Mathematical Modelling and Control

: This paper deals with three-dimensional di ﬀ erential system of nonlinear fractional order problem

where D α 0 + , D β 0 + , D γ 0 + are the standard Caputo fractional derivative, n − 1 < α, β, γ ≤ n, n ≥ 2 and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.
Keywords: fractional differential equation; coincidence degree theory; resonance

Introduction
In the recent years, the glorious developments have been envisaged in the field of fractional differential equations due to their applications being used in various fields such as blood flow phenomena, electro Chemistry of corrosion, industrial robotics, probability and Statistics and so on, refer [1][2][3][4][5][6][7]. In particular, the fractional derivative has been used in lot of physical applications such as propagation of fractional diffusive waves in viscoelastic solids [8], charge transmit-time dispersion amorphous semi-conductor [9] and a non-Markovian diffusion process with memory [10]. Recently, two-point BVP's for fractional differential equations have been studied in some papers (see [11,12]).
The existence of solutions to coupled systems of fractional differential equations has been given in papers [13][14][15][16].
Hu and Zhang [28] investigated the existence, uniqueness of solutions to integer higher-order nonlinear coupled fractional differential equations at resonance by the coincidence degree theory.
Hu [29] discussed the solution of a higher-order coupled system of nonlinear fractional differential equations with infinite-point boundary conditions by coincidence degree theory.
Motivated by the results mentioned above, the two point BVP's of system of higher-order fractional differential equations have been studied by some authors, to the best of our knowledge, no work has been done on the BVP of system involving three-dimensional differential system higher-order fractional differential equations with Caputo fractional derivative. Inspired by the aforementioned studies, in this manuscript, we establish sufficient conditions for the existence of solutions to the nonlinear fractional order three-dimensional differential system with BVP's of the form.
∈ (0, 1), with the boundary conditions, where D α 0 + , D β 0 + and D γ 0 + denote the standard Caputo fractional derivative, n − 1 < α, β, γ ≤ n, n ≥ 2. Boundary value problems being at resonance means that the associated Our main aim of this paper is to establish some new criteria for the existence of solutions of (1.1) and (1. (1.4) by the Mawhin's coincidence degree theory [30]. In Section 4, we illustrate the main result further by providing an example.

Preliminaries
This section starts with a quick review of the fractional calculus concepts that will be used in this work. So let's start with the Riemann-Liouville fractional integrals and derivatives definitions.
provided the right hand side is pointwise defined on R + . is given by where n − 1 < α ≤ n and Γ is the gamma function, such that the integral is pointwise defined on R + . Definition 2.3. [12] Assume that f is (n − 1)-times absolutely continuous function, the Caputo fractional derivative of order α > 0 of f is given by where n is the smallest integer greater than or equal to α, provided that the right side integral is pointwise defined on (0, +∞).

Main results
Our main result is as follows.
Obviously, X and Z are Banach spaces.
Hence, ( x, y, z) ∈ ImL. Then we get Similarly, we get that Then L is a Fredholm operator of index zero, the linear continuous projector operators P : X → X and Q : Z → Z can be defined as (3.5)
This shows that L is a Fredholm operator of index zero.
The proof is similar to Case 6, hence the details are omitted.
The proof is similar to Case 5, hence the details are omitted.
is bounded.
Remark 3.8. Suppose the second part of (H 3 ) holds, then the set is bounded.
Proof of the Theorem 3.1:

Conclusions
To provide sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems in order to ensure that the existence of solutions for the BVP's of fractional differential equation of the form (1.1) and (1.4). By using Mawhin's coincidence degree method we proved that the problem has atleast one solution. This paper provides an example to further illustrate the main result.