Institute of Mathematics, Jilin University, Changchun 130012, China

Institute of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130017, China

The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.

1. Introduction

In this paper we apply the lower and upper functions method to study the fourth-order nonlinear equation

with

This equation can be used to model the deformations of an elastic beam, and the type of boundary conditions considered depends on how the beam is supported at the two endpoints [1, 2]. We consider the separated boundary conditions

with

For the fourth-order differential equation

the authors in [3] obtained the existence of solutions with the assumption that

Motivated by the above works, we consider the existence of solutions when

The outline of this paper is as follows. In Section 2, we give the definition of lower and upper functions to problems (1.1) and (1.2) and obtain some a priori estimates. Section 3 will be devoted to the study of the existence of solutions. In Section 4, we give an example to illustrate the conclusions.

2. Definitions and A Priori Estimates

Upper and lower functions will be an important tool to obtain a priori bounds on

Definition 2.1.

The functions

define a pair of lower and upper functions of problems (1.1) and (1.2) if the following conditions are satisfied:

(i)

(ii)

(iii)

Remark 2.2.

By integration, from (iii) and (2.1), we obtain

that is, lower and upper functions, and their first derivatives are also well ordered.

To have an a priori estimate on

Definition 2.3.

Given a set

with

Lemma 2.4.

Let

and consider the set

Let

Then, for every

for

Proof.

Let

Assume that

If

If

Applying a convenient change of variable, we have, by (2.3) and (2.11),

Hence,

In a similar way, it can be proved that

Consider now the case

In a similar way, we may show that

Taking

Remark 2.5.

Observe that the estimation

3. Existence and Location Result

In the presence of an ordered pair of lower and upper functions, the existence and location results for problems (1.1) and (1.2) can be obtained.

Theorem 3.1.

Suppose that there exist lower and upper functions

If

for

where

for

Proof.

Define the auxiliary functions

For

with the boundary conditions

Take

Step 1.

Every solution

for

Assume, for contradiction, that the above estimate does not hold for

If

For

If

and

The case

By the boundary condition (3.7) there exists a

Step 2.

There is an

with

Consider the set

and for

In the following we will prove that the function

So, defining

Therefore,

Moreover, for

every solution

Define

The hypotheses of Lemma 2.4 are satisfied with

Step 3.

For

Define the operators

by

and for

with

Observe that

given by

For

By Steps 1 and 2, degree

The equation

and has only the trivial solution. Then, by the degree theory,

So the equation

has at least one solution

Step 4.

The function

The proof will be finished if the above function

Assume, for contradiction, that there is a

If

If

By Definition 2.1 this yields a contradiction

Then

Using an analogous technique, it can be deduced that

On the other hand, by (1.2),

that is,

Applying the same technique, we have

and then by Definition 2.1 (iii), (3.44) and (3.46), we obtain

that is,

Since, by (3.44),

and, therefore,

and so

The inequalities

4. An Example

The following example shows the applicability of Theorem 3.1 when

Example 4.1.

Consider now the problem

with

is continuous in

are, respectively, lower and upper functions of (4.1) and (4.2). Moreover, define

Then

Therefore, by Theorem 3.1, there is at least one solution

Notice that the function

does not satisfy the two-sided Nagumo condition.

Acknowledgments

The authors would like to thank the referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, and by the 985 Program of Jilin University.