Electronic Journal Of Qualitative Theory Of Differential Equations

We establish some Filippov type existence theorems for solutions of fractional semilinear differential inclusions involving Caputo’s fractional derivative in Banach spaces.


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 of fractional order ( [21,22,24] etc.).The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim ( [16]).Very recently several qualitative results for fractional differential inclusions were obtained in [1, 3, 7-12, 19, 23] etc.. Applied problems require definitions of fractional derivative allowing the utilization of physically interpretable initial conditions.Caputo's fractional derivative, originally introduced in [5] and afterwards adopted in the theory of linear visco elasticity, satisfies this demand.For a consistent bibliography on this topic, historical remarks and examples we refer to [1].
The study of theory of abstract differential equations with fractional derivatives in infinite dimensional spaces is also very recent.The main problem consists in how to introduce new concepts of mild solutions.One of the first paper on this topic is [15].In [20] it is showed that several papers on fractional differential equations in Banach spaces were incorrect and used an approach to treat these equations based on the theory of resolvent operators for integral equations.A suitable definition of mild solutions based on Laplace transform and probability density functions may be found in [26][27][28][29].
In this paper we study fractional semilinear differential inclusions of the form D r c x(t) ∈ Ax(t) + F (t, x(t)) t ∈ I, x(0) = x 0 (1.1) where I = [0, T ], X is a separable Banach space, A is the infinitesimal generator of a strongly continuous semigroup {T (t), t ≥ 0}, F (., .): I × X → P(X) is a set-valued map and D r c is the Caputo fractional derivative of order r ∈ (0, 1]. The aim of the present paper is twofold.On one hand, we show that Filippov's ideas ( [17]) can be suitably adapted in order to obtain the existence of a solution of problem (1.1).We recall that for a first order differential inclusion defined by a lipschitzian set-valued map with nonconvex values Filippov's theorem ( [17]) consists in proving the existence of o solution starting from a given "almost" solution.Moreover, the result provides an estimate between the starting "quasi" solution and the solution of the differential inclusion.On the other hand, we prove the existence of solutions continuously depending on a parameter for problem (1.1).This result may be interpreted as a continuous variant of Filippov's theorem for problem (1.1).The key tool in the proof of this theorem is a result of Bressan and Colombo ([4]) concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values.This result allows to obtain a continuous selection of the solution set of the problem considered.
Our results may be interpreted as extensions of previous results of Frankowska ( [18]) and Staicu ([25]) obtained for "classical" semilinear differential inclusions.
The paper is organized as follows: in Section 2 we briefly recall some preliminary results that we will use in the sequel and in Section 3 we prove the main results of the paper.

Preliminaries
In this section we sum up some basic facts that we are going to use later.
Let Let I be the interval [0, T ], T > 0, denote by L(I) the σ-algebra of all Lebesgue measurable subsets of I and let X be a real separable Banach space with the norm |.| and with the corresponding metric d(., .).Denote by B the closed unit ball in X. 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.
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.Recall that 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).
Let F (., .): I × X → P(X) be a set-valued map.Recall that F (., .) is called L(I) ⊗ B(X) measurable if for any closed subset C ⊂ X we have We recall next the following definitions.For more details, we refer to [21].
EJQTDE, 2012 No. 64, p. 3 b) The Riemann-Liouville derivative of order r of f (. b) The Caputo derivative of a constant is equal to zero.c) If f : I → X, with X a Banach space, then integrals which appears in Definition 2.1 are taken in Bochner's sense.

EJQTDE, 2012 No. 64, p. 4
The results summarized in the next lemmas will be used in the proof of our main results.
Lemma 2.5.( [18]) Let X be a separable Banach space, let H : I → P(X) be a measurable set-valued map with nonempty closed values and g, h Moreover, if F (., .): I × X → P(X) has nonempty closed values, F (., x) is measurable for any x ∈ X and x(.) ∈ C(I, X) then the set-valued map t → F (t, x(t)) is measurable.
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.
Then the multifunction G(.) : S → D(I, X) defined by  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.

The main results
In order to obtain a Filippov type existence result for problem (1.1) one need the following assumptions.Hypothesis 3.1.i) The operator A generates a strongly continuous semigroup {T (t), t ≥ 0} on a real separable Banach space X and there exists a constant M ≥ 1 such that sup t∈I |T (t)| ≤ M.
ii) F (., .): I × X → P(X) is a set-valued map with non-empty closed values and for all x ∈ X, F (., x) is measurable.
Then for any ε > 0 there exists (x(.), f (.)) a trajectory-selection pair of problem (1.1) such that EJQTDE, 2012 No. 64, p. 6 We claim that is enough to construct the sequences x n (.) ∈ C(I, X), f n (.) ∈ L 1 (I, X), n ≥ 1 with the following properties below Therefore {x n (.)} is a Cauchy sequence in the Banach space C(I, X).Thus, from (3.7) for almost all t ∈ I, the sequence {f n (t)} is Cauchy in X.Moreover, from (3.4) and the last inequality we have On the other hand, from (3.5), (3.7) and (3.8) we obtain for almost all (3.9) EJQTDE, 2012 No. 64, p. 7 Let x(.) ∈ C(I, X) be the limit of the Cauchy sequence x n (.).From (3.9) the sequence f n (.) is integrably bounded and we have already proved that for almost all t ∈ I, the sequence {f n (t)} is Cauchy in X.Take f (.) ∈ L 1 (I, X) with f (t) = lim n→∞ f n (t).
Using the fact that the values of F (., .)are closed we get that f (t) ∈ F (t, x(t)) a.e.(I).
One may write successively, Therefore, we may pass to the limit in (3.1) and we obtain that x(.) is a solution of problem (1.1) Finally, passing to the limit in (3.8) and (3.9) we obtained the desired estimations.
It remains to construct the sequences x n (.), f n (.) with the properties in (3.3)-(3.7).The construction will be done by induction.
Then there exists x(.) ∈ C(I, X) a solution of problem (1.1) such that Next we obtain a continuous version of Theorem 3.1.This result allows to provide a continuous selection of the solution set of problem (1.1).Hypothesis 3.4.i) The operator A generates a strongly continuous semigroup {T (t), t ≥ 0} on a real separable Banach space X and there exists a constant M ≥ 1 such that sup t∈I |T (t)| ≤ M.
We consider the set-valued maps G 0 (.), H 0 (.) defined, respectively, by T r ε(s).We define We shall construct, using the same idea as in [14], two sequences of approximations f p (.) : S → L 1 (I, X), x p (.) : S → C(I, X) with the following properties a) f p (.) : S → L 1 (I, X), x p (.) : Suppose we have already constructed f i (.), x i (.) satisfying a)-c) and define x p+1 (.) as in d).From c) and d) one has (3.10) On the other hand,

.11)
For any s ∈ S we define the set-valued maps We note that from (3.11) From (3.10), c) and d) we obtain Therefore f p (s)(.), u p (s)(.) are Cauchy sequences in the Banach space L 1 (I, X) and C(I, X), respectively.Let f (.) : S → L 1 (I, X), x(.) : S → C(I, X) be their limits.The function s → b(s) + ε(s) + Λ(s) is continuous, hence locally bounded.Therefore (3.13) implies that for every s ′ ∈ S the sequence f p (s ′ )(.) satisfies the Cauchy condition uniformly with respect to s ′ on some neighborhood of s.Hence, s → f (s)(.) is continuous from S into L 1 (I, X).
(Y, d) be a metric space.The Pompeiu-Hausdorff distance of the closed subsets A, B ⊂ Y is defined by d H (A, B) = max{d * (A, B), d * (B, A)}, d * (A, B) = sup{d(a, B); a ∈ A}, where d(x, B) = inf{d(x, y); y ∈ B}.With cl(A) we denote the closure of the set A ⊂ X.
the set H p+1 (s) is not empty.