The authors use the upper and lower solution method to study the existence of solutions of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation

It is well known that the upper and lower solution method is a powerful tool for proving existence results for boundary value problems. The upper and lower solution method has been used to deal with the multipoint boundary value problems for second-order ordinary differential equations [

We will develop the upper and lower solution method for the boundary value problem

In this section, we will give some preliminary considerations and some lemmas which are essential to our main results.

Suppose the functions

Because of Definition

Let

We assume throughout this paper the following.

There are lower and upper solutions

Function

Function

Function

Function

It is not difficult to obtain the following lemma.

The boundary value problem

It is easy to prove the following lemma similarly to [

Assume that

Assume that

By Lemma

Denote

Assume

Then

Now, define an operator

This shows that

In the following we prove that

Now, let

Further, by the definition of

Because there is a

Assume

By Lemma

For fixed

Define the following sets:

Clearly, sequences

Similar to the proof of Theorem

Assume

We all know it is difficult to find a solution of some nonlinear ordinary differential equation. But according to Theorem

Consider the following linear boundary value problem

It is easy to know that

Consider nonlinear boundary value problem

It is easy to verify that

In this paper, we study a nonlinear mixed two-point boundary value problem for a third-order nonlinear ordinary differential equation. Some new existence results are obtained by developing the upper and lower solution method. Furthermore, some applications are also presented.

The work is supported by the Fundamental Research Funds for the Central Universities (no. ZXH2012 K004) and Civil Aviation University of China Research Funds (no. 2012KYM05). The authors would like to thank the referees for their valuable comments.