Triple Positive Solutions for a Type of Second-Order Singular Boundary Problems

  • Jiemei Li1, 2Email author and

    Affiliated with

    • Jinxiang Wang1

      Affiliated with

      Boundary Value Problems20102010:376471

      DOI: 10.1155/2010/376471

      Received: 7 April 2010

      Accepted: 26 August 2010

      Published: 1 September 2010

      Abstract

      This paper deals with the existence of triple positive solutions for a type of second-order singular boundary problems with general differential operators. By using the Leggett-Williams fixed point theorem, we establish an existence criterion for at least three positive solutions with suitable growth conditions imposed on the nonlinear term.

      1. Introduction

      In this paper, we study the existence of triple positive solutions for the following second-order singular boundary value problems with general differential operators:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ1_HTML.gif
      (1.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq2_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq3_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ2_HTML.gif
      (1.2)

      It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq5_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq6_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq7_HTML.gif

      When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq8_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq10_HTML.gif , the two kinds of singular boundary value problems have been discussed extensively in the literature; see [110] and the references therein. Hence, the problem that we consider is more general and is different from those in previous work.

      Furthermore, we will see in the later that the presence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq11_HTML.gif brings us three main difficulties:

      (1) the Green's function cannot be explicitly expressed;

      (2) the equivalence between BVP (1.1) and its associated integral equation has to be proved;

      (3) the compactness of associated integral operator has to be verified.

      We will overcome the above mentioned difficulties in Section 2. Also, although the Leggett-William fixed point theorem is used extensively in the study of triple positive differential equations, the method has not been used to study this type of second-order singular boundary value problem with general differential operators. We are concerned with solving these problems in this paper.

      To state our main tool used in this paper, we give some definitions and notations.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq12_HTML.gif be a real Banach space with a cone http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq13_HTML.gif . A map http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq14_HTML.gif is said to be a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq15_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq16_HTML.gif is a continuous and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ3_HTML.gif
      (1.3)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq17_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq18_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq19_HTML.gif be two numbers such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq21_HTML.gif a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq22_HTML.gif . We define the following convex sets:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ4_HTML.gif
      (1.4)

      Theorem 1.1 (Leggett-Williams fixed point theorem).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq23_HTML.gif be completely continuous, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq24_HTML.gif be a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq25_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq26_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq27_HTML.gif . Suppose that there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq28_HTML.gif such that

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq30_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq31_HTML.gif ;

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq32_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq33_HTML.gif ;

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq34_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq35_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq36_HTML.gif .

      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq37_HTML.gif has at least three fixed points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq38_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq39_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq41_HTML.gif .

      Remark 1.2.

      We note the existence of triple positive solutions of other kind of boundary value problems; see He and Ge [11], Zhao et al. [12], Zhang and Liu [13], Graef et al. [14], and the references therein.

      The rest of the paper is organized as follows. In Section 2, we overcome the above-mentioned difficulties in this work. The main results are formulated and proved in Section 3. Finally, an example is presented to demonstrate the application of the main theorems in Section 4.

      2. Preliminaries and Lemmas

      Throughout this paper, we assume the following:

      (H1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq42_HTML.gif ;

      (H2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq43_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq44_HTML.gif ;

      (H3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq45_HTML.gif is continuous and does not vanish identically on any subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq46_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq47_HTML.gif ;

      (H4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq48_HTML.gif is continuous.

      Lemma 2.1.

      Suppose that (H1) and (H2) hold. Then

      (i) the initial value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ5_HTML.gif
      (2.1)

      has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq50_HTML.gif ;

      (ii) the initial value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ6_HTML.gif
      (2.2)

      has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq52_HTML.gif .

      Proof.

      We only prove (i). (ii) can be treated in the same way.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq54_HTML.gif is a solution of (2.1), that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ7_HTML.gif
      (2.3)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ8_HTML.gif
      (2.4)
      Multiplying both sides of (2.3) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq55_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ9_HTML.gif
      (2.5)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq57_HTML.gif , integrating (2.5) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq58_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ10_HTML.gif
      (2.6)
      Moreover, integrating (2.6) on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq59_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq60_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ11_HTML.gif
      (2.7)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ12_HTML.gif
      (2.8)
      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq61_HTML.gif , and (2.7) reduces to
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ13_HTML.gif
      (2.9)
      By using Fubini's theorem, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ14_HTML.gif
      (2.10)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ15_HTML.gif
      (2.11)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq62_HTML.gif is a solution of integral equation (2.11).

      Conversely, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq63_HTML.gif is a solution of (2.11) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq64_HTML.gif , by reversing the above argument we could deduce that the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq65_HTML.gif is a solution of (2.1) and satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq67_HTML.gif . Therefore, to prove that (2.1) has a unique solution, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq68_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq69_HTML.gif is equivalent to prove that (2.11) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq70_HTML.gif .

      To do this, we endow the following norm in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq71_HTML.gif :
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ16_HTML.gif
      (2.12)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq72_HTML.gif be operator defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ17_HTML.gif
      (2.13)
      Since
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ18_HTML.gif
      (2.14)
      then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq73_HTML.gif is well defined. Set
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ19_HTML.gif
      (2.15)
      Then, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq74_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ20_HTML.gif
      (2.16)
      and subsequently,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ21_HTML.gif
      (2.17)
      Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ22_HTML.gif
      (2.18)

      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq75_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq76_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq77_HTML.gif by Banach contraction principle. That is, (2.11) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq78_HTML.gif .

      Remark 2.2.

      Lemma 2.1 generalizes Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq79_HTML.gif of [1], where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq80_HTML.gif .

      Lemma 2.3.

      Suppose that (H1) and (H2) hold. Then

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq81_HTML.gif is nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq82_HTML.gif ;

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq83_HTML.gif is nonincreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq84_HTML.gif .

      Proof.

      We only prove (i). (ii) can be treated in the same way.

      Suppose on the contrary that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq85_HTML.gif is not nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq86_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq87_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ23_HTML.gif
      (2.19)
      This together with the equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq88_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ24_HTML.gif
      (2.20)

      which is a contradiction!

      Remark 2.4.

      From Lemmas 2.1 and 2.3, there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq89_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq91_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq92_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ25_HTML.gif
      (2.21)
      In fact, since
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ26_HTML.gif
      (2.22)
      we have that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq94_HTML.gif . Then, there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq96_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ27_HTML.gif
      (2.23)
      that is
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ28_HTML.gif
      (2.24)
      In the following, we will show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq97_HTML.gif . Suppose on the contrary, if there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq98_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ29_HTML.gif
      (2.25)

      then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq99_HTML.gif , which is a contradiction!

      The other inequality can be treated in the same manner.

      Lemma 2.5.

      Suppose that (H1), (H2), and (H3) hold. Then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ30_HTML.gif
      (2.26)

      Proof.

      We only prove the first equality; the other can be treated in the same way. From Remark 2.4 and (H3), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ31_HTML.gif
      (2.27)
      Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq100_HTML.gif of [2] together with the facts that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq101_HTML.gif and (H3) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ32_HTML.gif
      (2.28)
      Combining (2.27) and (2.28), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ33_HTML.gif
      (2.29)

      Lemma 2.6.

      Suppose that (H1), (H2), and (H3) hold. Then the problem
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ34_HTML.gif
      (2.30)
      has a unique solution
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ35_HTML.gif
      (2.31)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ36_HTML.gif
      (2.32)

      Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq102_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq103_HTML.gif .

      Proof.

      By Lemma 2.3 and (2.32), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ37_HTML.gif
      (2.33)

      This together with Remark 2.4 implies that the right side of (2.31) is well defined.

      Now we check that the function
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ38_HTML.gif
      (2.34)
      satisfies (2.30). In fact,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ39_HTML.gif
      (2.35)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ40_HTML.gif
      (2.36)
      Equation (2.34) and Lemma 2.5 imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ41_HTML.gif
      (2.37)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq104_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq105_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ42_HTML.gif
      (2.38)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq106_HTML.gif with the norm
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ43_HTML.gif
      (2.39)
      and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq107_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq108_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ44_HTML.gif
      (2.40)

      Lemma 2.7.

      Suppose that (H1)–(H3) hold and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq109_HTML.gif is a positive solution of (2.30). Then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ45_HTML.gif
      (2.41)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ46_HTML.gif
      (2.42)
      Furthermore, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq110_HTML.gif , there exists corresponding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq111_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ47_HTML.gif
      (2.43)

      Proof.

      In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq112_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ48_HTML.gif
      (2.44)
      and if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq113_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ49_HTML.gif
      (2.45)
      Combining this and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq114_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ50_HTML.gif
      (2.46)
      Take
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ51_HTML.gif
      (2.47)

      Then Lemma 2.3 guarantees that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq115_HTML.gif , and Lemma 2.7 guarantees that (2.43) holds.

      Remark 2.8.

      From Lemma 2.7 and Remark 2.4, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ52_HTML.gif
      (2.48)
      Now, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq116_HTML.gif , we can define the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq117_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ53_HTML.gif
      (2.49)

      Lemma 2.9.

      Let (H1)–(H4) hold. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq118_HTML.gif is a completely continuous operator.

      Proof.

      From (H3) and (H4) and Lemma 2.6, it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq119_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq120_HTML.gif is continuous by the Lebesgue http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq121_HTML.gif s dominated convergence theorem.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq122_HTML.gif be any bounded set. Then (H3) and (H4) imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq123_HTML.gif is a bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq124_HTML.gif .

      Since
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ54_HTML.gif
      (2.50)
      then this together with the similar proof of Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq125_HTML.gif of [2] yields
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ55_HTML.gif
      (2.51)

      From this fact, it is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq126_HTML.gif is equicontinuous. Therefore, by the Arzela-Ascoli theorem, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq127_HTML.gif is a completely continuous operator.

      3. Main Result

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq128_HTML.gif be nonnegative continuous concave functional defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ56_HTML.gif
      (3.1)

      We notice that, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq130_HTML.gif , and also that by Lemma 2.6, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq131_HTML.gif is a solution of (1.1) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq132_HTML.gif is a fixed point of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq133_HTML.gif .

      For convenience we introduce the following notations. Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ57_HTML.gif
      (3.2)

      Theorem 3.1.

      Assume that (H1)–(H4) hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq134_HTML.gif , and suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq135_HTML.gif satisfies the following conditions:

      (S1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq136_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq137_HTML.gif ;

      (S2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq138_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq139_HTML.gif ;

      (S3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq140_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq141_HTML.gif .

      Then the boundary value problem (1.1) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq142_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq143_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq144_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq145_HTML.gif .

      Proof.

      From Lemma 2.9, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq146_HTML.gif is a completely continuous operator. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq147_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq148_HTML.gif , and assumption (S3) implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq149_HTML.gif . Therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ58_HTML.gif
      (3.3)

      Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq150_HTML.gif . In the same way, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq151_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq152_HTML.gif . Therefore, condition (ii) of Leggett-williams fixed-point theorem holds.

      To check condition (i) of Leggett-Williams fixed-point theorem, choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq153_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq154_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq155_HTML.gif . so,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ59_HTML.gif
      (3.4)
      Hence, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq156_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq157_HTML.gif . From assumption (S2) and Remark 2.8, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ60_HTML.gif
      (3.5)
      Finally, we assert that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq159_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq160_HTML.gif . To see this, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq162_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ61_HTML.gif
      (3.6)

      To sum up, all the conditions of Leggett-williams fixed-point theorem are satisfied. Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq163_HTML.gif has at least three fixed points, that is, problem (1.1) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq164_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq165_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq167_HTML.gif .

      Theorem 3.2.

      Assume that (H1)–(H4) hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq168_HTML.gif , and suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq169_HTML.gif satisfies the following conditions:

      (A1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq170_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq171_HTML.gif ;

      (A2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq172_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq173_HTML.gif .

      Then the boundary value problem (1.1) has at least http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq174_HTML.gif positive solutions.

      Proof.

      When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq175_HTML.gif , it follows from condition (A1) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq176_HTML.gif , which means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq177_HTML.gif has at least one fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq178_HTML.gif by the Schauder fixed-point Theorem. When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq179_HTML.gif , it is clear that Theorem 3.1 holds (with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq180_HTML.gif ). Then we can obtain at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq181_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq182_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq183_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq184_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq185_HTML.gif . Following this way, we finish the proof by the induction method.

      4. Example

      Consider the following boundary value problem:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ62_HTML.gif
      (4.1)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ63_HTML.gif
      (4.2)
      Then, by computation, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ64_HTML.gif
      (4.3)
      Furthermore, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq186_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ65_HTML.gif
      (4.4)
      In fact, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq187_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq188_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq189_HTML.gif . It is easy to compute that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ66_HTML.gif
      (4.5)
      Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq190_HTML.gif , that is
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ67_HTML.gif
      (4.6)

      The other inequalities in (4.4) can be proved by the same method.

      Thus, we can choose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq191_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq193_HTML.gif . By computation, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ68_HTML.gif
      (4.7)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq194_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq195_HTML.gif . Then, we can compute
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ69_HTML.gif
      (4.8)
      Consequently,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ70_HTML.gif
      (4.9)
      Therefore, all the conditions of Theorem 3.1 are satisfied, then problem (4.1) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq196_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq197_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ71_HTML.gif
      (4.10)

      Declarations

      Acknowledgment

      The first author was partially supported by NNSF of China (10901075).

      Authors’ Affiliations

      (1)
      Department of Mathematics, Northwest Normal University
      (2)
      The School of Mathematics, Physics Software Engineering, Lanzhou Jiaotong University

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      Copyright

      © Jiemei Li and Jinxiang Wang. 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.