Monotone Positive Solution of Nonlinear Third-Order BVP with Integral Boundary Conditions

  • Jian-Ping Sun1Email author and

    Affiliated with

    • Hai-Bao Li1

      Affiliated with

      Boundary Value Problems20102010:874959

      DOI: 10.1155/2010/874959

      Received: 7 September 2010

      Accepted: 31 October 2010

      Published: 2 November 2010

      Abstract

      This paper is concerned with the following third-order boundary value problem with integral boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq1_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq2_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq4_HTML.gif . By using the Guo-Krasnoselskii fixed-point theorem, some sufficient conditions are obtained for the existence and nonexistence of monotone positive solution to the above problem.

      1. Introduction

      Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [1].

      Recently, third-order two-point or multipoint boundary value problems (BVPs for short) have attracted a lot of attention [217]. It is known that BVPs with integral boundary conditions cover multipoint BVPs as special cases. Although there are many excellent works on third-order two-point or multipoint BVPs, a little work has been done for third-order BVPs with integral boundary conditions. It is worth mentioning that, in 2007, Anderson and Tisdell [18] developed an interval of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq5_HTML.gif values whereby a positive solution exists for the following third-order BVP with integral boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ1_HTML.gif
      (1.1)
      by using the Guo-Krasnoselskii fixed-point theorem. In 2008, Graef and Yang [19] studied the third-order BVP with integral boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ2_HTML.gif
      (1.2)

      For second-order or fourth-order BVPs with integral boundary conditions, one can refer to [2024].

      In this paper, we are concerned with the following third-order BVP with integral boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ3_HTML.gif
      (1.3)

      Throughout this paper, we always assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq7_HTML.gif . Some sufficient conditions are established for the existence and nonexistence of monotone positive solution to the BVP (1.3). Here, a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq8_HTML.gif of the BVP (1.3) is said to be monotone and positive if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq11_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq12_HTML.gif . Our main tool is the following Guo-Krasnoselskii fixed-point theorem [25].

      Theorem 1.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq13_HTML.gif be a Banach space and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq14_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq15_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq17_HTML.gif are bounded open subsets of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq18_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq19_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq20_HTML.gif be a completely continuous operator such that either

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq21_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq23_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq24_HTML.gif , or

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq25_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq27_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq28_HTML.gif .

      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq29_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq30_HTML.gif .

      2. Preliminaries

      For convenience, we denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq31_HTML.gif .

      Lemma 2.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq32_HTML.gif . Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq33_HTML.gif , the BVP
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ4_HTML.gif
      (2.1)
      has a unique solution
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ5_HTML.gif
      (2.2)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ6_HTML.gif
      (2.3)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq34_HTML.gif be a solution of the BVP (2.1). Then, we may suppose that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ7_HTML.gif
      (2.4)
      By the boundary conditions in (2.1), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ8_HTML.gif
      (2.5)
      Therefore, the BVP (2.1) has a unique solution
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ9_HTML.gif
      (2.6)

      Lemma 2.2 (see [12]).

      For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq35_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ10_HTML.gif
      (2.7)

      Lemma 2.3 (see [26]).

      For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq36_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ11_HTML.gif
      (2.8)

      In the remainder of this paper, we always assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq39_HTML.gif .

      Lemma 2.4.

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq41_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq42_HTML.gif , then the unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq43_HTML.gif of the BVP (2.1) satisfies

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq45_HTML.gif ,

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq48_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq49_HTML.gif .

      Proof.

      Since (1) is obvious, we only need to prove (2). By (2.2), we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ12_HTML.gif
      (2.9)

      which indicates that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq50_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq51_HTML.gif .

      On the one hand, by (2.9) and Lemma 2.3, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ13_HTML.gif
      (2.10)
      On the other hand, in view of (2.2) and Lemma 2.2, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ14_HTML.gif
      (2.11)
      It follows from (2.10) and (2.11) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ15_HTML.gif
      (2.12)
      which together with Lemma 2.2 implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ16_HTML.gif
      (2.13)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq52_HTML.gif be equipped with the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq53_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq54_HTML.gif is a Banach space. If we denote
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ17_HTML.gif
      (2.14)
      then it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq55_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq56_HTML.gif . Now, we define an operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq57_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq58_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ18_HTML.gif
      (2.15)

      Obviously, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq59_HTML.gif is a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq60_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq61_HTML.gif is a monotone nonnegative solution of the BVP (1.3).

      Lemma 2.5.

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq62_HTML.gif is completely continuous.

      Proof.

      First, by Lemma 2.4, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq63_HTML.gif .

      Next, we assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq64_HTML.gif is a bounded set. Then there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq65_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq66_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq67_HTML.gif . Now, we will prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq68_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq69_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq70_HTML.gif . Then there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq71_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq72_HTML.gif . Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ19_HTML.gif
      (2.16)
      Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq73_HTML.gif , by Lemma 2.2, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ20_HTML.gif
      (2.17)
      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq74_HTML.gif is uniformly bounded. At the same time, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq75_HTML.gif , in view of Lemma 2.3, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ21_HTML.gif
      (2.18)
      which shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq76_HTML.gif is also uniformly bounded. This indicates that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq77_HTML.gif is equicontinuous. It follows from Arzela-Ascoli theorem that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq78_HTML.gif has a convergent subsequence in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq79_HTML.gif . Without loss of generality, we may assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq80_HTML.gif converges in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq81_HTML.gif . On the other hand, by the uniform continuity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq82_HTML.gif , we know that for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq83_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq84_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq85_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq86_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ22_HTML.gif
      (2.19)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq87_HTML.gif . Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq88_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq89_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq90_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ23_HTML.gif
      (2.20)

      which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq91_HTML.gif is equicontinuous. Again, by Arzela-Ascoli theorem, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq92_HTML.gif has a convergent subsequence in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq93_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq94_HTML.gif has a convergent subsequence in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq95_HTML.gif . Thus, we have shown that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq96_HTML.gif is a compact operator.

      Finally, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq97_HTML.gif is continuous. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq99_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq100_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq101_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq102_HTML.gif . Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ24_HTML.gif
      (2.21)
      Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq104_HTML.gif , in view of Lemmas 2.2 and 2.3, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ25_HTML.gif
      (2.22)
      By applying Lebesgue Dominated Convergence theorem, we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ26_HTML.gif
      (2.23)

      which indicates that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq105_HTML.gif is continuous. Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq106_HTML.gif is completely continuous.

      3. Main Results

      For convenience, we define
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ27_HTML.gif
      (3.1)

      Theorem 3.1.

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq107_HTML.gif , then the BVP (1.3) has at least one monotone positive solution.

      Proof.

      In view of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq108_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq109_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ28_HTML.gif
      (3.2)
      By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq110_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq111_HTML.gif so that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ29_HTML.gif
      (3.3)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq112_HTML.gif . Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq113_HTML.gif , in view of (3.2) and (3.3), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ30_HTML.gif
      (3.4)
      By integrating the above inequality on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq114_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ31_HTML.gif
      (3.5)
      which together with (3.4) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ32_HTML.gif
      (3.6)
      On the other hand, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq115_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq116_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ33_HTML.gif
      (3.7)
      By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq117_HTML.gif , we may choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq118_HTML.gif , so that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ34_HTML.gif
      (3.8)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq119_HTML.gif . Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq120_HTML.gif , in view of (3.7) and (3.8), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ35_HTML.gif
      (3.9)
      which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ36_HTML.gif
      (3.10)

      Therefore, it follows from (3.6), (3.10), and Theorem 1.1 that the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq121_HTML.gif has one fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq122_HTML.gif , which is a monotone positive solution of the BVP (1.3).

      Theorem 3.2.

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq123_HTML.gif , then the BVP (1.3) has at least one monotone positive solution.

      Proof.

      The proof is similar to that of Theorem 3.1 and is therefore omitted.

      Theorem 3.3.

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq124_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq126_HTML.gif , then the BVP (1.3) has no monotone positive solution.

      Proof.

      Suppose on the contrary that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq127_HTML.gif is a monotone positive solution of the BVP (1.3). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq129_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq130_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ37_HTML.gif
      (3.11)
      By integrating the above inequality on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq131_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ38_HTML.gif
      (3.12)
      which together with (3.11) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ39_HTML.gif
      (3.13)

      This is a contradiction. Therefore, the BVP (1.3) has no monotone positive solution.

      Similarly, we can prove the following theorem.

      Theorem 3.4.

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq132_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq134_HTML.gif , then the BVP (1.3) has no monotone positive solution.

      Example 3.5.

      Consider the following BVP:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ40_HTML.gif
      (3.14)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq136_HTML.gif , if we choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_IEq137_HTML.gif , then it is easy to compute that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ41_HTML.gif
      (3.15)
      which shows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F874959/MediaObjects/13661_2010_Article_962_Equ42_HTML.gif
      (3.16)

      So, it follows from Theorem 3.1 that the BVP (3.14) has at least one monotone positive solution.

      Declarations

      Acknowledgment

      This work was supported by the National Natural Science Foundation of China (10801068).

      Authors’ Affiliations

      (1)
      Department of Applied Mathematics, Lanzhou University of Technology

      References

      1. Gregus M: Third Order Linear Differential Equations, Mathematics and Its Applications. Reidel, Dordrecht, the Netherlands; 1987.View Article
      2. Anderson DR: Green's function for a third-order generalized right focal problem. Journal of Mathematical Analysis and Applications 2003,288(1):1-14. 10.1016/S0022-247X(03)00132-XMATHMathSciNetView Article
      3. Du Z, Ge W, Lin X: Existence of solutions for a class of third-order nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2004,294(1):104-112. 10.1016/j.jmaa.2004.02.001MATHMathSciNetView Article
      4. Feng Y: Solution and positive solution of a semilinear third-order equation. Journal of Applied Mathematics and Computing 2009,29(1-2):153-161. 10.1007/s12190-008-0121-9MathSciNetView Article
      5. Feng Y, Liu S: Solvability of a third-order two-point boundary value problem. Applied Mathematics Letters 2005,18(9):1034-1040. 10.1016/j.aml.2004.04.016MATHMathSciNetView Article
      6. Guo L-J, Sun J-P, Zhao Y-H: Existence of positive solutions for nonlinear third-order three-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2008,68(10):3151-3158. 10.1016/j.na.2007.03.008MATHMathSciNetView Article
      7. Henderson J, Tisdale CC: Five-point boundary value problems for third-order differential equations by solution matching. Mathematical and Computer Modelling 2005,42(1-2):133-137. 10.1016/j.mcm.2004.04.007MATHMathSciNetView Article
      8. Hopkins B, Kosmatov N: Third-order boundary value problems with sign-changing solutions. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):126-137. 10.1016/j.na.2006.05.003MATHMathSciNetView Article
      9. Liu Z, Debnath L, Kang SM: Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem. Computers & Mathematics with Applications 2008,55(3):356-367. 10.1016/j.camwa.2007.03.021MATHMathSciNetView Article
      10. Liu Z, Ume JS, Kang SM: Positive solutions of a singular nonlinear third order two-point boundary value problem. Journal of Mathematical Analysis and Applications 2007,326(1):589-601. 10.1016/j.jmaa.2006.03.030MATHMathSciNetView Article
      11. Ma R: Multiplicity results for a third order boundary value problem at resonance. Nonlinear Analysis: Theory, Methods & Applications 1998,32(4):493-499. 10.1016/S0362-546X(97)00494-XMATHMathSciNetView Article
      12. Sun Y: Positive solutions for third-order three-point nonhomogeneous boundary value problems. Applied Mathematics Letters 2009,22(1):45-51. 10.1016/j.aml.2008.02.002MATHMathSciNetView Article
      13. Sun Y: Positive solutions of singular third-order three-point boundary value problem. Journal of Mathematical Analysis and Applications 2005,306(2):589-603. 10.1016/j.jmaa.2004.10.029MATHMathSciNetView Article
      14. Yang B: Positive solutions of a third-order three-point boundary-value problem. Electronic Journal of Differential Equations 2008,2008(99):1-10.
      15. Yao Q: Positive solutions of singular third-order three-point boundary value problems. Journal of Mathematical Analysis and Applications 2009,354(1):207-212. 10.1016/j.jmaa.2008.12.057MATHMathSciNetView Article
      16. Yao Q: Successive iteration of positive solution for a discontinuous third-order boundary value problem. Computers & Mathematics with Applications 2007,53(5):741-749. 10.1016/j.camwa.2006.12.007MATHMathSciNetView Article
      17. Yao Q, Feng Y: The existence of solution for a third-order two-point boundary value problem. Applied Mathematics Letters 2002,15(2):227-232. 10.1016/S0893-9659(01)00122-7MATHMathSciNetView Article
      18. Anderson DR, Tisdell CC: Third-order nonlocal problems with sign-changing nonlinearity on time scales. Electronic Journal of Differential Equations 2007,2007(19):1-12.MathSciNetView Article
      19. Graef JR, Yang B: Positive solutions of a third order nonlocal boundary value problem. Discrete and Continuous Dynamical Systems. Series S 2008,1(1):89-97.MATHMathSciNet
      20. Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):364-371. 10.1016/j.na.2007.12.007MATHMathSciNetView Article
      21. Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. Journal of Computational and Applied Mathematics 2008,222(2):351-363. 10.1016/j.cam.2007.11.003MATHMathSciNetView Article
      22. Kong L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2010,72(5):2628-2638. 10.1016/j.na.2009.11.010MATHMathSciNetView Article
      23. Zhang X, Feng M, Ge W: Existence result of second-order differential equations with integral boundary conditions at resonance. Journal of Mathematical Analysis and Applications 2009,353(1):311-319. 10.1016/j.jmaa.2008.11.082MATHMathSciNetView Article
      24. Zhang X, Ge W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Computers & Mathematics with Applications 2009,58(2):203-215. 10.1016/j.camwa.2009.04.002MATHMathSciNetView Article
      25. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
      26. Erbe LH, Wang H: On the existence of positive solutions of ordinary differential equations. Proceedings of the American Mathematical Society 1994,120(3):743-748. 10.1090/S0002-9939-1994-1204373-9MATHMathSciNetView Article

      Copyright

      © The Author(s) Jian-Ping Sun and Hai-Bao Li. 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.