ON APPLICATIONS OF COUPLED FIXED-POINT THEOREM IN HYBRID DIFFERENTIAL EQUATIONS OF ARBITRARY ORDER

Currently in most of the research areas the rate of fractional differential equations has been increased due to wide range of application of fractional calculus theory in the problems of real life. In various scientific and engineering disciplines such as physics, mechanics, chemistry these applications can be found. It can be also used in control theory, optimization theory, signal processing, economics etc. [1-5]. Beside this, due to its existence in daily life situations, most of the authors are motivated towards the existence and uniqueness of solutions of fractional differential equations [6,7,9,10,].


INTRODUCTION AND PRELIMINARIES
Currently in most of the research areas the rate of fractional differential equations has been increased due to wide range of application of fractional calculus theory in the problems of real life.In various scientific and engineering disciplines such as physics, mechanics, chemistry these applications can be found.It can be also used in control theory, optimization theory, signal processing, economics etc. [1][2][3][4][5].Beside this, due to its existence in daily life situations, most of the authors are motivated towards the existence and uniqueness of solutions of fractional differential equations [6,7,9,10,].
Using different types of fixed point theorems such as Banach contraction principle, Schaefer fixed point theorem, and Leray-Schauder degree are studied in detail for existence and uniqueness of solutions to multi-points boundary value problems [13,[15][16][17]19,20].
Dhage and Lakshmikantham studied the existence and uniqueness theorems of the solution to the ordinary first-order hybrid differential equation with perturbation of first type  which are Lebegue integrable bounded by a Lebesgue integrable function on J [5].
Ammiet et al. [37], focus on the generalization of (1.1) by replacing the ordinary by fractional derivative in Riemann-Liouville sense.Furthermore, a researcher generalizes (1.2) by replacing the classical differentiation by fractional derivative in the Riemann-Liouville sense [38].
In other research, author discuss the existence of solutions to hybrid fractional differential equations in both types using the Caputo's fractional derivative instead of the classical one in both (1.1) and (1.2) [8].
Recently, existence of solutions to boundary value problems for coupled systems of fractional order differential equations have also attracted some attentions, we refer to [11,12,14,40].
Lately, researcher discussed a two-point boundary value problem for a coupled system of fractional differential equations [12].There also some researcher analyzed the solutions of coupled nonlinear fractional reaction-diffusion equations [40].Motivated with the above works, our purpose in this paper is to prove the existence of solution to the following system of fractional hybrid differential equations of order for )), ( holds almost everywhere on J Lemma 1.5 [28].The fractional differential equation of order has a unique solution of the form? where Lemma 1.6 [28].The following result holds for fractional differential equations Lemma 1.9 Assume that hypothesis )) x be a solution of the Cauchy problem.Since the Riemann- Liouville fractional integral q I is a monotone operator, thus we apply the fractional integral q I on both sides of , then Similarly, in this way in last if assume Similarly, by the same procedure, we get By rearranging the terms, we have  1) has a solution defined on J .
and a subset of X defined as Clearly S is bounded, closed and convex subset of the Banach space X.
Consider the following systems )) Systems (1.12) and (1.13) can be written as , such that By Arzel á -Ascoli theorem, S is compact and continuous operator on  Therefore, the system of integral equations ( 1) has a solution defined on J .

EXAMPLE Example 2.1
Consider the following Hybrid coupled system of FDEs by taking studied the existence and uniqueness theorems of the solution of the ordinary first-order hybrid differential equation with perturbation of second type[36].

3 C
of Theorem 1 has been hold.Thus, all condition of theorem is satisfied hence the operator fixed point on S ~.
and bounded subset of the Banach space X and A     ii.B is completely continuous; iii. .Now, Consider the following assumption (C0) The function ( nonlinear contraction on X with control gs, xt, I  xtds   s 1  t2  s 1 ds   Hence all the conditions of Theorem 1.10 are satisfied, which show that the FHDEs system has a solution in