Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 495972,13 pages doi:10.1155/2009/495972

Research Article

Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables

Zhixia Ma1 and Liguang Xu2

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, xlg132@126.com

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 "9-cone," we obtain the attracting and invariant sets of the equations.

Copyright © 2009 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 application 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 "9-cone," we obtain the attracting and invariant sets of the equations.

2. Preliminaries

Consider the impulsive infinite delay difference equation with continuous variable

Xi{t) = aiXiit - Ti) + ^ aijfj (xj (t - Ti)) bijgj (xj (t - T2))

j=i j=i

+ pij(t - s)hj(xj(s))ds + Ii, t/ tk, t > t0,

Xi(t) = Jik(Xi(t-)), t > to, t = tk, k = i,2,...,

where air Iir aij, and bij (i,j e N) are real constants, pij e Le (here, N and Le will be defined later), t1 and t2 are positive real numbers. tk (k = i, 2,...) is an impulsive sequence such that t1 <t2 < ■■■ ,limk= go. fj, gj,hj, and Jik: R ^ R are real-valued functions.

By a solution of (2.1), we mean a piecewise continuous real-valued function xi(t) defined on the interval (-g, g) which satisfies (2.1) for all t > t0.

In the sequel, by ®i we will denote the set of all continuous real-valued functions fa defined on an interval (-g,0], which satisfies the "compatibility condition"

fai(0) = aifai(-T1^) + aijfj (<pj (-T1^ bijgj (<pj (-T2)) ^ Pij (-s)hj (<pj (s)) ds + Ii.

j=1 j=i J -g

By the method of steps, one can easily see that, for any given initial function e Oi, there exists a unique solution xt (t), i e N, of (2.1) which satisfies the initial condition

xi(t + to) = fai(t), t e (-g,0], (2.3)

this function will be called the solution of the initial problem (2.1)-(2.3).

For convenience, we rewrite (2.1) and (2.3) into the following vector form

x(t) = Aox(t - Ti) + Af (x(t - Ti)) + Bg(x(t - T2))

P(t - s)h(x(s))ds + I, t = tk, t > to,

x(t) = Jk(x(t~)), t > to, t = tk, k = 1,2, x(to + 0) = 0(0), 0 e (-g,0],

where x(t) = (xi(t),... ,xn(t))T, Ao = diag{ai,.../a„}/ A = (atj )nxn, B = (bij )nxn, P (t) =

(Pij (t))nxn'1 = (h,...Jn)T, f(x) = (fi(xi ),...,fn(xn))T, g(x) = (gi(xi).....gn(xn))T, h(x) =

(hi(xi)/.../hn(Xn))T, Jk(x) = (Jik(x),...,Jnk(x))T, and <p = (<pi,...,<pn)T e in which ® = (®i.....On)T.

In what follows, we introduce some notations and recall some basic definitions. Let Rn(Rn) be the space of n-dimensional (nonnegative) real column vectors, Rmxn(Rmxn) be the set of m x n (nonnegative) real matrices, E be the n-dimensional unit matrix, and | ■ | be the Euclidean norm of Rn. For A,B e Rmxn or A,B e Rn, A > B (A < B,A > B,A < B) means that each pair of corresponding elements of A and B satisfies the inequality "> (<, > ,<)."Especially, A is called a nonnegative matrix if A > 0, and z is called a positive vector if z > 0. N = {i,2,...,n} and en = (i,i,...,i)T e Rn.

C [X, Y] denotes the space of continuous mappings from the topological space X to the

topological space Y. Especially, let C = C[(-ro,0],Rn]

PC [J, R"] =

f(s) is continuous for all but at most countable points s e J and at these points s e J, f (s+) and f(s~) exist, f (s) = f(s+)

where J c R is an interval, y(s+) and ) denote the right-hand and left-hand limits of the function y(s), respectively. Especially, let PC = PC[(-<x, 0],R"]

f (s) : R+ ^ R, where R+ = [0, œ)

f (s) is piecewise continuous and satisfies ekoS |f (s)\ds < œ, where 10 > 0 is constant

For x e R", $ e C ($ e PC), and A e R"x" we define

[x]+ = (|xi|.....\X"\)T, [$]+œ = ([h(t)]+œ.....[$"(t)]+œ)T,

m)]+œ = sup \$i(t + d)\, i e N, [A]+ = (K\)nxn,

de(-œ00]

and 9(A) denotes the spectral radius of A.

For any $ e C or $ e PC, we always assume that $ is bounded and introduce the following norm:

= suP \Hs) I-

Definition 2.1. The set S c PC is called a positive invariant set of (2.4), if for any initial value $ e S, the solution x(t,t0,$) e S, t > t0.

Definition 2.2. The set S c PC is called a global attracting set of (2.4), if for any initial value $ e PC, the solution x(t,t0,$) satisfies

dist(x(t,t0,$),S) 0, as t (2.9)

where dist($, S) = inff eSdist($, f), dist($, f) = sup0e(-OT 0] \$(0) - f(0) |, for f e PC.

Definition 2.3. System (2.4) is said to be globally exponentially stable if for any solution x(t, t0, $), there exist constants 0 and K0 > 0 such that

|x(t,to,$)|< K0\\4>\\e-t(t-t0), t > t0. (2.10)

Lemma 2.4 (See [13,14]). If M e R?xn and q(M) < 1, then (E - M)-1 > 0.

Lemma 2.5 (La Salle [14]). Suppose that M e Rnxn and g(M) < 1, then there exists a positive vector z such that (E - M)z > 0.

For M e Rn*n and g(M) < 1, we denote

Q?(M) = {z e Rn \ (E - M)z > 0,z> 0}, (2.11)

which is a nonempty set by Lemma 2.5, satisfying that k1zi n k2z2 e Q?(M) for any scalars ki > 0, k2 > 0, and vectors z1, z2 e Q^(M). So Q?(M) is a cone without vertex in R", we call it a "9-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 of (2.4).

Theorem 3.1. Let P = (pt])n*n,W = (wlj)n*n e Rn+*", I = (I1.....In)T e R+, and Q(t) =

(qtj(t))nxn, where 0 < qtj(t) e Le. Denote Q = (qtj)nxn = (J7qtj(t)dt)n*n and let q(P n W n Q) < 1 and u(t) e Rn be a solution of the following infinite delay difference inequality with the initial condition u(0) e PC[(-G,t0],Rn]:

¡•CO

u(t) < Pu(t - T1)n Wu(t - T2)n\ Q(s)u(t - s)ds n I, t > t0. (3.1)

(a) Then

u(t) < ze-X(i-i0) n(E - P - W - Qyh, t > t0, (3.2)

provided the initial conditions

u(0) < ze-X(6-i0) n(E - P - W - Qfh, 0 e (-*>, t],

where z = (zi, z2,. .. ,zn ) e q.q(P + W + Q) and the positive number X < X0 is determined by the following inequality:

^PeiT1 + WeXT2 + j Q(s)eXsds

eXsds)-E)z< 0.

(b) Then

u(t) < d(E - P - W - Q) I, t > ÎQ

provided the initial conditions

u(d) < d(E - P - W - Q) I, d > 1,6 e (-œ,to].

Proof. (a): Since g(P + W + Q) < 1 and P + W + Q e R+xn, then, by Lemma 2.5, there exists a positive vector z e q.q(P + W + Q) such that (E - (P + W + Q))z > 0. Using continuity and noting qij (t) e Le, we know that (3.4) has at least one positive solution X < X0, that is,

PijelT1 + WijelT2 +

¡•CO

\ qij (s)<

zj < zi, i e N.

Let N = (E - P - W - Q) I, N = (Ni,...,Nn)T, one can get that (E - P - W - Q)N = I, or

^ (pij + wij + jNj + Ii = Ni, i e N. j=i

To prove (3.2), we first prove, for any given e> 0, when u(d) < ze X(e to) + N,d e (-m, t0],

ui(t) < (1 + e) [;zie-X(t-t0) + N^ = y(t), t > t0, i e N. (3.9)

If (3.9) is not true, then there must be a t* > t0 and some integer r such that

ur(t*) > yr(t*), ui(t) < yi(t), t e (-œ, t*)r i e N.

(3.10)

By using (3.1), (3.7)-(3.10), and qtj(t) > 0, we have n

ur(i*) < £pj (1 + e) [zje-l(t"-T1-t0) + Nj\ j=1

+ 2Wrj (1 + e) [zje-A(t*-T2-t0) + Nj] j=i

+ X qrj(s)(1 + e) [zje-A(t*-s-t0) + Nj]ds + Ir

j=1J0 L J

= E\PrjelTl + WrjeXT2 + J qrj(s)eXsdsjzj(1 + e)e^A(f-to)

(3.11)

+ X (Prj + + tfrj)Nj(1 + e) + (1 + e)Ir - eIr

< (1 + e) [zre-A(i*-io) + Nr]

= y (f),

which contradicts the first equality of (3.10), and so (3.9) holds for all t > t0. Letting e ^ 0, then (3.2) holds, and the proof of part (a) is completed.

(b) For any given initial function: u(t0 + 0) = $(6), 0 e (-to,0], where $ e PC, there is a constant d > 1 such that [$] + < dN. To prove (3.5), we first prove that

u(t) < dN + A = (X1,...,Xn)T = x, t > to, (3.12)

where A = (E - P - W - Q) ene (e > 0 small enough), provided that the initial conditions satisfies [$] + < x.

If (3.12) is not true, then there must be a t* > t0 and some integer r such that

Ur (t*) >Xr, u(t) < x, t e (-<x>,f). (3.13)

By using (3.1), (3.8), (3.13) qij(t) > 0, and ç(P + W + Q) < 1, we obtain that

u(f) < (P + W + Q)x +1

= (P + W + Q) (dN + A) + I

< d[(P + W + Q)N + l] + (P + W + Q)A (3.14)

< dN + A = x,

which contradicts the first equality of (3.13), and so (3.12) holds for all t > t0. Letting e ^ 0, then (3.5) holds, and the proof of part (b) is completed. □

Remark 3.2. Suppose that Q(t) = 0 in part (a) of Theorem 3.1, then we get [15, Lemma 3].

In the following, we will obtain attracting and invariant sets of (2.4) by employing Theorem 3.1. Here, we firstly introduce the following assumptions.

(A1) For any x e R", there exist nonnegative diagonal matrices F, G, H such that

[/(x)]+ < F[x]+, [g(x)] + < G[x]+, [h(x)]+ < H[x]+. (3.15)

(A2) For any x e R", there exist nonnegative matrices Rk such that

[Jk(x)]+ < Rk[x]+, k = 1,2..........(3.16)

(A3) Let q(P + W + Q) < 1, where

¡•CO

P = [Ao]+ + [A]+F, W =[B]+G, Q ^ Q(s)ds, Q(s) = [P(s)]+H. (3.17)

(A4) There exists a constant j such that

^ lnY < j < X, k = 1,2,..., (3.18)

tk - tk-1

where the scalar X satisfies 0 < X < X0 and is determined by the following inequality

('eX(peXT1 + WeXT2 + j Q(s)eXsds^ - E^z < 0, (3.19)

where z = (zi,..., z„)J e Q.^(P + W + Q), and

Yk > 1, YkZ > Rkz, k = 1,2,.... (3.20)

(A5) Let

a = ^ lnOk< go, k = 1,2,..., (3.21)

where ak > 1 satisfy

Rk (E - P - W - Q)-1 [I ]+ < ak (E - P - W - Q)-1 [I ]+. (3.22)

Theorem 3.3. If(A1)-(A5) hold, then S = e PC | < ea(E - P - W - Q) } is a global attracting set of (2.4).

Proof. Since q(P + W + Q) < 1 and P,W,Q e W+xn, then, by Lemma 2.5, there exists a positive vector z e qq(P + W + Q) such that (E - (P + W + Q))z > 0. Using continuity and noting pij (t) e Le, we obtain that inequality (3.19) has at least one positive solution X < X0. From (2.4) and condition (A1), we have

[x(t)]+ < [A0x(t - n)]+ + [Af (x(t - n))]+ + [Bg(x(t - T2))]+

P(t - s)h(x(s))ds

< [A)] + [x(t - n)r + [A]+F[(x(t - n))]+ + [B]+G[(x(t - n)] + (3.23)

¡•CO

H [P(s)]+H[(x(t - s))]+ds + [I] +

¡•CO

= P [x(t - T1)]+ + W [(x(t - T2 ))]+ ^ I Q(s)[(x(t - s))]+ds +[I]+,

where tfc-1 < t <tk,k = 1,2, —

Since ç(P + W + Q) < 1 and P,W,Q e Rn+xn, then, by Lemma 2.4, we can get

(E - P - W - Q)-1 > 0, and so N = (E - P - N - Q)-1 [I]+ > 0.

For the initial conditions: x(t0 + 0) = $(0), 0 e (-œ, 0], where $ e PC, we have

[x(t)]+ < K0ze-X(t-t0) < K0ze-X(t-t0) + N, t e (-œ,k], (3.24)

K0 = ■ ,, z e qJP + W + Q). (3.25)

mm1<i<n{Zi] VV /

By the property of ç-cone and z e Q.ç(P + W + Q), we have k0z e Qç(P + W + Q). Then, all the conditions of part (a) of Theorem 3.1 are satisfied by (3.23), (3.24), and condition (A3), we derive that

[x(t)]+ < K0ze-X(i-i0) + N, t e [t0, h). (3.26)

Suppose for all i = 1,...,k, the inequalities

[x(t)]+ < Y0 ••• T<-1K0ze-X(t-t0) + 00 ••• o-N, t e [t-1, t)

(3.27)

hold, where Yo = 00 = i. Then, from (3.20), (3.22), (3.27), and (A2), the impulsive part of (2.4) satisfies that

[x(tk)]+= [Jk(x(tk))] + < Rk[x(tk)]+

< Rk [ro ••• Yk-1 Koze-X(tkkto) + 00 ■■■ Ok-iiv] (3.28)

< Yo ■■■ YkkijkKoze~X(tkkto) + 00 ••• 0k-i0kN.

This, together with (3.27), leads to

[x(t)]+ < Yo ■ ■■Yk-iYkKoze-X(t-to) + 00 ■ ■■Ok-oN, t e (-»,tk]. (3.29) By the property of 9-cone again, the vector

Yo ■■■Yk-iYkKoz e Q^P + W + Q). (3.30)

On the other hand,

[x(t)]+ < P [x(t k Ti)]++ W [(x(t k T2))]++ Q(t)[(x(t k s))]+ds + 00.....Ok [I] + , t = tk.

(3.3i)

It follows from (3.29)-(3.3i) and part (a) of Theorem 3.i that

[x(t)]+ < Yo ■■■ YkkiYkKoze-X(tkto) + 00 ■■■ 0kki0kN, t e [tk, tk+i). (3.32) By the mathematical induction, we can conclude that

[x(t)]+ < Yo ■■■YkkiKoze-X(tkto) + 00 ■■■0kkiN, t e [tk^tk), k = i,2,.... (3.33) From (3.i8) and (3.2i),

Yk < eY(tkkik-i), 00 ■■■ 0kki < ea, (3.34) we can use (3.33) to conclude that

[x(t)]+ < eY(iikio) ■ ■ ■ eY(ikkikikk2)Koze-X(ikio) + 00 ■■■ 0kkiN

< KozeY^-«e-^-W + e0N (3.35)

= Koze-^^ + e0N, t e [tk-i,tk), k = i,2,....

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

Theorem 3.4. If(A1)-(A3) with Rk < E hold, then S = {$ e PC | [$] + < (E - P - WV - Q) 1 [I] + } is a positive invariant set and also a global attracting set of (2.4).

Proof. For the initial conditions: x(t0 + s) = $(s), s e (-to, 0], where $ e S, we have

[x(t)] + < (E - P - W - Q)-1[I]+, t e (-TO,t0]. (3.36) By (3.36) and the part (b) of Theorem 3.1 with d = 1, we have

[x(t)]+ < (E - P - W - Q)-1[I]+, t e [tc,t1). (3.37) Suppose for all i = 1,...,k, the inequalities

[x(t)]+ < (E - P - W - Q)-1[I]+, t e [ti-1,ti), (3.38) hold. Then, from (A2) and Rk < E, the impulsive part of (2.4) satisfies that

[x(tk)]+ < [Jk(x(t-))] + < Rk[x(t-)]+ < E[x(t-)]+ < (E - P - W - Q)-1[I]+. (3.39) This, together with (3.36) and (3.38), leads to

[x(t)]+ < (E - P - W - Q)-1[I]+, t e (-TO,tk]. (3.40) It follows from (3.40) and the part (b) of Theorem 3.1 that

[x(t)]+ < (E - P - W - Q)-1[I]+, t e [tk,tk+1). (3.41) By the mathematical induction, we can conclude that

[x(t)]+ < (E - P - W - Q)-1[I]+, t e [tk-1,tk), k = 1,2..........(3.42)

Therefore, S = {$ e PC | [$]+ < (E - P - W - Q) [I] + } is a positive invariant set. Since Rk < E, a direct calculation shows that jk = ak = 1 and a = 0 in Theorem 3.3. It follows from Theorem 3.3 that the set S is also a global attracting set of (2.4). The proof is complete. □

For the case I = 0, we easily observe that x(t) = 0 is a solution of (2.4) from (Ai) and (A2). In the following, we give the attractivity of the zero solution and the proof is similar to that of Theorem 3.3.

Corollary 3.5. If (A1) - (A4) hold with I = 0, then the zero solution of (2.4) is globally exponentially stable.

Remark 3.6. If Jk(x) = x, that is, they have no impulses in (2.4), then by Theorem 3.4, we can obtain the following result.

Corollary 3.7. If (Ai) and (A3) hold, then S = e PC | [$] + < (£ - P - W - Q)-1 [!] + } 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:

1 1 1 Xl(f) = 4Xl(t - 1) + — Sin(xi(f - 1)) + 15X2(i - 1)

+ ^|x2(t - 2)| - e-6(t-s)\x1(s)\ds + 2

15 J-«

1 1 1 x2(t) = — x2(t - 1) + - sin(xi(t - 1)) + -x2(t - 1) 4 5 6

|x1(t - 2)| +C e^12(t-s)|x2(s)|ds + 3

J —00

/mfmk,

x1(tk) = a1kx^tk) - ß1kx2(tk) x2(tk) = ß2kx^t-) + a2kx^t-),

where aik and ßik are nonnegative constants, and the impulsive sequence tk (k = 1,2,...) satisfies: t1 < t2 < ••• ,limktk = to. For System (4.1), we have pn(s) = -e-6s, p22(s) = e-12s,p12(s) = p21(s) = 0. So, it is easy to check that pij (s) e Le, i,j = 1,2, provided that 0 < 10 < 1. In this example, we may let 10 = 0.1.

The parameters of (A1)-(A3) are as follows:

F = G = H =

1 0 0 1

(6; \ V 0 112 J

Q = • 1

12 15 11

■1 1 F = I 115

^5 12'

a1k ß1k ß2k a2k

.15 0 015

.1» ■1 l'

F + W + Q = I 2 3 1-Q V11J

It is easy to prove that g(P + W + Q)= 5/6 < 1 and

Qp(P + W + Q) = | (zi,Z2)T> 0

3zi <Z2 < 2zi (4.4)

Let z = (1,1)T e QÇ(P + W + Q) and 1 = 0.01 < 10 which satisfies the inequality

e4 Pe1 + We21 + I Q(s)e1sds) - E)z< 0. (4.5)

/•аэ

Let = max{a1k + ¡1k,a2k + ¡2k}, then jk satisfy jkz > Rkz, k = 1,2, —

Case 1. Let aik = «2k = (1/3)e1/25k, ¡ik = ¡2k = (2/3)e1/25k, and tk - tk-i = 5k, then

= e1/25k > 1, JnY^ = = < 0.0Q8 = r<X. (4.6)

lk ~ , tk - tk-1 5k 25k x 5k" 1 y '

Moreover, Ok = e1/25k > 1, a = ln Ok = E?=1 In e1/25k = 1/24. Clearly, all conditions of Theorem 3.3 are satisfied. So S = {^ e PC | [$] + < e1/24(E - P - W - Q)-1I} = (6e1/24,6e1/24)T is a global attracting set of (4.1).

Case 2. Let a1k = a2k = (1/3)e1/2k and ¡1k = ¡2k = 0, then Rk = (1/3)e1/2kE < E. Therefore, by

Theorem 3.4, S = {$ e PC | < N = (E - P - W - Q)-1I} = (6,6)T is a positive invariant set and also a global attracting set of (4.1).

Case 3. If I = 0 and let «1k = «2k = (1/3)e0 04k and ¡1k = ¡2k = (2/3)e0 04k, then

Yk = e0 04k > 1, J^ = = 0.008 = r<X. (4.7)

tk - tk-1 5k

Clearly, all conditions of Corollary 3.5 are satisfied. Therefore, by Corollary 3.5, the zero solution of (4.1) is globally exponentially stable.

Acknowledgment

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

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