Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables

doi:10.1155/2009/495972

Research Article

Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables

Zhixia Ma

and Liguang Xu

1College of Computer Science & Technology, Southwest University for Nationalities, Chengdu 610064, China

2Department 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.

However, besides the delay effect, an impulsive effect likewise exists in a wide variety of evolutionary process, in which states are changed abruptly at certain moments of time.

Recently, impulsive difference equations with discrete variable have attracted considerable attention. In particular, delay effect on the asymptotic behavior and other behaviors of impulsive difference equations with discrete variable has been extensively studied by many authors and various results are reported 8–12. However, to the best of our knowledge, very little has been done with the corresponding problems for impulsive delay difference equations with continuous variables. Motivated by the above discussions, the main aim of

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

x_it a_ix_it−τ₁ n j1

a_ijf_j

x_jt−τ₁ ⁿ

j1

b_ijg_j

x_jt−τ₂

_t

−∞p_ijt−shj

x_js

dsI_i, t /t_k, t≥t₀, x_it J_ik

x_i t⁻

, t≥t₀, tt_k, k1,2, . . . ,

2.1

whereai,Ii,aij, andbij i, j∈ Nare real constants,pij ∈L^ehere,NandL^ewill be defined later,τ1andτ2are positive real numbers.tk k1,2, . . .is an impulsive sequence such that t₁< t₂<· · ·,lim_{k→ ∞}t_k∞.f_j,g_j,h_j, andJ_ik:R → Rare real-valued functions.

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 functionsφ_i defined on an interval−∞,0, which satisfies the “compatibility condition”

φi0 aiφi−τ1

n j1

aijfj

φj−τ1 ⁿ

j1

bijgj

φj−τ2

₀

−∞pij−shj

φjs dsIi.

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

x_itt₀ φ_it, 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 xt A₀xt−τ₁ Afxt−τ₁ Bgxt−τ₂

_t

−∞Pt−shxsdsI, t /tk, t≥t0, xt J_k

x t⁻

, t≥t₀, tt_k, k1,2, . . . , xt0θ φθ, θ∈−∞,0,

2.4

where xt x1t, . . . , xnt^T, A0 diag{a1, . . . , an}, A aij_n×n,B bij_n×n, Pt pijt_n×n,I I1, . . . , I_n^T,fx f1x1, . . . , fnxn^T,gx g1x1, . . . , gnxn^T,hx h1x1, . . . , hnxn^T,J_kx J1kx, . . . , Jnkx^T, andφ φ1, . . . , φ_n^T ∈Φ, in whichΦ Φ1, . . . ,Φn^T.

In what follows, we introduce some notations and recall some basic definitions. Let RⁿRⁿbe the space ofn-dimensionalnonnegativereal column vectors,R^m×nR^m×n be the set of m×nnonnegative real matrices, E be the n-dimensional unit matrix, and| · | be the Euclidean norm ofRⁿ. For A, B ∈ R^m×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.N^Δ{1,2, . . . , n}ande_n 1,1, . . . ,1^T ∈Rⁿ.

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

P CJ,Rⁿ

⎧⎪

⎪⎨

⎪⎪

⎩ψ:J−→Rⁿ

ψsis continuous for all but at most countable pointss∈Jand at these points s∈J, ψsand ψs⁻exist, ψs ψs

⎫⎪

⎪⎬

⎪⎪

⎭, 2.5

whereJ⊂Ris an interval,ψsandψs⁻denote the right-hand and left-hand limits of the functionψs, respectively. Especially, letP C^ΔP C−∞,0,Rⁿ

L^e

⎧⎪

⎨

⎪⎩

ψs:R → R, whereR 0,∞

ψsis piecewise continuous and satisfies _∞

0

e^λ0s ψsds <∞, whereλ0 >0 is constant

⎫⎪

⎬

⎪⎭. 2.6

Forx∈Rⁿ,φ∈Cφ∈P C, andA∈R^n×nwe define x |x1|, . . . ,|xn|^T, φ_∞

φ1t

∞, . . . ,φnt_∞^T, φit_∞ sup

θ∈−∞,0

φ_itθ, i∈ N, A a_ij

n×n, 2.7

andAdenotes the spectral radius ofA.

For anyφ ∈ Cor φ ∈ P C, we always assume thatφ is bounded and introduce the following norm:

φ sup

−∞<θ≤0

φs. 2.8 Definition 2.1. The setS⊂P Cis called a positive invariant set of2.4, if for any initial value φ∈S, the solutionxt, t0, φ∈S,t≥t₀.

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

dist x

t, t0, φ , S

−→0, ast−→∞, 2.9

where distφ, S inf_ψ∈Sdistφ, ψ, distφ, ψ sup_{θ∈−∞,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∈R^n×n andM<1, thenE−M⁻¹≥0.

Lemma 2.5La Salle14. Suppose thatM ∈ R^n×n andM < 1, then there exists a positive vectorzsuch thatE−Mz >0.

ForM∈R^n×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 ∈ R^n×n , I I1, . . . , I_n^T ∈ Rⁿ, and Qt qijt_n×n, where 0≤q_ijt∈L^e. DenoteQ q_ijn×n^Δ _∞

0q_ijtdt_n×nand letPWQ <1 and ut ∈ Rⁿ be a solution of the following infinite delay difference inequality with the initial conditionuθ∈P C−∞, t0,Rⁿ:

ut≤P ut−τ₁ Wut−τ_{2 ∞}

0

Qsut−sdsI, t≥t₀. 3.1

(a) Then

ut≤ze^−λt−t0 E−P−W−Q ⁻¹I, t≥t₀, 3.2 provided the initial conditions

uθ≤ze^−λθ−t0 E−P−W−Q ⁻¹I, θ∈−∞, t0, 3.3

wherez z1, z2, . . . , zn^T ∈ΩPWQ and the positive numberλ≤λ0is determined by the following inequality:

e^λ

P e^λτ1We^λτ2 _∞

0

Qse^λsds

−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 ∈R^n×n , then, byLemma 2.5, there exists a positive vectorz∈ΩP WQ such thatE−PWQz > 0. Using continuity and notingq_ijt∈L^e, we know that3.4has at least one positive solutionλ≤λ₀, that is,

n j1

p_ije^λτ1w_ije^λτ2 _∞

0

q_ijse^λsds

z_j≤z_i, i∈ N. 3.7

LetN^Δ E−P−W−Q ⁻¹I,N N1, . . . , N_n^T, one can get thatE−P−W−QN I, or

n j1

pijwijqij

NjIiNi, i∈ N. 3.8

To prove3.2, we first prove, for any givenε >0, whenuθ≤ze^−λθ−t0N, θ∈−∞, t0,

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

u_rt^∗> y_rt^∗, u_it≤y_it, t∈−∞, t^∗, i∈ N. 3.10

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

urt^∗≤ⁿ

j1

prj1ε

zje^{−λt∗−τ1−t0}Nj

ⁿ

j1

wrj1ε

zje^{−λt∗−τ2−t0} Nj

ⁿ

j1

_∞

0

qrjs1ε

zje^{−λt∗−s−t0}Nj

dsIr

ⁿ

j1

prje^λτ1wrje^λτ2 _∞

0

qrjse^λsds

zj1εe^{−λt∗−t0}

ⁿ

j1

prjwrjqrj

Nj1ε 1εIr−εIr

≤1ε

z_re^{−λt∗−t0}N_r y_rt^∗,

3.11

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

bFor any given initial function:ut0θ φθ,θ∈−∞,0, whereφ∈P C, there is a constantd≥1 such thatφ_∞≤dN. To prove3.5, we first prove that

ut≤dN Λ x^Δ 1, . . . , x_n^T x, t≥t₀, 3.12

whereΛ E−P−W−Q ⁻¹enεε > 0 small enough, provided that the initial conditions satisfiesφ_∞≤x.

If3.12is not true, then there must be at^∗> t₀and some integerrsuch that

u_rt^∗> x_r, ut≤x, t∈−∞, t^∗. 3.13

By using3.1,3.8,3.13qijt≥0, andPWQ <1, we obtain that ut^∗≤

PWQ xI

PWQ

dN Λ I

≤d

PWQ NI

PWQ Λ

≤dN Λ x,

3.14

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 lnγ_k

tk−t_k−1 ≤γ < λ, k1,2, . . . , 3.18 where the scalarλsatisfies 0< λ≤λ₀and is determined by the following inequality

e^λ

P e ^λτ1We ^λτ2 _∞

0

Qse^λsds

−E

z≤0, 3.19

wherez z1, . . . , zn^T ∈ΩPWQ, and

γk≥1, γkz≥Rkz, k1,2, . . . . 3.20 A5Let

σ^∞

k1

lnσ_k<∞, k1,2, . . . , 3.21

whereσ_k≥1 satisfy

RkE−P−W−Q ⁻¹I≤σkE−P−W−Q ⁻¹I. 3.22

Theorem 3.3. If (A1)–(A5) hold, thenS{φ∈P C|φ_∞≤e^σE−P−W−Q ⁻¹I}is a global attracting set of 2.4.

Proof. SincePWQ <1 andP , W, Q ∈R^n×n , then, byLemma 2.5, there exists a positive vectorz ∈ ΩPWQ such thatE−PWQz > 0. Using continuity and noting pijt∈L^e, we obtain that inequality3.19has at least one positive solutionλ≤λ0.

From2.4and conditionA1, we have xt≤A0xt−τ₁

Afxt−τ₁

Bgxt−τ₂

t

−∞Pt−shxsds

I

≤A0xt−τ1 AFxt−τ1 BGxt−τ2

_∞

0

PsHxt−sds I

Pxt−τ1Wxt −τ2 _∞

0

Qsxt−sds I,

3.23

wheret_k−1≤t < t_k, k1,2, . . . .

Since P W Q < 1 and P , W, Q ∈ R^n×n , then, by Lemma 2.4, we can get E−P−W−Q ⁻¹≥0, and soN^Δ 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

min_1≤i≤n{zi}, z∈Ω

PWQ

. 3.25

By the property of-cone andz ∈ΩPWQ, we have κ₀z∈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

xt≤γ0· · ·γ_ι−1κ0ze^−λt−t0σ0· · ·σ_ι−1N, t∈tι−1, tι, 3.27

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

xtk Jk

x t⁻_k

≤Rkx t⁻_k

≤R_k

γ₀· · ·γ_k−1κ₀ze^−λtk−t0σ₀· · ·σ_k−1N

≤γ0· · ·γ_k−1γkκ0ze^−λtk−t0σ0· · ·σ_k−1σkN.

3.28

This, together with3.27, leads to

xt ≤γ₀· · ·γ_k−1γ_kκ₀ze^−λt−t0σ₀· · ·σ_k−1σ_kN, t∈−∞, tk. 3.29 By the property of-cone again, the vector

γ0· · ·γ_k−1γkκ0z∈Ω

PWQ

. 3.30

On the other hand,

xt≤Pxt−τ₁Wxt −τ_{2 ∞}

0

Qtxt−sdsσ₀, . . . , σ_kI, t /t_k. 3.31

It follows from3.29–3.31and partaofTheorem 3.1that

xt≤γ0· · ·γ_k−1γkκ0ze^−λt−t0σ0· · ·σ_k−1σkN, t∈tk, t_k1. 3.32

By the mathematical induction, we can conclude that

xt≤γ₀· · ·γ_k−1κ₀ze^−λt−t0σ₀· · ·σ_k−1N, t∈t_k−1, t_k, k 1,2, . . . . 3.33

From3.18and3.21,

γ_k≤e^{γtk−tk−1}, σ₀· · ·σ_k−1≤e^σ, 3.34 we can use3.33to conclude that

xt≤e^γt1−t0· · ·e^{γtk−1−tk−2}κ₀ze^−λt−t0σ₀· · ·σ_k−1N

≤κ₀ze^γt−t0e^−λt−t0e^σN

κ₀ze^{−λ−γt−t0}e^σN, t∈tk−1, t_k, k1,2, . . . .

3.35

This implies that the conclusion of the theorem holds and the proof is complete.

Theorem 3.4. If (A1)–(A3) withRk≤Ehold, thenS{φ∈P C|φ_∞≤E−P−W−Q ⁻¹I} 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, fromA2andR_k≤E, the impulsive part of2.4satisfies that xtk≤

J_k x

t⁻_k

≤R_kx t⁻_k

≤Ex t⁻_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, t_k1. 3.41

By the mathematical induction, we can conclude that

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

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

Corollary 3.5. IfA1−A4hold withI0, then the zero solution of2.4is globally exponentially stable.

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 ⁻¹I}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:

x₁t 1

4x₁t−1 1

12sinx1t−1 1

15x₂t−1 4

15|x2t−2| − _t

−∞e^−6t−s|x1s|ds2 x2t −1

4x2t−1 1

5sinx1t−1 1

6x2t−1 2

15|x1t−2|

_t

−∞e^−12t−s|x2s|ds3

, m /m_k, 4.1

with

x₁tk α_1kx₁ t⁻_k

−β_1kx₂ t⁻_k x2tk β2kx1

t⁻_k

α2kx2

t⁻_k

, 4.2

where αik and βik are nonnegative constants, and the impulsive sequence tkk 1,2, . . . satisfies:t₁ < t₂ < · · ·,lim_{k→ ∞}t_k ∞. For System4.1, we havep₁₁s −e^−6s,p₂₂s e^−12s, p₁₂s p₂₁s 0. So, it is easy to check that p_ijs ∈ L^e,i, j 1,2, provided that 0< λ0<1. In this example, we may letλ00.1.

The parameters ofA1–A3are as follows:

A0

⎛

⎜⎝ 1 4 0 0 −1

4

⎞

⎟⎠, A

⎛

⎜⎝ 1 12

1 1 15 5

1 6

⎞

⎟⎠, B

⎛

⎜⎝ 0 4 2 15 15 0

⎞

⎟⎠,

FGH

%1 0 0 1

&

, P

⎛

⎜⎝ 1 3

1 1 15 5

5 12

⎞

⎟⎠, W

⎛

⎜⎝0 4 2 15 15 0

⎞

⎟⎠,

Q

⎛

⎜⎝ 1 6 0 0 1 12

⎞

⎟⎠, Rk

%α1k β1k

β_2k α_2k

&

, PWQ

⎛

⎜⎝ 1 2

1 1 3 3

1 2

⎞

⎟⎠.

4.3

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

PWQ

'

z1, z2^T>0 2

3z1< z2 < 3 2z1

(

. 4.4

Letz 1,1^T∈ΩPWQ andλ0.01< λ₀which satisfies the inequality

e^λ

P e ^λWe ^2λ _∞

0

Qse^λsds

−E

z <0. 4.5

Letγkmax{α1kβ1k, α2kβ2k}, thenγksatisfyγkz≥Rkz,k1,2, . . . .

Case 1. Letα1kα2k 1/3e^1/25k,β1kβ2k 2/3e^1/25k, andtk−t_k−15k, then

γ_ke^1/25k ≥1, lnγ_k

t_k−t_k−1 lne^1/25k

5k 1

25^k×5k ≤0.008γ < λ. 4.6 Moreover, σk e^1/25k ≥ 1,σ )_∞

k1lnσk )_∞

k1lne^1/25k 1/24. Clearly, all conditions of Theorem 3.3 are satisfied. So S {φ ∈ P C | φ_∞ ≤ e^1/24E−P−W−Q ⁻¹I}

6e^1/24,6e^1/24Tis a global attracting set of4.1.

Case 2. Letα_1kα_2k 1/3e^1/2k andβ_1k β_2k0, thenR_k 1/3e^1/2kE≤E. Therefore, by Theorem 3.4,S{φ∈P C|φ_∞≤N E−P−W−Q ⁻¹I} 6,6^T is a positive invariant set and also a global attracting set of4.1.

Case 3. IfI0 and letα_1kα_2k 1/3e^0.04kandβ_1kβ_2k 2/3e^0.04k, then

γke^0.04k≥1, lnγ_k

t_k−t_k−1 lne^0.04k

5k 0.008γ < λ. 4.7

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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