# Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses

- Jianli Li
^{1}Email author and - Juan J Nieto
^{2}

**2009**:834158

**DOI: **10.1155/2009/834158

© J. Li and J.J. Nieto 2009

**Received: **7 March 2009

**Accepted: **25 April 2009

**Published: **4 June 2009

## Abstract

We consider a multi-point boundary value problem on the half-line with impulses. By using a fixed-point theorem due to Avery and Peterson, the existence of at least three positive solutions is obtained.

## 1. Introduction

Impulsive differential equations are a basic tool to study evolution processes that are subjected to abrupt changes in their state. For instance, many biological, physical, and engineering applications exhibit impulsive effects (see [1–3]). It should be noted that recent progress in the development of the qualitative theory of impulsive differential equations has been stimulated primarily by a number of interesting applied problems [4–24].

where , , , , and , and satisfy

() , , and when is bounded, and are bounded on ;

Boundary value problems on the half-line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and there are many results in this area, see [8, 13, 14, 20, 25–27], for example.

They showed the existence at least three positive solutions for (1.4) by using a fixed point theorem in a cone due to Avery-Peterson [28].

Yan [20], by using Leray-Schauder theorem and fixed point index theory presents some results on the existence for the boundary value problems on the half-line with impulses and infinite delay.

However to the best knowledge of the authors, there is no paper concerned with the existence of three positive solutions to multipoint boundary value problems of impulsive differential equation on infinite interval so far. Motivated by [20, 25], in this paper, we aim to investigate the existence of triple positive solutions for BVP (1.1). The method chosen in this paper is a fixed point technique due to Avery and Peterson [28].

## 2. Preliminaries

In this section, we give some definitions and results that we will use in the rest of the paper.

Definition 2.1.

To prove our main results, we need the following fixed point theorem due to Avery and Peterson in [28].

Theorem 2.2.

is completely continuous and there exist positive numbers , and with such that

## 3. Some Lemmas

Define is continuous at each , left continuous at , exists, .

By a solution of (1.1) we mean a function in satisfying the relations in (1.1).

Lemma 3.1.

The proof is similar to Lemma in [9], and here we omit it.

Lemma 3.2 (see [20, Theorem ]).

Let . Then is compact in , if the following conditions hold:

(b)the functions belonging to are piecewise equicontinuous on any interval of ;

(c)the functions from are equiconvergent, that is, given , there corresponds such that for any and .

Lemma 3.3.

Proof.

So is quasi-equicontinuous on any compact interval of .

That is (3.11) holds. By Lemma 3.2, is relatively compact. In sum, is completely continuous.

## 4. Existence of Three Positive Solutions

The main result of this paper is the following.

Theorem 4.1.

Assume hold. Let , , and suppose that satisfy the following conditions:

Proof.

Step 1.

Thus (4.5) holds.

Step 2.

This shows the condition (i) in Theorem 2.2 is satisfied.

Step 3.

Hence, condition (ii) in Theorem 2.2 is satisfied.

Step 4.

## 5. An Example

## Declarations

### Acknowledgments

This work is supported by the NNSF of China (no. 60671066), A project supported by Scientific Research Fund of Hunan Provincial Education Department (07B041) and Program for Young Excellent Talents in Hunan Normal University, The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PR.

## Authors’ Affiliations

## References

- Benchohra M, Henderson J, Ntouyas S:
*Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications*.*Volume 2*. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366.View ArticleGoogle Scholar - 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 - Zavalishchin ST, Sesekin AN:
*Dynamic Impulse Systems: Theory and Application, Mathematics and Its Applications*.*Volume 394*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xii+256.View ArticleGoogle Scholar - Belley J-M, Virgilio M: Periodic Liénard-type delay equations with state-dependent impulses.
*Nonlinear Analysis: Theory, Methods & Applications*2006, 64(3):568–589. 10.1016/j.na.2005.06.025MATHMathSciNetView ArticleGoogle Scholar - Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations.
*Bulletin of the London Mathematical Society*2008, 40(1):143–150. 10.1112/blms/bdm110MATHMathSciNetView ArticleGoogle Scholar - Cardinali T, Rubbioni P: Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(1):73–84. 10.1016/j.na.2007.05.001MATHMathSciNetView ArticleGoogle Scholar - Di Piazza L, Satco B: A new result on impulsive differential equations involving non-absolutely convergent integrals.
*Journal of Mathematical Analysis and Applications*2009, 352(2):954–963. 10.1016/j.jmaa.2008.11.048MATHMathSciNetView ArticleGoogle Scholar - Guo D: Existence of positive solutions for
th-order nonlinear impulsive singular integro-differential equations in Banach spaces.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 68(9):2727–2740. 10.1016/j.na.2007.02.019MATHMathSciNetView ArticleGoogle Scholar - Gao S, Chen L, Nieto JJ, Torres A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence.
*Vaccine*2006, 24(35–36):6037–6045. 10.1016/j.vaccine.2006.05.018View ArticleGoogle Scholar - Jiao J, Chen L, Nieto JJ, Torres A: Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey.
*Applied Mathematics and Mechanics*2008, 29(5):653–663. 10.1007/s10483-008-0509-xMATHMathSciNetView ArticleGoogle Scholar - Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations.
*Journal of Mathematical Analysis and Applications*2007, 325(1):226–236. 10.1016/j.jmaa.2005.04.005MATHMathSciNetView ArticleGoogle Scholar - Luo Z, Shen J: Stability of impulsive functional differential equations via the Liapunov functional.
*Applied Mathematics Letters*2009, 22(2):163–169. 10.1016/j.aml.2008.03.004MathSciNetView ArticleGoogle 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 - Liang S, Zhang J: The existence of three positive solutions for some nonlinear boundary value problems on the half-line.
*Positivity*2009, 13(2):443–457. 10.1007/s11117-008-2213-zMATHMathSciNetView ArticleGoogle Scholar - Nieto JJ, O'Regan D: Variational approach to impulsive differential equations.
*Nonlinear Analysis: Real World Applications*2009, 10(2):680–690. 10.1016/j.nonrwa.2007.10.022MATHMathSciNetView ArticleGoogle Scholar - Nieto JJ: Impulsive resonance periodic problems of first order.
*Applied Mathematics Letters*2002, 15(4):489–493. 10.1016/S0893-9659(01)00163-XMATHMathSciNetView ArticleGoogle Scholar - Stamov GTr: On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model.
*Applied Mathematics Letters*2009, 22(4):516–520. 10.1016/j.aml.2008.07.002MATHMathSciNetView ArticleGoogle Scholar - Wang JR, Xiang X, Wei W, Chen Q: Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces.
*Fixed Point Theory and Applications*2008, 2008:-15.Google Scholar - Xian X, O'Regan D, Agarwal RP: Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions.
*Boundary Value Problems*2008, 2008:-21.Google Scholar - Yan B: Boundary value problems on the half-line with impulses and infinite delay.
*Journal of Mathematical Analysis and Applications*2001, 259(1):94–114. 10.1006/jmaa.2000.7392MATHMathSciNetView ArticleGoogle Scholar - Yan J, Zhao A, Nieto JJ: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems.
*Mathematical and Computer Modelling*2004, 40(5–6):509–518. 10.1016/j.mcm.2003.12.011MATHMathSciNetView ArticleGoogle Scholar - Zhang H, Chen L, Nieto JJ: A delayed epidemic model with stage-structure and pulses for pest management strategy.
*Nonlinear Analysis: Real World Applications*2008, 9(4):1714–1726. 10.1016/j.nonrwa.2007.05.004MATHMathSciNetView ArticleGoogle Scholar - Zhang X, Shuai Z, Wang K: Optimal impulsive harvesting policy for single population.
*Nonlinear Analysis: Real World Applications*2003, 4(4):639–651. 10.1016/S1468-1218(02)00084-6MATHMathSciNetView ArticleGoogle Scholar - Zeng G, Wang F, Nieto JJ: Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response.
*Advances in Complex Systems*2008, 11(1):77–97. 10.1142/S0219525908001519MATHMathSciNetView ArticleGoogle Scholar - Lian H, Pang H, Ge W: Triple positive solutions for boundary value problems on infinite intervals.
*Nonlinear Analysis: Theory, Methods & Applications*2007, 67(7):2199–2207. 10.1016/j.na.2006.09.016MATHMathSciNetView ArticleGoogle 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)00431-9MATHMathSciNetView ArticleGoogle Scholar - O'Regan D:
*Theory of Singular Boundary Value Problems*. World Scientific, River Edge, NJ, USA; 1994:xii+154.MATHView ArticleGoogle 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.MATHMathSciNetView ArticleGoogle Scholar

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