_{1}

^{*}

This paper presents sufficient conditions for the existence of positive solutions for the fourth-order boundary value problem system with p-Laplacian operator. The existence of single or multiple positive solutions for the system is showed through the fixed point index theory in cones under some assumptions.

In this paper, we are concerned with the existence and multiplicity of positive solutions for the system (BVP):

where

Several papers ([

Motivated by the results mentioned above, here we establish some sufficient conditions for the existence of to (BVP) (1.1) under certain suitable weak conditions. The main results in this paper improve and generalize the results by others.

The following fixed-point index theorem in cones is fundamental.

Theorem A [

1) If for

2) If for

In this paper, let

ce with the norm

Suppose

Obviously,

Define a cone

define an integral operator

Let us list the following assumptions for convenience.

Lemma 2.1

It is easy to see that

Lemma 2.2 Suppose that

Lemma 2.3 Suppose that

Proof Firstly, assume

Then

Secondly, suppose

Due to the continuity of

Theorem, then

Lastly, since

Then for all

So

Therefore,

For convenience we denote

Theorem 3.1 Suppose that

Then the system (1.1) has at least one positive solution

Proof By Lemma 2.3, we know

Hence,

erefore

On the other hand, from

Hence,

If

Therefore it follows from the fixed-point theorem that

Theorem 3.2 Suppose that

Then the system (1.1) has at least one positive solution

Proof By lemma 2.3, we know

Let

Hence,

If

always may set

On the other hand, from

Case (i). Suppose that _{i} > 0 satisfying

we get

Hence,

then

Case (ii). Suppose that

2. Let

Hence,

so

Therefore it follows from the fixed-point theorem that

Remark 3.1 Note that if

Remak 3.2 When

Theorem 3.3 Suppose that

Proof. Choosing

Hence,

On the other hand, From (H_{6}), if

Hence,

If

we always may set

Therefore it follows from the fixed-point theorem that

Theorem 3.4 Suppose that

The proofs are similar to that of Theorem 3.2 and are omitted.

Theorem 3.5 Assume that

Then the system (1.1) has at least two positive solutions

Theorem 3.6 Assume that

Remark 3.3 Under suitable weak conditions, the multiplicity results for fourth-order singular boundary value problem with

Zengxia Cai, (2015) The Existence and Multiplicity of Solutions for Singular Boundary Value Systems with p-Laplacian. Journal of Applied Mathematics and Physics,03,411-416. doi: 10.4236/jamp.2015.34052