϑdsdϰdϑ+∫0ι∫0ϰ∫0ϑfr,s,ι,νr,s,ιdrdsdτ.In the next example we use Theorems17 and 21, to find the solution of given fuzzy differential inclusion.Example 22.
Consider the hyperbolic PDI(57)∂3ω∂ι∂ϰ∂ϑ∈ϝι,ϰ,ϑ,ωαω,with boundary conditions (58)ωι,ϰ,0=sin⁡ι+ϰ,ωι,0,ϑ=sin⁡ι+ϑω0,ϰ,ϑ=sin⁡ϰ+ϑwhere ϝ:0,12×R×R→E1 defined by (59)ϝι,ϰ,ϑ,ω=cos⁡π-ι-ϰ-ϑe-2ι;2e-2ιSwhere e-2ϰ+ϖ;2e-2ϰ+ϖS is a symmetric triangular fuzzy number with compact support e-2ϰ+ϖ,2e-2ϰ+ϖ. For any α∈0,1, the α-level set is (60)ϝι,ϰ,ϑ,ωα=cos⁡π-ι-ϰ-ϑ1+0.5αe-2ι,2-0.5αe-2ι.Clearly ϝ is convex and bounded. Using Theorem 17, there exists a continuous selection f(ι,ϰ,ϑ,ω)∈ϝ(ι,ϰ,ϑ,ω(ι,ϰ,ϑ))γωι,ϰ,ϑ for each (ι,ϰ,ϑ,ω)∈Ω, such that (61)-cos⁡ι+ϰ+ϑ=fι,ϰ,ϑ,ωwhich implies (62)-cos⁡ι+ϰ+ϑ=∂3ω∂ι∂ϰ∂ϑ.Hence the solution of ∂3ω/∂ι∂ϰ∂ϑ+cos⁡ι+ϰ+ϑ=0 is (63)ωι,ϰ,ϑ=sin⁡ι+ϰ+ϑ.Now for α∈0,1 consider (64)hDϝι,ϰ,ϑ,ωα,ϝι,ϰ,ϑ,να=cos⁡π-ι-ϰ-ϑe-2ιhD1+0.5α,2-0.5α,1+0.5α,2-0.5α=0≤e-2ιω-νUsing Theorem 21, there exists a continuous selection f(ι,ϰ,ϑ,ω)∈ϝι,ϰ,ϑ,να for each (ι,ϰ,ϑ,ω)∈Ω, such that (65)-cos⁡ι+ϰ+ϑ=fι,ϰ,ϑ,ωwhich implies (66)-cos⁡ι+ϰ+ϑ=∂3ω∂ι∂ϰ∂ϑ.Hence the solution of ∂3ω/∂ι∂ϰ∂ϑ+cos⁡ι+ϰ+ϑ=0 is (67)ωι,ϰ,ϑ=sin⁡ι+ϰ+ϑ.



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*Source: 2161230-2019-07-16.xml*