Existence Results for a Nonlocal Coupled System of Sequential Fractional Differential Equations Involving ψ-Hilfer Fractional Derivatives

Department of Mechanical Engineering Technology, College of Industrial Engineering Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Department of Electronics Engineering Technology, College of Industrial Engineering Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand Intelligent and Nonlinear Dynamic Innovations, Department of Mathematics, Faculty of Applied Science, KingMongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand


Introduction
Fractional calculus is an emerging field in applied mathematics that deals with derivatives and integrals of arbitrary orders. One of the most important advantages of fractional order models in comparison with integer order ones is that fractional integrals and derivatives are a powerful tool for the description of memory and hereditary properties of some materials. For details and applications, we refer the reader to the texts [1][2][3][4][5][6]. There are some different definitions of fractional derivatives, from the most popular of Riemann-Liouville and Caputo type fractional derivatives, to the other ones such as Hadamard fractional derivative and the Erdeyl-Kober fractional derivative. A generalization of both Riemann-Liouville and Caputo derivatives was given by Hil-fer in [7], which is known as the Hilfer fractional derivative D α,β xðtÞ of order α and a type β ∈ ½0, 1: Some properties and applications of the Hilfer derivative can be found in [8,9] and references cited therein.
Initial value problems involving Hilfer fractional derivatives were studied by several authors (see, for example, [10][11][12]). Nonlocal boundary value problems for Hilfer fractional differential equation have been discussed in [13,14]. Coupled systems for Hilfer fractional differential equations with nonlocal integral boundary conditions were studied in [15].
The fractional derivative with another function, in the Hilfer sense, called ψ-Hilfer fractional derivative, has been introduced in [16], which unifies several different fractional operators. For some recent results on existence and uniqueness of initial value problems and results on Ulam-Hyers-Rassias stability, see [17][18][19] and references therein. Recently, in [20], the authors extended the results in [13] to ψ-Hilfer nonlocal implicit fractional boundary value problems. For recent results in ψ-Hilfer fractional derivative, we refer to [21][22][23] and references cited therein.
In [24], the authors initiated the study of existence and uniqueness of solutions for a new class of boundary value problems of sequential ψ-Hilfer-type fractional differential equations with multipoint boundary conditions of the form where H D α,β;ψ is the ψ-Hilfer fractional derivative of order α, Existence and uniqueness results were proved by using classical fixed point theorems. The Banach's fixed point theorem was used to obtain the uniqueness result, while nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem are applied to obtain the existence results for the problem (1).
In this paper, we investigate the existence and uniqueness criteria for the solutions of the following nonlocal coupled system of sequential ψ-Hilfer fractional derivative of the form where H D α,β;ψ , H D p,q;ψ are the ψ-Hilfer fractional derivatives of orders α and p, 1 < α, p ≤ 2, and two parameters β, q, 0 ≤ β, q ≤ 1, given constants k, ν, λ i , μ j ∈ ℝ, a ≥ 0, and the points In order to study the problem (2), we convert it into an equivalent fixed point problem and then we use Banach's fixed point theorem to prove the uniqueness of its solutions, while by applying the Leray-Schauder alternative [25], we obtain the existence result.
The remaining part of the article is structured as follows: Section 3 contains the main results for the problem (2). Examples illustrating the existence and uniqueness results are also included. We recall the related background material in Section 2, in which also we establish a lemma regarding a linear variant of the problem (2).

Preliminaries
Here, some notations and definitions of fractional calculus are reminded [1]. Definition 1. The Riemann-Liouville fractional integral of order ς > 0 for a continuous function is defined by provided the right-hand side exists on ða, ∞Þ.
Definition 2. The Riemann-Liouville fractional derivative of order ς > 0 of a continuous function is defined by where n = ½ς + 1 denotes the integer part of real number ς and D = d/dt, provided the right-hand side is point-wise defined on ða, ∞Þ.
Definition 3. The Caputo fractional derivative of order ς > 0 of a continuous function is defined by where the right-hand side is point-wise defined on ða, ∞Þ.
where f ½n−k The following lemma deals with a linear variant of the system (2). Lemma 11. Let γ = α + 2β − αβ, δ = p + 2q − pq, and h, z ∈ Cð½a, b, ℝÞ be given functions. Then, the unique solution of ψ-Hilfer the fractional differential linear system is given by where and it is assumed that Proof. Assume that x is a solution of the nonlocal boundary value problem (15) on ½a, b. Operating fractional integral I α;ψ on both sides of the first equation in (15) and using 3 Advances in Mathematical Physics Lemma 10, we obtain for t ∈ ½a, b, Hence, using the fact that ð1 − βÞð2 − αÞ = 2 − γ, we have where c 1 = H D γ−1,β;ψ xðtÞj t=a and c 2 = I 2−γ;ψ xðtÞj t=a : From the first boundary condition xðaÞ = 0, we can obtain c 2 = 0, since lim t⟶a ðt − aÞ γ−2 = ∞: Then, we get By a similar way, we obtain where d 1 is an arbitrary constant. From the second boundary conditions xðbÞ = ∑ m−2 i=1 λ i yð θ i Þ and yðbÞ = ∑ n−2 j=1 μ j xðζ j Þ, we get the system where Solving the system (24), we find that Substituting the value of c 1 , d 1 in (22) and (23) yields the solution (16) and (17). The converse follows by direct computation. This completes the proof.
Lemma 12 (Leray-Schauder alternative). Let F : E ⟶ E be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let Then, either the set EðFÞ is unbounded, or F has at least one fixed point.
Theorem 13. Assume that f , g : ½a, b × ℝ × ℝ ⟶ ℝ are continuous functions, and there exist real constants p i , q i ≥ 0 , ði = 1, 2Þ and p 0 , q 0 > 0 such that ∀x i , y i ∈ ℝ, ði = 1, 2Þ, If where X i , Y i , F i , G i , i = 1, 2 are given by (30)-(37); then, the system (2) has at least one solution on ½a, b: Proof. The operator S is continuous, by the continuity of functions f and g: We will show that the operator S : W × W ⟶ W × W is completely continuous. Let Z r = fðx, yÞ ∈ W × W : ∥ðx, yÞ∥≤rg be bounded set. Then, there exist positive constants L i , i = 1, 2 such that |f ðt, xðtÞ, yðtÞÞ | ≤ L 1 , |gðt, xðtÞ, yðtÞÞ| ≤ L 2 ,∀ðx, yÞ ∈ Z r : Then, for any ðx, yÞ ∈ Z r , we have which implies that Similarly, it can be shown that From the above inequalities, it follows that the operator S is uniformly bounded, since Next, we show that S is equicontinuous. Let t 1 , t 2 ∈ ½a, b with t 1 < t 2 : Then, we have

Advances in Mathematical Physics
Analogously, we can obtain Therefore, the operator Sðx, yÞ is equicontinuous, and thus, the operator Sðx, yÞ is completely continuous.
Finally, it will be verified that the set E = fðx, yÞ ∈ W × W | ðx, yÞ = λSðx, yÞ, 0 ≤ λ ≤ 1g is bounded. Let ðx, yÞ ∈ E with ðx, yÞ = λSðx, yÞ: For any t ∈ ½a, b, we have Then, Hence, we have which imply that Consequently, which proves that E is bounded. Thus, the operator S, by Lemma 12, has at least one fixed point. Hence, the boundary value problem (2) has at least one solution. The proof is complete.
The uniqueness of solutions of the system (2) is proved in the next theorem, via Banach's contraction mapping principle.
Theorem 14. Assume that f , g : ½a, b × ℝ × ℝ ⟶ ℝ are continuous functions, and there exist positive constants P , Q such that for all t ∈ ½a, b and u i , v i ∈ ℝ, i = 1, 2, we have Then, the system (2) has a unique solution on ½a, b, provided that where X i , Y i , F i , G i , i = 1, 2 are given by (30)-(37).

Conclusion
We investigated the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations involving Hilfer fractional derivative with coupled nonlocal multipoint boundary conditions by applying the framework of fixed point theorems. The existence of a unique solution is obtained via Banach's fixed point theorem, while the existence result is proved by using Leray-Schauder alternative. The results obtained in the present paper are new and significantly contribute to the existing literature on the topic.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.