Impulsive Periodic Boundary Value Problems for Dynamic Equations on Time Scale

doi:10.1155/2009/603271

Research Article

Impulsive Periodic Boundary Value Problems for

Dynamic Equations on Time Scale

Eric R. Kaufmann

Department of Mathematics & Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA

Correspondence should be addressed to Eric R. Kaufmann,[email protected]

Received 31 March 2009; Accepted 20 May 2009

Recommended by Victoria Otero-Espinar

LetTbe a periodic time scale with periodpsuch that 0, ti, T mp ∈ T, i 1,2, . . . , n, m ∈ N, and 0 < ti < ti1. Assume each ti is dense. Using Schaeffer’s theorem, we show that the

impulsive dynamic equationyΔt −atyσt ft, yt, t∈T, yt_i yt−_i Iti, yti, i 1,2, . . . , n, y0 yT,whereyt±_i limt→t±

iyt,yti yt −

i, andyΔis theΔ-derivative onT,

has a solution.

Copyrightq2009 Eric R. Kaufmann. 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.

1. Introduction

Due to their importance in numerous application, for example, physics, population dynamics, industrial robotics, optimal control, and other areas, many authors are studying

dynamic equations with impulse effects; see1–19 and references therein.

The primary motivation for this work are the papers by Kaufmann et al.9 and Li et

al.12 . In9 , the authors used a fixed point theorem due to Krasnosel’ski˘ıto establish the

existence theorems for the impulsive dynamic equation:

yΔ_{t −aty}σ_{t ft, yt, t∈0, T ∩T,}

yt

i

yt− i

Iti, yti, i1,2, . . . , n,

y0 0,

1.1

whereyt±_i limt→t±

In12 , the authors gave sufficient conditions for the existence of solutions for the impulsive periodic boundary value problem equation:

u_{t λut ft, ut,}

ut

k

ut− k

Ikutk, k1,2, . . . , p,

u0 uT,

1.2

whereλ ∈ R, λ /0, T > 0, and 0 t0 < t1 < · · · < tp < tp1 T. This paper extends and

generalized the above results to dynamic equations on time scales.

We assume the reader is familiar with the notation and basic results for dynamic

equations on time scales. While the books20,21 are indispensable resources for those who

study dynamic equations on time scales, these manuscripts do not explicitly cover the concept of periodicity. The following definitions are essential in our analysis.

Definition 1.1see8 . We say that a time scaleTisperiodicif there exist ap >0 such that if

t∈T, thent±p∈T. ForT/R, the smallest positivepis called theperiodof the time scale.

Example 1.2. The following time scales are periodic:

1ThZhas periodph,

2TR,

3T∞_k−∞2k−1h,2kh , h >0 has periodp2h,

4T{tk−qm:k∈Z, m∈N0}, where 0< q <1, has periodp1.

Remark 1.3. All periodic time scales are unbounded above and below.

Definition 1.4. LetT/Rbe a periodic time scale with periodp. We say that the functionf :

T → Ris periodic with periodTif there exists a natural numbernsuch thatT np,ft±T

ftfor allt∈TandT is the smallest number such thatft±T ft.

IfTR, we say thatfis periodic with periodT >0 ifTis the smallest positive number

such thatft±T ftfor allt∈T.

Remark 1.5. If T is a periodic time scale with period p, then σt ± np σt ± np.

Consequently, the graininess functionμsatisfiesμt±np σt±np−t±np σt−tμt

and so, is a periodic function with periodp.

LetTbe a periodic time scale with period psuch that 0, ti, T ∈ T, fori 1,2, . . . , n,

where T mp for some m ∈ N, 0 < ti < ti1, and assume that each ti is dense inT for

eachi 1,2, . . . , n. We show the existence of solutions for the nonlinear periodic impulsive dynamic equation:

yΔ_{t −aty}σ_{t ft, yt, t∈T,}

yt

i

yt− i

Iti, yti, i1,2, . . . , n,

y0 yT,

1.3

whereyt±_i limt→t±

iyt,andyti yt

−

i. Define0, T {t∈T: 0≤t≤T}and note that

InSection 2we present some preliminary ideas that will be used in the remainder of

the paper. InSection 3we give sufficient conditions for the existence of at least one solution

of the nonlinear problem1.3.

2. Preliminaries

In this section we present some important concepts found in 20, 21 that will be used

throughout the paper. We also define the space in which we seek solutions, state Schaeffer’s

theorem, and invert the linearized dynamic equation.

A functionp:T → Ris said to beregressiveprovided 1μtpt/0 for allt∈Tκ. The

set of all regressive rd-continuous functionsf:T → Ris denoted byR.

Letp∈ Randμt/0 for allt∈T. Theexponential functiononT, defined by

ept, s exp

t

s 1

μz Log

1μzpzΔz

, 2.1

is the solution to the initial value problem yΔ pty, ys 1. Other properties of the

exponential function are given in the following lemma,20, Theorem 2.36.

Lemma 2.1. Letp∈ R. Then

ie0t, s≡1andept, t≡1; iiepσt, s 1μtptept, s;

iii1/ept, s e pt, swhere, pt −pt/1μtpt; ivept, s 1/eps, t e ps, t;

vept, seps, r ept, r; vi 1/ep·, sΔ−pt/eσp·, s.

Definetn1≡Tand letJ0 0, t1 . Fori1,2, . . . , n, letJi ti, ti1 . Define

PCy:T−→R|yt±T yt, y∈CJi, yt±_i exist andyt−_iyti, i1, . . . , n ,

2.2

and

PC1_{y:T → R|yt±T yt, y∈C}1_J

i, i1, . . . , n

2.3

whereCJiis the space of all real-valued continuous functions onJi, andC1Jiis the

space of all continuously delta-differentiable functions onJi. The setPC is a Banach space

when it is endowed with the supremum norm:

umax

0≤i≤n{ui}, 2.4

We employ Schaeffer’s fixed point theorem, see 22 , to prove the existence of a periodic solution.

Theorem 2.2Schaeffer’s Theorem. LetSbe a normed linear space and let the operatorF:S → S

be compact. Define

HF y∈S|yμFy, μ∈0,1 . 2.5

Then either

ithe setHFis unbounded, or

iithe operatorFhas a fixed point inS.

The following conditions hold throughout the paper:

Aa∈ Ris periodic with periodT;atT atfor allt∈T.

Ff ∈CT×R,Rand for allt∈T,ftT, ytT ft, yt.

Furthermore, to ensure that the boundary value problem is not at resonance, we assume that

ηe _aT,0<1.

Consider the linear boundary value problem:

yΔ_{t −aty}σ_{t ζt, t∈T,}

yt

i

yt− i

Iti, yti, i1,2, . . . , n,

y0 yT,

2.6

whereζ∈PC. Our first result inverts the operator2.6.

Lemma 2.3. The functiony∈PC1_{is a solution of2.6if and only ify∈PCis a solution of}

yt T

0

Gt, sζsΔsn i1

Gt, tiIti, yti, 2.7

where

Gt, s 1

1−η

⎧ ⎨ ⎩

e _at, s, 0≤s≤t≤T,

ηe _at, s, 0≤t < s≤T. 2.8

Proof. It is easy to see that ify∈PC1_{is a solution of2.6, then fort∈0, T ,we have}

yt e _at,0y0

_t

0

e _at, sζsΔs n {i|ti≤t}

Apply the periodic boundary conditiony0 yTto obtain

We can rewrite this equation as follows:

yt e at,0

3. The Nonlinear Problem

In this section we give sufficient conditions for the existence of periodic solutions of1.3. To

this end, define the operatorN:PC → PCby

Nyt T

0

Gt, sfs, ysΔsn i1

Gt, tiIti, yti. 3.1

Thenyis a solution of1.3if and only ifyis a fixed point ofN. A standard application of

the Arzel`a-Ascoli theorem yields thatNis compact.

Our first result is an existence and uniqueness theorem.

Theorem 3.1. Suppose there exist constantsEi, i1, . . . , n,andLfor which

ft, y−ft, x≤Ly−x, ∀t∈T, 3.2

and

Iti, yti−Iti, xti≤Eiyti−xti, i1,2, . . . , n, 3.3

and such that

max t∈0,T

L

T

0

|e at, s|Δs n

i1

Ei|e at, ti|

<1−η. 3.4

Then there exists a unique solution to1.3.

Proof. We will show that there exists a unique solution yt of 3.1. By Lemma 2.3 this

solution is the unique solution of1.3.

Lety, x∈PC. Then for allt∈T

Lyt−Lxt≤ T

0

|Gt, s|fs, ys−fs, xsΔs

n i1

|Gt, ti|Iti, yti−Iti, xti

≤ y−x

1−η

L

_T

0

e _at, sΔsn i1

Ei|e at, ti|

<y−x.

3.5

Hence,Ly−Lx ≤ y−x. By the Contraction Mapping Principal, there exists a unique

Consequently,

Schaeffer’s theorem, the operatorNhas a fixed point. This fixed point is a solution of1.3

and the proof is complete.

In our next theorem we assume thatf andI are sublinear at infinity with respect to

the second variable.

Theorem 3.3. Assume that

F1lim|y| → ∞ft, y/y 0, uniformly, and Ilim|y| → ∞It, y/y 0, uniformly.

Then there exists at least one solution of the boundary value problem1.3.

Proof. Suppose that the set

as k → ∞, which contradicts vk 1 for all k. Thus the set HN is bounded. By

Theorem 2.2, the operatorN : PC → PC has a fixed point. This fixed point is a solution

of1.3and the proof is complete.

The following corollary is an immediate consequences ofTheorem 3.3

Corollary 3.4. Assume thatfandIare bounded. Then there exists at least one solution of 1.3.

References

1 D. D. Ba˘ınov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Application, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989.

2 M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “On first order impulsive dynamic equations on time scales,”Journal of Difference Equations and Applications, vol. 10, no. 6, pp. 541–548, 2004.

3 F. Geng, Y. Xu, and D. Zhu, “Periodic boundary value problems for first-order impulsive dynamic equations on time scales,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 4074– 4087, 2008.

4 J. R. Graef and A. Ouahab, “Extremal solutions for nonresonance impulsive functional dynamic equations on time scales,”Applied Mathematics and Computation, vol. 196, no. 1, pp. 333–339, 2008. 5 D. Guo, “Existence of positive solutions for nth-order nonlinear impulsive singular

integro-differential equations in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2727–2740, 2008.

6 D. Guo, “A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space,”Applied Mathematics and Computation, vol. 125, no. 1, pp. 59–77, 2002.

7 D. Guo, “Multiple positive solutions of a boundary value problem for nth-order impulsive integro-differential equations in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 4, pp. 618–641, 2005.

8 E. R. Kaufmann and Y. N. Raffoul, “Periodic solutions for a neutral nonlinear dynamical equation on a time scale,”Journal of Mathematical Analysis and Applications, vol. 319, no. 1, pp. 315–325, 2006. 9 E. R. Kaufmann, N. Kosmatov, and Y. N. Raffoul, “Impulsive dynamic equations on a time scale,”

Electronic Journal of Differential Equations, vol. 2008, no. 67, pp. 1–9, 2008.

10 E. R. Kaufmann, N. Kosmatov, and Y. N. Raffoul, “A second-order boundary value problem with impulsive effects on an unbounded domain,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 9, pp. 2924–2929, 2008.

11 V. Lakshmikantham, D. D. Ba˘ınov, and P. S. Simeonov,Theory of Impulsive Differential Equations, vol. 6 ofSeries in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1994.

12 J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 226–236, 2007. 13 X. Liu, “Nonlinear boundary value problems for first order impulsive integro-differential equations,”

Applicable Analysis, vol. 36, no. 1-2, pp. 119–130, 1990.

14 J. J. Nieto, “Basic theory for nonresonance impulsive periodic problems of first order,”Journal of Mathematical Analysis and Applications, vol. 205, no. 2, pp. 423–433, 1997.

15 C. Pierson-Gorez, “Impulsive differential equations of first order with periodic boundary conditions,” Differential Equations and Dynamical Systems, vol. 1, no. 3, pp. 185–196, 1993.

16 Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 1997.

17 A. M. Samo˘ılenko and N. A. Perestyuk,Impulsive Differential Equations, vol. 14 ofWorld Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995. 18 A. S. Vatsala and Y. Sun, “Periodic boundary value problems of impulsive differential equations,”

Applicable Analysis, vol. 44, no. 3-4, pp. 145–158, 1992.

20 M. Bohner and A. Peterson,Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨auser, Boston, Mass, USA, 2001.

21 M. Bohner and A. Peterson,Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2003.