Vol.08 No.02(2017), Article ID:74404,16 pages
Existence of Solutions of Three-Dimensional Fractional Differential Systems
Vadivel Sadhasivam, Jayapal Kavitha, Muthusamy Deepa
Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College (Affli. to Periyar University), Rasipuram, India
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: January 6, 2017; Accepted: February 24, 2017; Published: February 27, 2017
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 Differential Equations, Boundary Value Problem, Coincidence Degree Theory
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  -  . For nonlinear fractional boundary value problem, many fixed point theorems were applied to investigate the existence of solutions as in references     . 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  -  ). The main motivation for investigating the fractional boundary value problem arises from fractional advection-dispersion equation.
Hu et al.  investigated the two-point boundary value problem for fractional differential equations of the following form
where is the Caputo fractional differential operator, , and is continuous.
In  , 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
where are the Caputo fractional derivatives, , and is continuous.
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 are the standard Caputo fractional derivatives, , and is continuous.
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 and the Caputo fractional derivative.
Let X and Y be real Banach spaces and let be a Fred- holm operator with index zero if and ImL is closed in Y and there exist continuous projectors such that
It follows that
is invertible. Here denotes the inverse of.
If is an open bounded subset of X, and, then the map will be called L-compact on, if is bounded and is compact, where I is the identity operator.
Lemma 1.  Let be a Fredholm operator with index zero and be L-compact on. Assume that the following condi- tions are satisfied.
1. for every;
2. for every;
3., where is a projection such that.
Then the operator equation has at least one solution in .
Definition 1.  The Riemann-Liouville fractional integral of order of a function on the half-axis is given by
provided the right hand side is pointwise defined on
Definition 2.  Assume that is -times absolutely continuous function, the Caputo fractional derivative of order of x is given by
where n is the smallest integer greater than or equal to, provided that the right side integral is pointwise defined on.
Lemma 2.  Let and. If, then
where, here n is the smallest integer greater
than or equal to.
In this paper, let us take with the norm and with the norm, where. Then we denote with the norm and with the norm. Clearly, both and are Banach spaces.
Define the operators by
Define the operator by
Let the Nemytski operator be defined as
where is defined by
is defined by
and is defined by
Then Neumann boundary value problem (1) is equivalent to the operator equation
3. Main Results
In this section, we begin with the following theorem on existence of solutions for
Neumann boundary value problem (1).
Theorem 1. Let be continuous. Assume that
(H1) there exist nonnegative functions with
such that for all
(H2) there exists a constant such that for all either
(H3) there exists a constant such that for every satisfying either
Then Neumann boundary value problem (1) has at least one solution.
Lemma 3. Let L be defined by (2). Then
Proof. By Lemma 2, has the solution
From the boundary conditions, we have
For, there exists such that. By using the Lemma 2, we get
Then, we have
By the boundary value conditions of (1), we can get that x satisfies
On the other hand, suppose and satisfies Let then and. Hence,. Then we get
Similarly, we have
Lemma 4. Let L be defined by (2). Then L is a Fredholm operator of index zero, and are the linear continuous projector opera- tors can be defined as
Further more, the operator can be written by
Proof. Clearly, and. It follows that , we have. By using simple calculation, we get that. Then we have
For, we have
By the definition of, we get
Similarly, we can show that and. Thus, we can get .
where . It follows that and, we get. It is clear that
Hence L is a Fredholm operator of index zero.
From the definitions of P and, we will prove that is the inverse of. Infact, for, we have
Moreover, for, we have and
which together with the boundary condition yields that
From (5) and (6), we get.
Lemma 5. Assume is an open bounded subset such that , then N is L-compact on.
Proof. By the continuity of f1, f2 and f3, we can get and are bounded. By the Arzela-Ascoli theorem, we will prove that is equicontinuous.
From the continuity of f1, f2 and f3, there exist constants such that for all.
Furthermore, for, we have
Similarly, we can show that
Since and are uniformly continuous on [0, 1], we have is equicontinuous. Thus is compact.
Lemma 6. Assume that hold, then the set
Proof. Let, then. By (4), we get
Then, by integral mean value theorem, there exist constants such that and Then we get
From, we get and. Hence we have
Similarly, we can show that
By, we get
Similarly, we have
Combining (13) with (12), we get
Combining (14) with (11), we get
Thus, from and (14), we get
From (8), (9) and (10), we have
Hence is bounded.
Lemma 7. Assume that holds, then the set
Proof. For, we have. Then from,
From imply that. Thus, we get
Therefore is bounded.
Lemma 8. Assume that the first part of holds, then the set
Proof. For, we have and
If, then by, we get. If, then . For, we obtain. Otherwise, if or or, from, one has
which contradict to (15) or (16) or (17). Hence, is bounded.
Remark 1 Suppose the second part of holds, then the set
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. By Lemma 6 and Lemma 7, we obtain
(1) for every;
(2) for every.
By Lemma 8 (or Remark 1), we get for. Therefore
Thus, the condition (3) of Lemma 1 is satisfied. By Lemma 1, we obtain has at least one solution in. Hence Neumann boundary value problem (1) has at least one solution. This completes the proof.
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
Now let us compute from.
From the above inequality, we get Also,
We get,. And we get, . Choose. Also,
where and. All the condi-
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
Now let us compute from.
From the above inequality, we get Also,
Here,. We get,. Choose.
where and. Hence all the condi-
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.
Cite this paper
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
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