^{1}

^{2}

^{3, 4}

^{4}

^{1}

^{2}

^{3}

^{4}

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation

In this paper, we consider the existence of multiple positive solutions for the following fourth-order singular Sturm-Liouville boundary value problem involving a perturbed term

Equation (

Recently, for the case where the nonlinearity

In this paper, we focus on the particularly difficult and interesting situation, when (

Our main tool used for the analysis here is known as Guo-Krasnoselskii’s fixed point theorem, for the convenience of the reader, we now state it as follows.

Let

The following definition introduces global Carathéodory’s conditions imposed on a map.

A map

for each

for a.e.

there exists a

The following lemmas play an important role in proving our main results.

Let

Set

As [

For any

It follows from the monotonicity of

Also, it is well known the Green function for the boundary value problem

In order to obtain existence of positive solutions to problem (

Let

It follows from (

Conversely, if

In the rest of the paper, we always suppose that the following assumptions hold.

It follows from (B1), (B3) and from the monotonicity of

Assume (B3) is satisfied. Then, the boundary value problem

First,

Define a modified function

If

In fact, if

Thus, the BVP (

The basic space used in this paper is

Assume that (B1)–(B3) hold. Then,

For any fixed

On the other hand, since

Next, for any

On the other hand, from (

At the end, according to the Ascoli-Arzela Theorem, using standard arguments, one can show

Suppose (B1)–(B3) hold. In addition, assume that the following conditions are satisfied.

There exists a constant

There exists a constant

Let

It follows from (S1) that

On the other hand, let

It follows from (S2), (

Next, let us choose

Now let

It follows from

As for (

Let

Suppose (B1)–(B3) hold. In addition, assume that the following conditions are satisfied.

There exists a constant

There exists a constant

(S6)

Firstly, let

It follows from (S4), (

Next, by (S5), we have

Let

It follows from (S5) that

On the other hand, choose a large enough real number

From (S6), there exists

By Lemma

Noticing that

In fact, let

The corresponding Green’s functions can be written by

Now, take

Choose

On the other hand, we take

The authors were supported financially by the National Natural Science Foundation of China (11071141) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).