Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses

  • Jianli Li1Email author and

    Affiliated with

    • Juan J Nieto2

      Affiliated with

      Boundary Value Problems20092009:834158

      DOI: 10.1155/2009/834158

      Received: 7 March 2009

      Accepted: 25 April 2009

      Published: 4 June 2009

      Abstract

      We consider a multi-point boundary value problem on the half-line with impulses. By using a fixed-point theorem due to Avery and Peterson, the existence of at least three positive solutions is obtained.

      1. Introduction

      Impulsive differential equations are a basic tool to study evolution processes that are subjected to abrupt changes in their state. For instance, many biological, physical, and engineering applications exhibit impulsive effects (see [13]). It should be noted that recent progress in the development of the qualitative theory of impulsive differential equations has been stimulated primarily by a number of interesting applied problems [424].

      In this paper, we consider the existence of multiple positive solutions of the following impulsive boundary value problem (for short BVP) on a half-line:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ1_HTML.gif
      (11)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq1_HTML.gif ,  http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq4_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq5_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq6_HTML.gif satisfy

      () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq8_HTML.gif ;

      () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq10_HTML.gif ,  http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq11_HTML.gif , and when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq12_HTML.gif is bounded, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq14_HTML.gif are bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq15_HTML.gif ;

      () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq17_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq18_HTML.gif is not identically zero on any compact subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq19_HTML.gif . Furthermore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq20_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ2_HTML.gif
      (12)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ3_HTML.gif
      (13)

      Boundary value problems on the half-line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and there are many results in this area, see [8, 13, 14, 20, 2527], for example.

      Lian et al. [25] studied the following boundary value problem of second-order differential equation with a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq21_HTML.gif -Laplacian operator on a half-line:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ4_HTML.gif
      (14)

      They showed the existence at least three positive solutions for (1.4) by using a fixed point theorem in a cone due to Avery-Peterson [28].

      Yan [20], by using Leray-Schauder theorem and fixed point index theory presents some results on the existence for the boundary value problems on the half-line with impulses and infinite delay.

      However to the best knowledge of the authors, there is no paper concerned with the existence of three positive solutions to multipoint boundary value problems of impulsive differential equation on infinite interval so far. Motivated by [20, 25], in this paper, we aim to investigate the existence of triple positive solutions for BVP (1.1). The method chosen in this paper is a fixed point technique due to Avery and Peterson [28].

      2. Preliminaries

      In this section, we give some definitions and results that we will use in the rest of the paper.

      Definition 2.1.

      Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq22_HTML.gif is a cone in a Banach. The map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq23_HTML.gif is a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq24_HTML.gif provided http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq25_HTML.gif is continuous and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ5_HTML.gif
      (21)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq26_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq27_HTML.gif . Similarly, the map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq28_HTML.gif is a nonnegative continuous convex functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq29_HTML.gif provided http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq30_HTML.gif is continuous and
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ6_HTML.gif
      (22)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq31_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq32_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq33_HTML.gif be nonnegative, continuous, convex functionals on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq35_HTML.gif be a nonnegative, continuous, concave functionals on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq36_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq37_HTML.gif be a nonnegative continuous functionals on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq38_HTML.gif . Then, for positive real numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq39_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq40_HTML.gif , we define the convex sets
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ7_HTML.gif
      (23)
      and the closed set
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ8_HTML.gif
      (24)

      To prove our main results, we need the following fixed point theorem due to Avery and Peterson in [28].

      Theorem 2.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq41_HTML.gif be a cone in a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq42_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq44_HTML.gif be nonnegative continuous convex functionals on a cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq46_HTML.gif be a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq47_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq48_HTML.gif be a nonnegative continuous functional on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq49_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq50_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq51_HTML.gif , such that for some positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq53_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ9_HTML.gif
      (25)
      for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq54_HTML.gif . Suppose
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ10_HTML.gif
      (26)

      is completely continuous and there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq55_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq56_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq57_HTML.gif such that

      (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq59_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq60_HTML.gif ;

      (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq61_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq62_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq63_HTML.gif ;

      (iii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq65_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq66_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq67_HTML.gif

      Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq68_HTML.gif has at least three fixed points http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq69_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ11_HTML.gif
      (27)

      3. Some Lemmas

      Define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq70_HTML.gif is continuous at each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq71_HTML.gif , left continuous at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq73_HTML.gif exists, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq74_HTML.gif .

      By a solution of (1.1) we mean a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq75_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq76_HTML.gif satisfying the relations in (1.1).

      Lemma 3.1.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq77_HTML.gif is a solution of (1.1) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq78_HTML.gif is a solution of the following equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ12_HTML.gif
      (31)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq79_HTML.gif is defined as (1.3).

      The proof is similar to Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq80_HTML.gif in [9], and here we omit it.

      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq81_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq82_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ13_HTML.gif
      (32)
      It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq83_HTML.gif . Consider the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq84_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ14_HTML.gif
      (33)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq85_HTML.gif is a Banach space, equipped with the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq86_HTML.gif . Define the cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq87_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ15_HTML.gif
      (34)

      Lemma 3.2 (see [20, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq88_HTML.gif ]).

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq89_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq90_HTML.gif is compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq91_HTML.gif , if the following conditions hold:

      (a) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq92_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq93_HTML.gif ;

      (b)the functions belonging to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq94_HTML.gif are piecewise equicontinuous on any interval of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq95_HTML.gif ;

      (c)the functions from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq96_HTML.gif are equiconvergent, that is, given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq97_HTML.gif , there corresponds http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq98_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq99_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq101_HTML.gif .

      Lemma 3.3.

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq102_HTML.gif is completely continuous.

      Proof.

      Firstly, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq103_HTML.gif , from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq104_HTML.gif , it is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq105_HTML.gif is well defined, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq106_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq107_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq108_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ16_HTML.gif
      (35)
      so
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ17_HTML.gif
      (36)

      which shows http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq109_HTML.gif .

      Now we prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq110_HTML.gif is continuous and compact, respectively. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq111_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq112_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq113_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq114_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq115_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq116_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq117_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq118_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq119_HTML.gif , and we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ18_HTML.gif
      (37)
      Therefore by the Lebesgue dominated convergence theorem and continuity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq121_HTML.gif , one arrives at
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ19_HTML.gif
      (38)

      Therefore http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq122_HTML.gif is continuous.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq123_HTML.gif be any bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq124_HTML.gif . Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq125_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq126_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq127_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq128_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq129_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ20_HTML.gif
      (39)

      So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq130_HTML.gif is bounded.

      Moreover, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq132_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq133_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ21_HTML.gif
      (310)

      So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq134_HTML.gif is quasi-equicontinuous on any compact interval of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq135_HTML.gif .

      Finally, we prove for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq136_HTML.gif , there exists sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq137_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ22_HTML.gif
      (311)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq138_HTML.gif , we can choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq139_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ23_HTML.gif
      (312)
      For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq140_HTML.gif , it follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ24_HTML.gif
      (313)

      That is (3.11) holds. By Lemma 3.2, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq141_HTML.gif is relatively compact. In sum, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq142_HTML.gif is completely continuous.

      4. Existence of Three Positive Solutions

      Let the nonnegative continuous concave functional http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq143_HTML.gif , the nonnegative continuous convex functionals http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq144_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq145_HTML.gif , and the nonnegative continuous functionals http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq146_HTML.gif be defined on the cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq147_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ25_HTML.gif
      (41)
      For notational convenience, we denote by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ26_HTML.gif
      (42)

      The main result of this paper is the following.

      Theorem 4.1.

      Assume http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq148_HTML.gif hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq149_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq151_HTML.gif and suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq152_HTML.gif satisfy the following conditions:

      () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq154_HTML.gif

      () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq156_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq157_HTML.gif ,

      () http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq159_HTML.gif ,

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq160_HTML.gif . Then (1.1) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq162_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ27_HTML.gif
      (43)

      Proof.

      Step 1.

      From the definition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq163_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq164_HTML.gif , we easily show that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ28_HTML.gif
      (44)
      Next we will show that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ29_HTML.gif
      (45)
      In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq165_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ30_HTML.gif
      (46)
      From condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq166_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ31_HTML.gif
      (47)
      It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ32_HTML.gif
      (48)

      Thus (4.5) holds.

      Step 2.

      We show that condition (i) in Theorem 2.2 holds. Taking http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq167_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq169_HTML.gif , which shows http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq170_HTML.gif . Thus for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq171_HTML.gif , there is
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ33_HTML.gif
      (49)
      Hence by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq172_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ34_HTML.gif
      (410)
      Therefore we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ35_HTML.gif
      (411)

      This shows the condition (i) in Theorem 2.2 is satisfied.

      Step 3.

      We now prove (ii) in Theorem 2.2 holds. For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq173_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq174_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ36_HTML.gif
      (412)

      Hence, condition (ii) in Theorem 2.2 is satisfied.

      Step 4.

      Finally, we prove (iii) in Theorem 2.2 is satisfied. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq175_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq176_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq177_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq178_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ37_HTML.gif
      (413)
      by the condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq179_HTML.gif of this theorem,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ38_HTML.gif
      (414)
      Thus condition (iii) in Theorem 2.2 holds. Therefore an application of Theorem 2.2 implies the boundary value problem (1.1) has at least three positive solutions such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ39_HTML.gif
      (415)

      5. An Example

      Now we consider the following boundary value problem
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ40_HTML.gif
      (51)

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq180_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq183_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq184_HTML.gif . If taking http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq185_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq186_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq187_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq188_HTML.gif satisfies the following:

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq190_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq191_HTML.gif ;

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq192_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq193_HTML.gif ;

      (3) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq195_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq196_HTML.gif .

      Then all conditions of Theorem 4.1 hold, so by Theorem 4.1, boundary value problem (5.1) has at least three positive solutions.

      Declarations

      Acknowledgments

      This work is supported by the NNSF of China (no. 60671066), A project supported by Scientific Research Fund of Hunan Provincial Education Department (07B041) and Program for Young Excellent Talents in Hunan Normal University, The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PR.

      Authors’ Affiliations

      (1)
      Department of Mathematics, Hunan Normal University
      (2)
      Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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      © J. Li and J.J. Nieto 2009

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