Open Access

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

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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ1_HTML.gif
(11)

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

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

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

() https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq17_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq19_HTML.gif . Furthermore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq20_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ2_HTML.gif
(12)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq21_HTML.gif -Laplacian operator on a half-line:
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq24_HTML.gif provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq25_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ5_HTML.gif
(21)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq26_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq27_HTML.gif . Similarly, the map https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq29_HTML.gif provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq30_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ6_HTML.gif
(22)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq33_HTML.gif be nonnegative, continuous, convex functionals on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq34_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq36_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq37_HTML.gif be a nonnegative continuous functionals on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq38_HTML.gif . Then, for positive real numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq39_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq40_HTML.gif , we define the convex sets
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ7_HTML.gif
(23)
and the closed set
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq42_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq43_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq45_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq47_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq48_HTML.gif be a nonnegative continuous functional on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq49_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq50_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq51_HTML.gif , such that for some positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq53_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ9_HTML.gif
(25)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq54_HTML.gif . Suppose
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq55_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq56_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq57_HTML.gif such that

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

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

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

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

3. Some Lemmas

Define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq70_HTML.gif is continuous at each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq71_HTML.gif , left continuous at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq72_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq73_HTML.gif exists, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq75_HTML.gif in https://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.

https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ12_HTML.gif
(31)

where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq81_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq82_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ13_HTML.gif
(32)
It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq83_HTML.gif . Consider the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq84_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ14_HTML.gif
(33)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq86_HTML.gif . Define the cone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq87_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq88_HTML.gif ]).

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

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

(b)the functions belonging to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq95_HTML.gif ;

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

Lemma 3.3.

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

Proof.

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

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

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

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

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

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

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

So https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq135_HTML.gif .

Finally, we prove for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq136_HTML.gif , there exists sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq137_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ22_HTML.gif
(311)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq138_HTML.gif , we can choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq139_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ23_HTML.gif
(312)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq140_HTML.gif , it follows that
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq141_HTML.gif is relatively compact. In sum, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq143_HTML.gif , the nonnegative continuous convex functionals https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq145_HTML.gif , and the nonnegative continuous functionals https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq146_HTML.gif be defined on the cone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq147_HTML.gif by
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq148_HTML.gif hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq149_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq151_HTML.gif and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq152_HTML.gif satisfy the following conditions:

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

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

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

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq162_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq163_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq164_HTML.gif , we easily show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ28_HTML.gif
(44)
Next we will show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ29_HTML.gif
(45)
In fact, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq165_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ30_HTML.gif
(46)
From condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq166_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ31_HTML.gif
(47)
It follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq167_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq169_HTML.gif , which shows https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq170_HTML.gif . Thus for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq171_HTML.gif , there is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ33_HTML.gif
(49)
Hence by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq172_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ34_HTML.gif
(410)
Therefore we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq173_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq174_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq175_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq176_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq177_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq178_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ37_HTML.gif
(413)
by the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_IEq179_HTML.gif of this theorem,
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F834158/MediaObjects/13661_2009_Article_881_Equ40_HTML.gif
(51)

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

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

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

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

© J. Li and J.J. Nieto 2009

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