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

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## 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 Open image in new window Our nonlinearity Open image in new window may be singular at Open image in new window and/or Open image in new window .

### 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, 2, 3, 4] and references therein).

where Open image in new window is a parameter and Open image in new window satisfies the following hypothesis:

Typical functions that satisfy the above sublinear hypothesis ( Open image in new window ) are those taking the form

where 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 , Open image in new window , Open image in new window . The hypothesis ( Open image in new window ) 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, 8, 9, 10, 11, 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 Open image in new window in Open image in new window 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 Open image in new window may depend on the first-order derivative of the unknown function Open image in new window , and the second one is that the nonlinearity Open image in new window may be singular at Open image in new window and/or Open image in new window .

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 Open image in new window . Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory.

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

These differ from our problem because Open image in new window in (1.4) cannot be singular at Open image in new window , Open image in new window and the nonlinearity Open image in new window 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 Open image in new window 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, 22, 23]. In [5, 6, 22, 23] they considered the case that Open image in new window depends on even order derivatives of Open image in new window . Although the nonlinearity Open image in new window in [21] depends on the first-order derivative, where the nonlinearity Open image in new window is increasing with respect to the unknown function Open image in new window . Papers [24, 25] derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity Open image in new window does not depend on the derivatives of the unknown functions, and Open image in new window is decreasing with respect to Open image in new window .

Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated (see [26, 27, 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 Open image in new window in [26] and Open image in new window in [27] are not singular at Open image in new window , Open image in new window , Open image in new window . 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 Open image in new window in BVP (1.1) may be singular at Open image in new window and/or Open image in new window , 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 Open image in new window .

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

Lemma 1.1 (see [31]).

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

Then, Open image in new window has a fixed point in Open image in new window .

Lemma 1.2 (see [32]).

Suppose also that Open image in new window is a family of connected subsets of Open image in new window , satisfying the following conditions:

Open image in new window and Open image in new window for each Open image in new window .

(2)For any two given numbers Open image in new window and Open image in new window with Open image in new window , Open image in new window is a relatively compact set of Open image in new window .

where Open image in new window there exists a sequence Open image in new window such that Open image in new window .

## 2. Some Preliminaries and Lemmas

*Banach*space, where Open image in new window Define

Lemma 2.1.

Open image in new window , Open image in new window , and Open image in new window have the following properties:

(1) Open image in new window , Open image in new window , Open image in new window , for all Open image in new window .

(2) Open image in new window , Open image in new window , Open image in new window (or Open image in new window ), for all Open image in new window .

(3) Open image in new window , Open image in new window , Open image in new window , for all Open image in new window .

(4) Open image in new window , Open image in new window , Open image in new window , for all Open image in new window .

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 Open image in new window .

Therefore, property (4) holds.

Lemma 2.2.

Proof.

Thus, (2.10) holds.

Lemma 2.3.

hold. Then Open image in new window .

Proof.

Thus, Open image in new window is well defined on Open image in new window .

Therefore, Open image in new window follows from (2.21).

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

Lemma 2.4.

Suppose that ( Open image in new window ) and (2.17) hold. Then for any Open image in new window , Open image in new window is completely continuous.

Proof.

First of all, notice that Open image in new window maps Open image in new window into Open image in new window by Lemma 2.3.

Thus, Open image in new window is bounded on Open image in new window .

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

From (2.25), (2.26), and the absolute continuity of integral function, it follows that Open image in new window is equicontinuous.

Therefore, Open image in new window is relatively compact, that is, Open image in new window is a compact operator on Open image in new window .

*Lebesgue*dominated convergence theorem, it follows that

Thus, Open image in new window is continuous on Open image in new window . Therefore, Open image in new window 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 ( Open image in new window ) holds, then BVP (1.1) has at least one positive solution for any Open image in new window if and only if the integral inequality (2.17) holds.

Proof.

Let Open image in new window let Open image in new window and let Open image in new window then (3.1) holds.

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

where Open image in new window .

Thus, (3.17) holds.

where Open image in new window .

This implies that (3.20) holds.

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

## 4. Unbounded Connected Branch of Positive Solutions

then, by Theorem 3.1, Open image in new window for any Open image in new window .

Theorem 4.1.

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

(i)for any Open image in new window , and

Proof.

We now prove our conclusion by the following several steps.

Therefore, Open image in new window has no positive solution in Open image in new window . As a consequence, Open image in new window is bounded.

By the complete continuity of Open image in new window , Open image in new window is compact.

If Open image in new window , then taking Open image in new window .

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

Therefore, Theorem 4.1 holds and the proof is complete.

## Notes

### 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).

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