Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq1_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ1_HTML.gif
(1.1)
where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq4_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq6_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq7_HTML.gif In this paper, nonlinear terms https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq9_HTML.gif may be singular at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq10_HTML.gif , and/or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq11_HTML.gif , where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq13_HTML.gif . By singularity, we mean that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq14_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq15_HTML.gif or https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq18_HTML.gif but also at https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ3_HTML.gif
(2.1)
Evidently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq20_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq21_HTML.gif is a Banach space with norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ4_HTML.gif
(2.2)
and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ5_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ6_HTML.gif
(2.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq23_HTML.gif with norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ7_HTML.gif
(2.5)

Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq25_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq26_HTML.gif be a normal cone in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq27_HTML.gif with normal constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq28_HTML.gif which defines a partial ordering in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq29_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq30_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq32_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq33_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq34_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq35_HTML.gif if and only if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq37_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq38_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq39_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq40_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq41_HTML.gif , we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq42_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq43_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq45_HTML.gif . It is clear, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq46_HTML.gif are cones in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq48_HTML.gif , respectively. A map https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq50_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq51_HTML.gif satisfies (1.2).

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq53_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq55_HTML.gif be all Lebesgue measurable functions from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq56_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq57_HTML.gif . Denote
https://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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq59_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq60_HTML.gif and there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq61_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq62_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ9_HTML.gif
(2.7)
uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq63_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ10_HTML.gif
(2.8)
(H2) For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq65_HTML.gif and countable bounded set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq66_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq67_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ11_HTML.gif
(2.9)
with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ12_HTML.gif
(2.10)

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

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq70_HTML.gif imply
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq72_HTML.gif . Evidently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq73_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq74_HTML.gif are closed convex sets in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq75_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq78_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ14_HTML.gif
(2.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ15_HTML.gif
(2.13)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq79_HTML.gif is satisfied, then operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq81_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq82_HTML.gif .

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ17_HTML.gif
(2.15)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ18_HTML.gif
(2.16)
By virtue of condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq83_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq84_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ19_HTML.gif
(2.17)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ20_HTML.gif
(2.18)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ21_HTML.gif
(2.19)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq85_HTML.gif , we have, by (2.19)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ22_HTML.gif
(2.20)
which together with condition https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ23_HTML.gif
(2.21)
Thus, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq87_HTML.gif implies that
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ27_HTML.gif
(2.25)
Hence,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ29_HTML.gif
(2.27)
So, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ30_HTML.gif
(2.28)
So, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ31_HTML.gif
(2.29)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq90_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq91_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq92_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq93_HTML.gif and we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ32_HTML.gif
(2.30)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ33_HTML.gif
(2.31)
Finally, we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq94_HTML.gif is continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq95_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq96_HTML.gif is a bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq97_HTML.gif . Thus, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq98_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq99_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq100_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ34_HTML.gif
(2.32)
It is clear,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ35_HTML.gif
(2.33)
and by (2.20),
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq102_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq103_HTML.gif . By the same method, we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq104_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq105_HTML.gif . Therefore, the continuity of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq107_HTML.gif is satisfied, then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq109_HTML.gif is a fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq110_HTML.gif .

Proof.

Suppose that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq112_HTML.gif integrating (1.2) from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq113_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq114_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq115_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ39_HTML.gif
(2.37)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ40_HTML.gif
(2.38)
Thus, we obtain
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ42_HTML.gif
(2.40)
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq116_HTML.gif and the integral https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq117_HTML.gif are convergent. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq118_HTML.gif is a fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq119_HTML.gif .

Conversely, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq120_HTML.gif is fixed point of operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq122_HTML.gif be satisfied, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq123_HTML.gif is a bounded set. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq124_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq126_HTML.gif and for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq127_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq128_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ45_HTML.gif
(2.43)

uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq129_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq131_HTML.gif , the proof for operator https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ46_HTML.gif
(2.44)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq133_HTML.gif we obtain by (2.44)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq134_HTML.gif that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq136_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq137_HTML.gif is bounded, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq138_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq139_HTML.gif . By (2.25), we get
https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq143_HTML.gif there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq144_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ49_HTML.gif
(2.47)

uniformly with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq145_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq147_HTML.gif there exists sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq148_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ50_HTML.gif
(2.48)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq149_HTML.gif as https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq152_HTML.gif be a bounded set in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq153_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq154_HTML.gif holds. Then
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq156_HTML.gif be a closed convex set of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq158_HTML.gif Assume that the continuous operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq159_HTML.gif has the following property: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq160_HTML.gif countable, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq161_HTML.gif is relatively compact. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq162_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq163_HTML.gif .

Lemma 2.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq164_HTML.gif is satisfied, then for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq165_HTML.gif imply that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq169_HTML.gif are bounded sets in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq170_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ52_HTML.gif
(2.50)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq171_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq173_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq175_HTML.gif be normal (fully regular) in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq176_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq177_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq178_HTML.gif is normal (fully regular) in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq180_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq182_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq183_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq184_HTML.gif

Proof.

By Lemma 2.1, operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq186_HTML.gif into https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq188_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq189_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq190_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq191_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq192_HTML.gif . Obviously, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq194_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq195_HTML.gif is not empty since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq197_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq198_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq199_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq200_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq201_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq202_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq203_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq204_HTML.gif . Then https://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),
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ54_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq206_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq208_HTML.gif is convergent uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq209_HTML.gif So, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq210_HTML.gif we can choose a sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq211_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ55_HTML.gif
(3.3)
Then, by [1, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq212_HTML.gif ], (2.44), (3.1), (3.3), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq213_HTML.gif , and Lemma 2.7, we obtain
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ58_HTML.gif
(3.6)

On the other hand, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq214_HTML.gif . Then, (3.5), (3.6), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq215_HTML.gif , and Lemma 2.7 imply https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq216_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq217_HTML.gif is relatively compact in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq219_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq220_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq222_HTML.gif be normal and conditions https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq225_HTML.gif for any positive solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq226_HTML.gif of BVP (1.2). Moreover, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq228_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq229_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq230_HTML.gif uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq232_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq233_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq234_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ59_HTML.gif
(3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ60_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ61_HTML.gif
(3.9)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq235_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq236_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ64_HTML.gif
(3.12)
which imply that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq237_HTML.gif . Similarly, we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq238_HTML.gif . Thus, https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq240_HTML.gif and
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ68_HTML.gif
(3.16)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq241_HTML.gif Then, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq243_HTML.gif and operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq244_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq245_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq246_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq247_HTML.gif is not empty since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq248_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq249_HTML.gif Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq250_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq252_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq253_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq254_HTML.gif So, there exists an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq255_HTML.gif and a subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq256_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq257_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq258_HTML.gif uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq259_HTML.gif Since that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq260_HTML.gif is normal and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq262_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq263_HTML.gif uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq264_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq265_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq266_HTML.gif are closed convex sets in space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq267_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq268_HTML.gif It is clear,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ69_HTML.gif
(3.17)
By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq269_HTML.gif and (3.16), we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq270_HTML.gif in (3.9), we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq271_HTML.gif in (3.10), we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq272_HTML.gif and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ74_HTML.gif
(3.22)
Similarly, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ75_HTML.gif
(3.23)
Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq275_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq276_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq277_HTML.gif It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq278_HTML.gif for any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq280_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq281_HTML.gif Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq282_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq284_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq285_HTML.gif that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq286_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq287_HTML.gif Hence, by induction, we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq288_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq290_HTML.gif be fully regular and conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq291_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq293_HTML.gif , the conclusion https://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 https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq296_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq297_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq298_HTML.gif

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq299_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq300_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq301_HTML.gif is a real Banach space. Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq302_HTML.gif . It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq303_HTML.gif is a normal cone in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq305_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq306_HTML.gif . In this situation, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq307_HTML.gif in which
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ78_HTML.gif
(4.2)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq308_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq309_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq310_HTML.gif . It is clear, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq311_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq312_HTML.gif . Notice that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq313_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq314_HTML.gif , by (4.2), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ79_HTML.gif
(4.3)
which imply https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq315_HTML.gif is satisfied for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq316_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ80_HTML.gif
(4.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq317_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq318_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ81_HTML.gif
(4.5)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ82_HTML.gif
(4.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ83_HTML.gif
(4.7)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ84_HTML.gif
(4.8)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq319_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq320_HTML.gif be given, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq321_HTML.gif be any sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq322_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq323_HTML.gif . By (4.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ85_HTML.gif
(4.9)
So, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq325_HTML.gif such that
https://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)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ87_HTML.gif
(4.11)
Hence https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq327_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq328_HTML.gif

For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq329_HTML.gif , we have by (4.6)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_Equ89_HTML.gif
(4.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq330_HTML.gif is between https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq331_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq332_HTML.gif . By (4.13), we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq333_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq334_HTML.gif , and we can also get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F978605/MediaObjects/13661_2009_Article_894_IEq335_HTML.gif holds for https://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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