We generalize the unbounded upper and lower solution method to a third-order ordinary differential equation on the half line subject to the Sturm-Liouville boundary conditions. By using such techniques and the Schäuder fixed point theorem, some criteria are presented for the existence of solutions and positive ones to the problem discussed.

Boundary value problems on infinite intervals, arising from the study of radially symmetric solutions of nonlinear elliptic equation [

The method of upper and lower solutions is a powerful technique to deal with the existence of boundary value problems (BVPs). In many cases, when given one pair of well-ordered lower and upper solution, nonlinear BVPs always have at least one solution in the closed interval. To obtain this kind of result, we can employ topological degree theory, the monotone iterative technique, or critical theory. For details, we refer the reader to see [

When the method of upper and lower solution is applied to the infinite interval problems, diagonalization process is always used; see [

In [

There are many results of third-order boundary value problems on finite interval; see [

We present here some definitions and lemmas which are essential in the proof of the main results.

A function

Given a positive function

Let

If

It is easy to verify that (

Let

all functions from

all functions from

all functions from

From the above results, we can obtain the following general criteria for the relative compactness of subsets in

Given

all functions from

the functions from

the functions from

Set

Set

In this section, we present the existence criteria for the existence of solutions and positive solutions of BVP (_{1}) and (H_{2}) here.

:

BVP (

: For any

Suppose condition (H_{1}) holds. And suppose further that the following condition holds:

there exists a constant

If

Let

Consider

Consider

There exists

Suppose that

Similarly if

Thus there exists

Condition (H_{3}) is necessary for an a priori estimation of

Suppose _{1})–(H_{3}) hold. Then BVP (

Let

We claim that

By the Schäuder fixed point theorem,

Consider

There exists

Consequently,

For finite interval problem, it is sharp to define the lower and upper solutions satisfying

If

Let _{2}) holds and the following conditions hold.

BVP (

For any

Then BVP (

Choose

This research is supported by the National Natural Science Foundation of China (nos. 11101385 and 60974145) and by the Fundamental Research Funds for the Central Universities.