Positive Solutions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq1_HTML.gif th-Order Nonlinear Impulsive Differential Equation with Nonlocal Boundary Conditions

  • Meiqiang Feng1Email author,

    Affiliated with

    • Xuemei Zhang2 and

      Affiliated with

      • Xiaozhong Yang2

        Affiliated with

        Boundary Value Problems20102011:456426

        DOI: 10.1155/2011/456426

        Received: 25 March 2010

        Accepted: 9 May 2010

        Published: 10 June 2010

        Abstract

        This paper is devoted to study the existence, nonexistence, and multiplicity of positive solutions for the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq2_HTML.gif th-order nonlocal boundary value problem with impulse effects. The arguments are based upon fixed point theorems in a cone. An example is worked out to demonstrate the main results.

        1. Introduction

        The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. [1]; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. [2]. The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations; see, for instance, [314] and their references.

        At the same time, a class of boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [15], semiconductor problems [16], hydrodynamic problems [17]. Such problems include two, three, and multipoint boundary value problems as special cases and attract much attention; see, for instance, [7, 8, 11, 1844] and references cited therein. In particular, we would like to mention some results of Eloe and Ahmad [19] and Pang et al. [22]. In [19], by applying the fixed point theorem in cones due to the work of Krasnosel'kii and Guo, Eloe and Ahmad established the existence of positive solutions of the following http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq3_HTML.gif th boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ1_HTML.gif
        (1.1)
        In [22], Pang et al. considered the expression and properties of Green's function for the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq4_HTML.gif th-order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq5_HTML.gif -point boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ2_HTML.gif
        (1.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq6_HTML.gif . Furthermore, they obtained the existence of positive solutions by means of fixed point index theory.

        Recently, Yang and Wei [23] and the author of [24] improved and generalized the results of Pang et al. [22] by using different methods, respectively.

        On the other hand, it is well known that fixed point theorem of cone expansion and compression of norm type has been applied to various boundary value problems to show the existence of positive solutions; for example, see [7, 8, 11, 19, 23, 24]. However, there are few papers investigating the existence of positive solutions of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq7_HTML.gif th impulsive differential equations by using the fixed point theorem of cone expansion and compression. The objective of the present paper is to fill this gap. Being directly inspired by [19, 22], using of the fixed point theorem of cone expansion and compression, this paper is devoted to study a class of nonlocal BVPs for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq8_HTML.gif th-order impulsive differential equations with fixed moments.

        Consider the following http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq9_HTML.gif th-order impulsive differential equations with integral boundary conditions:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ3_HTML.gif
        (1.3)

        Here http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq10_HTML.gif (where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq11_HTML.gif is fixed positive integer) are fixed points with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq12_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq14_HTML.gif represent the right-hand limit and left-hand limit of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq15_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq16_HTML.gif , respectively, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq17_HTML.gif is nonnegative.

        For the case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq18_HTML.gif , problem (1.3) reduces to the problem studied by Samoĭlenko and Perestyuk in [4]. By using the fixed point index theory in cones, the authors obtained some sufficient conditions for the existence of at least one or two positive solutions to the two-point BVPs.

        Motivated by the work above, in this paper we will extend the results of [4, 19, 2224] to problem (1.3). On the other hand, it is also interesting and important to discuss the existence of positive solutions for problem (1.3) when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq19_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq20_HTML.gif . Many difficulties occur when we deal with them; for example, the construction of cone and operator. So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem (1.3). Our argument is based on fixed point theory in cones [45].

        To obtain positive solutions of (1.3), the following fixed point theorem in cones is fundamental which can be found in [45, page 93].

        Lemma 1.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq22_HTML.gif be two bounded open sets in Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq23_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq25_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq26_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq27_HTML.gif and let operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq28_HTML.gif be completely continuous. Suppose that one of the following two conditions is satisfied:
        1. (i)

          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq29_HTML.gif ;

           
        2. (ii)

          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq30_HTML.gif .

           

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq31_HTML.gif has at least one fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq32_HTML.gif .

        2. Preliminaries

        In order to define the solution of problem (1.3), we will consider the following space.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq33_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ4_HTML.gif
        (2.1)
        Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq34_HTML.gif is a real Banach space with norm
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ5_HTML.gif
        (2.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq35_HTML.gif

        A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq36_HTML.gif is called a solution of problem (1.3) if it satisfies (1.3).

        To establish the existence of multiple positive solutions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq37_HTML.gif of problem (1.3), let us list the following assumptions:

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq39_HTML.gif ;

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq41_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq42_HTML.gif .

        Lemma 2.1.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq44_HTML.gif hold. Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq45_HTML.gif is a solution of problem (1.3) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq46_HTML.gif is a solution of the following impulsive integral equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ6_HTML.gif
        (2.3)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ7_HTML.gif
        (2.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ8_HTML.gif
        (2.5)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ9_HTML.gif
        (2.6)

        Proof.

        First suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq47_HTML.gif is a solution of problem (1.3). It is easy to see by integration of (1.3) that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ10_HTML.gif
        (2.7)
        Integrating again and by boundary conditions, we can get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ11_HTML.gif
        (2.8)
        Similarly, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ12_HTML.gif
        (2.9)
        Letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq48_HTML.gif in (2.9), we find
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ13_HTML.gif
        (2.10)
        Substituting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq49_HTML.gif and (2.10) into (2.9), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ14_HTML.gif
        (2.11)
        Multiplying (2.11) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq50_HTML.gif and integrating it, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ15_HTML.gif
        (2.12)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ16_HTML.gif
        (2.13)
        Then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ17_HTML.gif
        (2.14)

        Then, the proof of sufficient is complete.

        Conversely, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq51_HTML.gif is a solution of (2.3), direct differentiation of (2.3) implies that, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq52_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ18_HTML.gif
        (2.15)
        Evidently,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ19_HTML.gif
        (2.16)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ20_HTML.gif
        (2.17)

        So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq54_HTML.gif , and it is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq55_HTML.gif , and the lemma is proved.

        Similar to the proof of that from [22], we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq56_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq57_HTML.gif have the following properties.

        Proposition 2.2.

        The function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq58_HTML.gif defined by (2.5) satisfyong http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq59_HTML.gif is continuous for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq60_HTML.gif .

        Proposition 2.3.

        There exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq61_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ21_HTML.gif
        (2.18)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq62_HTML.gif is defined in (2.20).

        Proposition 2.4.

        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq63_HTML.gif , then one has

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq64_HTML.gif is continuous for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq65_HTML.gif ;

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq66_HTML.gif .

        Proof.

        From the properties of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq67_HTML.gif and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq68_HTML.gif , we can prove that the results of Proposition 2.4 hold.

        Proposition 2.5.

        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq69_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq70_HTML.gif defined by (2.4) satisfies

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq71_HTML.gif is continuous for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq72_HTML.gif ;

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq73_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq74_HTML.gif , and

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ22_HTML.gif
        (2.19)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq75_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ23_HTML.gif
        (2.20)

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq76_HTML.gif is defined in Proposition 2.3.

        Proof.
        1. (i)

          From Propositions 2.2 and 2.4, we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq77_HTML.gif is continuous for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq78_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq79_HTML.gif .

           
        2. (ii)

          From (ii) of Proposition 2.2 and (ii) of Proposition 2.4, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq80_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq81_HTML.gif .

           

        Now, we show that (2.19) holds.

        In fact, from Proposition 2.3, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ24_HTML.gif
        (2.21)

        Then the proof of Proposition 2.5 is completed.

        Remark 2.6.

        From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq82_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq83_HTML.gif .

        Lemma 2.7.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq85_HTML.gif hold. Then, the solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq86_HTML.gif of problem (1.3) satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq87_HTML.gif

        Proof.

        It is an immediate subsequence of the facts that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq88_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq89_HTML.gif .

        Remark 2.8.

        From (ii) of Proposition 2.5, one can find that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ25_HTML.gif
        (2.22)
        For the sake of applying Lemma 1.1, we construct a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq90_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq91_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ26_HTML.gif
        (2.23)
        Define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq92_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ27_HTML.gif
        (2.24)

        Lemma 2.9.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq94_HTML.gif hold. Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq95_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq96_HTML.gif is completely continuous.

        Proof.

        From Proposition 2.5 and (2.24), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ28_HTML.gif
        (2.25)

        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq97_HTML.gif .

        Next, by similar arguments to those in [8] one can prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq98_HTML.gif is completely continuous. So it is omitted, and the lemma is proved.

        3. Main Results

        Write
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ29_HTML.gif
        (3.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq99_HTML.gif denotes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq100_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq101_HTML.gif

        In this section, we apply Lemma 1.1 to establish the existence of positive solutions for BVP (1.3).

        Theorem 3.1.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq103_HTML.gif hold. In addition, letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq105_HTML.gif satisfy the following conditions:

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq108_HTML.gif ;

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq110_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq111_HTML.gif ,

        BVP (1.3) has at least one positive solution.

        Proof.

        Considering http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq112_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq113_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ30_HTML.gif
        (3.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq114_HTML.gif satisfy
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ31_HTML.gif
        (3.3)
        here
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ32_HTML.gif
        (3.4)
        Now, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq115_HTML.gif , we prove that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ33_HTML.gif
        (3.5)
        In fact, if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq116_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq117_HTML.gif . Noticing (3.2), then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ34_HTML.gif
        (3.6)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ35_HTML.gif
        (3.7)

        Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq118_HTML.gif , which is a contraction. Hence, (3.2) holds.

        Next, turning to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq119_HTML.gif . Case ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq120_HTML.gif ). http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq121_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq122_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ36_HTML.gif
        (3.8)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq123_HTML.gif . Choose
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ37_HTML.gif
        (3.9)
        We show that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ38_HTML.gif
        (3.10)
        In fact, if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq124_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq125_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ39_HTML.gif
        (3.11)
        This and (3.9) imply that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ40_HTML.gif
        (3.12)
        So, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ41_HTML.gif
        (3.13)
        that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ42_HTML.gif
        (3.14)
        It is easy to see that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ43_HTML.gif
        (3.15)

        In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq126_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq127_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq128_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq129_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq130_HTML.gif , which contracts http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq131_HTML.gif So, (3.15) holds. Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq132_HTML.gif , this is also a contraction. Hence, (3.10) holds.

        Case ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq133_HTML.gif ). http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq134_HTML.gif . There exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq135_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ44_HTML.gif
        (3.16)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq136_HTML.gif . If we define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq137_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq138_HTML.gif . Choose
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ45_HTML.gif
        (3.17)

        We prove that (3.10) holds.

        In fact, if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq139_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq140_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ46_HTML.gif
        (3.18)
        This and (3.17) imply that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ47_HTML.gif
        (3.19)
        So, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ48_HTML.gif
        (3.20)
        From (3.20), we obtain that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ49_HTML.gif
        (3.21)
        So, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ50_HTML.gif
        (3.22)
        From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq141_HTML.gif , we can find that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ51_HTML.gif
        (3.23)

        Similar to the proof in case ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq142_HTML.gif ), we can show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq143_HTML.gif . Then, from (3.23), we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq144_HTML.gif , which is a contraction. Hence, (3.10) holds.

        Applying (i) of Lemma 1.1 to (3.2) and (3.10) yields that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq145_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq146_HTML.gif . Thus, it follows that BVP (1.3) has at least one positive solution, and the theorem is proved.

        Theorem 3.2.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq147_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq148_HTML.gif hold. In addition, letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq150_HTML.gif satisfy the following conditions:

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq152_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq153_HTML.gif ;

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq155_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq156_HTML.gif ,

        BVP (1.3) has at least one positive solution.

        Proof.

        Considering http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq157_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq158_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq159_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq160_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq161_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq162_HTML.gif .

        Similar to the proof of (3.2), we can show that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ52_HTML.gif
        (3.24)
        Next, turning to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq163_HTML.gif . Under condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq164_HTML.gif , similar to the proof of (3.10), we can also show that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ53_HTML.gif
        (3.25)

        Applying (i) of Lemma 1.1 to (3.24) and (3.25) yields that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq165_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq166_HTML.gif . Thus, it follows that BVP (1.3) has one positive solution, and the theorem is proved.

        Theorem 3.3.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq167_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq168_HTML.gif hold. In addition, letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq169_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq170_HTML.gif satisfy the following condition:

        there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq172_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq174_HTML.gif implies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ54_HTML.gif
        (3.26)

        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq175_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq176_HTML.gif , BVP (1.3) has at least two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq178_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq179_HTML.gif .

        Proof.

        We choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq180_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq181_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq182_HTML.gif holds, similar to the proof of (3.2), we can prove that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ55_HTML.gif
        (3.27)
        If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq183_HTML.gif holds, similar to the proof of (3.24), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ56_HTML.gif
        (3.28)
        Finally, we show that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ57_HTML.gif
        (3.29)
        In fact, if there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq184_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq185_HTML.gif then by (2.23), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ58_HTML.gif
        (3.30)
        and it follows from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq186_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ59_HTML.gif
        (3.31)

        that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq187_HTML.gif , which is a contraction. Hence, (3.29) holds.

        Applying Lemma 1.1 to (3.27), (3.28), and (3.29) yields that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq188_HTML.gif has two fixed points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq189_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq190_HTML.gif . Thus it follows that BVP (1.3) has two positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq191_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq192_HTML.gif . The proof is complete.

        Our last results corresponds to the case when problem (1.3) has no positive solution. Write
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ60_HTML.gif
        (3.32)

        Theorem 3.4.

        Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq193_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq194_HTML.gif , then problem (1.3) has no positive solution.

        Proof.

        Assume to the contrary that problem (1.3) has a positive solution, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq195_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq196_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq197_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq198_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ61_HTML.gif
        (3.33)

        which is a contradiction, and this completes the proof.

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

        Example 3.5.

        Consider the following boundary value problem:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ62_HTML.gif
        (3.34)

        Conclusion.

        BVP (3.34) has at least one positive solution.

        Proof.

        BVP (3.34) can be regarded as a BVP of the form (1.3), where
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ63_HTML.gif
        (3.35)
        It is not difficult to see that conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq200_HTML.gif hold. In addition,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_Equ64_HTML.gif
        (3.36)

        Then, conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq201_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F456426/MediaObjects/13661_2010_Article_40_IEq202_HTML.gif of Theorem 3.1 hold. Hence, by Theorem 3.1, the conclusion follows, and the proof is complete.

        Declarations

        Acknowledgment

        This work is supported by the National Natural Science Foundation of China (10771065), the Natural Sciences Foundation of Heibei Province (A2007001027), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education(KM201010772018) and Beijing Municipal Education Commission(71D0911003). The authors thank the referee for his/her careful reading of the paper and useful suggestions.

        Authors’ Affiliations

        (1)
        School of Science, Beijing Information Science & Technology University
        (2)
        Department of Mathematics and Physics, North China Electric Power University

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        © Meiqiang Feng et al. 2011

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