# Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

- Lishan Liu
^{1, 2}Email author, - Xinan Hao
^{1}and - Yonghong Wu
^{2}

**2010**:236560

**DOI: **10.1155/2010/236560

© Lishan Liu et al. 2010

**Received: **14 May 2010

**Accepted: **11 October 2010

**Published: **18 October 2010

## Abstract

This paper investigates the second-order multipoint boundary value problem on the half-line , , , , , where , , , , and is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

## 1. Introduction

where , and is continuous, in which .

Yan et al. [17] established the results of existence and multiplicity of positive solutions to the BVP (1.3) by using lower and upper solutions technique.

Motivated by the above works, we will study the existence results of unbounded (positive) solution for second order multi-point BVP (1.1). Our main features are as follows. Firstly, BVP (1.1) depends on derivative, and the boundary conditions are more general. Secondly, we will study multi-point BVP on infinite intervals. Thirdly, we will obtain the unbounded (positive) solution to BVP (1.1). Obviously, with the boundary condition in (1.1), if the solution exists, it is unbounded. Hence, we extend and generalize the results of [16, 17] to some degree. The main tools used in this paper are Leray-Schauder nonlinear alternative and the fixed point index theory.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, the existence of unbounded solution is established. In Section 4, the existence and uniqueness of positive solution are obtained. Finally, we formulate two examples to illustrate the main results.

## 2. Preliminaries and Lemmas

then is a Banach space with the norm (see [17]).

The Arzela-Ascoli theorem fails to work in the Banach space due to the fact that the infinite interval is noncompact. The following compactness criterion will help us to resolve this problem.

Lemma 2.1 (see [17]).

Let . Then, is relatively compact in if the following conditions hold:

(a) is bounded in ;

(b) the functions belonging to and are locally equicontinuous on ;

(c) the functions from and are equiconvergent, at .

Throughout the paper we assume the following.

Lemma 2.2.

in which , and for .

Proof.

By using arguments similar to those used to prove Lemma 2.2 in [9], we conclude that (2.7) holds. This completes the proof.

Remark 2.3.

Let us first give the following result of completely continuous operator.

Lemma 2.4.

Supposing that and hold, then is completely continuous.

- (1)
First, we show that is well defined.

So, for any .

- (2)
We show that is continuous.

- (3)
We show that is relatively compact.

which implies that is uniformly bounded.

Since is arbitrary, then and are locally equicontinuous on .

(c) For , from (2.27), we have

which means that and are equiconvergent at . By Lemma 2.1, is relatively compact.

Therefore, is completely continuous. The proof is complete.

Let be Banach space, be a bounded open subset of , and be a completely continuous operator. Then either there exist such that , or there exists a fixed point .

## 3. Existence Result

In this section, we present the existence of an unbounded solution for BVP (1.1) by using the Leray-Schauder nonlinear alternative.

Theorem 3.1.

Suppose that conditions hold. Then BVP (1.1) has at least one unbounded solution.

Proof.

From Lemmas 2.2 and 2.4, BVP (1.1) has a solution if and only if is a fixed point of in . So, we only need to seek a fixed point of in .

which contradicts . By Lemma 2.5, has a fixed point . Letting , boundary conditions imply that is an unbounded solution of BVP (1.1).

## 4. Existence and Uniqueness of Positive Solution

In this section, we restrict the nonlinearity and discuss the existence and uniqueness of positive solution for BVP (1.1).

Lemma 4.1.

Suppose that and hold. Then, is completely continuous.

Proof.

Therefore, .

Theorem 4.2.

Suppose that conditions and hold and the following condition holds:

Then, BVP (1.1) has a unique unbounded positive solution.

Proof.

Therefore, for all , that is, for any . Then, Lemma 2.6 yields , which implies that has a fixed point . Let . Then, is an unbounded positive solution of BVP (1.1).

So, is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).

## 5. Examples

Example 5.1.

Then, , and it is easy to prove that is satisfied. By direct calculations, we can obtain that . By Theorem 3.1, BVP (5.1) has an unbounded solution.

Example 5.2.

Then, . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.

## Declarations

### Acknowledgments

The authors are grateful to the referees for valuable suggestions and comments. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 10771117) and the Natural Science Foundation of Shandong Province of China (Y2007A23, Y2008A24). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

## Authors’ Affiliations

## References

- Bicadze AV, Samarskiĭ AA:
**Some elementary generalizations of linear elliptic boundary value problems.***Doklady Akademii Nauk SSSR*1969,**185:**739-740.MathSciNetGoogle Scholar - II'in VA, Moiseev EI:
**Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects.***Differential Equations*1987,**23:**803-810.Google Scholar - II'in VA, Moiseev EI:
**Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator.***Differential Equations*1987,**23:**979-987.Google Scholar - Gupta CP:
**Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation.***Journal of Mathematical Analysis and Applications*1992,**168**(2):540-551. 10.1016/0022-247X(92)90179-HMATHMathSciNetView ArticleGoogle Scholar - Feng W, Webb JRL:
**Solvability of****-point boundary value problems with nonlinear growth.***Journal of Mathematical Analysis and Applications*1997,**212**(2):467-480. 10.1006/jmaa.1997.5520MATHMathSciNetView ArticleGoogle Scholar - Sun Y, Liu L:
**Solvability for a nonlinear second-order three-point boundary value problem.***Journal of Mathematical Analysis and Applications*2004,**296**(1):265-275. 10.1016/j.jmaa.2004.04.013MATHMathSciNetView ArticleGoogle Scholar - Sun Y:
**Positive solutions of nonlinear second-order****-point boundary value problem.***Nonlinear Analysis. Theory, Methods & Applications*2005,**61**(7):1283-1294. 10.1016/j.na.2005.01.105MATHMathSciNetView ArticleGoogle Scholar - Zhang X, Liu L, Wu C:
**Nontrivial solution of third-order nonlinear eigenvalue problems.***Applied Mathematics and Computation*2006,**176**(2):714-721. 10.1016/j.amc.2005.10.017MATHMathSciNetView ArticleGoogle Scholar - Cheung W-S, Ren J:
**Positive solution for****-point boundary value problems.***Journal of Mathematical Analysis and Applications*2005,**303**(2):565-575. 10.1016/j.jmaa.2004.08.056MATHMathSciNetView ArticleGoogle Scholar - Eloe PW, Ahmad B:
**Positive solutions of a nonlinear****th order boundary value problem with nonlocal conditions.***Applied Mathematics Letters*2005,**18**(5):521-527. 10.1016/j.aml.2004.05.009MATHMathSciNetView ArticleGoogle Scholar - Atkinson FV, Peletier LA:
**Ground states of**( )**and the Emden-Fowler equation.***Archive for Rational Mechanics and Analysis*1986,**93**(2):103-127. 10.1007/BF00279955MATHMathSciNetView ArticleGoogle Scholar - Erbe L, Schmitt K:
**On radial solutions of some semilinear elliptic equations.***Differential and Integral Equations*1988,**1**(1):71-78.MATHMathSciNetGoogle Scholar - Kawano N, Yanagida E, Yotsutani S:
**Structure theorems for positive radial solutions to**in .*Funkcialaj Ekvacioj*1993,**36**(3):557-579.MATHMathSciNetGoogle Scholar - Meehan M, O'Regan D:
**Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals.***Nonlinear Analysis. Theory, Methods & Applications*1999,**35:**355-387. 10.1016/S0362-546X(97)00719-0MATHMathSciNetView ArticleGoogle Scholar - Tian Y, Ge W:
**Positive solutions for multi-point boundary value problem on the half-line.***Journal of Mathematical Analysis and Applications*2007,**325**(2):1339-1349. 10.1016/j.jmaa.2006.02.075MATHMathSciNetView ArticleGoogle Scholar - Lian H, Ge W:
**Solvability for second-order three-point boundary value problems on a half-line.***Applied Mathematics Letters*2006,**19**(10):1000-1006. 10.1016/j.aml.2005.10.018MATHMathSciNetView ArticleGoogle Scholar - Yan B, O'Regan D, Agarwal RP:
**Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity.***Journal of Computational and Applied Mathematics*2006,**197**(2):365-386. 10.1016/j.cam.2005.11.010MATHMathSciNetView ArticleGoogle Scholar - Lian H, Ge W:
**Existence of positive solutions for Sturm-Liouville boundary value problems on the half-line.***Journal of Mathematical Analysis and Applications*2006,**321**(2):781-792. 10.1016/j.jmaa.2005.09.001MATHMathSciNetView ArticleGoogle Scholar - Baxley JV:
**Existence and uniqueness for nonlinear boundary value problems on infinite intervals.***Journal of Mathematical Analysis and Applications*1990,**147**(1):122-133. 10.1016/0022-247X(90)90388-VMATHMathSciNetView ArticleGoogle Scholar - Zima M:
**On positive solutions of boundary value problems on the half-line.***Journal of Mathematical Analysis and Applications*2001,**259**(1):127-136. 10.1006/jmaa.2000.7399MATHMathSciNetView ArticleGoogle Scholar - Bai C, Fang J:
**On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals.***Journal of Mathematical Analysis and Applications*2003,**282**(2):711-731. 10.1016/S0022-247X(03)00246-4MATHMathSciNetView ArticleGoogle Scholar - Hao Z-C, Liang J, Xiao T-J:
**Positive solutions of operator equations on half-line.***Journal of Mathematical Analysis and Applications*2006,**314**(2):423-435. 10.1016/j.jmaa.2005.04.004MATHMathSciNetView ArticleGoogle Scholar - Wang Y, Liu L, Wu Y:
**Positive solutions of singular boundary value problems on the half-line.***Applied Mathematics and Computation*2008,**197**(2):789-796. 10.1016/j.amc.2007.08.013MATHMathSciNetView ArticleGoogle Scholar - Kang P, Wei Z:
**Multiple positive solutions of multi-point boundary value problems on the half-line.***Applied Mathematics and Computation*2008,**196**(1):402-415. 10.1016/j.amc.2007.06.004MATHMathSciNetView ArticleGoogle Scholar - Zhang X:
**Successive iteration and positive solutions for a second-order multi-point boundary value problem on a half-line.***Computers & Mathematics with Applications*2009,**58**(3):528-535. 10.1016/j.camwa.2009.04.015MATHMathSciNetView ArticleGoogle Scholar - Sun Y, Sun Y, Debnath L:
**On the existence of positive solutions for singular boundary value problems on the half-line.***Applied Mathematics Letters*2009,**22**(5):806-812. 10.1016/j.aml.2008.07.009MATHMathSciNetView ArticleGoogle Scholar - Agarwal PR, O'Regan D:
*Infinite Interval Problems for Differential, Difference and Integral Equation*. Kluwer Academic, Dodrecht, The Netherlands; 2001.View ArticleGoogle Scholar - Guo DJ, Lakshmikantham V:
*Nonlinear problems in abstract cones*.*Volume 5*. Academic Press, New York, NY, USA; 1988:viii+275.MATHGoogle Scholar - Deimling K:
*Nonlinear Functional Analysis*. Springer, Berlin, Germany; 1985:xiv+450.MATHView ArticleGoogle Scholar

## Copyright

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.