The author investigates the fourth-order integral boundary value problem with two parameters

The theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can all be reduced to nonlocal problems with integral boundary conditions (see, e.g., [

Moreover, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoints and nonlocal boundary value problems as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention. To identify a few, we refer the reader to [

In the recent literature, several sorts of boundary value problems with integral boundary conditions have been studied further, see [

On the other hand, the fourth-order boundary value problem describe the deformations of an elastic beam in equilibrium state. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors; see, for example, [

Motivated by the above-mentioned works [

Let us begin with listing the following assumption conditions, which will be used in the sequel:

Let

Let

In this section, we shall give some important preliminary lemmas, which will be used in proving of our main results.

Suppose that (H2) holds, then there exist unique

Let

Put

(H3) Functions

Let

Let

Let

For any

Let

Denote operator B on

Define operator L:

We need the following Lemma.

Let (H2) holds. Assume that

(1) Assume

(2) Inversely, assume

We have also the following lemma.

Suppose (H3) holds. Then

In view of Lemma

On the other hand, from

By (

For any

Now, by Lemma

For the remainder of this section, we give the definition of positive solution.

By a positive solution of BVP (

We introduce now some notations, which will be used in the sequel.

Let

(H4) There exists a number

We are now in a position to state and prove our main results on the existence.

Suppose that (H1)–(H4) hold. If

By Lemma

We will show that

Hence, in view of Arzela-Ascoli theorem, we know that the operator

Now, we show that the operator

Now, from (

Now we set

In fact, for any

We shall deduce that for any

In fact, in view of Lemma

By (H4), we have

Now, we show that

If

If

If

If

If

If

Now, set

Thus, by (

Let

Now, from

Let

Let us take

Even in the case that

The author wishes to express thanks to the anonymous referees for their valuable suggestions and comments. He also would like to thank the Natural Science Foundation of Educational Committee of Hubei (D200722002) for their support.