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In this article, we consider the three-dimensional fractional differential system of the form together with the Neumann boundary conditions, where are the standard Caputo fractional derivatives, . A new result on the existence of solutions for a class of fractional differential system is obtained by using Mawhin’s coincidence degree theory. Suitable examples are given to illustrate the main results.

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 [

Hu et al. [

where

In [

where

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

together with the Neumann boundary conditions,

where

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.

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

Let X and Y be real Banach spaces and let

It follows that

is invertible. Here

If

Lemma 1. [

1.

2.

3.

Then the operator equation

Definition 1. [

provided the right hand side is pointwise defined on

Definition 2. [

where n is the smallest integer greater than or equal to

Lemma 2. [

where

than or equal to

In this paper, let us take

Define the operators

where

and

Define the operator

where

Let the Nemytski operator

where

and

Then Neumann boundary value problem (1) is equivalent to the operator equation

In this section, we begin with the following theorem on existence of solutions for

Neumann boundary value problem (1).

Theorem 1. Let

(H1) there exist nonnegative functions

such that for all

where

(H2) there exists a constant

or

(H3) there exists a constant

or

Then Neumann boundary value problem (1) has at least one solution.

Lemma 3. Let L be defined by (2). Then

and

Proof. By Lemma 2,

From the boundary conditions, we have

For

Then, we have

By the boundary value conditions of (1), we can get that x satisfies

On the other hand, suppose

Similarly, we have

and

Lemma 4. Let L be defined by (2). Then L is a Fredholm operator of index zero,

Further more, the operator

Proof. Clearly,

For

By the definition of

Similarly, we can show that

Let

where

Thus

Hence L is a Fredholm operator of index zero.

From the definitions of P and

Moreover, for

which together with the boundary condition

From (5) and (6), we get

Lemma 5. Assume

Proof. By the continuity of f_{1}, f_{2} and f_{3}, we can get

From the continuity of f_{1}, f_{2} and f_{3}, there exist constants

Furthermore, for

By

and

Similarly, we can show that

Since

Lemma 6. Assume that

is bounded.

Proof. Let

and

Then, by integral mean value theorem, there exist constants

From

We obtain

Similarly, we can show that

and

By

and

Then

and

So,

Similarly, we have

and

Combining (13) with (12), we get

Combining (14) with (11), we get

Thus, from

and

From (8), (9) and (10), we have

Hence

Lemma 7. Assume that

is bounded.

Proof. For

and

From

Therefore

Lemma 8. Assume that the first part of

is bounded.

Proof. For

and

If

or

or

which contradict to (15) or (16) or (17). Hence,

Remark 1 Suppose the second part of

is bounded.

Proof of the Theorem 1: Set

From the Lemma 4 and Lemma 5 we can get L is a Fredholm operator of index zero and N is L-compact on

(1)

(2)

Choose

By Lemma 8 (or Remark 1), we get

Thus, the condition (3) of Lemma 1 is satisfied. By Lemma 1, we obtain

In this section, we give two examples to illustrate our main results.

Example 1. Consider the following Neumann boundary value problem of fractional differential equation of the form

Here

Now let us compute

From the above inequality, we get

Here,

We get,

where

tions of Theorem 1 are satisfied. Hence, boundary value problem (18) has at least one solution.

Example 2. Consider the Neumann boundary value problem of fractional differential equation of the following form

Here

Now let us compute

From the above inequality, we get

Here,

Here,

Also,

where

tions of Theorem 1 are satisfied. Therefore, boundary value problem (19) has at least one solution.

We have investigated some existence results for three-dimensional fractional differential system with Neumann boundary condition. By using Mawhin’s coin- cidence degree theory, we established that the given boundary value problem admits at least one solution. We also presented examples to illustrate the main results.

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the manuscript.

Sadhasivam, V., Kavitha, J. and Deepa, M. (2017) Existence of Solutions of Three-Dimensional Fractional Differential Systems. Applied Mathematics, 8, 193-208. https://doi.org/10.4236/am.2017.82016