# On Coupled Systems of Time-Fractional Differential Problems by Using a New Fractional Derivative

Authors: Ahmed Alsaedi; Dumitru Baleanu; Sina Etemad; Shahram Rezapour
Journal: Journal of Function Spaces
 (2016)

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## Abstract

The existence of solutions for a coupled system of time-fractional differential equations including continuous functions and the Caputo-Fabrizio fractional derivative is examined. After that we investigated a coupled system of time-fractional differential inclusions including compact- and convex-valuedL1-Caratheodory multifunctions and the Caputo-Fabrizio fractional derivative.


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## Body

## 1. Introduction

The fractional calculus is nowadays an excellent mathematical tool which opens the gates for finding hidden aspects of the dynamics of the complex processes which appear naturally in many branches of science and engineering [1–6]. The methods and techniques of this type of calculus are continuously generalized and improved especially during the last few decades. We recall that the existence and multiplicity of positive solutions corresponding to singular fractional boundary value problems were discussed in [7]. Also, the existence results for several nonlinear fractional differential equations were reported in [8]. Besides, the existence of positive solutions corresponding to a coupled system of multiterm singular fractional integrodifferential boundary value problems was shown in [9]. Inventing new derivatives and applying them to study the dynamics of complex systems are an important priority for researchers. As a result, very recently, a new fractional derivative without singular kernel has been provided [10, 11]. By using the main results presented in these two new works, we present the next definition.Definition 1 (see [10]).
Theα order Caputo-Fabrizio time-fractional differential derivative of the function u is written as(1)DtαCFux,t=2-αMα21-α∫0texp-αt-s1-α∂u∂tds,t≥0,where M(α) represents a normalization function, 0<α<1, and u∈H1[(0,1)×(0,1)].Note thatDtαCFu(x,t)=0 whenever u is a constant function and the kernel has no singularity at t=s [10, 11]. Also, Losada and Nieto defined the new time-fractional integral based on the new definition of Caputo-Fabrizio fractional derivative [11]. By using this idea, we provide the notion of Caputo-Fabrizio time-fractional integral.Definition 2.
Theα order time-fractional integral of a function u has the form [11](2)ItαCFux,t=21-α2-αMαux,t+2α2-αMα∫0tux,sds,t≥0,where M(α) represents a normalization function and 0<α<1.Losada and Nieto showed thatM(α)=2/2-α for all 0≤α≤1 [11]. By substituting M(α) in (1), we obtain the definition of the time-fractional Caputo-Fabrizio derivative of order α for a function u as follows:(3)DtαCFux,t=11-α∫0texp-αt-s1-α∂u∂sds;x,t∈0,1×0,1.They proved that solution of DtαCFv(x,t)=g(x,t) is given by(4)vx,t=21-α2-αMαgx,t+2α2-αMα∫0tgx,sds+v0,0,where 0<α<1 and (x,t)∈[0,1]×[0,1] [11].The next step is to consider(X,d) being a metric space. Let us denote by P(X) and 2X the class of all subsets and the class of all nonempty subsets of X, respectively. Hence Pcl(X), Pbd(X), Pcv(X), Pcp(X), and Pcp,cv(X) are the class of all closed subsets, the class of all bounded subsets, the class of all convex subsets, the class of all compact subsets, and the class of all compact and convex subsets of X, respectively. We claim that u∈X is a fixed point of the multifunction F:X→2X whenever u∈Fu [12]. A multifunction F:[0,1]×[0,1]→Pcl(R) is called measurable whenever the function (x,t)↦d(w,F(x,t))=inf{w-v:v∈F(x,t)} is measurable for all w∈R [12]. The Pompeiu-Hausdorff metric Hd:2X×2X→[0,∞) is defined by (5)HdA,B=max⁡supa∈A⁡da,B,supb∈B⁡dA,b,such that d(A,b)=infa∈Ad(a,b) [13]. (CB(X),Hd) is a metric space and (CB(X),Hd) depicts a generalized metric space. Here CB(X) denotes the set of closed and bounded subsets of X and C(X) represents the set of closed subsets of X [12, 13]. We recall that F is said to be convex-valued (compact-valued) whenever Fu is convex (compact) set for each u∈X [12]. We mention that a multifunction F:X→C(X) is a contraction whenever there exists a constant γ∈(0,1) such that Hd(Fu,Fv)≤γd(u,v) for all u,v∈X [12]. In 1970, Covitz and Nadler proved that each closed-valued contractive multifunction on a complete metric space has a fixed point [14].Below we examine the existence of solutions for two coupled systems of nonlinear time-fractional differential equations and inclusions within Caputo-Fabrizio time-fractional derivative. First, we discuss the coupled system, namely,(6)DtαCFux,t=f1x,t,ux,t,vx,t,DtβCFvx,t=f2x,t,ux,t,vx,tsuch that(7)u0,0=0,v0,0=0,where 0<α<1, 0<β<1, (x,t)∈[0,1]×[0,1], and the mappings f1,f2:[0,1]×[0,1]×R×R→R are continuous functions. In addition, we discuss the existence of solutions for the coupled system of nonlinear time-fractional differential inclusions(8)DtαCFux,t∈F1x,t,ux,t,vx,t,DtβCFvx,t∈F2x,t,ux,t,vx,tsuch that(9)u0,0=0,v0,0=0,where F1,F2:[0,1]×[0,1]×R×R→P(R) are some multivalued maps.We say thatF:[0,1]×[0,1]×R×R→2R is a Caratheodory multifunction whenever (x,t)↦F(x,t,u1,u2) is measurable for all ui∈R and (u1,u2)↦F(x,t,u1,u2) is upper semicontinuous (u.s.c) for almost all (x,t)∈[0,1]×[0,1] and u1,u2∈X [12]. A Caratheodory multifunction F:[0,1]×[0,1]×R×R→2R is said to be an L1-Caratheodory whenever for each ρ>0 there exists ϕρ∈L1([0,1]×[0,1],R+) such that (10)Fx,t,u1,u2=supx,t∈0,1×0,1⁡s:s∈Fx,t,u1,u2≤ϕρx,tfor all |ui|≤ρ and for almost all (x,t)∈[0,1]×[0,1] [12]. The set of selections of Fi at ui is defined by (11)SFiui=wi∈L10,1×0,1,R:wix,t∈Fx,t,uix,t,ui′x,tfor  almost  allx,t∈0,1×0,1,for all ui,ui′∈CR([0,1]×[0,1]) for i=1,2. The sets SFi(ui) are nonempty for all ui∈CK([0,1]×[0,1]) whenever dimK<∞ [12, 15]. The graph of the multifunction F:X→Y is defined by the set Gr(F)={(x,y)∈X×Y:y∈F(x)} (see [12, 13]). We say that the graph Gr(F) of F:X→Pcl(Y) is a closed subset of X×Y whenever for all sequences {un}n∈N in X and {yn}n∈N in Y with un→u0, yn→y0, and yn∈F(un) for all n we have y0∈F(u0) [12]. Below we introduce the following results which will be required in our proofs.Theorem 3 (see [12]).
Suppose thatX is a Banach space, T:X→X is a completely continuous operator, and the set K={u∈X:u=λTu,forsomeλ∈[0,1]} is bounded. Then, T has a fixed point.Lemma 4 (see [12, Proposition 1.2]).
IfF:X→Pcl(Y) is upper semicontinuous, then Gr(F) is a closed subset of X×Y. If F is completely continuous with a closed graph, then it is upper semicontinuous.Lemma 5 (see [12]).
LetX be a separable Banach space and F:[0,1]×[0,1]×X×X→Pcp,cv(X) an L1-Caratheodory function. Then the operator Θ·SF:CX([0,1]×[0,1])→Pcp,cv(CX([0,1]×[0,1])) defined by u↦(Θ·SF)(u)=Θ(SF,u) is a closed graph operator, where Θ is a linear continuous mapping from L1([0,1]×[0,1],X) into CX([0,1]×[0,1]).Theorem 6 (see [12]).
LetE be a Banach space, C a closed convex subset of E, U an open subset of C, and 0∈U. Let us suppose that F:U¯→Pcp,cv(C) depicts an upper semicontinuous compact map, such that Pcp,cv(C) denotes the family of nonempty, compact convex subsets of C. Then either F admits a fixed point in U¯ or there exist u∈∂U and λ∈(0,1) such that u∈λF(u).


## 2. Main Results

First, we investigate the coupled system(12)DtαCFux,t=f1x,t,ux,t,vx,t,DtβCFvx,t=f2x,t,ux,t,vx,tequipped with the boundary value conditions u(0,0)=0 and v(0,0)=0, where f1,f2:[0,1]×[0,1]×X2→X are continuous mappings, α,β∈(0,1), x,t∈[0,1], and DtαCF and DtβCF are the Caputo-Fabrizio time-fractional derivatives. Now, consider the Banach space X={u:u∈CR([0,1]×[0,1])} endowed with the sup-norm uX=sup(x,t)∈[0,1]×[0,1]|u(x,t)|. Thus, the product space (X×X,·X×X) is also a Banach space via the product norm u,vX×X=uX+vX. First, we prove the next key lemma.Lemma 7.
Suppose thatf∈LX1([0,1]×[0,1]) and 0<α<1. The function u0∈CX([0,1]×[0,1]) is a solution for the time-fractional integral equation(13)ux,t=21-α2-αMαfx,t-f0,0+2α2-αMα∫0tfx,sdsif and only if u0 is a unique solution of the time-fractional differential equation(14)DtαCFux,t=fx,t,x,t∈0,1×0,1,u0,0=0.Proof.
A solution of initial value problem (14) is denoted by u0. As a result (DtαCFu0)(x,t)=f(x,t) and u0(0,0)=0. By integrating both sides we get (15)u0x,t-u00,0=21-α2-αMαfx,t-f0,0+2α2-αMα∫0tfx,sdsand so u0(x,t)=(2(1-α)/2-αMα)(f(x,t)-f0,0)+2α/2-αMα∫0t‍f(x,s)ds. This shows that u0 represents the solution of time-fractional integral equation (13). If u1 and u2 are two distinct solutions for initial value problem (14), then DtαCFu1x,t-DtαCFu2x,t=[DtαCFu1-u2](x,t)=0 and (u1-u2)(0,0)=0. By the property of the Caputo-Fabrizio time-fractional derivative in [11], we get u1=u2. Hence, u0 is a unique solution of initial value problem (14). Now, suppose that u0 is a solution of time-fractional integral equation (13). Then, we conclude that u0(x,t)=(21-α/(2-α)M(α))(f(x,t)-f(0,0))+(2α/2-αMα)∫0t‍f(x,s)ds. By using (4), one can see that this function represents a solution for initial value problem (14). Note that u0(0,0)=0.Now, we consider (1)-(2). For each (x,t)∈[0,1]×[0,1], define the operators T1,T2:X→X by(16)T1vx,t=21-α2-αMαf1x,t,ux,t,vx,t-21-α2-αMαf10,0,u0,0,v0,0+2α2-αMα∫0tf1x,s,ux,s,vx,sds,T2ux,t=21-β2-βMβf2x,t,ux,t,vx,t-21-β2-βMβf20,0,u0,0,v0,0+2β2-βMβ∫0tf2x,s,ux,s,vx,sdsand put(17)N1=4-2α2-αMα,N2=4-2β2-βMβ.Theorem 8.
Suppose thatf1,f2:[0,1]×[0,1]×X×X→X are the continuous mappings in system (6)-(7) and there exist positive constants L1 and L2 fulfilling f1x,t,u1,u2≤L1 and |f2(x,t,u1,u2)|≤L2 for all (x,t)∈[0,1]×[0,1] and u1,u2∈X. Then, system (6)-(7) possesses at least one solution.Proof.
Let the operatorsT1,T2:X→X defined by (16). We define the operator T:X×X→X×X by T(u,v)(x,t)≔((T1v)(x,t),(T2u)(x,t)) for all (x,t)∈[0,1]×[0,1]. Note that T is continuous because the mappings f1 and f2 are continuous. We prove that the operator T maps bounded sets into the bounded subsets of X×X. Let Ω be a bounded subset of X×X, (u,v)∈Ω, and (x,t)∈[0,1]×[0,1]. Then, we have(18)T1vx,t=21-α2-αMαf1x,t,ux,t,vx,t-21-α2-αMαf10,0,u0,0,v0,0+2α2-αMα∫0tf1x,s,ux,s,vx,sds≤21-α2-αMαf1x,t,ux,t,vx,t+21-α2-αMαf10,0,0,0+2α2-αMα∫0tf1x,s,ux,s,vx,sds≤L121-α2-αMα+21-α2-αMα+2α2-αMαt≤L141-α2-αMα+2α2-αMα≤L14-2α2-αMα=L1N1and so T1vx,tX≤L1N1. Also, we have(19)T2ux,t=21-β2-βMβf2x,t,ux,t,vx,t-21-β2-βMβf20,0,u0,0,v0,0+2β2-βMβ∫0tf2x,s,ux,s,vx,sds≤21-β2-βMβf2x,t,ux,t,vx,t+21-β2-βMβf20,0,0,0+2β2-βMβ∫0tf2x,s,ux,s,vx,sds≤L221-β2-βMβ+21-β2-βMβ+2β2-βMβt≤L241-β2-βMβ+2β2-βMβ≤L24-2β2-βMβ=L2N2and so T2ux,tX≤L2N2. Thus, Tu,vx,tX×X≤L1N1+L2N2. This shows that the operator T maps bounded sets into the bounded sets of X×X. Now, we show that the operator T is equicontinuous. Let (x,t1),(x,t2)∈[0,1]×[0,1] with t1<t2. Then, we have(20)T1vx,t2-T1vx,t1=21-α2-αMαf1x,t2,ux,t2,vx,t2-21-α2-αMαf10,0,u0,0,v0,0+2α2-αMα∫0t2f1x,s,ux,s,vx,sds-21-α2-αMαf1x,t1,ux,t1,vx,t1+21-α2-αMαf10,0,u0,0,v0,0-2α2-αMα∫0t1f1x,s,ux,s,vx,sds≤21-α2-αMαf1x,t2,ux,t2,vx,t2-f1x,t1,ux,t1,vx,t1+2α2-αMα∫t1t2f1x,s,ux,s,vx,sds≤21-α2-αMαf1x,t2,ux,t2,vx,t2-f1x,t1,ux,t1,vx,t1+2αL12-αMαt2-t1.This implies that T1vx,t2-T1vx,t1→0 whenever (x,t2)→(x,t1). By utilizing the Arzela-Ascoli theorem, T1 is completely continuous. Similarly, we have (21)T2ux,t2-T2ux,t1=21-β2-βMβf2x,t2,ux,t2,vx,t2-21-β2-βMβf20,0,u0,0,v0,0+2β2-βMβ∫0t2f2x,s,ux,s,vx,sds-21-β2-βMβf2x,t1,ux,t1,vx,t1+21-β2-βMβf20,0,u0,0,v0,0-2β2-βMβ∫0t1f2x,s,ux,s,vx,sds≤21-β2-βMβf2x,t2,ux,t2,vx,t2-f2x,t1,ux,t1,vx,t1+2β2-βMβ∫t1t2f2x,s,ux,s,vx,sds≤21-β2-βMβf2x,t2,ux,t2,vx,t2-f2x,t1,ux,t1,vx,t1+2βL22-βMβt2-t1.
Again, by utilizing the Arzela-Ascoli theorem we observe thatT2 is completely continuous. Therefore, we get Tu,vx,t2-Tu,vx,t2X×X→0 whenever (x,t2) tends to (x,t1). Thus, T is completely continuous. In the next step we prove that (22)Ω=u,v∈X×X:u,v=λTu,vfor  someλ∈0,1is bounded. Let (u,v) be an arbitrary element of Ω. Choose λ∈[0,1] fulfilling (u,v)=λT(u,v). Hence, v(x,t)=λ(T1v)(x,t) and u(x,t)=λ(T2u)(x,t) for all (x,t)∈[0,1]×[0,1]. Since (23)1λvx,t=T1vx,t≤L1N1,we get |v(x,t)|≤λL1N1 and so vx,tX≤λL1N1. Similarly, we prove that ux,tX≤λL2N2. Thus, u,vX×X≤λL1N1+L2N2 and so Ω is a bounded set. Now, by using Theorem 3, we get that T has a fixed point which is a solution for the coupled system of the time-fractional differential equations.Next we study the existence of solution for the coupled system of time-fractional differential inclusions(24)DtαCFux,t∈F1x,t,ux,t,vx,t,DtβCFvx,t∈F2x,t,ux,t,vx,twith the initial value conditions u(0,0)=0 and v(0,0)=0, where F1,F2:[0,1]×[0,1]×R×R→P(R) are some multivalued maps.Definition 9.
One says that(u1,u2)∈C([0,1]×[0,1],X)×C([0,1]×[0,1],X) is a solution for the system of the time-fractional differential inclusions whenever it satisfies the initial value conditions and there exists (w1,w2)∈L1([0,1]×[0,1])×L1([0,1]×[0,1]) such that wi(x,t)∈Fi(x,t,u(x,t),v(x,t)) for almost all (x,t)∈[0,1]×[0,1] and i=1,2 and also (25)uix,t=21-α2-αMαwix,t,ux,t,vx,t-21-α2-αMαwi0,0,u0,0,v0,0+2α2-αMα∫0twix,s,ux,s,vx,sds,for all (x,t)∈[0,1]×[0,1] and i=1,2.Theorem 10.
LetF1,F2:[0,1]×[0,1]×R×R→Pcp,cv(R) be L1-Caratheodory multifunctions. Suppose that there exist a nondecreasing bounded continuous map ψ:[0,∞)→(0,∞) and a continuous function p:[0,1]×[0,1]→(0,∞) such that Fi(x,t,ui(x,t),ui′(x,t))≤p(x,t)ψ(ui) for all (x,t)∈[0,1]×[0,1], ui,ui′∈X for i=1,2. Then, coupled system of time-fractional differential inclusions (8)-(9) has at least one solution.Proof.
Define the operatorN:X×X→2X×X by N(u1,u2)=N1(u1,u2)N2(u1,u2), where(26)N1u1,u2=h1∈X×X:there  existsv1∈SF1,u1such  thath1x,t=v1x,t∀x,t∈0,1×0,1,N2u1,u2=h2∈X×X:there  existsv2∈SF2,u2such  thath2x,t=v2x,t∀x,t∈0,1×0,1,h1x,t=21-α2-αMαv1x,t-21-α2-αMαv10,0+2α2-αMα∫0tv1x,sds,h2x,t=21-β2-βMβv2x,t-21-β2-βMβv20,0+2β2-βMβ∫0tv2x,sds.By Lemma 7, it is clear that each fixed point of the operator N is a solution for system of time-fractional differential inclusions (8). First, we prove that the multifunction N is convex-valued. Let (u1,u2)∈X×X, (h1,h2),(h1′,h2′)∈N(u1,u2). Choose vi,vi′∈SFi(u1,u2) such that (27)hix,t=21-α2-αMαvix,t-21-α2-αMαvi0,0+2α2-αMα∫0tvix,sds,hi′x,t=21-α2-αMαvi′x,t-21-α2-αMαvi′0,0+2α2-αMα∫0tvi′x,sdsfor almost all (x,t)∈[0,1]×[0,1] and i=1,2. Let 0≤λ≤1 be given. Then, we have (28)λhi+1-λhi′x,t=21-α2-αMαλvix,t+1-λvi′x,t-21-α2-αMαλvi0,0+1-λvi′0,0+2α2-αMα∫0tλvix,s+1-λvi′x,sdsfor i=1,2. Since the operator Fi has convex values, SFi(ui) is a convex set and [λhi+1-λhi′]∈Ni(u1,u2) for i=1,2. This implies that the operator N has convex values. Now, we prove that N maps bounded sets of X into bounded sets. Let r>0, Br={(u1,u2)∈X×X:(u1,u2)≤r} be a bounded subset of X×X, (h1,h2)∈N(u1,u2), and (u1,u2)∈Br. Then, there exists (v1,v2)∈SF1(u1)×SF2(u2) such that (29)h1x,t=21-α2-αMαv1x,t-21-α2-αMαv10,0+2α2-αMα∫0tv1x,sdsand h2(x,t)=(21-β/(2-β)Mβ)v2(x,t)-(2(1-β)/(2-β)M(β))v2(0,0) + 2β/(2-β)M(β)∫0t‍v2(x,s)ds for almost all (x,t)∈[0,1]×[0,1]. If p∞=sup(x,t)∈[0,1]×[0,1]px,t, then we obtain (30)h1x,t=21-α2-αMαv1x,t-21-α2-αMαv10,0+2α2-αMα∫0tv1x,sds≤21-α2-αMαv1x,t+21-α2-αMαv10,0+2α2-αMα∫0tv1x,sds≤px,tψu121-α2-αMα+21-α2-αMα+2α2-αMαt≤p∞ψu141-α2-αMα+2α2-αMα≤p∞ψu14-2α2-αMα=p∞ψu1N1,where the constant N1 is defined by (17). This implies that h1≤p∞ψ(u1)N1. Similarly, we get h2≤p∞ψ(u2)N2, where the constant N2 is defined by (17). Thus, (h1,h2)≤p∞ψ((u1,u2))(N1+N2). Now, we prove that N maps bounded sets into equicontinuous subsets of X×X. Let (u1,u2)∈Br and (x,t1),(x,t2)∈[0,1]×[0,1] with t1<t2. Then, we have(31)h1x,t2-h1x,t1=21-α2-αMαv1x,t2-21-α2-αMαv10,0+2α2-αMα∫0t2v1x,sds-21-α2-αMαv1x,t1+21-α2-αMαv10,0-2α2-αMα∫0t1v1x,sds≤21-α2-αMαv1x,t2-v1x,t1+2α2-αMα∫t1t2v1x,sds≤21-α2-αMαv1x,t2-v1x,t1+2αp∞ψu12-αMαt2-t1.By using a similar method, we obtain (32)h2x,t2-h2x,t1≤21-β2-βMβv2x,t2-v2x,t1+2βp∞ψu22-βMβt2-t1.Hence, |hi(x,t2)-hi(x,t1)|→0 as (x,t2)→(x,t1). By using the Arzela-Ascoli theorem we get that N is completely continuous. Here, we prove that N is upper semicontinuous. By using Lemma 4, N is upper semicontinuous whenever it has a closed graph. Since N is completely continuous, we must show that N has a closed graph.
Let{(u1n,u2n)} be a sequence in X×X with (u1n,u2n)→(u10,u20) and (h1n,h2n)∈N(u1n,u2n) with (h1n,h2n)→(h10,h20). We show that (h10,h20)∈N(u10,u20). For each (h1n,h2n)∈N(u1n,u2n), we can choose (v1n,v2n)∈SF1(u1n)×SF2(u2n) such that (33)h1nx,t=21-α2-αMαv1nx,t-21-α2-αMαv1n0,0+2α2-αMα∫0tv1nx,sdsand h2n(x,t)=(2(1-β)/(2-β)M(β))v2n(x,t)-(2(1-β)/(2-β)M(β))v2n(0,0)+(2β/(2-β)M(β))∫0t‍v2n(x,s)ds for all (x,t)∈[0,1]×[0,1]. It is sufficient to show that there exists (v10,v20)∈SF1(u10)×SF2(u20) such that (34)h10x,t=21-α2-αMαv10x,t-21-α2-αMαv100,0+2α2-αMα∫0tv10x,sds,and h20(x,t)=(2(1-β)/(2-β)M(β))v20(x,t)-(2(1-β)/(2-β)M(β))v20(0,0)+2β/2-βMβ∫0t‍v20(x,s)ds for all (x,t)∈[0,1]×[0,1]. Now, consider the linear operators Θ1,Θ2:L1([0,1]×[0,1],X)→C([0,1]×[0,1],X) defined by (35)Θ1vx,t=21-α2-αMαvx,t-21-α2-αMαv0,0+2α2-αMα∫0tvx,sdsand Θ2(v)(x,t)=(2(1-β)/(2-β)M(β))v(x,t)-(2(1-β)/(2-β)M(β))v(0,0)+(2β/(2-β)M(β))∫0t‍v(x,s)ds. Note that(36)h1nx,t-h10x,t=21-α2-αMαv1nx,t-v10x,t-21-α2-αMαv1n0,0-v100,0+2α2-αMα∫0tv1nx,s-v10x,sds⟶0,h2nx,t-h20x,t=21-β2-βMβv2nx,t-v20x,t-21-β2-βMβv2n0,0-v200,0+2β2-βMβ∫0tv2nx,s-v20x,sds⟶0.By using Lemma 5, we get that Θi·SFi is a closed graph operator for i=1,2. Also, we get hin(x,t)∈Θi(SFi(uin)) for all n. Since uin→ui0, we get (37)h10x,t=21-α2-αMαv10x,t-21-α2-αMαv100,0+2α2-αMα∫0tv10x,sdsand h20x,t=21-β/2-βMβv20x,t-21-β/2-βMβv200,0+(2β/(2-β)M(β))∫0t‍v20(x,s)ds for some vi0∈SFi(ui0)(i=1,2). Thus, N has a closed graph.
Now, we prove that there is an open setU⊆X with (u1,u2)∉N(u1,u2) for all λ∈(0,1) and (u1,u2)∈∂U. Let λ∈(0,1) and (u1,u2)∈λN(u1,u2). Then, there exists vi∈L1([0,1]×[0,1],R) with vi∈SFi(ui)(i=1,2) such that (38)u1x,t=21-α2-αMαv1x,t-21-α2-αMαv10,0+2α2-αMα∫0tv1x,sdsand u2(x,t)=(2(1-β)/(2-β)M(β))v2(x,t)-(2(1-β)/(2-β)M(β))v2(0,0)+2β/(2-β)M(β)∫0t‍v2(x,s)ds for all (x,t)∈[0,1]×[0,1]. By using the above computed values, we obtain ui≤p∞ψ(ui)∑i=1n‍Ni for i=1,2. This follows that ui/p∞ψui∑i=1n‍Ni≤1 for i=1,2. Choose Mi>0 with ui≠Mi in such a way that Mi/p∞ψ(ui)∑i=1n‍Ni>1 for i=1,2. Put U={(u1,u2)∈X×X:(u1,u2)<min{M1,M2}}. We note that the operator N:U¯→P(X) is upper semicontinuous and completely continuous. Also, we showed that there is no (u1,u2)∈∂U such that (u1,u2)∈λN(u1,u2) for some λ∈(0,1). Hence, with the help of Theorem 6, we get that N has a fixed point (u1,u2)∈U¯ which is a solution for time-fractional differential inclusion (8)-(9).



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*Source: 4626940-2016-01-04.xml*