Positive Solutions of Singular Multipoint Boundary Value Problems for Systems of Nonlinear Second-Order Differential Equations on Infinite Intervals in Banach Spaces

Boundary Value Problems20092009:978605

DOI: 10.1155/2009/978605

Received: 27 April 2009

Accepted: 12 June 2009

Published: 14 July 2009

Abstract

The cone theory together with Mönch fixed point theorem and a monotone iterative technique is used to investigate the positive solutions for some boundary problems for systems of nonlinear second-order differential equations with multipoint boundary value conditions on infinite intervals in Banach spaces. The conditions for the existence of positive solutions are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is also derived.

1. Introduction

In recent years, the theory of ordinary differential equations in Banach space has become a new important branch of investigation (see, e.g., [14] and references therein). By employing a fixed point theorem due to Sadovskii, Liu [5] investigated the existence of solutions for the following second-order two-point boundary value problems (BVP for short) on infinite intervals in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq1_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ1_HTML.gif
(1.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq2_HTML.gif On the other hand, the multipoint boundary value problems arising naturally from applied mathematics and physics have been studied so extensively in scalar case that there are many excellent results about the existence of positive solutions (see, i.e., [612] and references therein). However, to the best of our knowledge, only a few authors [5, 13, 14] have studied multipoint boundary value problems in Banach spaces and results for systems of second-order differential equation are rarely seen. Motivated by above papers, we consider the following singular http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq3_HTML.gif -point boundary value problem on an infinite interval in a Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq4_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq6_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq7_HTML.gif In this paper, nonlinear terms http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq9_HTML.gif may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq10_HTML.gif , and/or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq11_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq12_HTML.gif denotes the zero element of Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq13_HTML.gif . By singularity, we mean that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq14_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq15_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq16_HTML.gif

Very recently, by using Shauder fixed point theorem, Guo [15] obtained the existence of positive solutions for a class of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq17_HTML.gif th-order nonlinear impulsive singular integro-differential equations in a Banach space. Motivated by Guo's work, in this paper, we will use the cone theory and the Mönch fixed point theorem combined with a monotone iterative technique to investigate the positive solutions of BVP (1.2). The main features of the present paper are as follows. Firstly, compared with [5], the problem we discussed here is systems of multipoint boundary value problem and nonlinear term permits singularity not only at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq18_HTML.gif but also at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq19_HTML.gif . Secondly, compared with [15], the relative compact conditions we used are weaker. Furthermore, an iterative sequence for the solution under some normal type conditions is established which makes it very important and convenient in applications.

The rest of the paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is worked out to illustrate the main results.

2. Preliminaries and Several Lemmas

Let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ3_HTML.gif
(2.1)
Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq20_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq21_HTML.gif is a Banach space with norm
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ4_HTML.gif
(2.2)
and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq22_HTML.gif is also a Banach space with norm
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ5_HTML.gif
(2.3)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ6_HTML.gif
(2.4)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq23_HTML.gif with norm
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ7_HTML.gif
(2.5)

Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq24_HTML.gif is also a Banach space. The basic space using in this paper is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq25_HTML.gif .

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq26_HTML.gif be a normal cone in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq27_HTML.gif with normal constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq28_HTML.gif which defines a partial ordering in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq29_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq30_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq32_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq33_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq34_HTML.gif . So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq35_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq36_HTML.gif . For details on cone theory, see [4].

In what follows, we always assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq37_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq38_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq39_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq40_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq41_HTML.gif , we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq42_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq43_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq45_HTML.gif . It is clear, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq46_HTML.gif are cones in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq48_HTML.gif , respectively. A map http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq49_HTML.gif is called a positive solution of BVP (1.2) if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq51_HTML.gif satisfies (1.2).

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq52_HTML.gif denote the Kuratowski measure of noncompactness in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq54_HTML.gif , respectively. For details on the definition and properties of the measure of noncompactness, the reader is referred to [14]. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq55_HTML.gif be all Lebesgue measurable functions from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq56_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq57_HTML.gif . Denote
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ8_HTML.gif
(2.6)

Let us list some conditions for convenience.

(H1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq59_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq60_HTML.gif and there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq61_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq62_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ9_HTML.gif
(2.7)
uniformly for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq63_HTML.gif , and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ10_HTML.gif
(2.8)
(H2) For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq65_HTML.gif and countable bounded set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq66_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq67_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ11_HTML.gif
(2.9)
with
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ12_HTML.gif
(2.10)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq68_HTML.gif .

(H3) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq70_HTML.gif imply
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ13_HTML.gif
(2.11)

In what follows, we write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq72_HTML.gif . Evidently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq73_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq74_HTML.gif are closed convex sets in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq76_HTML.gif , respectively.

We will reduce BVP (1.2) to a system of integral equations in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq77_HTML.gif . To this end, we first consider operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq78_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ14_HTML.gif
(2.12)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ15_HTML.gif
(2.13)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ16_HTML.gif
(2.14)

Lemma 2.1.

If condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq79_HTML.gif is satisfied, then operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq80_HTML.gif defined by (2.12) is a continuous operator from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq81_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq82_HTML.gif .

Proof.

Let
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ17_HTML.gif
(2.15)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ18_HTML.gif
(2.16)
By virtue of condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq83_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq84_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ19_HTML.gif
(2.17)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ20_HTML.gif
(2.18)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ21_HTML.gif
(2.19)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq85_HTML.gif , we have, by (2.19)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ22_HTML.gif
(2.20)
which together with condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq86_HTML.gif implies the convergence of the infinite integral
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ23_HTML.gif
(2.21)
Thus, we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ24_HTML.gif
(2.22)
which together with (2.13) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq87_HTML.gif implies that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ25_HTML.gif
(2.23)
Therefore, by (2.15) and (2.20), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ26_HTML.gif
(2.24)
Differentiating (2.13), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ27_HTML.gif
(2.25)
Hence,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ28_HTML.gif
(2.26)
It follows from (2.24) and (2.25) that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ29_HTML.gif
(2.27)
So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq88_HTML.gif . On the other hand, it can be easily seen that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ30_HTML.gif
(2.28)
So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq89_HTML.gif . In the same way, we can easily get that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ31_HTML.gif
(2.29)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq90_HTML.gif Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq91_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq92_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq93_HTML.gif and we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ32_HTML.gif
(2.30)
where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ33_HTML.gif
(2.31)
Finally, we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq94_HTML.gif is continuous. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq95_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq96_HTML.gif is a bounded subset of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq97_HTML.gif . Thus, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq98_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq99_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq101_HTML.gif . Similar to (2.24) and (2.26), it is easy to have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ34_HTML.gif
(2.32)
It is clear,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ35_HTML.gif
(2.33)
and by (2.20),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ36_HTML.gif
(2.34)
It follows from (2.33) and (2.34) and the dominated convergence theorem that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ37_HTML.gif
(2.35)

It follows from (2.32) and (2.35) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq102_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq103_HTML.gif . By the same method, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq104_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq105_HTML.gif . Therefore, the continuity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq106_HTML.gif is proved.

Lemma 2.2.

If condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq107_HTML.gif is satisfied, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq108_HTML.gif is a solution of BVP (1.2) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq109_HTML.gif is a fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq110_HTML.gif .

Proof.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq111_HTML.gif is a solution of BVP (1.2). For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq112_HTML.gif integrating (1.2) from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq113_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq114_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ38_HTML.gif
(2.36)
Integrating (2.36) from 0 to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq115_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ39_HTML.gif
(2.37)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ40_HTML.gif
(2.38)
Thus, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ41_HTML.gif
(2.39)
which together with the boundary value conditions imply that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ42_HTML.gif
(2.40)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ43_HTML.gif
(2.41)
Substituting (2.40) and (2.41) into (2.37) and (2.38), respectively, we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ44_HTML.gif
(2.42)

It follows from Lemma 2.1 that the integral http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq116_HTML.gif and the integral http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq117_HTML.gif are convergent. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq118_HTML.gif is a fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq119_HTML.gif .

Conversely, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq120_HTML.gif is fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq121_HTML.gif , then direct differentiation gives the proof.

Lemma 2.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq122_HTML.gif be satisfied, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq123_HTML.gif is a bounded set. Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq124_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq125_HTML.gif are equicontinuous on any finite subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq126_HTML.gif and for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq127_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq128_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ45_HTML.gif
(2.43)

uniformly with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq129_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq130_HTML.gif

Proof.

We only give the proof for operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq131_HTML.gif , the proof for operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq132_HTML.gif can be given in a similar way. By (2.13), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ46_HTML.gif
(2.44)
For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq133_HTML.gif we obtain by (2.44)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ47_HTML.gif
(2.45)

Then, it is easy to see by (2.45) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq134_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq135_HTML.gif is equicontinuous on any finite subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq136_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq137_HTML.gif is bounded, there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq138_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq139_HTML.gif . By (2.25), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ48_HTML.gif
(2.46)

It follows from (2.46) and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq140_HTML.gif and the absolute continuity of Lebesgue integral that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq141_HTML.gif is equicontinuous on any finite subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq142_HTML.gif .

In the following, we are in position to show that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq143_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq144_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ49_HTML.gif
(2.47)

uniformly with respect to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq145_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq146_HTML.gif

Combining with (2.45), we need only to show that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq147_HTML.gif there exists sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq148_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ50_HTML.gif
(2.48)

for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq149_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq150_HTML.gif The rest part of the proof is very similar to Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq151_HTML.gif in [5], we omit the details.

Lemma 2.4.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq152_HTML.gif be a bounded set in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq153_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq154_HTML.gif holds. Then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ51_HTML.gif
(2.49)

Proof.

The proof is similar to that of Lemma http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq155_HTML.gif in [5], we omit it.

Lemma 2.5 (see [1, 2]).

Mönch Fixed-Point Theorem. Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq156_HTML.gif be a closed convex set of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq158_HTML.gif Assume that the continuous operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq159_HTML.gif has the following property: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq160_HTML.gif countable, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq161_HTML.gif is relatively compact. Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq162_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq163_HTML.gif .

Lemma 2.6.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq164_HTML.gif is satisfied, then for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq165_HTML.gif imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq166_HTML.gif

Proof.

It is easy to see that this lemma follows from (2.13), (2.25), and condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq167_HTML.gif . The proof is obvious.

Lemma 2.7 (see [16]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq168_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq169_HTML.gif are bounded sets in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq170_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ52_HTML.gif
(2.50)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq172_HTML.gif denote the Kuratowski measure of noncompactness in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq174_HTML.gif , respectively.

Lemma 2.8 (see [16]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq175_HTML.gif be normal (fully regular) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq176_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq177_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq178_HTML.gif is normal (fully regular) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq179_HTML.gif .

3. Main Results

Theorem 3.1.

If conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq181_HTML.gif are satisfied, then BVP (1.2) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq182_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq183_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq184_HTML.gif

Proof.

By Lemma 2.1, operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq185_HTML.gif defined by (2.13) is a continuous operator from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq186_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq187_HTML.gif , and, by Lemma 2.2, we need only to show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq188_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq189_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq190_HTML.gif . Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq191_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq192_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq193_HTML.gif is a bounded closed convex set in space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq194_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq195_HTML.gif is not empty since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq196_HTML.gif . It follows from (2.27) and (3.6) that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq197_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq198_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq199_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq200_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq201_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq202_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq203_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq204_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq205_HTML.gif We have, by (2.13) and (2.25),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ53_HTML.gif
(3.1)
By Lemma 2.4, we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ54_HTML.gif
(3.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq206_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq207_HTML.gif .

By (2.21), we know that the infinite integral http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq208_HTML.gif is convergent uniformly for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq209_HTML.gif So, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq210_HTML.gif we can choose a sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq211_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ55_HTML.gif
(3.3)
Then, by [1, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq212_HTML.gif ], (2.44), (3.1), (3.3), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq213_HTML.gif , and Lemma 2.7, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ56_HTML.gif
(3.4)
It follows from (3.2) and (3.4) that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ57_HTML.gif
(3.5)
In the same way, we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ58_HTML.gif
(3.6)

On the other hand, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq214_HTML.gif . Then, (3.5), (3.6), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq215_HTML.gif , and Lemma 2.7 imply http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq216_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq217_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq218_HTML.gif Hence, the Mönch fixed point theorem guarantees that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq219_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq220_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq221_HTML.gif . Thus, Theorem 3.1 is proved.

Theorem 3.2.

Let cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq222_HTML.gif be normal and conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq223_HTML.gif be satisfied. Then BVP (1.2) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq224_HTML.gif which is minimal in the sense that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq225_HTML.gif for any positive solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq226_HTML.gif of BVP (1.2). Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq227_HTML.gif and there exists a monotone iterative sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq228_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq229_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq230_HTML.gif uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq231_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq232_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq233_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq234_HTML.gif where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ59_HTML.gif
(3.7)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ60_HTML.gif
(3.8)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ61_HTML.gif
(3.9)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ62_HTML.gif
(3.10)

Proof.

From (3.7), one can see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq235_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ63_HTML.gif
(3.11)
By (3.7) and (3.11), we have that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq236_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ64_HTML.gif
(3.12)
which imply that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq237_HTML.gif . Similarly, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq238_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq239_HTML.gif . It follows from (2.13) and (3.9) that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ65_HTML.gif
(3.13)
By Lemma 2.1, we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq240_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ66_HTML.gif
(3.14)
By Lemma 2.6 and (3.13), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ67_HTML.gif
(3.15)
It follows from (3.14), by induction, that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ68_HTML.gif
(3.16)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq241_HTML.gif Then, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq242_HTML.gif is a bounded closed convex set in space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq243_HTML.gif and operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq244_HTML.gif maps http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq245_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq246_HTML.gif . Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq247_HTML.gif is not empty since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq248_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq249_HTML.gif Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq251_HTML.gif Similar to above proof of Theorem 3.1, we can obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq252_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq253_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq254_HTML.gif So, there exists an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq255_HTML.gif and a subsequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq256_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq257_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq258_HTML.gif uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq259_HTML.gif Since that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq260_HTML.gif is normal and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq261_HTML.gif is nondecreasing, it is easy to see that the entire sequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq262_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq263_HTML.gif uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq264_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq265_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq266_HTML.gif are closed convex sets in space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq267_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq268_HTML.gif It is clear,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ69_HTML.gif
(3.17)
By http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq269_HTML.gif and (3.16), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ70_HTML.gif
(3.18)
Noticing (3.17) and (3.18) and taking limit as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq270_HTML.gif in (3.9), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ71_HTML.gif
(3.19)
In the same way, taking limit as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq271_HTML.gif in (3.10), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ72_HTML.gif
(3.20)
which together with (3.19) and Lemma 2.2 implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq272_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq273_HTML.gif is a positive solution of BVP (1.2). Differentiating (3.9) twice, we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ73_HTML.gif
(3.21)
Hence, by (3.17), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ74_HTML.gif
(3.22)
Similarly, we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ75_HTML.gif
(3.23)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq274_HTML.gif be any positive solution of BVP (1.2). By Lemma 2.2, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq275_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq276_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq277_HTML.gif It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq278_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq279_HTML.gif So, by Lemma 2.6, we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq280_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq281_HTML.gif Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq282_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq283_HTML.gif Then, it follows from Lemma 2.6 that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq284_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq285_HTML.gif that is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq286_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq287_HTML.gif Hence, by induction, we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ76_HTML.gif
(3.24)

Now, taking limits in (3.24), we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq288_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq289_HTML.gif and the theorem is proved.

Theorem 3.3.

Let cone http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq290_HTML.gif be fully regular and conditions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq291_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq292_HTML.gif be satisfied. Then the conclusion of Theorem 3.2 holds.

Proof.

The proof is almost the same as that of Theorem 3.2. The only difference is that, instead of using condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq293_HTML.gif , the conclusion http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq294_HTML.gif is implied directly by (3.15) and (3.16), the full regularity of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq295_HTML.gif and Lemma 2.4.

4. An Example

Consider the infinite system of scalar singular second order three-point boundary value problems:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ77_HTML.gif
(4.1)

Proposition 4.1.

Infinite system (4.1) has a minimal positive solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq296_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq297_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq298_HTML.gif

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq299_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq300_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq301_HTML.gif is a real Banach space. Choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq302_HTML.gif . It is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq303_HTML.gif is a normal cone in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq304_HTML.gif with normal constants 1. Now we consider infinite system (4.1), which can be regarded as a BVP of form (1.2) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq305_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq306_HTML.gif . In this situation, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq307_HTML.gif in which
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ78_HTML.gif
(4.2)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq308_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq309_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq310_HTML.gif . It is clear, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq311_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq312_HTML.gif . Notice that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq313_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq314_HTML.gif , by (4.2), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ79_HTML.gif
(4.3)
which imply http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq315_HTML.gif is satisfied for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq316_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ80_HTML.gif
(4.4)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq317_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq318_HTML.gif where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ81_HTML.gif
(4.5)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ82_HTML.gif
(4.6)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ83_HTML.gif
(4.7)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ84_HTML.gif
(4.8)
Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq319_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq320_HTML.gif be given, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq321_HTML.gif be any sequence in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq322_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq323_HTML.gif . By (4.5), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ85_HTML.gif
(4.9)
So, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq324_HTML.gif is bounded and by the diagonal method together with the method of constructing subsequence, we can choose a subsequence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq325_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ86_HTML.gif
(4.10)
which implies by virtue of (4.9)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ87_HTML.gif
(4.11)
Hence http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq326_HTML.gif It is easy to see from (4.9)–(4.11) that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ88_HTML.gif
(4.12)

Thus, we have proved that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq327_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq328_HTML.gif

For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq329_HTML.gif , we have by (4.6)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ89_HTML.gif
(4.13)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq330_HTML.gif is between http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq331_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq332_HTML.gif . By (4.13), we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ90_HTML.gif
(4.14)
In the same way, we can prove that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq333_HTML.gif is relatively compact in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq334_HTML.gif , and we can also get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ91_HTML.gif
(4.15)

Thus, by (4.14) and (4.15), it is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq335_HTML.gif holds for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq336_HTML.gif . Thus, our conclusion follows from Theorem 3.1. This completes the proof.

Declarations

Acknowledgment

The project is supported financially by the National Natural Science Foundation of China (10671167) and the Natural Science Foundation of Liaocheng University (31805).

Authors’ Affiliations

(1)
School of Mathematics, Liaocheng University

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© Xingqiu Zhang. 2009

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