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This paper investigates the existence and nonexistence of positive solutions for a class of fourth-order nonlinear differential equation with integral boundary conditions. The associated Green's function for the fourth-order boundary value problems is first given, and the arguments are based on Krasnoselskii's fixed point theorem for operators on a cone.

In this paper, we consider the following fourth-order boundary value problems (BVPs) with integral boundary conditions

The existence of solutions for nonlinear higher-order nonlocal BVPs has been studied by several authors, for example, see [

For the case of

The following assumptions will stand throughout this paper:

The paper is organized as follows. In Section

In our main results, we will make use of the following lemmas.

Let

If

The general solution of

If

Letting

Letting

If

From (

For all

For all

Assume that (A1)–(A3) holds. If

By using Lemma

If (A3) holds, then one has the following three properties:

By using Lemma

By using by (

Let

For a fixed

If (A1)–(A3) hold, then

By the properties of Green's function, if

Clearly, the operator

For convenience, we introduce the following notations:

Assume that (A1)–(A3) hold. In addition, one supposes that one of the following conditions is satisfied

Then, the BVP (

Assume that (A1)–(A3) hold. In addition, one supposes that one of the following conditions is satisfied

Then, the BVP (

Considering

Then, for

Next, turning to

Let

Considering

Then, for

Next, turning to

Let

Let

Applying Lemma

Assume that (A1)–(A3) hold, as do the following two conditions:

there exists

Then, the BVP (

We choose

If

If

By (

Applying Lemma

Assume that (A1)–(A3) hold, as do the following two conditions:

there exist

Then, the BVP (

We choose

If

If

Applying Lemma

Consider the following fourth-order BVP

About the nonexistence of positive solution, we consider BVP (

If

About the existence of positive solution, we consider BVP (

If

About the multiplicity of positive solution, we consider BVP (

Furthermore, letting

In [

This work is partially supported by the National Natural Science Foundation of PR China (10971045), the Natural Science Foundation of Hebei Province (A2013208147) and (A2011208012), and the Foundation of Hebei University of Science and Technology (XL201246). The authors thank the referee for his/her careful reading of the paper and useful suggestions that have greatly improved this paper.