Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator

Boundary Value Problems20112011:827510

DOI: 10.1155/2011/827510

Received: 20 September 2010

Accepted: 19 January 2011

Published: 23 February 2011

Abstract

We consider the existence, multiplicity of positive solutions for the integral boundary value problem with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq1_HTML.gif -Laplacian http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq6_HTML.gif is an odd, increasing homeomorphism from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq7_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq8_HTML.gif . We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq9_HTML.gif is involved with the first-order derivative explicitly.

1. Introduction

We are interested in the existence of positive solutions for the integral boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq10_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq11_HTML.gif satisfy the following conditions.

(H1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq12_HTML.gif is an odd, increasing homeomorphism from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq13_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq14_HTML.gif , and there exist two increasing homeomorphisms http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq16_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq17_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq18_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ2_HTML.gif
(1.2)

Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq19_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq20_HTML.gif denotes the inverse of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq21_HTML.gif .

(H2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq22_HTML.gif is continuous. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq23_HTML.gif are nonnegative, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq25_HTML.gif .

The assumption (H1) on the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq26_HTML.gif was first introduced by Wang [1, 2], it covers two important cases: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq28_HTML.gif . The existence of positive solutions for two above cases received wide attention (see [310]). For example, Ji and Ge [4] studied the multiplicity of positive solutions for the multipoint boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ3_HTML.gif
(1.3)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq30_HTML.gif . They provided sufficient conditions for the existence of at least three positive solutions by using Avery-Peterson fixed point theorem. In [5], Feng et al. researched the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ4_HTML.gif
(1.4)

where the nonlinear term http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq31_HTML.gif does not depend on the first-order derivative and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq32_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq33_HTML.gif . They obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq34_HTML.gif by applying Krasnoselskii fixed point theorem.

As for integral boundary value problem, when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq35_HTML.gif is linear, the existence of positive solutions has been obtained (see [810]). In [8], the author investigated the positive solutions for the integral boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ5_HTML.gif
(1.5)

The main tools are the priori estimate method and the Leray-Schauder fixed point theorem. However, there are few papers dealing with the existence of positive solutions when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq36_HTML.gif satisfies (H1) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq37_HTML.gif depends on both http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq39_HTML.gif . This paper fills this gap in the literature. The aim of this paper is to establish some simple criteria for the existence of positive solutions of BVP(1.1). To get rid of the difficulty of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq40_HTML.gif depending on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq41_HTML.gif , we will define a special norm in Banach space (in Section 2).

This paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of one or two positive solutions for BVP(1.1) is established by applying the Krasnoselskii fixed point theorem. In Section 4, we give the existence of three positive solutions for BVP(1.1) by using a new fixed point theorem introduced by Avery and Peterson. In Section 5, we give some examples to illustrate our main results.

2. Preliminaries

The basic space used in this paper is a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq42_HTML.gif with norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq43_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq44_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq45_HTML.gif . Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ6_HTML.gif
(2.1)

It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq46_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq47_HTML.gif .

Lemma 2.1 (see [7]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq48_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq50_HTML.gif .

Lemma 2.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq51_HTML.gif , then there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq52_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq53_HTML.gif .

Proof.

The mean value theorem guarantees that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq54_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ7_HTML.gif
(2.2)
Moreover, the mean value theorem of differential guarantees that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq55_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ8_HTML.gif
(2.3)
So we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ9_HTML.gif
(2.4)

Denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq56_HTML.gif ; then the proof is complete.

Lemma 2.3.

Assume that (H1), (H2) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq57_HTML.gif is a solution of BVP(1.1), there exists a unique http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq58_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq59_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq61_HTML.gif .

Proof.

From the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq62_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq63_HTML.gif is strictly decreasing. It follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq64_HTML.gif is also strictly decreasing. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq65_HTML.gif is strictly concave on [0, 1]. Without loss of generality, we assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq66_HTML.gif . By the concavity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq67_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq69_HTML.gif . So we get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq70_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq71_HTML.gif , it is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq72_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq74_HTML.gif .

On the other hand, from the concavity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq75_HTML.gif , we know that there exists a unique http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq76_HTML.gif where the maximum is attained. By the boundary conditions and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq77_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq78_HTML.gif or 1, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq79_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq80_HTML.gif and then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq81_HTML.gif .

Lemma 2.4.

Assume that (H1), (H2) hold. Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq82_HTML.gif is a solution of BVP(1.1); then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ10_HTML.gif
(2.5)
or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ11_HTML.gif
(2.6)

Proof.

First, by integrating (1.1) on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq83_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ12_HTML.gif
(2.7)
then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ13_HTML.gif
(2.8)
Thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ14_HTML.gif
(2.9)
or
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ15_HTML.gif
(2.10)
According to the boundary condition, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ16_HTML.gif
(2.11)

By a similar argument in [5], http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq84_HTML.gif ; then the proof is completed.

Now we define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq85_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ17_HTML.gif
(2.12)

Lemma 2.5.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq86_HTML.gif is completely continuous.

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq87_HTML.gif ; then from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq88_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ18_HTML.gif
(2.13)

So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq89_HTML.gif is monotone decreasing continuous and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq90_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq91_HTML.gif is nonnegative and concave on [0, 1]. By computation, we can get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq92_HTML.gif . This shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq93_HTML.gif . The continuity of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq94_HTML.gif is obvious since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq95_HTML.gif is continuous. Next, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq96_HTML.gif is compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq97_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq98_HTML.gif be a bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq100_HTML.gif is a constant such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq101_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq102_HTML.gif . From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq103_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq104_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ19_HTML.gif
(2.14)

Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq105_HTML.gif is uniformly bounded and equicontinuous. So we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq106_HTML.gif is compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq107_HTML.gif . From (2.13), we know for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq108_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq109_HTML.gif , such that when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq110_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq111_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq112_HTML.gif is compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq113_HTML.gif ; it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq114_HTML.gif is compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq115_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq116_HTML.gif is compact on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq117_HTML.gif .

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq118_HTML.gif is completely continuous.

It is easy to prove that each fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq119_HTML.gif is a solution for BVP(1.1).

Lemma 2.6 (see [1]).

Assume that (H1) holds. Then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq120_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ20_HTML.gif
(2.15)

To obtain positive solution for BVP(1.1), the following definitions and fixed point theorems in a cone are very useful.

Definition 2.7.

The map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq121_HTML.gif is said to be a nonnegative continuous concave functional on a cone of a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq122_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq123_HTML.gif is continuous and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ21_HTML.gif
(2.16)
for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq124_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq125_HTML.gif . Similarly, we say the map http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq126_HTML.gif is a nonnegative continuous convex functional on a cone of a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq127_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq128_HTML.gif is continuous and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ22_HTML.gif
(2.17)

for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq129_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq130_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq132_HTML.gif be a nonnegative continuous convex functionals on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq133_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq134_HTML.gif a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq135_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq136_HTML.gif a nonnegative continuous functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq137_HTML.gif . Then for positive real number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq138_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq139_HTML.gif , we define the following convex sets:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ23_HTML.gif
(2.18)

Theorem 2.8 (see [11]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq140_HTML.gif be a real Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq141_HTML.gif a cone. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq142_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq143_HTML.gif are two bounded open sets in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq144_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq146_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq147_HTML.gif be completely continuous. Suppose that one of following two conditions is satisfied:

(1) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq149_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq151_HTML.gif ;

(2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq152_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq153_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq155_HTML.gif .

Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq156_HTML.gif has at least one fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq157_HTML.gif .

Theorem 2.9 (see [12]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq158_HTML.gif be a cone in a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq159_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq160_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq161_HTML.gif be a nonnegative continuous convex functionals on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq163_HTML.gif a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq164_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq165_HTML.gif a nonnegative continuous functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq166_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq167_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq168_HTML.gif , such that for positive number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq169_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq170_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ24_HTML.gif
(2.19)

for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq171_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq172_HTML.gif is completely continuous and there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq173_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq174_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq175_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq178_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq179_HTML.gif ;

() http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq181_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq182_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq183_HTML.gif ;

() http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq185_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq186_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq187_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq188_HTML.gif .

Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq189_HTML.gif has at least three fixed points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq190_HTML.gif , such that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq191_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq192_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq193_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq194_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq195_HTML.gif ,

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq196_HTML.gif .

3. The Existence of One or Two Positive Solutions

For convenience, we denote
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ25_HTML.gif
(3.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq197_HTML.gif denotes 0 or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq198_HTML.gif .

Theorem 3.1.

Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.
  1. (i)

    There exist two constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq199_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq200_HTML.gif such that

     

(a) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq201_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq202_HTML.gif and

(b) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq203_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq204_HTML.gif ;

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq205_HTML.gif ;

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq206_HTML.gif .

Then BVP(1.1) has at least one positive solution.

Proof.
  1. (i)

    Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq207_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq208_HTML.gif .

     
For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq209_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq210_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq211_HTML.gif , which implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq212_HTML.gif . Hence, by (2.12) and Lemma 2.6,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ26_HTML.gif
(3.2)
This implies that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ27_HTML.gif
(3.3)
Next, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq213_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq214_HTML.gif . Thus, by (2.12) and Lemma 2.6,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ28_HTML.gif
(3.4)
From (2.13), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ29_HTML.gif
(3.5)
This implies that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ30_HTML.gif
(3.6)
Therefore, by Theorem 2.8, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq215_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq216_HTML.gif . That is BVP(1.1) has at least one positive solution such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq217_HTML.gif .
  1. (ii)
    Considering http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq218_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq219_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ31_HTML.gif
    (3.7)
     
Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq220_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ32_HTML.gif
(3.8)

then for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq221_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq222_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq223_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq224_HTML.gif . In the following, we consider two cases.

Case 1 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq225_HTML.gif ).

In this case,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ33_HTML.gif
(3.9)

Case 2 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq226_HTML.gif ).

In this case,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ34_HTML.gif
(3.10)

Then it is similar to the proof of (3.6); we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq227_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq228_HTML.gif .

Next, turning to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq229_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq230_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ35_HTML.gif
(3.11)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq231_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq232_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq233_HTML.gif . So
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ36_HTML.gif
(3.12)
Then like in the proof of (3.3), we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq234_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq235_HTML.gif . Hence, BVP(1.1) has at least one positive solution such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq236_HTML.gif .
  1. (iii)

    The proof is similar to the (i) and (ii); here we omit it.

     

In the following, we present a result for the existence of at least two positive solutions of BVP(1.1).

Theorem 3.2.

Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.

(I) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq237_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq238_HTML.gif , and there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq239_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ37_HTML.gif
(3.13)
(II) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq240_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq241_HTML.gif , and there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq242_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ38_HTML.gif
(3.14)

Then BVP(1.1) has at least two positive solutions.

4. The Existence of Three Positive Solutions

In this section, we impose growth conditions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq243_HTML.gif which allow us to apply Theorem 2.9 of BVP(1.1).

Let the nonnegative continuous concave functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq244_HTML.gif , the nonnegative continuous convex functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq245_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq246_HTML.gif , and nonnegative continuous functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq247_HTML.gif be defined on cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq248_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ39_HTML.gif
(4.1)
By Lemmas 2.1 and 2.2, the functionals defined above satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ40_HTML.gif
(4.2)

for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq249_HTML.gif . Therefore, the condition (2.19) of Theorem 2.9 is satisfied.

Theorem 4.1.

Assume that (H1) and (H2) hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq250_HTML.gif and suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq251_HTML.gif satisfies the following conditions:

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq253_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq254_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq256_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq257_HTML.gif .

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq259_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq260_HTML.gif ;

Then BVP(1.1) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq261_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq262_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ41_HTML.gif
(4.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq263_HTML.gif defined as (3.1), http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq264_HTML.gif .

Proof.

We will show that all the conditions of Theorem 2.9 are satisfied.

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq265_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq266_HTML.gif . With Lemma 2.2 implying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq267_HTML.gif , so by ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq268_HTML.gif ), we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq269_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq270_HTML.gif . Thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ42_HTML.gif
(4.4)

This proves that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq271_HTML.gif .

To check condition ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq272_HTML.gif ) of Theorem 2.9, we choose
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ43_HTML.gif
(4.5)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ44_HTML.gif
(4.6)
Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq273_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq274_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq275_HTML.gif . Hence, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq276_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq277_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq278_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq279_HTML.gif . From assumption ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq280_HTML.gif ), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ45_HTML.gif
(4.7)
It is similar to the proof of assumption (i) of Theorem 3.1; we can easily get that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ46_HTML.gif
(4.8)

This shows that condition ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq281_HTML.gif ) of Theorem 2.9 is satisfied.

Secondly, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq282_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq283_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ47_HTML.gif
(4.9)

Thus condition ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq284_HTML.gif ) of Theorem 2.9 holds.

Finally, as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq285_HTML.gif , there holds http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq286_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq287_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq288_HTML.gif ; then by the assumption ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq289_HTML.gif ),
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ48_HTML.gif
(4.10)
So like in the proof of assumption (i) of Theorem 3.1, we can get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ49_HTML.gif
(4.11)

Hence condition ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq290_HTML.gif ) of Theorem 2.9 is also satisfied.

Thus BVP(1.1) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq291_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq292_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ50_HTML.gif
(4.12)

5. Examples

In this section, we give three examples as applications.

Example 5.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq293_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq294_HTML.gif . Now we consider the BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ51_HTML.gif
(5.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq295_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq296_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq297_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq298_HTML.gif . Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq299_HTML.gif . By calculations we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ52_HTML.gif
(5.2)
For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq300_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ53_HTML.gif
(5.3)
for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq301_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ54_HTML.gif
(5.4)

Hence, by Theorem 3.1, BVP(5.1) has at least one positive solution.

Example 5.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq302_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq303_HTML.gif . Consider the BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ55_HTML.gif
(5.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq304_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq305_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq306_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq307_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq308_HTML.gif . It easy to see
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ56_HTML.gif
(5.6)
Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq309_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq310_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq311_HTML.gif .
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ57_HTML.gif
(5.7)

Hence, by Theorem 3.2, BVP(5.5) has at least two positive solutions.

Example 5.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq312_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq313_HTML.gif ; consider the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ58_HTML.gif
(5.8)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ59_HTML.gif
(5.9)
Choosing http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq314_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq315_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq316_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq317_HTML.gif , then by calculations we obtain that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ60_HTML.gif
(5.10)
It is easy to check that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ61_HTML.gif
(5.11)
Thus, according to Theorem 4.1, BVP(5.8) has at least three positive solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq318_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq319_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ62_HTML.gif
(5.12)

Declarations

Acknowledgments

The research was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

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© Yonghong Ding. 2011

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