stability and periodicity for impulsive neural networks with delays

Shao and Zhang Advances in Difference Equations 2013, 2013:352http://www.advancesindifferenceequations.com/content/2013/1/352

RESEARCH Open Access

Stability and periodicity for impulsive neuralnetworks with delaysYuanfu Shao1* and Qianhong Zhang2

*Correspondence:[email protected] of Science, GuilinUniversity of Technology, GuilinGuangxi, 541004, P.R. ChinaFull list of author information isavailable at the end of the article

AbstractBy constructing a suitable Lyapunov functional and using some inequalities, weinvestigate the existence, uniqueness and global exponential stability of periodicsolutions for a class of generalized cellular neural networks with impulses andtime-varying delays. Finally, an illustrative example and simulations are given to showthe effectiveness of the main results.

Keywords: periodicity; stability; impulsive; delays

1 IntroductionThe dynamics of cellular neural networks has been deeply investigated due to its applica-bility in solving image processing, signal processing and pattern recognition problems [,]. Recently, the study of the existence and exponential stability of periodic solutions forneural networks has receivedmuch attention andmany known results have been obtained[–]. For example, authors [] considered the following neural networks with delays:

xi(t) = –ai(t)xi(t) +n∑j=

bij(t)fj(xj(t)

)+

n∑j=

cij(t)fj(xj

(t – τij(t)

))

+n∑j=

n∑l=

dijl(t)fj(xj(t)

)fl(xl(t)

)+

n∑j=

n∑l=

eijl(t)fj(t – σijl(t)

)

× fl(xl

(t – σijl(t)

))+ Ii(t), ()

when τij(t) = τij, σijl(t) = σijl are constants, a set of easily verifiable sufficient conditionsguaranteeing the existence and globally exponential stability of one periodic solution wasderived.However, in practice, impulsive effects are inevitably encountered in implementation of

networks, which can also be found in information science, electronics, automatic controlsystems and so on (see [, , –, , , –]). Thus it is necessary to study theimpulsive case of system ().Motivated by the discussion, in this paper, by employing some inequalities and con-

structing a suitable Lyapunov functional, we aim to investigate the existence and expo-nential stability of a periodic solution for a class of nonautonomous neural networks with

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impulses and time-varying delays as follows:


bij(t)fj(xj(t)

)+

n∑j=

cij(t)fj(xj

(t – τij(t)

))

+n∑j=

n∑l=

dijl(t)fj(xj(t)

)fl(xl(t)

)+

n∑j=

n∑l=


)

× fl(xl

(t – σijl(t)

))+ Ii(t), t �= tk ,

�xi(tk) = eik(xi

(t–k

)), t = tk ,

()

where i, j, l = , , . . . ,n, n corresponds to the number of units in the neural networks. xi(t)corresponds to the state variable at time t, fj is the activation of the neurons. Ii(t) is theexternal bias at time t. ≤ τij(t) ≤ τ , ≤ σijl(t)≤ σ corresponds to the transmission delayrespectively, τ = max τ+

ij , τ+ij = max≤t≤ω |τij(t)|, σ = maxσ +

ijl , σ +ilj = max≤t≤ω |σijl(t)|. The

fixed moments of time tk satisfy t < t < · · · , limk→∞ tk = ∞. �xi(tk) = xi(t+k ) – xi(t–k ),xi(tk) = xi(t+k ) and xi(t–k ) exists. eik(xi(t–k )) represents impulsive perturbations of the ith unitat tk , k = , , . . . .By appropriately choosing coefficients, system () contains many models as its special

cases, which were studied in [, , , , , ] respectively.Throughout this paper, for i, j, l = , , . . . ,n, k = , , . . . , ai(t), bij(t), cij(t), dijl(t), eijl(t),

τij(t), σijl(t), Ii(t) are all continuous ω-periodic functions and [,ω] ∩ {tk} = {t, t, . . . , tq}.Further, we suppose that:

(H) There exist positive constantsMi and Ni such that

∣∣fi(u) – fi(v)∣∣ ≤Mi|u – v|, ∣∣fi(u)∣∣ ≤Ni for all u, v ∈ R.

(H) There exists a constant Lik such that |eik(u) – eik(v)| ≤ Lik|u – v| for u, v ∈ R.(H) There exists a positive integer q such that tk+q = tk +ω, ei(k+q)(·) = eik(·).For convenience, we introduce the following notations:

a–i = mint∈[,ω]

ai(t) > , b+ij = max≤t≤ω

∣∣bij(t)∣∣, c+ij = max≤t≤ω

∣∣cij(t)∣∣,d+ijl = max

≤t≤ω

∣∣dijl(t)∣∣, e+ijl = max≤t≤ω

∣∣eijl(t)∣∣.This paper is organized as follows. In Section , preliminaries are introduced. In Sec-

tion , by constructing a suitable Lyapunov functional, the criteria ensuring global ex-ponential stability of a periodic solution for system () are established. In Section , anexample and simulations are shown to illustrate the validity of the main results. Finally,we conclude this paper with a brief discussion in Section .

2 PreliminariesFirstly, we introduce some definitions and lemmas. Let Rn be the space of n-dimensionalreal column vectors, and PC = {f : [–τ ∗, ] → Rn|f (s) is continuous for s /∈ (–τ ∗, ] ∩{tk}, f (s+) = f (s), f (s–) exists for s ∈ (–τ ∗, ] ∩ {tk},k = , , . . .}. PCt = {f : [–τ ∗, ] → Rn|f (s)



is continuous for s /∈ (–τ ∗, ]∩ {tk – t}, f (s+) = f (s), f (s–) exists for s ∈ (–τ ∗, ]∩ {tk – t},k =, , . . .}.A function x(t) = (x(t),x(t), . . . ,xn(t))T : [–τ ∗, +∞)→ Rn is called a solution of () with

the initial condition given by

x(s) = φ(s) for all s ∈ [–τ ∗,

],φ ∈ PC, ()

if x(t) ∈ PC[[, +∞),Rn] satisfies () and (), where τ ∗ = max≤i,j,l≤n{maxt∈[,∞)(|τij(t)|,|σijl(t)|)}. We denote a solution through φ by x(t,φ) or xt(φ), xt(s,φ) = x(t + s,φ) for alls ∈ [–τ ∗, ], t ≥ . Obviously, any solution x(t,φ) of () is continuous at t �= tk and right-hand continuous at t = tk , t ≥ . In addition, we define ‖φ‖ = sup–τ∗≤s≤(

∑ni= |φi(s)|r) r ,

where φ = (φ,φ, . . . ,φn) ∈ PC and r ≥ is a constant.

Definition System () is said to be globally exponentially stable if for any two solutionsx(t,φ) and y(t,ψ), there exist some constants r > andM ≥ such that

∣∣x(t,φ) – y(t,ψ)∣∣ ≤M‖φ –ψ‖e–rt

for all t > .

Definition Let f (t) : R → R be a continuous function, then the Dini right derivative off (t) is defined as

D+f (t)dt

= lim suph→+

f (t + h) – f (t)h

.

From Definition , we can easily obtain the following lemma.

Lemma [] Let f (t) be a continuous function on R. If f (t) is differentiable at t, then

D+∣∣f (t)∣∣ =⎧⎪⎨⎪⎩f (t) if f (t) > or f (t) = and f (t) > ;–f (t) if f (t) < or f (t) = and f (t) < ; if f (t) = and f (t) = .

Next, we introduce two important inequalities which play a key role in obtaining themain results.

Lemma (Beckenbach and Bellman) [, ] For a ≥ , bk ≥ , k = , , . . . ,m, the follow-ing inequality holds:

am∏k=

bqkk ≤ r

m∑k=

qkbrk +rar ,

where qk > are some constants and∑m

k= qk = r – , r > .

Lemma (Halanay inequality) [, ] Assume that p, q are constants satisfying p > q > .g(t) is a continuous nonnegative function on [t – τ , t] satisfying D+g(t) ≤ –pg(t) + q‖gt‖



for all t ≥ t, where ‖gt‖ = sup–τ≤s≤ |g(t + s)|. Then, for all t ≥ t, we have

g(t) ≤ ‖gt‖e–λ(t–t),

where λ is the unique positive root of the equation λ – p + qeλτ = .

3 Main resultsIn this section, we construct a suitable Lyapunov functional to study the existence andglobal exponential stability of periodic solutions of system ().

Theorem Suppose that (H)-(H) hold. Further,

(H) There exist r > , qk > , λi > , αki, βki, γki and ηki ∈ R (i = , , . . . ,n, k = , , . . . ,m)such that p > q > ,

∑mk= qk = r – , where

p = max≤i≤n

[ra–i –

n∑j=

b+jiλj

λiMr(–

∑mk= αki)

i –n∑j=

b+ijm∑k=

qkMrαkjqk

j –n∑j=

c+ijm∑k=

qkMrβkjqk

j

–n∑j=

n∑l=

(d+ijl + d+

ilj)Nl

m∑k=

qkMrγkjqkj –

n∑j=

n∑l=

(d+jil + d+

jli)λj

λiNlM

–∑m

k= γkii

–n∑j=

n∑l=

(e+ijl + e+ilj

)Nl

m∑k=

qkMrηkjqkj

],

q = max≤i≤n

[ n∑j=

c+jiλj

λiMr(–

∑mk= βki)

i +n∑j=

n∑l=

λj

λi

(e+jil + e+jli

)NlM

r(–∑m

k= ηki)i

].

(H) There is < μ < λ such that ln[(+Lk)eλτr ] ≤ μ(tk–tk–)

r , k = , , . . . , Lk = max≤i≤n{Lik},λ is the unique positive solution of the equation λ – p + qeλτ = .

Then system () has a unique ω-periodic solution which is globally exponentially stable.

Proof Let x(t) = (x(t),x(t), . . . ,xn(t))T and y(t) = (y(t), y(t), . . . , yn(t))T be two solutionsof () through φ and ψ , respectively, where φ,ψ ∈ PC, then we have

d(xi(t) – yi(t))dt

= –ai(t)(xi(t) – yi(t)

)+

n∑j=

bij(t)(fj(xj(t)

)– fj

(yj(t)

))+

n∑j=

cij(t)(fj(xj

(t – τij(t)

))– fj

(yj

(t – τij(t)

)))

+n∑j=

n∑l=

dijl(t)(fj(xj(t)

)fl(xl(t)

)– fj

(yj(t)

)fl(yl(t)

))

+n∑j=

n∑l=

eijl(t)(fj(xj

(t – σijl(t)

))fl(xl

(t – σill(t)

))

– fj(yj

(t – σijl(t)

))fl(yl

(t – σijl(t)

)))xi(tk) – yi(tk)

= xi(t–k

)– yi

(t–k

)+ eik

(xi

(t–k

))– eik

(yi

(t–k

)). ()



Let zi(t) = xi(t) – yi(t). Consider the following Lyapunov functional:

V (t) =n∑i=

λi∣∣xi(t) – yi(t)

∣∣r = n∑i=

λi∣∣zi(t)∣∣r .

Calculating the Dini upper right derivative ofV (t) along the solution of () at a continuouspoint t �= tk and applying Lemma , we have

D+(V (t) =n∑i=

λiD+∣∣xi(t) – yi(t)∣∣r = n∑

i=λir

∣∣xi(t) – yi(t)∣∣r–D+∣∣xi(t) – yi(t)

∣∣

=n∑i=

λir∣∣xi(t) – yi(t)

∣∣r– sgn(xi(t) – yi(t)

)

×{–ai(t)

(xi(t) – yi(t)

)+

n∑j=

bij(t)(fj(xj(t)

)– fj

(yj(t)

))

+n∑j=

cij(t)(fj(xj

(t – τij(t)

))– fj

(yj

(t – τij(t)

)))

+n∑j=

n∑l=

dijl(t)(fj(xj(t)

)fl(xl(t)

)– fj

(yj(t)

)fl(yl(t)

))

+n∑j=

n∑l=

eijl(t)(fj(xj

(t – σijl(t)

))fl(xl

(t – σijl(t)

))

– fj(yj

(t – σijl(t)

))fl(yl

(t – σijl(t)

)))}

≤n∑i=

λir{–ai(t)

∣∣xi(t) – yi(t)∣∣r + n∑

j=b+ijMj

∣∣xi(t) – yi(t)∣∣r–∣∣xj(t) – yj(t)

∣∣

+n∑j=

c+ijMj∣∣xi(t) – yi(t)

∣∣r–∣∣xj(t – τij(t))– yj

(t – τij(t)

)∣∣

+n∑j=

n∑l=

e+ijl[NlMj

∣∣xi(t) – yi(t)∣∣r–∣∣xj(t – σijl(t)

)– yj

(t – σijl(t)

)∣∣+NjMl

∣∣xi(t) – yi(t)∣∣r–∣∣xl(t – σijl(t)

)– yl

(t – σijl(t)

)∣∣]+

n∑j=

n∑l=

d+ijl[NlMj

∣∣xi(t) – yi(t)∣∣r–∣∣xj(t) – yj(t)

∣∣

+NjMl∣∣xi(t) – yi(t)

∣∣r–∣∣xl(t) – yl(t)∣∣]}

=n∑i=

λir{–ai(t)

∣∣zi(t)∣∣r + n∑j=

b+ijMj∣∣zi(t)∣∣r–∣∣zj(t)∣∣

+n∑j=

c+ijMj∣∣zi(t)∣∣r–∣∣zj(t – τij(t)

)∣∣



+n∑j=

n∑l=

e+ijl[NlMj

∣∣zi(t)∣∣r–∣∣zj(t – σijl(t))∣∣ +NjMl

∣∣zi(t)∣∣r–∣∣zl(t – σijl(t))∣∣]

+n∑j=

n∑l=

d+ijl[NlMj

∣∣zi(t)∣∣r–∣∣zj(t)∣∣ +NjMl∣∣zi(t)∣∣r–∣∣zl(t)∣∣]

}

=n∑i=

λir{–ai(t)

∣∣zi(t)∣∣r + n∑j=

b+ijM–

∑mk= αkj

j∣∣zj(t)∣∣ m∏

k=Mαkj

j∣∣zi(t)∣∣qk

+n∑j=

c+ijM–

∑mk= βkj

j∣∣zj(t – τij(t)

)∣∣ m∏k=

Mβkjj

∣∣zi(t)∣∣qk

+n∑j=

n∑l=

(d+ijl + d+

ilj)NlM

–∑m

k= γkjj

∣∣zj(t)∣∣ m∏k=

Mγkjj

∣∣zi(t)∣∣qk

+n∑j=

n∑l=

(e+ijl + e+ilj

)NlM

–∑m

k= ηkjj

∣∣zj(t – σijl(t))∣∣ m∏

k=Mηkj

j∣∣zi(t)∣∣qk

}

≤n∑i=

λi

{–ra–i

∣∣zi(t)∣∣r + n∑j=

b+ijMr(–

∑mk= αkj)

j∣∣zj(t)∣∣r + n∑

j=b+ij

m∑k=

qkMrαkjqk

j∣∣zi(t)∣∣r

+n∑j=

c+ijMr(–

∑mk= βkj)

j∣∣zj(t – τj(t)

)∣∣r + n∑j=

c+ijm∑k=

qkMrβkjqk

j∣∣zi(t)∣∣r

+n∑j=

n∑l=

(d+ijl + d+

ilj)Nl

[ m∑k=

qkMrγkjqkj

∣∣zi(t)∣∣r +Mr(–∑m

k= γkj)j

∣∣zj(t)∣∣r]

+n∑j=

n∑l=

(e+ijl + e+ilj

)Nl

[ m∑k=

qkMrηkjqkj

∣∣zi(t)∣∣r +Mr(–∑m

k= ηkj)j

∣∣zj(t – σijl(t))∣∣r]}

≤n∑i=

λi

[–ra–i +

n∑j=

b+jiλj

λiMr(–

∑mk= αki)

i +n∑j=

b+ijm∑k=

qkMrαkjqk

j

+n∑j=

c+ijm∑k=

qkMrβkjqk

j +n∑j=

n∑l=

(d+ijl + d+

ilj)Nl

m∑k=

qkMrγkjqkj

+n∑j=

n∑l=

(d+jil + d+

jli)λj

λiNlM

r(–∑m

k= γki)i

+n∑j=

n∑l=

(e+ijl + e+ilj

)Nl

m∑k=

qkMrηkjqkj

]∣∣zi(t)∣∣r

+n∑i=

λi

[ n∑j=

c+jiλj

λiMr(–

∑mk= βki)

i +n∑j=

n∑l=

λj

λi

(e+jil + e+jli

)NlM

r(–∑m

k= ηki)i

]‖zi‖r

≤ –pV (t) + q‖Vt‖. ()

For any t ∈ [t, t), by (H) and Lemma , we have

∣∣V (t)∣∣ ≤ ‖Vt‖e–λ(t–t), ()



i.e.,

∣∣V (t)∣∣/r ≤ ‖Vt‖/re–λ(t–t)/r .

Also, it follows from (H) that

∣∣V (t)∣∣/r ≤

{ n∑i=

λi(∣∣xi(t– )

– yi(t–

)∣∣ + Li∣∣xi(t– )

– yi(t–

)∣∣)r}/r

={ n∑

i=λi( + Li)r

∣∣xi(t– )– yi

(t–

)∣∣r}/r

≤ ( + L)‖Vt‖/re–λ(t–t)/r . ()

Therefore, from () and (), for any t ∈ [t, t], we have

∣∣V (t)∣∣/r ≤ ( + L)‖Vt‖/re–λ(t–t)/r . ()

Similar to (), for all t ∈ [t, t), we can derive that

∣∣V (t)∣∣/r ≤ ∥∥V (t)

∥∥/re–λ(t–t)/r

={

sup–τ∗≤s≤

∣∣V (t + s)∣∣}/r

e–λ(t–t)/r

≤ ( + L)‖Vt‖/re–λ(t–τ–t)/re–λ(t–t)/r ,

≤ ( + L)eλτ‖Vt‖/re–λ(t–t)/r .

Again by (H), then

∣∣V (t)∣∣/r ≤

{ n∑i=

λi(∣∣xi(t– )

– yi(t–

)∣∣ + Li∣∣xi(t– )

– yi(t–

)∣∣)r}/r

≤ ( + L)( + L)eλτ‖Vt‖/re–λ(t–t)/r .

Thus, for any t ∈ (t, t],

∣∣V (t)∣∣/r ≤ ( + L)( + L)eλτ‖Vt‖/re–λ(t–t)/r .

By mathematical induction, for all t ∈ [tk , tk–), we have

∣∣V (t)∣∣/r ≤ ( + L)( + L) · · · ( + Lk)ekλτ‖Vt‖/re–λ(t–t)/r .

According to (H), ( + Lk)eλτ /r ≤ eμ(tk–tk–)/r for all k = , , . . . , hence

∣∣V (t)∣∣/r ≤ eμ(t–t)/reμ(t–t)/r · · · eμ(tk–tk–)/r‖Vt‖/re–λ(t–t)/r

≤ ‖Vt‖/re–(λ–μ)(t–t)/r .



Therefore,

∣∣V (t)∣∣/r ≤ ‖Vt‖/re–(λ–μ)(t–t)/r for all t ≥ t,

i.e.,

∣∣x(t,φ) – y(t,ψ)∣∣ ≤ r

√λmax

λmine–(λ–μ)(t–t)/r‖φ –ψ‖ for all t ≥ t,

where λmax = max≤i≤n{λi}, λmin = min≤i≤n{λi}. This implies that system () is globallyexponentially stable.Next, we prove the existence of a periodic solution of (). Define a Poincáre mapping

T : PC → PCω as follows:

T(φ) = xω(φ) for any φ ∈ PC.

By the periodicity of (), we can derive that Tm(φ) = xkω(φ) for any integer m > . Fromthe above conclusion, we can choose a positive integerm such that r

√λmaxλmin

e–(λ–μ)(t–mω)/r < .According to the periodicity of {tk} and PC = PCω , we have

∥∥Tmφ – Tmψ∥∥ =

∥∥xmω(φ) – xmω(ψ)∥∥ < ‖φ –ψ‖.

Then operator Tm is a construction mapping in PC. Obviously, PC is a Banach space.By virtue of the Banach fixed point theorem, Tm has a unique fixed point φ∗ ∈ PC.On the other hand, Tm(Tφ∗) = T(Tmφ∗) = T(φ∗), then Tφ∗ is also a fixed point of Tm.By the uniqueness of fixed point, we obtain that Tφ∗ = φ∗. Therefore, system () hasa unique ω-periodic solution which is globally exponentially stable. The proof is com-plete. �

If there is no impulse, system () reduces to () studied in []. From the proof of Theo-rem , we have the following corollary.

Corollary Suppose that (H)-(H) hold.Then system () has a uniqueω-periodic solutionwhich is globally exponentially stable.

Remark In [], by using theMawhin continuation theorem and constructing Lyapunovfunctionals, the authors studied the existence and stability of system (), where the de-lays are constant. However, in this paper, for time-varying delays, by employing many dif-ferent analysis techniques from [], we establish the criteria ensuring the existence andexponential stability of a periodic solution of (). Furthermore, if we take r = , m = ,qk = , λi = , αki = βki = γki = ηki = /, then condition (H) is transformed into (H)′ asfollows:

(H)′ a–i –n∑i=

b+jiMi –n∑j=

c+jiMi –n∑j=

n∑l=

(d+jil +d+

jli)NlMi –

n∑j=

n∑l=

(e+jil + e+jli

)NlMi > ,



which is just the corresponding condition in []. One can derive the same results as [].That is, the criteria in [] are the special case of Theorem . Therefore, we extend andimprove the earlier results in this sense.If dijl(t) = , then system () is transformed into the following model studied in

[].


bij(t)fj(xj(t)

)

+n∑j=

cij(t)fj(xj

(t – τij(t)

))+

n∑j=

n∑l=

eijl(t)

× fj(t – σijl(t)

)fl(xl

(t – σijl(t)

))+ Ii(t), t �= tk , ()

�xi(tk) = eik(xi

(t–k

)), t = tk .

Using a similar proof as above, one can obtain the main result of []. It is omittedhere.If cij(t) = dijl(t) = eijl(t) = , then () is transformed into the following model:


bij(t)fj(xj(t)

)+ Ii(t), t �= tk ,

�xi(tk) = eik(xi

(t–k

)), t = tk .

()

Similarly, we can obtain the sufficient conditions of the existence and exponential stabilityof a periodic solution to system (), we omit it.If one takes r = ,m = , qk = , λi = , αki = /, then (H) is reduced to

(H)∗ a–i –n∑j=

b+jiMi > .

The following corollary can be derived.

Corollary Suppose that (H)-(H), (H)∗, (H) hold, then system () has a unique ex-ponentially stable ω-periodic solution.

Remark Corollary is just the result of []. In [], the impulses are required to belinear functions. However, without the above restrictions, we also establish the criteriaensuring the existence of an ω-periodic solution which is exponentially stable. Thus weextend and generalize the earlier results.

In addition, if dijl(t) = eijl(t) = , () reduces to the model studied in []. Similar resultscan be derived. In [], the delay functions are required to be differential, but we do notneed the restriction here.On the other hand, in real world, by appropriately choosing parameters r,m, qk , αki, βki,

γki, ηki, one can see thatmany known assumptions can be included as special cases of (H),hence the discussion is interesting and valuable.



Figure 1 The dynamics of system (11). (a) Time series of x1(t), (b) time series of x2(t), (c) phase portrait ofx1(t) and x2(t).

4 An illustrative example and simulationsIn this section, we give an example and simulations to show the validity of themain results.

Example Let


bij(t)fj(xj(t)

)+

n∑j=

cij(t)fj(xj

(t – τij(t)

))

+n∑j=

n∑l=

dijl(t)fj(xj(t)

)fl(xl(t)

)+

n∑j=

n∑l=


)

× fl(xl

(t – σijl(t)

))+ Ii(t), t �= tk ,

�xi(tk) = eik(xi

(t–k

)), t = tk ,

()



where

n = , fj(x) =|x + | – |x – |

, a(t) = + sin t,

a(t) = + cos t, τij(t) = π , σijl(t) = π , eik(xi

(t–k

))= –.xi(tk),

B(t) =(bij(t)

)× =

(sin t cos t

cos t/ sin t/

), C(t) =

(cij(t)

)× =

( cos t sin tsin t/ cos t/

),

dijl(t) = eijl(t) = , tk = kπ .

Take r = , m = , qk = , α = βki = γki = δki = /. It is easy to verify that conditions ofTheorem () hold. Therefore, by Theorem , system () has a unique π-periodic solution,which is globally exponentially stable. By numerical analysis, the conclusion can be showedclearly, see Figure .

5 ConclusionIn this paper, by using the Halanay inequality and constructing Lyapunov functions, weaddress the existence and exponential stability of periodic solutions for a class of gener-alized cellular neural networks with impulses and time-varying delays. Easily verifiablesufficient conditions are obtained. The main results extend and improve some previouslyknown results [, , , ]. The criteria possess many adjustable parameters which pro-vide flexibility for the design and analysis of a dynamical system. It is interesting and valu-able.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

Author details1College of Science, Guilin University of Technology, Guilin Guangxi, 541004, P.R. China. 2Guizhou Key Laboratory ofEconomics System, Guizhou College of Finance and Economics, Guiyang Guizhou, 550004, P.R. China.

AcknowledgementsThis paper is supported by National Natural Science Foundation of P.R. China (11161015, 11361012, 11161011), NaturalScience Foundation of Guangxi (2013GXNSFAA019003, 2013GXNSFDA019001), and partially supported by National HighTechnology Research and Development Program 863 of P.R. China.

Received: 10 August 2013 Accepted: 11 November 2013 Published: 03 Dec 2013

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10.1186/1687-1847-2013-352Cite this article as: Shao and Zhang: Stability and periodicity for impulsive neural networks with delays. Advances inDifference Equations 2013, 2013:352


http://dx.doi.org/10.1016/j.na.2007.03.024

http://dx.doi.org/10.1155/2009/491268

stability and periodicity for impulsive neural networks with delays

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