Electronic Journal Of Qualitative Theory Of Differential Equations

We study a boundary value problem for a fractional differential inclusion of order � 2 (1,2] with non-separated boundary conditions involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.


Introduction
In this paper we study the following problem The present paper is motivated by a recent paper of Ahmad and Ntouyas ( [1]) where it is studied problem (1.1)-(1.2) and several existence results for this problem are obtained using nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.For motivation, examples and recent developments on differential inclusions of fractional order (in particular, for problem (1.1)-(1.2))we refer the reader to [1] and the references therein.
The aim of our paper is twofold.On one hand, we show that Filippov's ideas ( [5]) can be suitably adapted in order to obtain the existence of solutions for problem (1.1)-(1.2).We recall that for a differential inclusion defined by a lipschitzian set-valued map with nonconvex values, Filippov's theorem ( [5]) consists in proving the existence of a solution starting from a given almost solution.Moreover, the result provides an estimate between the starting almost solution and the solution of the differential inclusion.
On the other hand, following the approach in [10] we prove the arcwise connectedness of the solution set of problem (1.1)-(1.2).The proof is based on a result ( [9,10]) concerning the arcwise connectedness of the fixed point set of a class of set-valued contractions.
The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel, Section 3 is devoted to the Filippov type existence theorem and in Section 4 we obtain the arcwise connectedness of the solution set.

Preliminaries
In what follows we denote by I the interval [0, T ], C(I, R) is the Banach space of all continuous functions from I to R with the norm ||x|| C = sup t∈I |x(t)| and L 1 (I, R) is the Banach space of integrable functions u(.) : I → R endowed with the norm ||u|| 1 = T 0 |u(t)|dt.Let (X, d) be a metric space.We recall that the Pompeiu-Hausdorff distance of the closed subsets A, B ⊂ X is defined by provided the right-hand side is pointwise defined on (0, ∞) and Γ(.) is the (Euler's) Gamma function defined by Γ where n = [α] + 1.It is assumed implicitly that f is n times differentiable whose n-th derivative is absolutely continuous.
Lemma 2.2.For a given integrable function f (.) : [0, T ] → R, the unique solution of the boundary problem is given by and the Green function is given by , 0 ≤ t ≤ s ≤ T.
Taking into account the definition of the Green's function, using the fact that Γ(α) = (α − 1)Γ(α − 1) and the inequality For simplicity we denote M :

A Filippov type existence result
First we recall a selection result which is a version of the celebrated Kuratowski and Ryll-Nardzewski selection theorem ( [8]).
Lemma 3.1.( [3]) Consider X a separable Banach space, B is the closed unit ball in X, H : I → P(X) is a set-valued map with nonempty closed values and g : I → X, L : I → R + are measurable functions.If then the set-valued map t → H(t)∩(g(t)+L(t)B) has a measurable selection.
In the sequel we assume the following conditions on F .Hypothesis 3.2.i) F : I × R → P(R) has nonempty closed 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 We are now ready to prove the main result of this section.Theorem 3.3.Assume that Hypothesis 3.2 is satisfied, assume that M||L|| 1 < 1 and let y ∈ C(I, R) be such that there exists q(.) Proof.The set-valued map t → F (t, y(t)) is measurable with closed values and the hypothesis that d(D α c y(t), F (t, y(t))) ≤ q(t) a.e.(I) is equivalent to It follows from Lemma 3.1 that there exists a measurable selection We claim that it is enough to construct the sequences Therefore {x n } n∈N is a Cauchy sequence in the Banach space C(I, R), hence converging uniformly to some x ∈ C(I, R).Therefore, by (3.5), for almost all t ∈ I, the sequence {f n (t)} n∈N is Cauchy in R. Let f be the pointwise limit of f n .
Moreover, one has On the other hand, from (3.2), (3.5) and (3.6) we obtain for almost all Hence the sequence f n is integrably bounded and therefore f ∈ L 1 (I, R).Using Lebesque's dominated convergence theorem and taking the limit in (3.3), (3.4) we deduce that x is a solution of (1.1).Finally, passing to the limit in (3.6) we obtained the desired estimate on x.
It remains to construct the sequences x n , f n with the properties in (3.3)-(3.5).The construction will be done by induction.
Since the first step is already realized, assume that for some N ≥ 1 we already constructed

e. (I).
We define x N +1 as in (3.ii) A less powerful Filippov type existence result for problem (1.1)-(1.2) may be obtained using fixed point techniques.More exactly, by applying the set-valued contraction principle in the space of derivatives of trajectories instead of the space of solutions (as usual, for example [1]) one may obtain (see, for example, [4] for this technique) that for any ε > 0 there exists x ε (.) a solution of (1.1)-(1.2) satisfying for all Obviously, the estimation in (3.1) is better than the one in (3.7).
iii) If the assumptions of of Theorem 3.3 are satisfied with y = 0, q = L, then Theorem 3.3 improves Theorem 3.3 in [1], since in addition our result provides an a priori estimate of the solution of the form 4 Arcwise connectedness of the solution set In this section we are concerned with the more general problem where F : I × R × R → P(R) and H : I × R → P(R).
We assume that F and H are closed-valued multifunctions Lipschitzian with respect to the second variable and F is contractive in the third variable.Obviously, the right-hand side of the differential inclusion in (4.1) is in general neither convex nor closed.We prove the arcwise connectedness of the solution set to (4.1)-(4.2).When F does not depend on the last variable (4.1) reduces to (1.1) and the result remains valid for problem (1.1)-(1.2).
Let Z be a metric space with the distance d Z .In what follows, when the product EJQTDE, 2011 No. 45, p. 7 Let X be a nonempty set and let F : X → P(Z) be a set-valued map with nonempty closed values.The range of F is the set F (X) = ∪ x∈X F (x).The multifunction F is called Hausdorff continuous if for any x 0 ∈ X and every ǫ > 0 there exists δ > 0 such that x ∈ X, d X (x, x 0 ) < δ implies D Z (F (x), F (x 0 )) < ǫ.
Let (T, F , µ) be a finite, positive, nonatomic measure space and let (X, |.| X ) be a Banach space.We recall that a set A ∈ F is called atom of µ if µ(A) = 0 and for any B ∈ F , B ⊂ A one has µ(B) = 0 or µ(B) = µ(A).µ is called nonatomic measure if F does not contains atoms of µ.For example, Lebesgue's measure on a given interval in R n is a nonatomic measure.
We denote by L 1 (T, X) the Banach space of all (equivalence classes of) Bochner integrable functions u : T → X endowed with the norm Next we recall some preliminary results that are the main tools in the proof of our result.
To simplify the notation we write E in place of L 1 (T, X).
ii) There exists l ∈ L 1 (I, R + ) such that, for every u, u ′ ∈ R,
3) The set S = ∪ c∈R 2 S(c) is arcwise connected in C(I, R).
Proof. 1) For c ∈ R 2 and u ∈ L 1 (I, R), set We prove that the multifunctions φ : Since u c is measurable and H satisfies Hypothesis 4.3 i) and ii), the multifunction t → H(t, u c (t)) is measurable and nonempty closed valued, hence it has a measurable selection.Therefore due to Hypothesis 4.3 iv), the set φ(c, u) is nonempty.The fact that the set φ(c, u) is closed and decomposable follows by simple computation.In the same way we obtain that ψ(c, u, v) is a nonempty closed decomposable set.
We next note that the function T : Since F ix(Γ(c, .)) is nonempty and arcwise connected in L 1 (I, R), the set S(c) has the same properties in C(I, R).

Example 4 . 6 .
Consider the following problem