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Using bifurcation techniques, we first prove a global bifurcation theorem for nonlinear second-order semipositone integral boundary value problems. Then the existence and multiplicity of nodal solutions of the above problems are obtained. Finally, an example is worked out to illustrate our main results.

In this paper, we consider the existence and multiplicity of nodal solutions for the following nonlinear second-order semipositone integral boundary value problems (BVP for short):

Boundary value problems with integral boundary conditions for ordinary differential equations arise in different areas of applied mathematics and physics. Moreover, they include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to [

In [

The purpose of this paper is to investigate the existence and multiplicity of sign-changing solutions of BVP (

Motivated by [

Now we give some notations and a global bifurcation theorem which will be used in Section

If

This paper is arranged as the follows: some preliminaries and some lemmas are given including the study of the eigenvalues and eigenfunctions of the linearization of BVP (

Let

For any

Note that

In the following, we give some information on the spectrum structure of the linear integral boundary value problem corresponding to BVP (

Define the operators

It is easy to prove the following lemma.

The linear operator

We now define a function

All the zeros of

Suppose that

Suppose that

Since

Now, (

For each integer

Thus,

For any fixed integer

Consider

Consider

Consider

Therefore,

As the proof of Lemma 4 in [

(1) For each

(2) The algebraic multiplicity of each positive eigenvalue

Suppose that

there exists a subsequence

From Lemmas

Define the operators

It is easy to see that

For

Let

for each

(1) For each

(2) Since

Suppose that

Since

If

Consider (

By (

Since

Similarly we can obtain that

Let

Let

Immediately, from Theorem

Suppose that all the conditions of Theorem

Consider the following nonlinear second-order integral boundary value problem:

By direct computation, it is easy to see that

Next, we check that all the conditions of Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research is supported by the Reward Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2011SF022), China.