On a fractional integro-differential inclusion

We study a Cauchy problem for a fractional integro-differential inclusion of order α ∈ (1, 2] involving a nonconvex set-valued map. Arcwise connectedness of the solution set is provided. Also we prove that the set of selections corresponding to the solutions of the problem considered is a retract of the space of integrable functions on a given interval.


Introduction
Differential equations with fractional order have recently proved to be strong tools in the modelling of many physical phenomena.As a consequence there was an intensive development of the theory of differential equations and inclusions of fractional order ( [12,13,14] etc.).Applied problems require definitions of fractional derivative allowing the utilization of physically interpretable initial conditions.Caputo's fractional derivative, originally introduced in [3] and afterwards adopted in the theory of linear visco elasticity, satisfies this demand.Very recently several qualitative results for fractional integro-differential equations were obtained in [6,11,17] etc.
The aim of this paper is twofold.On one hand, we prove the arcwise connectedness of the solution set of problem (1.1) when the set-valued map is Lipschitz in the second and third variable.On the other hand, under such type of hypotheses on the set-valued map we establish a more general topological property of the solution set of problem (1.1).Namely, we prove Email: acernea@fmi.unibuc.ro 2 A. Cernea that the set of selections of the set-valued map F that correspond to the solutions of problem (1.1) is a retract of L 1 ([0, T], R).Both results are essentially based on the results of Bressan and Colombo [1] concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values.
We note that in the classical case of differential inclusions similar results are obtained using various methods and tools ( [2,8] etc.).Our results may be interpreted as extensions of the results in [4,15,16] to fractional integro-differential inclusions.
The paper is organized as follows: in Section 2 we present the notations, definitions and the preliminary results to be used in the sequel and in Section 3 we prove our main results.

Preliminaries
Let T > 0, I := [0, T] and denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I. Let X be a real separable Banach space with the norm | • |.Denote 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 cl(A) the closure of A.
The distance between a point x ∈ X and a subset A ⊂ X is defined as usual by d(x, A) = inf{|x − a|; a ∈ A}.We recall that the Pompeiu-Hausdorff distance between the closed subsets 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| 1 = T 0 |x(t)| dt.We recall first several preliminary results we shall use in the sequel.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.
We denote by D(I, X) the family of all decomposable closed subsets of L 1 (I, X).
Next (S, d) is a separable metric space; we recall that a multifunction G : S → P (X) is said to be lower semicontinuous (l.s.c.) if for any closed subset C ⊂ X, the subset {s ∈ S; G(s) ⊂ C} is closed.The next lemmas may be found in [1].Lemma 2.1.If F : I → D(I, X) is a lower semicontinuous multifunction with closed nonempty and decomposable values then there exists f : Then the multifunction G : S → D(I, X) defined by has nonempty values.Then H has a continuous selection, i.e. there exists a continuous mapping h : S → L 1 (I, X) such that h(s) ∈ H(s) ∀s ∈ S. Definition 2.4.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 defined by where n = [α] + 1.It is assumed implicitly that f is n times differentiable and its n-th derivative is absolutely continuous.
We recall (e.g., [12]) that if α > 0 and In this case (x(•), f (•)) is called a trajectory-selection pair of problem (1.1).We shall use the following notations for the solution sets and for the selection sets of problem (1.1).

The main results
In order to prove our topological properties of the solution set of problem (1.1) we need the following hypotheses.
We use next the following notations Theorem 3.1.Assume that Hypothesis is satisfied and |I α M| < 1.
Taking into account Hypothesis one may write Next we use the following notation At the same time, from Hypothesis it follows that Fix δ > 0 and for m ∈ N we set δ m = m+1 m+2 δ.We shall prove next that there exists a continuous mapping g a.e. in I. Define T α , a.e. in I and, by (3.1), we find that G 1 (λ) is nonempty for any λ ∈ [0, 1].Moreover, since the mapping λ → p 0 (λ) is continuous, we apply Lemma 2.3 and we obtain the existence of a continuous mapping hence with properties a)-c).Define now We shall prove that for all m ≥ 1 and λ ∈ [0, 1] there exist x m (λ) ∈ C(I, R) and g m (λ) ∈ L 1 (I, R) with the following properties: Assume that we have already constructed g m (•) and x m (•) with i)-vi) and define As in the case m = 1 we obtain that Φ m+1 : [0, 1] → D(I, R) is lower semicontinuous.
From ii), v) and Hypothesis, for almost all t ∈ I, we have < p m+1 (λ).
For λ ∈ [0, 1] consider the set To prove that G m+1 (λ) is not empty we note first that r m := |I α M| m (δ m+1 − δ m ) > 0 and by Hypothesis and ii) one has Moreover, since Φ m+1 : [0, 1] → D(I, R) is lower semicontinuous and the maps λ → p m+1 (λ), λ → h m (λ) are continuous, we apply Lemma 2.3.and we obtain the existence of a continuous selection g m+1 of G m+1 .Therefore, and thus {x m (λ)} m∈N is a Cauchy sequence in the Banach space C(I, R), hence it converges to some function x(λ) ∈ C(I, R).
On a fractional integro-differential inclusion The function λ → T α−1 Γ(α) |p 0 (λ)| 1 + δ is continuous, so it is locally bounded.Therefore the Cauchy condition is satisfied by {x m (λ)} m∈N locally uniformly with respect to λ and this implies that the mapping λ → x(λ) is continuous from [0, 1] into C(I, R).Obviously, the convergence of the sequence {x m (λ)} to x(λ) in C(I, R) implies that g m (λ) converges to g(λ) in L 1 (I, R).
Finally, from ii), Hypothesis and from the fact that the values of F are closed we obtain that x(λ) ∈ S(ξ 0 , ξ 1 ).From i) and v) we have x(0) = x 0 , x(1) = x 1 and the proof is complete.
Then the multifunction Φ : L 1 (I, R) → P (L 1 (I, R)) is lower semicontinuous with closed decomposable and nonempty values.
Proof.According to (3.4), Lemma 2.2 and the continuity of p 0 we obtain that Ψ has closed decomposable and nonempty values and the same holds for the set-valued map Φ.