Existence and Uniqueness of Smooth Positive Solutions to a Class of Singular http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq1_HTML.gif -Point Boundary Value Problems

  • Xinsheng Du1Email author and

    Affiliated with

    • Zengqin Zhao1

      Affiliated with

      Boundary Value Problems20092009:191627

      DOI: 10.1155/2009/191627

      Received: 2 April 2009

      Accepted: 23 November 2009

      Published: 1 December 2009

      Abstract

      This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m-point boundary value problems of second-order ordinary differential equations. A necessary and sufficient condition for the existence and uniqueness of smooth positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq2_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq3_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq4_HTML.gif .

      1. Introduction and the Main Results

      In this paper,we will consider the existence and uniqueness of positive solutions to a class of second-order singular http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq5_HTML.gif -point boundary value problems of the following differential equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ1_HTML.gif
      (1.1)
      with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq6_HTML.gif are constants, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq8_HTML.gif satisfies the following hypothesis:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq10_HTML.gif is continuous, nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq11_HTML.gif , and nonincreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq12_HTML.gif for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq13_HTML.gif there exists a real number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq14_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq15_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ3_HTML.gif
      (1.3)

      there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq16_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq17_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq18_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ4_HTML.gif
      (1.4)

      Remark 1.1.

      (i) Inequality (1.3) implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ5_HTML.gif
      (1.5)

      Conversely, (1.5) implies (1.3).

      (ii) Inequality (1.4) implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ6_HTML.gif
      (1.6)

      Conversely, (1.6) implies (1.4).

      Remark 1.2.

      It follows from (1.3), (1.4) that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ7_HTML.gif
      (1.7)

      When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq19_HTML.gif is increasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq20_HTML.gif , singular nonlinear http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq21_HTML.gif -point boundary value problems have been extensively studied in the literature, see [13]. However, when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq22_HTML.gif is increasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq23_HTML.gif , and is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq24_HTML.gif , the study on it has proceeded very slowly. The purpose of this paper is to fill this gap. In addition, it is valuable to point out that the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq25_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq26_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq27_HTML.gif

      When referring to singularity we mean that the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq28_HTML.gif in (1.1) are allowed to be unbounded at the points http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq29_HTML.gif , and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq30_HTML.gif . A function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq31_HTML.gif is called a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq32_HTML.gif (positive) solution to (1.1) and (1.2) if it satisfies (1.1) and (1.2) ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq33_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq34_HTML.gif ). A http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq35_HTML.gif (positive) solution to (1.1) and (1.2) is called a smooth (positive) solution if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq37_HTML.gif both exist ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq38_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq39_HTML.gif ). Sometimes, we also call a smooth solution a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq40_HTML.gif solution. It is worth stating here that a nontrivial http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq41_HTML.gif nonnegative solution to the problem (1.1), (1.2) must be a positive solution. In fact, it is a nontrivial concave function satisfying (1.2) which, of course, cannot be equal to zero at any point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq42_HTML.gif

      To seek necessary and sufficient conditions for the existence of solutions to the above problems is important and interesting, but difficult. Thus, researches in this respect are rare up to now. In this paper, we will study the existence and uniqueness of smooth positive solutions to the second-order singular http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq43_HTML.gif -point boundary value problem (1.1) and (1.2). A necessary and sufficient condition for the existence of smooth positive solutions is given by constructing lower and upper solutions and with the maximal principle. Also, the uniqueness of the smooth positive solutions is studied.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq44_HTML.gif is called a lower solution to the problem (1.1), (1.2), if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq45_HTML.gif and satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ8_HTML.gif
      (1.8)

      Upper solution is defined by reversing the above inequality signs. If there exist a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq46_HTML.gif and an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq47_HTML.gif to problem (1.1), (1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq48_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq49_HTML.gif is called a couple of upper and lower solution to problem (1.1), (1.2).

      To prove the main result, we need the following maximal principle.

      Lemma 1.3 (maximal principle).

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq50_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq51_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq52_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq53_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq54_HTML.gif

      Proof.

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ9_HTML.gif
      (1.9)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ10_HTML.gif
      (1.10)

      then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq55_HTML.gif

      By integrating (1.9) twice and noting (1.10), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ11_HTML.gif
      (1.11)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ12_HTML.gif
      (1.12)

      In view of (1.11) and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq56_HTML.gif , we can obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq57_HTML.gif This completes the proof of Lemma 1.3.

      Now we state the main results of this paper as follows.

      Theorem 1.4.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq58_HTML.gif holds, then a necessary and sufficient condition for the problem (1.1) and (1.2) to have smooth positive solution is that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ13_HTML.gif
      (1.13)

      Theorem 1.5.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq59_HTML.gif and (1.13) hold, then the smooth positive solution to problem (1.1) and (1.2) is also the unique http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq60_HTML.gif positive solution.

      2. Proof of Theorem 1.4

      2.1. The Necessary Condition

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq61_HTML.gif is a smooth positive solution to the boundary value problem (1.1) and (1.2). We will show that (1.13) holds.

      It follows from
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ14_HTML.gif
      (2.1)
      that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq62_HTML.gif is nonincreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq63_HTML.gif Thus, by the Lebesgue theorem, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ15_HTML.gif
      (2.2)
      It is well known that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq64_HTML.gif can be stated as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ16_HTML.gif
      (2.3)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ17_HTML.gif
      (2.4)
      By (2.3) and (1.2) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ18_HTML.gif
      (2.5)
      therefore because of (2.3) and (2.5),
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ19_HTML.gif
      (2.6)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq65_HTML.gif is a smooth positive solution to (1.1) and (1.2), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ20_HTML.gif
      (2.7)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq66_HTML.gif From (2.6), (2.7) it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ21_HTML.gif
      (2.8)
      Without loss of generality we may assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq67_HTML.gif This together with the condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq68_HTML.gif implies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ22_HTML.gif
      (2.9)

      On the other hand, notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq69_HTML.gif is a smooth positive solution to (1.1), (1.2), we have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ23_HTML.gif
      (2.10)
      therefore, there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq70_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq71_HTML.gif Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq73_HTML.gif It follows from (1.7) that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ24_HTML.gif
      (2.11)
      Consequently http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq74_HTML.gif which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ25_HTML.gif
      (2.12)
      From (2.9) and (2.12) it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ26_HTML.gif
      (2.13)

      which is the required inequality.

      2.2. The Existence of Lower and Upper Solutions

      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq75_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq76_HTML.gif thus
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ27_HTML.gif
      (2.14)
      Otherwise, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq77_HTML.gif then there exists a real number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq78_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq79_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq80_HTML.gif this contradicts with the condition that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq81_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq82_HTML.gif By condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq83_HTML.gif and (2.14) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ28_HTML.gif
      (2.15)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ29_HTML.gif
      (2.16)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq84_HTML.gif

      Suppose that (1.13) holds. Let

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ30_HTML.gif
      (2.17)

      Since by (1.13), (2.17) we obviously have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ31_HTML.gif
      (2.18)
      and there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq85_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ32_HTML.gif
      (2.19)
      By (2.14) and (2.16) we see, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq86_HTML.gif is sufficiently small, then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ33_HTML.gif
      (2.20)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ34_HTML.gif
      (2.21)
      Then from (2.19) and (2.21) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ35_HTML.gif
      (2.22)

      Consequently, with the aid of (2.20), (2.22) and the condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq87_HTML.gif we have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ36_HTML.gif
      (2.23)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ37_HTML.gif
      (2.24)
      From (2.17), (2.21) it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ38_HTML.gif
      (2.25)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ39_HTML.gif
      (2.26)

      therefore, (2.23)–(2.26) imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq88_HTML.gif are lower and upper solutions to the problem (1.1) and (1.2), respectively.

      2.3. The Sufficient Condition

      First of all, we define a partial ordering in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq89_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq90_HTML.gif if and only if

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ40_HTML.gif
      (2.27)

      Then, we will define an auxiliary function. For all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq91_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ41_HTML.gif
      (2.28)

      By the assumption of Theorem 1.4, we have that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq92_HTML.gif is continuous.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq93_HTML.gif be a sequence satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq94_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq95_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq96_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq97_HTML.gif be a sequence satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ42_HTML.gif
      (2.29)

      For each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq98_HTML.gif let us consider the following nonsingular problem:

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ43_HTML.gif
      (2.30)

      Obviously, it follows from the proof of Lemma 1.3 that problem (2.30) is equivalent to the integral equation

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ44_HTML.gif
      (2.31)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq99_HTML.gif is defined in the proof of Lemma 1.3. It is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq100_HTML.gif is a completely continuous operator and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq101_HTML.gif is a bounded set. Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq102_HTML.gif is a solution to (2.30) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq103_HTML.gif Using the Schauder's fixed point theorem, we assert that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq104_HTML.gif has at least one fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq105_HTML.gif

      We claim that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ45_HTML.gif
      (2.32)

      From this it follows that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ46_HTML.gif
      (2.33)

      Suppose by contradiction that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq106_HTML.gif is not satisfied on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq107_HTML.gif . Let

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ47_HTML.gif
      (2.34)
      therefore
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ48_HTML.gif
      (2.35)

      Since by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq108_HTML.gif and (2.30) we obviously have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq109_HTML.gif

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ49_HTML.gif
      (2.36)

      So,when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq110_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq111_HTML.gif and

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ50_HTML.gif
      (2.37)

      Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq112_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq113_HTML.gif is an upper convex function in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq114_HTML.gif .

      By (2.30) and (2.36), for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq115_HTML.gif we have the following two cases:

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq116_HTML.gif

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq117_HTML.gif

      For case (i): it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq118_HTML.gif this is a contradiction.

      For case (ii): in this case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq119_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq120_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq121_HTML.gif , thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq122_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq123_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq124_HTML.gif From http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq125_HTML.gif we see http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq126_HTML.gif which is in contradiction with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq127_HTML.gif

      From this it follows that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq128_HTML.gif

      Similarly, we can verify that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq129_HTML.gif Consequently (2.32) holds.

      Using the method of [4] and [5, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq130_HTML.gif ], we can obtain that there is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq131_HTML.gif positive solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq132_HTML.gif to (1.1), (1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq133_HTML.gif and a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq134_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq135_HTML.gif on any compact subintervals of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq136_HTML.gif

      3. Proof of Theorem 1.5

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq137_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq138_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq139_HTML.gif positive solutions to (1.1) and (1.2), and at least one of them is a smooth positive solution. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq140_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq141_HTML.gif without loss of generality, we may assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq142_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq143_HTML.gif Let

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ51_HTML.gif
      (3.1)

      It follows from (3.1) that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ52_HTML.gif
      (3.2)

      By (1.2), it is easy to check that there exist the following two possible cases:

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq144_HTML.gif

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq145_HTML.gif

      Assume that case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq146_HTML.gif holds. By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq147_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq148_HTML.gif it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq149_HTML.gif exist (finite or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq150_HTML.gif ), moreover, one of them must be finite. The same conclusion is also valid for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq151_HTML.gif It follows from (3.2) that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ53_HTML.gif
      (3.3)

      consequently

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ54_HTML.gif
      (3.4)

      Similarly

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ55_HTML.gif
      (3.5)
      From (3.1), (3.4), and (3.5) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ56_HTML.gif
      (3.6)

      On the other hand, (3.2), (1.7), and condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq152_HTML.gif yield

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ57_HTML.gif
      (3.7)

      that is,

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ58_HTML.gif
      (3.8)

      therefore

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ59_HTML.gif
      (3.9)

      From this it follows that

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ60_HTML.gif
      (3.10)

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq153_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq154_HTML.gif then, by (3.6) we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq155_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq156_HTML.gif which imply that there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq157_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq158_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq159_HTML.gif It follows from (3.2) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq160_HTML.gif therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq161_HTML.gif Substituting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq162_HTML.gif into (1.1) and using condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq163_HTML.gif , we have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ61_HTML.gif
      (3.11)

      Noticing (3.11) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq164_HTML.gif we have

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ62_HTML.gif
      (3.12)

      which contradicts with the condition that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq165_HTML.gif Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq167_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq168_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq169_HTML.gif , which contradicts with (3.6). So case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq170_HTML.gif is impossible.

      By analogous methods, we can obtain a contradiction for case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq171_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq172_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq173_HTML.gif which implies that the result of Theorem 1.5 holds.

      4. Concerned Remarks and Applications

      Remark 4.1.

      The typical function satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq174_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq175_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq176_HTML.gif

      Remark 4.2.

      Condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq177_HTML.gif includes e-concave function (see [6]) as special case. For example, Liu and Yu [7] consider the existence and uniqueness of positive solution to a class of singular boundary value problem under the following condition:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ63_HTML.gif
      (4.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq178_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq179_HTML.gif is nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq180_HTML.gif , nonincreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq181_HTML.gif Clearly, condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq182_HTML.gif is weaker than the above condition (4.1).

      In fact, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq183_HTML.gif from (4.1) it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ64_HTML.gif
      (4.2)
      On the other hand, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq184_HTML.gif from (4.1) it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ65_HTML.gif
      (4.3)

      that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq185_HTML.gif

      In what follows, by using the results obtained in this paper, we study the boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ66_HTML.gif
      (4.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq186_HTML.gif We have the following theorem.

      Theorem 4.3.

      A necessary and sufficient condition for problem (4.4) to have smooth positive solution is that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ67_HTML.gif
      (4.5)

      Moreover, when the positive solution exists, it is unique.

      Remark 4.4.

      Consider (1.1) and the following singular http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq187_HTML.gif -point boundary value conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ68_HTML.gif
      (4.6)

      By analogous methods, we have the following results.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq188_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq189_HTML.gif positive solution to (1.1) and (4.6), then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq190_HTML.gif can be stated
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ69_HTML.gif
      (4.7)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq191_HTML.gif is defined in (2.4).

      Theorem 4.5.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq192_HTML.gif holds, then a necessary and sufficient condition for the problem (1.1) and (4.6) to have smooth positive solution is that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_Equ70_HTML.gif
      (4.8)

      Theorem 4.6.

      Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq193_HTML.gif and (4.8) hold, then the smooth positive solution to problem (1.1) and (4.6) is also unique http://static-content.springer.com/image/art%3A10.1155%2F2009%2F191627/MediaObjects/13661_2009_Article_833_IEq194_HTML.gif positive solution.

      Declarations

      Acknowledgment

      Research supported by the National Natural Science Foundation of China (10871116), the Natural Science Foundation of Shandong Province (Q2008A03) and the Doctoral Program Foundation of Education Ministry of China (200804460001).

      Authors’ Affiliations

      (1)
      School of Mathematics Sciences, Qufu Normal University

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      Copyright

      © X. Du and Z. Zhao. 2009

      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.