Existence of Positive Solutions for Nonlinear Eigenvalue Problems

  • Sheng-Ping Wang1Email author,

    Affiliated with

    • Fu-Hsiang Wong2 and

      Affiliated with

      • Fan-Kai Kung2

        Affiliated with

        Boundary Value Problems20102010:961496

        DOI: 10.1155/2010/961496

        Received: 2 June 2009

        Accepted: 2 February 2010

        Published: 8 March 2010

        Abstract

        We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq2_HTML.gif , with some suitable boundary conditions, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq3_HTML.gif is a parameter.

        1. Introduction

        We consider the existence of positive solutions of the following two-point boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ1_HTML.gif
        (BVP λa)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq5_HTML.gif are nonnegative constants, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq6_HTML.gif .

        In the last thirty years, there are many mathematician considered the boundary value problem (BVP λ ) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq7_HTML.gif , see, for example, Chu et al. [1], Chu et al. [2], Chu and Zhau [3], Chu and Jiang [4], Coffman and Marcus [5], Cohen and Keller [6], Erbe [7], Erbe et al. [8], Erbe and Wang [9], Guo and Lakshmikantham [10], Iffland [11], Njoku and Zanolin [12], Santanilla [13].

        In 1993, Wong [14] showed the following excellent result.

        Theorem 1 A (see [14]).

        Assume that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ2_HTML.gif
        (1.1)
        is an increasing function with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq8_HTML.gif . If there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq9_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ3_HTML.gif
        (1.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq10_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq11_HTML.gif , then, there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq12_HTML.gif such that the boundary value problem (BVP λ ) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq13_HTML.gif has a positive solution in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq14_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq15_HTML.gif , while there is no such solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq16_HTML.gif in which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq17_HTML.gif

        Seeing such facts, we cannot but ask "whether or not we can obtain a similar conclusion for the boundary value problem (BVP λ )." We give a confirm answer to the question.

        First, We observe the following statements.

        (1)Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ4_HTML.gif
        (1.3)

        on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq18_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq19_HTML.gif is the Green's function of the differential equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq20_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq21_HTML.gif with respect to the boundary value condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq22_HTML.gif .

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq23_HTML.gif , is a cone in the Banach space with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq24_HTML.gif .

        In order to discuss our main result, we need the follo wing useful lemmas which due to Lian et al. [15] and Guo and Lakshmikantham [10], respectively.

        Lemma 1 B (see [10]).

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq25_HTML.gif be defined as in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq26_HTML.gif . Then, we have the following results.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq28_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq30_HTML.gif )

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq32_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq34_HTML.gif )

        Lemma 1 C (see [10, Lemmas http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq36_HTML.gif ]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq37_HTML.gif be a real Banach space, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq38_HTML.gif be a cone. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq40_HTML.gif is completely continuous. Then

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq41_HTML.gif if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ5_HTML.gif
        (1.4)

        (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq42_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq43_HTML.gif is the fixed point index of a compact map http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq44_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq45_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq46_HTML.gif , with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq47_HTML.gif .

        2. Main Results

        Now, we can state and prove our main result.

        Theorem 2.1.

        Suppose that there exist two distinct positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq49_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq50_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq52_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ6_HTML.gif
        (2.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ7_HTML.gif
        (2.2)
        Then (BVP λ ) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq53_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq54_HTML.gif between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq56_HTML.gif if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ8_HTML.gif
        (2.3)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ9_HTML.gif
        (2.4)

        Proof.

        It is clear that (BVP λ ) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq57_HTML.gif if, and only if, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq58_HTML.gif is the solution of the operator equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ10_HTML.gif
        (2.5)
        It follows from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq59_HTML.gif in our observation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq60_HTML.gif and Lemma B that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ11_HTML.gif
        (2.6)
        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq61_HTML.gif , which implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq62_HTML.gif . Furthermore, it is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq63_HTML.gif is completely continuous. If there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq64_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq65_HTML.gif , then we obtain the desired result. Thus, we may assume that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ12_HTML.gif
        (2.7)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq67_HTML.gif . We now separate the rest proof into the following three steps.

        Step 1.

        It follows from the definitions of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq69_HTML.gif that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq70_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ13_HTML.gif
        (2.8)
        which implies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ14_HTML.gif
        (2.9)
        Hence, by (2.5),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ15_HTML.gif
        (2.10)
        which implies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ16_HTML.gif
        (2.11)
        Hence
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ17_HTML.gif
        (2.12)
        We now claim that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ18_HTML.gif
        (2.13)
        In fact, if there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq72_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq73_HTML.gif then, by (2.11),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ19_HTML.gif
        (2.14)
        which gives a contradiction. This proves that (2.13) holds. Thus, by Lemma C,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ20_HTML.gif
        (2.15)

        Step 2.

        First, we claim that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ21_HTML.gif
        (2.16)
        Suppose to the contrary that there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq75_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ22_HTML.gif
        (2.17)
        It is clear that (2.17) is equivalent to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ23_HTML.gif
        (2.18)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq77_HTML.gif it follows that there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq78_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ24_HTML.gif
        (2.19)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ25_HTML.gif
        (2.20)
        Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq79_HTML.gif . From http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq80_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq81_HTML.gif , we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq82_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq83_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq85_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq86_HTML.gif . It follows from
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ26_HTML.gif
        (2.21)
        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq87_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq88_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ27_HTML.gif
        (2.22)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ28_HTML.gif
        (2.23)
        Thus
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ29_HTML.gif
        (2.24)
        This contradiction implies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ30_HTML.gif
        (2.25)
        Therefore, by Lemma C,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ31_HTML.gif
        (2.26)

        Step 3.

        It follows from Steps (1) and (2) and the property of the fixed point index (see, for example, [10, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq89_HTML.gif ]) that the proof is complete.

        Remark 2.2.

        It follows from the conclusion of Theorem 2.1 that the positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq90_HTML.gif and nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq91_HTML.gif satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ32_HTML.gif
        (2.27)

        There are many functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq92_HTML.gif and positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq93_HTML.gif satisfying (2.27). For example, Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq95_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq96_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq97_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq98_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq99_HTML.gif and

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ33_HTML.gif
        (2.28)

        Remark 2.3.

        We now define
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ34_HTML.gif
        (2.29)
        A simple calculation shows that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ35_HTML.gif
        (2.30)

        Then, we have the following results.

        (i)Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq100_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq101_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq102_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq103_HTML.gif can be chosen small arbitrarily) such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ36_HTML.gif
        (2.31)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ37_HTML.gif
        (2.32)

        It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq104_HTML.gif .

        (ii)Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq105_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq106_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq107_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq108_HTML.gif can be chosen large arbitrarily) such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ38_HTML.gif
        (2.33)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ39_HTML.gif
        (2.34)

        which satisfies the hypothesis (2.1) of Theorem 2.1.

        (iii)Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq109_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq110_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq111_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq112_HTML.gif can be chosen small arbitrarily) such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ40_HTML.gif
        (2.35)
        Hence,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ41_HTML.gif
        (2.36)

        which satisfies the hypothesis (2.1) of Theorem 2.1.

        (iv)Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq113_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq114_HTML.gif , there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq115_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq116_HTML.gif can be chosen large arbitrarily) such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ42_HTML.gif
        (2.37)

        Hence, we have the following two cases.

        Case i.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq117_HTML.gif is bounded, say
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ43_HTML.gif
        (2.38)
        for some constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq118_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq119_HTML.gif (since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq120_HTML.gif can be chosen large arbitrarily, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq121_HTML.gif can be chosen large arbitrarily, too),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ44_HTML.gif
        (2.39)

        Case ii.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq122_HTML.gif is unbounded, then there exist a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq123_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq124_HTML.gif can be chosen large arbitrarily) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq125_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ45_HTML.gif
        (2.40)
        It follows from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq126_HTML.gif and (2.37) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ46_HTML.gif
        (2.41)

        By Cases (i), (ii) and Remark 2.2, we see that the hypothesis (2.2) of Theorem 2.1 is satisfied if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq127_HTML.gif .

        We immediately conclude the following corollaries.

        Corollary 2.4.

        (BVP λ ) has at least one positive solution for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq128_HTML.gif if one of the following conditions holds:

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq130_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq132_HTML.gif

        Proof.

        It follows from Remark 2.3 and Theorem 2.1 that the desired result holds, immediately.

        Corollary 2.5.

        Let

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq134_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq136_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq137_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq139_HTML.gif .

        Then, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq140_HTML.gif , (BVP λ ) has at least two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq142_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ47_HTML.gif
        (2.42)

        Proof.

        It follows from Remark 2.3 that there exist two real numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq143_HTML.gif satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ48_HTML.gif
        (2.43)
        Hence, by Theorem 2.1 and Remark 2.2, we see that for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq144_HTML.gif , there exist two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq146_HTML.gif of (BVP λ ) such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ49_HTML.gif
        (2.44)

        Thus, we complete the proof.

        Corollary 2.6.

        Let

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq148_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq150_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq151_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq152_HTML.gif .

        Then, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq153_HTML.gif , (BVP λ ) has at least two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq155_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ50_HTML.gif
        (2.45)

        Proof.

        It follows from Remark 2.3 that there exist two real numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq156_HTML.gif satisfying
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ51_HTML.gif
        (2.46)
        Hence, by Theorem 2.1 and Remark 2.2, we see that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq157_HTML.gif , (BVP λ ) has two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq159_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ52_HTML.gif
        (2.47)

        Thus, we completed the proof.

        3. Examples

        To illustrate the usage of our results, we present the following examples.

        Example 3.1.

        Consider the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ53_HTML.gif
        (BVP.1)
        Clearly,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ54_HTML.gif
        (3.1)

        If we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq160_HTML.gif , then it follows from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq161_HTML.gif of Corollary 2.4 that (BVP.1) has a solution if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq162_HTML.gif .

        Example 3.2.

        Consider the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ55_HTML.gif
        (BVP.2)
        Clearly,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ56_HTML.gif
        (3.2)

        If we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq163_HTML.gif , then it follows from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq164_HTML.gif of Corollary 2.4 that (BVP.2) has a solution if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq165_HTML.gif .

        Example 3.3.

        Consider the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ57_HTML.gif
        (BVP.3)
        Clearly, if we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq167_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_Equ58_HTML.gif
        (3.3)

        Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F961496/MediaObjects/13661_2009_Article_970_IEq168_HTML.gif .

        Authors’ Affiliations

        (1)
        Holistic Education Center, Cardinal Tien College of Healthcare and Management
        (2)
        Department of Mathematics, National Taipei University of Education

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        Copyright

        © Sheng-PingWang 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.