A Note on a Coupled System of Hilfer Fractional Differential Inclusions

A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have nonconvex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we avoid the applications of fixed point theorems as usual. An illustration of the results is given by a suitable example.


Introduction
The fractional derivative introduced by Hilfer in [1] was recently used in the study of many boundary value problems concerned with fractional derivatives. This fractional derivative generalizes both Riemann-Liouville and Caputo derivatives; in fact, this derivative is an interpolation between Riemann-Liouville and Caputo derivatives. Several properties and applications of the Hilfer fractional derivative may be found in [2]. Additionally, we recall that the literature is full of explanations and motivations for considering systems defined by fractional order derivatives (e.g., [3][4][5][6][7] etc.).
Recently, many papers in the literature have been devoted to the study of fractional differential equations and inclusions defined by the Hilfer fractional derivative (e.g., [8][9][10][11] etc.). We point out that a complete survey on this field of study may be found in [8]. Taking into account this new trend in research, our intention is to contribute to the development of this topic by establishing new results for a particular class of problems.
The present note is devoted to the following boundary value problem where H is the Hilfer fractional derivative of order α and type β, I ϕ is the Riemann-Liouville fractional integral of order ϕ > 0, α 1 , Our study is motivated by two recent papers [10,11]. In [10], problems (1) and (2) is studied in the single-valued case; namely, the right-hand side in (1) is given by single-valued maps. Existence and uniqueness results are provided by using well-known fixed point theorems: Banach, Leray-Schauder and Krasnoselskii. In [11], a "simple" (not coupled) set-valued problem as in (1) and (2) is studied, and existence results are also obtained by applying known set-valued fixed point results as Leray-Schauder and Nadler.
The purpose of our note is twofold. On one hand, we extend the study in [10] to the set-valued framework, and on the other hand, we generalize the study in [11] to the coupled case. The approach we present here avoids the applications of fixed point theorems and takes into account the case when the values of F 1 and F 2 are not convex; but these set-valued maps are assumed to be Lipschitz in the second and third variable. In this case, we establish an existence result for problems (1) and (2). Our result use Filippov's technique [12]; more exactly, the existence of solutions is obtained by starting from a pair of given "quasi"solutions. In addition, the result provides an estimate between the "quasi" solutions and the solutions obtained.
Even if the technique used here may be seen at other classes of coupled systems of fractional differential inclusions [13][14][15], to the best of our knowledge, the present paper is the first in literature which contains an existence result of Filippov type for coupled systems of Hilfer fractional differential inclusions.

Preliminaries
We set by I the interval [a, b]. We denote by C(I, R) the Banach space of all continuous functions x(.) : I → R endowed with the norm |x(.)| C = sup t∈I |x(t)| and by L 1 (I, R) the Banach space of all integrable functions x(.) : 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, provided the right-hand side is pointwise defined on (0, ∞).
The Caputo fractional derivative of order It is assumed implicitly that f is n times differentiable whose n-th derivative is absolutely continuous.
The Hilfer fractional derivative of order α ∈ (n − 1, n) and type β ∈ [0, 1] of a function As it was already recalled, this derivative interpolates between Riemann-Liouville and Caputo derivatives. When β = 0 the Hilfer fractional derivative gives Riemann-Liouville fractional derivative D α,0 H f (t) = d n dt n I n−α f (t) and when β = 1 the Hilfer fractional derivative gives Caputo fractional derivative D α,1 H f (t) = I n−α d n dt n f (t).
The next technical result proved in [10] considers a linear version of problems (1) and (2), for which an integral representation of the solution is provided.
Then, the solution of the system with boundary conditions (2) is given by where
In what follows, χ A (·) denotes the characteristic function of the set A ⊂ R.

Remark 1.
Let us introduce the following notations χ [a,z j ] (τ)), Then, the solutions (x 1 (.), x 2 (.)) in Lemma 1 may be put as Moreover, we have the following estimates: Finally, in the proof of our main result we need the following selection result for set-valued maps (e.g., [16]). It is, in fact, a variant of the well known selection theorem due to Kuratowski and Ryll-Nardzewski which, briefly, states that a measurable set-valued map with nonempty closed values admits a measurable selection.

Lemma 2.
Let Z be a separable Banach space, B its closed unit ball, A : I → P (Z) is a set-valued map whose values are nonempty closed and b : I → Z, c : I → R + are two measurable functions. If then the set-valued map t → A(t) ∩ (b(t) + c(t)B) admits a measurable selection.