Application of a tripled ﬁxed point theorem to investigate a nonlinear system of fractional order hybrid sequential integro-di ﬀ erential equations

: The goal of this manuscript is to study the existence theory of solution for a nonlinear boundary value problem of tripled system of fractional order hybrid sequential integro-di ﬀ erential equations. The analysis depends on some results from fractional calculus and ﬁxed point theory. As a result, we generalized Darbo’s ﬁxed point theorem to form an updated version of tripled ﬁxed point theorem to investigate the proposed system. Also, Hyres-Ulam and generalized Hyres-Ulam stabilities results are established for the considered system. For the illustration of our main results, we provide an example.


Introduction
Measure of non-compactness (MNC, in short) was initially introduced as a foremost tool to prove generalization of the cantor intersection theorem by Kuratowski [1] in 1930 . In functional analysis, MNC is a function which associates a number to a non-empty and bounded subset of metric spaces in such a way that a compact set gets measure zero and all non-compact sets have measure greater than zero. The MNC depends on the sets that to what extent they are apart from compactness. Sets which are far-away from compactness will have a greater non-zero value of MNC (see [2]). Darbo in 1955 continued the use of Kuratowski MNC for further analysis (see details in [1]). He presented a fixed point theorem (FPT) based on MNC which now in literature is a prominent result known by Darbo's fixed point theorem (DFPT). DFPT is very useful tool in fixed point theory which is obtained by generalizing Schauder's FPT. Also from Banach FPT, DFPT appends on the existence part, that is contraction of condensing operators. An operator which for any set produces such images which are themselves more compact than the considered set is known as condensing or densifying operator. In more broad sense, the properties of condensing operators are similar to the compact operators. MNC has wide range of applications in theory of fixed points and is helpful in investigation of integral, integro-differential, differential and other operator equations. Also the mentioned concept has been used very well to study integral, integro-differential and differential equations of fractional orders in Banach spaces (see [3]).
In last few decades, many researchers obtained various existence results for above mentioned equations by using MNC and other methods (we refer few papers as [4][5][6][7][8][9][10][11][12][13]). The authors [4] have examined boundary value problem (BVP) of hybrid sequential fractional integro-differential equation for existence of solution by using Krasnoselskii's FPT. Deep et al. [14] extended Darbo's FPT by using MNC in a Banach space. They used the extended result for the existence of solution to a tripled system of nonlinear equations containing triple integrals. Karakaya et al. [15] investigated the existence result by using MNC for tripled fixed point problem of a class of densifying operators. For the purpose of application they applied the existence result to a tripled system of differential equations.
On the other hand stability theory is an important aspect of the qualitative analysis. It is important from numerical and optimization point of view. In the existence literature, various concepts for stability theory has been used. Here we recall some important concept like Laypunove, exponential, Mittag-Leffler type stabilities which have been studied for various problems in FDEs. An important concept of stability which was introduced by Ulam in 1940 and explained by Hyers in 1941 for functional equations known as Hyers-Ulam (HU) stability (see [29]). The mentioned concept has been extended very well for various problems in fractional calculus. Sufficient conditions have been established for HU type stability by using the tools of nonlinear functional analysis. Here we refer few remarkable work as [30][31][32][33]. Further, the aforementioned concepts of stability has been extended to coupled hybrid FDEs. We refer some useful work as [34][35][36][37][38]. Inspired from the above discussion, we establish necessary and sufficient conditions for the existence HU and generalized (GHU) stability results for the proposed system. For this need, we utilize, some fundamental concept from nonlinear functional analysis.

Elementary definitions and results
In this portion, we present some fundamental definitions, results and properties for Caputo fractional derivative, Riemann-Liouville fractional integrals and results of MNC [1,2,9,14,15,18,19,27,28], which build up background knowledge for bringing forth the main results.
Definition 2.1. The Riemann-Liouville integral of fractional order δ > 0 of a function µ ∈ L 1 ([a, b], R + ) is defined by provided that the integral on right side converges.
Definition 2.2. The Caputo fractional derivative with order δ > 0 of a function µ ∈ C[a, b] is defined by which is known as semi-group property.
Throughout this paper, we assume B to be a Banach space, β B be the family of bounded subsets of B,B be closure of B and conv(B) be closed convex hull of B, then we proceed to the following results.    The following theorem is an important generalization of Darbo's fixed point theorem 2.8 proved by Aghajani et al. [6]. where Ψ : R + → R + is an upper semi-continuous, non-decreasing function and ∀y ∈ R + , Ψ(y) < y. Then T has at least one fixed point solution.

Main results
In this section, first we derive extension of the required theorem for existence theory of solution to the proposed problem. We prove our main result by using Theorem 2.15. Secondly, we evaluate an equivalent integral form of the system of HFSIDEs (1.1) with boundary conditions. The last aim of the section is to investigate existence of solution of system of HFSIDEs (1.1) by using our newly obtained fixed point theorem.
where for µ ∈ M and > 0, W r (µ, ) is the modulus of continuity of µ on the closed compact interval [0, r]. Now by Eq (3.1), we have for which if we take the supremum and use the fact Ψ i is non-decreasing, we get Since µ, υ and ω are arbitrary and Ψ i is non-decreasing, so Moreover, M 1 , M 2 and M 3 are subspaces of M, therefore lim sup Now using the last inequality with (3.3), then by Eq (3.2) we have Hence Theorem 2.15 is satisfied, consequently T i has a tripled fixed point solution.
Then for the BVP of HFSIDE An equivalent integral form is where G(y, s) and G j (y, s) are the Green's functions given by

HU and GHU stability
This section is dedicated to the study of HU stability and GHU stability for our proposed tripled system (1.1) of HFSIDEs. We take benefit from the definitions given in [39] to give definitions of HU stability and GHU stability for the desired investigation of stability analysis.
Hence bythe Definition 4.2, the solution of the problem (3.11) is GHU stable. Consequently, the solution of the tripled system (1.1) of HFSIDEs is GHU stable.

Examples
In this section, we present an example for the the tripled system BVP (1.1) of HFSIDEs to test our respective existence and stability results.
Then it is simple to show that hypothesis (H 1 )-(H 4 ) holds. Hence by Theorem 3.3 BVP tripled system (5.1) of HFSIDEs has atleast one solution. Furthermore, by Theorem 4.3 the solution of the system (5.1) is HU stable and by Theorem 4.4, the solution of the system (5.1) is GHU stable.

Conclusions
Varieties of existence and uniqueness results appears in a range from theoretical aspects in the literature of analysis. FDEs appears in mathematical modeling of different process and phenomenon in various fields like blood flow phenomena, electro-dynamics, visco-elasticity and biophysics. Modeling through systems of differential equations is an important class of bio-mathematics, physics, applied chemistry and many more areas. Also the area has been extended recently to FDEs as well. BVPs have many applications in engineering and physical sciences. Therefore, systems of BVP of FDEs have been investigated very well. In this paper, we have established sufficient conditions for the existence theory and HU and GHU stability analysis for the tripled system of HFSIDEs under boundary conditions. Our proposed system (1.1) can easily reduce to a system of fractional order Volterra integro-differential equations. The said equations have been studied in various articles which have lots of applications. For instance authors [40] have presented the model of physical system of fractional order Volterra integro-differential equations which is characterized by Levy jumps. The authors solved the Levy jumps problem by reduction of fractional order Schrodinger equation to fractional order Volterra integro-differential equations with hyper singular kernel. So the tripled system of HFSIDEs studied in this work has many applications for modelling different phenomena. Moreover in this study, as a result of Darbo's fixed point theorem and literature of MNC, we concluded a new fixed point result given as a Theorem 3.1. For the application purpose, we utilized Theorem 3.1 for the existence of solution to the considered tripled system (1.1) of hybrid fractional integro-differential equations. HU and GHU stabilities are also investigated for the problem (1.1). In last section, we presented an example which justify all our acquired results. In future, we can extend the above results for tripled systems of HFSIDEs under non-singular kernel type derivatives.