We study a boundary value problem to Langevin equation involving two fractional orders. The Banach fixed point theorem and Krasnoselskii's fixed point theorem are applied to establish the existence results.
1. Introduction
Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, and so forth. In consequence, the subject of fractional differential equations is gaining much importance and attention. For some recent developments on the subject, see [1–8] and the references therein.
Langevin equation is widely used to describe the evolution of physical phenomena in fluctuating environments. However, for systems in complex media, ordinary Langevin equation does not provide the correct description of the dynamics. One of the possible generalizations of Langevin equation is to replace the ordinary derivative by a fractional derivative in it. This gives rise to fractional Langevin equation, see for instance [9–12] and the references therein.
In this paper, we consider the following boundary value problem of Langevin equation with two different fractional orders:
where is a positive constant, , , , and are the Caputo fractional derivatives, is continuous, and is a real number.
The organization of this paper is as follows. In Section 2, we recall some definitions of fractional integral and derivative and preliminary results which will be used in this paper. In Section 3, we will consider the existence results for problem (1.1). In Section 4, we will give an example to ensure our main results.
2. Preliminaries
In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.
Definition 2.1.
The Caputo fractional derivative of order of a function , is defined as
where denotes the integer part of the real number .
Definition 2.2.
The RiemannLiouville fractional integral of order of a function , , is defined as
provided that the right side is pointwise defined on .
Definition 2.3.
The RiemannLiouville fractional derivative of order of a continuous function is given by
where and denotes the integer part of real number , provided that the right side is pointwise defined on .
Lemma 2.4 (see [8]).
Let , then the fractional differential equation has solution
where , , .
Lemma 2.5 (see [8]).
Let , then
for some , , .
Lemma 2.6.
The unique solution of the following boundary value problem
is given by
Proof.
Similar to the discussion of [9, equation (1.5)], the general solution of
can be written as
By the boundary conditions and , we obtain
Hence,
Lemma 2.7 (Krasnoselskiis fixed point theorem).
Let be a bounded closed convex subset of a Banach space , and let , be the operators such that
(i) whenever ,
(ii) is completely continuous,
(iii) is a contraction mapping.
Then there exists such that .
Lemma 2.8 (Hölder inequality).
Let , , , , then the following inequality holds:
3. Main Result
In this section, our aim is to discuss the existence and uniqueness of solutions to the problem (1.1).
Let be a Banach space of all continuous functions from with the norm .
Theorem 3.1.
Assume that
(H1) there exists a realvalued function for some such that
If
where , , , , then problem (1.1) has a unique solution.
Proof.
Define an operator by
Let and choose
where is such that .
Now we show that , where . For , by Hölder inequality, we have
Take notice of Beta functions:
We can get
Therefore, .
For and for each , based on Hölder inequality, we obtain
Since , consequently is a contraction. As a consequence of Banach fixed point theorem, we deduce that has a fixed point which is a solution of problem (1.1).
Corollary 3.2.
Assume that
(H1)′ There exists a constant such that
If
then problem (1.1) has a unique solution.
Theorem 3.3.
Suppose that (H1) and the following condition hold:
(H2) There exists a constant and a realvalued function such that
Then the problem (1.1) has at least one solution on if
Proof.
Let us fix
here, ; consider , then is a closed, bounded, and convex subset of Banach space . We define the operators and on as
For , based on Hölder inequality, we find that
Thus, , so .
For and for each , by the analogous argument to the proof of Theorem 3.1, we obtain
From the assumption
it follows that is a contraction mapping.
The continuity of implies that the operator is continuous. Also, is uniformly bounded on as
On the other hand, let , for all , setting
For each , we will prove that if and , then
In fact, we have
In the following, the proof is divided into two cases.
Case 1.
For , we have
Case 2.
for , , we have.
Therefore, is equicontinuous and the ArzelaAscoli theorem implies that is compact on , so the operator is completely continuous.
Thus, all the assumptions of Lemma 2.7 are satisfied and the conclusion of Lemma 2.7 implies that the boundary value problem (1.1) has at least one solution on .
Corollary 3.4.
Suppose that the condition (H1)′ hold and, assume that
Further assume that
(H2)′ there exists a constant such that
then problem (1.1) has at least one solution on .
4. Example
Let , , , . We consider the following boundary value problem
where
Because of , let , then , we have and . Further,
Then BVP (4.1) has a unique solution on according to Theorem 3.1.
On the other hand, we find that
Then BVP (4.1) has at least one solution on according to Theorem 3.3.
Acknowledgments
This work was supported by the Natural Science Foundation of China (10971173), the Natural Science Foundation of Hunan Province (10JJ3096), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province.
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