Open Access

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

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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq6_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq7_HTML.gif is continuous, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ2_HTML.gif
(1.2)

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

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

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

(H3) there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq19_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq20_HTML.gif and
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ6_HTML.gif
(1.6)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ7_HTML.gif
(1.7)
where https://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
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ9_HTML.gif
(P)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq23_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq24_HTML.gif and (or) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq25_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq26_HTML.gif is continuous, and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ10_HTML.gif
(1.9)

under the assumption

(H4) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq28_HTML.gif are nonnegative, and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ11_HTML.gif
(1.10)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ12_HTML.gif
(1.11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ13_HTML.gif
(1.12)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ14_HTML.gif
(1.13)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq30_HTML.gif ] where the special case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq31_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ16_HTML.gif
(1.15)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ17_HTML.gif
(1.16)

In this case, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq33_HTML.gif and the corresponding eigenfunction is https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq35_HTML.gif . Multiplying (1.15) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq36_HTML.gif and integrating from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq37_HTML.gif to https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq40_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq41_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq42_HTML.gif . A nonlinear mapping https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq43_HTML.gif is said to be positive if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq44_HTML.gif . It is said to be https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq45_HTML.gif -completely continuous if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq47_HTML.gif to precompact subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq48_HTML.gif . Finally, a positive linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq49_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq51_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq52_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq53_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq54_HTML.gif is a continuous linear operator on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq55_HTML.gif , denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq56_HTML.gif the spectral radius of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq57_HTML.gif . Define
https://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 https://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)

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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq68_HTML.gif with the spectral radius https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq70_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq71_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq72_HTML.gif locally uniformly in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq74_HTML.gif of
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ20_HTML.gif
(1.19)

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

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

Proof.

Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq82_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq84_HTML.gif ] that the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq85_HTML.gif in [14, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq86_HTML.gif ] reduces to a single point https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq89_HTML.gif , the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ23_HTML.gif
(2.1)
has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq90_HTML.gif which is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ24_HTML.gif
(2.2)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq91_HTML.gif , the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ26_HTML.gif
(2.4)
has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq92_HTML.gif which is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ27_HTML.gif
(2.5)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ29_HTML.gif
(2.7)

Let

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

Definition 2.4.

One says that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ30_HTML.gif
(2.8)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ32_HTML.gif
(2.10)

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

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ33_HTML.gif
(2.11)
For https://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
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq101_HTML.gif imply that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ37_HTML.gif
(2.15)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq102_HTML.gif , we may define the norm of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq103_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ38_HTML.gif
(2.16)

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

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

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

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

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

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

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

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

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

Lemma 2.5.

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

Proof.

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

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

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

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

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq147_HTML.gif , define a linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq148_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq149_HTML.gif be the spectral radius of https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq151_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq152_HTML.gif , with a positive eigenfunction https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq154_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq155_HTML.gif can be explicitly given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ57_HTML.gif
(2.35)

and the corresponding eigenfunction https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ58_HTML.gif
(2.36)

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

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

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

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

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

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

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

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq173_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq174_HTML.gif , from Lemma 2.3, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq175_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ65_HTML.gif
(2.43)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq176_HTML.gif , if https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq178_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq179_HTML.gif on https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq182_HTML.gif C]; [17, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq183_HTML.gif ]), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq184_HTML.gif has an algebraically simple eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq185_HTML.gif with an eigenfunction https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq187_HTML.gif with a positive eigenfunction of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq189_HTML.gif by setting
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ68_HTML.gif
(3.1)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq190_HTML.gif is compact.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq191_HTML.gif be such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ70_HTML.gif
(3.3)
Obviously,( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq192_HTML.gif )impliesthat
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ71_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ72_HTML.gif
(3.5)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ73_HTML.gif
(3.6)
then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq193_HTML.gif is nondecreasing and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ74_HTML.gif
(3.7)
Let us consider
https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq195_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq196_HTML.gif by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ79_HTML.gif
(3.12)
locally uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq197_HTML.gif . From https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq200_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq202_HTML.gif of the set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ80_HTML.gif
(3.13)

such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq204_HTML.gif yields a solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq206_HTML.gif crosses the hyperplane https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq208_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq209_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq210_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq211_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ81_HTML.gif
(3.14)

we note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq212_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq213_HTML.gif since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq216_HTML.gif .

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

In this case, we show that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq218_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ83_HTML.gif
(3.16)

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

From (3.16), we have that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq222_HTML.gif . We divide the equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ84_HTML.gif
(3.17)
by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq223_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq224_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq225_HTML.gif is bounded in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq227_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq228_HTML.gif with https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ85_HTML.gif
(3.18)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq230_HTML.gif Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ86_HTML.gif
(3.19)
where https://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,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq232_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq233_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq234_HTML.gif to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq236_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq237_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq239_HTML.gif has a linear minorant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq240_HTML.gif and there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq241_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq242_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq243_HTML.gif .

By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq244_HTML.gif ,there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq245_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq246_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ88_HTML.gif
(3.21)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq247_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ89_HTML.gif
(3.22)
then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq248_HTML.gif is a linear minorant of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq249_HTML.gif . Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ90_HTML.gif
(3.23)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq250_HTML.gif , here, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq252_HTML.gif .

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

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

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

Assume that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq257_HTML.gif , such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq258_HTML.gif ,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_Equ95_HTML.gif
(3.28)

Again https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq259_HTML.gif joins https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297578/MediaObjects/13661_2010_Article_35_IEq260_HTML.gif to https://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

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