- Research Article
- Open Access

# 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.

## Keywords

- Finite Interval
- Nonlocal Boundary
- Impulsive Effect
- Impulsive Differential Equation
- Multiple Positive Solution

## 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

## References

- Lakshmikantham V, Baĭnov DD, Simeonov PS:
*Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics*.*Volume 6*. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar - Baĭnov DD, Simeonov PS:
*Systems with Impulse Effect: Stability, Theory and Application, Ellis Horwood Series: Mathematics and Its Applications*. Ellis Horwood, Chichester, UK; 1989:255.Google Scholar - Samoĭlenko AM, Perestyuk NA:
*Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 14*. World Scientific, River Edge, NJ, USA; 1995:x+462.Google Scholar - Guo D: Multiple positive solutions of impulsive nonlinear Fredholm integral equations and applications.
*Journal of Mathematical Analysis and Applications*1993, 173(1):318-324. 10.1006/jmaa.1993.1069View ArticleMathSciNetGoogle Scholar - Guo D, Liu X: Multiple positive solutions of boundary-value problems for impulsive differential equations.
*Nonlinear Analysis: Theory, Methods & Applications*1995, 25(4):327-337. 10.1016/0362-546X(94)00175-HView ArticleMathSciNetGoogle Scholar - Guo D: Existence of solutions of boundary value problems for nonlinear second order impulsive differential equations in Banach spaces.
*Journal of Mathematical Analysis and Applications*1994, 181(2):407-421. 10.1006/jmaa.1994.1031View ArticleMathSciNetGoogle Scholar - Guo D: Periodic boundary value problems for second order impulsive integro-differential equations in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*1997, 28(6):983-997. 10.1016/S0362-546X(97)82855-6View ArticleMathSciNetGoogle Scholar - Liu X: Monotone iterative technique for impulsive differential equations in a Banach space.
*Journal of Mathematical and Physical Sciences*1990, 24(3):183-191.MathSciNetGoogle Scholar - Vatsala AS, Sun Y: Periodic boundary value problems of impulsive differential equations.
*Applicable Analysis*1992, 44(3-4):145-158. 10.1080/00036819208840074View ArticleMathSciNetGoogle Scholar - Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order.
*Journal of Mathematical Analysis and Applications*1997, 205(2):423-433. 10.1006/jmaa.1997.5207View ArticleMathSciNetGoogle Scholar - He Z, Ge W: Periodic boundary value problem for first order impulsive delay differential equations.
*Applied Mathematics and Computation*1999, 104(1):51-63. 10.1016/S0096-3003(98)10059-0View ArticleMathSciNetGoogle Scholar - Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Impulsive functional differential equations with variable times.
*Computers & Mathematics with Applications*2004, 47(10-11):1659-1665. 10.1016/j.camwa.2004.06.013View ArticleMathSciNetGoogle Scholar - Benchohra M, Ntouyas SK, Ouahab A: Existence results for second order boundary value problem of impulsive dynamic equations on time scales.
*Journal of Mathematical Analysis and Applications*2004, 296(1):65-73. 10.1016/j.jmaa.2004.02.057View ArticleMathSciNetGoogle Scholar - Nieto JJ, Rodríguez-López R: New comparison results for impulsive integro-differential equations and applications.
*Journal of Mathematical Analysis and Applications*2007, 328(2):1343-1368. 10.1016/j.jmaa.2006.06.029View ArticleMathSciNetGoogle Scholar - Nieto JJ, Rodríguez-López R: Monotone method for first-order functional differential equations.
*Computers & Mathematics with Applications*2006, 52(3-4):471-484. 10.1016/j.camwa.2006.01.012View ArticleMathSciNetGoogle Scholar - Nieto JJ, Rodríguez-López R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations.
*Journal of Mathematical Analysis and Applications*2006, 318(2):593-610. 10.1016/j.jmaa.2005.06.014View ArticleMathSciNetGoogle Scholar - Jankowski T: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments.
*Applied Mathematics and Computation*2008, 197(1):179-189. 10.1016/j.amc.2007.07.081View ArticleMathSciNetGoogle Scholar - Jankowski T: Positive solutions to second order four-point boundary value problems for impulsive differential equations.
*Applied Mathematics and Computation*2008, 202(2):550-561. 10.1016/j.amc.2008.02.040View ArticleMathSciNetGoogle Scholar - Jankowski T: Existence of solutions for second order impulsive differential equations with deviating arguments.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(6):1764-1774. 10.1016/j.na.2006.08.020View ArticleMathSciNetGoogle Scholar - Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations.
*Journal of Mathematical Analysis and Applications*2006, 321(2):501-514. 10.1016/j.jmaa.2005.07.076View ArticleMathSciNetGoogle Scholar - Ma Y, Sun J: Stability criteria for impulsive systems on time scales.
*Journal of Computational and Applied Mathematics*2008, 213(2):400-407. 10.1016/j.cam.2007.01.040View ArticleMathSciNetGoogle Scholar - He Z, Yu J: Periodic boundary value problem for first-order impulsive functional differential equations.
*Journal of Computational and Applied Mathematics*2002, 138(2):205-217. 10.1016/S0377-0427(01)00381-8View ArticleMathSciNetGoogle Scholar - Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations.
*Journal of Computational and Applied Mathematics*2009, 223(1):438-448. 10.1016/j.cam.2008.01.024View ArticleMathSciNetGoogle Scholar - Yan J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters.
*Journal of Mathematical Analysis and Applications*2007, 327(2):854-868. 10.1016/j.jmaa.2006.04.018View ArticleMathSciNetGoogle Scholar - Benchohra M, Ntouyas SK, Ouahab A: Extremal solutions of second order impulsive dynamic equations on time scales.
*Journal of Mathematical Analysis and Applications*2006, 324(1):425-434. 10.1016/j.jmaa.2005.12.028View ArticleMathSciNetGoogle Scholar - Benchohra M, Henderson J, Ntouyas SK:
*Impulsive Differential Equations and Inclusions*. Hindawi, New York, NY, USA; 2006.View ArticleGoogle Scholar - Li J, Nieto JJ: Existence of positive solutions for multipoint boundary value problem on the half-line with impulses.
*Boundary Value Problems*2009, 2009:-12.Google Scholar - Avery RI, Peterson AC: Three positive fixed points of nonlinear operators on ordered Banach spaces.
*Computers & Mathematics with Applications*2001, 42(3–5):313-322.View ArticleMathSciNetGoogle Scholar - Guo D, Liu XZ: Impulsive integro-differential equations on unbounded domain in a Banach space.
*Nonlinear Studies*1996, 3(1):49-57.MathSciNetGoogle Scholar - Guo D: Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space.
*Applied Mathematics and Computation*1999, 99(1):1-15. 10.1016/S0096-3003(97)10174-6View ArticleMathSciNetGoogle Scholar - Guo D: Second order impulsive integro-differential equations on unbounded domains in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*1999, 35(4):413-423. 10.1016/S0362-546X(97)00564-6View ArticleMathSciNetGoogle Scholar - Guo D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space.
*Applied Mathematics and Computation*2003, 143(2-3):233-249. 10.1016/S0096-3003(02)00356-9View ArticleMathSciNetGoogle Scholar - Li J, Shen J: Existence of positive solution for second-order impulsive boundary value problems on infinity intervals.
*Boundary Value Problems*2006, 2006:-11.Google Scholar - Ma R: Existence of positive solutions for second-order boundary value problems on infinity intervals.
*Applied Mathematics Letters*2003, 16(1):33-39. 10.1016/S0893-9659(02)00141-6View ArticleMathSciNetGoogle Scholar - Agarwal RP, O'Regan D: Boundary value problems of nonsingular type on the semi-infinite interval.
*The Tohoku Mathematical Journal*1999, 51(3):391-397. 10.2748/tmj/1178224769View ArticleMathSciNetGoogle Scholar - Constantin A: On an infinite interval boundary value problem.
*Annali di Matematica Pura ed Applicata. Serie Quarta*1999, 176: 379-394. 10.1007/BF02506002View ArticleMathSciNetGoogle Scholar - Liu Y: Existence and unboundedness of positive solutions for singular boundary value problems on half-line.
*Applied Mathematics and Computation*2003, 144(2-3):543-556. 10.1016/S0096-3003(02)00431-9View ArticleMathSciNetGoogle Scholar - Liu Y: Boundary value problems for second order differential equations on unbounded domains in a Banach space.
*Applied Mathematics and Computation*2003, 135(2-3):569-583. 10.1016/S0096-3003(02)00070-XView ArticleMathSciNetGoogle Scholar - Yan B, Liu Y: Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line.
*Applied Mathematics and Computation*2004, 147(3):629-644. 10.1016/S0096-3003(02)00801-9View ArticleMathSciNetGoogle Scholar

## Copyright

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