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

  • Liyuan Zhang1Email author and

    Affiliated with

    • Yukun An1

      Affiliated with

      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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq3_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq4_HTML.gif R+R+ is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq6_HTML.gif , and http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ1_HTML.gif
      (1.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq8_HTML.gif are some given real constants and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq9_HTML.gif is a continuous function on http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq12_HTML.gif is an even http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq13_HTML.gif periodic function with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq14_HTML.gif and odd with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq15_HTML.gif , in order to get the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq17_HTML.gif periodic extension http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq20_HTML.gif yields http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq22_HTML.gif be a continuous function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq23_HTML.gif . Suppose the following assumptions are held:

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

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq29_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq30_HTML.gif , uniformly with respect to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq32_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq33_HTML.gif , where http://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 http://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
      http://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
      http://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:
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ6_HTML.gif
      (1.6)
      http://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:

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

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

      Note.

      The set of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq38_HTML.gif which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq39_HTML.gif is nonempty. For instance, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq40_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq41_HTML.gif holds for http://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:
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ10_HTML.gif
      (2.1)
      Let
      http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq46_HTML.gif .

         

      Lemma 2.2.

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

      Proof.

      There are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq52_HTML.gif in the equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq53_HTML.gif . Since condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq54_HTML.gif holds, we have
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ13_HTML.gif
      (2.4)
      Therefore http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq57_HTML.gif hold if and only if
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq58_HTML.gif are all greater than http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq60_HTML.gif are less than http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq62_HTML.gif and one which is greater than http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq64_HTML.gif , then we have http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq66_HTML.gif , one gets
      http://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
      http://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 http://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
      http://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]).

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

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

         
      2. (ii)

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

         
      3. (iii)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq72_HTML.gif , where http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ18_HTML.gif
      (2.9)

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

      Let http://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:
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ21_HTML.gif
      (2.12)

      Lemma 2.4.

      Let http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ23_HTML.gif
      (2.14)
      and, therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ24_HTML.gif
      (2.15)
      that is,
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq81_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ27_HTML.gif
      (2.18)
      It is clear that http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq85_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq86_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ28_HTML.gif
      (2.19)

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

      Proof.

      For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq90_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq91_HTML.gif , then http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ29_HTML.gif
      (2.20)

      namely http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq93_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq94_HTML.gif . The complete continuity of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq96_HTML.gif be a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq97_HTML.gif be a closed convex cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq98_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq99_HTML.gif is a bounded open subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq100_HTML.gif with boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq101_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq102_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq103_HTML.gif be a completely continuous mapping. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq104_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq105_HTML.gif , then the fixed point index http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq107_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq108_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq109_HTML.gif .

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

      Lemma 2.6 (see [9]).

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

      Lemma 2.7 (see [9]).

      Let http://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)

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

         
      2. (ii)

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

         

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

      Lemma 2.8 (see [9]).

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

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

         
      2. (ii)

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

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

         
      2. (ii)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq140_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq143_HTML.gif , by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq144_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq146_HTML.gif so that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ30_HTML.gif
      (3.1)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq147_HTML.gif , we now prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq148_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq150_HTML.gif . In fact, if there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq151_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq152_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq153_HTML.gif , then, by definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq155_HTML.gif satisfies differential equation the following:
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq156_HTML.gif and integrating on http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ32_HTML.gif
      (3.3)
      By Lemma 2.4, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq158_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq159_HTML.gif . We see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq160_HTML.gif , which is a contradiction. Hence, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq162_HTML.gif . By Lemma 2.6 we have
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq164_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq166_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ34_HTML.gif
      (3.5)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq167_HTML.gif , then it is clear that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ35_HTML.gif
      (3.6)
      Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq168_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq169_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq170_HTML.gif , from (3.5) we see that
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ37_HTML.gif
      (3.8)
      Therefore,
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq172_HTML.gif is large enough, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq173_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq174_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq175_HTML.gif . In fact, if there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq176_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq177_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq178_HTML.gif , then http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ39_HTML.gif
      (3.10)
      Consequently, we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ40_HTML.gif
      (3.11)
      By Lemma 2.4,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ42_HTML.gif
      (3.13)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq181_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq183_HTML.gif , http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ44_HTML.gif
      (3.15)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq185_HTML.gif has a fixed point in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq187_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq188_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq189_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ45_HTML.gif
      (3.16)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq190_HTML.gif , then for every http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ46_HTML.gif
      (3.17)
      Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq192_HTML.gif . Next, we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq193_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq194_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq195_HTML.gif . In fact, if there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq196_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq197_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq198_HTML.gif , then http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ47_HTML.gif
      (3.18)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq200_HTML.gif , we see that http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq203_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq204_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq205_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ49_HTML.gif
      (3.20)
      Set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq206_HTML.gif , we obviously have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ50_HTML.gif
      (3.21)
      If there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq208_HTML.gif such that http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq210_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq211_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq212_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq213_HTML.gif , http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ53_HTML.gif
      (3.24)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq215_HTML.gif has a fixed point in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq217_HTML.gif and http://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)

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

         
      2. (ii)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq221_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq224_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq226_HTML.gif imply http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq227_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq228_HTML.gif .

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

      Theorem 4.1.

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq235_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq236_HTML.gif and http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq238_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq239_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq241_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq242_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq244_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq245_HTML.gif .

      We now prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq246_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq247_HTML.gif is satisfied. In fact, for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq248_HTML.gif , from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq249_HTML.gif we have
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ56_HTML.gif
      (4.3)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq250_HTML.gif has fixed points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq252_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq253_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq254_HTML.gif , respectively, which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq255_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq259_HTML.gif and http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq262_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq264_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq265_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq266_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq267_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq268_HTML.gif .

      We now prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq269_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq270_HTML.gif is satisfied. In fact, for every http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ57_HTML.gif
      (4.4)

      Therefore, http://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, http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ58_HTML.gif
      (4.5)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq274_HTML.gif has the fixed points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq275_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq276_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq277_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq278_HTML.gif , respectively, which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq279_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq282_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq283_HTML.gif , and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq284_HTML.gif that satisfies
      1. (i)

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

         
      2. (ii)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq288_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq289_HTML.gif and http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq291_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq292_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq293_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq295_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq296_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq297_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq298_HTML.gif implies http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_Equ60_HTML.gif
      (4.7)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq300_HTML.gif has the fixed points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq301_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq302_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq303_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq304_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq305_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq306_HTML.gif , which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq307_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F878131/MediaObjects/13661_2010_Article_964_IEq308_HTML.gif and http://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 http://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.