Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq1_HTML.gif Our nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq2_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq3_HTML.gif and/or https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ1_HTML.gif
(1.1)

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

() https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq8_HTML.gif , and there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq13_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq16_HTML.gif satisfies
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq17_HTML.gif ) are those taking the form

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

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq20_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq24_HTML.gif . The hypothesis ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq26_HTML.gif in https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq30_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq31_HTML.gif and/or https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq34_HTML.gif , we mean that the function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq38_HTML.gif , and/or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq39_HTML.gif . A function https://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) ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq41_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq43_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq44_HTML.gif ). For some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq45_HTML.gif , if the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq46_HTML.gif (1.1) has a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq47_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq48_HTML.gif is called an eigenvalue and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ4_HTML.gif
(1.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq50_HTML.gif is a closed subset and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq51_HTML.gif , https://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 https://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:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq56_HTML.gif and the nonlinearity https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq59_HTML.gif depends on even order derivatives of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq60_HTML.gif . Although the nonlinearity https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq65_HTML.gif is decreasing with respect to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq67_HTML.gif in [26] and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq68_HTML.gif in [27] are not singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq70_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq73_HTML.gif and/or https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq76_HTML.gif be a real Banach space, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq77_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq78_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq79_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq80_HTML.gif be bounded open sets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq81_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq82_HTML.gif . Suppose that https://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:

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

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

Lemma 1.2 (see [32]).

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

Suppose also that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq99_HTML.gif , satisfying the following conditions:

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

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

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

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

2. Some Preliminaries and Lemmas

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq114_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq115_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq116_HTML.gif is a Banach space, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq117_HTML.gif Define
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq118_HTML.gif is a cone of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq119_HTML.gif . Denote
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ9_HTML.gif
(2.2)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ10_HTML.gif
(2.3)
Then https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ11_HTML.gif
(2.4)

Lemma 2.1.

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

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

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

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

(4) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq137_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq139_HTML.gif , for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq141_HTML.gif , by (2.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ12_HTML.gif
(2.5)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq142_HTML.gif , by (2.4), we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq143_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq144_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq145_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ14_HTML.gif
(2.7)
if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq146_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq147_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq148_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ16_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ17_HTML.gif
(2.10)

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq149_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq151_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq152_HTML.gif , so
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq153_HTML.gif , we can obtain that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq154_HTML.gif , define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq155_HTML.gif by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ22_HTML.gif
(2.15)
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq156_HTML.gif ) and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ24_HTML.gif
(2.17)

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

Proof.

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

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq171_HTML.gif is well defined on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ28_HTML.gif
(2.21)

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

Obviously, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq176_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq177_HTML.gif .

Lemma 2.4.

Suppose that ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq179_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq181_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq182_HTML.gif into https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq184_HTML.gif is bounded. In fact, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq185_HTML.gif , by (2.10) we can get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ29_HTML.gif
(2.22)
Choose positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq186_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq187_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq188_HTML.gif . This, together with ( https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ30_HTML.gif
(2.23)

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

Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq192_HTML.gif is a compact operator on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq194_HTML.gif is equicontinuous for arbitrary bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq195_HTML.gif .

Since for each https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq199_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ31_HTML.gif
(2.24)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq200_HTML.gif . Notice that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ32_HTML.gif
(2.25)
Thus for any given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq201_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq202_HTML.gif and for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq203_HTML.gif , we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq204_HTML.gif is equicontinuous.

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

Finally, we show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq208_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq209_HTML.gif . Suppose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq211_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq212_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq215_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq216_HTML.gif uniformly, with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq217_HTML.gif . From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq218_HTML.gif , choose still positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq221_HTML.gif . Then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq222_HTML.gif is integrable on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ35_HTML.gif
(2.28)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq224_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq225_HTML.gif . Therefore, https://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 ( https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq230_HTML.gif . Then there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq231_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq232_HTML.gif ) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq233_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq234_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq236_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq237_HTML.gif , that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq238_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq239_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq240_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq241_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq242_HTML.gif . By the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq243_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq244_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ37_HTML.gif
(3.2)
On the other hand,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ38_HTML.gif
(3.3)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq245_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq246_HTML.gif and let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq248_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq249_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq250_HTML.gif . This, together with ( https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ39_HTML.gif
(3.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq252_HTML.gif . Hence, integrating (3.4) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq253_HTML.gif to 1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ40_HTML.gif
(3.5)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq254_HTML.gif increases on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq255_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ41_HTML.gif
(3.6)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ42_HTML.gif
(3.7)
Notice that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ43_HTML.gif
(3.8)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ44_HTML.gif
(3.9)
Thus,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ46_HTML.gif
(3.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq257_HTML.gif . So,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ48_HTML.gif
(3.13)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ49_HTML.gif
(3.14)
So,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq259_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ51_HTML.gif
(3.16)
For any fixed https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq260_HTML.gif , first of all, we prove
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ52_HTML.gif
(3.17)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq262_HTML.gif , then
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq263_HTML.gif ), we get
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ55_HTML.gif
(3.20)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq265_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq266_HTML.gif , we get
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq267_HTML.gif ), it follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq268_HTML.gif has a fixed point in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq270_HTML.gif for any https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq272_HTML.gif ) and (2.17). Let
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq273_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq274_HTML.gif .

Theorem 4.1.

Suppose ( https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq277_HTML.gif which comes from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq278_HTML.gif such that

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

(ii) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq281_HTML.gif is bounded. In fact, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ59_HTML.gif
(4.2)
then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq282_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq283_HTML.gif , we get
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq284_HTML.gif ), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ61_HTML.gif
(4.4)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ62_HTML.gif
(4.5)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq287_HTML.gif , we get
https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq288_HTML.gif ), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ64_HTML.gif
(4.7)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq289_HTML.gif has no positive solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq290_HTML.gif . As a consequence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq292_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq293_HTML.gif is compact.

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

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

It is obvious that in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq333_HTML.gif , the family of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq334_HTML.gif makes up an open covering of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq335_HTML.gif . Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq337_HTML.gif which also covers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq338_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq339_HTML.gif , then
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq341_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ70_HTML.gif
(4.13)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ71_HTML.gif
(4.14)
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq343_HTML.gif such that the connected branch https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq344_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq345_HTML.gif containing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq346_HTML.gif satisfies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq347_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq348_HTML.gif be the connected branch of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq349_HTML.gif including https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq350_HTML.gif , then this https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq352_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq353_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq354_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq355_HTML.gif . Noticing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq356_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq357_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq358_HTML.gif be the connected branch of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq359_HTML.gif including https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq360_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq361_HTML.gif for any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq363_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq364_HTML.gif , we have, by ( https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ73_HTML.gif
(4.16)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_Equ74_HTML.gif
(4.17)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq366_HTML.gif is given by (3.16). Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq367_HTML.gif in (4.16) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq368_HTML.gif in (4.17), we have
https://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

References

  1. Agarwal RP, O'Regan D: Nonlinear superlinear singular and nonsingular second order boundary value problems. Journal of Differential Equations 1998, 143(1):60-95. 10.1006/jdeq.1997.3353View ArticleMathSciNetGoogle Scholar
  2. Liu L, Kang P, Wu Y, Wiwatanapataphee B: Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(3):485-498. 10.1016/j.na.2006.11.014View ArticleMathSciNetGoogle Scholar
  3. O'Regan D: Theory of Singular Boundary Value Problems. World Scientific, River Edge, NJ, USA; 1994:xii+154.View ArticleGoogle Scholar
  4. Zhang Y: Positive solutions of singular sublinear Emden-Fowler boundary value problems. Journal of Mathematical Analysis and Applications 1994, 185(1):215-222. 10.1006/jmaa.1994.1243View ArticleMathSciNetGoogle Scholar
  5. Wei Z:Existence of positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq369_HTML.gif th-order singular sublinear boundary value problems. Journal of Mathematical Analysis and Applications 2005, 306(2):619-636. 10.1016/j.jmaa.2004.10.037View ArticleMathSciNetGoogle Scholar
  6. Wei Z, Pang C:The method of lower and upper solutions for fourth order singular https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq370_HTML.gif -point boundary value problems. Journal of Mathematical Analysis and Applications 2006, 322(2):675-692. 10.1016/j.jmaa.2005.09.064View ArticleMathSciNetGoogle Scholar
  7. Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. Journal of Mathematical Analysis and Applications 1986, 116(2):415-426. 10.1016/S0022-247X(86)80006-3View ArticleMathSciNetGoogle Scholar
  8. Agarwal RP: On fourth order boundary value problems arising in beam analysis. Differential and Integral Equations 1989, 2(1):91-110.MathSciNetGoogle Scholar
  9. Bai Z: The method of lower and upper solutions for a bending of an elastic beam equation. Journal of Mathematical Analysis and Applications 2000, 248(1):195-202. 10.1006/jmaa.2000.6887View ArticleMathSciNetGoogle Scholar
  10. Franco D, O'Regan D, Perán J: Fourth-order problems with nonlinear boundary conditions. Journal of Computational and Applied Mathematics 2005, 174(2):315-327. 10.1016/j.cam.2004.04.013View ArticleMathSciNetGoogle Scholar
  11. Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis 1988, 26(4):289-304. 10.1080/00036818808839715View ArticleMathSciNetGoogle Scholar
  12. Li Y: On the existence of positive solutions for the bending elastic beam equations. Applied Mathematics and Computation 2007, 189(1):821-827. 10.1016/j.amc.2006.11.144View ArticleMathSciNetGoogle Scholar
  13. Yao Q: Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(5-6):1570-1580. 10.1016/j.na.2007.07.002View ArticleGoogle Scholar
  14. Yao Q: Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2683-2694. 10.1016/j.na.2007.08.043View ArticleMathSciNetGoogle Scholar
  15. Yao Q: Monotonically iterative method of nonlinear cantilever beam equations. Applied Mathematics and Computation 2008, 205(1):432-437. 10.1016/j.amc.2008.08.044View ArticleMathSciNetGoogle Scholar
  16. Yao Q: Solvability of singular cantilever beam equation. Annals of Differential Equations 2008, 24(1):93-99.MathSciNetGoogle Scholar
  17. Yao QL: Positive solution to a singular equation for a beam which is simply supported at left and clamped at right by sliding clamps. Journal of Yunnan University. Natural Sciences 2009, 31(2):109-113.MathSciNetGoogle Scholar
  18. Yao QL: Existence and multiplicity of positive solutions to a class of nonlinear cantilever beam equations. Journal of Systems Science & Mathematical Sciences 2009, 29(1):63-69.Google Scholar
  19. Yao QL: Positive solutions to a class of singular elastic beam equations rigidly fixed at both ends. Journal of Wuhan University. Natural Science Edition 2009, 55(2):129-133.MathSciNetGoogle Scholar
  20. Yao Q: Existence of solution to a singular beam equation fixed at left and clamped at right by sliding clamps. Journal of Natural Science. Nanjing Normal University 2007, 9(1):1-5.MathSciNetGoogle Scholar
  21. Graef JR, Kong L: A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(11):2389-2412. 10.1016/j.na.2006.03.028View ArticleMathSciNetGoogle Scholar
  22. Xu Y, Li L, Debnath L: A necessary and sufficient condition for the existence of positive solutions of singular boundary value problems. Applied Mathematics Letters 2005, 18(8):881-889. 10.1016/j.aml.2004.07.029View ArticleMathSciNetGoogle Scholar
  23. Zhao J, Ge W: A necessary and sufficient condition for the existence of positive solutions to a kind of singular three-point boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3973-3980. 10.1016/j.na.2009.02.067View ArticleMathSciNetGoogle Scholar
  24. Zhao ZQ: Positive solutions of boundary value problems for nonlinear singular differential equations. Acta Mathematica Sinica 2000, 43(1):179-188.MathSciNetGoogle Scholar
  25. Zhao Z:On the existence of positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F594128/MediaObjects/13661_2010_Article_49_IEq371_HTML.gif -order singular boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(11):2553-2561. 10.1016/j.na.2005.09.003View ArticleMathSciNetGoogle Scholar
  26. Ma R, An Y: Global structure of positive solutions for nonlocal boundary value problems involving integral conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(10):4364-4376. 10.1016/j.na.2009.02.113View ArticleMathSciNetGoogle Scholar
  27. Ma R, Xu J: Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(1):113-122. 10.1016/j.na.2009.06.061View ArticleMathSciNetGoogle Scholar
  28. Ma RY, Thompson B: Nodal solutions for a nonlinear fourth-order eigenvalue problem. Acta Mathematica Sinica 2008, 24(1):27-34. 10.1007/s10114-007-1009-6View ArticleMathSciNetGoogle Scholar
  29. Dancer E: Global solutions branches for positive maps. Archive for Rational Mechanics and Analysis 1974, 55: 207-213. 10.1007/BF00281748View ArticleMathSciNetGoogle Scholar
  30. Rabinowitz PH: Some aspects of nonlinear eigenvalue problems. The Rocky Mountain Journal of Mathematics 1973, 3(2):161-202. 10.1216/RMJ-1973-3-2-161View ArticleMathSciNetGoogle Scholar
  31. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar
  32. Sun JX: A theorem in point set topology. Journal of Systems Science & Mathematical Sciences 1987, 7(2):148-150.Google Scholar

Copyright

© Huiqin Lu. 2011

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