Open Access

Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval

Boundary Value Problems20102011:684542

DOI: 10.1155/2011/684542

Received: 20 May 2010

Accepted: 19 July 2010

Published: 2 August 2010

Abstract

The cone theory and monotone iterative technique are used to investigate the minimal nonnegative solution of nonlocal boundary value problems for second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times. All the existing results obtained in previous papers on nonlocal boundary value problems are under the case of the boundary conditions with no impulsive effects or the boundary conditions with impulsive effects on a finite interval with a finite number of impulsive times, so our work is new. Meanwhile, an example is worked out to demonstrate the main results.

1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations. For an introduction of the basic theory of impulsive differential equations in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq1_HTML.gif see Lakshmikantham et al. [1], Bainov and Simeonov [2], and Samoĭlenko and Perestyuk [3] and the references therein.

Usually, we only consider the differential equation, integrodifferential equation, functional differential equations, or dynamic equations on time scales on a finite interval with a finite number of impulsive times. To identify a few, we refer the reader to [413] and references therein. In particular, we would like to mention some results of Guo and Liu [5] and Guo [6]. In [5], by using fixed-point index theory for cone mappings, Guo and Liu investigated the existence of multiple positive solutions of a boundary value problem for the following second-order impulsive differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq2_HTML.gif is a cone in the real Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq4_HTML.gif denotes the zero element of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq6_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq8_HTML.gif .

In [6], by using fixed-point theory, Guo established the existence of solutions of a boundary value problem for the following second-order impulsive differential equation in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq9_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq11_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq12_HTML.gif .

On the other hand, the readers can also find some recent developments and applications of the case that impulse effects on a finite interval with a finite number of impulsive times to a variety of problems from Nieto and Rodríguez-López [1416], Jankowski [1719], Lin and Jiang [20], Ma and Sun [21], He and Yu [22], Feng and Xie [23], Yan [24], Benchohra et al. [25], and Benchohra et al. [26].

Recently, in [27], Li and Nieto obtained some new results of the case that impulse effects on an infinite interval with a finite number of impulsive times. By using a fixed-point theorem due to Avery and Peterson [28], Li and Nieto considered the existence of multiple positive solutions of the following impulsive boundary value problem on an infinite interval:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq13_HTML.gif .

At the same time, we also notice that there has been increasing interest in studying nonlinear differential equation and impulsive integrodifferential equation on an infinite interval with an infinite number of impulsive times; to identify a few, we refer the reader to Guo and Liu [29], Guo [3032], and Li and Shen [33]. It is here worth mentioning the works by Guo [31]. In [31], Guo investigated the minimal nonnegative solution of the following initial value problem for a second order nonlinear impulsive integrodifferential equation of Volterra type on an infinite interval with an infinite number of impulsive times in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq14_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq15_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq16_HTML.gif is a cone of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq17_HTML.gif .

However, the corresponding theory for nonlocal boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times is not investigated till now. Now, in this paper, we will use the cone theory and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times.

Consider the following boundary value problem for second-order nonlinear impulsive differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ5_HTML.gif
(1.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq18_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq19_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq20_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq21_HTML.gif denotes the jump of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq22_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq23_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq25_HTML.gif represent the right-hand limit and left-hand limit of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq26_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq27_HTML.gif , respectively. https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq28_HTML.gif has a similar meaning for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq29_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ7_HTML.gif
(1.7)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq30_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq31_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ8_HTML.gif
(1.8)
Define a cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq32_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ9_HTML.gif
(1.9)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq33_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq34_HTML.gif is called a nonnegative solution of (1.5), if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq36_HTML.gif satisfies (1.5).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq37_HTML.gif , then boundary value problem (1.5) reduces to the following two point boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ10_HTML.gif
(1.10)

which has been intensively studied; see Ma [34], Agarwal and O'Regan [35], Constantin [36], Liu [37, 38], and Yan and Liu [39] for some references along this line.

The organization of this paper is as follows. In Section 2, we provide some necessary background. In Section 3, the main result of problem (1.5) will be stated and proved. In Section 4, we give an example to illustrate how the main results can be used in practice.

2. Preliminaries

To establish the existence of minimal nonnegative solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq38_HTML.gif of problem (1.5), let us list the following assumptions, which will stand throughout this paper.

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq39_HTML.gif Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq40_HTML.gif and there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq41_HTML.gif and nonnegative constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq42_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ11_HTML.gif
(2.1)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq43_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq44_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq45_HTML.gif

Lemma 2.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq46_HTML.gif holds. Then for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq50_HTML.gif are convergent.

Proof.

By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq51_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ12_HTML.gif
(2.2)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ13_HTML.gif
(2.3)

The proof is complete.

Lemma 2.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq52_HTML.gif holds. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq53_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq54_HTML.gif is a solution of problem (1.5) if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq55_HTML.gif is a solution of the following impulsive integral equation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ14_HTML.gif
(2.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ15_HTML.gif
(2.5)

Proof.

First, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq56_HTML.gif is a solution of problem (1.5). It is easy to see by integration of (1.5) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ16_HTML.gif
(2.6)
Taking limit for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq57_HTML.gif , by Lemma 2.1 and the boundary conditions, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ17_HTML.gif
(2.7)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ18_HTML.gif
(2.8)
Integrating (2.8), we can get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ19_HTML.gif
(2.9)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ20_HTML.gif
(2.10)

So we have (2.4).

Conversely, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq58_HTML.gif is a solution of (2.4). Evidently,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ21_HTML.gif
(2.11)
Direct differentiation of (2.4) implies, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq59_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ22_HTML.gif
(2.12)

So https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq60_HTML.gif . It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq61_HTML.gif . The proof of Lemma 2.2 is complete.

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq62_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ23_HTML.gif
(2.13)

Lemma 2.3.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq64_HTML.gif hold. Then operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq65_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq66_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq67_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ24_HTML.gif
(2.14)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ25_HTML.gif
(2.15)
Moreover, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq68_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq69_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ26_HTML.gif
(2.16)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq70_HTML.gif . From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq72_HTML.gif , we can obtain that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq73_HTML.gif is an operator from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq74_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq75_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ27_HTML.gif
(2.17)
Direct differentiation of (2.13) implies, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq76_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ28_HTML.gif
(2.18)

Thus we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq77_HTML.gif It follows that (2.14) is satisfied. Equation (2.16) is easily obtained by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq78_HTML.gif .

3. Main Result

In this section, we establish the existence of a minimal nonnegative solution for problem (1.5).

Theorem 3.1.

Let conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq79_HTML.gif be satisfied. Suppose further that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ29_HTML.gif
(3.1)
Then problem (1.5) has the minimal nonnegative solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq80_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq81_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq82_HTML.gif is defined as in Lemma 2.3. Here, the meaning of minimal nonnegative solution is that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq83_HTML.gif is an arbitrary nonnegative solution of (1.5), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq84_HTML.gif Moreover, if we let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq85_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq86_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ30_HTML.gif
(3.2)

and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq88_HTML.gif converge uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq89_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq90_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq91_HTML.gif , respectively.

Proof.

By Lemma 2.3 and the definition of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq92_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq93_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ31_HTML.gif
(3.3)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ32_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ33_HTML.gif
(3.5)
By (3.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ34_HTML.gif
(3.6)
From (3.4), (3.5), and (3.6), we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq95_HTML.gif exist. Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ35_HTML.gif
(3.7)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq96_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ36_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ37_HTML.gif
(3.9)
From (3.6), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ38_HTML.gif
(3.10)
It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq97_HTML.gif is equicontinuous on every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq98_HTML.gif Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq99_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq100_HTML.gif . Which together with (3.4) imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq101_HTML.gif converges uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq102_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq103_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq104_HTML.gif On the other hand, by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq105_HTML.gif (3.6), and (3.9), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ39_HTML.gif
(3.11)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq106_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq107_HTML.gif (M is a finite positive number), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq108_HTML.gif is equicontinuous on every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq109_HTML.gif Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq110_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq111_HTML.gif , which together with (3.5) imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq112_HTML.gif converges uniformly to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq113_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq114_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq115_HTML.gif From above, we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq116_HTML.gif exists and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq117_HTML.gif It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq118_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ40_HTML.gif
(3.12)

Now we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq119_HTML.gif

By the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq120_HTML.gif and the uniform convergence of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq121_HTML.gif , we know that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ41_HTML.gif
(3.13)
On the other hand, by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq122_HTML.gif and (3.6) and (3.12), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ42_HTML.gif
(3.14)
Combining this with the dominated convergence theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ43_HTML.gif
(3.15)
Moreover, we can see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ44_HTML.gif
(3.16)
Now taking limits from two sides of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq123_HTML.gif and using (3.15)–(3.16), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ45_HTML.gif
(3.17)

By Lemma 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq124_HTML.gif is a nonnegative solution of (1.5).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq125_HTML.gif is an arbitrary nonnegative solution of (1.5). Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq126_HTML.gif It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq127_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq128_HTML.gif By (2.16) we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq129_HTML.gif This means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq130_HTML.gif Taking limit, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq131_HTML.gif . The proof of Theorem 3.1 is complete.

4. Example

To illustrate how our main results can be used in practice, we present an example.

Example 4.1.

Consider the following boundary value problem of second-order impulsive differential equation on infinite interval
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ46_HTML.gif
(4.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ47_HTML.gif
(4.2)

Evidently, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq132_HTML.gif is not the solution of (4.1).

Conclusion

Problem (4.1) has minimal positive solution.

Proof.

It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq134_HTML.gif is satisfied.

By the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq135_HTML.gif we see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ48_HTML.gif
(4.3)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ49_HTML.gif
(4.4)
Then, we easily obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_Equ50_HTML.gif
(4.5)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq136_HTML.gif is satisfied and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F684542/MediaObjects/13661_2010_Article_52_IEq137_HTML.gif . By Theorem 3.1, it follows that problem (4.1) has a minimal positive solution.

Declarations

Acknowledgments

This work is supported by the National Natural Science Foundation of China (10771065), the Natural Sciences Foundation of Hebei Province (A2007001027), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education(KM201010772018), the 2010 level of scientific research of improving project (5028123900), and Beijing Municipal Education Commission(71D0911003). The authors thank the referee for his/her careful reading of the paper and useful suggestions.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, North China Electric Power University
(2)
School of Applied Science, Beijing Information Science and Technology University

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© Xuemei Zhang et al. 2011

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