# Minimal Nonnegative Solution of Nonlinear Impulsive Differential Equations on Infinite Interval

- Xuemei Zhang
^{1}Email author, - Xiaozhong Yang
^{1}and - Meiqiang Feng
^{2}

**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 see Lakshmikantham et al. [1], Bainov and Simeonov [2], and Samoĭlenko and Perestyuk [3] and the references therein.

where is a cone in the real Banach space , denotes the zero element of , for and .

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 [14–16], Jankowski [17–19], 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].

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.

where and represent the right-hand limit and left-hand limit of at , respectively. has a similar meaning for .

Let . is called a nonnegative solution of (1.5), if and satisfies (1.5).

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 of problem (1.5), let us list the following assumptions, which will stand throughout this paper.

Lemma 2.1.

Suppose that holds. Then for all , , and are convergent.

Proof.

The proof is complete.

Lemma 2.2.

Proof.

So we have (2.4).

So . It is easy to verify that . The proof of Lemma 2.2 is complete.

Lemma 2.3.

Proof.

Thus we have It follows that (2.14) is satisfied. Equation (2.16) is easily obtained by .

## 3. Main Result

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

Theorem 3.1.

and and converge uniformly to and on , respectively.

Proof.

By Lemma 2.2, is a nonnegative solution of (1.5).

Suppose that is an arbitrary nonnegative solution of (1.5). Then It is clear that . Suppose that By (2.16) we have This means that Taking limit, we have . 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.

Evidently, is not the solution of (4.1).

**Conclusion**

Problem (4.1) has minimal positive solution.

Proof.

It is clear that and is satisfied.

Thus, is satisfied and . 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

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