Unbounded Solutions of Second-Order Multipoint Boundary Value Problem on the Half-Line

  • Lishan Liu1, 2Email author,

    Affiliated with

    • Xinan Hao1 and

      Affiliated with

      • Yonghong Wu2

        Affiliated with

        Boundary Value Problems20102010:236560

        DOI: 10.1155/2010/236560

        Received: 14 May 2010

        Accepted: 11 October 2010

        Published: 18 October 2010

        Abstract

        This paper investigates the second-order multipoint boundary value problem on the half-line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq9_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq10_HTML.gif is continuous. We establish sufficient conditions to guarantee the existence of unbounded solution in a special function space by using nonlinear alternative of Leray-Schauder type. Under the condition that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq11_HTML.gif is nonnegative, the existence and uniqueness of unbounded positive solution are obtained based upon the fixed point index theory and Banach contraction mapping principle. Examples are also given to illustrate the main results.

        1. Introduction

        In this paper, we consider the following second-order multipoint boundary value problem on the half-line
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq12_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq13_HTML.gif is continuous, in which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq14_HTML.gif .

        The study of multipoint boundary value problems (BVPs) for second-order differential equations was initiated by Bicadze and Samarskĭ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multi-point BVPs due to their theoretical challenge and great application potential. Many results on the existence of (positive) solutions for multi-point BVPs have been obtained, and for more details the reader is referred to [510] and the references therein. The BVPs on the half-line arise naturally in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium [1113] and have been also widely studied [1427]. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq15_HTML.gif , BVP (1.1) reduces to the following three-point BVP on the half-line:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ2_HTML.gif
        (1.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq16_HTML.gif . Lian and Ge [16] only studied the solvability of BVP (1.2) by the Leray-Schauder continuation theorem. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq18_HTML.gif , and nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq19_HTML.gif is variable separable, BVP (1.1) reduces to the second order two-point BVP on the half-line
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ3_HTML.gif
        (1.3)

        Yan et al. [17] established the results of existence and multiplicity of positive solutions to the BVP (1.3) by using lower and upper solutions technique.

        Motivated by the above works, we will study the existence results of unbounded (positive) solution for second order multi-point BVP (1.1). Our main features are as follows. Firstly, BVP (1.1) depends on derivative, and the boundary conditions are more general. Secondly, we will study multi-point BVP on infinite intervals. Thirdly, we will obtain the unbounded (positive) solution to BVP (1.1). Obviously, with the boundary condition in (1.1), if the solution exists, it is unbounded. Hence, we extend and generalize the results of [16, 17] to some degree. The main tools used in this paper are Leray-Schauder nonlinear alternative and the fixed point index theory.

        The rest of the paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, the existence of unbounded solution is established. In Section 4, the existence and uniqueness of positive solution are obtained. Finally, we formulate two examples to illustrate the main results.

        2. Preliminaries and Lemmas

        Denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq20_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq21_HTML.gif . Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ4_HTML.gif
        (2.1)
        For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq22_HTML.gif , define
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ5_HTML.gif
        (2.2)

        then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq23_HTML.gif is a Banach space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq24_HTML.gif (see [17]).

        The Arzela-Ascoli theorem fails to work in the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq25_HTML.gif due to the fact that the infinite interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq26_HTML.gif is noncompact. The following compactness criterion will help us to resolve this problem.

        Lemma 2.1 (see [17]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq27_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq28_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq29_HTML.gif if the following conditions hold:

        (a) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq30_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq31_HTML.gif ;

        (b) the functions belonging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq33_HTML.gif are locally equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq34_HTML.gif ;

        (c) the functions from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq35_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq36_HTML.gif are equiconvergent, at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq37_HTML.gif .

        Throughout the paper we assume the following.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq39_HTML.gif , and there exist nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq40_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq41_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ6_HTML.gif
        (2.3)

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq43_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq45_HTML.gif , where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ7_HTML.gif
        (2.4)
        Denote
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ8_HTML.gif
        (2.5)

        Lemma 2.2.

        Supposing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq46_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq47_HTML.gif , then BVP
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ9_HTML.gif
        (2.6)
        has a unique solution
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ10_HTML.gif
        (2.7)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ11_HTML.gif
        (2.8)

        in which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq48_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq49_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq50_HTML.gif .

        Proof.

        Integrating the differential equation from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq51_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq52_HTML.gif , one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ12_HTML.gif
        (2.9)
        Then, integrating the above integral equation from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq53_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq54_HTML.gif , noticing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq56_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ13_HTML.gif
        (2.10)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq57_HTML.gif , it holds that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ14_HTML.gif
        (2.11)

        By using arguments similar to those used to prove Lemma 2.2 in [9], we conclude that (2.7) holds. This completes the proof.

        Now, BVP (1.1) is equivalent to
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ15_HTML.gif
        (2.12)
        Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq58_HTML.gif , (2.12) becomes
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ16_HTML.gif
        (2.13)
        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq59_HTML.gif , define operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq60_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ17_HTML.gif
        (2.14)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ18_HTML.gif
        (2.15)
        Set
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ19_HTML.gif
        (2.16)

        Remark 2.3.

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq61_HTML.gif is the Green function for the following associated homogeneous BVP on the half-line:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ20_HTML.gif
        (2.17)
        It is not difficult to testify that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ21_HTML.gif
        (2.18)

        Let us first give the following result of completely continuous operator.

        Lemma 2.4.

        Supposing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq62_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq63_HTML.gif hold, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq64_HTML.gif is completely continuous.

        Proof.
        1. (1)

          First, we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq65_HTML.gif is well defined.

           
        For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq66_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq67_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq68_HTML.gif . Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ22_HTML.gif
        (2.19)
        so
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ23_HTML.gif
        (2.20)
        Similarly,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ24_HTML.gif
        (2.21)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ25_HTML.gif
        (2.22)
        Further,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ26_HTML.gif
        (2.23)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ27_HTML.gif
        (2.24)
        On the other hand, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq69_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq70_HTML.gif , by Remark 2.3, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ28_HTML.gif
        (2.25)
        Hence, by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq71_HTML.gif , the Lebesgue dominated convergence theorem, and the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq72_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq73_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ29_HTML.gif
        (2.26)

        So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq74_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq75_HTML.gif .

        We can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq76_HTML.gif . In fact, by (2.23) and (2.24), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ30_HTML.gif
        (2.27)
        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq77_HTML.gif is well defined.
        1. (2)

          We show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq78_HTML.gif is continuous.

           
        Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq79_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq80_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq81_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq82_HTML.gif , and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq83_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq84_HTML.gif . The continuity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq85_HTML.gif implies that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ31_HTML.gif
        (2.28)
        as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq86_HTML.gif . Moreover, since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ32_HTML.gif
        (2.29)
        we have from the Lebesgue dominated convergence theorem that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ33_HTML.gif
        (2.30)
        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq87_HTML.gif is continuous.
        1. (3)

          We show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq88_HTML.gif is relatively compact.

           
        (a) Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq89_HTML.gif be a bounded subset. Then, there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq90_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq91_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq92_HTML.gif . By the similar proof of (2.20) and (2.22), if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq93_HTML.gif , one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ34_HTML.gif
        (2.31)

        which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq94_HTML.gif is uniformly bounded.

        (b) For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq95_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq96_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ35_HTML.gif
        (2.32)
        Thus, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq97_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq98_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq99_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ36_HTML.gif
        (2.33)

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq100_HTML.gif is arbitrary, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq102_HTML.gif are locally equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq103_HTML.gif .

        (c) For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq104_HTML.gif , from (2.27), we have

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ37_HTML.gif
        (2.34)

        which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq105_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq106_HTML.gif are equiconvergent at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq107_HTML.gif . By Lemma 2.1, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq108_HTML.gif is relatively compact.

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq109_HTML.gif is completely continuous. The proof is complete.

        Lemma 2.5 (see [28, 29]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq110_HTML.gif be Banach space, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq111_HTML.gif be a bounded open subset of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq112_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq113_HTML.gif be a completely continuous operator. Then either there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq115_HTML.gif , or there exists a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq116_HTML.gif .

        Lemma 2.6 (see [28, 29]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq117_HTML.gif be a bounded open set in real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq118_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq119_HTML.gif be a cone of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq120_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq121_HTML.gif be completely continuous. Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ38_HTML.gif
        (2.35)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ39_HTML.gif
        (2.36)

        3. Existence Result

        In this section, we present the existence of an unbounded solution for BVP (1.1) by using the Leray-Schauder nonlinear alternative.

        Theorem 3.1.

        Suppose that conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq122_HTML.gif hold. Then BVP (1.1) has at least one unbounded solution.

        Proof.

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq123_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq124_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq125_HTML.gif , a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq126_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq127_HTML.gif . Set
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ40_HTML.gif
        (3.1)

        From Lemmas 2.2 and 2.4, BVP (1.1) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq128_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq129_HTML.gif is a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq130_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq131_HTML.gif . So, we only need to seek a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq132_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq133_HTML.gif .

        Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq134_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq135_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ41_HTML.gif
        (3.2)
        Therefore,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ42_HTML.gif
        (3.3)

        which contradicts http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq136_HTML.gif . By Lemma 2.5, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq137_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq138_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq139_HTML.gif , boundary conditions imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq140_HTML.gif is an unbounded solution of BVP (1.1).

        4. Existence and Uniqueness of Positive Solution

        In this section, we restrict the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq141_HTML.gif and discuss the existence and uniqueness of positive solution for BVP (1.1).

        Define the cone http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq142_HTML.gif as follows:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ43_HTML.gif
        (4.1)

        Lemma 4.1.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq144_HTML.gif hold. Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq145_HTML.gif is completely continuous.

        Proof.

        Lemma 2.4 shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq146_HTML.gif is completely continuous, so we only need to prove http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq147_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq148_HTML.gif , and from Remark 2.3, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ44_HTML.gif
        (4.2)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ45_HTML.gif
        (4.3)

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq149_HTML.gif .

        Theorem 4.2.

        Suppose that conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq151_HTML.gif hold and the following condition holds:

        suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq153_HTML.gif and there exist nonnegative functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq154_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq155_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ46_HTML.gif
        (4.4)

        Then, BVP (1.1) has a unique unbounded positive solution.

        Proof.

        We first show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq156_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq157_HTML.gif . By (4.4), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ47_HTML.gif
        (4.5)
        By Lemma 4.1, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq158_HTML.gif is completely continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq159_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq160_HTML.gif . Set
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ48_HTML.gif
        (4.6)
        For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq161_HTML.gif , by (4.5), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ49_HTML.gif
        (4.7)

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq162_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq163_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq164_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq165_HTML.gif . Then, Lemma 2.6 yields http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq166_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq167_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq168_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq169_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq170_HTML.gif is an unbounded positive solution of BVP (1.1).

        Next, we show the uniqueness of positive solution for BVP (1.1). We will show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq171_HTML.gif is a contraction. In fact, by (4.4), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ50_HTML.gif
        (4.8)

        So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq172_HTML.gif is indeed a contraction. The Banach contraction mapping principle yields the uniqueness of positive solution to BVP (1.1).

        5. Examples

        Example 5.1.

        Consider the following BVP:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ51_HTML.gif
        (5.1)
        We have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ52_HTML.gif
        (5.2)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ53_HTML.gif
        (5.3)

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq173_HTML.gif , and it is easy to prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq174_HTML.gif is satisfied. By direct calculations, we can obtain that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq175_HTML.gif . By Theorem 3.1, BVP (5.1) has an unbounded solution.

        Example 5.2.

        Consider the following BVP:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ54_HTML.gif
        (5.4)
        In this case, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ55_HTML.gif
        (5.5)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_Equ56_HTML.gif
        (5.6)

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F236560/MediaObjects/13661_2010_Article_906_IEq176_HTML.gif . By Theorem 4.2, BVP (5.4) has a unique unbounded positive solution.

        Declarations

        Acknowledgments

        The authors are grateful to the referees for valuable suggestions and comments. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 10771117) and the Natural Science Foundation of Shandong Province of China (Y2007A23, Y2008A24). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

        Authors’ Affiliations

        (1)
        School of Mathematical Sciences, Qufu Normal University
        (2)
        Department of Mathematics and Statistics, Curtin University of Technology

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