Open Access

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

Boundary Value Problems20112011:516481

DOI: 10.1155/2011/516481

Received: 30 September 2010

Accepted: 26 February 2011

Published: 14 March 2011

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 [18] 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 [912] and the references therein.

In this paper, we consider the following boundary value problem of Langevin equation with two different fractional orders:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq1_HTML.gif is a positive constant, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq4_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq5_HTML.gif are the Caputo fractional derivatives, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq6_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq7_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq8_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq9_HTML.gif , is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ2_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq10_HTML.gif denotes the integer part of the real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq11_HTML.gif .

Definition 2.2.

The Riemann-Liouville fractional integral of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq12_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq14_HTML.gif , is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ3_HTML.gif
(2.2)

provided that the right side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq15_HTML.gif .

Definition 2.3.

The Riemann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq16_HTML.gif of a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq17_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ4_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq19_HTML.gif denotes the integer part of real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq20_HTML.gif , provided that the right side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq21_HTML.gif .

Lemma 2.4 (see [8]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq22_HTML.gif , then the fractional differential equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq23_HTML.gif has solution
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ5_HTML.gif
(2.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq26_HTML.gif .

Lemma 2.5 (see [8]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq27_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ6_HTML.gif
(2.5)

for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq30_HTML.gif .

Lemma 2.6.

The unique solution of the following boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ7_HTML.gif
(2.6)
is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ8_HTML.gif
(2.7)

Proof.

Similar to the discussion of [9, equation (1.5)], the general solution of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ9_HTML.gif
(2.8)
can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ10_HTML.gif
(2.9)
By the boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq32_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ11_HTML.gif
(2.10)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ12_HTML.gif
(2.11)

Lemma 2.7 (Krasnoselskii https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq33_HTML.gif s fixed point theorem).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq34_HTML.gif be a bounded closed convex subset of a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq35_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq37_HTML.gif be the operators such that
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq38_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq39_HTML.gif ,

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq40_HTML.gif is completely continuous,

     
  3. (iii)

    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq41_HTML.gif is a contraction mapping.

     

Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq42_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq43_HTML.gif .

Lemma 2.8 (Hölder inequality).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq46_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq47_HTML.gif , then the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ13_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq48_HTML.gif be a Banach space of all continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq49_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq50_HTML.gif .

Theorem 3.1.

Assume that

(H1) there exists a real-valued function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq51_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq52_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ14_HTML.gif
(3.1)
If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ15_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq56_HTML.gif , then problem (1.1) has a unique solution.

Proof.

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq57_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ16_HTML.gif
(3.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq58_HTML.gif and choose
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ17_HTML.gif
(3.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq59_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq60_HTML.gif .

Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq61_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq62_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq63_HTML.gif , by Hölder inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ18_HTML.gif
(3.5)
Take notice of Beta functions:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ19_HTML.gif
(3.6)
We can get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ20_HTML.gif
(3.7)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq64_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq65_HTML.gif and for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq66_HTML.gif , based on Hölder inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ21_HTML.gif
(3.8)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq67_HTML.gif , consequently https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq68_HTML.gif is a contraction. As a consequence of Banach fixed point theorem, we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq69_HTML.gif has a fixed point which is a solution of problem (1.1).

Corollary 3.2.

Assume that

(H1)′ There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq70_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ22_HTML.gif
(3.9)
If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ23_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq71_HTML.gif and a real-valued function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq72_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ24_HTML.gif
(3.11)
Then the problem (1.1) has at least one solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq73_HTML.gif if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ25_HTML.gif
(3.12)

Proof.

Let us fix
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ26_HTML.gif
(3.13)
here, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq74_HTML.gif ; consider https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq75_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq76_HTML.gif is a closed, bounded, and convex subset of Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq77_HTML.gif . We define the operators https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq79_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq80_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ27_HTML.gif
(3.14)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq81_HTML.gif , based on Hölder inequality, we find that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ28_HTML.gif
(3.15)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq82_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq83_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq84_HTML.gif and for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq85_HTML.gif , by the analogous argument to the proof of Theorem 3.1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ29_HTML.gif
(3.16)
From the assumption
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ30_HTML.gif
(3.17)

it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq86_HTML.gif is a contraction mapping.

The continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq87_HTML.gif implies that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq88_HTML.gif is continuous. Also, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq89_HTML.gif is uniformly bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq90_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ31_HTML.gif
(3.18)
On the other hand, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq91_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq92_HTML.gif , setting
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ32_HTML.gif
(3.19)
For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq93_HTML.gif , we will prove that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq95_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ33_HTML.gif
(3.20)
In fact, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ34_HTML.gif
(3.21)

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

Case 1.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq96_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ35_HTML.gif
(3.22)

Case 2.

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq98_HTML.gif , we have.
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ36_HTML.gif
(3.23)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq99_HTML.gif is equicontinuous and the Arzela-Ascoli theorem implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq100_HTML.gif is compact on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq101_HTML.gif , so the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq102_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq103_HTML.gif .

Corollary 3.4.

Suppose that the condition (H1)′ hold and, assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ37_HTML.gif
(3.24)

Further assume that

(H2)′ there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq104_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ38_HTML.gif
(3.25)

then problem (1.1) has at least one solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq105_HTML.gif .

4. Example

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq106_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq107_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq108_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq109_HTML.gif . We consider the following boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ39_HTML.gif
(4.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ40_HTML.gif
(4.2)
Because of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq110_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq111_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq112_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq114_HTML.gif . Further,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ41_HTML.gif
(4.3)

Then BVP (4.1) has a unique solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq115_HTML.gif according to Theorem 3.1.

On the other hand, we find that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ42_HTML.gif
(4.4)

Then BVP (4.1) has at least one solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq116_HTML.gif according to Theorem 3.3.

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Xiangnan University
(2)
School of Mathematics and Computational Science, Xiangtan University

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Copyright

© A. Chen and Y. Chen. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.