- Research Article
- Open Access

# Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations

- Huiqin Lu
^{1}Email author

**Received:**16 October 2010**Accepted:**27 January 2011**Published:**21 February 2011

## Abstract

This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from Our nonlinearity may be singular at and/or .

## Keywords

- Eigenvalue Problem
- Closed Subset
- Fixed Point Theorem
- Global Structure
- Point Index

## 1. Introduction

Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on. Therefore, singular boundary value problems have been investigated extensively in recent years (see [1–4] and references therein).

where is a parameter and satisfies the following hypothesis:

Typical functions that satisfy the above sublinear hypothesis ( ) are those taking the form

where , , , , , , . The hypothesis ( ) is similar to that in [5, 6].

Because of the extensive applications in mechanics and engineering, nonlinear fourth-order two-point boundary value problems have received wide attentions (see [7–12] and references therein). In mechanics, the boundary value problem (1.1) (BVP (1.1) for short) describes the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. The term in represents bending effect which is useful for the stability analysis of the beam. BVP (1.1) has two special features. The first one is that the nonlinearity may depend on the first-order derivative of the unknown function , and the second one is that the nonlinearity may be singular at and/or .

In this paper, we study the existence of positive solutions and the structure of positive solution set for the BVP (1.1). Firstly, we construct a special cone and present a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from . Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory.

By singularity of , we mean that the function in (1.1) is allowed to be unbounded at the points , , , and/or . A function is called a (positive) solution of the BVP (1.1) if it satisfies the BVP (1.1) ( for and for ). For some , if the (1.1) has a positive solution , then is called an eigenvalue and is called corresponding eigenfunction of the BVP (1.1).

These differ from our problem because in (1.4) cannot be singular at , and the nonlinearity in (1.5) does not depend on the derivatives of the unknown functions.

In this paper, we first establish a necessary and sufficient condition for the existence of positive solutions of BVP (1.1) for any by using the following Lemma 1.1. Efforts to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs by the lower and upper solution method can be found, for example, in [5, 6, 21–23]. In [5, 6, 22, 23] they considered the case that depends on even order derivatives of . Although the nonlinearity in [21] depends on the first-order derivative, where the nonlinearity is increasing with respect to the unknown function . Papers [24, 25] derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity does not depend on the derivatives of the unknown functions, and is decreasing with respect to .

Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated (see [26–28] and references therein). Ma and An [26] and Ma and Xu [27] discussed the global structure of positive solutions for the nonlinear eigenvalue problems and obtained the existence of an unbounded connected branch of positive solution set by using global bifurcation theorems (see [29, 30]). The terms in [26] and in [27] are not singular at , , . Yao [14] obtained one or two positive solutions to a singular elastic beam equation rigidly fixed at both ends by using Guo-Krasnosel'skii's fixed point theorem, but the global structure of positive solutions was not considered. Since the nonlinearity in BVP (1.1) may be singular at and/or , the global bifurcation theorems in [29, 30] do not apply to our problem here. In Section 4, we also investigate the global structure of positive solutions for BVP (1.1) by applying the following Lemma 1.2.

The paper is organized as follows: in the rest of this section, two known results are stated. In Section 2, some lemmas are stated and proved. In Section 3, we establish a necessary and sufficient condition for the existence of positive solutions. In Section 4, we prove that the closure of positive solution set possesses an unbounded connected branch which comes from .

Finally we state the following results which will be used in Sections 3 and 4, respectively.

Lemma 1.1 (see [31]).

Let be a real Banach space, let be a cone in , and let , be bounded open sets of , . Suppose that is completely continuous such that one of the following two conditions is satisfied:

Then, has a fixed point in .

Lemma 1.2 (see [32]).

Suppose also that is a family of connected subsets of , satisfying the following conditions:

and for each .

(2)For any two given numbers and with , is a relatively compact set of .

where there exists a sequence such that .

## 2. Some Preliminaries and Lemmas

Lemma 2.1.

, , and have the following properties:

(1) , , , for all .

(2) , , (or ), for all .

(3) , , , for all .

(4) , , , for all .

Proof.

From (2.4), it is easy to obtain the property (2.18).

Consequently, property (2) holds.

From property (2), it is easy to obtain property (3).

We next show that property (4) is true. From (2.4), we know that property (4) holds for .

Therefore, property (4) holds.

Lemma 2.2.

Proof.

Thus, (2.10) holds.

Lemma 2.3.

hold. Then .

Proof.

Thus, is well defined on .

Therefore, follows from (2.21).

Obviously, is a positive solution of BVP (1.1) if and only if is a positive fixed point of the integral operator in .

Lemma 2.4.

Suppose that ( ) and (2.17) hold. Then for any , is completely continuous.

Proof.

First of all, notice that maps into by Lemma 2.3.

Thus, is bounded on .

Now we show that is a compact operator on . By (2.23) and Ascoli-Arzela theorem, it suffices to show that is equicontinuous for arbitrary bounded subset .

From (2.25), (2.26), and the absolute continuity of integral function, it follows that is equicontinuous.

Therefore, is relatively compact, that is, is a compact operator on .

*Lebesgue*dominated convergence theorem, it follows that

Thus, is continuous on . Therefore, is completely continuous.

## 3. A Necessary and Sufficient Condition for Existence of Positive Solutions

In this section, by using the fixed point theorem of cone, we establish the following necessary and sufficient condition for the existence of positive solutions for BVP (1.1).

Theorem 3.1.

Suppose ( ) holds, then BVP (1.1) has at least one positive solution for any if and only if the integral inequality (2.17) holds.

Proof.

Let let and let then (3.1) holds.

This and (3.10) imply that (2.17) holds.

where .

Thus, (3.17) holds.

where .

This implies that (3.20) holds.

By Lemmas 1.1 and 2.4, (3.17), and (3.20), we obtain that has a fixed point in . Therefore, BVP (1.1) has a positive solution in for any .

## 4. Unbounded Connected Branch of Positive Solutions

then, by Theorem 3.1, for any .

Theorem 4.1.

Suppose ( ) and (2.17) hold, then the closure of positive solution set possesses an unbounded connected branch which comes from such that

(i)for any , and

(ii)

Proof.

We now prove our conclusion by the following several steps.

Therefore, has no positive solution in . As a consequence, is bounded.

By the complete continuity of , is compact.

If , then taking .

However, by the excision property and additivity of the fixed point index, we have from (4.12) and (4.14) that , which contradicts (4.15). Hence, there exists some such that the connected branch of containing satisfies that . Let be the connected branch of including , then this satisfies (4.9).

Therefore, Theorem 4.1 holds and the proof is complete.

## Declarations

### Acknowledgments

This work is carried out while the author is visiting the University of New England. The author thanks Professor Yihong Du for his valuable advices and the Department of Mathematics for providing research facilities. The author also thanks the anonymous referees for their carefully reading of the first draft of the manuscript and making many valuable suggestions. Research is supported by the NSFC (10871120) and HESTPSP (J09LA08).

## Authors’ Affiliations

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