differential inclusion

The existence of solutions for a nonlinear fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.


Introduction
Differential equations with fractional order have recently proved to be strong tools in the modelling of many physical phenomena; for a good bibliography on this topic we refer to [17].As a consequence there was an intensive development of the theory of differential equations of fractional order [2,15,20].The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim [11].Very recently several qualitative results for fractional differential inclusions were obtained in [3,6,7,8,9,13,18].
The present paper is motivated by a recent paper of Kaufmann and Yao [14], where it is considered problem (1.1)-(1.2) with F single valued and several existence results are provided.
The aim of our paper is to extend the study in [14] to the set-valued framework and to present some existence results for problem (1.1)-(1.2).Our results are essentially based on a nonlinear alternative of Leray-Schauder type, on Bressan-Colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and on Covitz and Nadler set-valued contraction principle.The methods used are standard, however their exposition in the framework of problem (1.1)-(1.2) is new.We note that our results extends the results in the literature obtained in the case a = 0 [18].
The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel and in Section 3 we prove our main results.

Preliminaries
In this section we sum up some basic facts that we are going to use later.
Let (X, d) be a metric space with the corresponding norm |.| and let I ⊂ R be a compact interval.Denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I, by P(X) the family of all nonempty subsets of X and by B(X) the family of all Borel subsets of X.If A ⊂ I then χ A : I → {0, 1} denotes the characteristic function of A. For any subset A ⊂ X we denote by A the closure of A.
Recall that the Pompeiu-Hausdorff distance of the closed subsets A, B ⊂ X is defined by As usual, we denote by C(I, X) the Banach space of all continuous functions x : I → X endowed with the norm |x| C = sup t∈I |x(t)| and by L 1 (I, X) the Banach space of all (Bochner) integrable functions x : I → X endowed with the norm |x| Consider T : X → P(X) a set-valued map.A point x ∈ X is called a fixed point for T if x ∈ T (x).T is said to be bounded on bounded sets if T (B) := ∪ x∈B T (x) is a bounded subset of X for all bounded sets B in X. T is said to be compact if T (B) is relatively compact for any bounded sets B in X. T is said to be totally compact if T (X) is a compact subset of X. T is said to be upper semicontinuous if for any open set D ⊂ X, the set {x ∈ X : T (x) ⊂ D} is open in X. T is called completely continuous if it is upper semicontinuous and totally bounded on X.
It is well known that a compact set-valued map T with nonempty compact values is upper semicontinuous if and only if T has a closed graph.
We recall the following nonlinear alternative of Leray-Schauder type and its consequences.
Theorem 2.1.[19] Let D and D be open and closed subsets in a normed linear space X such that 0 ∈ D and let T : D → P(X) be a completely continuous set-valued map with compact convex values.Then either i) the inclusion x ∈ T (x) has a solution, or ii) there exists x ∈ ∂D (the boundary of D) such that λx ∈ T (x) for some λ > 1.
Corollary 2.2.Let B r (0) and B r (0) be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : B r (0) → P(X) be a completely continuous set-valued map with compact convex values.Then either i) the inclusion x ∈ T (x) has a solution, or ii) there exists x ∈ X with |x| = r and λx ∈ T (x) for some λ > 1.
Corollary 2.3.Let B r (0) and B r (0) be the open and closed balls in a normed linear space X centered at the origin and of radius r and let T : B r (0) → X be a completely continuous single valued map with compact convex values.Then either i) the equation x = T (x) has a solution, or ii) there exists x ∈ X with |x| = r and x = λT (x) for some λ < 1.
We recall that a multifunction T : X → P(X) is said to be lower semicontinuous if for any closed subset C ⊂ X, the subset {s ∈ X : We say that F is of lower semicontinuous type if S F (.) is lower semicontinuous with closed and decomposable values.
Theorem 2.4.[4] Let S be a separable metric space and G : S → P(L 1 (I, R)) be a lower semicontinuous set-valued map with closed decomposable values.
Then G has a continuous selection (i.e., there exists a continuous mapping F is said to be L 1 -Carathéodory if for any l > 0 there exists h l ∈ L 1 (I, R) such that sup{|v| : v ∈ F (t, x)} ≤ h l (t) a.e.I, ∀x ∈ B l (0).Theorem 2.5.[16] Let X be a Banach space, let F : I × X → P(X) be a L 1 -Carathéodory set-valued map with S F = ∅ and let Γ : L 1 (I, X) → C(I, X) be a linear continuous mapping.
Then the set-valued map Γ • S F : C(I, X) → P(C(I, X)) defined by has compact convex values and has a closed graph in C(I, X) × C(I, X).
Consider a set valued map T on X with nonempty values in X. T is said to be a λ-contraction if there exists 0 < λ < 1 such that The set-valued contraction principle [10] states that if X is complete, and T : X → P(X) is a set valued contraction with nonempty closed values, then T has a fixed point, i.e. a point z ∈ X such that z ∈ T (z).Definition 2.6.a) The fractional integral of order α > 0 of a Lebesgue integrable function f : (0, ∞) → R is defined by provided the right-hand side is pointwise defined on (0, ∞) and Γ is the (Euler's) Gamma function.
Lemma 2.8.[14] For any f ∈ C(I, R) the unique solution of the boundary value problem Lx(t) + f (t) = 0 a.e.I, is solution of the integral equation where Note that G 1 (t, s) > 0 ∀t, s ∈ I (e.g., [1]) and

The main results
We are able now to present the existence results for problem (1.1)-(1.2).We consider first the case when F is convex valued.
Consider the set-valued map T : B r (0) → P(C(I, R)) defined by We show that T satisfies the hypotheses of Corollary 2.2.
EJQTDE, 2010 No. 78, p. 6 First, we show that T (x) ⊂ C(I, R) is convex for any x ∈ C(I, R).If v 1 , v 2 ∈ T (x) then there exist f 1 , f 2 ∈ S F (x) such that for any t ∈ I one has Let 0 ≤ α ≤ 1.Then for any t ∈ I we have The values of F are convex, thus S F (x) is a convex set and hence αv Secondly, we show that T is bounded on bounded sets of C(I, R).Let B ⊂ C(I, R) be a bounded set.Then there exist m > 0 such that i.e., T (B) is bounded.We show next that T maps bounded sets into equi-continuous sets.Let B ⊂ C(I, R) be a bounded set as before and v ∈ T (x) for some x ∈ B. There exists f ∈ S F (x) such that v(t) = 1 0 G 1 (t, s)f (s)ds − a 1 0 G 2 (t, s)x(s)ds.Then for any t, τ ∈ I we have We apply now Arzela-Ascoli's theorem we deduce that T is completely continuous on C(I, R).
In the next step of the proof we prove that T has a closed graph.Let We apply Theorem 2.5 to find that Γ • S F has closed graph and from the definition of Γ we get Therefore, T is upper semicontinuous and compact on B r (0).We apply Corollary 2.2 to deduce that either i) the inclusion x ∈ T (x) has a solution in B r (0), or ii) there exists x ∈ X with |x| C = r and λx ∈ T (x) for some λ > 1.
Assume that ii) is true.With the same arguments as in the second step of our proof we get r = |x| C ≤ G 0 |ϕ| 1 ψ(r) + |a|G 0 r which contradicts (3.1).Hence only i) is valid and theorem is proved.
We consider now the case when F is not necessarily convex valued.Our first existence result in this case is based on the Leray-Schauder alternative for single valued maps and on Bressan Colombo selection theorem.Hypothesis 3.3.i) F : I × R → P(R) has compact values, F is L(I) ⊗ B(R) measurable and x → F (t, x) is lower semicontinuous for almost all t ∈ I.
Then problem (1.1)-(1.2) has at least one solution on I.
Proof.We note first that if Hypothesis 3.3 is satisfied then F is of lower semicontinuous type (e.g., [12]).Therefore, we apply Theorem 2.4 to deduce that there exists f : EJQTDE, 2010 No. 78, p. 8 We consider the corresponding problem From Lebesgue's dominated convergence theorem and the continuity of f we obtain, for all t ∈ I i.e., T is continuous on B r (0).Repeating the arguments in the proof of Theorem 3.2 with corresponding modifications it follows that T is compact on B r (0).We apply Corollary 2.3 and we find that either i) the equation x = T (x) has a solution in B r (0), or ii) there exists x ∈ X with |x| C = r and x = λT (x) for some λ < 1.
As in the proof of Theorem 3.2 if the statement ii) holds true, then we obtain a contradiction to (3.1).Thus only the statement i) is true and problem (1.1) has a solution x ∈ C(I, R) with |x| C < r.EJQTDE, 2010 No. 78, p. 9 In order to obtain an existence result for problem (1.1)-(1.2) by using the set-valued contraction principle we introduce the following hypothesis on F .Hypothesis 3.5.i) F : I × R → P(R) has nonempty compact values and, for every x ∈ R, F (., x) is measurable.
ii) There exists L ∈ L 1 (I, R + ) such that for almost all t ∈ I, F (t, •) is L(t)-Lipschitz in the sense that Note that since the set-valued map F (., x(.)) is measurable with the measurable selection theorem (e.g., Theorem III. 6 in [5]) it admits a measurable selection f : I → R.Moreover, from Hypothesis 3.5 It is clear that the fixed points of T are solutions of problem (1.1)-(1.2).We shall prove that T fulfills the assumptions of Covitz Nadler contraction principle.
First, we note that since S F,x = ∅, T (x) = ∅ for any x ∈ C(I, R).
Secondly, we prove that T (x) is closed for any Since F has compact values and Hypothesis 3.5 is satisfied we may pass to a subsequence (if necessary) to get that f n converges to f ∈ L 1 (I, R) in L 1 (I, R).In particular, f ∈ S F,x and for any t ∈ I we have x n (t) → x * (t) =  Finally, we show that T is a contraction on C(I, R).Let x 1 , x 2 ∈ C(I, R) and v 1 ∈ T (x 1 ).Then there exist f 1 ∈ S F,x 1 such that v 1 (t) =

1 =
I |x(t)|dt.EJQTDE, 2010 No. 78, p. 2 A subset D ⊂ L 1 (I, X) is said to be decomposable if for any u, v ∈ D and any subset A ∈ L(I) one has uχ A + vχ B ∈ D, where B = I\A.