Existence of Solutions of Three-Dimensional Fractional Differential Systems

In this article, we consider the three-dimensional fractional differential system of the form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0


Introduction
Fractional calculus is a very effective tool in the modeling of many phenomena like control of dynamical systems, porous media, electro chemistry, viscoelasticity, electromagnetic and so on.The fractional theory and its applications are mentioned by many papers and monographs, we refer [1]- [9].For nonlinear fractional boundary value problem, many fixed point theorems were applied to investigate the existence of solutions as in references [10] [11] [12] [13].On the other hand, there is another effective approach, Mawhin's coincidence theory, which proves to be very useful for determining the existence of solutions for fractional order differential equations.In recent years, boundary value problems for fractional differential equations at resonance have been studied in many papers (see [14]- [21]).The main motivation for investigating the fractional boundary value problem arises from fractional advection-dispersion equation.
Hu et al. [22] investigated the two-point boundary value problem for fractional differential equations of the following form In [23], Hu et al. extended the above boundary value problem to the existence of solutions for the following coupled system of fractional differential equations of the form It seems that there has been no work done on the boundary value problem of system involving three nonlinear fractional differential equations.Motivated by the above observation, we investigate the following three-dimensional fractional differential system of the form

D u t f t v t v t t D v t f t w t w t t D w t f t u t u t t
together with the Neumann boundary conditions, are the standard Caputo fractional derivatives, 1 , , The main goal of this paper is to establish some new criteria for the existence of solutions of (1).The method is based on Mawhin's coincidence degree theory.
The results in this paper are generalized of the existing ones.

Preliminaries
In this section, we give the definitions of fractional derivatives and integrals and some notations which are useful throughout this paper.There are several kinds of definitions of fractional derivatives and integrals.In this paper, we use the Riemann-Liouville left sided definition on the half-axis R + and the Caputo frac- tional derivative.
Let X and Y be real Banach spaces and let If Ω is an open bounded subset of X, and dom L φ ∩ Ω ≠ , then the map QN Ω is bounded and ( ) ( ) Then the operator equation Lx Nx = has at least one solution in dom L ∩ Ω .Definition 1. [6] The Riemann-Liouville fractional integral of order on the half-axis R + 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 ( ) ( ) ( ) , here n is the smallest integer greater than or equal to α .
In this paper, let us take , where both X and Y are Banach spaces.Define the operators ( ) where , , , N w t f t w t w t ′ = , , .

N u t f t u t u t ′ =
Then Neumann boundary value problem (1) is equivalent to the operator equation

Main Results
In this section, we begin with the following theorem on existence of solutions for Neumann boundary value problem (1).
Then Neumann boundary value problem (1) has at least one solution.
Lemma 3. Let L be defined by (2).Then Ker Ker , Ker , Ker , , , , 0 , 0 , 0 , and Proof.By Lemma 2, ( ) From the boundary conditions, we have , there exists . By using the Lemma 2, we get By the boundary value conditions of (1), we can get that x satisfies ( ) ( ) On the other hand, suppose x Y ∈ and satisfies ( ) ( ) Similarly, we have Let L be defined by (2).Then L is a Fredholm operator of index zero, : are the linear continuous projector operators can be defined as , , , , 0 , 0 , 0 , .
Thus, we can get Hence L is a Fredholm operator of index zero.
From the definitions of P and P K , we will prove that P K is the inverse of , , , , .

Conclusion
We have investigated some existence results for three-dimensional fractional differential system with Neumann boundary condition.By using Mawhin's coincidence degree theory, we established that the given boundary value problem admits at least one solution.We also presented examples to illustrate the main results.
, then N is L-compact on Ω .Proof.By the continuity of f 1 , f 2 and f 3 , we can get ( ) the conditions of Theorem 1 are satisfied.Therefore, boundary value problem (19) has at least one solution.