Journal of Applied Mathematics

Volume 2013, Article ID 721909, 6 pages

http://dx.doi.org/10.1155/2013/721909

## Positive Solutions for Third-Order Boundary-Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

^{1}School of Electrical Engineering, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China^{2}Nanchang Institute of Science and Technology, Nanchang, Jiangxi 330108, China

Received 8 April 2013; Revised 2 June 2013; Accepted 9 June 2013

Academic Editor: Wei-Shih Du

Copyright © 2013 Yanping Guo and Fei Yang. 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.

#### Abstract

By using a fixed point theorem in a cone and the nonlocal third-order BVP's Green function, the existence of at least one positive solution for the third-order boundary-value problem with the integral boundary conditions , , , , and is considered, where is a nonnegative continuous function, , and The emphasis here is that depends on the first-order derivatives.

#### 1. Introduction

Third-order boundary-value problems for differential equation play a very important role in a variety of different areas of applied mathematics and physics. Recently, third-order boundary-value problems have been many scholars' research object. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can produce boundary-value problems with integral boundary conditions [1–3]. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer readers to the books of Corduneanu [4] and Agarwal and O'Regan [5].

Moreover, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary-value problems as special cases. Such kind of BVPs in Banach space has been studied by some researchers [6–8].

By the fixed point index theory in cones [9], Zhang et al. [10] investigated the multiplicity of positive solutions for a class of nonlinear boundary-value problems of fourth-order differential equations with integral boundary conditions in ordered Banach spaces. Feng et al. [11] investigated the existence and multiplicity of positive solutions for a class of nonlinear boundary-value problems of second-order differential equations with integral boundary conditions in ordered Banach spaces. Guo et al. [12] investigated the existence of positive solutions for the third-order boundary-value problems with integral boundary conditions and dependence on the second derivatives. In [13], by using the fixed point theorem of cone expansion and compression of norm type, Zhang and Ge proved the existence and multiplicity of symmetric positive solutions for the fourth-order boundary-value problems with integral boundary conditions. By using Krasnoselskii's fixed point theorem, Wang et al. [14] investigated the existence and nonexistence of positive solutions for a class of fourth-order nonlinear differential equation with integral boundary conditions where the arguments are based on Krasnoselskii's fixed point theorem for operators on a cone.

However, Zhao et al. [15] investigated the following third-order boundary-value problem with integral boundary conditions: under the assumptions, and is the zero element of , , and is nonnegative,where is a cone in the real Banach .

All the above works were done under the assumption that the first-order derivative is not involved explicitly in the nonlinear term . In this paper, we are concerned with the existence of positive solutions for the third-order boundary-value problem with the integral boundary conditions

Throughout, we assume , is continuous, , , and , where .

To show the existence of positive solutions for (3), we define two positive continuous convex functionals. Then, by using the fixed point theorem [16] in a cone and the nonlocal third-order BVP's Green function, we give some new criteria for the existence of positive solutions for (3).

#### 2. Preliminaries

Let be the Banach space equipped with the norm .

Lemma 1 (see [15]). *Suppose holds. Then for any , the problem
**
has a unique solution
**
where
*

Lemma 2 (see [15]). *For , one has .*

*Remark 3. *When , , it is easy to check that .

In addition, for , the maximum of occurs at .

Lemma 4 (see [15]). *Choose and ; then for all , , , one has
**
where .*

*Remark 5. *For , denote . Notice that is concave with respect to ; we have

Lemma 6 (see [15]). *Assume that holds; then *(i)*, , *(ii)*, , ,**where . *

Lemma 7. *If , , then the unique solution of problem (4) satisfies
*

*Proof. *By Lemmas 4 and 6 and (5), we get

For , we have

So,

The proof is completed.

Let be a Banach space and a cone. Suppose , are two continuous convex functionals satisfying , , for , , , for , and for , , , where is a constant.

Theorem 8 (see [16]). *Let , be constants and
**
two bounded open sets in . Set
**Assume is a completely continuous operator satisfying **, ; , ; **, ; ** there is a such that and , for all and . **Then has at least one fixed point in .*

#### 3. Main Results

Let be the Banach space equipped with the norm , and , is a cone in .

Define two continuous convex functionals and , for each ; then and , , for , ; for , , .

In the following, we denote

We will suppose that there are such that satisfies the following growth conditions: , for , , for , , for .

Let We denote

Lemma 9. *Suppose holds. Then is completely continuous. *

*Proof. *For , by Lemmas 2 and 4, it is obviously that .

By Lemma 7, we have
So, we can get .

In the following, we will show that is completely continuous.

At first we show that is continuous.

Let , , it satisfies , , and then there is a constant , such that ; then

By which is uniformly continuous on , we get

Next we show that is compact.

Let be bounded; then there is , such that . For , we have
where .

Consider

It is clear that is a bounded set in , because is uniformly continuous on , for , there exists , such that , and for , , .

For , we have

Therefore is equicontinuous. Using the Arzela-Ascoli theorem, a standard proof yields which is completely continuous.

Theorem 10. *Suppose hold. Then BVP (3) has at least one positive solution satisfying
*

*Proof. *Take
two bounded open sets in , and

By Lemma 9, is completely continuous, and there is a such that and for all and .

By , for and , we get

By Lemma 7, for and , there is , .

So, by , we get

For , we have

So,

By , for , we have

Theorem 8 implies there is such that . So, is a positive solution for BVP (3) satisfying

Thus, Theorem 10 is completed.

#### 4. Example

*Example 1. *Consider the following boundary-value problem
where

In this problem, we know that ; then we can get . Choose ; then .

Furthermore, we obtain

If we take , , and , then we get :
for ,
for ,
for .

Then all the conditions of Theorem 10 are satisfied. Therefore, by Theorem 10 we know that boundary-value problem (33) has at least one positive solution satisfying

#### Acknowledgments

The project is supported by the Natural Science Foundation of China (10971045) and the Natural Science Foundation of Hebei Province (A2013208147).

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