Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives

: In this research paper, we consider a class of a coupled system of fractional integrodifferential equations in the frame of Hilfer fractional derivatives with respect to another function. The existence and uniqueness results are obtained in weighted spaces by applying Schauder’s and Banach’s ﬁxed point theorems. The results reported here are more general than those found in the literature, and some special cases are presented. Furthermore, we discuss the Ulam–Hyers stability of the solution to the proposed system. Some examples are also constructed to illustrate and validate the main results.


Introduction
Recently, the theory of fractional differential equations (FDEs) has become an active space of exploration. This is because of its accurate outcomes compared with the classical differential equations (DEs). Indeed, fractional calculus has been improving the mathematical modeling of sundry phenomena in science and engineering, for more details, refer to the monographs [1][2][3][4][5]. The fundamental benefit of using fractional-order derivatives (FODs) rather than integer-order derivatives (IODs) is that IODs are local in nature, whereas FODs are global in nature. Numerous physical phenomena cannot be modeled for a single DE. To overcome this challenges, these kinds of phenomena can be given the assistance of coupled systems of DEs. As of late, coupled systems of FDEs have been investigated with various methodologies, see [6][7][8][9][10].
The existence and uniqueness results play a significant part in the theory of FDEs. The previously mentioned region has been investigated well for classical DEs. However, for FDEs, there are many theoretical aspects that need further investigation and exploration. The existence and uniqueness results of FDEs have been very much concentrated up by using Riemann-Liouville (R-L), Caputo, and Hilfer FDs, see [11][12][13][14].
Motivated by the above discussion, we investigate the existence, uniqueness, and H-U stability of the solutions of a coupled system involving aϑ-Hilfer FD of the type: where , and υ a , ω a ∈ R; (ii) D ρ 1 ,ρ 2 a + ,ϑ(κ) represents the ϑ-Hilfer FD of order ρ 1 and type ρ 2 .
We pay attention to the topic of a novel operator with respect to another function, as it covers many fractional systems that are special cases for various values of ϑ. More precisely, the existence, uniqueness, and U-H stability of solutions to the system (1) are obtained in weighted spaces by using standard fixed point theorems (Banach-type and Schauder type) along with Arzelà-Ascoli's theorem.
The content of this paper is organized as follows: Section 2 presents some required results and preliminaries about ϑ-Hilfer FD. Our main results for the system (1) are addressed in Section 3. Some examples to explain the acquired results are given in Section 4. In the end, we epitomize our study in the Conclusion section.

Theorem 2 ([39] (Schauder's Theorem)).
Let Ω be a non-empty closed and convex subset of a Banach space X . If T : Ω → Ω is a continuous such that T (Ω) is a relatively compact subset of X , then T has at least one fixed point in Ω.

Main Results
In this section, we establish the existence, uniqueness, and U-H stability results for the system (1). To obtain our principle results, we consider the following assumptions: Applying the integral I Similarly,

Uniqueness Result
Theorem 5. Assume that (Hy 1 ) holds. If max κ∈J {ζ 1 , ζ 2 } = ζ < 1, then the system (1) has a unique solution on J, where To demonstrate the desired result, we show that Π is a contraction. For each κ ∈ J and (υ, ω), (υ , ω ) ∈ S β , we have Since ζ < 1, Π is a contraction map. Thus, a unique solution exists on J for system (1) in view of Theorem 1, and this completes the proof.

U-H Stability Analysis
In this subsection, we discuss the U-H Stability of the considered system.

Conclusions
Recently, FDEs have attracted the interest of several researchers with prosperous applications, especially those involving generalized fractional operators. It is important that we investigate the fractional systems with generalized Hilfer derivatives since these derivatives cover many systems in the literature and they contain a kernel with different values that generates many special cases. As an additional contribution in this topic, existence, uniqueness, and U-H stability results of a coupled system for a new class of fractional integrodifferential equations in the generalized Hilfer sense are examined. The analysis of obtained results is based on applying Schauder's and Banach's fixed point theorems, and Arzelà-Ascoli's theorem.
It should be noted that in light of our obtained results, our use of the generalized Hilfer operator covers many systems associated with different values of the function ϑ and the parameter ρ 2 , as is the case in the Special Cases section. Acknowledgments: The authors thank the anonymous referees for their careful reading of the manuscript and their insightful comments, which have helped improve the quality of the manuscript. Moreover, the first author would like to thank the Department of Mathematics, College of Science, and the Deanship of Scientific Research at Qassim University for encouraging scientific research and supporting this work.

Conflicts of Interest:
The authors declare no conflict of interest.