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:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ1_HTML.gif
(1.1)

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

where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq12_HTML.gif of a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq14_HTML.gif , is defined as
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq16_HTML.gif of a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq17_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ4_HTML.gif
(2.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq18_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq21_HTML.gif .

Lemma 2.4 (see [8]).

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

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

Lemma 2.5 (see [8]).

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

for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq29_HTML.gif , http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ7_HTML.gif
(2.6)
is given by
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ9_HTML.gif
(2.8)
can be written as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ10_HTML.gif
(2.9)
By the boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq32_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ11_HTML.gif
(2.10)
Hence,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ12_HTML.gif
(2.11)

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

Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq35_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq36_HTML.gif , http://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)

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

     
  2. (ii)

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

     
  3. (iii)

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

Lemma 2.8 (Hölder inequality).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq47_HTML.gif , then the following inequality holds:
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq49_HTML.gif with the norm http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq51_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq52_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ14_HTML.gif
(3.1)
If
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ15_HTML.gif
(3.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq55_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq57_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ16_HTML.gif
(3.3)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq58_HTML.gif and choose
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ17_HTML.gif
(3.4)

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

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

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

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

Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq67_HTML.gif , consequently http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq70_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ22_HTML.gif
(3.9)
If
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq71_HTML.gif and a real-valued function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq72_HTML.gif such that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq73_HTML.gif if
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ25_HTML.gif
(3.12)

Proof.

Let us fix
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ26_HTML.gif
(3.13)
here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq74_HTML.gif ; consider http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq75_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq77_HTML.gif . We define the operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq79_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq80_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ27_HTML.gif
(3.14)
For http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ28_HTML.gif
(3.15)

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

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq84_HTML.gif and for each http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ29_HTML.gif
(3.16)
From the assumption
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ30_HTML.gif
(3.17)

it follows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq87_HTML.gif implies that the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq88_HTML.gif is continuous. Also, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq89_HTML.gif is uniformly bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq90_HTML.gif as
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq91_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq92_HTML.gif , setting
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ32_HTML.gif
(3.19)
For each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq93_HTML.gif , we will prove that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq95_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ33_HTML.gif
(3.20)
In fact, we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq96_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ35_HTML.gif
(3.22)

Case 2.

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

Therefore, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq100_HTML.gif is compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq101_HTML.gif , so the operator http://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 http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq104_HTML.gif such that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq105_HTML.gif .

4. Example

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq106_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq107_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq108_HTML.gif , http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ39_HTML.gif
(4.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_Equ40_HTML.gif
(4.2)
Because of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq110_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq111_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq112_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F516481/MediaObjects/13661_2010_Article_43_IEq114_HTML.gif . Further,
http://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 http://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
http://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 http://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

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