Fractional functional differential inclusions with finite delay

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Abstract

In this paper, we present fractional versions of the Filippov theorem and the Filippov–Wazewski theorem, as well as an existence result, compactness of the solution set and Hausdorff continuity of operator solutions for functional differential inclusions with fractional order, Dαy(t)F(t,yt),a.e.t[0,b],0<α<1,y(t)=ϕ(t),t[r,0], where J=[0,b],Dα is the standard Riemann–Liouville fractional derivative, and F is a set-valued map.

Introduction

This paper deals with the existence of solutions, for initial value problems (IVP for short), for fractional differential inclusions with infinite delay Dαy(t)F(t,yt),a.e.tJ[0,b],0<α<1,y(t)=ϕ(t),t[r,0], where Dα is the standard Riemann-Liouville fractional derivative,  F:J×C([r,0],R)P(R) is a multivalued map with compact values, P(R) is the family of all nonempty subsets of R, ϕC([r,0],R), and ϕ(0)=0.

For any function y defined on [r,b] and any tJ, we denote by yt the element of C([r,0],R) defined by yt(θ)=y(t+θ),θ[r,0]. Here yt() represents the history of the state from time tr up to the present time t.

Differential equations with fractional order have been recently proved to be valuable tools in the modeling of many physical phenomena [10], [20], [21], [36], [37], [3], [9], [18], [31], [32], [41], [43].

The arbitrary (fractional) order integral (Riemann–Liouville) operator is a singular integral operator, and for details on the arbitrary fractional order differential operator, see the references [14], [15], [17]. There has been a significant development in fractional differential equations in recent years; see the monographs of Miller and Ross [40], Samko et al. [45] and the papers of Diethelm et al. [10], [12], [13], El-Sayed [16], Mainardi [36], Momani and Hadid [38], Momai et al. [39], Podlubny et al. [44], and Yu and Gao [48], [49].

Very recently, some basic theory for initial value problems of fractional differential equations involving the Riemann–Liouville differential operator was discussed by Benchohra et al. [5] and Lakshmikantham and Vastala [33], [34], [35].

El-Sayed and Ibrahim [17] and Benchohra et al. [6] initiated the study of fractional multivalued differential inclusions.

In the case where α(1,2], existence results for fractional boundary value problems and a relaxation theorem were studied by Ouahab [42].

Our goal in this paper is to complement and extend some of the above results by giving some existence results and a relaxation theorem for Problem (1) and (2). Also,we prove that the set of solutions is compact.

Section snippets

Preliminaries

Throughout this paper we will use the following notations: P(X)={YX:Y}, Pcl(X)={YP(X):Y closed}, Pb(X)={YP(X):Y bounded}, Pb(X)={YP(X):Ycompact}.

Let (X,d) be a metric space induced from the normed space (X,||). Consider Hd:P(X)×P(X)R+{} given by Hd(A,B)=max{supaAd(a,B),supbBd(A,b)}, where d(A,b)=infaAd(a,b),d(a,B)=infbBd(a,b). Then (Pb,cl(X),Hd) is a metric space and (Pcl(X),Hd) is a generalized metric space; see [30].

Definition 2.1

A multivalued operator N:XPcl(X) is called

  • (a)

    γ-Lipschitz if

Filippov’s theorem

Now, we present a Filippov-type result for the problem (1) and (2). Let ψC([r,0],R),gL1(J,R+), and xC([r,b],R) be a solution of the problem Dαx(t)=g(t),a.e.tJ[0,b],x(t)=ψ(t)t[r,0],ψ(0)=0, where |Iαg|<.

Theorem 3.1

Assume the following conditions:

  • (H1)

    F:J×C([r,0],R)Pcp(R) is a function satisfying:

    • (a)

      For all uC([r,0],R),tF(t,u) is measurable;

    • (b)

      The map tγ(t)=d(Dαx,F(t,xt))g(t) is integrable where g(t) is defined in problem(3)(4).

  • (H2)

    There exists a function pL1(J,R+) such thatHd(F(t,u),F(t,z))p(t)uz

Relaxation theorem

In this section, we examine to what extent the convexification of the right-hand side of the inclusion introduces new solutions. More precisely, we want to find out if the solutions of the nonconvex problem are dense in those of the convex one. Such a result is known in the literature as a Relaxation theorem and has important implications in optimal control theory. It is well-known that in order to have optimal state-control pairs, the system has to satisfy certain convexity requirements. If

Existence results

In this section, we present the existence result for the problem (1), (2). Our considerations are based on the following fixed point theorem for contractive multivalued operators given by Covitz and Nadler [8] (see also Deimling, [11] Theorem 11.1).

Lemma 5.1

Let (X,d) be a complete metric space. If G:XPcl(X) is a contraction, then FixN .

For our results, we introduce some hypotheses:
  • (A1)

    F:[0,b]×C([r,0],R)Pcp(R); tF(t,x) is measurable, for each xC([r,0],R).

  • (A2)

    There exists a function pL1([0,b],R+)

Some properties of solution sets

We shall examine the properties of the map S[0,b]:C0P(C([r,0],R)) defined by, S[0,b](ϕ)={yΩ:yis a solution of the problem(1)- (2)}, where C0={ϕC([r,0])|ϕ(0)=0}.

Lemma 6.1

Assume that all the conditions ofTheorem 5.2are satisfied. Then the set-valued map S[0,b]:C0P(C([r,b],R)) is Lipschitz, with Lipschitz constant equal to1+|Iαp|2Γ(α)(Γ(α)|Iαp|).

Proof

It is clear that C0 is a closed set in C([r,0],R). We show that Hd(S[0,b](ϕ),S[0,b](ψ))Lϕψ,for allϕ,ψC0. Let ϕ,ψC0 and yS[0,b](ψ). Since F is a

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