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

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 .

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).

This paper investigates the following fourth-order nonlinear singular eigenvalue problem:

(1.1)

where is a parameter and satisfies the following hypothesis:

(), and there exist constants , , , , such that for any , , satisfies

(1.2)

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

(1.3)

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).

The existence of positive solutions of BVPs has been studied by several authors in the literature; for example, see [7–20] and the references therein. Yao [15, 18] studied the following BVP:

(1.4)

where is a closed subset and , . In [15], he obtained a sufficient condition for the existence of positive solutions of (1.4) by using the monotonically iterative technique. In [13, 18], he applied Guo-Krasnosel'skii's fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP (1.4) and the following BVP:

(1.5)

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]).

Let be a metric space and . Let and satisfy

(1.6)

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 .

Then there exists a connected branch of such that

(1.7)

where there exists a sequence such that .

2. Some Preliminaries and Lemmas

Let , , then is a Banach space, where Define

(2.1)

It is easy to conclude that is a cone of . Denote

(2.2)

Let

(2.3)

Then is the Green function of homogeneous boundary value problem

(2.4)

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).

We now prove that property (2) is true. For , by (2.4), we have

(2.5)

For , by (2.4), we have

(2.6)

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 .

For , if , then

(2.7)

if , then

(2.8)

Therefore, property (4) holds.

Lemma 2.2.

Assume that , then and

(2.9)

(2.10)

Proof.

Assume that , then , , , so

(2.11)

Therefore, (2.9) holds. From (2.9), we get

(2.12)

By (2.9) and the definition of , we can obtain that

(2.13)

Thus, (2.10) holds.

For any fixed , define an operator by

(2.14)

Then, it is easy to know that

(2.15)

(2.16)

Lemma 2.3.

Suppose that () and

(2.17)

hold. Then .

Proof.

From (), for any , , , we easily obtain the following inequalities:

(2.18)

For every , , choose positive numbers min . It follows from (), (2.10), Lemma 2.1, and (2.17) that

(2.19)

Similar to (2.19), from (), (2.10), Lemma 2.1, and (2.17), for every , , we have

(2.20)

Thus, is well defined on .

From (2.4) and (2.14)–(2.16), it is easy to know that

(2.21)

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.

Next, we show that is bounded. In fact, for any , by (2.10) we can get

(2.22)

Choose positive numbers , , . This, together with (), (2.22), (2.16), and Lemma 2.1 yields that

(2.23)

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 .

Since for each , (2.22) holds, we may choose still positive numbers , , . Then

(2.24)

where . Notice that

(2.25)

Thus for any given with and for any , we get

(2.26)

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 .

Finally, we show that is continuous on . Suppose , and . Then , and as uniformly, with respect to . From , choose still positive numbers , , . Then

(2.27)

By (2.17), we know that is integrable on . Thus, from the Lebesgue dominated convergence theorem, it follows that

(2.28)

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.

Suppose first that be a positive solution of BVP (1.1) for any fixed . Then there exist constants () with , such that

(3.1)

In fact, it follows from , and , that for and , for . By the concavity of and , we have

(3.2)

On the other hand,

(3.3)

Let letand let then (3.1) holds.

Choose positive numbers , , . This, together with (), (1.2), and (2.18) yields that

(3.4)

where . Hence, integrating (3.4) from to 1, we obtain

(3.5)

Since increases on , we get

(3.6)

that is,

(3.7)

Notice that , integrating (3.7) from 0 to 1, we have

(3.8)

That is,

(3.9)

Thus,

(3.10)

By an argument similar to the one used in deriving (3.5), we can obtain

(3.11)

where . So,

(3.12)

Integrating (3.12) from 0 to 1, we have

(3.13)

That is,

(3.14)

So,

(3.15)

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

Now assume that (2.17) holds, we will show that BVP (1.1) has at least one positive solution for any . By (2.17), there exists a sufficient small such that

(3.16)

For any fixed , first of all, we prove

(3.17)

where .

Let , then

(3.18)

From Lemma 2.1, (3.18), and (), we get

(3.19)

Thus, (3.17) holds.

Next, we claim that

(3.20)

where .

Let , then for , we get

(3.21)

Therefore, by Lemma 2.1 and (), it follows that

(3.22)

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

In this section, we study the global continua results under the hypotheses () and (2.17). Let

(4.1)

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.

First, we prove that for arbitrarily given is bounded. In fact, let

(4.2)

then for and , we get

(4.3)

Therefore, by Lemma 2.1 and (), it follows that

(4.4)

Let

(4.5)

where is given by (3.16). Then for and , we get

(4.6)

Therefore, by Lemma 2.1 and (), it follows that

(4.7)

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

By the complete continuity of , is compact.

Second, we choose sequences and satisfy

(4.8)

We are to prove that for any positive integer , there exists a connected branch of satisfying

(4.9)

Let be fixed, suppose that for any , the connected branch of , passing through , leads to . Since is compact, there exists a bounded open subset of such that , , and , where and later denote the closure and boundary of with respect to . If , then and are two disjoint closed subsets of . Since is a compact metric space, there are two disjoint compact subsets and of such that , , and . Evidently, . Denoting by the -neighborhood of and letting , then it follows that

(4.10)

If , then taking .

It is obvious that in , the family of makes up an open covering of . Since is a compact set, there exists a finite subfamily which also covers . Let , then

(4.11)

Hence, by the homotopy invariance of the fixed point index, we obtain

(4.12)

By the first step of this proof, the construction of , (4.4), and (4.7), it follows easily that there exist such that

(4.13)

(4.14)

(4.15)

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).

By Lemma 1.2, there exists a connected branch of such that for any . Noticing , we have . Let be the connected branch of including , then for any . Similar to (4.4) and (4.7), for any , , we have, by (), (4.2), (4.3), (4.5), (4.6), and Lemma 2.1,

(4.16)

(4.17)

where is given by (3.16). Let in (4.16) and in (4.17), we have

(4.18)

Therefore, Theorem 4.1 holds and the proof is complete.