Open Access

Multiple Positive Solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq1_HTML.gif th Order Multipoint Boundary Value Problem

Boundary Value Problems20102010:708376

DOI: 10.1155/2010/708376

Received: 22 January 2010

Accepted: 3 June 2010

Published: 20 June 2010

Abstract

We study the existence of multiple positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq2_HTML.gif th-order multipoint boundary value problem. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq6_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq9_HTML.gif . We obtained the existence of multiple positive solutions by applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature.

1. Introduction

The existence of positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq10_HTML.gif th-order multipoint boundary problems has been studied by some authors (see [1, 2]). In [1], Pang et al. studied the expression and properties of Green's funtion and obtained the existence of at least one positive solution for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq11_HTML.gif th-order differential equations by applying means of fixed point index theory:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq12_HTML.gif

By using the fixed point theorems of cone expansion and compression of norm type, Yang and Wei in [2] also obtained the existence of at least one positive solutions for the BVP (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq13_HTML.gif . This work is motivated by Ma (see [3]). This method is simpler than that of [1]. In addition, Eloe and Ahmad in [4] had solved successfully the existence of positive solutions to the BVP (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq14_HTML.gif = 1. Hao et al. in [5] had discussed the existence of at least two positive solutions for the BVP (1.1) by applying the Krasonse'skii-Guo fixed point theorem on cone expansion and compression if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq15_HTML.gif = 1. However, there are few papers dealing with the existence of multiple positive solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq16_HTML.gif th-order multipoint boundary value problem.

In this paper, we study the existence of at least two positive solutions associated with the BVP (1.1) by applying the fixed point theorems of cone expansion and compression of norm type if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq17_HTML.gif and the existence of at least three positive solutions for BVP (1.1) by using Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature and essentially improve and generalize some well-known results (see [18]).

The rest of the paper is organized as follows. In Section 2, we present several lemmas. In Section 3, we give some preliminaries and the fixed point theorems of cone expansion and compression of norm type. The existence of at least two positive solutions for the BVP (1.1) is formulated and proved in Section 4. In Section 5, we give Leggett-Williams fixed-point theorem and obtain the existence of at least three positive solutions for the BVP (1.1).

2. Several Lemmas

Definition 2.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq18_HTML.gif is said to be a position of the BVP (1.1) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq19_HTML.gif satisfies the following:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq21_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq23_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq24_HTML.gif and satisfies boundary value conditions (1.1);

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq26_HTML.gif hold for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq27_HTML.gif

Lemma 2.2 (see [1]).

Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ2_HTML.gif
(2.1)
then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq28_HTML.gif , the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ3_HTML.gif
(2.2)
has a unique solution:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ4_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ5_HTML.gif
(2.4)

Lemma 2.3 (see [1]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq29_HTML.gif ; Green's function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq30_HTML.gif defined by (2.4) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ6_HTML.gif
(2.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq31_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ7_HTML.gif
(2.6)

We omit the proof Lemma 2.3 here and you can see the detail of Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq32_HTML.gif in [1].

Lemma 2.4 (see [2]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq33_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq34_HTML.gif ; the unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq35_HTML.gif of the BVP (2.2)

satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ8_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq36_HTML.gif is defined by Lemma 2.3, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq37_HTML.gif .

3. Preliminaries

In this section, we give some preliminaries for discussing the existence of multiple positive solutions of the BVP (1.1) in the next. In real Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq38_HTML.gif in which the norm is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ9_HTML.gif
(3.1)
set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ10_HTML.gif
(3.2)

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq39_HTML.gif is a positive cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq40_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq41_HTML.gif is from Lemma 2.3.

For convenience, we make the following assumptions:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq43_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq44_HTML.gif does not vanish identically, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq45_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq47_HTML.gif is continuous;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq49_HTML.gif

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ11_HTML.gif
(3.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq50_HTML.gif is defined by (2.4).

From Lemmas 2.2–2.4, we have the following result.

Lemma 3.1 (see [2]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq51_HTML.gif are satisfied, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq52_HTML.gif is a completely continuous operator, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq53_HTML.gif , and the fixed points of the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq54_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq55_HTML.gif are the positive solutions of the BVP (1.1).

For convenience, one introduces the following notations. Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ12_HTML.gif
(3.4)

Problem 1.

Inspired by the work of the paper [2], whether we can obtain a similar conclusion or not, if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ13_HTML.gif
(3.5)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ14_HTML.gif
(3.6)

The aim of the following section is to establish some simple criteria for the existence of multiple positive solutions of the BVP (1.1), which gives a positive answer to the questions stated above. The key tool in our approach is the following fixed point theorem, which is a useful method to prove the existence of positive solutions for differential equations, for example [25, 9].

Lemma 3.2 (see [10, 11]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq56_HTML.gif is a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq57_HTML.gif is cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq58_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq59_HTML.gif be two bounded open sets in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq60_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq61_HTML.gif . Let operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq62_HTML.gif be completely continuous. Suppose that one of two conditions holds:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq63_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq64_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq65_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq66_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq67_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq68_HTML.gif .

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq69_HTML.gif has at least one fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq70_HTML.gif

4. The Existence of Two Positive Solutions

Theorem 4.1.

Suppose that the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq71_HTML.gif are satisfied and the following assumptions hold:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq73_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq75_HTML.gif ;

()There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq77_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq78_HTML.gif .

Then the BVP (1.1) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq80_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ15_HTML.gif
(4.1)

Proof.

At first, it follows from the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq81_HTML.gif that we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq82_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ16_HTML.gif
(4.2)
Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq83_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq84_HTML.gif ; from (3.3) and (2.4) and Lemma 2.4, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq85_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ17_HTML.gif
(4.3)
Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ18_HTML.gif
(4.4)
Further, it follows from the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq86_HTML.gif that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq87_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ19_HTML.gif
(4.5)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq88_HTML.gif , set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq89_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq90_HTML.gif and Lemma 2.4 imply
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ20_HTML.gif
(4.6)
and by the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq91_HTML.gif , (2.4), (3.3), and Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ21_HTML.gif
(4.7)
Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ22_HTML.gif
(4.8)
Finally, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq93_HTML.gif . By (2.3), (3.3), and the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq94_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ23_HTML.gif
(4.9)
which implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ24_HTML.gif
(4.10)
Thus from (4.4)–(4.10) and Lemmas 3.1 and 3.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq95_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq96_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq97_HTML.gif and a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq98_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq99_HTML.gif . Both are positive solutions of BVP (1.1) and satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ25_HTML.gif
(4.11)

The proof is complete.

Corollary 4.2.

Suppose that the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq100_HTML.gif are satisfied and the following assumptions hold:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq102_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq104_HTML.gif ;

()there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq106_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq107_HTML.gif

Then the BVP (1.1) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq109_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ26_HTML.gif
(4.12)

Proof.

From the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq110_HTML.gif , there exist sufficiently big positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq111_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ27_HTML.gif
(4.13)
by the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq112_HTML.gif ; so all the conditions of Theorem 4.1 are satisfied; by an application of Theorem 4.1, the BVP (1.1) has two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq113_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq114_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ28_HTML.gif
(4.14)

Theorem 4.3.

Suppose that the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq115_HTML.gif are satisfied and the following assumptions hold:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq117_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq119_HTML.gif ;

()there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq121_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq122_HTML.gif .

Then the BVP (1.1) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq124_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ29_HTML.gif
(4.15)

Proof.

It follows from the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq125_HTML.gif that we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq126_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ30_HTML.gif
(4.16)
Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq127_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq128_HTML.gif ; from (3.2) and (2.4), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq129_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ31_HTML.gif
(4.17)
Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ32_HTML.gif
(4.18)

It follows from the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq130_HTML.gif that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq131_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq132_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq133_HTML.gif and we consider two cases.

Case i.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq134_HTML.gif is unbounded; there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq135_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq136_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq137_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq139_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ33_HTML.gif
(4.19)

Case ii.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq140_HTML.gif is bounded, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq141_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq142_HTML.gif , taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq143_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq145_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ34_HTML.gif
(4.20)
Hence, in either case, we always may set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq146_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ35_HTML.gif
(4.21)
Finally, set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq147_HTML.gif ; then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq148_HTML.gif and Lemma 2.4 imply
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ36_HTML.gif
(4.22)
and by the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq149_HTML.gif , (2.4), and (3.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ37_HTML.gif
(4.23)
Hence, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ38_HTML.gif
(4.24)
From (4.18)–(4.24) and Lemmas 3.1 and 3.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq150_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq151_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq152_HTML.gif and a fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq153_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq154_HTML.gif . Both are positive solutions of the BVP(1.1) and satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ39_HTML.gif
(4.25)

The proof is complete.

Corollary 4.4.

Suppose that the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq155_HTML.gif are satisfied and the following assumptions hold:

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq157_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq159_HTML.gif ;

()there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq161_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq162_HTML.gif .

Then BVP (1.1) has at least two positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq164_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ40_HTML.gif
(4.26)

The proof of Corollary 4.4 is similar to that of Corollary 4.2; so we omit it.

5. The Existence of Three Positive Solutions

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq165_HTML.gif be a real Banach space with cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq166_HTML.gif . A map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq167_HTML.gif is said to be a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq168_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq169_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ41_HTML.gif
(5.1)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq171_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq172_HTML.gif be two numbers such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq173_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq174_HTML.gif be a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq175_HTML.gif . We define the following convex sets:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ42_HTML.gif
(5.2)

Lemma 5.1 (see [12]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq176_HTML.gif be completely continuous and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq177_HTML.gif be a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq178_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq179_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq180_HTML.gif . Suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq181_HTML.gif such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq183_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq184_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq185_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq186_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq187_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq188_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq189_HTML.gif

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq190_HTML.gif has at least three fixed points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq191_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq192_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ43_HTML.gif
(5.3)

Now, we establish the existence conditions of three positive solutions for the BVP (1.1).

Theorem 5.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq193_HTML.gif hold and there exist numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq194_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq195_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq196_HTML.gif such that the following conditions are satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq198_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq200_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq202_HTML.gif

where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ44_HTML.gif
(5.4)

Then the boundary value problem (1.1) has at least three positive solutions.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq203_HTML.gif be defined by (3.2) and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq204_HTML.gif be defined by (3.3). For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq205_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ45_HTML.gif
(5.5)

Then it is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq206_HTML.gif is a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq207_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq208_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq209_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq210_HTML.gif is completely continuous.

First, we prove that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq211_HTML.gif holds, then there exists a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq213_HTML.gif To do this, by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq214_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq216_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ46_HTML.gif
(5.6)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ47_HTML.gif
(5.7)
it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq217_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq218_HTML.gif . Take
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ48_HTML.gif
(5.8)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq219_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ49_HTML.gif
(5.9)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ50_HTML.gif
(5.10)

Hence (5.10) show that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq220_HTML.gif holds, then there exists a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq221_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq222_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq223_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq224_HTML.gif .

Now we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq226_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq227_HTML.gif . In fact, take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq228_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq229_HTML.gif . Moreover, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq230_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq231_HTML.gif , and we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ51_HTML.gif
(5.11)
Therfore, by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq232_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ52_HTML.gif
(5.12)
Next, we assert that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq233_HTML.gif . In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq234_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq235_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ53_HTML.gif
(5.13)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq236_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq237_HTML.gif .

Finally, we assert that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq238_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq239_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq240_HTML.gif . To see this, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq241_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq242_HTML.gif ,then we have from Lemma 2.3 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ54_HTML.gif
(5.14)
So we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ55_HTML.gif
(5.15)
To sum up (5.10) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq243_HTML.gif (5.15), all the conditions of Lemma 5.1 are satisfied by taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq244_HTML.gif . Hence, A has at least three fixed points; that is, BVP (1.1) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq245_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_IEq246_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F708376/MediaObjects/13661_2010_Article_948_Equ56_HTML.gif
(5.16)

The proof is complete.

Declarations

Acknowledgments

The authors are grateful to the referee's valuable comments and suggestions. The project is supported by the Natural Science Foundation of Anhui Province (KJ2010B226), The Excellent Youth Foundation of Anhui Province Office of Education (2009SQRZ169), and the Natural Science Foundation of Suzhou University (2009yzk17)

Authors’ Affiliations

(1)
Department of Mathematics, Suzhou University
(2)
School of Mathematics, Shandong University
(3)
School of Sciences, Shandong Jianzhu University

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© Y. Li and Z.Wei. 2010

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