STABILITY ANALYSIS OF A NONLOCAL FRACTIONAL IMPULSIVE COUPLED EVOLUTION DIFFERENTIAL EQUATION

This work is committed to establish the necessary assumptions related with the existence and uniqueness of solutions to a nonlocal coupled impulsive fractional diﬀerential equation. We attain our main results by the use of Krasnoselskii’s ﬁxed point theorem and Banach contraction principle. Additionally, we create a framework for studying the Hyers–Ulam stability of the considered problem. For the applications of theoretical result, we discuss an example at the end.


Introduction
In the last few decades, the theory of fractional differential equations (FDEs) has become one of the most attractive research area for finding new results. The reason behind this attractiveness is the fact that it precisely describes a large number of nonlinear phenomena in different branches of science and engineering like, viscoelasticity, control hypothesis, speculation, fluid dynamics, hydrodynamics, aerodynamics, information processing, system networking, picture processing etc. It is also a useful instrument for the depiction of memory and inherited properties of many materials and processes. As a result, FDEs theory gained a significant development in recent years, for details we refer the reader to [1,[7][8][9]11,12,15,19,20,22,25,[39][40][41][42].
Qualitative analysis of solutions to dynamical systems is a great tool for analyzing its different behaviors. Among these properties, surety of existence and uniqueness of solutions to the given dynamical systems is a challenging task for mathematicians. The aforementioned properties has been explored well for integer order differential equations (DEs). However, for FDEs there are many aspects that requires further investigations. The literature devoted to the existence and uniqueness of solutions has been marvelously studied by adapting Riemann-Liouville and Caputo factional derivatives, for more details we recommend [2,16,26,44,45].
In the study of dynamical systems, stability analysis is a basic requirement for the applicability of results. In stability theory, especially Ulam's stability, which was first established by Ulam [30], in 1940 and extended by Hyers [10] to DEs plays a pivot role. Many mathematicians further worked on the Hyers result in different directions, as can be seen in [3, 13, 14, 17, 18, 21, 23, 24, 27-29, 33, 34, 38, 46-52].
Many targets have been achieved about stability analysis of integer order DEs, but for FDEs only few monographs are devoted. Recently, Wang et. al in [36] studied Ulam's type stability of different kinds for FDEs. In [37], the authors studied the aforementioned stabilities for: Impulsive FDEs play a significant role in the applied models, for details see [4,32]. As pointed out in [4], the theory of initial and boundary value problems (BVPs) for the nonlinear impulsive FDEs is still in the early stage. In [4], the authors studied the following impulsive hybrid BVPs of FDEs: Motivated by the above mentioned work, in this article our target is to investigate the existence, uniqueness and Hyers-Ulam stability for the following system of FDEs , . . , t m } and χ 1 (·), χ 2 (·) , are linear and bounded operators on R. Furthermore, I k andĨ k are the impulsive operators. The nonlinear func- , ω (t − k ) left limits, respectively. The manuscript is organized as follows: In Section 2, we give essential definitions, lemmas and theorems. In Section 3, we develop suitable conditions for the existence and uniqueness of solution to (1.1), using Krasnoselskii's fixed point theorem and Banach contraction principle. In Section 4, we built up generalized results according to which problem (1.1) satisfies the conditions of Hyers-Ulam stability. In Section 5, we verify our results by discussing a particular example.

Preliminaries
In this part, we assemble some fundamental facts, definitions and lemmas used throughout this article, for detail reader should study [1,15,22].
Definition 2.1 ( [15]). Let δ ∈ R + , then the arbitrary order integral in the Riemann-Liouville sense for a function p : J → R is given as such that the integral on the right side is pointwise defined on R + .
It is to be noted that the integral on the right hand side is pointwise defined on R + .
Definition 2.3 (Urs [31], Definition 2). Consider a Banach space E such that Φ 1 , Φ 2 : E → E be two operators. Then the operator system provided by , is called Hyers-Ulam stable if we can find constants C i=1,2,3,4 > 0 such that for each j=1,2 > 0 and each solution ( z, ω) ∈ E of the inequalities given by there exists a solution (z,ω) ∈ E of system (2.1) which satisfy Definition 2.4. If the matrix H * ∈ C m×m has eigenvalues µ j , for j = 1, 2, . . . , m, then ρ(H * ) (spectral radius) is defined by and if the matrix converges to 0, then the fixed points consequential to the operational system (2.1) are Hyers-Ulam stable.

Existence Results
Before coming to the main result, we follow some restrictions.
(H 1 ). The bounded linear operators χ j,j=1,2 : D(χ j ) → R + are closed and for any where M p , M q , M p,p and M q,q are positive constants.
(H 2 ). The functions φ, ϕ : J×R×R → R are continuous such that ∀ (z, ω), (z,ω) ∈ E and t ∈ J, there exist M φ , M ϕ > 0, satisfying (H 4 ). I k ,Ĩ k : R → R are continuous, and there exist constants l I , lĨ > 0 for any supplemented with the boundary conditions in (1.1) is equivalent to the solution of the following integral equations and Proof. First we consider For t ∈ [0, t 1 ], the use of I α on each side of (3.3), gives Differentiating (3.4) with respect to t, we obtain

Applying the initial conditions, we gain
Now for t ∈ (t 1 , t 2 ] and using I α on (3.3), we get Differentiating (3.6) with respect to t, we have Using the initial conditions, we get Which gives (3.7) Similarly using ∆(z (t 1 )) =Ĩ 1 (z(t 1 )) = z (t Putting (3.7) and (3.8) in (3.6), we get In identical way for any t ∈ (t k , 1), we gain By differentiating with respect to t, we get Substituting the value of b 1 in (3.5), we obtain (3.1). On the same process, we can obtain (3.2). The proof is complete. .

Hyers-Ulam Stability
This section is devoted to the investigate the Hyers-Ulam stability for the solution of (1.1).
Theorem 4.1. Suppose that the hypothesis (H 1 ) to (H 7 ) and Λ < 1 hold along with the condition that the matrix H * is converging to 0. Then the solutions of (1.1) are Hyers-Ulam stable.
Proof. In view of Theorem 3.3, we have (4.1) From (4.1), we obtain the following inequality where Since H * converges to 0, thus (1.1) is Hyers-Ulam stable.

Example
For supporting our theoretical results, we discuss a particular example.

Conclusion
In this manuscript, we exercise the Arzelä-Ascoli theorem, Banach contraction principle and Krasnoselskii's fixed point theorem to attain the necessary criteria for the existence as well as uniqueness of the solution to considered switched coupled impulsive FDEs system given in (1.1). Similarly under particular assumptions and conditions, we have established the Hyers-Ulam stability result of the solution of the considered problem (1.1). From the obtained results, we conclude that such a method is very powerful, effectual and suitable for the solutions of nonlinear FDEs.