Open Access

Multiple Positive Solutions of the Singular Boundary Value Problem for Second-Order Impulsive Differential Equations on the Half-Line

Boundary Value Problems20102010:281908

DOI: 10.1155/2010/281908

Received: 17 November 2009

Accepted: 21 February 2010

Published: 29 March 2010

Abstract

This paper uses a fixed point theorem in cones to investigate the multiple positive solutions of a boundary value problem for second-order impulsive singular differential equations on the half-line. The conditions for the existence of multiple positive solutions are established.

1. Introduction

Consider the following nonlinear singular Sturm-Liouville boundary value problems for second-order impulsive differential equation on thehalf-line:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq6_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq7_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq9_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq10_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq11_HTML.gif , in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq12_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq13_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq15_HTML.gif are, respectively, the left and right limits of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq16_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq19_HTML.gif .

The theory of singular impulsive differential equations has been emerging as an important area of investigation in recent years. For the theory and classical results, we refer the monographs to [1, 2] and the papers [319] to readers. We point out that in a second-order differential equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq20_HTML.gif , one usually considers impulses in the position https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq21_HTML.gif and the velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq22_HTML.gif . However, in the motion of spacecraft one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position [20]. The impulses only on the velocity occur also in impulsive mechanics [21].

In recent paper [3], by using the Krasnoselskii's fixed point theorem, Kaufmann has discussed the existence of solutions for some second-order boundary value problem with impulsive effects on an unbounded domain. In [22] Sun et al. and [23] Liu et al., respectively, discussed the existence and multiple positive solutions for singular Sturm-Liouville boundary value problems for second-order differential equation on the half-line. But the Multiple positive solutions of this case with both singularity and impulses are not to be studied. The aim of this paper is to fill up this gap.

The rest of the paper is organized as follows. In Section 2, we give several important lemmas. The main theorems are formulated and proved in Section 3. And in Section 4, we give an example to demonstrate the application of our results.

2. Several Lemmas

Lemma 2.1 (see [23]).

If conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq24_HTML.gif are satisfied, then the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ2_HTML.gif
(2.1)
has a unique solution for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq25_HTML.gif . Moreover, this unique solution can be expressed in the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ3_HTML.gif
(2.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq26_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ4_HTML.gif
(2.3)

Remark 2.2.

It is easy to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq27_HTML.gif has the following properties:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq28_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq29_HTML.gif

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq30_HTML.gif is continuous differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq31_HTML.gif , except https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq32_HTML.gif ,

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq33_HTML.gif ,

(4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq34_HTML.gif ,

(5) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq35_HTML.gif ,

(6) for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq38_HTML.gif , where

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ5_HTML.gif
(2.4)

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq39_HTML.gif .

For the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq40_HTML.gif , and the corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq41_HTML.gif in Remark 2.2, we define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq42_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq43_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq46_HTML.gif exist, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq47_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq48_HTML.gif    =   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq49_HTML.gif    exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq50_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq53_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq54_HTML.gif is a Banach space with the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq55_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq56_HTML.gif is a positive cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq57_HTML.gif . For details of the cone theory, see [1]. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq58_HTML.gif is called a positive solution of BVP (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq59_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq61_HTML.gif satisfies (1.1).

As we know that the Ascoli-Arzela Theorem does not hold in infinite interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq62_HTML.gif , we need the following compactness criterion:

Lemma 2.3 (see [22]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq63_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq64_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq65_HTML.gif if the following conditions hold.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq66_HTML.gif is uniformly bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq67_HTML.gif .

(ii) The functions from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq68_HTML.gif are equicontinuous on any compact interval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq69_HTML.gif .

(iii) The functions from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq70_HTML.gif are equiconvergent, that is, for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq71_HTML.gif , there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq72_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq73_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq75_HTML.gif .

The main tool of this work is a fixed point theorem in cones.

Lemma 2.4 (see [4]).

Let X be a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq76_HTML.gif is a positive cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq77_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq78_HTML.gif are open subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq79_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq80_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq81_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq82_HTML.gif be a completely continuous operator such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq83_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq84_HTML.gif .

(ii) there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq85_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq86_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq88_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq89_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq90_HTML.gif .

Remark 2.5.

If (i) is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq91_HTML.gif and (ii) is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq92_HTML.gif , then Lemma 2.4 is still true.

Lemma 2.6 (see [3]).

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq93_HTML.gif is a solution of the BVP (1.1) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq94_HTML.gif satisfies the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ6_HTML.gif
(2.5)

The proof of this result is based on the properties of the Green function, so we omit it as elementary.

Define

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ7_HTML.gif
(2.6)

Obviously, the BVP (1.1) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq95_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq96_HTML.gif is a fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq97_HTML.gif defined by (2.6).

Let us list some conditions as follows.

(A1) There exist two nonnegative functions: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq100_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq101_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq103_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq104_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq106_HTML.gif , are continuous.

(A2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq108_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq109_HTML.gif

Lemma 2.7.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq110_HTML.gif are satisfied, then for any bounded open set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq111_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq112_HTML.gif is a completely continuous operator.

Proof.

For any bounded open set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq113_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq114_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq115_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq116_HTML.gif .

First, we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq117_HTML.gif is well defined. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq118_HTML.gif . From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq119_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq120_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq123_HTML.gif and

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ8_HTML.gif
(2.7)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq124_HTML.gif is well defined. For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq125_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ9_HTML.gif
(2.8)
Thus, by the Lebesgue dominated convergence theorem and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq126_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq127_HTML.gif , we have, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq129_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ10_HTML.gif
(2.9)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq130_HTML.gif . By the property (3) of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq131_HTML.gif , it is easy to get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq132_HTML.gif .

On the other hand, by (2.6) we have, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq134_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ11_HTML.gif
(2.10)
Then by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq135_HTML.gif , the property (5) of Remark 2.2 and the Lebesgue dominated convergence theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ12_HTML.gif
(2.11)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq136_HTML.gif .

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq137_HTML.gif , we get

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ13_HTML.gif
(2.12)
So
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ14_HTML.gif
(2.13)
On the other hand, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq138_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ15_HTML.gif
(2.14)

Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq139_HTML.gif .

Next, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq140_HTML.gif is continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq141_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq142_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq143_HTML.gif We prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq144_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq145_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq146_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq147_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ16_HTML.gif
(2.15)
On the other hand, by the continuities of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq148_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq149_HTML.gif and the continuities of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq150_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq151_HTML.gif , for the above https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq152_HTML.gif , there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq153_HTML.gif such that, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq155_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ17_HTML.gif
(2.16)
From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq156_HTML.gif , for the above https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq157_HTML.gif , there exists a sufficiently large number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq158_HTML.gif such that, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq159_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ18_HTML.gif
(2.17)
Therefore, by (2.15)–(2.17), we have, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq160_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ19_HTML.gif
(2.18)

This implies that the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq161_HTML.gif is continuous.

Finally we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq162_HTML.gif is a compact operator. In fact for any bounded set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq163_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq164_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq165_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq166_HTML.gif . Hence, we obtain

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ20_HTML.gif
(2.19)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq167_HTML.gif is uniformly bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq168_HTML.gif .

Given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq169_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq170_HTML.gif , as the proof of (2.9), we can get that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq171_HTML.gif are equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq172_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq173_HTML.gif is arbitrary, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq174_HTML.gif are locally equicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq175_HTML.gif . By (2.6), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq176_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq177_HTML.gif , and the Lebesgue dominated convergence theorem, we have

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ21_HTML.gif
(2.20)

Hence, the functions from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq178_HTML.gif are equiconvergent. By Lemma 2.3, we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq179_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq180_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq181_HTML.gif is completely continuous. This completed the proof of Lemma 2.7.

3. Main Results

For convenience and simplicity in the following discussion, we use the following notations:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ22_HTML.gif
(3.1)

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq182_HTML.gif hold. Then the BVP (1.1) has at least two positive solutions satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq183_HTML.gif if the following conditions hold:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq185_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq186_HTML.gif

(H2) there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq188_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq189_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq190_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq191_HTML.gif .

Proof.

By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq193_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq194_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq195_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ23_HTML.gif
(3.2)
Define the open sets
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ24_HTML.gif
(3.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq196_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq197_HTML.gif . Now we prove that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ25_HTML.gif
(3.4)
If not, then there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq199_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq200_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq201_HTML.gif then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq202_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ26_HTML.gif
(3.5)

This implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq203_HTML.gif , a contradiction. Therefore, (3.4) holds.

That by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq205_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq206_HTML.gif there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq207_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ27_HTML.gif
(3.6)

Define the open sets:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ28_HTML.gif
(3.7)
As the proof of (3.4), we can get that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ29_HTML.gif
(3.8)
On the other hand, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq208_HTML.gif , choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq209_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq210_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ30_HTML.gif
(3.9)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq211_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq212_HTML.gif , for the above https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq213_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq214_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq215_HTML.gif ; thus, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ31_HTML.gif
(3.10)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ32_HTML.gif
(3.11)
Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq217_HTML.gif , we can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ33_HTML.gif
(3.12)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq218_HTML.gif

Thus, we can obtain the existence of two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq219_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq220_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq221_HTML.gif by using Lemma 2.4 and Remark 2.5, respectively.

Using a similar proof of Theorem 3.1, we can get the following conclusions.

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq222_HTML.gif hold. Then the BVP (1.1) has at least two positive solutions satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq223_HTML.gif if the following conditions hold:

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq225_HTML.gif

(H4) there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq227_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq228_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq229_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq230_HTML.gif .

Corollary 3.3.

In Theorems 3.1 and 3.2, if conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq232_HTML.gif are replaced by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq234_HTML.gif , respectively, then the conclusions also hold.

(H1*) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq236_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq237_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq238_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq239_HTML.gif

(H2*) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq242_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq243_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq244_HTML.gif .

Remark 3.4.

Notice that, in the above conclusions, we suppose that the singularity only exist in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq245_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq246_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq247_HTML.gif . If we permit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq248_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq249_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq250_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq251_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq252_HTML.gif , then the discussion will be much more complex. Now we state the corresponding results.

Let us define the following.

(A1*) There exist four nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq254_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq255_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq256_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq257_HTML.gif is nondecreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq258_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq259_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq260_HTML.gif , are continuous.

(A2*) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq263_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq264_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq266_HTML.gif

Theorem 3.5.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq267_HTML.gif hold, then the BVP (1.1) has at least two positive solutions satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq268_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq269_HTML.gif hold.

Proof.

Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq270_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq271_HTML.gif . We only need to proove https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq272_HTML.gif is a completely continuous operator. Then the rest of the proof is the same as that Theorem 3.1. Notice that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ34_HTML.gif
(3.13)

and change https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq273_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq274_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq275_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq276_HTML.gif , then the same as the proof of Lemma 2.7, it is easy to compute that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq277_HTML.gif is a completely continuous operator.

Corresponding to Theorem 3.2 and Corollary 3.3, there are Theorem 3.6 and Corollary 3.7. We just list here without proof.

Theorem 3.6.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq278_HTML.gif hold, then the BVP (1.1) has at least two positive solutions satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq279_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq280_HTML.gif hold.

Corollary 3.7.

In Theorems 3.5 and 3.6, if conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq281_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq282_HTML.gif are replaced by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq283_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq284_HTML.gif , respectively, then the conclusions also hold.

4. Example

To illustrate how our main results can be used in practice we present the following example.

Example 4.1.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ35_HTML.gif
(4.1)

Conclusion 1.

BVP (4.1) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq285_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq286_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq287_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq288_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq289_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq290_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq291_HTML.gif . Then by simple computation we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ36_HTML.gif
(4.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq292_HTML.gif . Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq293_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ37_HTML.gif
(4.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq294_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq295_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq296_HTML.gif are satisfied. It is easy to get that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq297_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq298_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ38_HTML.gif
(4.4)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq299_HTML.gif are satisfied. Therefore, by Corollary 3.3, problem (4.1) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq300_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq301_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq302_HTML.gif . The proof is completed.

Declarations

Acknowledgment

This work is supported by the National Nature Science Foundation of P. R.China (10871063) and Scientific Research Fund of Hunan Provincial Education Department (07A038), partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project no.PGIDIT06PXIB207023PR.

Authors’ Affiliations

(1)
Department of Mathematics, Hunan Normal University
(2)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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© Jing Xiao et al. 2010

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