Stability for delayed generalized 2D discrete logistic systems

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LOGISTIC SYSTEMS

CHUAN JUN TIAN AND GUANRONG CHEN

Received 28 August 2003 and in revised form 19 February 2004

This paper is concerned with delayed generalized 2D discrete logistic systems of the form

xm+1,n=f(m,n,xm,n,xm,n+1,xm−σ,n−τ), whereσandτare positive integers, f :N20×R3→ Ris a real function, which contains the logistic map as a special case, andmandnare nonnegative integers, whereN0= {0, 1,...}andR=(−∞,∞). Some suﬃcient conditions for this system to be stable and exponentially stable are derived.

1. Introduction

In engineering applications, particularly in the fields of digital filtering, imaging, and spa-tial dynamical systems, 2D discrete systems have been a subject of focus for investigation (see, e.g., [1,2,3,4,5,6] and the references cited therein). In this paper, we consider the delayed generalized 2D discrete systems of the form

xm+1,n=f

m,n,xm,n,xm,n+1,xm−σ,n−τ, (1.1)

whereσ andτare positive integers,mandnare nonnegative integers, and f :N20×R3→ Ris a real function containing the logistic map as a special case, whereR=(−∞,∞), R3₌_R_×_R_×_R,_N

0= {0, 1,...}, andN20=N0×N0= {(m,n)|m,n=0, 1,...}. Obviously, if

f(m,n,x,y,z)≡µm,nx(1−x)−am,ny, (1.2) f(m,n,x,y,z)≡µm,nx(1−z)−am,ny, (1.3)

f(m,n,x,y,z)≡1−µx2−ay, (1.4)

or

f(m,n,x,y,z)≡bm,nx−am,ny−pm,nz, (1.5)

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then system (1.1) becomes, respectively,

xm+1,n+am,nxm,n+1=µm,nxm,n

1−xm,n

, (1.6)

xm+1,n+am,nxm,n+1=µm,nxm,n

1−xm−σ,n−τ, (1.7) xm+1,n+axm,n+1=1−µxm,n2, (1.8)

or

xm+1,n+am,nxm,n+1−bm,nxm,n+pm,nxm−σ,n−τ=0. (1.9)

Systems (1.6), (1.7), and (1.8) are regular 2D discrete logistic systems of diﬀerent forms, and particularly system (1.9) has been studied in the literature [2,4,5,6].

Ifam,n=0,µm,n=µ, andn=n0is fixed, then system (1.6) becomes the 1D logistic system

xm+1,n0=µxm,n0

1−xm,n0

, (1.10)

whereµis a parameter. System (1.10) has been intensively investigated in the literature. Hence, system (1.1) is quite general.

This paper is concerned with the stability of solutions of system (1.1), in which some suﬃcient conditions for the stability and exponential stability of system (1.1) will be de-rived.

Let Nt= {t,t+ 1,t+ 2,...} for any t∈Z, and Ω=N−σ×N−τ\N1×N0. It is obvi-ous that for any given functionφ= {φm,n}defined onΩ, it is easy to construct by in-duction a double sequence{xm,n}that equals the initial conditionφonΩand satisfies

(1.1) onN1×N0. In fact, from (1.1), one can calculate successively a solution sequence:

x1,0,x1,1,x2,0,x1,2,x2,1,x3,0,..., by using the initial conditions, which is said to be a solution of system (1.1) with the initial conditionφ.

Definition 1.1. Letx∗∈Rbe a constant. Ifx∗is a root of the equation

x−f(m,n,x,x,x)=0 for any (m,n)∈N20, (1.11)

thenx∗is said to be a fixed point or equilibrium point of system (1.1). The set of all fixed points of system (1.1) is called a fixed plane or equilibrium plane of the system.

It is easy to see thatx∗=0 is a fixed point of systems (1.6), (1.7), and (1.9), and

x∗₌₍₋₍_a_{+ 1)}_±₍_a_{+ 1)}2_{+ 4}µ₎/₂µ_{are two fixed points of system (1.8).}

Letx∗be a fixed point of system (1.1), letφ= {φm,n}be a function defined onΩ, and let

φx∗=supφm_,_n−x∗: (m,n)∈Ω. (1.12)

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Definition 1.2. Letx∗∈Rbe a fixed point of system (1.1). If, for anyε >0, there exists a positive constantδ >0 such that for any given bounded functionφ= {φm,n}defined on Ω,φ∈Sδ(x∗) implies that the solutionx= {xm,n} of system (1.1) with the initial

conditionφsatisfies

_xm_,_n₋_x∗_{< ε}_, _∀₍_m_,_n₎_∈_N

1×N0, (1.13)

then system (1.1) is said to be stable about the fixed pointx∗.

Definition 1.3. Letx∗∈Rbe a fixed point of system (1.1). If there exist positive constants

M >0 andξ∈(0, 1) such that for any given constantδ∈(0,M) and any given bounded functionφ= {φm,n}defined onΩ,φ∈Sδ(x∗) implies that the solution{xm,n}of system

(1.1) with the initial conditionφsatisfies

_xm_,_n₋_x∗_{< Mξ}m+n_, ₍_m_,_n₎_∈_N

1×N0, (1.14)

then system (1.1) is said to be DB-exponentially stable about the fixed pointx∗, where D means double variables and B means bounded initial condition.

Definition 1.4. Letx∗∈Rbe a fixed point of system (1.1). If there exist positive constants

M >0 andξ∈(0, 1) such that for any given bounded numberδ∈(0,M) and any given bounded functionφ= {φm,n}defined onΩ,φ∈Sδ(x∗) implies that the solution{xm,n}

of system (1.1) with the initial conditionφsatisfies

_xm_,_n₋_x∗_{< Mξ}m_, ₍_m_,_n₎_∈_N

1×N0, (1.15)

then system (1.1) is said to be SB-exponentially stable about the fixed pointx∗, where S means single variable and B means bounded initial condition.

Obviously, if system (1.1) is DB-exponentially stable, then it is SB-exponentially stable.

Definition 1.5. Let f(m,n,x,y,z) be a function defined onN2

0×Dand let (x0,y0,z0)∈D be a fixed inner point, whereD⊂R3_{. If, for any positive constant}_{ε >}_{0, there exists a} constantδ >0 such that for any|x−x0|< δ,|y−y0|< δ, and|z−z0|< δ,

_f₍_m_,_n_,_x_,_y_,_z₎₋_f

m,n,x0,y0,z0< ε for any (m,n)∈N20, (1.16)

thenf(m,n,x,y,z) is said to be uniformly continuous at the point (x0,y0,z0) (overmand

n). If the partial derivative functions fx(m,n,x,y,z), fy(m,n,x,y,z), and fz(m,n,x,y,z)

are all uniformly continuous at (x0,y0,z0), then f(m,n,x,y,z) is said to be uniformly continuously diﬀerentiable at (x0,y0,z0).

LetDbe an open subset ofR3_{. If} _f₍_m_,_n_,_x_,_y_,_z_{) is uniformly continuous at any point} (x,y,z)∈D, then it is said to be uniformly continuous onD.

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2. Stability conditions, the functiong(t) is continuously diﬀerentiable on [0, 1]. Hence, from the mean value theorem, there exists a constantt0∈(0, 1) such thatg(1)−g(0)=g (t0), that

is,Lemma 2.1holds. The proof is completed.

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For any suﬃciently small numberε >0, without loss of generality, letε < Mandδ=ε, and letφ= {φm,n}be a given bounded function defined onΩwhich satisfies|φm,n−x∗|<

δfor all (m,n)∈Ω. Let the sequence{xm,n}be a solution of system (1.1) with the initial

conditionφ. In view of (1.1) and the inequalities _x_0,0₋_x∗_≤_{δ < M}_, _x

0,1−x∗≤δ < M,

_x₋_σ_,₋_τ₋_x∗_≤_{δ < M}_, (2.6)

it follows from (2.3), (2.4), andLemma 2.1that there exists a constant

t0=t

0, 0,x0,0,x0,1,x−σ,−τ∈(0, 1), (2.7)

such that

_x_1,0₋_x∗₌_f_{0, 0,}_x

0,0,x0,1,x−σ,−τ−f0, 0,x∗,x∗,x∗

≤fx0, 0,λ,η,θx0,0−x∗+fy0, 0,λ,η,θx0,1−x∗ +fz(0, 0,λ,η,θ)x−σ,−τ−x∗

≤δ≤ε < M,

(2.8)

whereλ=x∗+t0(x0,0−x∗), η=x∗+t0(x0,1−x∗), andθ=x∗+t0(x−σ,−τ−x∗). Simi-larly, from (1.1), (2.3), and (2.4), one has

_x₁_.₁₋_x∗₌_f_{0, 1,}_x

0,1,x0,2,x−σ,1−τ−f0, 1,x∗,x∗,x∗≤ε < M. (2.9) In general, for any integern≥0,|x1,n−x∗| ≤ε < M.

Assume that for a certain integerk≥1,

|xi,n−x∗| ≤ε < M for anyi∈ {1, 2,...,k},n≥0. (2.10)

Then, it follows from (1.1), (2.3), and (2.4) that there exists a constant

t0=t

k,n,xk,n,xk,n+1,xk−σ,n−τ∈(0, 1), (2.11)

such that

_xk_+1,_n₋_x∗₌_f_k_,_n_,_xk

,n,xk,n+1,xk−σ,n−τ−fk,n,x∗,x∗,x∗

≤fxk,n,λ,η,θxk,n−x∗+fy(k,n,λ,η,θ)xk,n+1−x∗ +fz(k,n,λ,η,θ)xk−σ,n−τ−x∗

≤fx(k,n,λ,η,θ)+fy(k,n,λ,η,θ)+fz(k,n,λ,η,θ)·ε≤ε,

(2.12)

where λ=x∗₊_t₀₍_xk_,_n₋_x∗_), _η₌_x∗₊_t₀₍_xk_,_n₊₁₋_x∗_{), and}_θ₌_x∗₊_t₀₍_xk₋_σ_,_n₋_τ₋_x∗_). Hence, by induction,|xm,n−x∗| ≤εfor any (m,n)∈N1×N0, that is, system (1.1) is

stable. The proof is completed.

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Theorem2.3. Assume thatx∗is a fixed point of system (1.1), and the function f(m,n,x,y,

z)is continuously diﬀerentiable onR3 _{for any fixed} _m_and_n_{. Further, assume that there} exists an open subsetD⊂R3 _{such that}₍_x∗_,_x∗_,_x∗₎_∈_D_{and, for any}₍_m_,_n₎_∈_N2

0and any (x,y,z)∈D,

_f_x₍_m_,_n_,_x_,_y_,_z₎₊_f_y₍_m_,_n_,_x_,_y_,_z₎₊_f_z₍_m_,_n_,_x_,_y_,_z₎_≤₁_. _(2.13)

Then system (1.1) is stable.

From Theorems2.2and2.3, one obtains the following results.

Corollary2.4. Assume that there exists a constantr∈(0, 1)such that

_µm_,_n₊_am_,_n_≤_r _∀_m_≥_0,_n_≥₀_. _(2.14)

Then systems (1.6) and (1.7) are both stable.

In fact, system (1.6) is a special case of system (1.1) when

f(m,n,x,y,z)≡µm,nx(1−x)−am,ny. (2.15)

In view of (2.14), it is obvious that the function f(m,n,x,y,z)is both continuously diﬀ eren-tiable onR3_{for any fixed}₍_m_,_n₎_∈_N2

0and uniformly continuously diﬀerentiable at the point (0, 0, 0). Sincex∗=0is a fixed point of systems (1.6) and (1.7),

fxm,n,x∗,x∗,x∗=µm,n, fym,n,x∗,x∗,x∗= −am,n,

fzm,n,x∗,x∗,x∗=0.

(2.16)

Hence (2.14) implies (2.2). ByTheorem 2.2, systems (1.6) and (1.7) are both stable.

Corollary2.5. System (1.8) has fixed pointsx∗=(−(a+ 1)±(a+ 1)2_{+ 4}µ₎/₂µ_{. Assume} that there exists a constantr∈(0, 1)such that

2µ·x∗₊_|_a_{| ≤}_r. _(2.17)

Corollary2.6. Assume that

_am_,_n₊_bm_,_n₊_pm_,_n_≤₁ _∀_m_≥_0,_n_≥₀_. _(2.18)

Define four subsets ofN0×N0as follows:

B1=

(i,j)|0≤i≤σ, 0≤j < τ, B2=

(i,j)|0≤i≤σ, j≥τ,

B3=

(i,j)|i > σ, 0≤j < τ, B4=

(i,j)|i > σ, j≥τ. (2.19)

Obviously,B1is a finite set,B2,B3, andB4are infinite sets,B1,B2,B3, andB4are mutually disjoint, andN2

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Theorem2.7. Assume thatx∗is a fixed point of system (1.1), the function f(m,n,x,y,z) is both continuously diﬀerentiable onR3_{for any}₍_m_,_n₎_∈_N2

0 and uniformly continuously diﬀerentiable at the point(x∗,x∗,x∗)∈R3_{, and there exists a constant}_r_∈_{(0, 1)}_{such that} for any(m,n)∈B3,

_f_x

m,n,x∗,x∗,x∗+fym,n,x∗,x∗,x∗+r−mfzm,n,x∗,x∗,x∗≤r, (2.20)

and for(m,n)∈B1∪B2∪B4, _f_x

m,n,x∗,x∗,x∗+fym,n,x∗,x∗,x∗+r−σfzm,n,x∗,x∗,x∗≤r. (2.21)

Then, system (1.1) is SB-exponentially stable.

Proof. From the given conditions, there exist two positive constants,M >0 andξ∈(r, 1), such that (2.4) holds and, for any (m,n)∈B3,

_f_x₍_m_,_n_,_x_,_y_,_z₎₊_f_y₍_m_,_n_,_x_,_y_,_z₎₊_ξ−m_f

z(m,n,x,y,z)≤ξ, (2.22)

and for (m,n)∈B1∪B2∪B4,

_f_x₍_m_,_n_,_x_,_y_,_z₎₊_f_y₍_m_,_n_,_x_,_y_,_z₎₊_ξ−σ_f

z(m,n,x,y,z)≤ξ, (2.23)

for|x−x∗|< M,|y−x∗|< M, and|z−x∗|< M.

Letδ∈(0,M) be a given constant and letφ= {φm,n}be a given bounded function defined onΩwhich satisfies|φm,n−x∗|< δfor all (m,n)∈Ω. Let the sequence{xm,n}be

a solution of system (1.1) with the initial conditionφ. In view of (1.1) and the following inequalities:

_x_0,0₋_x∗_≤_{δ < M}_, _x

0,1−x∗≤δ < M,

_x₋_σ_,₋_τ₋_x∗_≤_{δ < M}_, (2.24)

it follows from (2.4), (2.23), andLemma 2.1that there exists a constant

t0=t

0, 0,x0,0,x0,1,x−σ,−τ∈(0, 1), (2.25)

such that

_x_1,0₋_x∗₌_f_{0, 0,}_x

0,0,x0,1,x−σ,−τ−f0, 0,x∗,x∗,x∗

≤fx0, 0,λ,η,θx0,0−x∗+fy(0, 0,λ,η,θ)x0,1−x∗ +fz(0, 0,λ,η,θ)x−σ,−τ−x∗

≤Mξ,

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whereλ=x∗+t0(x0,0−x∗), η=x∗+t0(x0,1−x∗), andθ=x∗+t0(x−σ,−τ−x∗). Simi-larly, from (1.1), (2.4), and (2.23), one has

_x₁_.₁₋_x∗₌_f_{0, 1,}_x

0,1,x0,2,x−σ,1−τ−f0, 1,x∗,x∗,x∗≤Mξ. (2.27) In general, for any integern≥0,|x1,n−x∗| ≤Mξ.

Assume that for a certain integerk∈ {1,...,σ},

_xi_,_n₋_x∗_≤_Mξi _{for any}_i_{∈ {}_{1, 2,}_..._,_k_}_,_n_≥₀_. _(2.28)

Then (k,n)∈B1∪B2∪B4and (k−σ,n−τ)∈Ω. From (2.4) and (2.23), one obtains _xk_+1,_n₋_x∗₌_f_k_,_n_,_xk

,n,xk,n+1,xk−σ,n−τ−fk,n,x∗,x∗,x∗

≤fx(k,n,λ,η,θ)xk,n−x∗+fy(k,n,λ,η,θ)xk,n+1−x∗ +fz(k,n,λ,η,θ)xk−σ,n−τ−x∗

≤fx(k,n,λ,η,θ)·Mξk+fy(k,n,λ,η,θ)·Mξk

+f_z(k,n,λ,η,θ)·M ≤Mξk+1_.

(2.29)

By induction,|xm,n−x∗| ≤Mξmfor anym∈ {1, 2,...,σ+ 1}andn≥0. Assume that for a certain integerk≥σ+ 1,

_xi_,_n₋_x∗_≤_Mξi _{for any}_i_{∈ {}_{1, 2,}_..._,_k_}_,_n_≥₀_. _(2.30)

If n∈ {0, 1,...,τ−1}, then (k,n)∈B3 and (k−σ,n−τ)∈Ω. Hence, from (1.1), (2.4), (2.22), and Lemma 2.1, there exists a constantt0=t(k,n,xk,n,xk,n+1,xk−σ,n−τ)∈

(0, 1) such that

_xk_+1,_n₋_x∗_≤_f

x(k,n,λ,η,θ)·Mξk+fy(k,n,λ,η,θ)·Mξk

+fz(k,n,λ,η,θ)·M ≤Mξk+1_,

(2.31)

whereλ=x∗₊_t

0(xk,n−x∗),η=x∗+t0(xk,n+1−x∗), andθ=x∗+t0(xk−σ,n−τ−x∗).

Ifn≥τ, then (k,n)∈B1∪B2∪B4and (k−σ,n−τ)∈/ Ω. Hence, from (2.4), (2.23), and the assumption, one has

_xk_+1,_n₋_x∗_≤_f

x(k,n,λ,η,θ)·Mξk+fy(k,n,λ,η,θ)·Mξk

+fz(k,n,λ,η,θ)·Mξk−σ ≤Mξk+1_.

(2.32)

By induction,|xm,n−x∗| ≤Mξm for any (m,n)∈N1×N0, that is, system (1.1) is

SB-exponentially stable. The proof is completed.

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Corollary2.8. Assume that there exists a constantr∈(0, 1)such that

_µm_,_n₊_am_,_n_≤_r _∀_m_≥_0,_n_≥₀_. _(2.33)

Then systems (1.6) and (1.7) are both SB-exponentially stable.

Corollary2.9. System (1.8) has fixed pointsx∗=(−(a+ 1)±(a+ 1)2_{+ 4}µ₎/₂µ_{. Assume} that there exists a constantr∈(0, 1)such that

2µ·x∗₊_|_a_{| ≤}_r. _(2.34)

Then system (1.8) is SB-exponentially stable.

Corollary2.10. Assume that there exists a constantr∈(0, 1)such that for(m,n)∈B3, _am_,_n₊_bm_,_n₊_r−m_pm

,n≤r, (2.35)

and for(m,n)∈B1∪B2∪B4,

_am_,_n₊_bm_,_n₊_r−σ_pm

,n≤r. (2.36)

Then system (1.9) is SB-exponentially stable.

Let

D1=(m,n) : 1≤m≤σ, 0≤n < τ, D2=(m,n) :m > σ, 0≤n < τ,

D3=(m,n) : 1≤m≤σ,n≥τ, D4=(m,n) :m > σ,n≥τ.

(2.37)

Obviously,D1,D2,D3, andD4are mutually disjoint, andN1×N0=D1∪D2∪D3∪D4. Theorem2.11. Assume thatx∗is a fixed point of system (1.1), and f(m,n,x,y,z)is both continuously diﬀerentiable onR3_{for any}₍_m_,_n₎_∈_N2

0and uniformly continuously diﬀeren-tiable at(x∗,x∗,x∗)∈R3_{. Further, assume that there exist a constant}_r_∈_{(0, 1)}_{and an open} subsetD⊂R3_with₍_x∗_,_x∗_,_x∗₎_∈_D_{such that for any}₍_x_,_y_,_z₎_∈_D_{and any}_n_≥₀_,

_f_x_(0,_n_,_x_,_y_,_z₎₊_f_y_(0,_n_,_x_,_y_,_z₎₊_f_z_(0,_n_,_x_,_y_,_z₎_≤_rn+1_, _(2.38)

and for all(m,n)∈D1∪D2∪D3, _f_x

m,n,x∗,x∗,x∗+rfym,n,x∗,x∗,x∗+r−m−nfzm,n,x∗,x∗,x∗≤r,

(2.39)

and for(m,n)∈D4, _f_x

m,n,x∗,x∗,x∗+rfym,n,x∗,x∗,x∗|+r−σ−τfzm,n,x∗,x∗,x∗≤r.

(2.40)

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Proof. From the given conditions and from (2.38), (2.39), and (2.40), there exist positive constantsM >0 andξ∈(r, 1) such that (2.4) holds and, for alln≥0,

_f_x_(0,_n_,_x_,_y_,_z₎₊_f_y_(0,_n_,_x_,_y_,_z₎₊_f_z_(0,_n_,_x_,_y_,_z₎_≤_ξn+1_, _(2.41)

and for all (m,n)∈D1∪D2∪D3,

_f_x₍_m_,_n_,_x_,_y_,_z₎₊_ξ_f_y₍_m_,_n_,_x_,_y_,_z₎₊_ξ−m−n_f

z(m,n,x,y,z)≤ξ, (2.42)

and for all (m,n)∈D4,

_f_x₍_m_,_n_,_x_,_y_,_z₎₊_ξ_f_y₍_m_,_n_,_x_,_y_,_z₎₊_ξ−σ−τ_f

z(m,n,x,y,z)≤ξ, (2.43)

for|x−x∗_|_{< M}_,_|_y₋_x∗_|_{< M}_{, and}_|_z₋_x∗_|_{< M}_.

Letδ∈(0,M) be a constant and letφ= {φm,n}be a given bounded function defined on

Ωwhich satisfies|φm,n−x∗|< δfor all (m,n)∈Ω. Let the sequence{xm,n}be a solution of system (1.1) with the initial conditionφ. In view of (1.1) and the inequalities

_x_0,_n₋_x∗_≤_{δ < M}_, _x

0,n+1−x∗≤δ < M,

_x₋_σ_,_n₋_τ₋_x∗_≤_{δ < M}_, (2.44)

it follows from (2.4) and (2.41) that there exists a constantt0,n=t(0,n,x0,n,x0,n+1,x−σ,n−τ) ∈(0, 1) such that for anyn∈N0,

_x_1,_n₋_x∗₌_f_0,_n_,_x

0,n,x0,n+1,x−σ,n−τ−f0,n,x∗,x∗,x∗

≤fx0,n,λ0,n,η0,n,θ0,nx0,n−x∗ +f_y0,n,λ0,n,η0,n,θ0,nx0,n+1−x∗ +fz0,n,λ0,n,η0,n,θ0,nx−σ,n−τ−x∗ ≤Mξn+1_,

(2.45)

where

λ0,n=x∗+t0,nx0,n−x∗, η0,n=x∗+t0,nx0,n+1−x∗

,

θ0,n=x∗+t0,nx−σ,n−τ−x∗. (2.46)

Assume that for somem∈ {1, 2,...,σ},

_xi_,_j₋_x∗_≤_Mξi+j_, _∀₁_≤_i_≤_m_{and all} _j_∈_N

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Then, for alln≥0, one has (m−σ,n−τ)∈Ωand (m,n)∈D1∪D2∪D3. Hence, it fol-lows from (2.4) and (2.42) that there exists a constanttm,n=t(m,n,xm,n,xm,n+1,xm−σ,n−τ) ∈(0, 1) such that

_xm_+1,_n₋_x∗₌_f_m_,_n_,_xm

,n,xm,n+1,xm−σ,n−τ−fm,n,x∗,x∗,x∗ ≤fxm,n,λm,n,ηm,n,θm,nxm,n−x∗

+fym,n,λm,n,ηm,n,θm,nxm,n+1−x∗ +fzm,n,λm,n,ηm,n,θm,nxm−σ,n−τ−x∗

≤fxm,n,λm,n,ηm,n,θm,n+ξfym,n,λm,n,ηm,n,θm,n

+ξ−m−nfzm,n,λm,n,ηm,n,θm,n×Mξm+n ≤Mξm+n+1_,

(2.48)

where

λm,n=x∗+tm,n

xm,n−x∗

, ηm,n=x∗+tm,n

xm,n+1−x∗

,

θm,n=x∗+tm,n

xm−σ,n−τ−x∗. (2.49)

By induction,|xm,n−x∗| ≤Mξm+nfor allm∈ {1, 2,...,σ+ 1}and alln≥0.

Assume that for somem≥σ+ 1,

_xi_,_n₋_x∗_≤_Mξi+n_, _∀₁_≤_i_≤_m_{and all}_n_∈_N

0. (2.50)

Then, ifn∈ {0, 1,...,τ−1}, then (m,n)∈D1∪D2∪D3 and (m−σ,n−τ)∈Ω. From (2.4) and (2.42), one has

_xm_+1,_n₋_x∗_≤_f

xm,n,λm,n,ηm,n,θm,nxm,n−x∗

+fym,n,λm,n,ηm,n,θm,nxm,n+1−x∗ +fzm,n,λm,n,ηm,n,θm,nxm−σ,n−τ−x∗

≤fxm,n,λm,n,ηm,n,θm,n+ξfym,n,λm,n,ηm,n,θm,n +ξ−m−nfzm,n,λm,n,ηm,n,θm,n×Mξm+n

≤Mξm+n+1_.

(2.51)

Ifn≥τ, then (m,n)∈D4 and (m−σ,n−τ)∈N1×N0. From (2.4), (2.43), and the as-sumption,

_xm_+1,_n₋_x∗_≤_f

x(m,n,λm,n,ηm,n,θm,n+ξfym,n,λm,n,ηm,n,θm,n +ξ−σ−τfxm,n,λm,n,ηm,n,θm,n×Mξm+n

≤Mξm+n+1_.

(2.52)

By induction,|xm,n−x∗| ≤Mξm+nfor any (m,n)∈N1×N0, that is, system (1.1) is

DB-exponentially stable. The proof is completed.

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Corollary2.12. Assume that there exist two constantsr∈(0, 1)andC∈(0, 1)such that _µ_0,_n₊_a_0,_n_≤_Crn+1_, _∀_n_≥_0,

_µm_,_n₊_r_am_,_n_≤_r_, _{for any}₍_m_,_n₎_∈_N₁_×_N₀_. (2.53)

Then, systems (1.6) and (1.7) are both DB-exponentially stable.

Corollary2.13. Assume that there exists a constantr∈(0, 1)such that _a_0,_n₊_b_0,_n₊_p_0,_n_≤_rn+1_, _{for any}_n_≥_0,

ram,n+bm,n+r−m−npm,n≤r, for any(m,n)∈D1∪D2∪D3,

ram,n+bm,n+r−σ−τpm,n≤r, for any(m,n)∈D4.

(2.54)

Then, system (1.9) is DB-exponentially stable.

Acknowledgment

This research was partially supported by the Hong Kong Research Grants Council un-der the CERG Grant CityU 1004/02E, and partially supported by the NNSF of China (60374017) and the NSF of Shenzhen.

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[3] S. T. Liu and G. Chen,Asymptotic behavior of delay 2-D discrete logistic systems, IEEE Trans. Circuits Systems I Fund. Theory Appl.49(2002), no. 11, 1677–1682.

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Chuan Jun Tian: College of Information Engineering, Shenzhen University, Shenzhen 518060, China

E-mail address:[email protected]

Guanrong Chen: Department of Electronic Engineering, City University of Hong Kong, Hong Kong