COUPLED SYSTEMS OF HILFER FRACTIONAL DIFFERENTIAL INCLUSIONS IN BANACH SPACES the coupled system of Hilfer fractional diﬀerential inclusions

. This paper deals with some existence results in Banach spaces for Hilfer and Hilfer-Hadamard fractional diﬀerential inclusions. The main tools used in the proofs are M¨onch’s ﬁxed point theorem and the concept of a measure of noncompactness.

1. Introduction. Fractional differential equations and inclusions appear in several areas such as engineering, mathematics, bio-engineering, physics, and other applied sciences [19,32]. For some fundamental results in the theory of fractional calculus and fractional differential equations, we refer the reader to the monographs of Abbas et al. [4,5], Kilbas et al. [23], Samko et al. [31], Zhou [35], as well as the papers by Abbas et al. [1,2], Benchohra et al. [10], Lakshmikantham et al. [24,25,26] and the references therein. Recently, considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations with Hilfer fractional derivative; see [14,15,19,21,33,34] and the references therein.
Recently, in [3,8,9,11,12,16,17] the authors applied the measure of noncompactness to some classes of Riemann-Liouville or Caputo fractional differential equations in Banach spaces. In this paper we discuss the existence of solutions to the coupled system of Hilfer fractional differential inclusions with the initial conditions where T > 0, α i ∈ (0, 1), β i ∈ [0, 1], γ i = α i + β i − α i β i , E is a real (or complex) separable Banach space with a norm · , P(E) is the family of all nonempty subsets of E, φ i ∈ E, F i : I × E × E → P(E), i = 1, 2, are given multivalued maps, I 1−γi 0 is the left-sided mixed Riemann-Liouville integral of order 1 − γ i , and D αi,βi 0 is the Hilfer derivative operator of order α i and type β i .
We also consider the coupled system of Hilfer-Hadamard fractional differential inclusions ( H D α1,β1 1 u)(t) ∈ G 1 (t, u(t), v(t))), with the initial conditions where is the left-hand mixed Hadamard integral of order 1 − γ i , and H D αi,βi 1 is the Hilfer-Hadamard fractional derivative of order α i and type β i .

2.
Preliminaries. Let C := C(I) be the Banach space of all continuous functions w from I into E with the supremum (uniform) norm w ∞ := sup t∈I w(t) .
As usual, AC(I) denotes the space of absolutely continuous functions from I into E. We denote by AC 1 (I) the space defined by By L 1 (I), we denote the space of measurable functions v : I → E that are Bochner integrable and normed by Let L ∞ (I) be the Banach space of measurable functions v : I → R that are essentially bounded and equipped with the norm v L ∞ = inf{c > 0 : |v(t)| ≤ c, a.e. t ∈ I}.
By C γ (I) and C 1 γ (I), we denote the weighted spaces of continuous functions defined by with the norm w Cγ := sup t∈I t 1−γ w(t) , and C 1 γ (I) = {w ∈ C : with the norm w C 1 γ := w ∞ + w Cγ . Also, by C := C γ1 × C γ2 we denote the product weighted space with the norm Let P cl (E) = {A ∈ P(E) : A is closed}, P c (E) = {A ∈ P(E) : A is convex}, and P cp,c (E) = {A ∈ P(E) : A is compact and convex}. If there exists x ∈ E such that x ∈ G(x) then the multivalued map G : E → P(E) has a fixed point. The symbol F ixG stands for the set of fixed point of G. If for every y ∈ E, the function t → d(y, G(t)) = inf{|y − z| : z ∈ G(t)}, then a multivalued map G : J → P cl (E) is said to be measurable.
Definition 2.1. Let X and Y be two sets. The graph of a set-valued map N : For more details about multivalued functions, see for instance [6,7,13,20].
is upper semicontinuous for a. e. t ∈ I, then the multivalued function F : I × E → P(E) is Carathéodory.
For each u ∈ C(I), we defined the set of selections of F by We next give some results and properties of the fractional calculus. Notice that for all r, r 1 , r 2 > 0 and each w ∈ C, we have I r 0 w ∈ C, and (I r1 Let r ∈ (0, 1], γ ∈ [0, 1), and w ∈ C 1−γ (I). Then the next expression leads to the left inverse operator as follows: 1−γ (I), then the following composition is proved in [31]: t r−1 f or all t ∈ (0, T ].
is the Caputo fractional derivative of the function w of order r ∈ (0, 1]. In [19], Hilfer studied applications of a generalized fractional operator with the Riemann-Liouville and the Caputo derivatives as specific cases (see also [21,33]).
1. The operator (D α,β 0 w)(t) can be written as Moreover, the parameter γ satisfies 2. The special case of (5) with β = 0 coincides with the Riemann-Liouville derivative, and with β = 1, it coincides with the Caputo derivative. In addition, w exists and is in L 1 (I), then Furthermore, if w ∈ C γ (I) and I Then the linear Cauchy problem has a unique solution given by From the above corollary, we have the following lemma.
. Then solving the system (1)-(2) is equivalent to the finding the solutions of the system of integral The symbol M X will stand for the class of all bounded subsets of a metric space X.
Definition 2.9 ([9]). Let E be a Banach space and let Ω E denote the family of bounded subsets of E. If

Properties.
(1) µ(M ) = 0 if and only if M is compact (M is relatively compact).
Lemma 2.11 ([27]). Let I be a compact real interval, let F be a Carathéodory multivalued map, and let Θ be a continuous linear map from L 1 (I) → C(I). Then the operator is a closed graph operator in C(I) × C(I).
We now recall the set-valued version of Mönch's fixed point theorem.
Theorem 2.12 ( [28]). Let E be Banach space, K ⊂ E be a closed and convex set, U be a relatively open subset of K, and N : U → P c (K). Assume that N maps compact sets into relatively compact sets, graph(N ) is closed, and for some x 0 ∈ U , we have: for all x ∈ U \U and λ ∈ (0, 1). Then there exists x ∈ U with x ∈ N (x).

3.
Coupled system of Hilfer fractional differential inclusions. First, we define what we mean by a solution of the system (1)-(2). Definition 3.1. By a solution of the system (1)-(2) we mean a pair of measurable functions (u, v) ∈ C that satisfy conditions (2) and the inclusions (1) on I.
In the sequel, we will need the following conditions. (H 1 ) The multivalued maps F i : for a.e. t ∈ I and u, v ∈ E. (H 3 ) For each bounded and measurable set B i ⊂ C γi , i = 1, 2, and for each t ∈ I, we have We now prove our main result in this section on the existence of solutions to the system (1)-(2). Proof. Define the multivalued operators N 1 : C γ1 → P(C γ1 ) and N 2 : C γ2 → P(C γ2 ) by Consider the continuous operator N : C → P(C) defined by Clearly, the fixed points of N are solutions of the system (1)- (2). We shall show that the multivalued operator N satisfies all the assumptions of Theorem 2.12. The proof will be given in several steps.
Step N (u, v), then there exist w 1 , w 2 ∈ S F •u and z 1 , z 2 ∈ S F •v such that for each t ∈ I, we have Let 0 ≤ λ ≤ 1; then, for each t ∈ I, Since S F1•u is convex (because F 1 has convex values), we have λh 1 + (1 − λ)h 2 ∈ N 1 (u). Also, for each t ∈ I, we have Step 2. For each compact M ⊂ C, N (M ) is relatively compact. Let (h n , k n ) be any sequence in N (M ) with M ⊂ C and M compact. To apply the Arzelà-Ascoli compactness criterion on C, we will show that (h n , k n ) has a convergent subsequence. Since (h n , k n ) ∈ N (M ) there exist (u n , v n ) ∈ M , w n ∈ S F1•un , and z n ∈ S F2•vn such that Using Theorem 2.10 and the properties of the Kuratowski measure of noncompactness, we have and On the other hand, since M is compact, the sets  (6) and (7) we obtain that {h n (t) : n ≥ 1} and {k n (t) : n ≥ 1} are relatively compact for each t ∈ I.
For each t 1 , t 2 ∈ I with t 1 < t 2 , we have Similarly, As t 1 −→ t 2 , the right-hand sides of the inequalities (8) and (9) tend to zero. This shows that {(h n , k n ) : n ≥ 1} is equicontinuous. Consequently, {(h n , k n ) : n ≥ 1} is relatively compact in C.
Step 5. A priori estimate. Let (u, v) ∈ C be such that (u, v) ∈ λN (u, v) for some λ ∈ (0, 1). Then for each t ∈ I, we have On the other hand, and similarly, Thus, Condition (ii) in Theorem 2.12 is satisfied by our choice of the open set U. From Steps 1-5 and Theorem 2.12, we conclude that N has at least one fixed point (u, v) ∈ C which in turn is a solution of the system (1)-(2).

Coupled system of Hilfer-Hadamard fractional differential inclusions.
In this section, we study the existence of solutions for the system (3) (ln t) 1−r w(t) .
Let C := C γ1,ln × C γ2,ln be the product weighted space with the norm We recall some definitions and properties of Hadamard fractional integration and differentiation. We refer to [23] for more detailed analysis.
. Then the problem (3) is equivalent to the Volterra integral equation where v ∈ S F •u .
We now give without proof an existence result for system (3)-(4).