Stability Results for a Coupled System of Impulsive Fractional Differential Equations

: In this paper, we establish sufﬁcient conditions for the existence, uniqueness and Ulam– Hyers stability of the solutions of a coupled system of nonlinear fractional impulsive differential equations. The existence and uniqueness results are carried out via Banach contraction principle and Schauder’s ﬁxed point theorem. The main theoretical results are well illustrated with the help of an example.

It is well known that the effects of a pulse cannot be ignored in many processes and phenomena. For example, in biological systems such as heart beats, blood flows, mechanical systems with impact, population dynamical systems and so on. Thus, researchers used differential equations with impulses to describe the aforesaid kinds of phenomena. Therefore, many mathematicians studied impulsive FDEs with different boundary conditions; see [32][33][34][35][36][37][38][39][40] and references cited therein.
In fields such as numerical analysis, optimization theory, and nonlinear analysis, we mostly deal with the approximate solutions and hence we need to check how close these solutions are to the actual solutions of the related system. For this purpose, many approaches can be used, but the approach of Ulam-Hyers stability is a simple and easy one. The aforesaid stability was first initiated by Ulam in 1940 and then was confirmed by Hyers in 1941 [41,42]. That's why this stability is known as Ulam-Hyers stability. In 1978 [43], Rassias generalized the Ulam-Hyers stability by considering variables. Thereafter, mathematicians extended the work mentioned above to functional, differential, integrals and FDEs; for more information about the topic, the reader is recommended to [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59].
The remaining article is organized as follows: In Section 2, we give some definitions and lemmas related to fractional calculus. In Section 3, we establish our main results about the existence and uniqueness of solutions for the proposed system (1). In Section 4, we study the Ulam-Hyers stability. In Section 5, we provide an example to support our main results.

Definition 2.
(see [60]) The Riemann-Liouville fractional integral of order α > 0 for a function x : (0, ∞) → R is defined as provided that the right side is pointwise defined on (0, ∞), where Γ is the Euler Gamma function.

Lemma 2.
(see [60]) If α, β > 0, t ∈ J, then, for x(t), we have Lemma 3. (Banach contraction principle, see [59]) If X is real Banach space and W : X −→ X is a contraction mapping, then W has a unique fixed point in X.

Theorem 1.
(Schauder fixed point theorem, see [59]) If ω is a closed bounded convex subset of a Banach space X and W : ω −→ ω is completely continuous, then W has at least one fixed point in ω.
For the sake of convenience, we introduce the Banach space as follows: Let J = [0, 1], J = J/{t 1 , t 2 , . . . , t n }. Define the set by It is easy to verify that X is a Banach space equipped with the norm: Similarly, we can define a set Y = PC(J), which is a Banach space endowed with the defined norm: Furthermore, we define the Banach space Y = X × Y with the norms (x, y) = x 0 + y 0 and (x, y) = max x 0 , y 0 .
, y(t)) satisfy all the equations and boundary value conditions of the system (1).

Main Results
In this section, we use fixed point theorems to prove the existence of solutions to problem (1). According to Lemmas 4 and 5, we define operator W : where . Thus, solving problem (1) is equivalent to obtain a fixed point of the operator W. Next, we have to prove the uniqueness of solutions of problem (1). Theorem 2. Let the following conditions (M 1 ) − (M 3 ) hold, and then the boundary value problem (1) has a unique solution. (M 1 ) : For all t ∈ J and x j , y j ∈ R (j = 1, 2) there exists some positive constants µ j , µ j (j = 1, 2) such that (M 3 ) : Proof. By using the Banach contraction principle, we can prove that W, defined by (17), has a fixed point. Before proving the main result first, we will prove the contraction. When t ∈ J, from (17) and When t ∈ t m , t m+1 , then Utilizing (M 1 ) and (M 2 ) in (19) and taking the maximum, we get In the same fashion, we can obtain and Thus, from (18)- (22) and (M 3 ), we infer that W is a contraction mapping. According to Lemma 3, W has a fixed point (x * (t), y * (t)) ∈ Y , which is unique. Therefore, problem (1) has a unique solution (x * (t), y * (t)). Proof. For the sake of simplicity, let us denote and R υ = max{( 1 + 1, 1 ξ + 1)}. Define the operator W, as in (17), and a closed ball of Banach space Y as follows:

Remark 2. A function (x, y) ∈ Y is a solution of the inequality (29), if and only if there exist functions
h ,ˆ w ∈ Y and a sequence m ,ˆ m , m = 1, 2, . . . , n depending on (x, y), such that Similarly, one can easily state such a remark for the inequality (30).
Proof. Let (x, y) ∈ Y be any solution of the inequality (28) and let (ζ, χ) ∈ Y be the unique solution of the following: By Lemma 2.4, we have Since (x, y) is a solution of the inequality (28) and t ∈ J; hence, by Remark 1, we obtain For t ∈ [0, t 1 ], we have For computational convenience, we use s(t) for the sum of terms which are free of ; then, (34) becomes By utilizing Remark 1, we get Thus, (35) becomes Let Using (36) in (37), we have Utilizing (M 1 ) and (M 2 ), we get After some calculation and rearrangement in (38), we get where In addition, for t ∈ t m , t m+1 , we have For computational convenience, we use s * 1 (t) for the sum of terms which are free of , so we have By utilizing Remark 1, we get For computational convenience, let .
Thus, (40) becomes Using (M 1 ) and (M 2 ), we have After some calculation and rearrangement in (43), we get where .
On the similar fashion, for t ∈ [0, t 1 ], and utilizing (M 1 ) − (M 2 ), we can find where s 2 (t) are those terms which are free of and where In addition, for t ∈ t m , t m+1 , 1 m n, we can get where s * 2 (t) are those terms which are free of and .
In addition, where .

Conclusions
In the above study, we have successfully built up existence theory for the solutions of system (1). The required analysis has been developed with the help of the Banach contraction principle and Schauder fixed point theorem. We found that the fractional order coupled system is additionally complicated and challenging as compared to the single FDEs. We also concluded that, if we increase the order or boundary conditions, then the end result turns into extra accurate. Our results are new and fascinating. Our methods can be used to study the existence of solutions for the high order or multiple-point boundary value systems of a nonlinear coupled system of FDEs. Furthermore, we have presented different kinds of Ulam-Hyers stability results for the solution of the considered system (1). In addition, we have presented our main theoretical results with the help of an example. In the future, this concept can be extended to more applied and complicated problems of applied nature. The obtained results can be used in fields like numerical analysis and managerial sciences including business mathematics and economics, etc.