R E S E A R C H
Open Access
Existence and exponential stability of
anti-periodic solutions for HCNNs with
time-varying leakage delays
Aiping Zhang
**Correspondence:
[email protected] School of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, P.R. China
Abstract
This paper is concerned with a class of high-order cellular neural networks (HCNNs) model with time-varying delays in the leakage terms. By using the Lyapunov functional method and differential inequality techniques, we establish sufficient conditions on the existence and exponential stability of anti-periodic solutions for the model. Our results complement some recent ones.
MSC: 34C25; 34D40
Keywords: high-order cellular neural networks; anti-periodic solution; exponential stability; time-varying delay; leakage term
1 Introduction
In the past decade, high-order cellular neural networks (HCNNs) have attracted much attention due to their wide range of applications in many fields such as signal and image processing, pattern recognition, optimization, and many other subjects. There have been extensive results on the problem of global stability of periodic solutions and anti-periodic solutions of HCNNs in the literature (see [–]). Recently, some attention has been paid to neural networks with time delay in the leakage (or ‘forgetting’) term (see [–]). In particular, Xu [] considered the existence and exponential stability of the anti-periodic solutions for the following HCNNs with time-varying delays in the leakage terms:
xi(t) = –ci(t)xit–δi(t)
+
n
j=
aij(t)fjxjt–τij(t)
+
n
j=
n
l=
bijl(t)gjxjt–αijl(t)
glxlt–βijl(t)
+
n
j=
n
l=
dijl(t)
∞
σijl(u)hj
xj(t–u)du
∞
νijl(u)hl
xl(t–u)du
+Ii(t), i= , , . . . ,n, (.)
in whichncorresponds to the number of units in a neural network,xi(t) corresponds to the state vector of theith unit at the timet,ci(t) represents the rate with which theith unit will reset its potential to the resting state in isolation when disconnected from the
network and external inputs,aij(t),bijl(t) anddijl(t) are the first- and second-order con-nection weights of the neural network,δi(t)≥ corresponds to the time-varying leakage
delays,αijl(t)≥,βijl(t)≥ andτij(t)≥ correspond to the transmission delays,σijl(u)
andνijl(u) correspond to the transmission delay kernels,Ii(t) denotes the external inputs
at timet,fj,gjandhjare the activation functions of signal transmission.
The initial conditions associated with system (.) are of the form
xi(s) =ϕi(s), s∈(–∞, ],i= , , . . . ,n, (.)
whereϕi(·) denotes a real-valued bounded continuous function defined on (–∞, ].
Under some suitable conditions on coefficients of (.), the author in [] derived some new sufficient conditions ensuring that all solutions of system (.) converge ex-ponentially to the anti-periodic solution, but the result leaves room for improvement. In fact, in the proof of Lemma ., the expression xi(t) –xi(t–δi(t)) =
t t–δi(t)x
i(u)du
was used, and the author replaced xi(u) by the right-hand side of equation (.). The case thatt–δi(t) < is possible, so the integration
t t–δi(t)x
i(u)dushould be handled as
t t–δi(t)x
i(u)du=
t–δi(t)x
i(u)du+
t
xi(u)du, for
t
xi(u)ducan be replaced by the
right-hand side of equation (.), but fort–δ
i(t)x
i(u)ducannot be replaced by the right-hand
side of equation (.). A similar error also occurs in Lemma . of []. For this reason, the course of proof in Lemmas . and . is not true. Motivated by this, we shall give a new proof to ensure the existence and exponential stability of the anti-periodic solutions for system (.). Moreover, an example is also provided to illustrate the effectiveness of our results.
Letu(t) :R→Rbe continuous int.u(t) is said to beT-anti-periodic onRif
u(t+T) = –u(t) for allt∈R.
Throughout this paper, fori,j,l= , , . . . ,n, it will be assumed thatci,Ii,aij,bijl,dijl:R→R
andδi,τij,αijl,βijl :R→[, +∞) are bounded continuous functions,σijl,νijl: [, +∞)→ Rare continuous functions, ciis bounded above and below by positive constants, δiis a bounded continuous function,|σijl(t)|eκt and|νijl(t)|eκt are integrable on [, +∞) for a
certain positive constantκ, and
ci(t+T) =ci(t), aij(t+T)fj(v) = –aij(t)fj(–v), (.)
bijl(t+T)gj(vj)gl(vl) = –bijl(t)gj(–vj)gl(–vl), (.)
dijl(t+T)
∞
σijl(u)hj
vj(t–u)du
∞
νijl(u)hl
vl(t–u)du
= –dijl(t)
∞
σijl(u)hj
–vj(t–u)du
∞
νijl(u)hl
–vl(t–u)du, (.)
δi(t+T) =δi(t), τij(t+T) =τij(t), Ii(t+T) = –Ii(t), (.)
αijl(t+T) =αijl(t), βijl(t+T) =βijl(t), (.)
For bounded continuous functionsf, we set
f–=inf
t∈Rf(t), f
+=sup
t∈R
f(t).
In order to investigate the anti-periodic solution of HCNNs (.), we also give some usual assumptions.
(H) There exist nonnegative constantsLfj,L g j,Lhj,M
g
j andMjhsuch that
fj(u) –fj(v)≤Lfj|u–v|, gj(u) –gj(v)≤Lgj|u–v|, hj(u) –hj(v)≤Lhj|u–v|
and
gj(u)≤Mgj, hj(u)≤Mhj,
whereu,v∈R,j= , , . . . ,n.
(H) For allt> andi∈ {, , . . . ,n}, there exist positive constantsξ,ξ, . . . ,ξnandη∗
such thatc+iδi+< , and
–ci(t) – c+iδ+i–ci(t) – –δi(t)cit–δi(t)
–c+
iδ+i ξi
+
n
j=
aij(t)Lfj
–c+
jδj+ ξj
+
n
j=
n
l=
bijl(t)Lg j
–c+jδ+j ξjM
g l +M
g j
–c+lδ+l ξlL
g l
+
n
j=
n
l=
dijl(t)
∞
σijl(u)Lhjdu
–c+jδj+ξj
∞
νijl(u)duMhl
+Mjh
∞
σijl(u)du
∞
νijl(u)Lhldu
–c+
jδ+j ξj
< –η∗.
2 Preliminary lemmas and main results
Lemma . Let(H)and(H)hold.Suppose that x(t) = (x(t),x(t), . . . ,xn(t))Tis a solution
of system(.)with the initial conditions
ϕi(t) –
t t–δi(t)
ci(s)ϕi(s)ds
<ξi
γ
η∗, t∈(–∞, ],i= , , . . . ,n, (.)
where
γ =max
i∈In
n
j=
a+ijfj()+
n
j=
n
l=
b+ijlgj()Mgl
+
n
j=
n
l=
d+ijl
∞
σijl(u)duhj()
∞
νijl(u)duMhl +Ii+
Then
xi(t) –
t t–δi(t)
ci(s)xi(s)ds<ξi γ
η∗ (.)
and
xi(t)≤ ξi γ η∗ –c+
iδi+
(.)
for all t≥,i= , , . . . ,n.
Proof Suppose (.) holds. Then, for a givenˆt≥ andi∈Jn={, , . . . ,n}, we have
xi(t)≤xi(t) –
t t–δi(t)
ci(s)xi(s)ds+
t t–δi(t)
ci(s)xi(s)ds
<ξi γ
η∗ +c
+
iδ+i sup s∈(–∞,ˆt]
xi(s) for allt∈(–∞,ˆt]
and
xi(t)≤ sup
s∈(–∞,ˆt]
xi(s)
<ξi γ
η∗ +c
+
iδ+i sup s∈(–∞,ˆt]
xi(s) for allt∈(–∞,ˆt],
which combined with (H) implies that (.) holds. Therefore, it suffices to prove (.). We achieve this by way of contradiction. Let
Xi(t) =xi(t) –
t t–δi(t)
ci(s)xi(s)ds.
Suppose that (.) does not hold. Then there existi∈Jnandt∗> such that
Xi
(t∗)=ξi γ
η∗ and (.) holds for allt∈(–∞,t∗) andi∈Jn. (.)
It follows that (.) holds for allt∈(–∞,t∗) andi∈Jn. From (.), we have
d dtXi(t)
=xi(t) –ci(t)xi(t) –
–δi (t)
ci
t–δi(t)
xi
t–δi(t)
= –ci(t)xi(t) –
–δi(t)ci
t–δi(t)
xi
t–δi(t)
+
–ci(t)xi
t–δi(t)
+
n
j=
aij(t)fj
xjt–τij(t)
+
n
j=
n
l=
bijl(t)gj
xjt–αijl(t)
glxlt–βijl(t)
+
n
j=
aij(t∗)L f j
–c+
jδj+ ξj+
n
j=
n
l=
bijl(t∗)L g j
ξj
–c+
jδj+ Mgl
+
n
j=
n
l=
dijl(t∗)
∞
σijl(u)duL h j
ξj
–c+jδj+
∞
νijl(u)M h ldu
γ
η∗ +γ
< –η∗γ
η∗+γ
= .
This contradicts with D–|X
i(t∗)| ≥ and hence (.) is proved. This completes the
proof.
Remark . In view of the boundedness of this solution in Lemma ., from the theory of functional differential equations with infinite delay in [], it follows that the solution of system (.) with initial conditions satisfying (.) can be defined on [, +∞).
Lemma . Suppose that (H)-(H) are true. Let x∗(t) = (x∗(t),x∗(t), . . . ,x∗n(t))T be the solution of system (.) with initial value ϕ∗ = (ϕ∗(t),ϕ∗(t), . . . ,ϕn∗(t))T, and let x(t) =
(x(t),x(t), . . . ,xn(t))Tbe the solution of system(.)with initial valueϕ= (ϕ(t),ϕ(t), . . . ,
ϕn(t))T.Then there exists a positive constant r such that
xi(t) –x∗i(t) =Oe–rt, i∈Jn.
Proof In view of (H), using a similar argument as that in the proof of (.) in [], we can chooseκ>r> andη> such thatci(t) >r, and
–ci(t) –r – c+iδi+–ci(t)erδi(t)– –δ i(t)
cit–δi(t)
–c+
iδi+ ξi
+
n
j=
aij(t)Ljferτij(t)
–c+
jδ+j ξj
+
n
j=
n
l=
bijl(t)
Lgjerαijl(t)
–c+
jδj+
ξjMgl +Mgjerβijl(t)
–c+
lδ+l ξlLgl
+
n
j=
n
l=
dijl(t)
∞
σijl(u)eruLhjdu
–c+jδ+j ξj
∞
νijl(u)duMhl
+Mjh
∞
σijl(u)du
∞
νijl(u)eruLhl du
–c+
jδj+ ξj
< –η, t≥,i∈Jn. (.)
Lety(t) =x(t) –x∗(t). Then
yi(t)
= –ci(t)yit–δi(t)
+
n
j=
aij(t)fjyjt–τij(t)
+x∗jt–τij(t)
–fjx∗jt–τij(t)
×glylt–βijl(t)
+x∗lt–βijl(t)
–gjx∗jt–αijl(t)
glx∗lt–βijl(t)
+
n
j=
n
l=
dijl(t)
∞
σijl(u)hj
yj(t–u) +x∗j(t–u)du
×
∞
νijl(u)hl
yl(t–u) +x∗l(t–u)
du
–
∞
σijl(u)hj
x∗j(t–u)du
∞
νijl(u)hl
x∗l(t–u)du
, (.)
where
Yi(t) =ertyi(t) –
t t–δi(t)
ci(s)ersyi(s)ds, i∈Jn.
Denote
M=max
≤i≤n
sup
s∈(–∞,]
Yi(s).
There existsK> such that
Yi(t)≤M<Kξi for allt∈(–∞, ] andi∈In.
We claim that
Yi(t)<Kξi for allt> andi∈Jn. (.)
Otherwise, there existi∈Jnandθ> such that
Yi(θ)=Kξi and Yj(t)<Kξj for allt∈(–∞,θ) andj∈Jn.
It follows that fort∈(–∞,θ] andj∈Jn,
ertyj(t)≤ertyj(t) –
t t–δj(t)
cj(s)ersyj(s)ds+
t t–δj(t)
cj(s)ersyj(s)ds ≤Kξj+c+jδ+j sup
s∈(–∞,θ]
ersyj(s) (.)
and hence
ertyj(t)≤ sup
s∈(–∞,θ]
ersyj(s)≤ Kξj –c+
jδj+
. (.)
Then, for the upper left derivative of|Yi(t)|, from (.), (.), (.) and (H), we have
≤D–Yi(θ)
≤–ci(θ) –rYi(θ) +–ci(θ) –r
θ
θ–δi(θ)
+
which is a contradiction. This proves (.), which produces
Remark . Ifx∗(t) = (x∗(t),x∗(t), . . . ,x∗n(t))T is theT-anti-periodic solution of system
(.), it follows from Lemma . thatx∗(t) is globally exponentially stable.
Theorem . Suppose that(H)and(H)are satisfied.Then system(.)has exactly one
T -anti-periodic solution x∗(t).Moreover,x∗(t)is globally exponentially stable.
Proof The proof proceeds in the same way as in Theorem . in [].
3 Example and remark
In this section, some examples and remarks are provided to demonstrate the effectiveness of our results.
Example . Consider the following HCNNs with time-varying delays in the leakage terms:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x(t) = –.x(t–, cost) +sintf(x(t– )) + sintf(x(t– )) + sintg
(x(t– )) + sintg(x(t– ))g(x(t– )) +
sintg
(x(t– )) + sint∞e–uh
(x(t–u))du
∞ e
–uh
(x(t–u))du+sint,
x(t) = –.x(t–, cost) + sintf(x(t– )) +sintf(x(t– )) + sintg
(x(t– )) + sintg(x(t– ))g(x(t– )) +
sin tg
(x(t– ))
+ sint∞e–uh(x(t–u))du
∞ e
–uh
(x(t–u))du+ sint,
(.)
where
fi(x) =
|x|+cosx, gi(x) =hi(x) =|arctanx|+cosx, ci(t) = ., Ii(t) =isint, i= , ,
δ(t) =δ(t) = ,cos
t, a (t) =
sint, a(t) = sin
t,
a(t) = sin
t, a (t) =
sin
t, b
(t) =b(t) =b(t) = sint,
b(t) =b(t) =b(t) = sin
t, d (t) =
sint, d(t) = sin
t.
Note that
Lfi = , Lgi =Lih= , Mgi =Mhi =π
+ , i= , .
Therefore,
–ci(t) – c+iδ+i–ci(t) – –δi(t)cit–δi(t)
–c+
iδ+i ξi
+
n
j=
aij(t)Lfj
–c+
jδj+ ξj
+
n
j=
n
l=
bijl(t)
Lgj
–c+
jδ+j
ξjMlg+Mgj
–c+
lδ+l ξlLgl
+
n
j=
n
l=
dijl(t)
∞
σijl(u)Lhjdu
–c+
jδj+ ξj
∞
νijl(u)duMhl
+Mjh
∞
σijl(u)du
∞
νijl(u)Lhldu
–c+
jδ+j ξj
< –
.×
– ×.× ,
– .× ,
×
– .×,
+
+
×
– .×,
+ ×
×××
π
+
×
– .× ,
+
×××
π
+
×
– .×,
< –., t≥,ξi= ,i= , ,
which implies that system (.) satisfies all the conditions in Theorem .. Hence, system (.) has exactly oneπ-anti-periodic solution. Moreover, theπ-anti-periodic solution is globally exponentially stable.
Remark . Since
t–δi(t) =t–
,cos
t<
is possible for somet> ,i= , , one can find that the results in [] and the references therein cannot be applicable to prove that all solutions of HRNNs (.) converge exponen-tially to the anti-periodic solution. In this present paper, the expression
xi(t) –xit–δi(t)
=
t t–δi(t)
xi(u)du
has not been used in the proof of Theorem .. In particular, by introducing two new transformations
Xi(t) =xi(t) –
t t–δi(t)
ci(s)xi(s)ds
and
Yi(t) =ertyi(t) –
t t–δi(t)
ci(s)ersyi(s)ds, i∈Jn,
we employ a novel proof to establish some criteria to guarantee the global exponential stability of the anti-periodic solution for HRNNs with leakage delays. Moreover, we also find that Theorem . of [] holds under the following additional conditions:
This implies that the results of this paper are new and complement the corresponding ones in [].
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work was supported by the Scientific Research Fund of Hunan Provincial Natural Science Foundation of China (Grant No. 12JJ3007) and the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant No. Y6110436).
Received: 27 March 2013 Accepted: 21 May 2013 Published: 10 June 2013 References
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