A generalized neutral-type inclusion problem in the frame of the generalized Caputo fractional derivatives

In this paper, we study the existence of solutions for a generalized sequential Caputo-type fractional neutral differential inclusion with generalized integral conditions. The used fractional operator has the generalized kernel in the format of (ϑ(t)−ϑ(s))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$( \vartheta (t)-\vartheta (s)) $\end{document} along with differential operator 1ϑ′(t)ddt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{\vartheta '(t)}\,\frac{\mathrm{d}}{\mathrm{d}t}$\end{document}. We obtain existence results for two cases of convex-valued and nonconvex-valued multifunctions in two separated sections. We derive our findings by means of the fixed point principles in the context of the set-valued analysis. We give two suitable examples to validate the theoretical results.


Introduction
Fractional differential inclusions as a generalization of fractional differential equations are established to be of considerable interest and value in optimizations and stochastic processes [1]. Fractional differential inclusions additionally help us study dynamical systems in which speeds are not remarkably specific by the condition of the system, regardless of relying upon it. In recent periods the theory of fractional differential equations has gained a lot of interest in all areas of mathematics; see [2][3][4]. Also, fractional differential equations and fractional differential inclusions appear naturally in a variety of scientific fields and have a wide range of applications; see [5][6][7][8]. Almeida [9] introduced a new operator called the ψ-Caputo fractional derivative combining a fractional operator with other different types of fractional derivatives and thus opened a new window to modern and complicated applications.
Throughout the years, many researchers have been interested in discussing the existence of solutions for fractional differential equations and fractional differential inclusions involving various types of fractional derivatives; see .
Motivated by the aforementioned works and inspired by [9], we prove the existence of solutions to the following nonlinear neutral fractional differential inclusion involving ϑ-Caputo fractional derivative with ϑ-Riemann-Liouville fractional integral boundary conditions: where t ∈ [0, T), and H : We obtain the desired results for the suggested ϑ-Caputo inclusion FBVP (3) involving convex and nonconvex set-valued maps using some well-known fixed point theorems. We also construct two examples to validate our results. Reported findings are new in the frame of the generalized sequential Caputo fractional derivatives implemented on a novel neutral-type generalized fractional differential inclusion.
Observe that our problem (3) involves a general structure and is reduced to an Erdelyi-Kober-type (and Hadamard-type) inclusion problem when we take ϑ(t) = t η (and ϑ(t) = log(t), respectively). Moreover, problem (3) is more general than problem (2). This paper is organized as follows. Some fundamentals ideas of fractional calculus and theory of multifunctions are presented in Sect. 2. The main results on the existence of solutions to the ϑ-Caputo inclusion problem (3) using some fixed point theorems are obtained in Sect. 3. Two examples are provided in Sect. 4. In the final section, we give conclusive remarks.

Lemma 2.6 ([38]) Let
and K, q ∈ C. Then the solution of linear-type problem is given by

Multifunction theory
We present some concepts regarding the multifunctions (set-valued maps) [41]. For this aim, consider the Banach space (C, · ) and S : C → P(C) as a multifunction that: is bounded with respect to ϕ for any bounded set B ⊂ C, that is, (III) is measurable whenever for each η ∈ R, the function is measurable.
For other notions such as the complete continuity or upper semicontinuity (u.s.c.), see [41]. Furthermore, the set of selections of H is given by Next, we define P δ (C) = W ∈ P(C) : W = ∅ and has property δ , where P cl , P c , P b , and P cp denote the classes of all closed, convex, bounded, and compact sets in C.
for every k * ≤ l and for almost all t ∈ J T .
The forthcoming lemmas are required to attain the desired outcomes in the current research study.

Lemma 2.8 ([42]) Let C and S be two Banach spaces, and let
is completely continuous and has a closed graph, then S is u.s.c.

) be linear and continuous. Then
Theorem 2.10 (Nonlinear alternative for contractive maps [42]) Let C be a Banach space, and let D be a bounded neighborhood of 0 ∈ C. Let 1 : C → P cp,c (C) and 2 : D → P cp,c (C) be two set-valued operators satisfying: (i) 1 is a contraction, and (ii) 2 is u.s.c. and compact. IfS = 1 + 2 , then either (a)S has a fixed-point in D, or (b) there exist ϕ ∈ ∂D and μ ∈ (0, 1) such that ϕ ∈ μS(ϕ).
Theorem 2.11 (Nadler-Covitz fixed point theorem [44]) Let C be a complete metric space. If H : C → P cl (C) is a contraction, then H has a fixed point.

Existence results for set-valued problems
In this section, we establish the main existence theorems. and
Proof First, to switch the neutral-type fractional differential inclusion (3) into a fixedpoint problem, we defineS : C → P(C) as for ω ∈ R H,ϕ . Consider two operators 1 : C→C and 2 : C → P(C) defined as Obviously,S = 1 + 2 . In what follows, we will show that the operators satisfy the hypotheses of the nonlinear alternative for contractive maps (Theorem 2.10). First, we define the bounded set and show that 1 and 2 define the set-valued operators 1 , 2 : B c → P cp,c (C). To do this, we show that 1 and 2 are compact and convex-valued. We consider two steps.

is bounded on bounded sets of C.
Let B c be bounded in C. For φ ∈ 2 (ϕ) and ϕ ∈ B c , there exists ω ∈ R H,ϕ such that Under assumption (Hyp2), for any t ∈ J T , we have Step 2. 2 maps bounded sets of C into equicontinuous sets. Let ϕ ∈ B c and φ ∈ 2 (ϕ). Then there is a function ω ∈ R H,ϕ such that Then As t 1 → t 2 , we obtain Hence 2 (B c ) is equicontinuous. From steps 1-2, by the Arzelà-Ascoli theorem, 2 is completely continuous.

Case 2: nonconvex-valued multifunctions
In this section, we obtain another existence criterion for ϑ-Caputo fractional differential inclusion (3) under new assumptions. We will show our desired existence with a nonconvex-valued multifunction by using a theorem of Nadler and Covitz (Theorem 2.11). Consider (C, d) as a metric space. Consider  d(b,c). Then (P b,cl (C), H d ) is a metric space (see [45]).

Theorem 3.4 Consider
Then the neutral-type fractional differential inclusion (3) has one solution on J T if where ζ 1 , ζ 2 are given in (7).
Proof By virtue of assumption (Hyp6) and Theorem III.6 in [46], H has a measurable selection and thus, R H,ϕ = ∅. In the sequel, we prove that the operatorS : C → P(C) defined in (9) satisfies the assumptions of Nadler and Covitz fixed-point theorem (Theorem 2.11).
To Prove the closedness ofS(ϕ) for all ϕ ∈ C, let {u n } n≥0 ∈S(ϕ) be such that u n → u (n → ∞) in C. In such a case, u ∈ C and there is ω n ∈ R H,ϕ n such that K T, ϕ(T) + I b 1 +b 2 +b 3 ;ϑ 0+ ω n (T) .
Accordingly, there exists a subsequence ω n which converges to ω in L 1 (J T , R), because H has compact values. As a result, ω ∈ R H,ϕ and we get Hence u ∈S(ϕ).
As a sequel, we obtain Therefore Similarly, interchanging the roles of ϕ and ϕ, we get BecauseS is a contraction, we deduce that it has a fixed-point, which is a solution of (3) by the Covitz-Nadler theorem, and the proof is completed.

Examples
In this section, we consider some particular cases of BVPs consisting of fractional differential inclusions to validate the existence results. Consider the fractional differential inclusions of the form ⎧ ⎨ for t ∈ (0, T).
The following examples are instances of fractional differential inclusions in the particular cases of (12).

Conclusive remarks
Generalized fractional operators are a generalization of the standard operators with special kernels. Besides, fixed point theorems play a key role in studying the qualitative properties of the solutions to certain fractional dynamical equations representing complex systems and chaotic systems. In this paper, we investigated the existence results by assuming two cases where the set-valued map has convex or nonconvex values of (3) in the frame of power law with generalized kernel. We employed some nonlinear analysis techniques. Along with the use of generalized fractional operators, we established a nonlinear alternative for contractive maps in the case of the convex multifunctions and the Nadler-Covitz fixed point theorem in relation to contractions in the case of nonconvex-valued multifunctions. We gave simulative examples to illustrate the theoretical results.
As a future work, we will try to extend the existing FBVP in the present paper to a general structure with the Mittag-Leffler power law [47] and for ψ-Hilfer fractional operator [48].