Existence of Solutions to Nonlinear Langevin Equation Involving Two Fractional Orders with Boundary Value Conditions

Abstract

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:

(1.1)

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

(2.1)

where denotes the integer part of the real number .

Definition 2.2.

The Riemann-Liouville fractional integral of order of a function , , is defined as

(2.2)

provided that the right side is pointwise defined on .

Definition 2.3.

The Riemann-Liouville fractional derivative of order of a continuous function is given by

(2.3)

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

(2.4)

where , , .

Lemma 2.5 (see [8]).

Let , then

(2.5)

for some , , .

Lemma 2.6.

The unique solution of the following boundary value problem

(2.6)

is given by

(2.7)

Proof.

Similar to the discussion of [9, equation (1.5)], the general solution of

(2.8)

can be written as

(2.9)

By the boundary conditions and , we obtain

(2.10)

Hence,

(2.11)

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:

(2.12)

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 real-valued function for some such that

(3.1)

If

(3.2)

where , , , , then problem (1.1) has a unique solution.

Proof.

Define an operator by

(3.3)

Let and choose

(3.4)

where is such that .

Now we show that , where . For , by Hölder inequality, we have

(3.5)

Take notice of Beta functions:

(3.6)

We can get

(3.7)

Therefore, .

For and for each , based on Hölder inequality, we obtain

(3.8)

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

(3.9)

If

(3.10)

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 real-valued function such that

(3.11)

Then the problem (1.1) has at least one solution on if

(3.12)

Proof.

Let us fix

(3.13)

here, ; consider , then is a closed, bounded, and convex subset of Banach space . We define the operators and on as

(3.14)

For , based on Hölder inequality, we find that

(3.15)

Thus, , so .

For and for each , by the analogous argument to the proof of Theorem 3.1, we obtain

(3.16)

From the assumption

(3.17)

it follows that is a contraction mapping.

The continuity of implies that the operator is continuous. Also, is uniformly bounded on as

(3.18)

On the other hand, let , for all , setting

(3.19)

For each , we will prove that if and , then

(3.20)

In fact, we have

(3.21)

In the following, the proof is divided into two cases.

Case 1.

For , we have

(3.22)

Case 2.

for , , we have.

(3.23)

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 .

Corollary 3.4.

Suppose that the condition (H1)′ hold and, assume that

(3.24)

Further assume that

(H2)′ there exists a constant such that

(3.25)

then problem (1.1) has at least one solution on .

4. Example

Let , , , . We consider the following boundary value problem

(4.1)

where

(4.2)

Because of , let , then , we have and . Further,

(4.3)

Then BVP (4.1) has a unique solution on according to Theorem 3.1.

On the other hand, we find that

(4.4)

Then BVP (4.1) has at least one solution on according to Theorem 3.3.