Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq1_HTML.gif -Laplacian https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq5_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq6_HTML.gif is an odd, increasing homeomorphism from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq7_HTML.gif onto https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ1_HTML.gif
(1.1)

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

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

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

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

The assumption (H1) on the function https://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: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq27_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ3_HTML.gif
(1.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq29_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ4_HTML.gif
(1.4)

where the nonlinear term https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq32_HTML.gif , https://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 https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq36_HTML.gif satisfies (H1) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq37_HTML.gif depends on both https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq38_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq40_HTML.gif depending on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq42_HTML.gif with norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq43_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq44_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq45_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ6_HTML.gif
(2.1)

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

Lemma 2.1 (see [7]).

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

Lemma 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq51_HTML.gif , then there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq52_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq54_HTML.gif , such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq55_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ8_HTML.gif
(2.3)
So we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ9_HTML.gif
(2.4)

Denote https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq58_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq59_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq61_HTML.gif .

Proof.

From the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq62_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq63_HTML.gif is strictly decreasing. It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq64_HTML.gif is also strictly decreasing. Thus, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq66_HTML.gif . By the concavity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq67_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq69_HTML.gif . So we get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq70_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq71_HTML.gif , it is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq72_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq73_HTML.gif , https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq77_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq78_HTML.gif or 1, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq79_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq80_HTML.gif and then https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ10_HTML.gif
(2.5)
or
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq83_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ12_HTML.gif
(2.7)
then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ13_HTML.gif
(2.8)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ14_HTML.gif
(2.9)
or
https://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
https://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], https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq85_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ17_HTML.gif
(2.12)

Lemma 2.5.

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

Proof.

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

So https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq89_HTML.gif is monotone decreasing continuous and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq90_HTML.gif . Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq92_HTML.gif . This shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq93_HTML.gif . The continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq94_HTML.gif is obvious since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq95_HTML.gif is continuous. Next, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq96_HTML.gif is compact on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq97_HTML.gif .

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

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

Thus, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq120_HTML.gif ,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq122_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq123_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ21_HTML.gif
(2.16)
for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq124_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq125_HTML.gif . Similarly, we say the map https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq127_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq128_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ22_HTML.gif
(2.17)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq131_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq133_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq134_HTML.gif a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq135_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq136_HTML.gif a nonnegative continuous functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq137_HTML.gif . Then for positive real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq138_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq139_HTML.gif , we define the following convex sets:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq140_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq141_HTML.gif a cone. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq143_HTML.gif are two bounded open sets in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq144_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq146_HTML.gif . Let https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq148_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq149_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq151_HTML.gif ;

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

Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq157_HTML.gif .

Theorem 2.9 (see [12]).

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

for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq171_HTML.gif . Suppose https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq173_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq174_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq175_HTML.gif such that

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

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

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

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

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

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

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

https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ25_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq197_HTML.gif denotes 0 or https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq199_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq200_HTML.gif such that

     

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

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

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

(iii) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq207_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq208_HTML.gif .

     
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq209_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq210_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq211_HTML.gif , which implies https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ26_HTML.gif
(3.2)
This implies that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ27_HTML.gif
(3.3)
Next, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq213_HTML.gif , we have https://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,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ29_HTML.gif
(3.5)
This implies that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq215_HTML.gif has a fixed point in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq217_HTML.gif .
  1. (ii)
    Considering https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq218_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq219_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ31_HTML.gif
    (3.7)
     
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq220_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ32_HTML.gif
(3.8)

then for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq221_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq222_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq223_HTML.gif , we have https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq225_HTML.gif ).

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

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

In this case,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq227_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq228_HTML.gif .

Next, turning to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq229_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq230_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ35_HTML.gif
(3.11)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq231_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq232_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq233_HTML.gif . So
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq234_HTML.gif for https://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 https://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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq237_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq238_HTML.gif , and there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq239_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ37_HTML.gif
(3.13)
(II) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq240_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq241_HTML.gif , and there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq242_HTML.gif such that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq244_HTML.gif , the nonnegative continuous convex functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq245_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq246_HTML.gif , and nonnegative continuous functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq247_HTML.gif be defined on cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq248_HTML.gif by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ40_HTML.gif
(4.2)

for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq250_HTML.gif and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq251_HTML.gif satisfies the following conditions:

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq259_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq261_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq262_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ41_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq263_HTML.gif defined as (3.1), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq265_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq266_HTML.gif . With Lemma 2.2 implying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq267_HTML.gif , so by ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq268_HTML.gif ), we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq269_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq270_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ42_HTML.gif
(4.4)

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

To check condition ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq272_HTML.gif ) of Theorem 2.9, we choose
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ43_HTML.gif
(4.5)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ44_HTML.gif
(4.6)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq273_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq274_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq275_HTML.gif . Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq276_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq277_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq278_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq279_HTML.gif . From assumption ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq280_HTML.gif ), we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ46_HTML.gif
(4.8)

This shows that condition ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq282_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq283_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ47_HTML.gif
(4.9)

Thus condition ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq285_HTML.gif , there holds https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq286_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq287_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq288_HTML.gif ; then by the assumption ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq289_HTML.gif ),
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ49_HTML.gif
(4.11)

Hence condition ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq291_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq292_HTML.gif satisfying
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq293_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq294_HTML.gif . Now we consider the BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ51_HTML.gif
(5.1)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq297_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq298_HTML.gif . Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq299_HTML.gif . By calculations we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ52_HTML.gif
(5.2)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq300_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ53_HTML.gif
(5.3)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq301_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq302_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq303_HTML.gif . Consider the BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ55_HTML.gif
(5.5)

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

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq306_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq307_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq308_HTML.gif . It easy to see
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ56_HTML.gif
(5.6)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq309_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq310_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq311_HTML.gif .
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq312_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq313_HTML.gif ; consider the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ58_HTML.gif
(5.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_Equ59_HTML.gif
(5.9)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq314_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq315_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq316_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq317_HTML.gif , then by calculations we obtain that
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq318_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F827510/MediaObjects/13661_2010_Article_59_IEq319_HTML.gif satisfying
https://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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Copyright

© Yonghong Ding. 2011

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