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

Boundary Value Problems20102011:867615

DOI: 10.1155/2011/867615

Received: 13 July 2010

Accepted: 25 November 2010

Published: 6 December 2010

Abstract

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

1. Introduction

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

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

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

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

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

2. Preliminaries

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

Definition 2.1 (see [8]).

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

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

Definition 2.2 (see [8]).

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

Definition 2.3 (see [8]).

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

Definition 2.4 (see [8]).

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

Lemma 2.5 (see [8]).

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

Lemma 2.6.

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

Proof.

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

So Lemma 2.6 is proved.

Lemma 2.7.

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

Proof.

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

The proof of sufficiency is complete.

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

The proof of Lemma 2.7 is complete.

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

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

Lemma 2.8.

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

Proof.

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

Lemma 2.9.

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

Proof.

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

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

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

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

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

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

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

Lemma 2.10 (see [19]).

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

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

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

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

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

3. Main Result

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

Theorem 3.1.

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

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

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

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

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

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

Proof.

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

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

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

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

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

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

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

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

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

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

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

Theorem 3.2.

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

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

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

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

Proof.

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

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

4. An Example

Example 4.1.

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

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

Proof.

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

Moreover, we have

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

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

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

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

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

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

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Department of Mathematics, Yunnan University

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Copyright

© Y. Li and T. Zhang. 2011

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