Positive Solutions for Fourth-Order Singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq1_HTML.gif -Laplacian Differential Equations with Integral Boundary Conditions

  • Xingqiu Zhang1, 2Email author and

    Affiliated with

    • Yujun Cui3

      Affiliated with

      Boundary Value Problems20102010:862079

      DOI: 10.1155/2010/862079

      Received: 7 April 2010

      Accepted: 12 August 2010

      Published: 19 August 2010

      Abstract

      By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq2_HTML.gif as well as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq3_HTML.gif positive solutions for fourth-order singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq4_HTML.gif -Laplacian differential equations with integral boundary conditions. Our nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq5_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq7_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq8_HTML.gif . The dual results for the other integral boundary condition are also given.

      1. Introduction

      In this paper, we consider the existence of positive solutions for the following nonlinear fourth-order singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq9_HTML.gif -Laplacian differential equations with integral boundary conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq17_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq18_HTML.gif is nonnegative. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq20_HTML.gif . Throughout this paper, we always assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq22_HTML.gif and nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq23_HTML.gif satisfies the following hypothesis:

      (H) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq24_HTML.gif is continuous, nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq25_HTML.gif and nonincreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq26_HTML.gif for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq27_HTML.gif , and there exists a real number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq28_HTML.gif such that, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq29_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ2_HTML.gif
      (1.2)
      there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq32_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq33_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ3_HTML.gif
      (1.3)

      Remark 1.1.

      Condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq34_HTML.gif is used to discuss the existence and uniqueness of smooth positive solutions in [1].

      (i)Inequality (1.2) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ4_HTML.gif
      (1.4)

      Conversely, (1.4) implies (1.2).

      (ii)Inequality (1.3) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ5_HTML.gif
      (1.5)

      Conversely, (1.5) implies (1.3).

      Remark 1.2.

      Typical functions that satisfy condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq35_HTML.gif are those taking the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq36_HTML.gif = http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq37_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq38_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq40_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq41_HTML.gif .

      Remark 1.3.

      It follows from (1.2) and (1.3) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ6_HTML.gif
      (1.6)

      Boundary value problems with integral boundary conditions arise in variety of different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. They include two point, three point, and nonlocal boundary value problems (see [25]) as special cases and have attracted much attention of many researchers, such as Gallardo, Karakostas, Tsamatos, Lomtatidze, Malaguti, Yang, Zhang, and Feng (see [613], e.g.). For more information about the general theory of integral equations and their relation to boundary value problems, the reader is referred to the book by Corduneanu [14] and Agarwal and O'Regan [15].

      Recently, Zhang et al. [13] studied the existence and nonexistence of symmetric positive solutions for the following nonlinear fourth-order boundary value problems:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ7_HTML.gif
      (1.7)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq46_HTML.gif is nonnegative, symmetric on the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq48_HTML.gif is continuous, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq49_HTML.gif are nonnegative, symmetric on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq50_HTML.gif .

      To seek necessary and sufficient conditions for the existence of solutions to the ordinary differential equations is important and interesting, but difficult. Professors Wei [16, 17], Du and Zhao [18], Graef and Kong [19], Zhang and Liu [20], and others have done much excellent work under some suitable conditions in this direction. To the author's knowledge, there are no necessary and sufficient conditions available in the literature for the existence of solutions for integral boundary value problem (1.1). Motivated by above papers, the purpose of this paper is to fill this gap. It is worth pointing out that the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq51_HTML.gif permits singularity not only at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq52_HTML.gif but also at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq53_HTML.gif . By singularity, we mean that the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq54_HTML.gif is allowed to be unbounded at the points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq56_HTML.gif .

      2. Preliminaries and Several Lemmas

      A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq58_HTML.gif is called a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq59_HTML.gif (positive) solution of BVP (1.1) if it satisfies (1.1) ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq60_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq61_HTML.gif ). A http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq62_HTML.gif (positive) solution of (1.1) is called a psuedo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq63_HTML.gif (positive) solution if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq65_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq66_HTML.gif . Denote that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ8_HTML.gif
      (2.1)

      Definition 2.1.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq67_HTML.gif is called a lower solution of BVP (1.1) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq68_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ9_HTML.gif
      (2.2)

      Definition 2.2.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq69_HTML.gif is called an upper solution of BVP (1.1) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq70_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ10_HTML.gif
      (2.3)
      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq71_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ11_HTML.gif
      (2.4)

      To prove the main results, we need the following maximum principle.

      Lemma 2.3 (Maximum principle).

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq72_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq74_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq75_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq77_HTML.gif

      Proof.

      Set
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ12_HTML.gif
      (2.5)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ13_HTML.gif
      (2.6)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ14_HTML.gif
      (2.7)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ15_HTML.gif
      (2.8)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ16_HTML.gif
      (2.9)
      then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq78_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq80_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq81_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ17_HTML.gif
      (2.10)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ18_HTML.gif
      (2.11)
      then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ19_HTML.gif
      (2.12)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ20_HTML.gif
      (2.13)
      By integration of (2.12), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ21_HTML.gif
      (2.14)
      Integrating again, we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ22_HTML.gif
      (2.15)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq82_HTML.gif in (2.15), we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ23_HTML.gif
      (2.16)
      Substituting (2.13) and (2.16) into (2.15), we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ24_HTML.gif
      (2.17)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ25_HTML.gif
      (2.18)
      Notice that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ26_HTML.gif
      (2.19)
      therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ27_HTML.gif
      (2.20)
      Substituting (2.20) into (2.17), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ28_HTML.gif
      (2.21)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ29_HTML.gif
      (2.22)
      Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq83_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq85_HTML.gif . From (2.21), it is easily seen that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq86_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq87_HTML.gif By (2.11), we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq88_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq89_HTML.gif Thus, we have proved that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq90_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq91_HTML.gif . Similarly, the solution of (2.5) and (2.7) can be expressed by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ30_HTML.gif
      (2.23)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ31_HTML.gif
      (2.24)

      By (2.23), we can get that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq92_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq93_HTML.gif

      Lemma 2.4.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq94_HTML.gif holds. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq95_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq96_HTML.gif positive solution of BVP (1.1). Then there exist two constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq97_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ32_HTML.gif
      (2.25)

      Proof.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq98_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq99_HTML.gif positive solution of BVP (1.1). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq100_HTML.gif can be stated as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ33_HTML.gif
      (2.26)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ34_HTML.gif
      (2.27)
      It is easy to see that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ35_HTML.gif
      (2.28)
      By (2.26), for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq101_HTML.gif , we have that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ36_HTML.gif
      (2.29)
      From (2.26) and (2.27), we get that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ37_HTML.gif
      (2.30)
      Setting
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ38_HTML.gif
      (2.31)

      then from (2.29) and (2.30), we have (2.25).

      Lemma 2.5.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq102_HTML.gif holds. And assume that there exist lower and upper solutions of BVP (1.1), respectively, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq104_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq105_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq106_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq107_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq108_HTML.gif . Then BVP (1.1) has at least one http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq109_HTML.gif positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq110_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq111_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq112_HTML.gif . If, in addition, there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq113_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ39_HTML.gif
      (2.32)

      then the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq114_HTML.gif of BVP (1.1) is a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq115_HTML.gif positive solution.

      Proof.

      For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq116_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq118_HTML.gif , we defined an auxiliary function
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ40_HTML.gif
      (2.33)

      By condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq119_HTML.gif , we have that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq120_HTML.gif is continuous.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq121_HTML.gif be sequences satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq124_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq125_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq126_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq127_HTML.gif , be sequences satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ41_HTML.gif
      (2.34)
      For each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq128_HTML.gif , consider the following nonsingular problem:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ42_HTML.gif
      (2.35)
      For convenience, we define linear operators as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ43_HTML.gif
      (2.36)
      By the proof of Lemma 2.3, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq129_HTML.gif is a solution of problem (2.35) if and only if it is the fixed point of the following operator equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ44_HTML.gif
      (2.37)

      By (2.33), it is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq130_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq131_HTML.gif is a bounded set. Moreover, by the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq132_HTML.gif , we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq133_HTML.gif is a compact operator and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq134_HTML.gif is a relatively compact set. So, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq135_HTML.gif is a completely continuous operator. In addition, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq136_HTML.gif is a solution of (2.35) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq137_HTML.gif is a fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq138_HTML.gif . Using the Shauder's fixed point theorem, we assert that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq139_HTML.gif has at least one fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq140_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq141_HTML.gif , we can get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq142_HTML.gif

      We claim that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ45_HTML.gif
      (2.38)
      From this it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ46_HTML.gif
      (2.39)
      Indeed, suppose by contradiction that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq143_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq144_HTML.gif . By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq145_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ47_HTML.gif
      (2.40)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ48_HTML.gif
      (2.41)
      On the other hand, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq146_HTML.gif is an upper solution of (1.1), we also have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ49_HTML.gif
      (2.42)
      Then setting
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ50_HTML.gif
      (2.43)
      By (2.41) and (2.42), we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ51_HTML.gif
      (2.44)
      By Lemma 2.3, we can conclude that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ52_HTML.gif
      (2.45)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ53_HTML.gif
      (2.46)
      Set
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ54_HTML.gif
      (2.47)
      Then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ55_HTML.gif
      (2.48)
      By Lemma 2.3, we can conclude that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ56_HTML.gif
      (2.49)

      which contradicts the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq147_HTML.gif Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq148_HTML.gif is impossible.

      Similarly, we can show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq149_HTML.gif So, we have shown that (2.38) holds.

      Using the method of [21] and Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq150_HTML.gif .2 in [22], we can obtain that there is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq151_HTML.gif positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq152_HTML.gif of (1.1) such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq153_HTML.gif and a subsequence of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq154_HTML.gif converging to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq155_HTML.gif on any compact subintervals of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq156_HTML.gif .

      In addition, if (2.32) holds, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq157_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq158_HTML.gif is absolutely integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq159_HTML.gif . This implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq160_HTML.gif is a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq161_HTML.gif positive solution of (1.1).

      3. The Main Results

      Theorem 3.1.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq162_HTML.gif holds, then a necessary and sufficient condition for BVP (1.1) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq163_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ57_HTML.gif
      (3.1)

      Proof.

      The proof is divided into two parts, necessity and suffeciency.

      Necessity.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq164_HTML.gif is a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq165_HTML.gif positive solution of (1.1). Then both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq167_HTML.gif exist. By Lemma 2.4, there exist two constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq168_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ58_HTML.gif
      (3.2)
      Without loss of generality, we may assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq169_HTML.gif . This together with condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq170_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ59_HTML.gif
      (3.3)
      On the other hand, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq171_HTML.gif is a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq172_HTML.gif positive solution of (1.1), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ60_HTML.gif
      (3.4)
      Otherwise, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq173_HTML.gif . By the proof of Lemma 2.3, we have that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq174_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq175_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq176_HTML.gif which contradicts that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq177_HTML.gif is a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq178_HTML.gif positive solution. Therefore, there exists a positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq179_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq180_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq181_HTML.gif . By (1.6) we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ61_HTML.gif
      (3.5)
      Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq182_HTML.gif , which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ62_HTML.gif
      (3.6)
      It follows from (3.3) and (3.6) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ63_HTML.gif
      (3.7)

      which is the desired inequality.

      Sufficiency.

      First, we prove the existence of a pair of upper and lower solutions. Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq183_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq184_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ64_HTML.gif
      (3.8)
      Otherwise, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq185_HTML.gif , then there exists a real number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq186_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq187_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq188_HTML.gif , which contradicts the condition that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq189_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq190_HTML.gif . In view of condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq191_HTML.gif and (3.8), we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ65_HTML.gif
      (3.9)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ66_HTML.gif
      (3.10)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq192_HTML.gif .

      Suppose that (3.1) holds. Firstly, we define the linear operators http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq193_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq194_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ67_HTML.gif
      (3.11)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ68_HTML.gif
      (3.12)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq195_HTML.gif is given by (2.27). Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ69_HTML.gif
      (3.13)
      It is easy to know from (3.11) and (3.12) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq196_HTML.gif By Lemma 2.4, we know that there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq197_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ70_HTML.gif
      (3.14)
      Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq198_HTML.gif sufficiently small, then by (3.10), we get that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq199_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ71_HTML.gif
      (3.15)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ72_HTML.gif
      (3.16)
      Thus, from (3.14) and (3.16), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ73_HTML.gif
      (3.17)
      Considering http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq200_HTML.gif , it follows from (3.15), (3.17), and condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq201_HTML.gif that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ74_HTML.gif
      (3.18)
      From (3.13) and (3.16), it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ75_HTML.gif
      (3.19)

      Thus, we have shown that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq203_HTML.gif are lower and upper solutions of BVP (1.1), respectively.

      Additionally, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq204_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq205_HTML.gif , by (3.17) and condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq206_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ76_HTML.gif
      (3.20)

      From (3.1), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq207_HTML.gif So, it follows from Lemma 2.5 that BVP (1.1) admits a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq208_HTML.gif positive solution such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq209_HTML.gif

      Remark 3.2.

      Lin et al. [23, 24] considered the existence and uniqueness of solutions for some fourth-order and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq210_HTML.gif conjugate boundary value problems when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq211_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ77_HTML.gif
      (3.21)

      under the following condition:

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq213_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq215_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq216_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ78_HTML.gif
      (3.22)

      Lei et al. [25] and Liu and Yu [26] investigated the existence and uniqueness of positive solutions to singular boundary value problems under the following condition:

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq218_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq220_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq222_HTML.gif is nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq223_HTML.gif and nonincreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq224_HTML.gif .

      Obviously, (3.21)-(3.22) imply condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq225_HTML.gif and condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq226_HTML.gif implies condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq227_HTML.gif . So, condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq228_HTML.gif is weaker than conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq229_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq230_HTML.gif . Thus, functions considered in this paper are wider than those in [2326].

      In the following, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq231_HTML.gif admits the form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq232_HTML.gif , that is, nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq233_HTML.gif is not mixed monotone on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq234_HTML.gif , but monotone with respect http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq235_HTML.gif , BVP (1.1) becomes
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ79_HTML.gif
      (3.23)

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq236_HTML.gif satisfies one of the following:

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq238_HTML.gif is continuous, nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq239_HTML.gif , for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq240_HTML.gif , there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq241_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq242_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq243_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq244_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ80_HTML.gif
      (3.24)

      Theorem 3.3.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq245_HTML.gif holds, then a necessary and sufficient condition for BVP (3.23) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq246_HTML.gif positive solution is that the following integral condition holds
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ81_HTML.gif
      (3.25)

      Proof.

      The proof is similar to that of Theorem 3.1; we omit the details.

      Theorem 3.4.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq247_HTML.gif holds, then a necessary and sufficient condition for problem (3.23) to have a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq248_HTML.gif positive solution is that the following integral condition holds
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ82_HTML.gif
      (3.26)

      Proof.

      The proof is divided into two parts, necessity and suffeciency.

      Necessity.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq249_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq250_HTML.gif positive solution of BVP (3.23). By Lemma 2.4, there exist two constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq252_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq253_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ83_HTML.gif
      (3.27)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq254_HTML.gif be a constant such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq255_HTML.gif . By condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq256_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ84_HTML.gif
      (3.28)
      By virtue of (3.28), we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ85_HTML.gif
      (3.29)
      By boundary value condition, we know that there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq257_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ86_HTML.gif
      (3.30)
      For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq258_HTML.gif by integration of (3.29), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ87_HTML.gif
      (3.31)
      Integrating (3.31), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ88_HTML.gif
      (3.32)
      Exchanging the order of integration, we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ89_HTML.gif
      (3.33)
      Similarly, by integration of (3.29), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ90_HTML.gif
      (3.34)
      Equations (3.33) and (3.34) imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ91_HTML.gif
      (3.35)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq259_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq260_HTML.gif positive solution of BVP (1.1), there exists a positive http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq261_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq262_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq263_HTML.gif . On the other hand, choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq264_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq265_HTML.gif . By condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq266_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ92_HTML.gif
      (3.36)
      Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq267_HTML.gif , which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ93_HTML.gif
      (3.37)
      It follows from (3.35) and (3.37) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ94_HTML.gif
      (3.38)

      which is the desired inequality.

      Sufficiency.

      Suppose that (3.26) holds. Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ95_HTML.gif
      (3.39)
      It is easy to know, from (3.11) and (3.26), that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ96_HTML.gif
      (3.40)
      Thus, (3.12), (3.39), and (3.40) imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq268_HTML.gif By Lemma 2.4, we know that there exists a positive number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq269_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ97_HTML.gif
      (3.41)
      Take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq270_HTML.gif sufficiently small, then by (3.10), we get that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq271_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ98_HTML.gif
      (3.42)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ99_HTML.gif
      (3.43)
      Thus, from (3.41) and (3.43), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ100_HTML.gif
      (3.44)
      Notice that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq272_HTML.gif , it follows from (3.42)–(3.44) and condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq273_HTML.gif that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ101_HTML.gif
      (3.45)
      From (3.39) and (3.43), it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ102_HTML.gif
      (3.46)

      Thus, we have shown that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq274_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq275_HTML.gif are lower and upper solutions of BVP (1.1), respectively.

      From the first conclusion of Lemma 2.5, we conclude that problem (1.1) has at least one http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq276_HTML.gif positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq277_HTML.gif .

      4. Dual Results

      Consider the fourth-order singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq278_HTML.gif -Laplacian differential equations with integral conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ103_HTML.gif
      (4.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ104_HTML.gif
      (4.2)
      Firstly, we define the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq279_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ105_HTML.gif
      (4.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq280_HTML.gif is given by (2.27).

      By analogous methods, we have the following results.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq281_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq282_HTML.gif positive solution of problem (4.1). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq283_HTML.gif can be expressed by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ106_HTML.gif
      (4.4)

      Theorem 4.1.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq284_HTML.gif holds, then a necessary and sufficient condition for (4.1) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq285_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ107_HTML.gif
      (4.5)

      Theorem 4.2.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq286_HTML.gif holds, then a necessary and sufficient condition for problem (4.2) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq287_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ108_HTML.gif
      (4.6)

      Theorem 4.3.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq288_HTML.gif holds, then a necessary and sufficient condition for problem (4.2) to have a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq289_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ109_HTML.gif
      (4.7)
      Consider the fourth-order singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq290_HTML.gif -Laplacian differential equations with integral conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ110_HTML.gif
      (4.8)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ111_HTML.gif
      (4.9)
      Define the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq291_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ112_HTML.gif
      (4.10)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq292_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq293_HTML.gif positive solution of problem (4.8). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq294_HTML.gif can be expressed by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ113_HTML.gif
      (4.11)

      Theorem 4.4.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq295_HTML.gif holds, then a necessary and sufficient condition for problem (4.8) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq296_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ114_HTML.gif
      (4.12)

      Theorem 4.5.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq297_HTML.gif holds, then a necessary and sufficient condition for problem (4.9) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq298_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ115_HTML.gif
      (4.13)

      Theorem 4.6.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq299_HTML.gif holds, then a necessary and sufficient condition for problem (4.9) to have a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq300_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ116_HTML.gif
      (4.14)
      Consider the fourth-order singular http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq301_HTML.gif -Laplacian differential equations with integral conditions:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ117_HTML.gif
      (4.15)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ118_HTML.gif
      (4.16)
      Define the linear operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq302_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ119_HTML.gif
      (4.17)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq303_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq304_HTML.gif positive solution of problem (4.15). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq305_HTML.gif can be expressed by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ120_HTML.gif
      (4.18)

      Theorem 4.7.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq306_HTML.gif holds, then a necessary and sufficient condition for problem (4.15) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq307_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ121_HTML.gif
      (4.19)

      Theorem 4.8.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq308_HTML.gif holds, then a necessary and sufficient condition for problem (4.16) to have a pseudo- http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq309_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ122_HTML.gif
      (4.20)

      Theorem 4.9.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq310_HTML.gif holds, then a necessary and sufficient condition for problem (4.16) to have a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_IEq311_HTML.gif positive solution is that the following integral condition holds:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F862079/MediaObjects/13661_2010_Article_960_Equ123_HTML.gif
      (4.21)

      Declarations

      Acknowledgments

      The project is supported financially by a Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA53) and the National Natural Science Foundation of China (no. 10971179).

      Authors’ Affiliations

      (1)
      Department of Mathematics, Huazhong University of Science and Technology
      (2)
      Department of Mathematics, Liaocheng University
      (3)
      Department of Applied Mathematics, Shandong University of Science and Technology

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      Copyright

      © The Author(s) Xingqiu Zhang and Yujun Cui. 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.