Positive Solutions of Singular Initial-Boundary Value Problems to Second-Order Functional Differential Equations

  • Fengfei Jin1 and

    Affiliated with

    • Baoqiang Yan1Email author

      Affiliated with

      Boundary Value Problems20082008:457028

      DOI: 10.1155/2008/457028

      Received: 23 August 2007

      Accepted: 5 August 2008

      Published: 26 August 2008

      Abstract

      Positive solutions to the singular initial-boundary value problems http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq1_HTML.gif are obtained by applying the Schauder fixed-point theorem, where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq2_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq4_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq6_HTML.gif . As an application, an example is given to demonstrate our result.

      1. Introduction

      Recently, in [14], Erbe, Kong, Jiang, Wang, and Weng considered the following singular functional differential equations:
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq7_HTML.gif and the existence of positive solutions to (1.1) is obtained. When http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq8_HTML.gif in (1.1), Agarwal and O'Regan in [5], Lin and Xu in [6] discussed the existence of positive solutions to (1.1) also. We notice that the nonlinearities http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq9_HTML.gif in all the above-mentioned references depend on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq10_HTML.gif .

      The more difficult case is that the term http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq11_HTML.gif depends on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq12_HTML.gif for second-order functional differential equations with delay. When http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq13_HTML.gif has no singularity at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq15_HTML.gif , there are many results on the following (1.2) (see [79] and references therein). Up to now, to our knowledge, there are fewer results on (1.2) when the term http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq16_HTML.gif is allowed to possess singularity for the term http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq17_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq19_HTML.gif , which is of more actual significance.

      In this paper, motivated by above results, we consider the second-order initial-boundary value problems:
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq20_HTML.gif . By Leray-Schauder fixed-point theorem, the existence of positive solutions to (1.2) is obtained when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq21_HTML.gif is singular at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq23_HTML.gif .

      For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq25_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq27_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq28_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq29_HTML.gif are Banach spaces. Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq31_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq33_HTML.gif are cones in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq35_HTML.gif respectively. Now, we give a new definition.

      Deffinition.

      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq36_HTML.gif is said to be singular at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq37_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq38_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq39_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq40_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq41_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq42_HTML.gif is said to be singular at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq43_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq44_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq45_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq46_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq47_HTML.gif .

      And one defines some functions which one has to use in this paper.

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ3_HTML.gif
      (1.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq48_HTML.gif is a Green's function. It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq49_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq51_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq52_HTML.gif

      We now introduce the definition of a solution to IBVP(1.2).

      Deffinition.

      A function http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq53_HTML.gif is said to be a solution to IBVP(1.2) if it satisfies the following conditions:

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq54_HTML.gif is continuous and nonnegative on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq55_HTML.gif ;

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq56_HTML.gif ;

      (3) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq58_HTML.gif exist on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq59_HTML.gif ;

      (4) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq60_HTML.gif is Lebesgue integrable on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq61_HTML.gif ;

      (5) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq62_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq63_HTML.gif .

      Furthermore, a solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq64_HTML.gif is said to be positive if http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq65_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq66_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq67_HTML.gif be a solution to IBVP(1.2). Then, it can be represented as
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ4_HTML.gif
      (1.4)
      It is clear that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ5_HTML.gif
      (1.5)

      for all solutions, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq68_HTML.gif , to IBVP(1.2), where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq69_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq70_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq71_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq72_HTML.gif throughout this paper. Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq73_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq74_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq75_HTML.gif .

      Throughout this paper, we assume the following hypotheses hold.

      (H1) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq76_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq77_HTML.gif .

      (H2) There exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq78_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ6_HTML.gif
      (1.6)

      Lemma 1.3.

      Assume that (H 1 )-(H 2 ) hold, then there exists a , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ7_HTML.gif
      (1.7)

      for all solutions, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq80_HTML.gif , to (1.2).

      Proof.

      Suppose that the claim is false. (1.5) guarantees that there exists a sequence http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq81_HTML.gif of solutions to IBVP(1.2) such that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ8_HTML.gif
      (1.8)
      Without loss of generality, we may assume that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ9_HTML.gif
      (1.9)
      From http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq82_HTML.gif and (1.5), it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ10_HTML.gif
      (1.10)

      which contradicts the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq83_HTML.gif and hence the claim is true provided http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq84_HTML.gif is suitably small.

      Remark 1.4.

      The following inequality
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ11_HTML.gif
      (1.11)

      holds provided that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq85_HTML.gif is sufficiently small, where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq86_HTML.gif is in Lemma 1.3.

      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq87_HTML.gif There exist a nonnegative continuous function http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq88_HTML.gif defined on (0,1) and two nonnegative continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq89_HTML.gif defined on, respectively, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq90_HTML.gif , such that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ12_HTML.gif
      (1.12)
      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq92_HTML.gif ) satisfy
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ13_HTML.gif
      (1.13)
      Furthermore, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq93_HTML.gif is nonincreasing and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq94_HTML.gif is nondecreasing, that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ14_HTML.gif
      (1.14)

      Lemma 1.5 (see [7]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq95_HTML.gif be the Banach space and let X be any nonempty, convex, closed, and bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq96_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq97_HTML.gif is a continuous mapping of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq98_HTML.gif into itself and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq99_HTML.gif is relatively compact, then the mapping http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq100_HTML.gif has at least one fixed point (i.e., there exists an http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq101_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq102_HTML.gif ) .

      Using Lemma 1.5, we present the existence of at least one positive solution to (1.2) when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq103_HTML.gif is singular at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq105_HTML.gif (notice the new Definition 1.1). To some extent, our paper complements and generalizes these in [16, 810].

      2. Main Results

      Theorem 2.1.

      Assume that (H 1 )–(H 3 ) hold. Then, the IBVP( 1.2 ) has at least one positive solution.

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq106_HTML.gif , we can choose an http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq107_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ15_HTML.gif
      (2.1)
      where the positive number http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq108_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ16_HTML.gif
      (2.2)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ17_HTML.gif
      (2.3)
      For each http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq109_HTML.gif , we define http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq110_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ18_HTML.gif
      (2.4)

      It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq111_HTML.gif satisfies the hypotheses http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq113_HTML.gif .

      We now consider the modified initial-boundary value problem:
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ19_HTML.gif
      (2.5)
      We claim that for all solutions, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq114_HTML.gif , to IBVP(2.5),
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ20_HTML.gif
      (2.6)
      Suppose that the claim is false. Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq115_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ21_HTML.gif
      (2.7)

      Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq116_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq117_HTML.gif , there are the following three cases.

      Case 1.

      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq118_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq119_HTML.gif .

      The solution of IBVP(2.5) can be represented as (notice http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq120_HTML.gif Remark 1.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ22_HTML.gif
      (2.8)

      which contradicts (2.7).

      Case 2.

      There exists a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq121_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq123_HTML.gif .

      In this case, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ23_HTML.gif
      (2.9)

      which contradicts (2.7).

      Case 3.

      There exists a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq124_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq126_HTML.gif .

      From (1.5), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ24_HTML.gif
      (2.10)

      which contradicts (2.7).

      So we have
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ25_HTML.gif
      (2.11)

      To prove the existence of positive solutions to IBVP(2.5), we seek to transform (2.5) into an integral equation via the use of Green's function and then find a positive solution by using Lemma 1.5.

      Define a nonempty convex and closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq127_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ26_HTML.gif
      (2.12)
      Then, we define an operator http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq128_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ27_HTML.gif
      (2.13)
      From http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq129_HTML.gif and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq130_HTML.gif , we have, for every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq131_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ28_HTML.gif
      (2.14)
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ29_HTML.gif
      (2.15)

      Together with the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq132_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq133_HTML.gif .

      Also,
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ30_HTML.gif
      (2.16)
      is continuous in (0,1), and
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ31_HTML.gif
      (2.17)
      From http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq134_HTML.gif and (2.15), we can get
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ32_HTML.gif
      (2.18)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq135_HTML.gif is integrable on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq136_HTML.gif .

      Now, we claim that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq137_HTML.gif is equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq138_HTML.gif . We will prove the claim. For any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq139_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ33_HTML.gif
      (2.19)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq140_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq142_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq143_HTML.gif , there is a http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq144_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ34_HTML.gif
      (2.20)
      By (2.6), we have, for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq145_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ35_HTML.gif
      (2.21)

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq146_HTML.gif is a constant number.

      Put http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq147_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq148_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ36_HTML.gif
      (2.22)
      Set http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq149_HTML.gif . Then for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq150_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ37_HTML.gif
      (2.23)

      Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq151_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq152_HTML.gif , the above inequality holds for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq153_HTML.gif .

      Thus, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq154_HTML.gif is a relative compact subset of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq155_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq156_HTML.gif is a compact operator.

      We are now going to prove that the mapping http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq157_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq158_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq159_HTML.gif be arbitrarily chosen and let http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq160_HTML.gif converge to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq161_HTML.gif uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq162_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq163_HTML.gif . Now, we claim that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq164_HTML.gif converge to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq165_HTML.gif uniformly as http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq166_HTML.gif . From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq167_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ38_HTML.gif
      (2.24)
      Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ39_HTML.gif
      (2.25)

      that is, the claim is true.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq168_HTML.gif is continuous with respect to http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq169_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq170_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ40_HTML.gif
      (2.26)
      for each fixed http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq171_HTML.gif . From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq173_HTML.gif , we know that
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ41_HTML.gif
      (2.27)
      and hence
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ42_HTML.gif
      (2.28)
      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq174_HTML.gif is a Lebesgue integrable function defined on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq175_HTML.gif because of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq176_HTML.gif . Consequently, we apply the dominated convergence theorem to get
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ43_HTML.gif
      (2.29)

      which shows that the mapping http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq177_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq178_HTML.gif .

      Then from Lemma 1.5, we get that there exists at least one positive solution, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq179_HTML.gif , to IBVP(2.5) in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq180_HTML.gif . The solution can be represented by (1.4), where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq181_HTML.gif is replaced with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq182_HTML.gif . So, (2.6) holds. Furthermore, from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq183_HTML.gif , we can get
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ44_HTML.gif
      (2.30)

      Thus, the solution of IBVP(2.5) is also the one of (1.2). The proof is complete.

      3. Application

      Example 3.1.

      Consider the singular IBVP(3.1):
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ45_HTML.gif
      (3.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq184_HTML.gif .

      4. Conclusion

      Equation (3.1) has at least one positive solution.

      Now, we will check that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq185_HTML.gif hold in (3.1).

      In IBVP(3.1), http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq186_HTML.gif . It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq187_HTML.gif is continuous and singular at http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq188_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq189_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq190_HTML.gif we choose
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ46_HTML.gif
      (4.1)
      when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq191_HTML.gif ; by simple computation, we can get
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ47_HTML.gif
      (4.2)

      It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq192_HTML.gif is nonincreasing and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq193_HTML.gif is nondecreasing.

      Now, we check http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq194_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq195_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq196_HTML.gif (notice the definition of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq197_HTML.gif ), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ48_HTML.gif
      (4.3)
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ49_HTML.gif
      (4.4)
      We define
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ50_HTML.gif
      (4.5)

      Now, we will prove that there exists http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq198_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq199_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq200_HTML.gif .

      Obviously,
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ51_HTML.gif
      (4.6)
      Put http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq201_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ52_HTML.gif
      (4.7)

      From the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq202_HTML.gif , we can find http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq203_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq204_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq205_HTML.gif . Then, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq206_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq207_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq208_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq209_HTML.gif .

      Furthermore, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ53_HTML.gif
      (4.8)
      Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_Equ54_HTML.gif
      (4.9)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F457028/MediaObjects/13661_2007_Article_806_IEq210_HTML.gif holds.

      So, from Theorem 2.1, IBVP(3.1) has at least one positive solution.

      Declarations

      Acknowledgments

      The research was supported by NNSF of China (10571111) and the fund of Shandong Education Committee (J07WH08).

      Authors’ Affiliations

      (1)
      School of Mathematics Sciences, Shandong Normal University

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      © F. Jin and B. Yan. 2008

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