Solvability of Fractional Differential Inclusion with a Generalized Caputo Derivative

This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviatedadvanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.


Introduction
The history of the theory of fractional calculus goes back to 1695 when Leibniz sent a question to L-Hôpital [1]. Although in the starter fractional calculus had an efflorescence as a mathematical analysis idea, nowadays, its use has also sawing into many other subjects of engineering and science such as biology, physics, mechanics, chemistry, and bioengineering [2][3][4][5].
It is known that differential inclusions are more general than differential equations and various phenomena of science, control, and engineering are successfully modeled as fractional differential inclusions [6,7].
Recently, fractional differential inclusions with nonlocal conditions have attracted the attention of many researchers. In 2011, El-Sayed et al. [8] established the solvability of the ordinary differential inclusion with deviated-advanced nonlocal condition.
In the few past years, there has been important works in fractional differential inclusions with other types of nonlocal conditions. Detailedly, in 2015, Wang et al. [9] established the existence of solutions for the Caputo fractional differential inclusions involving nonlocal conditions. In the second year, Lian et al. [10] established the solvability of the frac-tional differential inclusions with nonlocal conditions by using the measure of noncompactness and several-valued fixed-point approach. In 2019, Castaing et al. [11] studied the solvability of a new class of the Riemann-Liouville fractional differential inclusion with nonlocal integral conditions in a separable Banach space.
Þ, c i , d j > 0, t i , τ j ∈ 0, 1 ð Þ,∀i = 1, ⋯, l 1 , j = 1, ⋯, l 2 : In the above-cited monographs, the Caputo and Riemann-Liouville derivatives were utilized. In 2017, Almeida [12] obtained the new generalized Caputo fractional derivative, that is, a Caputo-kind operator of a function with respect to another function. Indeed, this fractional operator is more general than Riemann-Liouville, Hadamard, Erdely Kober, and Caputo operator kinds. More details about the generalized Caputo fractional operator are found in [13,14]. Since then, generalized fractional operators draw increasing attention due to their advantages, because the generalized fractional operators will give us new opportunities to improve the theoretical results and to model a lot of real-life events. In 2019, Promsakon et al. [15] established the solvability of a new class of impulsive fractional boundary value problems involving the generalized Caputo fractional derivative. In 2020, Belmor et al. [16] investigated the solvability of fractional differential inclusion including the generalized Caputo derivative with integral nonlocal conditions. There are other works that showed interest in the generalized Caputo operators; we mention for example [17][18][19][20].
Nowadays, Herzallah and Radwan [21] studied the fractional version of the system (1) with the classical Caputo operator, namely Motivated by the above-cited contributions, in particular systems (1) and (2), we propose a new fractional differential inclusion involving generalized Caputo operator, given by where * D α 0 + ,Q is the generalized Caputo derivative w.r.t. the function Q such that α ∈ ð0, 1Þ, AðtÞ: DðAÞ ⊆ ℝ ⟶ ℝ is linear bounded operator and F : I × ℝ ⟶ Pðℝ + Þ. We show the existence of solution for the proposed system (3). The proposed system (3) is more flexible since it allows us to choose fractional derivative depending on the particular established phenomenon. Therefore, the tools of generalized fractional differential inclusions facilitate the investigation of optimal controls and stochastic processing, in particular, modeling of control processes that are considered by selecting a trial function [7]. Moreover, nonlocal conditions give more accurate measurements, precise results, and efficient effect than the classical boundary conditions. An outline of this paper is as follows. In Section 2, some bases and results are given needed in the sequel. In Section 3, we study the solvability of the generalized system (3). In Section 4, we apply the abstract results in order to establish the existence of solution for some illustrative examples.

Preliminaries
In this part, we recall some definitions and theorems that will be used later. Let ðE, k:k E Þ be a Banach space and P ðEÞ = fZ ⊂ E : Z ≠ ∅g. Now, throughout this paper, let Let W ⊆ E. The fixed point of set-valued map Ψ : W ⟶ PðEÞ is a point ω ∈ W such that ω ∈ ΨðωÞ. The graph of Ψ is defined as A selection of Ψ is a single-valued map ψ : W ⟶ E such that GðψÞ ⊆ GðΨÞ.
Therefore, Ψ is completely continuous if Ψ (W) is relatively compact for each W ∈ P bd ðWÞ. In fact, if Ψ is completely continuous with nonempty compact values, then Ψ is upper semicontinuous (u.s.c., for short) if and only if G (Ψ) is closed.
An important role is played by the fixed-point principle to obtain the solvability of various types of operator equations (see, for example, [25][26][27][28][29]). We will apply the following fixed-point theorem to obtain the main results.

Journal of Function Spaces
Further, the generalized Caputo derivative can be defined via the generalized Riemann-Liouville fractional derivative as [13] where η The following lemma, which concerns some properties of generalized fractional operators, plays a key role in the sequel. Lemma 8. [13]. Suppose that η : (3) if η ∈ C n ðI, ℝÞ and α ∈ ð0, 1Þ, then

Main Results
The differential inclusions using fractional derivatives have been proven to be of major interest to the academic community, not only mathematicians but also researchers in other fields. There is a motivating way to obtain the solvability of the differential inclusions; this way is representing the solution by integral equation. The solvability of system (3) will be established under the following hypotheses: (H1) For all ηðtÞ ∈ I, there exists L A ∈ ð0,∞Þ such that (H5) There exists a function p ∈ L 1 ðI, R + Þ and K > 0 such that kFðt, ηðtÞÞk ≤ KpðtÞ, and there exists M * > 0 such that where The integral representation of the system (3) will be given in the following lemma. Lemma 9. Let the hypotheses (H1)-(H2) hold. Suppose that ψ : I ⟶ CðI, ℝÞ, then the solution η(t) of the following is given by where σ = 1/ð∑ Proof. Applying the operator J α 0 + ,Q on both sides of equation (21). Then, by Lemma 8, we get Therefore, we obtain

Journal of Function Spaces
Putting t = θðt i Þ in equation (24), we get Thus, we have Putting t = ϕðτ j Þ in equation (24), we get Thus, we have Hence, we obtain Substituting equation (29) into equation (24), we obtain the result.
Step 3. ϒ is u.s.c. We only need to show that ϒ has a closed graph to be u.s.c. Let η n ⟶ b η and h n ⟶ĥ where h n ∈ ϒ η n .We need to show thatĥ ∈ ϒ b η.Associated with h n ∈ ϒ η n ,there exists ψ n ∈ S 1 F,η n such that for all t ∈ I, we have h n t ð Þ = σβ 〠

Conclusions
In this paper, we established the solvability of fractional differential inclusions involving the generalized Caputo operator by applying Leray-Schauder's alternative approach with the help of the Lagrange mean-value classical theorem. The proposed system studied in the present work is more practical and more generalized. The results given in this paper extended and developed some previous works. We presented some examples to illustrate the solvability results.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.