Positive Solutions for Third-Order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq1_HTML.gif -Laplacian Functional Dynamic Equations on Time Scales

  • Changxiu Song1Email author and

    Affiliated with

    • Xuejun Gao1

      Affiliated with

      Boundary Value Problems20102011:279752

      DOI: 10.1155/2011/279752

      Received: 31 March 2010

      Accepted: 9 December 2010

      Published: 15 December 2010

      Abstract

      The authors study the boundary value problems for a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq2_HTML.gif -Laplacian functional dynamic equation on a time scale, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq8_HTML.gif . By using the twin fixed-point theorem, sufficient conditions are established for the existence of twin positive solutions.

      1. Introduction

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq9_HTML.gif be a closed nonempty subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq10_HTML.gif , and let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq11_HTML.gif have the subspace topology inherited from the Euclidean topology on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq12_HTML.gif . In some of the current literature, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq13_HTML.gif is called a time scale (or measure chain). For notation, we shall use the convention that, for each interval of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq14_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq16_HTML.gif will denote time scales interval, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq17_HTML.gif .

      In this paper, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq18_HTML.gif be a time scale such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq19_HTML.gif , 0, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq20_HTML.gif . We are concerned with the existence of positive solutions of the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq21_HTML.gif -Laplacian dynamic equation on a time scale
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq22_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq23_HTML.gif -Laplacian operator, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq26_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq27_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq28_HTML.gif and

      the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq30_HTML.gif is continuous,

      the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq32_HTML.gif is left dense continuous (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq33_HTML.gif and does not vanish identically on any closed subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq34_HTML.gif . Here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq35_HTML.gif denotes the set of all left dense continuous functions from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq36_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq37_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq39_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq40_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq42_HTML.gif is continuous, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq43_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq44_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq46_HTML.gif is continuous and satisfies that there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq47_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ2_HTML.gif
      (1.2)

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq48_HTML.gif -Laplacian problems with two-, three-, m-point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example see [14] and references therein. However, there are not many concerning the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq49_HTML.gif -Laplacian problems on time scales, especially for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq50_HTML.gif -Laplacian functional dynamic equations on time scales.

      The motivations for the present work stems from many recent investigations in [58] and references therein. Especially, Kaufmann and Raffoul [8] considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions. In this paper, we apply the twin fixed-point theorem to obtain at least two positive solutions of boundary value problem (BVP for short) (1.1) when growth conditions are imposed on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq51_HTML.gif . Finally, we present two corollaries, which show that under the assumptions that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq52_HTML.gif is superlinear or sublinear, BVP (1.1) has at least two positive solutions.

      Given a nonnegative continuous functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq53_HTML.gif on a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq54_HTML.gif of a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq55_HTML.gif , we define for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq56_HTML.gif the sets
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ3_HTML.gif
      (1.3)

      The following twin fixed-point lemma due to [9] will play an important role in the proof of our results.

      Lemma 1.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq57_HTML.gif be a real Banach space, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq58_HTML.gif a cone of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq59_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq61_HTML.gif two nonnegative increasing continuous functionals, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq62_HTML.gif a nonnegative continuous functional, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq63_HTML.gif . Suppose that there are two positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq65_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ4_HTML.gif
      (1.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq66_HTML.gif is completely continuous. There are positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq67_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ5_HTML.gif
      (1.5)

      and

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq68_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq69_HTML.gif ,

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq70_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq71_HTML.gif ,

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq73_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq74_HTML.gif .

      Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq75_HTML.gif has at least two fixed points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq77_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ6_HTML.gif
      (1.6)

      2. Positive Solutions

      We note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq78_HTML.gif is a solution of (1.1) if and only if
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ7_HTML.gif
      (2.1)

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq79_HTML.gif be endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq80_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq81_HTML.gif is concave and nonnegative valued on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq82_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq83_HTML.gif .

      Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq84_HTML.gif is a Banach space with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq86_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq87_HTML.gif . For each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq88_HTML.gif , extend http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq89_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq90_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq91_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq92_HTML.gif .

      Define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq93_HTML.gif as
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ8_HTML.gif
      (2.2)
      We seek a fixed point, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq94_HTML.gif , of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq95_HTML.gif in the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq96_HTML.gif . Define
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ9_HTML.gif
      (2.3)

      Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq97_HTML.gif denotes a positive solution of BVP (1.1).

      It follows from (2.2) that

      Lemma 2.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq98_HTML.gif be defined by (2.2). If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq99_HTML.gif , then

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq100_HTML.gif .

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq101_HTML.gif is completely continuous.

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq102_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq103_HTML.gif .

      (iv) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq104_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq105_HTML.gif .

      The proof is similar to the proofs of Lemma  2.3 and Theorem  3.1 in [7], and is omitted.

      Fix http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq106_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq107_HTML.gif , and set
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ10_HTML.gif
      (2.4)

      Throughout this paper, we assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq109_HTML.gif .

      Now, we define the nonnegative, increasing, continuous functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq111_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq112_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq113_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ11_HTML.gif
      (2.5)
      We have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ12_HTML.gif
      (2.6)
      Then,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ13_HTML.gif
      (2.7)
      We also see that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ14_HTML.gif
      (2.8)
      For the notational convenience, we denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq114_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq117_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ15_HTML.gif
      (2.9)

      Theorem 2.2.

      Suppose that there are positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq118_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ16_HTML.gif
      (2.10)

      Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq119_HTML.gif satisfies the following conditions:

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq121_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq122_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq123_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq125_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq126_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq127_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ17_HTML.gif
      (2.11)

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq129_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq130_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq131_HTML.gif .

      Then, BVP (1.1) has at least two positive solutions of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ18_HTML.gif
      (2.12)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq132_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq134_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq135_HTML.gif .

      Proof.

      By the definition of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq136_HTML.gif and its properties, it suffices to show that the conditions of Lemma 1.1 hold with respect to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq137_HTML.gif .

      First, we verify that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq138_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq139_HTML.gif .

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq140_HTML.gif , one gets http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq141_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq142_HTML.gif . Recalling that (2.7), we know http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq143_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq144_HTML.gif . Then, we get
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ19_HTML.gif
      (2.13)

      Secondly, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq145_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq146_HTML.gif .

      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq147_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq148_HTML.gif , it holds that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq149_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq150_HTML.gif , and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq151_HTML.gif implies
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ20_HTML.gif
      (2.14)
      Then,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ21_HTML.gif
      (2.15)
      So, we have
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ22_HTML.gif
      (2.16)
      Finally, we show that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ23_HTML.gif
      (2.17)
      It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq152_HTML.gif . On the other hand, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq153_HTML.gif and (2.7) imply
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ24_HTML.gif
      (2.18)
      Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ25_HTML.gif
      (2.19)
      By Lemma 1.1, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq154_HTML.gif has at least two different fixed points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq156_HTML.gif satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ26_HTML.gif
      (2.20)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ27_HTML.gif
      (2.21)

      which are twin positive solutions of BVP (1.1). The proof is complete.

      In analogy to Theorem 2.2, we have the following result.

      Theorem 2.3.

      Suppose that there are positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq157_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ28_HTML.gif
      (2.22)

      Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq158_HTML.gif satisfies the following conditions: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq159_HTML.gif

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq161_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq162_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq163_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ29_HTML.gif
      (2.23)

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq165_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq166_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq167_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq169_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq170_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq171_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ30_HTML.gif
      (2.24)
      Then, BVP (1.1) has at least two positive solutions of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ31_HTML.gif
      (2.25)

      Now, we give theorems, which may be considered as the corollaries of Theorems 2.2 and 2.3.

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ32_HTML.gif
      (2.26)
      and choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq173_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq174_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ33_HTML.gif
      (2.27)

      From above, we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq175_HTML.gif .

      Theorem 2.4.

      If the following conditions are satisfied:

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq177_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq178_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq179_HTML.gif ,

      there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq181_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq182_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ34_HTML.gif
      (2.28)
      Then, BVP (1.1) has at least two positive solutions of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ35_HTML.gif
      (2.29)

      Proof.

      First, choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq183_HTML.gif , one gets
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ36_HTML.gif
      (2.30)
      Secondly, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq184_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq185_HTML.gif sufficiently small such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ37_HTML.gif
      (2.31)
      Without loss of generality, suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq186_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq187_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq188_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq189_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq191_HTML.gif . Thus,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ38_HTML.gif
      (2.32)
      Thirdly, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq192_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq193_HTML.gif sufficiently large such that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ39_HTML.gif
      (2.33)
      Without loss of generality, suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq194_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq195_HTML.gif . Then,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ40_HTML.gif
      (2.34)

      We get now http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq196_HTML.gif , and then the conditions in Theorem 2.2 are all satisfied. By Theorem 2.2, BVP (1.1) has at least two positive solutions. The proof is complete.

      Theorem 2.5.

      If the following conditions are satisfied:

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq198_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq199_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq200_HTML.gif ,

      there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq202_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq203_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ41_HTML.gif
      (2.35)
      Then, BVP (1.1) has at least two positive solutions of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ42_HTML.gif
      (2.36)

      The proof is similar to that of Theorem 2.4 and we omitted it.

      The following Corollaries are obvious.

      Corollary 2.6.

      If the following conditions are satisfied:

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq205_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq206_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq207_HTML.gif ,

      there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq209_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq210_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ43_HTML.gif
      (2.37)
      Then, BVP (1.1) has at least two positive solutions of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ44_HTML.gif
      (2.38)

      Corollary 2.7.

      If the following conditions are satisfied:

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq212_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq214_HTML.gif ;

      there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq216_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq217_HTML.gif , one has
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ45_HTML.gif
      (2.39)
      Then, BVP (1.1) has at least two positive solutions of the form
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ46_HTML.gif
      (2.40)

      3. Example

      Example 3.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq218_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq221_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq222_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq223_HTML.gif .

      We consider the following boundary value problem:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ47_HTML.gif
      (3.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq224_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq225_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq226_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq227_HTML.gif .

      Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq228_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq229_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq231_HTML.gif , direct calculation shows that
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ48_HTML.gif
      (3.2)

      Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq232_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq233_HTML.gif satisfies

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq235_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq236_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq237_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ49_HTML.gif
      (3.3)

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq239_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq240_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq241_HTML.gif ,

      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq243_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq244_HTML.gif , uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq245_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ50_HTML.gif
      (3.4)

      Then all conditions of Theorem 2.3 hold. Thus, with Theorem 2.3, the BVP (3.1) has at least two positive solutions.

      Declarations

      Acknowledgment

      This paper is supported by Grants nos. (10871052) and (10901060) from the NNSF of China, and by Grant (no. 10151009001000032) from the NSF of Guangdong.

      Authors’ Affiliations

      (1)
      School of Applied Mathematics, Guangdong University of Technology

      References

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      © C. Song and X. Gao. 2011

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