ON COUPLED SYSTEM OF NONLINEAR HYBRID DIFFERENTIAL EQUATION WITH ARBITRARY ORDER

Nonlinear differential equations are crucial tools in the modeling of nonlinear real phenomena corresponding to a great variety of events, in relation with several fields of the physical sciences and technology [1]. For instance, they appear in the study of the air motion or the fluid dynamics, electricity, electromagnetism or the control of nonlinear process, among others [2]. The resolution of nonlinear differential equations requires, in general, the development of differential techniques in order to deduce the existence and other essential properties of the solutions. There are still many open problems related the solvability of nonlinear system, apart from the fact this is a field where advances are continuously taking place.

satisfy certain conditions.The proof of the existence theorem is based on a coupled fixed-point theorem of Krasnoselskii type, which extends a fixed-point theorem of Burton.Finally, our results are illustrated by a concrete example.

INTRODUCTION
Nonlinear differential equations are crucial tools in the modeling of nonlinear real phenomena corresponding to a great variety of events, in relation with several fields of the physical sciences and technology [1].For instance, they appear in the study of the air motion or the fluid dynamics, electricity, electromagnetism or the control of nonlinear process, among others [2].The resolution of nonlinear differential equations requires, in general, the development of differential techniques in order to deduce the existence and other essential properties of the solutions.There are still many open problems related the solvability of nonlinear system, apart from the fact this is a field where advances are continuously taking place.
Perturbation techniques are useful in the nonlinear analysis for studying the dynamical systems represented by nonlinear differential and integral equations.Evidently, some differential equations representing a certain dynamical system have no analytical solution, so the perturbation of such problems can be helpful.The perturbed differential equations are categorized into various types.An important type of these such perturbations is called a hybrid differential equation (i.e quadratic perturbation of nonlinear differential equation), and the references therein [3].
Recently, the hybrid differential equations have been much more attractive, and then there have been many works on the theory of hybrid differential equations [4][5][6][7].Additionally, hybrid fixed point theory can be used to develop the existence theory for the hybrid equations.We refer to the article [8][9][10][11][12].Dhage and Jadhav discussed the following firstorder hybrid differential equation with linear perturbation of second type: They proved the existence of the maximal and minimal solution for this equation [13].Furthermore, they established some basic results concerning the strict and non-strict differential inequalities.
Indeed, the fractional differential equations have recently been intensively used in modeling of several phenomena and have been studied by many researchers in recent years, therefore they seem to deserve an independent study of their theory parallel to the theory of ordinary differential equations [14][15][16][17][18][19][20][21][22][23].The following some problems using the differential operator in Caputo's sense were studied by some authors for existence of solutions given by . A studied the following two points boundary value problem for fractional differential equations with different boundary conditions . Motivated by the work cited above, in this paper, we study the following hybrid system of fractional differential equations with linear perturbation given by Where D stands for Cupoto fractional derivative of order ,  where satisfy certain conditions.We study existence of at least one solution to the aforesaid problem using coupled fixed-point theorem of Burton type and its extension to receive the required results [24].We also provide a concrete example for the demonstration of main results.

PRELIMINARIES
Here, in this section we give some fundamental definitions and results from fractional calculus and topological degree theory.For further detailed study, we refer to [1,3,4,28].Let such that: i.The map We need some precise definitions of the basic concepts.The following is a discussion of some of the concepts we will need.

Definition
The non-integer order integral of order

Definition
Let  be a positive number such that C Then the Caputo fractional order derivative of f is defined as The following is a fixed-point theorem in Banach spaces due to Burton [1].

Lemma 2.4 [1]
Let S be a nonempty, closed, convex, and bounded subset of a Banach space X and let has a solution in S. Now we recall the definition of a coupled fixed point for a bivariate mapping.Definition 2.5 [26] An element equipped with the supremum norm . Clearly it is a Banach space with respect to point-wise operations and the supremum norm.Now, by applying Theorem 2.6, we study the existence of solution for the FHDEs system (1.1) under the following general assumptions.

(H0)
The function There exists a constant There exist a continuous function As a consequence of Lemma 2.3, we have the following Lemma which is useful in the existence result.Theorem 3.1 [23] Assume that hypothesis Now we are going to prove the following existence theorem for the FHDEs of system (1.1).Theorem 3.2 Assume that hypotheses Matrix Science Mathematic, 1(2) : 11-16.

Then the unique solution of the boundary value problem is given by
) So, the equation and is transformed into the system of operator equations as by hypothesis (H1) we have ) This show that A is a non-linear contraction on X with a control function  S Let n x be a sequence in S converging to a point .

S x 
Then Taking supremum to both hand sides we get

Theorem 2 . 6
By a solution of the FHDEs system, we mean a function of absolutely continuous real-valued functions defined on .J Now, we prove a coupled fixed point theorem which is generalization of Lemma 2.4 of Dhage.Let S be a nonempty, closed, convex and bounded subset of the Banach space X and the FHDEs of system has a solution defined on .J Proof.Set ) , ( R J C X  and a subset S of X defined by Clearly S is a nonempty, convex, closed and bounded subset of the Banach space .article: Sajad Ali Khan, Kamal Shah, Rahmat Ali Khan (2017).On Coupled System Of Nonlinear Hybrid Differential Equa tion With Arbitrary Order.
show that B is compact and continuous operator on .

1
Consider the following coupled system of HFDEs