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

  • Jing Xiao1,

    Affiliated with

    • JuanJ Nieto2 and

      Affiliated with

      • Zhiguo Luo1Email author

        Affiliated with

        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:

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq6_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq7_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq9_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq10_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq11_HTML.gif , in which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq12_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq13_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq14_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq16_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq18_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq21_HTML.gif and the velocity http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq23_HTML.gif and http://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
        http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ3_HTML.gif
        (2.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq26_HTML.gif is defined by
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq27_HTML.gif has the following properties:

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

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

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

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

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

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

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

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

        For the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq40_HTML.gif , and the corresponding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq41_HTML.gif in Remark 2.2, we define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq42_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq43_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq45_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq46_HTML.gif exist, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq47_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq48_HTML.gif    =   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq49_HTML.gif    exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq50_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq51_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq53_HTML.gif . It is easy to see that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq55_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq56_HTML.gif is a positive cone in http://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]. http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq59_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq60_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq63_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq64_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq65_HTML.gif if the following conditions hold.

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

        (ii) The functions from http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq69_HTML.gif .

        (iii) The functions from http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq71_HTML.gif , there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq72_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq73_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq74_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq76_HTML.gif is a positive cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq77_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq78_HTML.gif are open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq79_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq80_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq81_HTML.gif . Let http://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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq83_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq84_HTML.gif .

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

        Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq89_HTML.gif has a fixed point in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq91_HTML.gif and (ii) is satisfied for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq94_HTML.gif satisfies the equation
        http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq95_HTML.gif if and only if http://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 http://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: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq99_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq100_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq101_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq103_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq104_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq106_HTML.gif , are continuous.

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

        Lemma 2.7.

        If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq111_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq113_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq115_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq116_HTML.gif .

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

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ8_HTML.gif
        (2.7)
        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq124_HTML.gif is well defined. For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq125_HTML.gif , we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq126_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq127_HTML.gif , we have, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq129_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ10_HTML.gif
        (2.9)

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq130_HTML.gif . By the property (3) of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq131_HTML.gif , it is easy to get http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq134_HTML.gif ,

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ11_HTML.gif
        (2.10)
        Then by http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ12_HTML.gif
        (2.11)

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

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

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ13_HTML.gif
        (2.12)
        So
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq138_HTML.gif we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ15_HTML.gif
        (2.14)

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

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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq148_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq149_HTML.gif and the continuities of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq150_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq151_HTML.gif , for the above http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq152_HTML.gif , there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq153_HTML.gif such that, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq155_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ17_HTML.gif
        (2.16)
        From http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq156_HTML.gif , for the above http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq157_HTML.gif , there exists a sufficiently large number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq158_HTML.gif such that, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq159_HTML.gif , we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq160_HTML.gif ,
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq161_HTML.gif is continuous.

        Finally we show that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq163_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq164_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq165_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq166_HTML.gif . Hence, we obtain

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

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

        Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq169_HTML.gif , for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq171_HTML.gif are equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq172_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq173_HTML.gif is arbitrary, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq174_HTML.gif are locally equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq175_HTML.gif . By (2.6), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq176_HTML.gif , http://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

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

        Hence, the functions from http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq179_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq180_HTML.gif . Therefore, http://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:

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

        Theorem 3.1.

        Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq183_HTML.gif if the following conditions hold:

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

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

        Proof.

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

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ23_HTML.gif
        (3.2)
        Define the open sets
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ24_HTML.gif
        (3.3)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq196_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq197_HTML.gif . Now we prove that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq199_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq200_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq201_HTML.gif then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq202_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ26_HTML.gif
        (3.5)

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

        Define the open sets:

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

        Therefore, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq219_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq220_HTML.gif satisfying http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq223_HTML.gif if the following conditions hold:

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

        (H4) there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq227_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq228_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq229_HTML.gif , a.e. http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq231_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq232_HTML.gif are replaced by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq233_HTML.gif and http://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*) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq236_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq237_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq238_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq239_HTML.gif

        (H2*) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq241_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq242_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq243_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq245_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq246_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq247_HTML.gif . If we permit http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq248_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq249_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq251_HTML.gif as http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq254_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq255_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq256_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq257_HTML.gif is nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq258_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq259_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq260_HTML.gif , are continuous.

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

        Theorem 3.5.

        Suppose http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq268_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq269_HTML.gif hold.

        Proof.

        Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq270_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq271_HTML.gif . We only need to proove http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_Equ34_HTML.gif
        (3.13)

        and change http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq273_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq274_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq275_HTML.gif , http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq279_HTML.gif , if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq281_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq282_HTML.gif are replaced by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq283_HTML.gif and http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq285_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq286_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq287_HTML.gif .

        Proof.

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

        Hence, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq300_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F281908/MediaObjects/13661_2009_Article_912_IEq301_HTML.gif satisfying http://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

        This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.