- Research Article
- Open Access

# Existence and Multiplicity of Positive Solutions of a Boundary-Value Problem for Sixth-Order ODE with Three Parameters

- Liyuan Zhang
^{1}Email author and - Yukun An
^{1}

**2010**:878131

https://doi.org/10.1155/2010/878131

© The Author(s) Liyuan Zhang and Yukun An. 2010

**Received:**13 May 2010**Accepted:**14 August 2010**Published:**18 August 2010

## Abstract

We study the existence and multiplicity of positive solutions of the following boundary-value problem:
,
,
, where
**R**^{+} → **R**^{+} is continuous,
,
, and
satisfy some suitable assumptions.

## Keywords

- Fixed Point Theorem
- Bistable System
- Phase Front
- Fixed Point Index
- Parabolic Differential Equation

## 1. Introduction

This equation arose in the formation of the spatial periodic patterns in bistable systems and is also a model for describing the behaviour of phase fronts in materials that are undergoing a transition between the liquid and solid state. When it was studied by Gardner and Jones [1] as well as by Caginalp and Fife [2].

If is an even periodic function with respect to and odd with respect to , in order to get the stationary spatial periodic solutions of (1.2), one turns to study the two points boundary-value problem (1.1). The periodic extension of the odd extension of the solution of problems (1.1) to the interval yields spatial periodic solutions of(1.2)

Gyulov et al. [3] have studied the existence and multiplicity of nontrivial solutions of BVP (1.1). They gained the following results.

Theorem 1.1.

Let be a continuous function and . Suppose the following assumptions are held:

as , uniformly with respect to in bounded intervals,

as , uniformly with respect to in bounded intervals,

then problem (1.1) has at least two nontrivial solutions provided that there exists a natural number such that , where is the symbol of the linear differential operator .

proposed in 1977.

The methods used in those papers are variational method and linking theorems.

On the other hand, The positive solutions of fourth-order boundary value problems (1.5) have been studied extensively by using the fixed point theorem of cone extension or compression. Here, we mention Li's paper [7], in which the author decomposes the fourth-order differential operator into the product of two second-order differential operators to obtain Green's function and then used the fixed point theorem of cone extension or compression to study the problem.

and then we assume the following conditions throughout:

is continuous,

Note.

The set of which satisfies is nonempty. For instance, if , then holds for .

## 2. Preliminaries

Lemma 2.1 (see [8]).

- (1)
Equation (2.1) has a triple root if ,

- (2)
Equation (2.1) has a real root and two mutually conjugate imaginary roots if ,

- (3)
Equation (2.1) has three real roots, two of which are reroots if ,

- (4)
Equation (2.1) has three unequal real roots if .

Lemma 2.2.

Let be the roots of the polynomial . Suppose that condition holds, then are real and greater than .

Proof.

Therefore, the equation has three real roots in reply to Lemma 2.1.

Then, we only prove that the system of inequalities (2.5) holds if and only if are all greater than .

which is a contradiction. Hence, the assumption is false. The proof is completed.

Lemma 2.3 (see [7]).

- (i)
,

- (ii)
, where is a constant,

- (iii)
, where is a constant.

then . Let be the maximum norm of , and let be the cone of all nonnegative functions in .

Lemma 2.4.

Proof.

The proof is completed.

where , we have the following.

Lemma 2.5.

Having , is completely continuous.

Proof.

namely . Therefore, . The complete continuity of is obvious.

The main results of this paper are based on the theory of fixed point index in cones [9]. Let be a Banach space and be a closed convex cone in . Assume that is a bounded open subset of with boundary , and . Let be a completely continuous mapping. If for every , then the fixed point index is well defined. We have that if , then has a fixed point in .

Let and for every .

Lemma 2.6 (see [9]).

Let be a completely continuous mapping. If for every and , then .

Lemma 2.7 (see [9]).

- (i)
,

- (ii)
for every and ,

then .

Lemma 2.8 (see [9]).

- (i)
If for every , then .

- (ii)
If for every , then .

## 3. Existence

We are now going to state our existence results.

Theorem 3.1.

- (i)
, ,

- (ii)
, ,

the BVP(1.6)-(1.7) has at least one positive solution.

Proof.

To prove Theorem 3.1, we just show that the mapping defined by (2.18) has a nonzero fixed point in the cases, respectively.

On the other hand, since , there exist and such that

Therefore, has a fixed point in , which is the positive solution of BVP(1.6)-(1.7).

On the other hand, since , there exist and such that

Therefore, has a fixed point in , which is the positive solution of BVP(1.6)-(1.7). The proof is completed.

From Theorem 3.1, we immediately obtain the following.

Corollary 3.2.

- (i)
, ,

- (ii)
, ,

the BVP(1.6)-(1.7) has at least one positive solution.

## 4. Multiplicity

Next, we study the multiplicity of positive solutions of BVP(1.6)-(1.7) and assume in this section that

(*H*3) there is a
such that
and
imply
, where
.

(*H*4) there is a
such that
and
imply
, where
.

Theorem 4.1.

If and and is satisfied, then BVP(1.6)-(1.7) has at least two positive solutions: and such that .

Proof.

According to the proof of Theorem 3.1, there exists , such that implies and implies .

Therefore, has fixed points and in and , respectively, which means that and are positive solutions of BVP(1.6)-(1.7) and . The proof is completed.

Theorem 4.2.

If and and is satisfied, then BVP(1.6)-(1.7) has at least two positive solutions: and such that .

Proof.

According to the proof of Theorem 3.1, there exists , such that implies and implies .

Therefore, , according to (i) of Lemma 2.8, .

Therefore, has the fixed points and in and , respectively, which means that and are positive solutions of BVP(1.6)-(1.7) and . The proof is completed.

Theorem 4.3.

- (i)
if and ,

- (ii)
if and

then BVP(1.6)-(1.7) has at least three positive solutions: , and such that .

Proof.

According to the proof of Theorem 3.1, there exists , such that implies and implies .

Therefore, has the fixed points , and in , and , which means that , and are positive solutions of BVP(1.6)-(1.7) and . The proof is completed.

## Authors’ Affiliations

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