Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables

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doi:10.1155/2009/495972

Research Article

Asymptotic Behavior of Impulsive Infinite Delay

Difference Equations with Continuous Variables

Zhixia Ma

1

_{and Liguang Xu}

2

1_{College of Computer Science & Technology, Southwest University for Nationalities,}

Chengdu 610064, China

2_{Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China}

Correspondence should be addressed to Liguang Xu,[email protected]

Received 3 June 2009; Accepted 2 August 2009

Recommended by Mouﬀak Benchohra

A class of impulsive infinite delay diﬀerence equations with continuous variables is considered. By establishing an infinite delay diﬀerence inequality with impulsive initial conditions and using the properties of “-cone,” we obtain the attracting and invariant sets of the equations.

Copyrightq2009 Z. Ma and L. Xu. 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

Difference equations with continuous variables are difference equations in which the unknown function is a function of a continuous variable 1. These equations appear as natural descriptions of observed evolution phenomena in many branches of the natural sciences see, e.g., 2, 3. The book mentioned in 3 presents an exposition of some unusual properties of difference equations, specially, of difference equations with continuous variables. In the recent years, the asymptotic behavior and other behavior of delay difference equations with continuous variables have received much attention due to its potential appli-cation in various fields such as numerical analysis, control theory, finite mathematics, and computer science. Many results have appeared in the literatures; see, for example,1,4–7.

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this paper is to study the asymptotic behavior of impulsive infinite delay diﬀerence equations with continuous variables. By establishing an infinite delay diﬀerence inequality with impulsive initial conditions and using the properties of “-cone,” we obtain the attracting and invariant sets of the equations.

2. Preliminaries

Consider the impulsive infinite delay diﬀerence equation with continuous variable

xit aixit−τ1

By a solution of 2.1, we mean a piecewise continuous real-valued function xit defined on the interval−∞,∞which satisfies2.1for allt≥t0.

In the sequel, byΦi we will denote the set of all continuous real-valued functionsφi defined on an interval−∞,0, which satisfies the “compatibility condition”

φi0 aiφi−τ1

By the method of steps, one can easily see that, for any given initial functionφi ∈Φi, there exists a unique solutionxit, i∈ N, of2.1which satisfies the initial condition

xitt0 φit, t∈−∞,0, 2.3

this function will be called the solution of the initial problem2.1–2.3.

For convenience, we rewrite2.1and2.3into the following vector form

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where xt x1t, . . . , xntT, A0 diag{a1, . . . , an}, A aij_n_×_n,B bij_n_×_n, Pt pijt_n_×_n,I I1, . . . , InT,fx f1x1, . . . , fnxnT,gx g1x1, . . . , gnxnT,hx h1x1, . . . , hnxnT,Jkx J1kx, . . . , JnkxT, andφ φ1, . . . , φnT ∈Φ, in whichΦ Φ1, . . . ,ΦnT.

In what follows, we introduce some notations and recall some basic definitions. Let Rn_Rn

be the space ofn-dimensionalnonnegativereal column vectors,Rm×nRm×nbe the

set of m×nnonnegative real matrices, E be the n-dimensional unit matrix, and| · | be the Euclidean norm ofRn. For A, B ∈ Rm×n or A, B ∈ Rn,A ≥ BA ≤ B, A > B, A < B

means that each pair of corresponding elements ofAandBsatisfies the inequality “≥≤, > , <.”Especially,Ais called a nonnegative matrix ifA≥0, andzis called a positive vector if

z >0.NΔ{1,2, . . . , n}anden 1,1, . . . ,1T ∈Rn.

CX, Ydenotes the space of continuous mappings from the topological spaceXto the topological spaceY. Especially, letCΔC−∞,0,Rn

countable pointss∈Jand at these points

s∈J, ψsand ψs−exist, ψs ψs

ψsis piecewise continuous and satisfies

_∞

Definition 2.1. The setS⊂P Cis called a positive invariant set of2.4, if for any initial value

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Definition 2.2. The setS ⊂P Cis called a global attracting set of2.4, if for any initial value

φ∈P C, the solutionxt, t0, φsatisfies

distxt, t0, φ

, S−→0, ast−→∞, 2.9

where distφ, S infψ∈Sdistφ, ψ, distφ, ψ supθ∈−∞,0|φθ−ψθ|, forψ∈P C.

Definition 2.3. System 2.4 is said to be globally exponentially stable if for any solution

xt, t0, φ, there exist constantsξ >0 andκ0 >0 such that

_x

t, t0, φ≤κ0φe−ξt−t0, t≥t0. 2.10

Lemma 2.4See13,14. IfM∈Rn×n

andM<1, thenE−M−1≥0. Lemma 2.5La Salle14. Suppose thatM ∈ Rn×n

andM < 1, then there exists a positive vectorzsuch thatE−Mz >0.

ForM∈Rn×n

andM<1, we denote

ΩM {z∈Rn |E−Mz >0, z >0}, 2.11

which is a nonempty set byLemma 2.5, satisfying thatk1z1k2z2 ∈ΩMfor any scalars

k1 >0,k2>0, and vectorsz1, z2 ∈ΩM. SoΩMis a cone without vertex inRn, we call it a “-cone”12.

3. Main Results

In this section, we will first establish an infinite delay diﬀerence inequality with impulsive initial conditions and then give the attracting and invariant sets of2.4.

Theorem 3.1. Let P pij_n_×_n, W wij_n_×_n ∈ Rn×n, I I1, . . . , InT ∈ Rn, and Qt

qijt_n_×_n, where0≤qijt∈Le. DenoteQ qij_n_×_nΔ

_∞

0qijtdtn×nand letPWQ<1 and ut ∈ Rn _{be a solution of the following infinite delay di}_ﬀ_{erence inequality with the initial} conditionuθ∈P C−∞, t0,Rn:

ut≤P ut−τ1 Wut−τ2

_∞

0

Qsut−sdsI, t≥t0. 3.1

(a) Then

ut≤ze−λt−t0 _E−_P−_W−_Q−1_I, _t≥_t

0, 3.2

provided the initial conditions

uθ≤ze−λθ−t0 _E−_P−_W−_Q−1_I, _θ∈−∞_{, t}

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wherez z1, z2, . . . , znT ∈ΩPWQand the positive numberλ≤λ0is determined by the following inequality:

eλ

P eλτ1_Weλτ2

_∞

0

Qseλsds

−E

z≤0. 3.4

(b) Then

ut≤dE−P−W−Q−1I, t≥t0, 3.5

provided the initial conditions

uθ≤dE−P−W−Q−1I, d≥1, θ∈−∞, t0. 3.6

Proof. a: SincePWQ<1 andPWQ ∈Rn×n

, then, byLemma 2.5, there exists a

positive vectorz∈ΩP WQsuch thatE−PWQz >0. Using continuity and notingqijt∈Le, we know that3.4has at least one positive solutionλ≤λ0, that is,

n

j1

pijeλτ1wijeλτ2

_∞

0

qijseλsds

zj≤zi, i∈ N. 3.7

LetNΔ E−P−W−Q−1I,N N1, . . . , NnT, one can get thatE−P−W−QNI, or

n

j1

pijwijqij

NjIiNi, i∈ N. 3.8

To prove3.2, we first prove, for any givenε >0, whenuθ≤ze−λθ−t0_{N, θ}∈−∞_{, t} 0,

uit≤1ε

zie−λt−t0Ni

_Δ

yit, t≥t0, i∈ N. 3.9

If3.9is not true, then there must be at∗> t0and some integerrsuch that

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By using3.1,3.7–3.10, andqijt≥0, we have

which contradicts the first equality of3.10, and so3.9holds for allt≥ t0. Lettingε → 0, then3.2holds, and the proof of partais completed.

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which contradicts the first equality of3.13, and so3.12holds for allt≥t0. Lettingε → 0, then3.5holds, and the proof of partbis completed.

Remark 3.2. Suppose thatQt 0 in partaofTheorem 3.1, then we get15, Lemma 3.

In the following, we will obtain attracting and invariant sets of2.4by employing Theorem 3.1. Here, we firstly introduce the following assumptions.

A1For anyx∈Rn, there exist nonnegative diagonal matricesF, G, Hsuch that

fx≤Fx, gx≤Gx, hx≤Hx. 3.15

A2For anyx∈Rn, there exist nonnegative matricesRksuch that

Jkx≤Rkx, k1,2, . . . . 3.16

A3LetPWQ<1, where

P A0 AF, W BG, Q

_∞

0

Qsds, Qs PsH. 3.17

A4There exists a constantγsuch that

lnγk

tk−tk−1 ≤

γ < λ, k1,2, . . . , 3.18

where the scalarλsatisfies 0< λ≤λ0and is determined by the following inequality

eλ

P eλτ1_We λτ2

_∞

0

Qseλsds

−E

z≤0, 3.19

wherez z1, . . . , znT ∈ΩPWQ, and

γk≥1, γkz≥Rkz, k1,2, . . . . 3.20

A5Let

σ

∞

k1

lnσk<∞, k1,2, . . . , 3.21

whereσk≥1 satisfy

RkE−P−W−Q

−1

I≤σkE−P−W−Q

−1

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Theorem 3.3. If (A1)–(A5) hold, thenS{φ∈P C|φ_∞ ≤eσE−P−W−Q

−1

I}is a global attracting set of 2.4.

Proof. SincePWQ<1 andP , W, Q ∈Rn×n

, then, byLemma 2.5, there exists a positive

vectorz ∈ ΩPWQsuch thatE−PWQz > 0. Using continuity and noting

pijt∈Le, we obtain that inequality3.19has at least one positive solutionλ≤λ0. From2.4and conditionA1, we have

xt≤A0xt−τ1

Afxt−τ1

_Bg_x_t₋_τ

2

t

−∞Pt−shxsds

I

≤A0xt−τ1 AFxt−τ1 BGxt−τ2

_∞

0

PsHxt−sds I

Pxt−τ1Wxt−τ2

_∞

0

Qsxt−sds I,

3.23

wheretk−1≤t < tk, k1,2, . . . .

Since P W Q < 1 and P , W, Q ∈ Rn×n

, then, by Lemma 2.4, we can get

E−P−W−Q−1≥0, and soNΔ E−P−W−Q−1I ≥0.

For the initial conditions:xt0θ φθ,θ∈−∞,0, whereφ∈P C, we have

xt ≤κ0ze−λt−t0≤κ0ze−λt−t0N, t∈−∞, t0, 3.24

where

κ0

φ

min1≤i≤n{zi}

, z∈Ω

PWQ. 3.25

By the property of-cone andz ∈ΩPWQ, we haveκ0z∈ΩPWQ. Then, all the conditions of partaofTheorem 3.1are satisfied by3.23,3.24, and conditionA3, we derive that

xt≤κ0ze−λt−t0N, t∈t0, t1. 3.26

Suppose for allι1, . . . , k, the inequalities

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hold, whereγ0σ01. Then, from3.20,3.22,3.27, andA2, the impulsive part of2.4 satisfies that

xtk

Jk

xt−_k ≤Rkx

t−_k

≤Rk

γ0· · ·γk−1κ0ze−λtk−t0σ0· · ·σk−1N

≤γ0· · ·γk−1γkκ0ze−λtk−t0σ0· · ·σk−1σkN.

3.28

This, together with3.27, leads to

xt ≤γ0· · ·γk−1γkκ0ze−λt−t0σ0· · ·σk−1σkN, t∈−∞, tk. 3.29

By the property of-cone again, the vector

γ0· · ·γk−1γkκ0z∈Ω

PWQ. 3.30

On the other hand,

xt≤Pxt−τ1Wxt−τ2

_∞

0

Qtxt−sdsσ0, . . . , σkI, t /tk. 3.31

It follows from3.29–3.31and partaofTheorem 3.1that

xt≤γ0· · ·γk−1γkκ0ze−λt−t0σ0· · ·σk−1σkN, t∈tk, tk1. 3.32

By the mathematical induction, we can conclude that

xt≤γ0· · ·γk−1κ0ze−λt−t0σ0· · ·σk−1N, t∈tk−1, tk, k 1,2, . . . . 3.33

From3.18and3.21,

γk≤eγtk−tk−1, σ0· · ·σk−1≤eσ, 3.34

we can use3.33to conclude that

xt≤eγt1−t0· · ·_eγtk−1−tk−2_κ

0ze−λt−t0σ0· · ·σk−1N

≤κ0zeγt−t0e−λt−t0eσN

κ0ze−λ−γt−t0eσN, t∈tk−1, tk, k1,2, . . . .

3.35

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Theorem 3.4. If (A1)–(A3) withRk≤Ehold, thenS{φ∈P C|φ_∞≤E−P−W−Q

−1 I}

is a positive invariant set and also a global attracting set of 2.4.

Proof. For the initial conditions:xt0s φs,s∈−∞,0, whereφ∈S, we have

xt≤E−P−W−Q−1I, t∈−∞, t0. 3.36

By3.36and the partbofTheorem 3.1withd1, we have

xt≤E−P−W−Q−1I, t∈t0, t1. 3.37

Suppose for allι1, . . . , k, the inequalities

xt≤E−P−W−Q−1I, t∈tι−1, tι, 3.38

hold. Then, fromA2andRk≤E, the impulsive part of2.4satisfies that

xtk≤

Jk

xt−_k≤Rkx

t−_k≤Ext−_k≤E−P−W−Q−1I. 3.39

This, together with3.36and3.38, leads to

xt≤E−P−W−Q−1I, t∈−∞, tk. 3.40

It follows from3.40and the partbofTheorem 3.1that

xt≤E−P−W−Q−1I, t∈tk, tk1. 3.41

By the mathematical induction, we can conclude that

xt ≤E−P−W−Q−1I, t∈tk−1, tk, k1,2, . . . . 3.42

Therefore, S {φ ∈ P C | φ_∞ ≤ E−P−W−Q−1I}is a positive invariant set. Since

Rk ≤E, a direct calculation shows thatγk σk 1 andσ0 inTheorem 3.3. It follows from Theorem 3.3that the setSis also a global attracting set of2.4. The proof is complete.

For the caseI 0, we easily observe thatxt≡0 is a solution of2.4fromA1and A2. In the following, we give the attractivity of the zero solution and the proof is similar to that ofTheorem 3.3.

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Remark 3.6. IfJkx x, that is, they have no impulses in2.4, then byTheorem 3.4, we can obtain the following result.

Corollary 3.7. IfA1andA3hold, thenS {φ ∈ P C |φ_∞ ≤ E−P−W−Q

−1

I}is a positive invariant set and also a global attracting set of 2.4.

4. Illustrative Example

The following illustrative example will demonstrate the eﬀectiveness of our results.

Example 4.1. Consider the following impulsive infinite delay diﬀerence equations:

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It is easy to prove thatPWQ 5/6<1 and

1/24. Clearly, all conditions of Theorem 3.3 are satisfied. So S {φ ∈ P C | φ_∞ ≤ e1/24_E₋_P₋_W₋_Q−1_I_} set and also a global attracting set of4.1.

Case 3. IfI0 and letα1kα2k 1/3e0.04kandβ1kβ2k 2/3e0.04k, then

Clearly, all conditions of Corollary 3.5 are satisfied. Therefore, by Corollary 3.5, the zero solution of4.1is globally exponentially stable.

Acknowledgment

The work is supported by the National Natural Science Foundation of China under Grant 10671133.

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