Some properties of implicit impulsive coupled system via φ-Hilfer fractional operator

The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system of φ-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as Leray–Schauder alternative and Banach, we obtain the existence and uniqueness of solutions. Further, by mathematical analysis technique we investigate the Ulam–Hyers (UH) and generalized UH (GUH) stability of solutions. Finally, we provide a pertinent example to corroborate the results obtained.


Introduction
Fractional differential equations (FDEs) have attracted the interest of researchers from various disciplines as they are a useful tool in modeling the dynamics of numerous physical systems and have applications in many fields of applied sciences, engineering and technical sciences, and so on. For further details, see [26,36,38,40]. There are various definitions of fractional calculus (FC) used in FDEs for modeling and describing the memory accurately. Among the famous operators of this calculus, there are Riemann-Liouville, Riemann, Grünwald-Letnikov, Caputo, Hilfer, and Hadamard, which are the most used.
In recent years, the impulsive fractional differential equations have become an important and successful tool in modeling some physical phenomena that have sudden changes and have discontinuous jumps by imposing impulsive conditions on the fractional differ-ential equations at discontinuity points. For applications and recent work, we refer the readers to [8,9,12,13,17,27,28,32,44].
On the other side, the study of coupled systems involving FDEs is also important as such systems occur in various problems of applied nature. For some theoretical works on coupled systems of FDEs, we refer to series of papers [11,16,19,20,23,30]. The topic of system stability is one of the most important qualitative characteristics of a solution, but to our knowledge, the results on UH and UHR stability of solutions for implicit impulsive coupled systems are very few in the literature.
represent the right and left limits of ϑ(σ ) at σ ∈ (σ k , σ k+1 ], k = 0, 1, . . . , m. The coupled systems of ϕ-Hilfer FDEs with impulsive conditions considered in this work are a wider class of coupled systems of BVPs that incorporates the BVPs for FDEs involving the most broadly used Riemann-Liouville and Caputo fractional derivatives. Regardless of this, the coupled systems (1.1) for various values of a function ϕ and parameter p include coupled systems of FDEs involving the Hilfer, Hadamard, Katugampola, and many other fractional derivative operators.
• If p = 0, then system (1.1) reduces to an implicit impulsive coupled system with the ϕ-Riemann-Liouville fractional derivative.
The major contribution of this paper is obtaining an equivalent fractional integral equation of the proposed system and establishing the existence, uniqueness, and UH and GUH stability of a solution for an implicit impulsive coupled system with ϕ-Hilfer FD. Our analysis relies on the Banach and Leray-Schauder fixed point theorems. Though we use the standard methodology to obtain our results, its exposition to the proposed system is new. The acquired results obtained in this paper are more general and cover many parallel problems that contain particular cases of functions because our proposed system contains a global fractional derivative that integrates many classic fractional derivatives. Moreover, the results obtained in this work can be extended to n-tuple fractional systems (FSs). Our results include the results of Almalahi et al. [15], Abdo et al. [6], and Kharade et al. [35] and will be a useful contribution to the existing literature on this topic. This paper is organized as follows. In Sect. 2, we render the rudimentary definitions and prove some lemmas and present some concepts of fixed point theorems. In Sect. 3, we prove the existence and uniqueness of solutions for impulsive implicit coupled system (1.1). In Sect. 4, we discuss the stability by means of mathematical analysis techniques. In Sect. 5, we give a pertinent example illustrating our results. Concluding remarks are presented in the last section.

Background material and auxiliary results
In this part, we give important definitions and auxiliary lemmas pertinent to our main results.
Let J := [0, T] and J := (0, T]. Let R = C(J) be the Banach space of continuous functions u : J → R with the norm u = max{|u(σ )| : σ ∈ J}. Clearly, R is a Banach space with this norm, and hence the product space R × R is also a Banach space with the norm (u, ϑ) = u + ϑ .
We define the space PC(J) of piecewise continuous functions u : J → R by Obviously, PC(J) is a Banach space endowed with the norm Definition 2.1 ([36]) Let y > 0 and f ∈ L 1 (J). Then the generalized RL fractional integral of a function f of order y with respect to ϕ is defined as

Definition 2.2 ([41])
Let n -1 < y < n ∈ N, and let f , ϕ ∈ PC n (J). Then the generalized Hilfer fractional derivative of a function f of order y and type 0 ≤ p ≤ 1 with respect to ϕ is defined as 36,41]) Let y, p > 0 and δ > 0. Then
Theorem 2.8 ([29] (Banach fixed point theorem)) Let X be a Banach space, let K ⊂ X be closed, and let : K → K be a strict contraction, that is, (x) -(y) ≤ L xy for some 0 < L < 1 and all x, y ∈ K . Then has a fixed point in K .

Existence of solution
In this section, we consider a general coupled system of Hilfer FDEs (1.1) involving an arbitrary function ϕ. To demonstrate our main results, we introduce the following hypotheses.
(H 1 ) The functions f , g : J × R × R → R are continuous, and there exist constant numbers f , g , f , g > 0 such that for all (u, ϑ), ( u, ϑ) ∈ R × R, (H 2 ) The functions f , g : J × R × R → R are continuous functions such that for each (u, ϑ) ∈ R, there exist nondecreasing continuous linear functions ω f , ω g : (H 3 ) The functions Z k : R → R are continuous, and there exists a constant L Z > 0 such that In the following, we will apply the Theorem 2.7 to obtain an existence result for system (1.1).

Theorem 3.1 Assume that (H 1 )-(H 3 ) hold. If
then problem (1.1) has at least one solution on J.
Proof Define the closed ball set We will prove that the operator defined by (2.7) has a fixed point by using Theorem 2.7. For this, we divide the proof into three steps.
In the following theorem, we prove the uniqueness of solutions to system (1.1) by using Theorem 2.8. , then system (1.1) has a unique solution.
Proof Consider the closed ball B R defined in Theorem 3.1. First, we show that (B R ) ⊂ B R . By the first step in Theorem 3.1 we have (B R ) ⊂ B R . Next, we need to prove that is a contraction map. Indeed, for (u, ϑ), ( u, ϑ) ∈ B R and σ ∈ J, we obtain and, consequently, we obtain By the same way we obtain From (3.7) and (3.8) it follows that Thus the operator is a contraction. So by Theorem 2.8 system (1.1) has a unique solution.

Concluding remarks
We obtained the existence, uniqueness, and UH stability of solutions for a new problem of ϕ-Hilfer FDEs with impulse conditions. Our investigations were based on the reduction of FDEs to FIEs and application the standard Leray-Schauder and Banach fixed point theorems. The acquired results in this paper are more general and cover many of the parallel problems that contain particular cases of the function ϕ, because our proposed system contains a global fractional derivative that integrates many classic fractional derivatives; for instance, for various values of a function ϕ and parameter p, the coupled system (1.1) includes coupled systems of FDEs involving the Hilfer, Hadamard, Katugampola, and many other fractional derivative operators, which are described in the introduction.