Boundary Value Problems

, 2010:236560 | Cite as

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

Open Access
Research Article
Part of the following topical collections:
  1. Degenerate and Singular Differential Operators with Applications to Boundary Value Problems

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

In this paper, we consider the following second-order multipoint boundary value problem on the half-line

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

The study of multipoint boundary value problems (BVPs) for second-order differential equations was initiated by Bicadze and Samarskĭ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multi-point BVPs due to their theoretical challenge and great application potential. Many results on the existence of (positive) solutions for multi-point BVPs have been obtained, and for more details the reader is referred to [5, 6, 7, 8, 9, 10] and the references therein. The BVPs on the half-line arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium [11, 12, 13] and have been also widely studied [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. When Open image in new window , BVP (1.1) reduces to the following three-point BVP on the half-line:
where Open image in new window . Lian and Ge [16] only studied the solvability of BVP (1.2) by the Leray-Schauder continuation theorem. When Open image in new window , Open image in new window , and nonlinearity Open image in new window is variable separable, BVP (1.1) reduces to the second order two-point BVP on the half-line

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.

Suppose that Open image in new window , and there exist nonnegative functions Open image in new window with Open image in new window such that

Open image in new window

Lemma 2.2.

has a unique solution

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

Proof.

Integrating the differential equation from Open image in new window to Open image in new window , one has
Then, integrating the above integral equation from Open image in new window to Open image in new window , noticing that Open image in new window and Open image in new window , we have

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

Now, BVP (1.1) is equivalent to
Letting Open image in new window , (2.12) becomes

Remark 2.3.

Open image in new window is the Green function for the following associated homogeneous BVP on the half-line:
It is not difficult to testify that

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.

Proof.
  1. (1)

    First, we show that Open image in new window is well defined.

     
Similarly,
Further,
On the other hand, for any Open image in new window and Open image in new window , by Remark 2.3, we have
Hence, by Open image in new window , the Lebesgue dominated convergence theorem, and the continuity of Open image in new window , for any Open image in new window , we have

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

We can show that Open image in new window . In fact, by (2.23) and (2.24), we obtain
Hence, Open image in new window is well defined.
  1. (2)

    We show that Open image in new window is continuous.

     
we have from the Lebesgue dominated convergence theorem that
Thus, Open image in new window is continuous.
  1. (3)

    We show that Open image in new window is relatively compact.

     
(a) Let Open image in new window be a bounded subset. Then, there exists Open image in new window such that Open image in new window for all Open image in new window . By the similar proof of (2.20) and (2.22), if Open image in new window , one has

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.

Lemma 2.5 (see [28, 29]).

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 .

Lemma 2.6 (see [28, 29]).

Let Open image in new window be a bounded open set in real Banach space Open image in new window , let Open image in new window be a cone of Open image in new window , and let Open image in new window be completely continuous. Suppose that

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 .

Therefore,

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).

Define the cone Open image in new window as follows:

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.

Lemma 2.4 shows that Open image in new window is completely continuous, so we only need to prove Open image in new window . Since Open image in new window , and from Remark 2.3, we have

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:

suppose that Open image in new window and there exist nonnegative functions Open image in new window with Open image in new window such that

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

Proof.

We first show that Open image in new window implies Open image in new window . By (4.4), we have
By Lemma 4.1, Open image in new window is completely continuous. Let Open image in new window . Then, Open image in new window . Set
For any Open image in new window , by (4.5), we have

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).

Next, we show the uniqueness of positive solution for BVP (1.1). We will show that Open image in new window is a contraction. In fact, by (4.4), we have

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.

Consider the following BVP:

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.

Consider the following BVP:
In this case, we have

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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Copyright information

© Lishan Liu et al. 2010

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.

Authors and Affiliations

  1. 1.School of Mathematical SciencesQufu Normal UniversityQufuChina
  2. 2.Department of Mathematics and StatisticsCurtin University of TechnologyPerthAustralia

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