Global Exponential Stability of Delayed Cohen Grossberg BAM Neural Networks with Impulses on Time Scales

Volume 2009, Article ID 491268,17pages doi:10.1155/2009/491268

Research Article

Global Exponential Stability of

Delayed Cohen-Grossberg BAM Neural Networks

with Impulses on Time Scales

Yongkun Li,

_{Yuchun Hua,}

_{and Yu Fei}

1_{Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China} 2_{School of Statistics and Mathematics, Yunnan University of Finance and Economics,}

Kunming, Yunnan 650221, China

Correspondence should be addressed to Yongkun Li,[email protected]

Received 18 April 2009; Accepted 14 July 2009

Recommended by Patricia J. Y. Wong

Based on the theory of calculus on time scales, the homeomorphism theory, Lyapunov functional method, and some analysis techniques, sufficient conditions are obtained for the existence, uniqueness, and global exponential stability of the equilibrium point of Cohen-Grossberg bidirectional associative memoryBAMneural networks with distributed delays and impulses on time scales. This is the first time applying the time-scale calculus theory to unify the discrete-time and continuous-discrete-time Cohen-Grossberg BAM neural network with impulses under the same framework.

Copyrightq2009 Yongkun Li et al. 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

often are subject to impulsive perturbations which can affect dynamical behaviors of the systems just as time delays. Therefore, it is necessary to consider both impulsive effect and delay effect on the stability of neural networks.

As is well known, both continuous and discrete systems are very important in implementation and applications. However, it is troublesome to study the stability for continuous and discrete systems, respectively. Therefore, it is worth studying a new method, such as the time-scale theory, which can unify the continuous and discrete situations.

Motivated by the above discussions, the objective of this paper is to study the global exponential stability of the following Cohen-Grossberg bidirectional associative memory networks with impulses and time delays on time scales:

xΔ_i t −aixit ⎡

⎣bixit− m

j 1 p0_jifj

yj

t−τji

−m

j 1 p1

ji _∞

0

hijsfj

yjt−s

Δsri ⎤

⎦, t≥0, t /tk, t∈T,

Δxitk Ikxitk, i 1,2, . . . , n, k 1,2, . . . ,

yΔ_jt −cj

yjt

dj

yjt

−n

i 1 q0_ijgi

xi

t−σij

−n

i 1 q1_ij

_∞

0

kijsgixit−sΔssj

, t≥0, t /tk, t∈T,

Δyjtk Jk

yjtk

, j 1,2, . . . , m, k 1,2, . . . ,

1.1

where T is a time scale; Ik, Jk : R → Rare continuous, xit, yjt are the states of the

ith neuron from the neural field FX and the jth neuron from the neural field FY at time

t, respectively;fj, gi denote the activation functions of thejth neuron from FY and the ith

neuron from FX, respectively;ri and sj are constants, which denote the external inputs on

theith neuron fromFX and thejth neuron fromFY, respectively;τjiandσij correspond to

the transmission delays; aixit and cjyjt represent amplification functions; bixit

and djyjt are appropriately behaved functions such that the solutions of system 1.1

remain bounded;p0

ji, p1ji, q0ij,andq1ijdenote the connection strengths which correspond to the

neuronal gains associated with the neuronal activations;IiandJjdenote the external inputs.

For each intervalIofR, we denote that by IT IT,Δxitk xit_k−xit−_k,Δyjtk yjt_k−

yjt−_k are the impulses at moments tk, and xit_k, xit−_k, yjt_k, yjt−_k i 1,2, . . . , n, j

1,2, . . . , mrepresent the right and left limits ofxit_kandyjt_kin the sense of time scales;

0< t1< t2<· · ·< tk → ∞is a strictly increasing sequence.

The system1.1is supplement with initial values given by

xis ϕis, s∈−∞,0T, i 1,2, . . . , n,

yjs ψjs, s∈−∞,0_T, j 1,2, . . . , m,

1.2

As usual in the theory of impulsive differential equations, at the points of discontinuity

tkof the solutiont → x1t, x2t, . . . , xnt, y1t, y2t, . . . , ymtTwe assume that

xitk xi

t−_k, yjtk yj

t−_k, xΔ_i tk xΔi

t−_k, y_jΔtk yΔj

t−_k, 1.3

fori 1,2, . . . , n, j 1,2, . . . , m.

The organization of the rest of this paper is as follows. In Section 2, we introduce some notations and definitions, and state some preliminary results which are needed in later sections. In Section 3, by means of homeomorphism theory, we study the existence and uniqueness of the equilibrium point of system 1.1. In Section 4, by constructing a suitable Lyapunov function, we establish the exponential stability of the equilibrium of1.1. InSection 5, we present an example to illustrate the feasibility and effectiveness of our results obtained in previous sections.

2. Preliminaries

In this section, we will cite some definitions and lemmas which will be used in the proofs of our main results.

LetTbe a nonempty closed subsettime scaleofR. The forward and backward jump operatorsσ, ρ:T → Tand the graininessμ:T → Rare defined, respectively, by

σt inf{s∈T:s > t}, ρt sup{s∈T:s < t}, μt σt−t. 2.1

A pointt ∈ Tis called left dense ift > infTand ρt t, left scattered ifρt < t, right dense ift < supTandσt t, and right scattered ifσt> t. IfThas a left-scattered maximumm, thenTk _{T\ {m}; otherwise T}k _{T. IfThas a right-scattered minimumm,}

thenTk T\ {m}; otherwiseTk T.

A functionf :T → Ris right dense continuous provided that it is continuous at right dense point inTand its left-side limits exist at left-dense points in T. Iff is continuous at each right dense point and each left-dense point, thenf is said to be a continuous function onT. The set of continuous functionsf:T → Rwill be denoted byCT.

For y : T → R andt ∈ Tk_{, we define the delta derivative of yt, y}Δ_{tto be the}

numberif it existswith the property that for a givenε > 0, there exists a neighborhoodU

oftsuch that

yσt−ys−yΔtσt−s< ε|σt−s| 2.2

for alls∈U.

Ifyis continuous, thenyis right dense continuous, andyis delta differentiable att, thenyis continuous att.

Letybe right dense continuous. IfyΔt yt, then we define the delta integral by

t

a

Definition 2.1see22. For eacht∈T, letNbe a neighborhood oft, then, forV ∈CrdT×

Rn_{,R, define DV}Δ_{t, xt to mean that, givenε > 0, there exists a right neighborhood}

Nε⊂Nof tsuch that

Vσt, xσt−Vs, xσt−μt, sft, xt μt, s < D

_VΔ_{t, xt ε 2.4}

for eachs∈Nε,s > t, whereμt, s≡σt−s. Iftis rd andVt, xtis continuous att, this

reduce to

DVΔt, xt Vσt, xσt−Vt, xσt

σt−t . 2.5

Definition 2.2see23. Ifa∈T,supT ∞, andfis right dense continuous ona,∞, then we define the improper integral by

_∞

a

ftΔt lim

b→ ∞

_b

a

ftΔt 2.6

provided that this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges.

A functionr :T → Ris called regressive if

1μtrt/0 2.7

for allt∈Tk.

Ifris regressive function, then the generalized exponential functioner is defined by

ert, s exp

t

s

ξμτrτΔτ

fors, t∈T, 2.8

with the cylinder transformation

ξhz ⎧ ⎪ ⎨ ⎪ ⎩

Log1hz

h , ifh /0, z, ifh 0.

2.9

Letp, q:T → Rbe two regressive functions, then we define

p⊕q: pqμpq, pq: p⊕q, p: p

1μp. 2.10

Lemma 2.3see24. Assume thatp, q:T → Rare two regressive functions, then

ie0t, s≡1andept, t≡1

iiepσt, s 1μtptept, s

iii ept, σs ept, s/1μsps

iv1/ept, s e pt, s

vept, s 1/eps, t e ps, t

viept, seps, r ept, r

viiept, seqt, s ep⊕qt, s

viiiept, s/eqt, s ep qt, s.

Definition 2.4. The equilibrium point u∗ x₁∗, x₂∗, . . . , x_n∗, y₁∗, y∗₂, . . . , y∗_mT of system 1.1 is said to be exponentially stable if there exists a positive constant α such that for every δ ∈ T, there exists N Nδ ≥ 1 such that the solu-tion ut x1t, x2t, . . . , xnt, y1t, y2t, . . . , ymtT of 1.1 with initial value

ϕ1s, ϕ2s, . . . , ϕns, ψ1s, ψ2s, . . . , ψmsTsatisfies

u−u∗ ≤Ne₋αt, δ

⎡ ⎣n

i 1

max

δ∈−∞,0T

ϕiδ−x∗i m

j 1

max

δ∈−∞,0T

ψjδ−yj∗ ⎤

⎦. 2.11

Lemma 2.5see25. IfHx∈CRnm,Rnmsatisfies the following conditions:

iHxis injective onRnm, iiH → ∞asx → ∞,

thenHxis a homeomorphism ofRnmonto itself.

Forz x1, x2, . . . , xn, y1, y2, . . . , ymT∈Rnm, we define the norm as

z

n

i 1

|xi| m

j 1

yj. 2.12

Throughout this paper, we assume that

H1ai, cj ∈CT,R, and satisfy 0 < a_i ≤aix ≤ ai,0 < c_j ≤ cjx ≤ cj, ∀x ∈R, i

1,2, . . . , n, j 1,2, . . . , m;

H2the activation functionsfj, gi ∈ CR,Rand there exist positive constantsMj, Ni

such that

_fjx−fj

y≤Mjx−y, gix−gi

y≤Nix−y, 2.13

H3bi, dj ∈ CR,R, bi0 dj0 0, i 1,2, . . . , n, j 1,2, . . . , m,and there exist

positive constantsηi, ωjsuch that

bix−bi

y x−y ≥ηi,

djx−dj

y

x−y ≥ωj, ∀x /y; 2.14

H4the kernelshjiandkijdefined on0,∞Tare nonnegative continuous integral func-tions such that∞₀hjisΔs 1,

_∞

0 shjisΔs < ∞,

_∞

0 kijsΔs 1,

_∞

0 skijsΔs <

∞.

3. Existence and Uniqueness of the Equilibrium

In this section, using homeomorphism theory, we will study the existence and uniqueness of the equilibrium point of system1.1.

An equilibrium point of1.1is a constant vectorx∗₁, x∗₂, . . . , x_n∗, y∗₁, y₂∗, . . . , y_m∗T ∈Rnm

which satisfies the system

ai

x∗_i

⎡ ⎣bi

x_i∗−

m

j 1

p0

jip1ji

fj

y_j∗ri ⎤

⎦ 0, i 1,2, . . . , n,

cj

y∗_jdj

y∗_j−

n

i 1

q0_ijq1_ijgi

x_i∗sj

0, j 1,2, . . . , m,

3.1

where the impulsive jumpsIk·, Jk·satisfy

Ik

x_i∗ 0, i 1,2, . . . , n, Jk

y∗_j 0, j 1,2, . . . , m. 3.2

From the assumptionsH1andH4, it follows that

bi

x∗_i

m

j 1

p0_jip1_jifj

y∗_jri, i 1,2, . . . , n,

dj

y_j∗

n

i 1

q0

ijq1ij

gi

x_i∗sj, j 1,2, . . . , m.

3.3

Noting that ifb−_i1·, d−_j1· exist and activation functionsfj·and gj·are bounded, then

Theorem 3.1. Assume thatH2andH3hold. Suppose further that for eachi 1,2, . . . , n, j

1,2, . . . , m, the following inequalities are satisfied:

ηi> m

j 1

q0_ijq1_ijNi, ωj> n

i 1

p0_jip1_jiMj. 3.4

Then there exists a unique equilibrium point of system1.1.

Proof. Consider a mappingΦ:Rnm → Rnmdefined by

Φiz bixi− m

j 1

p0_jip1_jifj

yj

ri, i 1,2, . . . , n,

Φiz dj

yj

−n

i 1

q0_ijq1_ijgixi sj, j 1,2, . . . , m,

3.5

wherez x1, x2, . . . , xn, y1, y2, . . . , ymT ∈ Rnm,Φz Φ1z, . . . ,Φnz, . . . ,ΦnmzT ∈

Rnm_{. First, we want to show that Φ is an injective mapping on R}nm_{. By contradiction,}

suppose that there exists a distinct z, z ∈ Rnm such that Φz Φz, where z

x1, x2, . . . , xn, y1, y2, . . . , ymT ∈ Rnm andz x1, x2. . . , xn, y1, y2, . . . , ym

T _{∈ Rnm. Then}

it follows from3.5that

bixi−bixi m

j 1

p0

jip1ji

fj

yj

−fj

y_j, i 1,2, . . . , n,

dj

yj

−dj

y_j

n

i 1

q0

ijq1ij

gixi−gjxi

, j 1,2, . . . , m.

3.6

In view ofH2-H3and3.6, we have

n

i 1

ηi|xi−xi| ≤ n

i 1

m

j 1

p0_jip1_jiMjyj−y_j,

m

j 1

ωjyj−y_j≤ m

j 1

n

i 1

q_ij0 q1_ijNi|xi−xi|.

3.7

Thus, we can obtain

n

i 1

⎡ ⎣ηi−

m

j 1

q0_ijq1_ijNi ⎤

⎦_|xi−xi| m

j 1

ωj− n

i 1

p0_jip1_jiMj

yj−yj≤0. 3.8

It follows from3.4and3.8that|xi−xi| 0 and|yj−yj| 0,i 1,2, . . . , n, j 1,2, . . . , m.

Then we will proveΦis a homeomorphism onRnm. For convenience, we letΦ z

Φz−Φ0, where

Φiz bixi− m

j 1

p0_jip1_jifj

yj

−fj0

, i 1,2, . . . , n,

Φnjz dj

yj

−n

i 1

q_ij0 q1_ijgixi−gi0

, j 1,2, . . . , m.

3.9

We assert thatΦ → ∞asz → ∞. Otherwise there is a sequence{zv_{}such thatz}v _→

∞andΦ zv_{is bounded asv → ∞, wherez}v _xv

1, xv2, . . . , xvn, y1v, yv2, . . . , ymvT ∈ Rnm.

Noting that

n

i 1

sgnxv_ibi

xv_i−Φizv

n

i 1

sgnxv_i

m

j 1

p0_jip_ji1fj

y_jv−fj0

≤n

i 1

m

j 1

p_ji0 p_ji1Mjyvj,

m

j 1

sgnyv_jdj

y_jv−Φnjzv

m

j 1

sgnyv_j

n

i 1

q0_ijq1_ijgi

xv_i−gi0

≤m

j 1

n

i 1

q0_ijq_ij1Nixvi,

3.10

we have

n

i 1

sgnx_ivbi

x_iv−Φizv

m

j 1

sgnyv_jdj

yv_j−Φnjzv

≤n

i 1

m

j 1

p0_jip1_jiMjyjv m

j 1

n

i 1

q0_ijq_ij1Nixvi.

3.11

On the other hand, we have

n

i 1

sgnx_ivbi

x_iv−Φizv

m

j 1

sgnyv_jdj

yv_j−Φnjzv

≥n

i 1

ηixvi− n

i 1

_Φ izv

m

j 1

ωjyvj− m

j 1

_Φ

njzv.

It follows from3.11and3.12that

Θ ⎛ ⎝n

i 1

xv_i

m

j 1

y_jv

⎞ ⎠_≤n

i 1

_Φ izv

m

j 1

_Φ

njzv, 3.13

where

Θ min

⎧ ⎨ ⎩min1≤i≤n

⎧ ⎨ ⎩ηi−

m

j 1

q0_ijq1_ijNi ⎫ ⎬ ⎭,1min≤j≤m

ωj− n

i 1

p0_jip_ji1Mj ⎫_⎬

⎭>0. 3.14

That is

zv ≤ 1

Θ%%% Φzv%%%, 3.15

which contradicts our choice of {zv_{}. Hence, Φ satisfies Φ → ∞ as z → ∞.}

By Lemma 2.5, Φ is a homeomorphism on Rnm and there exists a unique point z∗

x∗₁, x∗₂, . . . , x_n∗, y₁∗, y∗₂, . . . , y∗_mT such thatΦz∗ 0. From the definition ofΦ, we know that

z∗ x∗₁, x∗₂, . . . , x∗_n, y₁∗, y∗₂, . . . , y_m∗Tis the unique equilibrium point of1.1.

4. Global Exponential Stability of the Equilibrium

In this section, we will construct some suitable Lyapunov functions to derive the sufficient conditions which ensure that the equilibrium of1.1is globally exponentially stable.

Theorem 4.1. Assume that (H1)–(H4) hold, suppose further that

H5for eachi 1,2, . . . , n, j 1,2, . . . , m, the following inequalities are satisfied:

a_iηi> m

j 1

cjq0ijq1ij

Ni, cjωj> n

i 1

aip0jip1ji

Mj 4.1

H6the impulsive operatorsIikxitandJjkyjtsatisfy

Iikxitk −γik

xitk−x∗i

, 0< γik<2, i 1, . . . , n, k∈Z,

Jjk

yjtk

−γjk

yjtk−y∗j

, 0<γjk <2, j 1, . . . , m, k∈Z.

4.2

Then the unique equilibrium point of system1.1is globally exponentially stable.

Proof. According toTheorem 3.1, we know that 1.1 has a unique equilibrium pointz∗

Suppose thatzt x1t, x2t, . . . , xnt, y1t, y2t, . . . , ymtT is an arbitrary solution of

1.1. Letuit xit−x∗_i, vjt yjt−y∗_j, t≥0, then system1.1can be rewritten as

uΔ_i t −aiuit ⎡

⎣biuit− m

j 1 p0_jifj

vj

t−τji

−m

j 1 p_ji1

_∞

0

hjisfj

vjt−s

Δs−ri ⎤ ⎦,

i 1,2, . . . , n, t >0, t /tk, t∈T,

vΔ_jt −cj

vjt

dj

vjt

−n

i 1 q0_ijgi

ui

t−σij

−n

i 1 q1_ij

_∞

0

kijsgiuit−sΔs−sj

,

j 1,2, . . . , m, t >0, t /tk, t∈T,

4.3

where, fori 1,2, . . . , n, j 1,2, . . . , m,

aiuit ai

uit x∗i

, biuit bi

uit x∗i

−bi

x∗_i,

cj

vjt

cj

vjt yj∗

, fivit fj

vjt y∗j

−fj

y_j∗,

dj

vjt

dj

vjt yj∗

−bi

y_j∗, giuit gi

uit x∗i

−bi

x∗_i.

4.4

Also, for allt tk, k∈Z, i 1,2, . . . , n, j 1,2, . . . , m,

ui

t_k xi

t_k−x∗_i xitk Ikxitk−xi∗ _1−γik

xitk−xi∗≤xitk−xi∗≤ |uitk|, vj

t_k yj

t_k−y_j∗ yjtk Ik

yjtk

−y_j∗

1−γjk

yjtk−yj∗≤yjtk−y∗j≤vjtk.

Hence byH1andH2, we have

D|uit|Δ≤ −a_iηi|uit|ai m

j 1

p0_jiMjvj

t−τji

ai m

j 1

p_ji1Mj

_∞

0

hjisvjt−sΔs, i 1,2, . . . , n,

4.6

DvjtΔ≤ −c_jvjtcj n

i 1

q_ij0Niui

t−σij

cj n

i 1

q1_ijNi

_∞

0

kijs|uit−s|Δs, j 1,2, . . . , m.

4.7

Also, fori 1,2, . . . , n,

xitk0−x∗itk0 xitk Iikxitk−x∗itk−Iik

x∗_itk

1−γik

xitk−x∗itk

, k∈Z,

4.8

thus

xi

t_k−x∗_it_k 1−γikxitk−x∗itk

≤xitk−x∗itk, i 1, . . . , n, k∈Z.

4.9

Similarly, we have

yj

t_k−y∗_jt_k 1−γ_jk∗yjtk−y∗jtk

≤yjtk−y∗jtk, j 1,2, . . . , m, k∈Z.

4.10

LetGiandG∗j be defined by

Giεi a_iηi−εi− m

j 1

cjq0_ijNieεi

σt, t−σij

−m

j 1

cjqij1Ni _∞

0

kijseεiσt, t−sΔs, i 1,2, . . . , n,

G∗_jξj

c_jωj−ξj− n

i 1

aip0_jiMjeξj

σt, t−τji

−n

i 1

aip1_jiMj _∞

0

hjiseξjσt, t−sΔs, j 1,2, . . . , m,

respectively, whereεi, ξj∈0,∞. ByH5, we have

Gi0 aiηi− m

j 1

cjq0ijq1ij

Ni>0, i 1,2, . . . , n,

G∗_j0 c_jωj− n

i 1

aip0jip1ji

Mj>0, j 1,2, . . . , m.

4.12

Since Gi, G∗j are continuous on0,∞andGiεi → −∞, G∗jξj → −∞, asεi → ∞, ξj →

∞, there existε∗_i, ξ_j∗>0 such thatGiε∗_i 0, G∗_jξ∗_j 0 andGiεi>0, forεi ∈0, ε∗_i, G∗_jξj>

0, forξj∈0, ξ∗_j. By choosingα min1≤i≤n,1≤j≤m{ε∗_i, ξ_j∗}, we obtain

Giα a_iηi−α− m

j 1

cjq0_ijNieα

σt, t−σij

−m

j 1

cjq1_ijNi _∞

0

kijseασt, t−sΔs

≥0, i 1,2, . . . , n,

G∗_jα c_jωj−α− n

i 1

aip0jiMjeα

σt, t−τji

−n

i 1

aipji1Mj _∞

0

hjiseασt, t−sΔs

≥0, j 1,2, . . . , m,

4.13

Denote

μit eαt, δ|uit|, t∈R, i 1,2, . . . , n, 4.14

νjt eαt, δvjt, t∈R, j 1,2, . . . , m, 4.15

whereδ ∈ −∞,0_T. Fort > 0, t /tk, k ∈Z, i 1,2, . . . , n, j 1,2, . . . , m,it follows from

4.6–4.15, we can obtain

DμΔ_i t αeαt, δ|uit|eασt, δD|uit|Δ

≤αeαt, δ|uit|eασt, δ

×

⎡

⎣−a_iηi|uit|ai m

j 1

p0_jiMjvj

t−τji

ai m

j 1

p1

jiMj _∞

0

≤ −a_iηi−α

μit ai m

j 1

p_ji0Mjeα

σt, t−τji

νj

t−τji

ai m

j 1

p_ji1Mj

_∞

0

hjiseασt, t−sνjt−sΔs,

Dν_jΔt≤ −c_jωj−α

νjt cj n

i 1

q0_ijNieα

σt, t−σij

μi

t−σij

ci n

i 1

q_ij1Ni

_∞

0

kijseασt, t−sμit−sΔs.

4.16

Also,

μi

t_k≤μitk, νj

t_k≤νjtk, i 1,2, . . . , n, j 1,2, . . . , m, k∈Z. 4.17

Define a Lyapunov function

Vt

n

i 1

⎡

⎣μit ai m

j 1

p0_jiMjeα

σt, t−τji t

t−τji

νjsΔs

ai m

j 1

p1_jiMj

_∞

0

hjiseασt, t−s t

t−s

νjzΔzΔs ⎤ ⎦

m

j 1

νjt cj n

i 1

q_ij0Nieα

σt, t−σij t

t−σij

μisΔs

cj n

i 1

q_ij1Ni

_∞

0

kijseασt, t−s _t

t−s

μizΔzΔs

.

4.18

And we note thatVt>0 fort >0 andV0>0. Calculating theΔ-derivatives ofV, we get

DVΔt≤

n

i 1

⎡

⎣₋a_iηi−η

μit ai m

j 1

p0

jiMjeα

σt, t−τji

νjt

ai m

j 1

p1_jiMj

_∞

0

hjiseασt, t−sνjtΔs ⎤ ⎦

m

j 1

−c_jωj−η

νjt cj n

i 1

q0_ijNieα

σt, t−σij

μit

cj n

i 1

q_ij1Ni

_∞

0

−n

i 1

⎡

⎣a_iηi−η− m

j 1

cjq0ijNieη

σt, t−σij

−m

j 1

cjqij1Ni _∞

0

kijseασt, t−sΔs ⎤ ⎦μit

−m

j 1

c_jωj−η− n

i 1

aip0_jiMjeη

σt, t−τji

−n

i 1

aip1jiMj _∞

0

hjiseασt, t−sΔs

νjt

−n

i 1 Gi

ημit− m

j 1

G∗_jηνjt

≤0, t >0, t /tk, t∈T, k∈Z.

4.19

Also,

Vt_k

n

i 1

⎡ ⎣μi

t_kai m

j 1

p0_jiMjeα

σt_k, t_k−τji t_k

t_k−τji

νjsΔs

ai m

j 1

p_ji1Mj

_∞

0

hjiseα

σt_k, t_k−s

t_k

t_k−s

νjzΔzΔs ⎤ ⎦

m

j 1

νj

t_k

n

i 1

q0_ijNieα

σt_k, t_k−σij t_k

t_k−σij

μisΔs

cj n

i 1

q1

ijNi _∞

0

kijseα

σt_k, t_k−s

t_k

t_k−s

μizΔzΔs

≤n

i 1

⎡

⎣μitk ai m

j 1

p0_jiMjeα

σtk, tk−τji tk

tk−τji

νjsΔs

ai m

j 1

p_ji1Mj

_∞

0

hjiseασtk, tk−s _tk

tk−s

νjzΔzΔs ⎤ ⎦

m

j 1

νjtk cj n

i 1

q0_ijNieα

σtk, tk−σij tk

tk−σij

μisΔs

cj n

i 1

q1_ijNi

_∞

0

kijseασtk, tk−s tk

tk−s

μizΔzΔs

Vtk, k∈Z.

It follows thatVt≤V0fort >0 and hence, fort >0, we can obtain

n

i 1 μit

m

j 1 νjt≤

n

i 1

⎡ ⎣μi0

m

j 1

p0_jiMjeα

σ0,−τji 0

0−τji

νjsΔs

ai m

j 1

p1_jiMj

_∞

0

hjiseασ0,0−s 0

0−s

νjzΔzΔs ⎤ ⎦

m

j 1

νj0 n

i 1

q_ij0Nieα

σ0,0−σij 0

0−σij

μisΔs

cj n

i 1

q1_ijNi

_∞

0

kijseασ0,0−s ₀

0−s

μizΔzΔs

.

4.21

In view of4.14-4.15and the previous inequality, we have

n

i 1

_xit−x∗

it m

j 1

yjt−yj∗t

≤e αt, δ

⎡ ⎣n

i 1

⎛ ⎝1

m

j 1

ciq0ijNieα

σ0,−σij

σij

m

i 1

ciq1ijNi _∞

0

kijseασ0,−ssΔs &

max

δ∈−∞,0T

ϕiδ−x∗iδ

m

j 1

'

1

n

i 1

aip0jiMjeα

σ0,−τji

τji

n

i 1

aipji1Mj _∞

0

hjiseασ0,−ssΔs &

max

δ∈−∞,0T

ψjδ−y∗jδ ⎤ ⎦

≤Ne αt, δ

⎡ ⎣n

i 1

max

δ∈−∞,0T

ϕiδ−x∗iδ m

j 1

max

δ∈−∞,0T

ψjδ−yj∗δ ⎤ ⎦, 4.22 where N max ⎧ ⎨ ⎩1 m

j 1

ciq_ij0Nieα

σ0,−σij

σij

m

j 1

ciqij1Ni _∞

0

kijseασ0,−ssΔs,1 n

i 1

aipji0Mjeα

σ0,−τji

τji

n

i 1

aipji1Mj _∞

0

hjiseασ0,−ssΔs

≥1.

4.23

5. An Example

In this section, we give an example to illustrate our results.

Consider the following Cohen-Grossberg BAM neural networks system with dis-tributed delays and impulses:

xΔ₁t −

(

11

3cosx1t

)

×

*

5x1t−

1 4sin

2y1t−1

−

_∞

0

1 4e

−s_sin2y

1t−s

Δsr1

+

,

t >0, t /tk, t∈T,

Δx1tk I1x1tk, k 1,2, . . . ,

yΔt −

(

11

3siny1t

)

×

*

3y1t−

1

4cos2x1t−1−

_∞

0

1 4e

−s_cos2x

1t−sΔss1

+

,

t >0, t /tk, t∈T,

Δy1tk J1

y1tk

, k 1,2, . . . ,

5.1

whereT R, γ1k 1 1/2sin1k,γ1k 1 2/3cos 2k, k∈Z. A simple computation

shows thata c 2/3, a c 4/3, M1 N1 1, η1 5, ω1 3, p011 p110 q110 q110

1/4,0< γ1k,γ1k<2. It is easy to check that all conditions of Theorems3.1and4.1are satisfied.

Hence,5.1has a unique equilibrium point, which is globally exponentially stable.

6. Conclusion

Using the time-scale calculus theory, the homeomorphism theory and the Lyapunov functional method, some sufficient conditions are obtained to ensure the existence and the global exponential stability of the unique equilibrium point of Cohen-Grossberg BAM neural networks with distributed delays and impulses on time scales. This is the first time applying the time-scale calculus theory to unify and improve impulsive Cohen-Grossberg BAM neural networks with distributed delays on time scales under the same framework. The sufficient conditions we obtained can easily be checked in practice by simple algebraic methods.

Acknowledgments

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