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

*Boundary Value Problems*
**volume 2011**, Article number: 684542 (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.

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 [4–13] 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:

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

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

where , , 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].

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:

where .

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 [30–32], 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 :

where as is a cone of .

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:

where with . denotes the jump of at , that is,

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

Let

Let with the norm where

Define a cone by

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

If , then boundary value problem (1.5) reduces to the following two point boundary value problem:

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.

Suppose that and there exist and nonnegative constants such that

for

Lemma 2.1.

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

Proof.

By we have

Thus,

The proof is complete.

Lemma 2.2.

Suppose that holds. If , then is a solution of problem (1.5) if and only if is a solution of the following impulsive integral equation:

where

Proof.

First, suppose that is a solution of problem (1.5). It is easy to see by integration of (1.5) that

Taking limit for , by Lemma 2.1 and the boundary conditions, we have

Thus,

Integrating (2.8), we can get

It follows that

So we have (2.4).

Conversely, suppose that is a solution of (2.4). Evidently,

Direct differentiation of (2.4) implies, for

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

Define an operator

Lemma 2.3.

Assume that and hold. Then operator maps into , and

where

Moreover, for with one has

Proof.

Let . From the definition of and , we can obtain that is an operator from into , and

Direct differentiation of (2.13) implies, for

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.

Let conditions be satisfied. Suppose further that

Then problem (1.5) has the minimal nonnegative solution with , where is defined as in Lemma 2.3. Here, the meaning of minimal nonnegative solution is that if is an arbitrary nonnegative solution of (1.5), then Moreover, if we let then with

and and converge uniformly to and on , respectively.

Proof.

By Lemma 2.3 and the definition of operator , we have , and

By (3.3), we have

From (3.4), (3.5), and (3.6), we know that and exist. Suppose that

By the definition of we have

From (3.6), we obtain

It follows that is equicontinuous on every Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to on . Which together with (3.4) imply that converges uniformly to on , and On the other hand, by (3.6), and (3.9), we have

Since is bounded on (M is a finite positive number), is equicontinuous on every Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to on , which together with (3.5) imply that converges uniformly to on , and From above, we know that exists and It follows that and

Now we prove that

By the continuity of and the uniform convergence of , we know that

On the other hand, by and (3.6) and (3.12), we have

Combining this with the dominated convergence theorem, we have

Moreover, we can see that

Now taking limits from two sides of and using (3.15)–(3.16), we have

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.

Consider the following boundary value problem of second-order impulsive differential equation on infinite interval

where

Evidently, is not the solution of (4.1).

**Conclusion**

Problem (4.1) has minimal positive solution.

Proof.

It is clear that and is satisfied.

By the inequality we see that

Let

Then, we easily obtain that

Thus, is satisfied and . By Theorem 3.1, it follows that problem (4.1) has a minimal positive solution.

## References

- 1.
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. - 2.
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. - 3.
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. - 4.
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.1993.1069 - 5.
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-H - 6.
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.1994.1031 - 7.
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)82855-6 - 8.
Liu X: Monotone iterative technique for impulsive differential equations in a Banach space.

*Journal of Mathematical and Physical Sciences*1990, 24(3):183-191. - 9.
Vatsala AS, Sun Y: Periodic boundary value problems of impulsive differential equations.

*Applicable Analysis*1992, 44(3-4):145-158. 10.1080/00036819208840074 - 10.
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.1997.5207 - 11.
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)10059-0 - 12.
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.2004.06.013 - 13.
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.2004.02.057 - 14.
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.2006.06.029 - 15.
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.2006.01.012 - 16.
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.2005.06.014 - 17.
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.2007.07.081 - 18.
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.2008.02.040 - 19.
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.2006.08.020 - 20.
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.2005.07.076 - 21.
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.2007.01.040 - 22.
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)00381-8 - 23.
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.2008.01.024 - 24.
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.2006.04.018 - 25.
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.2005.12.028 - 26.
Benchohra M, Henderson J, Ntouyas SK:

*Impulsive Differential Equations and Inclusions*. Hindawi, New York, NY, USA; 2006. - 27.
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. - 28.
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. - 29.
Guo D, Liu XZ: Impulsive integro-differential equations on unbounded domain in a Banach space.

*Nonlinear Studies*1996, 3(1):49-57. - 30.
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)10174-6 - 31.
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)00564-6 - 32.
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)00356-9 - 33.
Li J, Shen J: Existence of positive solution for second-order impulsive boundary value problems on infinity intervals.

*Boundary Value Problems*2006, 2006:-11. - 34.
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)00141-6 - 35.
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/1178224769 - 36.
Constantin A: On an infinite interval boundary value problem.

*Annali di Matematica Pura ed Applicata. Serie Quarta*1999, 176: 379-394. 10.1007/BF02506002 - 37.
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)00431-9 - 38.
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-X - 39.
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)00801-9

## 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.

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### Keywords

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