Qualitative analysis of coupled system of sequential fractional integrodi ﬀ erential equations

: In this article we explore the existence, uniqueness, and stability for a coupled system of sequential fractional integrodi ﬀ erential equations involving ψ –Hilfer fractional derivative with multi– point boundary conditions. For the uniqueness result we use Banach ﬁxed point theorem, and the Leray–Schauder alternative to obtain the existence result. Further, we investigate various kinds of stability such as Hyers–Ulam stability. Examples are provided to verify our results.


Introduction
Fractional calculus is an intransitive branch of applied mathematical techniques that deals with integrals and derivatives of essentially arbitrary orders. Recently, it gained considerable importance and admiration due to its widespread applications in viscoelasticity, biology, fluid dynamics, hydrodynamics, chemistry, control hypotheses, speculation, aerodynamics, information processing system, image processing, etc. [9,10,12]. A significant feature of fractional order systems in differentiation with integer order ones is that fractional derivatives (FDs) and integrals have nonlocal nature that helps to trace the hereditary and memory characteristics of the related materials and processes under investigation [2,11,16,19].
Often, it is quite tough to obtain the appropriate solutions of the fractional differential equations (FDEs). Due to this problem, the qualitative presumptions of differential equations (DEs) play a significant role both in ordinary differential equations (ODEs) and FDEs. For boundary value problems (BVPs) of FDEs, the existence of solutions is a main requirement. Moreover, uniqueness of solutions is a significant factor for the more particular action of solutions. From the last few years, these qualitative properties are investigated with different approaches. The qualitative analysis of DEs represents the behavior of solutions of complicated phenomena. In addition, stability analysis of dynamical system of an integer as well as fractional order is very important in various fields of science and engineering. The concept of Hyers-Ulam (HU)-type stability, that began with the seminar work, from 1941 has gained a lot of attention. As a matter of fact, HU-type stability has been taken up by a number of mathematicians and the study of this area has grown to be one of the central subjects in the mathematical analysis. Stability analysis, particularly stability in terms of Ulam and Rassias, is an essential component of the qualitative theory of DEs, as shown by the prior results [8,22]. This type of stability can be treated with different approaches [15,17,18,[23][24][25][26].
One of the important approaches is the fixed point (FP) approach. FP theory is an important tool in nonlinear analysis. Particularly, obtaining the existence results for a variety of mathematical problems. Although there are many methods to analyze, under suitable conditions, the existence and uniqueness (EU) of solution of numerous problems with initial conditions, boundary conditions, integral boundary conditions, nonlinear boundary conditions and periodic boundary conditions for FDEs [3,4,27,28]. Applications of FP theory in terms of stability analysis of DEs can be found in [14]. There are various definitions of FDs the most popular of Caputo and Riemann-Liouville (RL). A generalization of both Caputo and RL was given by Hilfer [6], known as the Hilfer FD. Some applications and properties of Hilfer derivative can be found in [7]. The FD in the sense of Hilfer, called ψ-Hilfer FD, has been introduced in [21], which unify various fractional operators.
Abbas et al. [1], investigated the existence and attractively of solution for the following problem: is the ψ-Hilfer FD of order β and type ρ, g : R + × R → R * and f : R + × R → R are continuous functions.
Zhou et al.
[29], explored the existence and stability of solution to the following nonlinear ψ-Hilfer fractional integrodifferential equation
To analyze problem (1.2), we transform it to an analogous FP problem and establish the uniqueness of its solutions using Banach's FP theorem, while obtaining the existence result using the Leray-Schauder alternative [5].
if and only if and and it is assumed that Proof. Assume that is a solution of the nonlocal BVP (2.1) on [a, b]. Operating fractional integral on both sides of first equation I β;ψ a + and using Lemma 2.2, we obtain for ζ ∈ [a, b], Hence, using the fact that From the first boundary condition (a) = 0 we can obtain c 2 = 0, since lim ζ→a (ζ − a) γ−2 = ∞. Then we get By a similar way we obtain Solving the system (2.7), we find that Substituting the value of c 1 , d 1 in (2.5)

Existence and uniqueness results
We introduce the space Obviously (H, · ) is a Banach space. Then the product space (H × H, ( , ω) ) is also a Banach space equipped with norm ( , ω) = + ω .

AIMS Mathematics
Volume 7, Issue 5, 8012-8034. and Consequently, which proves that E is bounded. Thus, the operator S, by Lemma 3.1, has at least one fixed point. As a result, there is at least one solution to the BVP (1.2).
The next theorem uses Banach's contraction mapping principle to show the uniqueness of the system's solution (1.2).
Using the above estimates, we obtain Hence Similarly, In consequence, it follows that Which shows that SZ r ⊂ Z r .
We prove that the operator S is contraction. For ( 1 , ω 1 ), ( 2 , ω 2 ) ∈ H × H and for any ζ ∈ [a, b], we get and consequently we obtain Similarly. we obtain It follows from above two inequalities (3.12) and (3.13) that As a result, the operator S is a contraction. As a result of Banach's FP theorem, the operator S has a unique FP, which is the unique solution of problem (1.2).

Hyers-Ulam stability
We introduce the HU stability idea for problem (1.2) in this part. The definitions that follow are taken from [23].

Example
Consider the following system