Applied Mathematics
Vol.08 No.02(2017), Article ID:74404,16 pages
10.4236/am.2017.82016
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).
http://creativecommons.org/licenses/by/4.0/



Received: January 6, 2017; Accepted: February 24, 2017; Published: February 27, 2017
ABSTRACT
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.
Keywords:
Fractional Differential Equations, Boundary Value Problem, Coincidence Degree Theory

1. 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
where
is the Caputo fractional differential operator,
, and
is continuous.
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
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
(1)
together with the Neumann boundary conditions,
where
are the standard Caputo fractional derivatives,
Math_19#, 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.
2. 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
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. [14] 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. [6] 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. [6] 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. [6] 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
where
and
Define the operator
by
(2)
where
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
where
;
(H2) there exists a constant
such that for all
either
or
(H3) there exists a constant
such that for every
satisfying
either
or
Then Neumann boundary value problem (1) has at least one solution.
Lemma 3. Let L be defined by (2). Then
(3)
and
(4)
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
and
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
.
Let
,
where
. It follows that
and
, we get
. It is clear that
Thus
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
(5)
Moreover, for
, we have
and
which together with the boundary condition
yields that
(6)
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
By
and
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
is bounded.
Proof. Let
, then
. By (4), we get
and
Then, by integral mean value theorem, there exist constants
such that
and
Then we get
From
, we get
and
. Hence we have
(7)
We obtain
(8)
Similarly, we can show that
(9)
and
(10)
By
, we get
and
Then
and
So,
(12)
Similarly, we have
(12)
and
(13)
Combining (13) with (12), we get
(14)
Combining (14) with (11), we get
Thus, from
and (14), we get
and
From (8), (9) and (10), we have
Hence
is bounded.
Lemma 7. Assume that
holds, then the set
is bounded.
Proof. For
, we have
. Then from
,
and
From
imply that
. Thus, we get
Therefore
is bounded.
Lemma 8. Assume that the first part of
holds, then the set
is bounded.
Proof. For
, we have
and
(15)
(16)
and
(17)
If
, then by
, we get
. If
, then
. For
, we obtain
. Otherwise, if
or
or
, from
, one has
or
or
which contradict to (15) or (16) or (17). Hence,
is bounded.
Remark 1 Suppose the second part of
holds, then the set
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
. By Lemma 6 and Lemma 7, we obtain
(1)
for every
;
(2)
for every
.
Choose
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.
4. Examples
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
(18)
Here
. Moreover,
Now let us compute
from
.
From the above inequality, we get
Also,
Here,
. Finally,
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
(19)
Here
. Moreover,
Now let us compute
from
.
From the above inequality, we get
Also,
Here,
. Similarly,
Here,
. We get,
. Choose
.
Also,
where
and
. Hence all the condi-
tions of Theorem 1 are satisfied. Therefore, boundary value problem (19) has at least one solution.
5. Conclusion
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.
Acknowledgements
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
References
- 1. Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.
- 2. Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon.
- 3. Hilfer, Z. (2000) Appliations of Fractional Calculus in Physics. World Scientific, Singapore. https://doi.org/10.1142/3779
- 4. Metzler, R. and Klafter, J. (2000) Boundary Value Problems for Fractional Diffusion Equations. Physics A, 278, 107-125. https://doi.org/10.1016/S0378-4371(99)00503-8
- 5. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
- 6. Lakshmikantham, V., Leela, S. and Vasundhara Devi, J. (2009) Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge.
- 7. Mainardi, A. (2010) Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London. https://doi.org/10.1142/p614
- 8. Abbas, S., Benchora, M. and N’Guerekata, G.M. (2012) Topics in Fractional Differential Equations. Springer, New York. https://doi.org/10.1007/978-1-4614-4036-9
- 9. Zhou, Y. (2014) Basic Theory of Fractional Differential Equations. World Scientific, Singapore. https://doi.org/10.1142/9069
- 10. Ahmad, B. and Nieto, J.J. (2009) Existence Results for a Coupled System of Nonlinear Fractional Differential Equations with Three-Point Boundary Conditions. Computers and Mathematics with Applications, 58, 1838-1843.
https://doi.org/10.1016/j.camwa.2009.07.091
- 11. Liu, Y., Ahmad, B. and Agarwal, R.P. (2013) Existence of Solutions for a Coupled System of Nonlinear Fractional Differential Equations with Fractional Boundary Conditions on the Half-Line. Advances in Difference Equations, 2013, 46.
https://doi.org/10.1186/1687-1847-2013-46
- 12. Aphithana, A., Ntouyas, S.K. and Tariboon, J. (2015) Existence and Uniqueness of Symmetric Solutions for Fractional Differential Equations with Multi-Point Fractional Integral Conditions. Boundary Value Problems, 2015, 68.
https://doi.org/10.1186/s13661-015-0329-1
- 13. Wang, Y. (2016) Positive Solutions for Fractional Differential Equation Involving the Riemann-Stieltjes Integral Conditions with Two Parameters. Journal of Nonlinear Science and Applications, 9, 5733-5740.
- 14. Mawhin, J. (1993) Topological Degree and Boundary Value Problems for Nonlinear Differential Equations in Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, 1537, 74-142. https://doi.org/10.1007/BFb0085076
- 15. Kosmatov, N. (2010) A Boundary Value Problem of Fractional Order at Resonance. Electronic Journal of Differential Equations, 135, 1-10.
- 16. Bai, Z. and Zhang, Y. (2010) The Existence of Solutions for a Fractional Multi-Point Boundary Value Problem. Computers and Mathematics with Applications, 60, 2364-2372. https://doi.org/10.1016/j.camwa.2010.08.030
- 17. Wang, G., Liu, W., Zhu, S. and Zheng, T. (2011) Existence Results for a Coupled System of Nonlinear Fractional 2m-Point Boundary Value Problems at Resonance. Advances in Difference Equations, 44, 1-17. https://doi.org/10.1155/2011/783726
- 18. Zhang, Y., Bai, Z. and Feng, T. (2011) Existence Results for a Coupled System of Nonlinear Fractional Three-Point Boundary Value Problems at Resonance. Computers and Mathematics with Applications, 61, 1032-1047.
https://doi.org/10.1016/j.camwa.2010.12.053
- 19. Jiang, W. (2012) Solvability for a Coupled System of Fractional Differential Equations at Resonance. Nonlinear Analysis, 13, 2285-2292.
https://doi.org/10.1016/j.nonrwa.2012.01.023
- 20. Hu, Z., Liu, W. and Rui, W. (2012) Existence of Solutions for a Coupled System of Fractional Differential Equations. Springer, Berlin, 1-15.
https://doi.org/10.1186/1687-2770-2012-98
- 21. Hu, L. (2016) On the Existence of Positive Solutions for Fractional Differential Inclusions at Resonance. SpringerPlus, 5, 957.
https://doi.org/10.1186/s40064-016-2665-8
- 22. Hu, Z., Liu, W. and Chen, T. (2011) Two-Point Boundary Value Problems for Fractional Differential Equations at Resonance. Bulletin of the Malaysian Mathematical Society Series, 3, 747-755.
- 23. Hu, Z., Liu, W. and Chen, T. (2012) Existence of Solutions for a Coupled System of Fractional Differential Equations at Resonance. Boundary Value Problems, 2012, 98. https://doi.org/10.1186/1687-2770-2012-98