Abstract

This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced 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.

1. 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 [25].

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 fractional 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.

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 [1720].

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 is the generalized Caputo derivative w.r.t. the function such that , is linear bounded operator and . 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.

2. Preliminaries

In this part, we recall some definitions and theorems that will be used later. Let be a Banach space and . Now, throughout this paper, let

Let . The fixed point of set-valued map is a point such that . The graph of is defined as

A selection of is a single-valued map such that .

is closed (convex) valued if is closed (convex) for each , and is bounded on bounded sets if is bounded for each , that is, .

Therefore, Ψ is completely continuous if Ψ (W) is relatively compact for each . 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.

Let ; then is Banach with the norm .

Definition 1 [22]. A multivalued function is called L1Carathéodory if(i) is measurable for each ωE,(ii) is u.s.c. for almost all tI,(iii)for each , there exists such thatand almost every .

Definition 2 [23]. Let be the Banach space of all continuous functions with the norm . Therefore, let be the Banach space of all −differentiable maps with .

Definition 3 [24]. For every , define the family of −selection of as Therefore, is a nonempty set.

Lemma 4 [24]. Let be a −Carathéodory-multivalued function and be a continuous linear mapping. Then is a closed graph operator in .

An important role is played by the fixed-point principle to obtain the solvability of various types of operator equations (see, for example, [2529]). We will apply the following fixed-point theorem to obtain the main results.

Theorem 5 [30]. Let be a convex closed subset of , be an open subset of , and is u.s.c. and compact operator. Then, either (1)Ψ has a fixed point in , or(2)there exists and with .

Next, we outline some definitions of the generalized fractional operators [12, 13]. For more details about fractional operators, the readers are also referred to [1, 31].

Definition 6. Let be an increasing function having a derivative such that for all . The left generalized Riemann–Liouville fractional integral of order for some of an integrable function w.r.t. the function is given by [12]

Choosing and replacing in (8), we have the Hadamard fractional integral, given by [32]

Choosing and replacing in (8), we get the classical Riemann–Liouville integral, given by [1]

The left generalized Riemann–Liouville fractional derivative of order for some of an integrable function w.r.t. the function is given by [12]

Choosing and replacing in (11), we have the Hadamard fractional derivative, given by [32]

Choosing and replacing in (11), we get classical Riemann–Liouville derivative, given by [31]

Definition 7. Let for some and be an increasing mapping such that for all . Consider be an integrable function. The left generalized Caputo fractional derivative of order , w.r.t. the function Q is given by [12]

Choosing and replacing in (14), we obtain the classical Caputo fractional derivative. Choosing and replacing in (14), we have the Caputo-Hadamard fractional derivative, given by [32]

Further, the generalized Caputo derivative can be defined via the generalized Riemann–Liouville fractional derivative as [13] where .

The following lemma, which concerns some properties of generalized fractional operators, plays a key role in the sequel.

Lemma 8. [13]. Suppose that , then (1)if , then ,(2)if , then (3)if and , then

3. 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 , there exists such that .

(H2) .

(H3) The function is L1−Carathéodory and has nonempty, convex, and compact values.

(H4) The functions are continuous such that and for all .

(H5) There exists a function and such that , and there exists 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 , then the solution η(t) of the following problem is given by where .

Proof. Applying the operator on both sides of equation (21). Then, by Lemma 8, we get

Therefore, we obtain

Putting in equation (24), we get

Thus, we have

Putting in equation (24), we get

Thus, we have

Hence, we obtain

Substituting equation (29) into equation (24), we obtain the result.

We note that a function is called a solution for system (3) if there exists a map such that a.e. on I and η(t) is given by where .

Now, we establish the solvability of problem (3).

Theorem 10. Suppose that the hypotheses (H1)-(H5) are satisfied; then the system (3) has at least one solution.

Proof. By hypothesis (H3) and Lemma 4, there exists a single-value map . Define the multivalued operator as Then for every and , there exists such that Now, we can obtain the proof in 4 steps.
Step 1. is convex for all . Let ; then there exists such that for every , we get Let , then From (H3), we have takes convex values. Hence, . Thus, is convex.
Step 2. is completely continuous. First, we will prove that is bounded. Let . Define and let . From (H4), we have that for all . Therefore, we get Hence, sends bounded sets to bounded sets in .
Secondly, we will prove sends into equicontinuous sets of . Let such that .
Then, for all and , we have Hence, we get Next, by the Lagrange mean-value classical theorem, we obtain where . As , . Thus, is equicontinuous. From the Arzela-Ascoli theorem, we get is completely continuous.
Step 3. is u.s.c. We only need to show that has a closed graph to be u.s.c. Let and where .We need to show that .Associated with ,there exists such that for all , we have We want to show that there exists such that for each , we get Define the linear continuous operator by Hence, we get Thus, as . From Lemma 4, we can see is a closed graph in and . Since , then satisfies equation (41) for some . Thus, is an u.s.c.
Step 4. There exists an open set such that for some and .
Let and . Then, there exists with such that for all , we have As in the proof of Step 2, we get that Hence, we get From (H5), there exists such that . Let . Thus, there is no such that for . Hence, is u.s.c. From Theorem 5, we deduce that has a fixed point which is a solution of the system (3).

Theorem 11. Assume that (H1)-(H4) hold. In addition, suppose
(H6) there exists a nondecreasing continuous function and such that for all , and there exists such that Then, the system (3) has at least one solution on I.

Proof. Define the multivalued operator as in equation (31) of Theorem 10.
Step 1. is convex for all . By doing the same steps as in the proof of Theorem 10, we can get that is convex for all .
Step 2. is completely continuous. First, we will prove that is bounded. Let . Define and let . From (H4), we have that for all . Therefore, we get Hence, sends bounded sets to bounded sets in .
Secondly, we will prove sends into equicontinuous sets of . Let such that .
Then, for all and , we have Hence, we get In view of continuity of , we have as . Thus, is completely continuous.
Step 3. is u.s.c. As in the proof of Theorem 10, we have that is an u.s.c.
Step 4. There exists an open set such that for some and .
Let and . Then, there exists with such that for all , we have satisfies (44). As in the proof of Step 2, we have that Hence, we get From (H6), there exists such that . Let . Thus, there is no such that for . Hence, is u.s.c. From Theorem 5, we deduce that has a fixed point which is a solution of the system (3).

Putting in (3), we have the following result.

Corollary 12. Assume that (H1)-(H4) hold. In addition, suppose
(H7) there exists a nondecreasing continuous function and such that for all , and there exists such that Then the system (2) has at least one solution on I.

Choosing in system (3) and changing the interval with the interval in (3) and in all conditions (H1)-(H4), we obtain the following Caputo–Hadamard fractional system

The following result is a direct consequence of Theorem 11.

Corollary 13. Assume that (H1)-(H4) hold on . In addition, suppose
(H8) there exists a nondecreasing continuous function and such that for all , and there exists such that

Then, the system (57) has at least one solution on .

4. Applications

In the following examples, we point to how applied the abstract results in particular systems.

Example 14. Consider the following generalized fractional differential inclusion

Let . Here, we get

From given information, we obtain that . Hence, . In addition, it has . Thus, and . Then, there exists a constant such that satisfying the inequality of (H5).

By Theorem 10, we know the system (60) has at least one solution.

Example 15. Consider the following generalized fractional differential inclusion

Let . Here, we get ,

Thus, , , , , , , and . Therefore, . In addition, it has Further, , and . The condition (H6) of Theorem 11 is satisfied with . Consequently, all the hypotheses of Theorem 11 are satisfied. Thus, the problem (62) has at least one solution.

5. 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.