Existence of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions

  • Ruyun Ma1Email author and

    Affiliated with

    • Tianlan Chen1

      Affiliated with

      Boundary Value Problems20102011:297578

      DOI: 10.1155/2011/297578

      Received: 5 May 2010

      Accepted: 7 July 2010

      Published: 3 August 2010

      Abstract

      We study the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq6_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq7_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq8_HTML.gif are nonnegative. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques.

      1. Introduction

      The deformations of an elastic beam in an equilibrium state, whose both ends are simple supported, can be described by the fourth-order boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ1_HTML.gif
      (1.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq9_HTML.gif is continuous; see Gupta [1, 2]. In the past twenty more years, the existence of solutions and positive solutions of these kinds of problems and the Lidstone problem has been extensively studied; see [39] and the references therein. In [3], Ma was concerned with the existence of positive solutions of (1.1) and (1.2) under the assumptions:

      (H1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq10_HTML.gif is continuous and there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq11_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq12_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ3_HTML.gif
      (1.3)
      uniformly for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq13_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ4_HTML.gif
      (1.4)

      uniformly for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq14_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq15_HTML.gif ;

      (H2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq16_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq17_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq18_HTML.gif ;

      (H3) there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq19_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq20_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ5_HTML.gif
      (1.5)

      Ma proved the following.

      Theorem A (see [3, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq21_HTML.gif ]).

      Let (H1), (H2), and (H3) hold. Assume that either
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ6_HTML.gif
      (1.6)
      or
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ7_HTML.gif
      (1.7)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq22_HTML.gif denotes the first generalized eigenvalue of the generalized eigenvalue problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ8_HTML.gif
      (1.8)

      Then (1.1) and (1.2) have at least one positive solution.

      At the same time, we notice that a class of boundary value problems with integral boundary conditions appeared in heat conduction, chemical engineering underground water flow, thermoelasticity, and plasma physics. Such a kind of problems include two-point, three-point, multipoint and nonlocal boundary value problems as special cases and attracting the attention of a few readers; see [1013] and the references therein. For example, In particular, Zhang and Ge [10] used Guo-Krasnoselskii fixed-point theorem to study existence and nonexistence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ9_HTML.gif
      (P)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq23_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq24_HTML.gif and (or) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq25_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq26_HTML.gif is continuous, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq27_HTML.gif are nonnegative.

      Motivated by [3, 10], in this paper, we consider the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ10_HTML.gif
      (1.9)

      under the assumption

      (H4) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq28_HTML.gif are nonnegative, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq29_HTML.gif . The main result of this paper is the following.

      Theorem 1.1.

      Let (H1), (H2), (H3), and (H4) hold. Assume that either
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ11_HTML.gif
      (1.10)
      or
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ12_HTML.gif
      (1.11)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ13_HTML.gif
      (1.12)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ14_HTML.gif
      (1.13)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ15_HTML.gif
      (1.14)

      Then (1.9) has at least one positive solution.

      Remark 1.2.

      Theorem 1.1 generalizes [3, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq30_HTML.gif ] where the special case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq32_HTML.gif was treated.

      Remark 1.3.

      Zhang and Ge [10] proved existence and nonexistence of positive solutions via Guo-Krasnoselskii fixed-point theorem under some conditions which do not involve the eigenvalues of (1.12)–(1.14). While our Theorem 1.1 is established under (1.10) or (1.11) which is related to the eigenvalues of (1.12)–(1.14). Moreover, (1.10) and (1.11) are optimal. Let us consider the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ16_HTML.gif
      (1.15)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ17_HTML.gif
      (1.16)

      In this case, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq33_HTML.gif and the corresponding eigenfunction is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq34_HTML.gif . However, (1.15) and (1.16) has no positive solution. (In fact, suppose on the contrary that (1.15) and (1.16) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq35_HTML.gif . Multiplying (1.15) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq36_HTML.gif and integrating from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq37_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq38_HTML.gif , we get a desired contradiction!).

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq39_HTML.gif is a real Banach space with norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq40_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq41_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq42_HTML.gif . A nonlinear mapping http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq43_HTML.gif is said to be positive if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq44_HTML.gif . It is said to be http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq45_HTML.gif -completely continuous if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq46_HTML.gif is continuous and maps bounded subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq47_HTML.gif to precompact subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq48_HTML.gif . Finally, a positive linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq49_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq50_HTML.gif is said to be a linear minorant for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq51_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq52_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq53_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq54_HTML.gif is a continuous linear operator on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq55_HTML.gif , denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq56_HTML.gif the spectral radius of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq57_HTML.gif . Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ18_HTML.gif
      (1.17)

      The following lemma will play a very important role in the proof of our main results, which is essentially a consequence of Dancer [14, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq58_HTML.gif ].

      Lemma 1.4.

      Assume that
      1. (i)

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq59_HTML.gif has nonempty interior and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq60_HTML.gif ;

         
      2. (ii)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq61_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq62_HTML.gif -completely continuous and positive, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq63_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq65_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq66_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ19_HTML.gif
        (1.18)
         

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq67_HTML.gif is a strongly positive linear compact operator on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq68_HTML.gif with the spectral radius http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq70_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq71_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq72_HTML.gif locally uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq73_HTML.gif .

      Then there exists an unbounded connected subset http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq74_HTML.gif of
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ20_HTML.gif
      (1.19)

      such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq75_HTML.gif .

      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq76_HTML.gif has a linear minorant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq77_HTML.gif and there exists a
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ21_HTML.gif
      (1.20)
      such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq79_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq80_HTML.gif can be chosen in
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ22_HTML.gif
      (1.21)

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq81_HTML.gif is a strongly positive compact endomorphism of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq83_HTML.gif has nonempty interior, we have from Amann [15, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq84_HTML.gif ] that the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq85_HTML.gif in [14, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq86_HTML.gif ] reduces to a single point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq87_HTML.gif . Now the desired result is a consequence of Dancer [14, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq88_HTML.gif ].

      The rest of the paper is arranged as follows. In Section 2, we state and prove some preliminary results about the spectrum of (1.12)–(1.14). Finally, in Section 3, we proved our main result.

      2. Generalized Eigenvalues

      Lemma 2.1 (see [10]).

      Assume that (H4) holds. Then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq89_HTML.gif , the boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ23_HTML.gif
      (2.1)
      has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq90_HTML.gif which is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ24_HTML.gif
      (2.2)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ25_HTML.gif
      (2.3)

      Lemma 2.2 (see [10]).

      Assume that (H4) holds. Then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq91_HTML.gif , the boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ26_HTML.gif
      (2.4)
      has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq92_HTML.gif which is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ27_HTML.gif
      (2.5)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ28_HTML.gif
      (2.6)

      Lemma 2.3 (see [10]).

      Assume that (H4) holds. Then one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ29_HTML.gif
      (2.7)

      Let

      (H5) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq93_HTML.gif be two given constants with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq94_HTML.gif .

      Definition 2.4.

      One says that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq95_HTML.gif is a generalized eigenvalue of linear problem
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ30_HTML.gif
      (2.8)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ31_HTML.gif
      (2.9)

      if (2.8) and (2.9) have nontrivial solutions.

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ32_HTML.gif
      (2.10)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq96_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq97_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq98_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq99_HTML.gif .

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ33_HTML.gif
      (2.11)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq100_HTML.gif , from Lemma 2.1, it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ34_HTML.gif
      (2.12)
      By simple calculations, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ35_HTML.gif
      (2.13)
      Combining this with (H4), we conclude that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ36_HTML.gif
      (2.14)
      This together with (2.12) and the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq101_HTML.gif imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ37_HTML.gif
      (2.15)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq102_HTML.gif , we may define the norm of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq103_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ38_HTML.gif
      (2.16)

      We claim that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq104_HTML.gif is a Banach space.

      In fact, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq105_HTML.gif be a Cauchy sequence, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq106_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq107_HTML.gif . From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq108_HTML.gif , it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ39_HTML.gif
      (2.17)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq109_HTML.gif is a normal in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq110_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq111_HTML.gif . Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ40_HTML.gif
      (2.18)
      By the completeness of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq112_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq113_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ41_HTML.gif
      (2.19)
      From the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq114_HTML.gif , we have that for arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq115_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq116_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ42_HTML.gif
      (2.20)
      and subsequently,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ43_HTML.gif
      (2.21)
      Fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq117_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq118_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ44_HTML.gif
      (2.22)
      This is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ45_HTML.gif
      (2.23)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq119_HTML.gif is a Banach space.

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ46_HTML.gif
      (2.24)

      Then the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq120_HTML.gif is normal and nonempty interior http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq122_HTML.gif .

      In fact, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq123_HTML.gif , it follows from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq124_HTML.gif that

      (1) there exist real number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq125_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ47_HTML.gif
      (2.25)

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq127_HTML.gif .

      From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq128_HTML.gif and (H4), we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq129_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq130_HTML.gif . Moreover,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ48_HTML.gif
      (2.26)
      and subsequently,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ49_HTML.gif
      (2.27)
      for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq131_HTML.gif . We may take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq132_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ50_HTML.gif
      (2.28)
      Now, let us define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ51_HTML.gif
      (2.29)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq133_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ52_HTML.gif
      (2.30)

      Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq134_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq135_HTML.gif .

      Lemma 2.5.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq136_HTML.gif holds. Then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq137_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ53_HTML.gif
      (2.31)

      Proof.

      In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq138_HTML.gif , we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ54_HTML.gif
      (2.32)

      From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq139_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq140_HTML.gif , and so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq141_HTML.gif , and accordingly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq142_HTML.gif .

      We have from the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq143_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq144_HTML.gif , that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ55_HTML.gif
      (2.33)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq145_HTML.gif , and consequently http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq146_HTML.gif .

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq147_HTML.gif , define a linear operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq148_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ56_HTML.gif
      (2.34)

      Theorem 2.6.

      Assume that (H4) and (H5) hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq149_HTML.gif be the spectral radius of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq150_HTML.gif . Then (2.8) and (2.9) has an algebraically simple eigenvalue, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq151_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq152_HTML.gif , with a positive eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq153_HTML.gif . Moreover, there is no other eigenvalue with a positive eigenfunction.

      Remark 2.7.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq154_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq155_HTML.gif can be explicitly given by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ57_HTML.gif
      (2.35)

      and the corresponding eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq156_HTML.gif .

      Proof of Theorem 2.6.

      From Lemma 2.2, it is easy to check that (2.8) and (2.9) is equivalent to the integral equation
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ58_HTML.gif
      (2.36)

      We claim that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq157_HTML.gif .

      In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq158_HTML.gif , we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ59_HTML.gif
      (2.37)
      Since
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ60_HTML.gif
      (2.38)
      and for some constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq159_HTML.gif , it concludes that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ61_HTML.gif
      (2.39)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ62_HTML.gif
      (2.40)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq160_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq161_HTML.gif , and accordingly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq162_HTML.gif .

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq163_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq164_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq165_HTML.gif , and accordingly
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ63_HTML.gif
      (2.41)

      Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq166_HTML.gif , and accordingly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq167_HTML.gif .

      Now, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq168_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq169_HTML.gif is compactly embedded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq170_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq171_HTML.gif is compact.

      Next, we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq172_HTML.gif is positive.

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq173_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq174_HTML.gif , from Lemma 2.3, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ64_HTML.gif
      (2.42)
      Combining this with (2.39), there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq175_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ65_HTML.gif
      (2.43)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq176_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq177_HTML.gif , applying a similar proof process of (2.43), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ66_HTML.gif
      (2.44)
      Combining this with (2.39), there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq178_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ67_HTML.gif
      (2.45)

      This together with (2.9) and (H4) imply http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq179_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq180_HTML.gif .

      Therefore, it follows from (2.43) and (2.45) that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq181_HTML.gif .

      Now, by the Krein-Rutman theorem ([16, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq182_HTML.gif C]; [17, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq183_HTML.gif ]), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq184_HTML.gif has an algebraically simple eigenvalue http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq185_HTML.gif with an eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq186_HTML.gif . Moreover, there is no other eigenvalue with a positive eigenfunction. Correspondingly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq187_HTML.gif with a positive eigenfunction of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq188_HTML.gif , is a simple eigenvalue of (2.8) and (2.9). Moreover, for (2.8) and (2.9), there is no other eigenvalue with a positive eigenfunction.

      3. The Proof of the Main Result

      Before proving Theorem 1.1, we denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq189_HTML.gif by setting
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ68_HTML.gif
      (3.1)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ69_HTML.gif
      (3.2)

      It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq190_HTML.gif is compact.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq191_HTML.gif be such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ70_HTML.gif
      (3.3)
      Obviously,( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq192_HTML.gif )impliesthat
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ71_HTML.gif
      (3.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ72_HTML.gif
      (3.5)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ73_HTML.gif
      (3.6)
      then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq193_HTML.gif is nondecreasing and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ74_HTML.gif
      (3.7)
      Let us consider
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ75_HTML.gif
      (3.8)
      as a bifurcation problem from the trivial solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq194_HTML.gif . It is to easy to check that (3.8) can be converted to the equivalent equation
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ76_HTML.gif
      (3.9)
      From the proof process of Theorem 2.6, the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq195_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ77_HTML.gif
      (3.10)
      is compact and strongly positive. Define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq196_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ78_HTML.gif
      (3.11)
      then we have from (3.4) and Lemma 2.5 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ79_HTML.gif
      (3.12)
      locally uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq197_HTML.gif . From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq198_HTML.gif and Theorem 2.6 (with obvious changes), it follows that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq199_HTML.gif is a nontrivial solution of (3.8) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq200_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq201_HTML.gif . Combining this with Lemma 1.4, we conclude that there exists an unbounded connected subset http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq202_HTML.gif of the set
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ80_HTML.gif
      (3.13)

      such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq203_HTML.gif .

      Proof of Theorem 1.1.

      It is clear that any solution of the form http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq204_HTML.gif yields a solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq205_HTML.gif of (1.9). We will show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq206_HTML.gif crosses the hyperplane http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq207_HTML.gif . To do this, it is enough to show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq208_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq209_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq210_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq211_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ81_HTML.gif
      (3.14)

      we note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq212_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq213_HTML.gif since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq214_HTML.gif is the only solution of (3.8) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq216_HTML.gif .

      Case 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq217_HTML.gif ).

      In this case, we show that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ82_HTML.gif
      (3.15)

      We divide the proof into two steps.

      Step 1.

      We show that if there exists a constant number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq218_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ83_HTML.gif
      (3.16)

      then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq219_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq220_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq221_HTML.gif .

      From (3.16), we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq222_HTML.gif . We divide the equation
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ84_HTML.gif
      (3.17)
      by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq223_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq224_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq225_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq226_HTML.gif , choosing a subsequence and relabeling if necessary, we see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq227_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq228_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq229_HTML.gif . Moreover, we have from (3.7) and Lemma 2.5 that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ85_HTML.gif
      (3.18)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq230_HTML.gif Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ86_HTML.gif
      (3.19)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq231_HTML.gif , again choosing a subsequence and relabeling if necessary. Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ87_HTML.gif
      (3.20)

      This together with Theorem 2.6 imply that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq232_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq233_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq234_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq235_HTML.gif .

      Step 2.

      We show that there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq236_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq237_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq238_HTML.gif .

      By Lemma 1.4, we only need to show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq239_HTML.gif has a linear minorant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq240_HTML.gif and there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq241_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq242_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq243_HTML.gif .

      By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq244_HTML.gif ,there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq245_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq246_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ88_HTML.gif
      (3.21)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq247_HTML.gif , let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ89_HTML.gif
      (3.22)
      then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq248_HTML.gif is a linear minorant of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq249_HTML.gif . Moreover,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ90_HTML.gif
      (3.23)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ91_HTML.gif
      (3.24)

      Combining this with (2.39), we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq250_HTML.gif , here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq251_HTML.gif . Therefore, we have that from Lemma 1.4 that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq252_HTML.gif .

      Case 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq253_HTML.gif ).

      In this case, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq254_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ92_HTML.gif
      (3.25)
      and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq255_HTML.gif then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ93_HTML.gif
      (3.26)

      and, moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq256_HTML.gif .

      Assume that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq257_HTML.gif , such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq258_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ94_HTML.gif
      (3.27)
      Applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ95_HTML.gif
      (3.28)

      Again http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq259_HTML.gif joins http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq260_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq261_HTML.gif and the result follows.

      Declarations

      Acknowledgments

      The authors are very grateful to the anonymous referees for their valuable suggestions. This paper was supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Northwest Normal University

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      Copyright

      © R. Ma and T. Chen. 2011

      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.