Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations

Boundary Value Problems20112011:594128

DOI: 10.1155/2011/594128

Received: 16 October 2010

Accepted: 27 January 2011

Published: 21 February 2011

Abstract

This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq1_HTML.gif Our nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq2_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq3_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq4_HTML.gif .

1. Introduction

Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on. Therefore, singular boundary value problems have been investigated extensively in recent years (see [14] and references therein).

This paper investigates the following fourth-order nonlinear singular eigenvalue problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq5_HTML.gif is a parameter and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq6_HTML.gif satisfies the following hypothesis:

() http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq8_HTML.gif , and there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq13_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq16_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ2_HTML.gif
(1.2)

Typical functions that satisfy the above sublinear hypothesis ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq17_HTML.gif ) are those taking the form

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ3_HTML.gif
(1.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq20_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq22_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq24_HTML.gif . The hypothesis ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq25_HTML.gif ) is similar to that in [5, 6].

Because of the extensive applications in mechanics and engineering, nonlinear fourth-order two-point boundary value problems have received wide attentions (see [712] and references therein). In mechanics, the boundary value problem (1.1) (BVP (1.1) for short) describes the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. The term http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq26_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq27_HTML.gif represents bending effect which is useful for the stability analysis of the beam. BVP (1.1) has two special features. The first one is that the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq28_HTML.gif may depend on the first-order derivative of the unknown function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq29_HTML.gif , and the second one is that the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq30_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq31_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq32_HTML.gif .

In this paper, we study the existence of positive solutions and the structure of positive solution set for the BVP (1.1). Firstly, we construct a special cone and present a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq33_HTML.gif . Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory.

By singularity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq34_HTML.gif , we mean that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq35_HTML.gif in (1.1) is allowed to be unbounded at the points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq38_HTML.gif , and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq39_HTML.gif . A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq40_HTML.gif is called a (positive) solution of the BVP (1.1) if it satisfies the BVP (1.1) ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq41_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq43_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq44_HTML.gif ). For some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq45_HTML.gif , if the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq46_HTML.gif (1.1) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq47_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq48_HTML.gif is called an eigenvalue and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq49_HTML.gif is called corresponding eigenfunction of the BVP (1.1).

The existence of positive solutions of BVPs has been studied by several authors in the literature; for example, see [720] and the references therein. Yao [15, 18] studied the following BVP:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ4_HTML.gif
(1.4)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq50_HTML.gif is a closed subset and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq51_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq52_HTML.gif . In [15], he obtained a sufficient condition for the existence of positive solutions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq53_HTML.gif (1.4) by using the monotonically iterative technique. In [13, 18], he applied Guo-Krasnosel'skii's fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP (1.4) and the following BVP:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ5_HTML.gif
(1.5)

These differ from our problem because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq54_HTML.gif in (1.4) cannot be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq55_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq56_HTML.gif and the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq57_HTML.gif in (1.5) does not depend on the derivatives of the unknown functions.

In this paper, we first establish a necessary and sufficient condition for the existence of positive solutions of BVP (1.1) for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq58_HTML.gif by using the following Lemma 1.1. Efforts to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs by the lower and upper solution method can be found, for example, in [5, 6, 2123]. In [5, 6, 22, 23] they considered the case that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq59_HTML.gif depends on even order derivatives of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq60_HTML.gif . Although the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq61_HTML.gif in [21] depends on the first-order derivative, where the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq62_HTML.gif is increasing with respect to the unknown function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq63_HTML.gif . Papers [24, 25] derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq64_HTML.gif does not depend on the derivatives of the unknown functions, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq65_HTML.gif is decreasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq66_HTML.gif .

Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated (see [2628] and references therein). Ma and An [26] and Ma and Xu [27] discussed the global structure of positive solutions for the nonlinear eigenvalue problems and obtained the existence of an unbounded connected branch of positive solution set by using global bifurcation theorems (see [29, 30]). The terms http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq67_HTML.gif in [26] and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq68_HTML.gif in [27] are not singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq71_HTML.gif . Yao [14] obtained one or two positive solutions to a singular elastic beam equation rigidly fixed at both ends by using Guo-Krasnosel'skii's fixed point theorem, but the global structure of positive solutions was not considered. Since the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq72_HTML.gif in BVP (1.1) may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq73_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq74_HTML.gif , the global bifurcation theorems in [29, 30] do not apply to our problem here. In Section 4, we also investigate the global structure of positive solutions for BVP (1.1) by applying the following Lemma 1.2.

The paper is organized as follows: in the rest of this section, two known results are stated. In Section 2, some lemmas are stated and proved. In Section 3, we establish a necessary and sufficient condition for the existence of positive solutions. In Section 4, we prove that the closure of positive solution set possesses an unbounded connected branch which comes from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq75_HTML.gif .

Finally we state the following results which will be used in Sections 3 and 4, respectively.

Lemma 1.1 (see [31]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq76_HTML.gif be a real Banach space, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq77_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq78_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq80_HTML.gif be bounded open sets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq82_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq83_HTML.gif is completely continuous such that one of the following two conditions is satisfied:

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq84_HTML.gif

Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq92_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq93_HTML.gif .

Lemma 1.2 (see [32]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq94_HTML.gif be a metric space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq95_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq96_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq97_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ6_HTML.gif
(1.6)

Suppose also that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq98_HTML.gif is a family of connected subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq99_HTML.gif , satisfying the following conditions:

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq102_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq103_HTML.gif .

(2)For any two given numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq105_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq106_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq107_HTML.gif is a relatively compact set of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq108_HTML.gif .

Then there exists a connected branch http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq109_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq110_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ7_HTML.gif
(1.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq111_HTML.gif there exists a sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq112_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq113_HTML.gif .

2. Some Preliminaries and Lemmas

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq114_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq115_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq116_HTML.gif is a Banach space, where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq117_HTML.gif Define
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ8_HTML.gif
(2.1)
It is easy to conclude that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq118_HTML.gif is a cone of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq119_HTML.gif . Denote
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ9_HTML.gif
(2.2)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ10_HTML.gif
(2.3)
Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq120_HTML.gif is the Green function of homogeneous boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ11_HTML.gif
(2.4)

Lemma 2.1.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq122_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq123_HTML.gif have the following properties:

(1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq124_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq126_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq127_HTML.gif .

(2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq130_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq131_HTML.gif ), for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq132_HTML.gif .

(3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq133_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq134_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq135_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq136_HTML.gif .

(4) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq137_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq139_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq140_HTML.gif .

Proof.

From (2.4), it is easy to obtain the property (2.18).

We now prove that property (2) is true. For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq141_HTML.gif , by (2.4), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ12_HTML.gif
(2.5)
For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq142_HTML.gif , by (2.4), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ13_HTML.gif
(2.6)

Consequently, property (2) holds.

From property (2), it is easy to obtain property (3).

We next show that property (4) is true. From (2.4), we know that property (4) holds for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq143_HTML.gif .

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq144_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq145_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ14_HTML.gif
(2.7)
if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq146_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ15_HTML.gif
(2.8)

Therefore, property (4) holds.

Lemma 2.2.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq147_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq148_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ16_HTML.gif
(2.9)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ17_HTML.gif
(2.10)

Proof.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq149_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq151_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq152_HTML.gif , so
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ18_HTML.gif
(2.11)
Therefore, (2.9) holds. From (2.9), we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ19_HTML.gif
(2.12)
By (2.9) and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq153_HTML.gif , we can obtain that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ20_HTML.gif
(2.13)

Thus, (2.10) holds.

For any fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq154_HTML.gif , define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq155_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ21_HTML.gif
(2.14)
Then, it is easy to know that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ22_HTML.gif
(2.15)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ23_HTML.gif
(2.16)

Lemma 2.3.

Suppose that ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq156_HTML.gif ) and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ24_HTML.gif
(2.17)

hold. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq157_HTML.gif .

Proof.

From ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq158_HTML.gif ), for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq159_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq160_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq161_HTML.gif , we easily obtain the following inequalities:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ25_HTML.gif
(2.18)
For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq163_HTML.gif , choose positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq164_HTML.gif min http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq165_HTML.gif . It follows from ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq167_HTML.gif ), (2.10), Lemma 2.1, and (2.17) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ26_HTML.gif
(2.19)
Similar to (2.19), from ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq168_HTML.gif ), (2.10), Lemma 2.1, and (2.17), for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq170_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ27_HTML.gif
(2.20)

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq171_HTML.gif is well defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq172_HTML.gif .

From (2.4) and (2.14)–(2.16), it is easy to know that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ28_HTML.gif
(2.21)

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq173_HTML.gif follows from (2.21).

Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq174_HTML.gif is a positive solution of BVP (1.1) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq175_HTML.gif is a positive fixed point of the integral operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq176_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq177_HTML.gif .

Lemma 2.4.

Suppose that ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq178_HTML.gif ) and (2.17) hold. Then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq179_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq180_HTML.gif is completely continuous.

Proof.

First of all, notice that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq181_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq182_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq183_HTML.gif by Lemma 2.3.

Next, we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq184_HTML.gif is bounded. In fact, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq185_HTML.gif , by (2.10) we can get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ29_HTML.gif
(2.22)
Choose positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq186_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq187_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq188_HTML.gif . This, together with ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq189_HTML.gif ), (2.22), (2.16), and Lemma 2.1 yields that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ30_HTML.gif
(2.23)

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq190_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq191_HTML.gif .

Now we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq192_HTML.gif is a compact operator on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq193_HTML.gif . By (2.23) and Ascoli-Arzela theorem, it suffices to show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq194_HTML.gif is equicontinuous for arbitrary bounded subset http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq195_HTML.gif .

Since for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq196_HTML.gif , (2.22) holds, we may choose still positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq198_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq199_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ31_HTML.gif
(2.24)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq200_HTML.gif . Notice that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ32_HTML.gif
(2.25)
Thus for any given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq201_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq202_HTML.gif and for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq203_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ33_HTML.gif
(2.26)

From (2.25), (2.26), and the absolute continuity of integral function, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq204_HTML.gif is equicontinuous.

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq205_HTML.gif is relatively compact, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq206_HTML.gif is a compact operator on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq207_HTML.gif .

Finally, we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq208_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq209_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq210_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq211_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq212_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq215_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq216_HTML.gif uniformly, with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq217_HTML.gif . From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq218_HTML.gif , choose still positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq221_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ34_HTML.gif
(2.27)
By (2.17), we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq222_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq223_HTML.gif . Thus, from the Lebesgue dominated convergence theorem, it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ35_HTML.gif
(2.28)

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq224_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq225_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq226_HTML.gif is completely continuous.

3. A Necessary and Sufficient Condition for Existence of Positive Solutions

In this section, by using the fixed point theorem of cone, we establish the following necessary and sufficient condition for the existence of positive solutions for BVP (1.1).

Theorem 3.1.

Suppose ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq227_HTML.gif ) holds, then BVP (1.1) has at least one positive solution for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq228_HTML.gif if and only if the integral inequality (2.17) holds.

Proof.

Suppose first that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq229_HTML.gif be a positive solution of BVP (1.1) for any fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq230_HTML.gif . Then there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq231_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq232_HTML.gif ) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq233_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq234_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ36_HTML.gif
(3.1)
In fact, it follows from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq236_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq237_HTML.gif , that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq238_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq240_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq241_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq242_HTML.gif . By the concavity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq244_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ37_HTML.gif
(3.2)
On the other hand,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ38_HTML.gif
(3.3)

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq245_HTML.gif let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq246_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq247_HTML.gif then (3.1) holds.

Choose positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq248_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq249_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq250_HTML.gif . This, together with ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq251_HTML.gif ), (1.2), and (2.18) yields that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ39_HTML.gif
(3.4)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq252_HTML.gif . Hence, integrating (3.4) from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq253_HTML.gif to 1, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ40_HTML.gif
(3.5)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq254_HTML.gif increases on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq255_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ41_HTML.gif
(3.6)
that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ42_HTML.gif
(3.7)
Notice that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq256_HTML.gif , integrating (3.7) from 0 to 1, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ43_HTML.gif
(3.8)
That is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ44_HTML.gif
(3.9)
Thus,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ45_HTML.gif
(3.10)
By an argument similar to the one used in deriving (3.5), we can obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ46_HTML.gif
(3.11)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq257_HTML.gif . So,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ47_HTML.gif
(3.12)
Integrating (3.12) from 0 to 1, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ48_HTML.gif
(3.13)
That is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ49_HTML.gif
(3.14)
So,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ50_HTML.gif
(3.15)

This and (3.10) imply that (2.17) holds.

Now assume that (2.17) holds, we will show that BVP (1.1) has at least one positive solution for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq258_HTML.gif . By (2.17), there exists a sufficient small http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq259_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ51_HTML.gif
(3.16)
For any fixed http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq260_HTML.gif , first of all, we prove
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ52_HTML.gif
(3.17)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq261_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq262_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ53_HTML.gif
(3.18)
From Lemma 2.1, (3.18), and ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq263_HTML.gif ), we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ54_HTML.gif
(3.19)

Thus, (3.17) holds.

Next, we claim that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ55_HTML.gif
(3.20)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq264_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq265_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq266_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ56_HTML.gif
(3.21)
Therefore, by Lemma 2.1 and ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq267_HTML.gif ), it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ57_HTML.gif
(3.22)

This implies that (3.20) holds.

By Lemmas 1.1 and 2.4, (3.17), and (3.20), we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq268_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq269_HTML.gif . Therefore, BVP (1.1) has a positive solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq270_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq271_HTML.gif .

4. Unbounded Connected Branch of Positive Solutions

In this section, we study the global continua results under the hypotheses ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq272_HTML.gif ) and (2.17). Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ58_HTML.gif
(4.1)

then, by Theorem 3.1, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq273_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq274_HTML.gif .

Theorem 4.1.

Suppose ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq275_HTML.gif ) and (2.17) hold, then the closure http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq276_HTML.gif of positive solution set possesses an unbounded connected branch http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq277_HTML.gif which comes from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq278_HTML.gif such that

(i)for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq279_HTML.gif , and

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq280_HTML.gif

Proof.

We now prove our conclusion by the following several steps.

First, we prove that for arbitrarily given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq281_HTML.gif is bounded. In fact, let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ59_HTML.gif
(4.2)
then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq282_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq283_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ60_HTML.gif
(4.3)
Therefore, by Lemma 2.1 and ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq284_HTML.gif ), it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ61_HTML.gif
(4.4)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ62_HTML.gif
(4.5)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq285_HTML.gif is given by (3.16). Then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq286_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq287_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ63_HTML.gif
(4.6)
Therefore, by Lemma 2.1 and ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq288_HTML.gif ), it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ64_HTML.gif
(4.7)

Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq289_HTML.gif has no positive solution in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq290_HTML.gif . As a consequence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq291_HTML.gif is bounded.

By the complete continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq292_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq293_HTML.gif is compact.

Second, we choose sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq294_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq295_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ65_HTML.gif
(4.8)
We are to prove that for any positive integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq296_HTML.gif , there exists a connected branch http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq297_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq298_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ66_HTML.gif
(4.9)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq299_HTML.gif be fixed, suppose that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq300_HTML.gif , the connected branch http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq301_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq302_HTML.gif , passing through http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq303_HTML.gif , leads to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq304_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq305_HTML.gif is compact, there exists a bounded open subset http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq306_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq307_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq308_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq309_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq310_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq311_HTML.gif and later http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq312_HTML.gif denote the closure and boundary of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq313_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq314_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq315_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq316_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq317_HTML.gif are two disjoint closed subsets of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq318_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq319_HTML.gif is a compact metric space, there are two disjoint compact subsets http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq320_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq321_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq322_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq323_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq324_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq325_HTML.gif . Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq326_HTML.gif . Denoting by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq327_HTML.gif the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq328_HTML.gif -neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq329_HTML.gif and letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq330_HTML.gif , then it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ67_HTML.gif
(4.10)

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq331_HTML.gif , then taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq332_HTML.gif .

It is obvious that in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq333_HTML.gif , the family of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq334_HTML.gif makes up an open covering of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq335_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq336_HTML.gif is a compact set, there exists a finite subfamily http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq337_HTML.gif which also covers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq338_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq339_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ68_HTML.gif
(4.11)
Hence, by the homotopy invariance of the fixed point index, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ69_HTML.gif
(4.12)
By the first step of this proof, the construction of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq340_HTML.gif , (4.4), and (4.7), it follows easily that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq341_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ70_HTML.gif
(4.13)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ71_HTML.gif
(4.14)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ72_HTML.gif
(4.15)

However, by the excision property and additivity of the fixed point index, we have from (4.12) and (4.14) that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq342_HTML.gif , which contradicts (4.15). Hence, there exists some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq343_HTML.gif such that the connected branch http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq344_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq345_HTML.gif containing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq346_HTML.gif satisfies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq347_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq348_HTML.gif be the connected branch of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq349_HTML.gif including http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq350_HTML.gif , then this http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq351_HTML.gif satisfies (4.9).

By Lemma 1.2, there exists a connected branch http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq352_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq353_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq354_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq355_HTML.gif . Noticing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq356_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq357_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq358_HTML.gif be the connected branch of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq359_HTML.gif including http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq360_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq361_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq362_HTML.gif . Similar to (4.4) and (4.7), for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq363_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq364_HTML.gif , we have, by ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq365_HTML.gif ), (4.2), (4.3), (4.5), (4.6), and Lemma 2.1,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ73_HTML.gif
(4.16)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ74_HTML.gif
(4.17)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq366_HTML.gif is given by (3.16). Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq367_HTML.gif in (4.16) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq368_HTML.gif in (4.17), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ75_HTML.gif
(4.18)

Therefore, Theorem 4.1 holds and the proof is complete.

Declarations

Acknowledgments

This work is carried out while the author is visiting the University of New England. The author thanks Professor Yihong Du for his valuable advices and the Department of Mathematics for providing research facilities. The author also thanks the anonymous referees for their carefully reading of the first draft of the manuscript and making many valuable suggestions. Research is supported by the NSFC (10871120) and HESTPSP (J09LA08).

Authors’ Affiliations

(1)
School of Mathematical Sciences, Shandong Normal University

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© Huiqin Lu. 2011

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