Open Access

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

Boundary Value Problems20102010:878131

DOI: 10.1155/2010/878131

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: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq4_HTML.gif R+R+ is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq6_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq7_HTML.gif satisfy some suitable assumptions.

1. Introduction

The following boundary-value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq8_HTML.gif are some given real constants and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq9_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq10_HTML.gif , is motivated by the study for stationary solutions of the sixth-order parabolic differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ2_HTML.gif
(1.2)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq11_HTML.gif it was studied by Gardner and Jones [1] as well as by Caginalp and Fife [2].

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq12_HTML.gif is an even https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq13_HTML.gif periodic function with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq14_HTML.gif and odd with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq15_HTML.gif , in order to get the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq16_HTML.gif stationary spatial periodic solutions of (1.2), one turns to study the two points boundary-value problem (1.1). The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq17_HTML.gif periodic extension https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq18_HTML.gif of the odd extension of the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq19_HTML.gif of problems (1.1) to the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq20_HTML.gif yields https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq21_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq22_HTML.gif be a continuous function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq23_HTML.gif . Suppose the following assumptions are held:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq25_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq26_HTML.gif , uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq27_HTML.gif in bounded intervals,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq29_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq30_HTML.gif , uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq31_HTML.gif in bounded intervals,

then problem (1.1) has at least two nontrivial solutions provided that there exists a natural number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq32_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq33_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq34_HTML.gif is the symbol of the linear differential operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq35_HTML.gif .

At the same time, in investigating such spatial patterns, some other high-order parabolic differential equations appear, such as the extended Fisher-Kolmogorov (EFK) equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ3_HTML.gif
(1.3)
proposed by Coullet, Elphick, and Repaux in 1987 as well as by Dee and Van Saarlos in 1988 and Swift-Hohenberg (SH) equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ4_HTML.gif
(1.4)

proposed in 1977.

In much the same way, the existence of spatial periodic solutions of both the EFK equation and the SH equation was studied by Peletier and Troy [4], Peletier and Rottschäfer [5], Tersian and Chaparova [6], and other authors. More precisely, in those papers, the authors studied the following fourth-order boundary-value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ5_HTML.gif
(1.5)

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.

The purpose of this paper is using the idea of [7] to investigate BVP for sixth-order equations. We will discuss the existence and multiplicity of positive solutions of the boundary-value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ7_HTML.gif
(1.7)

and then we assume the following conditions throughout:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq36_HTML.gif is continuous,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq37_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ8_HTML.gif
(1.8)

Note.

The set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq38_HTML.gif which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq39_HTML.gif is nonempty. For instance, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq40_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq41_HTML.gif holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq42_HTML.gif .

To be convenient, we introduce the following notations:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ9_HTML.gif
(1.9)

2. Preliminaries

Lemma 2.1 (see [8]).

Set the cubic equation with one variable as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ10_HTML.gif
(2.1)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ11_HTML.gif
(2.2)
one has the following:
  1. (1)

    Equation (2.1) has a triple root if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq43_HTML.gif ,

     
  2. (2)

    Equation (2.1) has a real root and two mutually conjugate imaginary roots if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq44_HTML.gif ,

     
  3. (3)

    Equation (2.1) has three real roots, two of which are reroots if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq45_HTML.gif ,

     
  4. (4)

    Equation (2.1) has three unequal real roots if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq46_HTML.gif .

     

Lemma 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq47_HTML.gif be the roots of the polynomial https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq48_HTML.gif . Suppose that condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq49_HTML.gif holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq50_HTML.gif are real and greater than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq51_HTML.gif .

Proof.

There are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq52_HTML.gif in the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq53_HTML.gif . Since condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq54_HTML.gif holds, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ12_HTML.gif
(2.3)

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

By Vieta theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ13_HTML.gif
(2.4)
Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq57_HTML.gif hold if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ14_HTML.gif
(2.5)

Then, we only prove that the system of inequalities (2.5) holds if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq58_HTML.gif are all greater than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq59_HTML.gif .

In fact, the sufficiency is obvious, we just prove the necessity. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq60_HTML.gif are less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq61_HTML.gif . By the first inequality of (2.5), there exist two roots which are less than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq62_HTML.gif and one which is greater than https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq63_HTML.gif . Without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq64_HTML.gif , then we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq65_HTML.gif . Multiplying the second inequality of (2.5) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq66_HTML.gif , one gets
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ15_HTML.gif
(2.6)
Compare with the third inequality of (2.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ16_HTML.gif
(2.7)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq67_HTML.gif be Green's function of the linear boundary-value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ17_HTML.gif
(2.8)

Lemma 2.3 (see [7]).

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq68_HTML.gif has the following properties:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq69_HTML.gif ,

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq70_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq71_HTML.gif is a constant,

     
  3. (iii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq72_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq73_HTML.gif is a constant.

     
One denotes the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ18_HTML.gif
(2.9)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq74_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq75_HTML.gif be the maximum norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq76_HTML.gif ,  and let   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq77_HTML.gif be the cone of all nonnegative functions in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq78_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq79_HTML.gif , then one considers linear boundary-value problem (LBVP) as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ19_HTML.gif
(2.10)
with the boundary condition (1.7). Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ20_HTML.gif
(2.11)
the solution of LBVP (2.10)–(1.7) can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ21_HTML.gif
(2.12)

Lemma 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq80_HTML.gif , then the solution of LBVP(2.10)–(1.7) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ22_HTML.gif
(2.13)

Proof.

From (2.12) and (ii) of Lemma 2.3, it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ23_HTML.gif
(2.14)
and, therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ24_HTML.gif
(2.15)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ25_HTML.gif
(2.16)
Using (iii) of Lemma 2.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ26_HTML.gif
(2.17)

The proof is completed.

We now define a mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq81_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ27_HTML.gif
(2.18)
It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq82_HTML.gif is completely continuous. By Lemma 2.4, the positive solution of BVP(1.6)-(1.7) is equivalent to nontrivial fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq83_HTML.gif . We will find the nonzero fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq84_HTML.gif by using the fixed point index theory in cones. For this, one chooses the subcone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq85_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq86_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ28_HTML.gif
(2.19)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq87_HTML.gif , we have the following.

Lemma 2.5.

Having https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq88_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq89_HTML.gif is completely continuous.

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq90_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq91_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq92_HTML.gif is the solution of LBVP(2.10)–(1.7). By Lemma 2.4, one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ29_HTML.gif
(2.20)

namely https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq93_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq94_HTML.gif . The complete continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq95_HTML.gif is obvious.

The main results of this paper are based on the theory of fixed point index in cones [9]. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq96_HTML.gif be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq97_HTML.gif be a closed convex cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq98_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq99_HTML.gif is a bounded open subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq100_HTML.gif with boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq101_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq102_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq103_HTML.gif be a completely continuous mapping. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq104_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq105_HTML.gif , then the fixed point index https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq106_HTML.gif is well defined. We have that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq107_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq108_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq109_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq111_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq112_HTML.gif .

Lemma 2.6 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq113_HTML.gif be a completely continuous mapping. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq114_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq116_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq117_HTML.gif .

Lemma 2.7 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq118_HTML.gif be a completely continuous mapping. Suppose that the following two conditions are satisfied:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq119_HTML.gif ,

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq120_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq122_HTML.gif ,

     

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq123_HTML.gif .

Lemma 2.8 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq124_HTML.gif be a Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq125_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq126_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq127_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq128_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq129_HTML.gif is a completely continuous mapping such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq130_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq131_HTML.gif .
  1. (i)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq132_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq133_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq134_HTML.gif .

     
  2. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq135_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq136_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq137_HTML.gif .

     

3. Existence

We are now going to state our existence results.

Theorem 3.1.

Assume that (H1) and (H2) hold, then in each of the following case:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq139_HTML.gif ,

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq141_HTML.gif ,

     

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

Proof.

To prove Theorem 3.1, we just show that the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq142_HTML.gif defined by (2.18) has a nonzero fixed point in the cases, respectively.

Case(i): since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq143_HTML.gif , by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq144_HTML.gif , we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq146_HTML.gif so that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ30_HTML.gif
(3.1)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq147_HTML.gif , we now prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq148_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq150_HTML.gif . In fact, if there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq152_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq153_HTML.gif , then, by definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq155_HTML.gif satisfies differential equation the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ31_HTML.gif
(3.2)
and boundary condition (1.7). Multiplying (3.2) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq156_HTML.gif and integrating on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq157_HTML.gif , then using integration by parts in the left side, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ32_HTML.gif
(3.3)
By Lemma 2.4, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq158_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq159_HTML.gif . We see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq160_HTML.gif , which is a contradiction. Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq161_HTML.gif satisfies the hypotheses of Lemma 2.6, in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq162_HTML.gif . By Lemma 2.6 we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ33_HTML.gif
(3.4)

On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq164_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq166_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ34_HTML.gif
(3.5)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq167_HTML.gif , then it is clear that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ35_HTML.gif
(3.6)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq168_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq169_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq170_HTML.gif , from (3.5) we see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ36_HTML.gif
(3.7)
By Lemma 2.5, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ37_HTML.gif
(3.8)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ38_HTML.gif
(3.9)
from which we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq171_HTML.gif , namely the hypotheses (i) of Lemma 2.7 holds. Next, we show that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq172_HTML.gif is large enough, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq173_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq175_HTML.gif . In fact, if there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq177_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq178_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq179_HTML.gif satisfies (3.2) and boundary condition (1.7). Multiplying (3.2) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq180_HTML.gif and integrating, from (3.6) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ39_HTML.gif
(3.10)
Consequently, we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ40_HTML.gif
(3.11)
By Lemma 2.4,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ41_HTML.gif
(3.12)
from which and from (3.11) we get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ42_HTML.gif
(3.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq181_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq183_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq184_HTML.gif . Hence, hypothesis (ii) of Lemma 2.7 also holds. By Lemma 2.7, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ43_HTML.gif
(3.14)
Now, by the additivity of fixed point index, combine (3.4) and (3.14) to conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ44_HTML.gif
(3.15)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq185_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq186_HTML.gif , which is the positive solution of BVP(1.6)-(1.7).

Case (ii): since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq187_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq189_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ45_HTML.gif
(3.16)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq190_HTML.gif , then for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq191_HTML.gif , through the argument used in (3.9), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ46_HTML.gif
(3.17)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq192_HTML.gif . Next, we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq193_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq195_HTML.gif . In fact, if there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq197_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq198_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq199_HTML.gif satisfies (3.2) and boundary (1.7). From (3.2) and (3.16), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ47_HTML.gif
(3.18)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq200_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq201_HTML.gif , which is a contradiction. Hence, by Lemma 2.7, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ48_HTML.gif
(3.19)

On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq203_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq205_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ49_HTML.gif
(3.20)
Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq206_HTML.gif , we obviously have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ50_HTML.gif
(3.21)
If there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq207_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq208_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq209_HTML.gif , then (3.2) is valid. From (3.2) and (3.21), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ51_HTML.gif
(3.22)
By the proof of (3.13), we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq210_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq211_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq214_HTML.gif . Therefore, by Lemma 2.6, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ52_HTML.gif
(3.23)
From (3.19) and (3.23), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ53_HTML.gif
(3.24)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq215_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq216_HTML.gif , 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.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq218_HTML.gif hold, then in each of the following cases:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq220_HTML.gif ,

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq222_HTML.gif ,

     

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

(H3) there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq224_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq226_HTML.gif imply https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq227_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq228_HTML.gif .

(H4) there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq230_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq232_HTML.gif imply https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq233_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq234_HTML.gif .

Theorem 4.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq235_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq236_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq237_HTML.gif is satisfied, then BVP(1.6)-(1.7) has at least two positive solutions: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq238_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq239_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq240_HTML.gif .

Proof.

According to the proof of Theorem 3.1, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq241_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq242_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq243_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq244_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq245_HTML.gif .

We now prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq246_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq247_HTML.gif is satisfied. In fact, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq248_HTML.gif , from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq249_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ54_HTML.gif
(4.1)
From (ii) of Lemma 2.8, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ55_HTML.gif
(4.2)
Combining (3.14) and (3.19), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ56_HTML.gif
(4.3)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq250_HTML.gif has fixed points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq252_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq253_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq254_HTML.gif , respectively, which means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq255_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq256_HTML.gif are positive solutions of BVP(1.6)-(1.7) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq257_HTML.gif . The proof is completed.

Theorem 4.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq258_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq259_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq260_HTML.gif is satisfied, then BVP(1.6)-(1.7) has at least two positive solutions: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq262_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq263_HTML.gif .

Proof.

According to the proof of Theorem 3.1, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq264_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq265_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq266_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq267_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq268_HTML.gif .

We now prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq269_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq270_HTML.gif is satisfied. In fact, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq271_HTML.gif , from the proof of (i) of Theorem 3.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ57_HTML.gif
(4.4)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq272_HTML.gif , according to (i) of Lemma 2.8, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq273_HTML.gif .

Combining (3.4) and (3.23), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ58_HTML.gif
(4.5)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq274_HTML.gif has the fixed points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq275_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq276_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq277_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq278_HTML.gif , respectively, which means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq279_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq280_HTML.gif are positive solutions of BVP(1.6)-(1.7) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq281_HTML.gif . The proof is completed.

Theorem 4.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq282_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq283_HTML.gif , and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq284_HTML.gif that satisfies
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq285_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq287_HTML.gif ,

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq288_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq289_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq290_HTML.gif

     

then BVP(1.6)-(1.7) has at least three positive solutions: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq291_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq293_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq294_HTML.gif .

Proof.

According to the proof of Theorem 3.1, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq295_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq296_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq297_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq298_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq299_HTML.gif .

From the proof of Theorems 4.1 and 4.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ59_HTML.gif
(4.6)
Combining the four afore-mentioned equations, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ60_HTML.gif
(4.7)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq300_HTML.gif has the fixed points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq301_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq302_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq303_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq304_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq305_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq306_HTML.gif , which means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq307_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq308_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq309_HTML.gif are positive solutions of BVP(1.6)-(1.7) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq310_HTML.gif . The proof is completed.

Authors’ Affiliations

(1)
Nanjing University of Aeronautics and Astronautics

References

  1. Gardner RA, Jones CKRT: Traveling waves of a perturbed diffusion equation arising in a phase field model. Indiana University Mathematics Journal 1990,39(4):1197-1222. 10.1512/iumj.1990.39.39054MATHMathSciNetView Article
  2. Caginalp G, Fife PC: Higher-order phase field models and detailed anisotropy. Physical Review. B 1986,34(7):4940-4943. 10.1103/PhysRevB.34.4940MathSciNetView Article
  3. Gyulov T, Morosanu G, Tersian S: Existence for a semilinear sixth-order ODE. Journal of Mathematical Analysis and Applications 2006,321(1):86-98. 10.1016/j.jmaa.2005.08.007MATHMathSciNetView Article
  4. Peletier LA, Troy WC: Spatial Patterns, Progress in Nonlinear Differential Equations and their Applications. Volume 45. Birkhäuser Boston, Boston, Mass, USA; 2001:xvi+341.
  5. Peletier LA, Rottschäfer V: Large time behaviour of solutions of the Swift-Hohenberg equation. Comptes Rendus Mathématique. Académie des Sciences. Paris 2003,336(3):225-230.MATHView Article
  6. Tersian S, Chaparova J: Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations. Journal of Mathematical Analysis and Applications 2001,260(2):490-506. 10.1006/jmaa.2001.7470MATHMathSciNetView Article
  7. Li Y: Positive solutions of fourth-order boundary value problems with two parameters. Journal of Mathematical Analysis and Applications 2003,281(2):477-484. 10.1016/S0022-247X(03)00131-8MATHMathSciNetView Article
  8. Fan S: The new root formula and criterion of cubic equation. Journal of Hainan Normal University 1989, 2: 91-98.
  9. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.

Copyright

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

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.