- Research Article
- Open Access

# Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter

- Da-Bin Wang
^{1}Email author and - Wen Guan
^{1}

**2009**:830247

https://doi.org/10.1155/2009/830247

© D.-B.Wang and W.Guan. 2009

**Received:**13 February 2009**Accepted:**14 May 2009**Published:**22 June 2009

## Abstract

By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.

## Keywords

- Dynamic Equation
- Fixed Point Theorem
- Real Banach Space
- Scale Interval
- Impulsive Differential Equation

## 1. Introduction

Let be a time scale, that is, is a nonempty closed subset of . Let be fixed and be points in , an interval denoting time scales interval, that is, Other types of intervals are defined similarly. Some definitions concerning time scales can be found in [1–5].

where is a positive parameter, , is right-dense continuous, , and for each and represent the right and left limits of at .

where is a time scale which has at least finitely-many right-dense points, is regressive and right-dense continuous, is given function, . The paper [21] obtained the existence of one solution to problem (1.2) by using the nonlinear alternative of Leray-Schauder type.

They proved the existence of one solution to the problem (1.3) by applying Schaefer's fixed point theorem and the nonlinear alternative of Leray-Schauder type.

In [26], Li and Shen studied the problem (1.3). Some existence results to problem (1.3) are established by using a fixed point theorem, which is due to Krasnoselskii and Zabreiko, and the Leggett-Williams fixed point theorem.

In [27], the first author studied the problem (1.1) when . The existence of positive solutions to the problem (1.1) was obtained by means of the well-known Guo-Krasnoselskii fixed point theorem.

By using the fixed point index, some existence, multiplicity and nonexistence criteria of positive solutions to the problem (1.4) were obtained for suitable .

Motivated by the results mentioned above, in this paper, we shall show that the problem (1.1) has at least three positive solutions for suitable by using the Leggett-Williams fixed point theorem [29]. We note that for the case and problem (1.1) reduces to the problem studied by [30].

In the remainder of this section, we state the following theorem, which are crucial to our proof.

Theorem 1.1 (see [29]).

## 2. Preliminaries

Throughout the rest of this paper, we always assume that the points of impulse are right-dense for each

where is the restriction of to and

with the norm Then X is a Banach space.

Definition 2.1.

and the periodic boundary condition

Lemma 2.2.

Proof.

Since the method is similar to that of in [27, Lemma 3.1], we omit it here.

Lemma 2.3.

Proof.

It is obvious, so we omit it here.

Let

where It is not difficult to verify that is a cone in

By [27, Lemmas 3.3 and 3.4], it is easy to see that is completely continuous.

## 3. Main Result

Notation 1.

Theorem 3.1.

Assume that there exists a number such that the following conditions:

Proof.

Let it is easy to see that is a nonnegative continuous concave functional on such that

First, we assert that there exists such that is completely continuous.

In fact, by the condition
of (H_{2}), there exist
and
such that

Take then the set is a bounded set. According to that is completely continuous, then maps bounded sets into bounded sets and there exists a number such that

If we deduce that is completely continuous. If then from (3.4), we know that for any and hold. Then we have is completely continuous. Take then and are completely continuous.

Second, we assert that and for all

In fact, take so Moreover, for then and we have

Third, we assert that there exist such that if

Indeed, by the condition
of (H_{2}), there exist
and
such that

Finally, we assert that if and

To sum up, all the hypotheses of Theorem 1.1 are satisfied by taking Hence has at least three fixed points, that is, the problem (1.1) has at least three positive solutions and such that

Corollary 3.2.

Using (H_{3})
instead of (H_{2}) in Theorem 3.1, the conclusion of Theorem 3.1 remains true.

## 4. Example

## Declarations

### Acknowledgment

The authors express their gratitude to the anonymous referee for his/her valuable suggestions.

## Authors’ Affiliations

## References

- Agarwal RP, Bohner M:
**Basic calculus on time scales and some of its applications.***Results in Mathematics*1999,**35**(1-2):3-22.MathSciNetView ArticleMATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Hilger S:
**Analysis on measure chains-a unified approach to continuous and discrete calculus.***Results in Mathematics*1990,**18**(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Sivasundaram S, Kaymakcalan B:
*Dynamic Systems on Measure Chains, Mathematics and Its Applications*.*Volume 370*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.View ArticleGoogle Scholar - Baĭnov DD, Simeonov PS:
*Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood Series: Mathematics and Its Applications*. Ellis Horwood, Chichester, UK; 1989:255.Google Scholar - Bainov DD, Simeonov PS:
*Impulsive Differential Equations: Periodic Solutions and Applications*. Longman Scientific and Technical, Harlow, UK; 1993.MATHGoogle 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 - Agarwal RP, O'Regan D:
**Multiple nonnegative solutions for second order impulsive differential equations.***Applied Mathematics and Computation*2000,**114**(1):51-59. 10.1016/S0096-3003(99)00074-0MathSciNetView ArticleMATHGoogle 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)00381-8MathSciNetView ArticleMATHGoogle Scholar - He Z, Zhang X:
**Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions.***Applied Mathematics and Computation*2004,**156**(3):605-620. 10.1016/j.amc.2003.08.013MathSciNetView ArticleMATHGoogle Scholar - Li J-L, Shen J-H:
**Existence of positive periodic solutions to a class of functional differential equations with impulses.***Mathematica Applicata*2004,**17**(3):456-463.MathSciNetMATHGoogle 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.2005.04.005MathSciNetView ArticleMATHGoogle Scholar - Li J, Shen J:
**Positive solutions for first order difference equations with impulses.***International Journal of Difference Equations*2006,**1**(2):225-239.MathSciNetMATHGoogle Scholar - Li Y, Fan X, Zhao L:
**Positive periodic solutions of functional differential equations with impulses and a parameter.***Computers & Mathematics with Applications*2008,**56**(10):2556-2560. 10.1016/j.camwa.2008.05.007MathSciNetView ArticleMATHGoogle 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.1997.5207MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ:
**Impulsive resonance periodic problems of first order.***Applied Mathematics Letters*2002,**15**(4):489-493. 10.1016/S0893-9659(01)00163-XMathSciNetView ArticleMATHGoogle Scholar - Nieto JJ:
**Periodic boundary value problems for first-order impulsive ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2002,**51**(7):1223-1232. 10.1016/S0362-546X(01)00889-6MathSciNetView ArticleMATHGoogle Scholar - Vatsala AS, Sun Y:
**Periodic boundary value problems of impulsive differential equations.***Applicable Analysis*1992,**44**(3-4):145-158. 10.1080/00036819208840074MathSciNetView ArticleMATHGoogle Scholar - Belarbi A, Benchohra M, Ouahab A:
**Existence results for impulsive dynamic inclusions on time scales.***Electronic Journal of Qualitative Theory of Differential Equations*2005,**2005**(12):1-22.MathSciNetView ArticleGoogle Scholar - Benchohra M, Henderson J, Ntouyas SK, Ouahab A:
**On first order impulsive dynamic equations on time scales.***Journal of Difference Equations and Applications*2004,**10**(6):541-548. 10.1080/10236190410001667986MathSciNetView ArticleMATHGoogle 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.2004.02.057MathSciNetView ArticleMATHGoogle Scholar - Geng F, Xu Y, Zhu D:
**Periodic boundary value problems for first-order impulsive dynamic equations on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(11):4074-4087. 10.1016/j.na.2007.10.038MathSciNetView ArticleMATHGoogle Scholar - Graef JR, Ouahab A:
**Extremal solutions for nonresonance impulsive functional dynamic equations on time scales.***Applied Mathematics and Computation*2008,**196**(1):333-339. 10.1016/j.amc.2007.05.056MathSciNetView ArticleMATHGoogle Scholar - Henderson J:
**Double solutions of impulsive dynamic boundary value problems on a time scale.***Journal of Difference Equations and Applications*2002,**8**(4):345-356. 10.1080/1026190290017405MathSciNetView ArticleMATHGoogle Scholar - Li J, Shen J:
**Existence results for second-order impulsive boundary value problems on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(4):1648-1655. 10.1016/j.na.2008.02.047MathSciNetView ArticleMATHGoogle Scholar - Wang D-B:
**Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales.***Computers & Mathematics with Applications*2008,**56**(6):1496-1504. 10.1016/j.camwa.2008.02.038MathSciNetView ArticleMATHGoogle Scholar - Sun J-P, Li W-T:
**Positive solutions to nonlinear first-order PBVPs with parameter on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(3):1133-1145. 10.1016/j.na.2008.02.007MathSciNetView ArticleMATHGoogle Scholar - Leggett RW, Williams LR:
**Multiple positive fixed points of nonlinear operators on ordered Banach spaces.***Indiana University Mathematics Journal*1979,**28**(4):673-688. 10.1512/iumj.1979.28.28046MathSciNetView ArticleMATHGoogle Scholar - Sun J-P, Li W-T:
**Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales.***Computers & Mathematics with Applications*2007,**54**(6):861-871. 10.1016/j.camwa.2007.03.009MathSciNetView ArticleMATHGoogle Scholar

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