Multiple Positive Solutions for Second-Order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq1_HTML.gif -Laplacian Dynamic Equations with Integral Boundary Conditions

  • Yongkun Li1Email author and

    Affiliated with

    • Tianwei Zhang1

      Affiliated with

      Boundary Value Problems20102011:867615

      DOI: 10.1155/2011/867615

      Received: 13 July 2010

      Accepted: 25 November 2010

      Published: 6 December 2010

      Abstract

      We are concerned with the following second-order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq2_HTML.gif -Laplacian dynamic equations on time scales http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq4_HTML.gif , with integral boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq6_HTML.gif . By using Legget-Williams fixed point theorem, some criteria for the existence of at least three positive solutions are established. An example is presented to illustrate the main result.

      1. Introduction

      Boundary value problems with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq7_HTML.gif -Laplacian have received a lot of attention in recent years. They often occur in the study of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq8_HTML.gif -dimensional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq9_HTML.gif -Laplacian equation, non-Newtonian fluid theory, and the turbulent flow of gas in porous medium [17]. Many works have been carried out to discuss the existence of solutions or positive solutions and multiple solutions for the local or nonlocal boundary value problems.

      On the other hand, the study of dynamic equations on time scales goes back to its founder Stefan Hilger [8] and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models, we refer to [810]. In addition, the study of BVPs on time scales has received a lot of attention in the literature, with the pioneering existence results to be found in [1116].

      However, existence results are not available for dynamic equations on time scales with integral boundary conditions. Motivated by above, we aim at studying the second-order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq10_HTML.gif -Laplacian dynamic equations on time scales in the form of
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ1_HTML.gif
      (1.1)
      with integral boundary condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ2_HTML.gif
      (1.2)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq11_HTML.gif is positive parameter, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq12_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq13_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq16_HTML.gif is the delta derivative, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq17_HTML.gif is the nabla derivative, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq18_HTML.gif is a time scale which is a nonempty closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq19_HTML.gif with the topology and ordering inherited from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq20_HTML.gif , 0 and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq21_HTML.gif are points in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq22_HTML.gif , an interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq24_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq25_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq26_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq28_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq29_HTML.gif , and where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq30_HTML.gif .

      The main purpose of this paper is to establish some sufficient conditions for the existence of at least three positive solutions for BVPs (1.1)-(1.2) by using Legget-Williams fixed point theorem. This paper is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using Legget-Williams fixed point theorem, we establish sufficient conditions for the existence of at least three positive solutions for BVPs (1.1)-(1.2). An illustrative example is given in Section 4.

      2. Preliminaries

      In this section, we will first recall some basic definitions and lemmas which are used in what follows.

      Definition 2.1 (see [8]).

      A time scale http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq31_HTML.gif is an arbitrary nonempty closed subset of the real set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq32_HTML.gif with the topology and ordering inherited from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq33_HTML.gif . The forward and backward jump operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq34_HTML.gif and the graininess http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq35_HTML.gif are defined, respectively, by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ3_HTML.gif
      (2.1)

      The point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq36_HTML.gif is called left-dense, left-scattered, right-dense, or right-scattered if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq38_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq39_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq40_HTML.gif , respectively. Points that are right-dense and left-dense at the same time are called dense. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq41_HTML.gif has a left-scattered maximum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq42_HTML.gif , defined http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq43_HTML.gif ; otherwise, set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq44_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq45_HTML.gif has a right-scattered minimum http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq46_HTML.gif , defined http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq47_HTML.gif ; otherwise, set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq48_HTML.gif .

      Definition 2.2 (see [8]).

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq50_HTML.gif , then the delta derivative of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq51_HTML.gif at the point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq52_HTML.gif is defined to be the number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq53_HTML.gif (provided it exists) with the property that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq54_HTML.gif , there is a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq55_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq56_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ4_HTML.gif
      (2.2)
      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq58_HTML.gif , then the nabla derivative of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq59_HTML.gif at the point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq60_HTML.gif is defined to be the number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq61_HTML.gif (provided it exists) with the property that for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq62_HTML.gif , there is a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq63_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq64_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ5_HTML.gif
      (2.3)

      Definition 2.3 (see [8]).

      A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq65_HTML.gif is rd-continuous provided it is continuous at each right-dense point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq66_HTML.gif and has a left-sided limit at each left-dense point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq67_HTML.gif . The set of rd-continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq68_HTML.gif will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq69_HTML.gif . A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq70_HTML.gif is left-dense continuous (i.e., ld-continuous) if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq71_HTML.gif is continuous at each left-dense point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq72_HTML.gif and its right-sided limit exists (finite) at each right-dense point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq73_HTML.gif . The set of left-dense continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq74_HTML.gif will be denoted by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq75_HTML.gif .

      Definition 2.4 (see [8]).

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq76_HTML.gif , then we define the delta integral by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ6_HTML.gif
      (2.4)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq77_HTML.gif , then we define the nabla integral by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ7_HTML.gif
      (2.5)

      Lemma 2.5 (see [8]).

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq79_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ8_HTML.gif
      (2.6)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq81_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ9_HTML.gif
      (2.7)
      Let the Banach space
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ10_HTML.gif
      (2.8)
      be endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq82_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ11_HTML.gif
      (2.9)
      and choose a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq83_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ12_HTML.gif
      (2.10)

      Lemma 2.6.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq84_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq85_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq86_HTML.gif .

      Proof.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq87_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq88_HTML.gif . It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ13_HTML.gif
      (2.11)
      With http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq90_HTML.gif , one obtains
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ14_HTML.gif
      (2.12)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ15_HTML.gif
      (2.13)
      From (2.11)–(2.13), we can get that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ16_HTML.gif
      (2.14)

      So Lemma 2.6 is proved.

      Lemma 2.7.

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq91_HTML.gif is a solution of BVPs (1.1)-(1.2) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq92_HTML.gif is a solution of the following integral equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ17_HTML.gif
      (2.15)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ18_HTML.gif
      (2.16)

      Proof.

      First assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq93_HTML.gif is a solution of BVPs (1.1)-(1.2); then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ19_HTML.gif
      (2.17)
      That is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ20_HTML.gif
      (2.18)
      Integrating (2.18) from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq94_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq95_HTML.gif , it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ21_HTML.gif
      (2.19)
      Together with (2.19) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq96_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ22_HTML.gif
      (2.20)
      Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ23_HTML.gif
      (2.21)
      namely,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ24_HTML.gif
      (2.22)
      Substituting (2.22) into (2.19), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ25_HTML.gif
      (2.23)

      The proof of sufficiency is complete.

      Conversely, assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq97_HTML.gif is a solution of the following integral equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ26_HTML.gif
      (2.24)
      It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ27_HTML.gif
      (2.25)
      So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq98_HTML.gif . Furthermore, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ28_HTML.gif
      (2.26)
      which imply that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ29_HTML.gif
      (2.27)

      The proof of Lemma 2.7 is complete.

      Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq99_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ30_HTML.gif
      (2.28)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq100_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq101_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq102_HTML.gif .

      Lemma 2.8.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq103_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq104_HTML.gif .

      Proof.

      It is easily obtained from the second part of the proof in Lemma 2.7. The proof is complete.

      Lemma 2.9.

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq105_HTML.gif is complete continuous.

      Proof.

      First, we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq106_HTML.gif maps bounded set into itself. Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq107_HTML.gif is a positive constant and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq108_HTML.gif . Note that the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq109_HTML.gif guarantees that there is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq110_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq111_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq112_HTML.gif . So we get from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq113_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq114_HTML.gif that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ31_HTML.gif
      (2.29)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ32_HTML.gif
      (2.30)
      That is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq115_HTML.gif is uniformly bounded. In addition, notice that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ33_HTML.gif
      (2.31)
      which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ34_HTML.gif
      (2.32)
      which implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ35_HTML.gif
      (2.33)
      That is,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ36_HTML.gif
      (2.34)

      So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq116_HTML.gif is equicontinuous for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq117_HTML.gif . Using Arzela-Ascoli theorem on time scales [17], we obtain that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq118_HTML.gif is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales [18], it is easy to prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq119_HTML.gif is continuous. Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq120_HTML.gif is complete continuous. The proof of this lemma is complete.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq122_HTML.gif be nonnegative continuous convex functionals on a pone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq124_HTML.gif a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq125_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq126_HTML.gif positive numbers with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq127_HTML.gif we defined the following convex sets:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ37_HTML.gif
      (2.35)

      and introduce two assumptions with regard to the functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq129_HTML.gif as follows:

      (H1) there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq130_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq131_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq132_HTML.gif ;

      (H2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq133_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq135_HTML.gif .

      The following fixed point theorem duo to Bai and Ge is crucial in the arguments of our main result.

      Lemma 2.10 (see [19]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq136_HTML.gif be Banach space, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq137_HTML.gif a cone, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq139_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq140_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq141_HTML.gif are nonnegative continuous convex functionals satisfying (H1) and (H2), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq142_HTML.gif is a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq143_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq144_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq145_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq146_HTML.gif is a complete continuous operator. Suppose

      (C1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq148_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq149_HTML.gif ;

      (C2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq151_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq152_HTML.gif ;

      (C3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq153_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq154_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq155_HTML.gif .

      Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq156_HTML.gif has at least three fixed points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq157_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ38_HTML.gif
      (2.36)

      3. Main Result

      In this section, we will give sufficient conditions for the existence of at least three positive solutions to BVPs (1.1)-(1.2).

      Theorem 3.1.

      Suppose that there are positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq159_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq160_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq161_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq163_HTML.gif such that the following conditions are satisfied.

      (H3) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq164_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq165_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ39_HTML.gif
      (3.1)

      (H4) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq166_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq167_HTML.gif .

      (H5) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq168_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq169_HTML.gif , where

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ40_HTML.gif
      (3.2)

      Then BVPs (1.1)-(1.2) have at least three positive solutions.

      Proof.

      By the definition of the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq170_HTML.gif and its properties, it suffices to show that the conditions of Lemma 2.10 hold with respect to the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq171_HTML.gif .

      Let the nonnegative continuous convex functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq173_HTML.gif and the nonnegative continuous concave functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq174_HTML.gif be defined on the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq175_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ41_HTML.gif
      (3.3)

      Then it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq176_HTML.gif and (H1)-(H2) hold.

      First of all, we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq177_HTML.gif . In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq178_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ42_HTML.gif
      (3.4)
      and assumption (H3) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ43_HTML.gif
      (3.5)
      On the other hand, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq179_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq180_HTML.gif ; thus
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ44_HTML.gif
      (3.6)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq181_HTML.gif .

      In the same way, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq182_HTML.gif , then assumption (H4) implies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ45_HTML.gif
      (3.7)

      As in the argument above, we can get that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq183_HTML.gif . Thus, condition (C2) of Lemma 2.10 holds.

      To check condition (C1) in Lemma 2.10. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq184_HTML.gif . We choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq185_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq186_HTML.gif . It is easy to see that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ46_HTML.gif
      (3.8)
      Consequently,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ47_HTML.gif
      (3.9)
      Hence, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq187_HTML.gif , there are
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ48_HTML.gif
      (3.10)
      In view of assumption (H5), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ49_HTML.gif
      (3.11)
      It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ50_HTML.gif
      (3.12)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq188_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq189_HTML.gif . So condition (C1) in Lemma 2.10 is satisfied.

      Finally, we show that (C3) in Lemma 2.10 holds. In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq191_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ51_HTML.gif
      (3.13)

      Thus by Lemma 2.10 and the assumption that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq192_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq193_HTML.gif , BVPs (1.1)-(1.2) have at least three positive solutions. The proof is complete.

      Theorem 3.2.

      Suppose that there are positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq195_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq196_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq198_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq199_HTML.gif such that (H3)-(H4) and the following condition are satisfied.

      (H6) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq200_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq201_HTML.gif , where

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ52_HTML.gif
      (3.14)

      Then BVPs (1.1)-(1.2) have at least three positive solutions.

      Proof.

      Let the nonnegative continuous convex functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq202_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq203_HTML.gif be defined on the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq204_HTML.gif as Theorem 3.1 and the nonnegative continuous concave functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq205_HTML.gif be defined on the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq206_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ53_HTML.gif
      (3.15)
      We will show that condition (C1) in Lemma 2.10 holds. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq207_HTML.gif . We choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq208_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq209_HTML.gif . It is easy to see that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ54_HTML.gif
      (3.16)
      Consequently,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ55_HTML.gif
      (3.17)
      Hence, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq210_HTML.gif , there are
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ56_HTML.gif
      (3.18)
      In view of assumption (H6), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ57_HTML.gif
      (3.19)
      It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ58_HTML.gif
      (3.20)

      Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq211_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq212_HTML.gif . So condition (C1) in Lemma 2.10 is satisfied. Using a similar proof to Theorem 3.1, the other conditions in Lemma 2.10 are satisfied. By Lemma 2.10, BVPs (1.1)-(1.2) have at least three positive solutions. The proof is complete.

      4. An Example

      Example 4.1.

      Consider the following second-order Laplacian dynamic equations on time scales
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ59_HTML.gif
      (4.1)
      with integral boundary condition
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ60_HTML.gif
      (4.2)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ61_HTML.gif
      (4.3)

      Then BVPs (4.1)-(4.2) have at least three positive solutions.

      Proof.

      Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq214_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq215_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq216_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq217_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq218_HTML.gif . It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ62_HTML.gif
      (4.4)
      From (4.1)-(4.2), it is easy to obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ63_HTML.gif
      (4.5)
      Hence, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ64_HTML.gif
      (4.6)

      Moreover, we have

      (H3) for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq219_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ65_HTML.gif
      (4.7)

      (H4) for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq220_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ66_HTML.gif
      (4.8)

      (H5)for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq221_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_Equ67_HTML.gif
      (4.9)

      Therefore, conditions (H3)–(H5) in Theorem 3.1 are satisfied. Further, it is easy to verify that the other conditions in Theorem 3.1 hold. By Theorem 3.1, BVPs (4.1)-(4.2) have at least three positive solutions. The proof is complete.

      Declarations

      Acknowledgment

      This work is supported the by the National Natural Sciences Foundation of China under Grant no. 10971183.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Yunnan University

      References

      1. O'Regan D: Some general existence principles and results for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq222_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq223_HTML.gif . SIAM Journal on Mathematical Analysis 1993, 24(3):648–668. 10.1137/0524040MATHMathSciNetView Article
      2. del Pino M, Drábek P, Manásevich R: The Fredholm alternative at the first eigenvalue for the one-dimensional p -Laplacian. Journal of Differential Equations 1999, 151(2):386–419. 10.1006/jdeq.1998.3506MATHMathSciNetView Article
      3. Cabada A, Pouso RL: Existence results for the problem http://static-content.springer.com/image/art%3A10.1155%2F2011%2F867615/MediaObjects/13661_2010_Article_62_IEq224_HTML.gif with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 1999, 35(2):221–231. 10.1016/S0362-546X(98)00009-1MATHMathSciNetView Article
      4. Lü H, Zhong C: A note on singular nonlinear boundary value problems for the one-dimensional p -Laplacian. Applied Mathematics Letters 2001, 14(2):189–194. 10.1016/S0893-9659(00)00134-8MATHMathSciNetView Article
      5. Feng W, Webb JRL: Solvability of three point boundary value problems at resonance. Nonlinear Analysis: Theory, Methods & Applications 1997, 30(6):3227–3238. 10.1016/S0362-546X(96)00118-6MATHMathSciNetView Article
      6. Gupta CP: A non-resonant multi-point boundary-value problem for a p -Laplacian type operator. In Proceedings of the 5th Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), Electron. J. Differ. Equ. Conf.. Volume 10. Southwest Texas State University; 2003:143–152.
      7. Tian Y, Ge W: Periodic solutions of non-autonomous second-order systems with a p -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(1):192–203. 10.1016/j.na.2005.11.020MATHMathSciNetView Article
      8. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View Article
      9. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATH
      10. Agarwal RP, Bohner M, Li W-T: Nonoscillation and Oscillation: Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 267. Marcel Dekker, New York, NY, USA; 2004:viii+376.
      11. Sun H-R, Tang L-T, Wang Y-H: Eigenvalue problem for p -Laplacian three-point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2007, 331(1):248–262. 10.1016/j.jmaa.2006.08.080MATHMathSciNetView Article
      12. Jiang L, Zhou Z: Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(4):1376–1388. 10.1016/j.na.2007.06.034MATHMathSciNetView Article
      13. Tian Y, Ge W: Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(9):2833–2842. 10.1016/j.na.2007.08.054MATHMathSciNetView Article
      14. Aykut Hamal N, Yoruk Fulya: Positive solutions of nonlinear m -point boundary value problems on time scales. Journal of Computational and Applied Mathematics 2009, 231(1):92–105. 10.1016/j.cam.2009.02.003MATHMathSciNetView Article
      15. Sun H-R: Triple positive solutions for p -Laplacian m -point boundary value problem on time scales. Computers & Mathematics with Applications 2009, 58(9):1736–1741. 10.1016/j.camwa.2009.07.083MATHMathSciNetView Article
      16. Yang Y, Meng F: Positive solutions of the singular semipositone boundary value problem on time scales. Mathematical and Computer Modelling 2010, 52(3–4):481–489. 10.1016/j.mcm.2010.03.045MATHMathSciNetView Article
      17. Agarwal RP, Bohner M, Rehák P: Half-linear dynamic equations. In Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday. Volume 1. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:1–57.
      18. Aulbach B, Neidhart L: Integration on measure chains. In Proceedings of the 6th International Conference on Difference Equations, Boca Raton, Fla, USA. CRC Press; 2004:239–252.
      19. Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems. Computers & Mathematics with Applications 2004, 48(5–6):699–707. 10.1016/j.camwa.2004.03.002MATHMathSciNetView Article

      Copyright

      © Y. Li and T. Zhang. 2011

      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.