Open Access

Triple Positive Solutions for a Type of Second-Order Singular Boundary Problems

Boundary Value Problems20102010:376471

DOI: 10.1155/2010/376471

Received: 7 April 2010

Accepted: 26 August 2010

Published: 1 September 2010

Abstract

This paper deals with the existence of triple positive solutions for a type of second-order singular boundary problems with general differential operators. By using the Leggett-Williams fixed point theorem, we establish an existence criterion for at least three positive solutions with suitable growth conditions imposed on the nonlinear term.

1. Introduction

In this paper, we study the existence of triple positive solutions for the following second-order singular boundary value problems with general differential operators:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq2_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq3_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ2_HTML.gif
(1.2)

It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq5_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq6_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq7_HTML.gif

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq8_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq10_HTML.gif , the two kinds of singular boundary value problems have been discussed extensively in the literature; see [110] and the references therein. Hence, the problem that we consider is more general and is different from those in previous work.

Furthermore, we will see in the later that the presence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq11_HTML.gif brings us three main difficulties:

(1) the Green's function cannot be explicitly expressed;

(2) the equivalence between BVP (1.1) and its associated integral equation has to be proved;

(3) the compactness of associated integral operator has to be verified.

We will overcome the above mentioned difficulties in Section 2. Also, although the Leggett-William fixed point theorem is used extensively in the study of triple positive differential equations, the method has not been used to study this type of second-order singular boundary value problem with general differential operators. We are concerned with solving these problems in this paper.

To state our main tool used in this paper, we give some definitions and notations.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq12_HTML.gif be a real Banach space with a cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq13_HTML.gif . A map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq14_HTML.gif is said to be a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq15_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq16_HTML.gif is a continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ3_HTML.gif
(1.3)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq18_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq19_HTML.gif be two numbers such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq21_HTML.gif a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq22_HTML.gif . We define the following convex sets:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ4_HTML.gif
(1.4)

Theorem 1.1 (Leggett-Williams fixed point theorem).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq23_HTML.gif be completely continuous, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq24_HTML.gif be a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq25_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq26_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq27_HTML.gif . Suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq28_HTML.gif such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq30_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq31_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq32_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq33_HTML.gif ;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq34_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq35_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq36_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq37_HTML.gif has at least three fixed points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq38_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq39_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq40_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq41_HTML.gif .

Remark 1.2.

We note the existence of triple positive solutions of other kind of boundary value problems; see He and Ge [11], Zhao et al. [12], Zhang and Liu [13], Graef et al. [14], and the references therein.

The rest of the paper is organized as follows. In Section 2, we overcome the above-mentioned difficulties in this work. The main results are formulated and proved in Section 3. Finally, an example is presented to demonstrate the application of the main theorems in Section 4.

2. Preliminaries and Lemmas

Throughout this paper, we assume the following:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq42_HTML.gif ;

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq43_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq44_HTML.gif ;

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq45_HTML.gif is continuous and does not vanish identically on any subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq46_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq47_HTML.gif ;

(H4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq48_HTML.gif is continuous.

Lemma 2.1.

Suppose that (H1) and (H2) hold. Then

(i) the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ5_HTML.gif
(2.1)

has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq50_HTML.gif ;

(ii) the initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ6_HTML.gif
(2.2)

has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq52_HTML.gif .

Proof.

We only prove (i). (ii) can be treated in the same way.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq54_HTML.gif is a solution of (2.1), that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ7_HTML.gif
(2.3)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ8_HTML.gif
(2.4)
Multiplying both sides of (2.3) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq55_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ9_HTML.gif
(2.5)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq57_HTML.gif , integrating (2.5) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq58_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ10_HTML.gif
(2.6)
Moreover, integrating (2.6) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq59_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq60_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ11_HTML.gif
(2.7)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ12_HTML.gif
(2.8)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq61_HTML.gif , and (2.7) reduces to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ13_HTML.gif
(2.9)
By using Fubini's theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ14_HTML.gif
(2.10)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ15_HTML.gif
(2.11)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq62_HTML.gif is a solution of integral equation (2.11).

Conversely, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq63_HTML.gif is a solution of (2.11) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq64_HTML.gif , by reversing the above argument we could deduce that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq65_HTML.gif is a solution of (2.1) and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq67_HTML.gif . Therefore, to prove that (2.1) has a unique solution, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq68_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq69_HTML.gif is equivalent to prove that (2.11) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq70_HTML.gif .

To do this, we endow the following norm in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq71_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ16_HTML.gif
(2.12)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq72_HTML.gif be operator defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ17_HTML.gif
(2.13)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ18_HTML.gif
(2.14)
then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq73_HTML.gif is well defined. Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ19_HTML.gif
(2.15)
Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq74_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ20_HTML.gif
(2.16)
and subsequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ21_HTML.gif
(2.17)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ22_HTML.gif
(2.18)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq76_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq77_HTML.gif by Banach contraction principle. That is, (2.11) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq78_HTML.gif .

Remark 2.2.

Lemma 2.1 generalizes Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq79_HTML.gif of [1], where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq80_HTML.gif .

Lemma 2.3.

Suppose that (H1) and (H2) hold. Then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq81_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq82_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq83_HTML.gif is nonincreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq84_HTML.gif .

Proof.

We only prove (i). (ii) can be treated in the same way.

Suppose on the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq85_HTML.gif is not nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq86_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq87_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ23_HTML.gif
(2.19)
This together with the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq88_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ24_HTML.gif
(2.20)

which is a contradiction!

Remark 2.4.

From Lemmas 2.1 and 2.3, there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq89_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq90_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq91_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq92_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ25_HTML.gif
(2.21)
In fact, since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ26_HTML.gif
(2.22)
we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq94_HTML.gif . Then, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq96_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ27_HTML.gif
(2.23)
that is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ28_HTML.gif
(2.24)
In the following, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq97_HTML.gif . Suppose on the contrary, if there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq98_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ29_HTML.gif
(2.25)

then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq99_HTML.gif , which is a contradiction!

The other inequality can be treated in the same manner.

Lemma 2.5.

Suppose that (H1), (H2), and (H3) hold. Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ30_HTML.gif
(2.26)

Proof.

We only prove the first equality; the other can be treated in the same way. From Remark 2.4 and (H3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ31_HTML.gif
(2.27)
Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq100_HTML.gif of [2] together with the facts that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq101_HTML.gif and (H3) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ32_HTML.gif
(2.28)
Combining (2.27) and (2.28), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ33_HTML.gif
(2.29)

Lemma 2.6.

Suppose that (H1), (H2), and (H3) hold. Then the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ34_HTML.gif
(2.30)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ35_HTML.gif
(2.31)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ36_HTML.gif
(2.32)

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq102_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq103_HTML.gif .

Proof.

By Lemma 2.3 and (2.32), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ37_HTML.gif
(2.33)

This together with Remark 2.4 implies that the right side of (2.31) is well defined.

Now we check that the function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ38_HTML.gif
(2.34)
satisfies (2.30). In fact,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ39_HTML.gif
(2.35)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ40_HTML.gif
(2.36)
Equation (2.34) and Lemma 2.5 imply that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ41_HTML.gif
(2.37)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq104_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq105_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ42_HTML.gif
(2.38)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq106_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ43_HTML.gif
(2.39)
and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq107_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq108_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ44_HTML.gif
(2.40)

Lemma 2.7.

Suppose that (H1)–(H3) hold and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq109_HTML.gif is a positive solution of (2.30). Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ45_HTML.gif
(2.41)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ46_HTML.gif
(2.42)
Furthermore, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq110_HTML.gif , there exists corresponding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq111_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ47_HTML.gif
(2.43)

Proof.

In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq112_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ48_HTML.gif
(2.44)
and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq113_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ49_HTML.gif
(2.45)
Combining this and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq114_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ50_HTML.gif
(2.46)
Take
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ51_HTML.gif
(2.47)

Then Lemma 2.3 guarantees that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq115_HTML.gif , and Lemma 2.7 guarantees that (2.43) holds.

Remark 2.8.

From Lemma 2.7 and Remark 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ52_HTML.gif
(2.48)
Now, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq116_HTML.gif , we can define the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq117_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ53_HTML.gif
(2.49)

Lemma 2.9.

Let (H1)–(H4) hold. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq118_HTML.gif is a completely continuous operator.

Proof.

From (H3) and (H4) and Lemma 2.6, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq119_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq120_HTML.gif is continuous by the Lebesgue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq121_HTML.gif s dominated convergence theorem.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq122_HTML.gif be any bounded set. Then (H3) and (H4) imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq123_HTML.gif is a bounded set in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq124_HTML.gif .

Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ54_HTML.gif
(2.50)
then this together with the similar proof of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq125_HTML.gif of [2] yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ55_HTML.gif
(2.51)

From this fact, it is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq126_HTML.gif is equicontinuous. Therefore, by the Arzela-Ascoli theorem, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq127_HTML.gif is a completely continuous operator.

3. Main Result

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq128_HTML.gif be nonnegative continuous concave functional defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ56_HTML.gif
(3.1)

We notice that, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq130_HTML.gif , and also that by Lemma 2.6, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq131_HTML.gif is a solution of (1.1) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq132_HTML.gif is a fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq133_HTML.gif .

For convenience we introduce the following notations. Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ57_HTML.gif
(3.2)

Theorem 3.1.

Assume that (H1)–(H4) hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq134_HTML.gif , and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq135_HTML.gif satisfies the following conditions:

(S1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq136_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq137_HTML.gif ;

(S2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq138_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq139_HTML.gif ;

(S3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq140_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq141_HTML.gif .

Then the boundary value problem (1.1) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq142_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq143_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq144_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq145_HTML.gif .

Proof.

From Lemma 2.9, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq146_HTML.gif is a completely continuous operator. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq147_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq148_HTML.gif , and assumption (S3) implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq149_HTML.gif . Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ58_HTML.gif
(3.3)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq150_HTML.gif . In the same way, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq151_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq152_HTML.gif . Therefore, condition (ii) of Leggett-williams fixed-point theorem holds.

To check condition (i) of Leggett-Williams fixed-point theorem, choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq153_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq155_HTML.gif . so,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ59_HTML.gif
(3.4)
Hence, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq156_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq157_HTML.gif . From assumption (S2) and Remark 2.8, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ60_HTML.gif
(3.5)
Finally, we assert that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq159_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq160_HTML.gif . To see this, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq162_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ61_HTML.gif
(3.6)

To sum up, all the conditions of Leggett-williams fixed-point theorem are satisfied. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq163_HTML.gif has at least three fixed points, that is, problem (1.1) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq164_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq165_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq167_HTML.gif .

Theorem 3.2.

Assume that (H1)–(H4) hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq168_HTML.gif , and suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq169_HTML.gif satisfies the following conditions:

(A1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq170_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq171_HTML.gif ;

(A2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq172_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq173_HTML.gif .

Then the boundary value problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq174_HTML.gif positive solutions.

Proof.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq175_HTML.gif , it follows from condition (A1) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq176_HTML.gif , which means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq177_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq178_HTML.gif by the Schauder fixed-point Theorem. When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq179_HTML.gif , it is clear that Theorem 3.1 holds (with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq180_HTML.gif ). Then we can obtain at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq181_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq182_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq184_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq185_HTML.gif . Following this way, we finish the proof by the induction method.

4. Example

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ62_HTML.gif
(4.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ63_HTML.gif
(4.2)
Then, by computation, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ64_HTML.gif
(4.3)
Furthermore, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq186_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ65_HTML.gif
(4.4)
In fact, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq187_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq188_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq189_HTML.gif . It is easy to compute that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ66_HTML.gif
(4.5)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq190_HTML.gif , that is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ67_HTML.gif
(4.6)

The other inequalities in (4.4) can be proved by the same method.

Thus, we can choose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq191_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq193_HTML.gif . By computation, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ68_HTML.gif
(4.7)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq194_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq195_HTML.gif . Then, we can compute
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ69_HTML.gif
(4.8)
Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ70_HTML.gif
(4.9)
Therefore, all the conditions of Theorem 3.1 are satisfied, then problem (4.1) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq196_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_IEq197_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F376471/MediaObjects/13661_2010_Article_920_Equ71_HTML.gif
(4.10)

Declarations

Acknowledgment

The first author was partially supported by NNSF of China (10901075).

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University
(2)
The School of Mathematics, Physics Software Engineering, Lanzhou Jiaotong University

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Copyright

© Jiemei Li and Jinxiang Wang. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.