Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions
© A. Chen and Y. Chen. 2011
Received: 30 September 2010
Accepted: 26 February 2011
Published: 14 March 2011
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.
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.
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.
In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.
where denotes the integer part of the real number .
provided that the right side is pointwise defined on .
where and denotes the integer part of real number , provided that the right side is pointwise defined on .
Lemma 2.4 (see ).
where , , .
Lemma 2.5 (see ).
for some , , .
Lemma 2.7 (Krasnoselskii s fixed point theorem).
is completely continuous,
is a contraction mapping.
Then there exists such that .
Lemma 2.8 (Hölder inequality).
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 .
where , , , , then problem (1.1) has a unique solution.
where is such that .
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).
then problem (1.1) has a unique solution.
Suppose that (H1) and the following condition hold:
Thus, , so .
it follows that is a contraction mapping.
In the following, the proof is divided into two cases.
Therefore, is equicontinuous and the Arzela-Ascoli 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 .
Further assume that
then problem (1.1) has at least one solution on .
Then BVP (4.1) has a unique solution on according to Theorem 3.1.
Then BVP (4.1) has at least one solution on according to Theorem 3.3.
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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