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

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## Abstract

This paper investigates the second-order multipoint boundary value problem on the half-line Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window 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 Open image in new window 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.

### Keywords

Real Banach Space Point Index Nonlinear Elliptic Equation Lebesgue Dominate Convergence Theorem Continuation Theorem## 1. Introduction

where Open image in new window , and Open image in new window is continuous, in which Open image in new window .

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 Open image in new window is a Banach space with the norm Open image in new window (see [17]).

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

Lemma 2.1 (see [17]).

Let Open image in new window . Then, Open image in new window is relatively compact in Open image in new window if the following conditions hold:

(a) Open image in new window is bounded in Open image in new window ;

(b) the functions belonging to Open image in new window and Open image in new window are locally equicontinuous on Open image in new window ;

(c) the functions from Open image in new window and Open image in new window are equiconvergent, at Open image in new window .

Throughout the paper we assume the following.

Lemma 2.2.

in which Open image in new window , and Open image in new window for Open image in new window .

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 Open image in new window and Open image in new window hold, then Open image in new window is completely continuous.

- (1)
First, we show that Open image in new window is well defined.

So, Open image in new window for any Open image in new window .

- (2)
We show that Open image in new window is continuous.

- (3)
We show that Open image in new window is relatively compact.

which implies that Open image in new window is uniformly bounded.

Since Open image in new window is arbitrary, then Open image in new window and Open image in new window are locally equicontinuous on Open image in new window .

(c) For Open image in new window , from (2.27), we have

which means that Open image in new window and Open image in new window are equiconvergent at Open image in new window . By Lemma 2.1, Open image in new window is relatively compact.

Therefore, Open image in new window is completely continuous. The proof is complete.

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

## 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 Open image in new window 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 Open image in new window if and only if Open image in new window is a fixed point of Open image in new window in Open image in new window . So, we only need to seek a fixed point of Open image in new window in Open image in new window .

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

## 4. Existence and Uniqueness of Positive Solution

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

Lemma 4.1.

Suppose that Open image in new window and Open image in new window hold. Then, Open image in new window is completely continuous.

Proof.

Therefore, Open image in new window .

Theorem 4.2.

Suppose that conditions Open image in new window and Open image in new window hold and the following condition holds:

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

Proof.

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

So, Open image in new window 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, Open image in new window , and it is easy to prove that Open image in new window is satisfied. By direct calculations, we can obtain that Open image in new window . By Theorem 3.1, BVP (5.1) has an unbounded solution.

Example 5.2.

Then, Open image in new window . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.

## Notes

### 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.

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