On coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives

Abstract: In this paper, we investigate the existence, uniqueness and stability of coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives. To prove the existence and uniqueness results for afore mentioned system, we use the techniques of Kransnoselskii’s type fixed point theorem. Furthermore, different kinds of Ulam stabilities are discussed along with examples, to demonstrate the validity of main results.


Introduction
Fractional order differential equations are the generalizations of the classical integer order differential equations. The idea about the fractional order derivative was introduced at the end of the sixteenth century (1695) when Leibniz used the notation d n dσ n for n th order derivative. By writing a letter to him, L'Hospital asked the question: what would be the result if n = 1 2 ? Leibniz answered in such words, "An apparent Paradox, from which one day useful consequences will be drawn", and this question became the foundation of fractional calculus. Fractional calculus has become a speedily developing area and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine and engineering to biological sciences. Progressively, fractional differential equations play a very important role in fields such as thermodynamics, statistical physics, nonlinear oscillation of earthquakes, viscoelasticity, defence, optics, control, signal processing, electrical circuits, astronomy etc. There are some outstanding articles which provide the main theoretical tools for the qualitative analysis of this research field, and at the same time, shows the interconnection as well as the distinction between integral models, classical and fractional differential equations, see [14,16,18,19,22,25,26,28,30] Impulsive fractional differential equations are used to describe both physical, social sciences and many dynamical systems such as evolution processes pharmacotherapy. There are two types of impulsive fractional differential equations the first one is instantaneous impulsive fractional differential equations while the other one is non-instantaneous impulsive fractional differential equations. In last few decades, the theory of impulsive fractional differential equations are well utilized in medicine, mechanical engineering, ecology, biology and astronomy etc. There are some remarkable monographs [3,6,8,15,20,23,33,34], considering fractional differential equations with impulses.
The most preferable research area in the field of fractional differential equations (F DE s), which received great attention from the researchers is the theory regarding the existence of solutions. Many researchers developed some interesting results about the existence of solutions of different boundary value problems (BVPs) using different fixed point theorems. For details we refer the reader to [2, 7, 9-11, 13, 27]. Most of the time, it is quite intricate to find the exact solutions of nonlinear differential equations, in such a situation different approximation techniques are introduced. The difference between exact and approximate solutions is nowadays dealt with using Hyers-Ulam (HU) type stabilities, which were first introduced in 1940 by Ulam [29] and then answered by Hyers in the following year in the context of Banach spaces. Many researchers investigated HU type stabilities for different problems with different approaches [12, 17, 31, 35-37, 39, 40].
Wang et al. [32], presented stability of the following coupled system of implicit fractional integro-differential equations having anti-periodic boundary conditions: Motivated by the above work, we focus our attention on the following coupled impulsive fractional integro-differential equations with Riemann-Liouville derivatives of the form: , v(σ − k ) are the right limits and left limits respectively, E j , E * j , E k , E * k : R → R are continuous functions, and D α , I α are the α-order Riemann-Liouville fractional derivative and integral operators respectively.
The remaining article is arranged as follows: In Section 2, we present some basic definitions, theorems, and lemmas that will be used in our main results. In Section 3, we use suitable cases for the existence and uniqueness of solution for the proposed system (1.1) using Kransnoselskii's type fixed point theorem. In Section 4, we discuss different kinds of stabilities in the sense of Ulam under certain conditions. In Section 5, an example is given to support the main results.

Auxiliary results
In this section, we present some basics notations, definitions, and results that are used in the whole article.
Let T > 0, ω = [0, T]. The Banach space of all continuous functions from ω into R is denoted by C(ω, R) with the norm u = sup {|u(σ)| : σ ∈ ω} and the product of these spaces is also a Banach space with the norm The piecewise continuous functions with 1 < α, β ≤ 2 are denoted as follows: respectively. Their product ϑ = ϑ 1 × ϑ 2 is also a Banach space with the norm (u, v) ϑ = u ϑ 1 + v ϑ 2 .
[1] The Riemann-Liouville fractional integral of order α > 0 for a function u : R + → R is defined as provided that integral on the right side exists.
[α] denotes the integer part of the real number α. For more properties, the reader may refer to [1].

Ulam's Stabilities and Remarks
The following definitions and remarks are taken from [21,24].

Existence and uniqueness
In this section, we discuss the existence and uniqueness of solution of the proposed system (1.1).
Here we use Kransnoselskii's fixed point theorem to show that the operator 1 + 2 has at least one fixed point. Therefore, we choose a closed ball . Theorem 3.2. If hypotheses (H 1 )-(H 4 ) are hold, then the given system (1.1) has at least one solution.
In the same way, we have * * Thus, 2 is equicontinuous. So 2 is relatively compact on ϑ r . Hence, by the Arzelà-Ascoli Theorem, 2 is compact on ϑ r . Thus all the condition of Theorem 2.1 are satisfied. So the given system (1.1) has at least one solution.
then the given system (1.1) has unique solution.
In view of Theorem 3.2, we have for z = 1, 2, . . . , q. Hence This implies that the operator ϕ is a contraction. Therefore, (1.1) has a unique solution.

Ulam's stability analysis
In this section, we study different kinds of stabilities, like HU, generalized HU, HUR, and generalized HUR stability of the proposed system.
then the unique solution of the coupled system (1.1) is HU stable and consequently generalized HU stable.
For the next result, we assume the following: (H 5 ) Let there exists two nondecreasing functions w α , w β ∈ C(ω, R + ) such that then the unique solution of the given system (1.1) is HUR stable and accordingly generalized HUR stable.
Proof. With the help of Definitions 2.5 and 2.6, we can achieve our result doing the same steps as in Theorem 4.1.

Conclusion
In this article, we used the Kransnoselskii's fixed point theorem and acquired the necessary cases for the existence and uniqueness of solution for the given fractional integro-differential Eqs (1.1). Furthermore, under specific assumptions and conditions, we proved different kinds of Ulam's stability of system (1.1). The concept of Ulam's stability is very important because it gives a relationship between approximate and exact solutions, so our results may be very helpful in approximation theory and numerical analysis. The mentioned stability is rarely investigated for impulsive fractional integrodifferential equations. Finally, we illustrated the main results by giving a suitable example.