Volume 2010, Article ID 132790,19pages doi:10.1155/2010/132790
Research Article
Existence and Stability of Antiperiodic Solution for
a Class of Generalized Neural Networks with
Impulses and Arbitrary Delays on Time Scales
Yongkun Li, Erliang Xu, and Tianwei Zhang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China Correspondence should be addressed to Tianwei Zhang,[email protected]
Received 14 June 2010; Accepted 16 August 2010
Academic Editor: Kok Lay Teo
Copyrightq2010 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.
By using coincidence degree theory and Lyapunov functions, we study the existence and global exponential stability of antiperiodic solutions for a class of generalized neural networks with impulses and arbitrary delays on time scales. Some completely new sufficient conditions are established. Finally, an example is given to illustrate our results. These results are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses .
1. Introduction
In this paper, we consider the following generalized neural networks with impulses and arbitrary delays on time scales:
xΔt At, xtBt, xt Ft, xt , t∈T, t /tk,
Δxtk x
tk−xt−kIkxtk, ttk, k∈N,
1.1
where Tis an ω/2-periodic time scale and if t ∈ T, θ ∈ E, thentθ ∈ T, E is a subset of R− −∞,0 , At, xt diaga1t, x1t, a2t, x2t, . . . , ant, xnt, Bt, xt
b1t, x1t, b2t, x2t, . . . , bnt, xntT, Ft, xt f1t, xt, . . . , fnt, xtT, fit, xt
fit, x1t, x2t, . . . , xnt, xitθ xit θ, t ∈ T, θ ∈ E, i 1,2, . . . , n, andxtk, xt−k
represent the right and left limits ofxtkin the sense of time scales,{tl}is a sequence of real
numbers such that 0< t1< t2<· · ·< tn → ∞asl → ∞. There exists a positive integerqsuch
assume that0, ω/2T∩ {tl :l ∈N}{t1, t2, . . . , tq}. For each intervalIofR, we denote that
ITI∩T, especially, we denote thatTT∩0,∞.
System1.1includes many neural continuous and discrete time networks1–9 . For examples, the high-order Hopfield neural networks with impulses and delayssee8 :
xit −aixit ⎡
⎣bixit− n
j1 aijtgj
xjt
−n j1
bijtgj
xj
t−τjt
−n j1
n
l1 bijltgj
xj
t−τjt
glxlt−τlt Iit ⎤
⎦, t /tk,
1.2
Δxitk xi
tk−xi
t−keikxitk, i1,2, . . . , n, k1,2, . . . , 1.3
the Cohen-Grossberg neural networks with bounded and unbounded delayssee9 :
xit −aixit ⎡
⎣bixit− n
j1 cijtfj
xjt
−n j1
cijtgj
xj
t−τijt
−n j1
dijthj ∞
0
Kijuxjt−udu
Iit ⎤
⎦, t /tk,
1.4
Δxitk xi
tk−xi
t−klikxitk, i1,2, . . . , n, k1,2, . . . , 1.5
and so on.
Arising from problems in applied sciences, it is well known that anti-periodic problems of nonlinear differential equations have been extensively studied by many authors during the past twenty years; see 10–21 and references cited therein. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems
22,23 , and anti-periodic wavelets are discussed in24 .
Recently, several authors 25–30 have investigated the anti-periodic problems of neural networks without impulse by similar analytic skills. However, to the best of our knowledge, there are few papers published on the existence of anti-periodic solutions to neural networks with impulse.
The main purpose of this paper is to study the existence and global exponential stability of anti-periodic solutions of system1.1by using the method of coincidence degree theory and Lyapunov functions.
The initial conditions associated with system1.1are of the form
x0φ, that is, xiθ φiθ, θ∈E, i1,2, . . . , n. 1.6
Throughout this paper, we assume that
H1ait, u∈CT×R,R, aitω/2,−u ait, u,and there exist positive constants
am
H2bit, u ∈ CT×R,R, bitω/2,−u −bit, u. There exist positive constantsμi
andLb
i such that
∂bit, u
∂u ≥μi, |bit, u−bit, v| ≤L
b
i|u−v|, bit,0 0, 1.7
for allt∈T, u, v∈R, i1,2, . . . , n;
H3fi ∈CT×Rn,R, fitω/2,−u −fit, u, fori1,2, . . . , n. There exist positive
constantscisuch that
fit, x1t, . . . , xnt−fi
t, y1t, . . . , ynt≤ci n
j1
xjt−yjt, 1.8
for allt, x1t, . . . , xnt,t, y1t, . . . , ynt∈T×Rnandfit,0, . . . ,0 0, i1,2, . . . , n;
H4Iik ∈CR,Rand there exist positive constantsLIiksuch that
|Iiku−Iikv| ≤LIik|u−v|, 1.9
for all u, v∈ R, k∈ N, i1, 2, . . . , n.
For convenience, we introduce the following notation:
hM max
t∈0,ωT
|ht|, hm min
t∈0,ωT|ht|, h 2 ω
0
|ht|2Δt1/2, 1.10
wherehis anω-periodic function.
The organization of this paper is as follows. InSection 2, we introduce some definitions and lemmas. InSection 3, by using the method of coincidence degree theory, we obtain the existence of the anti-periodic solutions of system 1.1. InSection 4, we give the criteria of global exponential stability of the anti-periodic solutions of system 1.1. In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections3and4. The conclusions are drawn inSection 6.
2. Preliminaries
In this section, we will first recall some basic definitions and lemmas which can be found in books31,32 .
Definition 2.1see31 . A time scaleTis an arbitrary nonempty closed subset of real numbers
R. The forward and backward jump operatorsσ,ρ:T → Tand the graininessμ:T → R are defined, respectively, by
Definition 2.2see31 . A functionf :T → Ris called right-dense continuous provided it is continuous at right-dense point ofTand left-side limit existsfiniteat left-dense point of
T. The set of all right-dense continuous functions onTwill be denoted byCrd CrdT CrdT,R. Iff is continuous at each right-dense and left-dense point, thenf is said to be a
continuous function onT, the set of continuous function will be denoted byCT.
Definition 2.3see31 . Forx:T → R, one defines the delta derivative ofxt, xΔtto be
the numberif it existswith the property that for a givenε >0, there exists a neighborhood
Uoftsuch that
xσt−xt −xΔtσt−s ≤ε|σt−s|, 2.2
for alls∈U.
Definition 2.4see31 . IfFΔt ft, then one defines the delta integral by
t
a
fsΔsFt−Fa. 2.3
Definition 2.5see33 . For eacht∈T, letNbe a neighborhood oft. Then, one defines the
generalized derivativeor dini derivative,DuΔtto mean that, givenε > 0, there exists a right neighborhoodNε⊂Noftsuch that
uσt−us
μt, s < DuΔt ε, 2.4
for eachs∈Nε,s > t, whereμt, s σt−s.
In casetis right-scattered andutis continuous att, this reduces to
DuΔt uσt−ut
σt−t . 2.5
Similar to34, we will give the definition of anti-periodic function on a time scale as following.
Definition 2.6. Let T/Rbe a periodic time scale with periodp. One says that the function
f : T → Ris ω/2-anti-periodic if there exists a natural number nsuch that ω/2 np,
ftω/2 −ftfor allt∈Tandωis the smallest number such thatftω/2 −ft. IfT R, one says thatf isω/2-anti-periodic ifω/2 is the smallest positive number such thatftω/2 −ftfor allt∈T.
Definition 2.7see31 . A function p : T → Ris called regressive if 1μtpt/0 for all
t ∈ Tk, where μt σt−t is the graininess function. Ifp is regressive and right-dense
continuous function, then the generalized exponential functionepis defined by
ept, s exp t
s
ξμτpτΔτ
fors, t∈T, with the cylinder transformation
ξhz ⎧ ⎪ ⎨ ⎪ ⎩
Log1hz
h , if h /0, z, if h0.
2.7
Letp, q:T → Rbe two regressive functions, we define
p⊕q:pqμpq, p:− p
1μp, pqp⊕
q. 2.8
Then the generalized exponential function has the following properties.
Lemma 2.8see31,32 . Assume thatp, q:T → Rare two regressive functions, then
ie0t, s≡1andept, t≡1;
iiepσt, s 1μtptept, s;
iiiept, σs ept, s/1μsps;
iv1/ept, s ept, s;
vept, s 1/eps, t eps, t;
viept, seps, r ept, r;
viieΔp·, s pep·, s.
Lemma 2.9see31 . Assume thatf,g:T → Rare delta differentiable att∈Tk. Then
fgΔt fΔtgt fσtgΔt ftgΔt fΔtgσt. 2.9
The following lemmas can be found in35,36 , respectively.
Lemma 2.10. Lett1, t2∈0, w T. Ifx:T → Risω-periodic, then
xt≤xt1
ω
0
xΔsΔs, xt≥xt2−
ω
0
xΔsΔs. 2.10
Lemma 2.11. Leta, b∈T. For rd-continuous functionsf, g:a, b → R,one has
b
a
ftgtΔt≤b a
ft2Δt
1/2b
a
gt2Δt
1/2
. 2.11
for any solutionxt x1t, x2t, . . . , xntT of system 1.1with the initial valueφt
φ1t, φ2t, . . . , φntT ∈CET,Rn, such that
n
i1
xit−xi∗t≤Met, αφ−x∗, 2.12
where
φ−x∗
n
i1
sup
s∈ET
φis−x∗is, α∈ET. 2.13
The following continuation theorem of coincidence degree theory is crucial in the arguments of our main results.
Lemma 2.13see37 . LetX,Xbe two Banach spaces,Ω ⊂ Xbe open bounded and symmetric
with 0 ∈ Ω. Suppose that L : DL ⊂ X → Yis a linear Fredholm operator of index zero with
DL∩ Ω/∅andN:Ω → Yis L-compact. Further, one also assumes that
HLx−Nx /λ−Lx−N−xfor allx∈DL∩∂Ω, λ∈0,1 .
Then the equationLxNxhas at least one solution onDL∩Ω.
3. Existence of Antiperiodic Solutions
In this section, by usingLemma 2.13, we will study the existence of at least one anti-periodic solution of1.1.
Theorem 3.1. Assume thatH1–H4hold. Suppose further that
H5E eijn×nis a nonsingularMmatrix, where, fori, j1,2, . . . , n,
eij ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ωami −ω2ami aMi Lbi −ωami 2q
k1 LIik− 1
μi 2q
k1
LIik− 1 μi ωa
m i
aMi ωci, ij,
− 1
μi
ωam i
aM
i ωci, i /j.
3.1
Then system1.1has at least oneω/2-anti-periodic solution.
Proof. Let Ck0, ω;t
1, . . . , tq, tq1, . . . , t2q T {x : 0, ω T → Rnm|xkt is a piecewise
continuous map with first-class discontinuity points in0, ω T∩{tk}, and at each discontinuity
point it is continuous on the left}. Take
X
x∈C0, ω;t1, . . . , tq, tq1, . . . , t2q
T:x
tω
2
−xt, ∀t∈0,ω
2
T
,
YX×Rn×q
are two Banach spaces with the norms
x X
n
i1
|xi|0, z Y x Xy, 3.3
respectively, where|xi|0maxt∈0,ωT|xit|,i1, . . . , n, · is any norm ofRn×q.
Set
L: Dom L∩X−→Y, x−→xΔ,Δxt1, . . . ,Δx
tq
, 3.4
where
DomL
x∈C10, ω;t1, . . . , t2q
T:x
tω
2
−xt, ∀t∈0,ω
2
T
,
N:X−→Y,
Nx
⎛ ⎜ ⎜ ⎜ ⎝
⎛ ⎜ ⎜ ⎜ ⎝
A1t
.. .
Ant ⎞ ⎟ ⎟ ⎟ ⎠,
⎛ ⎜ ⎜ ⎜ ⎝
I11x1t1
.. .
In1xnt1
⎞ ⎟ ⎟ ⎟ ⎠, . . . ,
⎛ ⎜ ⎜ ⎜ ⎝
I1q
x1
tq
.. .
Inq
xn
tq
⎞ ⎟ ⎟ ⎟ ⎠
⎞ ⎟ ⎟ ⎟ ⎠,
3.5
where
Ait ait, xit
bit, xit fit, xt
, i1,2, . . . , n.
3.6
It is easy to see that
KerL{0}, ImL
zf, C1, . . . , Cq
∈Y:
ω
0
fsΔs0
Y. 3.7
Thus, dim KerL0codim ImL, andLis a linear Fredholm mapping of index zero. Define the projectorsP:X → KerLandQ:Y → Yby
P x
ω
0
xsΔs0, 3.8
QzQf, C1, . . . , Cq
1 ω
ω
0
fsΔs,0, . . . ,0
respectively. It is not difficult to show thatP andQare continuous projectors such that
ImP KerL, ImLKerQImI−Q. 3.10
Further, letL−1P L|DomL∩KerP and the generalized inverseKP L−1P is given by
KPz t
0
fsΔs
t>tk Ck−1
2
ω/2
0
fsΔs−1
2
q
k1
Ck, 3.11
in whichCqi−Cifor all 1≤i≤q.
Similar to the proof of Theorem 3.1 in 38 , it is not difficult to show that QNΩ,
KPI −QNΩare relatively compact for any open bounded setΩ ⊂ X. Therefore,N is
L-compact onΩfor any open bounded setΩ⊂X.
Corresponding to the operator equationLx−Nx λ−Lx−N−x, λ ∈ 0,1 , we have
xΔt 1
1λGt, x− λ
1λGt,−x, t∈T
, t /t
k,
Δxtk 1
1λIkxtk− λ
1λIk−xtk, ttk, k∈N,
3.12
or
xΔi t 1
1λGit, x− λ
1λGit,−x, t∈T
, t /t
k,
Δxitk 1
1λIikxitk− λ
1λIik−xitk, ttk, i1,2, . . . , n, k∈N,
3.13
where
Git, x ait, xit
bit, xit fit, xt
,
Git,−x ait,−xit
bit,−xit fit,−xt
, i1,2, . . . , n.
Sett0t0 0, t2q1ω, in view of3.13,H1–H4andLemma 2.11, we obtain that
ω
0
xiΔtΔt 2q1
k1
tk
tk−1
xΔi tΔt 2q
k1
|Δxitk|
≤ ω
0
11λGit, x−
λ
1λGit,−x
Δt
2q
k1
11λIikxitk−
λ
1λIik−xitk
≤ $
1 1λ
λ
1λ
% ω
0
max{|Git, x|,|Git,−x|}Δt
$
1 1λ
λ
1λ
%2q
k1
max{|Iikxitk|,|Iik−xitk|}
≤ ω
0
max&ait, xit
bit, xit fit, xt, ait,−xit
bit,−xit fit,−xt'Δt
2q
k1
max{|Iikxitk|,|Iik−xitk|}
≤aMi
$ω
0
max{|bit, xit−bit,0|,|bit,−xit−bit,0|}Δt
ω
0
max&fit, x1t, . . . , xnt−fit,0, . . . ,0,
fit,−x1t, . . . ,−xnt−fit,0, . . . ,0'Δt %
2q
k1
max{|Iikxitk−Iik0|,|Iik−xitk−Iik0|} 2q
k1
|Iik0|
≤aMi
⎡ ⎣Lbi
ω
0
|xit|Δt ω
0 ci
n
j1
xjtΔt⎤⎦2q
k1
LIik|xi|0 2q
k1
|Iik0|
≤aMi Lbi√ω xi 2aMi ci n
j1
xj2 √
ω 2q
k1
LIik|xi|0 2q
k1
|Iik0|,
3.15
wherei1,2, . . . , n. Integrating3.13from 0 toω, we have fromH1–H4that
ω
0
$
ait, xitbit, xit
1λ −
λait,−xitbit,−xit
1λ
%
ω
0
$
ait, xitbit, xit
1λ
λait, xitbit, xit
1λ
%
Δt
ω
0
ait, xitbit, xitΔt
1
1λ
ω
0
ait, xitfit, xtΔt− λ
1λ
ω
0
ait,−xitfit,−xtΔt
1
1λ 2q
k1
Iikxitk−
λ
1λ 2q
k1
Iik−xitk
≤aM i
ω
0
max&fit, x1t, . . . , xnt−fit,0, . . . ,0, fit,−x1t, . . . ,−xnt−fit,0, . . . ,0'Δt
2q
k1
max{|Iikxitk−Iik0|,|Iik−xitk−Iik0|} 2q
k1
|Iik0|
≤aMi ci n
j1
xj2 √
ω 2q
k1
LIik|xi|0 2q
k1
|Iik0|, i1,2, . . . , n,
3.16
byH2, we obtain that
ω
0
ait, xitxitΔt ≤ 1
μi
aMi ci n
j1
xj2 √
ω 1 μi
2q
k1
LIik|xi|0
1
μi 2q
k1
|Iik0|, 3.17
wherei1,2, . . . , n. FromLemma 2.10, for anyti1, ti2∈0, ω T, i1,2, . . . , n, we have
ω
0
ait, xitxitΔt≤ ω
0
ait, xitxi
ti1Δt
ω
0
ait, xit ω
0
xiΔtΔt
Δt,
3.18
ω
0
ait, xitxitΔt≥ ω
0
ait, xitxi
ti2Δt−
ω
0
ait, xit ω
0
xiΔtΔt
Δt.
3.19
Dividing by(0ωait, xitΔton the both sides of3.18and3.19, respectively, we obtain that
xi
ti1≥ (ω 1 0 ait, xitΔt
ω
0
ait, xitxitΔt− ω
0
xΔi tΔt, i1,2, . . . , n,
xi
ti2≤ (ω 1 0 ait, xitΔt
ω
0
ait, xitxitΔt ω
0
xΔi tΔt, i1,2, . . . , n.
Letti, ti ∈0, ω T, such thatxiti max
t∈0,ωTxit, xiti mint∈0,ωTxit, by the arbitrariness
ofti1, ti2in view of3.15,3.17,3.20, we have
xi
ti1≥ (ω 1 0 ait, xitΔt
ω
0
ait, xitxitΔt− ω
0
xΔi tΔt
≥ −(ω 1 0 ait, xitΔt
ω
0
ait, xitxitΔt −
ω
0
xΔi tΔt
≥ − 1
ωami
⎡ ⎣1 μi aM i n
j1 cixj2
√
ω 1 μi
2q
k1 LI
ik|xi|0
1
μi 2q
k1
|Iik0| ⎤ ⎦
− ⎡
⎣aMi Lbi√ωxj2aMi ci n
j1
xj2 √
ω 2q
k1
LIik|xi|0 2q
k1
|Iik0| ⎤ ⎦, xi ti 2
≤ (ω 1
0 ait, xitΔt
ω
0
ait, xitxitΔt ω
0
xΔi tΔt
≤ (ω 1
0 ait, xitΔt
ω
0
ait, xitxitΔt
ω
0
xΔi tΔt
≤ 1 ωam i ⎡ ⎣1 μi
aMi ci n
j1
xj2 √
ω 1 μi
2q
k1
LIik|xi|0
1
μi 2q
k1
|Iik0| ⎤ ⎦
⎡
⎣aMi Lbi√ωxj2aMi ci n
j1
xj2 √
ω 2q
k1
LIik|xi|0 2q
k1
|Iik0| ⎤ ⎦
3.21
wherei1,2, . . . , n. Thus, we have from3.21that
|xi|0 max t∈0,ωT
|xit|
≤ 1
ωami
⎡ ⎣1
μia M i ci
n
j1
xj
2
√
ω 1 μi
2q
k1
LIik|xi|0
1
μi 2q
k1
|Iik0| ⎤ ⎦
⎡
⎣aMi Lbi√ω xi 2aMi ci n
j1
xj 2 √ ω 2q
k1
LIik|xi|0 2q
k1
|Iik0| ⎤ ⎦,
3.22
wherei1,2, . . . , n. In addition, we have that
xi 2 ω
0
|xis|Δs 1/2
≤√ω max
t∈0,ωT
|xit| √
By3.22, we obtain that,
ωami |xi|0≤
⎡ ⎣1
μia M i ci
n
j1
xj
2
√
ω 1 μi
2q
k1
LIik|xi|0
1
μi 2q
k1
|Iik0| ⎤ ⎦
ωam i
⎡ ⎣aM
i Lbi √
ω xi 2aMi ci n
j1
xj2 √
ω 2q
k1 LI
ik |xi|0 2q
k1
|Iik0| ⎤ ⎦ ≤ ⎡ ⎣1 μi aM i ωci
n
j1
xj0
1
μi 2q
k1
LIik|xi|0
1
μi 2q
k1
|Iik0| ⎤ ⎦
ωami
⎡
⎣aMi Lbiω|xi|0aMi ωci n
j1
xj0 2q
k1
LIik|xi|0 2q
k1
|Iik0| ⎤ ⎦,
3.24
wherei1,2, . . . , n. That is,
)
ωami −ω2ami aMi Lbi −ωami 2q
k1 LIik− 1
μi 2q
k1 LIik
* |xi|0−
$
1
μi
ωami
aMi ωci %
xj0
≤ 1
μi 2q
k1
|Iik0|ωami 2q
k1
|Iik0|Di, i1,2, . . . , n.
3.25
Denote that,
|x|0 |x1|0,|x2|0, . . . ,|xn|0T, D D1, D2, . . . , DnT. 3.26
Then3.25can be rewritten in the matrix form
E|x|0 ≤D. 3.27
From the conditions ofTheorem 3.1,Eis a nonsingularMmatrix, therefore,
|x|0≤E−1DM1, M2, . . . , MnT. 3.28
Let
M
n
i1 Mi1
Clearly, Mis independent ofλ. 3.29
Take
It is clear thatΩsatisfies all the requirements inLemma 2.13and conditionHis satisfied. In view of all the discussions above, we conclude fromLemma 2.13that system1.1has at least oneω/2-anti-periodic solution. This completes the proof.
4. Global Exponential Stability of Antiperiodic Solution
Suppose thatx∗t x1∗t, x2∗t, . . . , x∗ntTis anω/2-anti-periodic solution of system1.1. In this section, we will construct some suitable Lyapunov functions to study the global exponential stability of this anti-periodic solution.
Theorem 4.1. Assume thatH1–H5hold. Suppose further that H6there exist positive constantsLai such that
|ait, u−ait, v| ≤Lai|u−v|, ∀u, v∈R, i1,2, . . . , n; 4.1
H7for allu, v∈R, i1,2, . . . , n, there exist positive constantsLabi such that
ait, ubit, u−ait, vbit, v u−v≤0, i1,2, . . . , n,
|ait, ubit, u−ait, vbit, v| ≥Labi |u−v|, i1,2, . . . , n;
4.2
H8there areω-periodic functionsritsuch thatrit supu∈R|fit, u|, i1,2, . . . , n;
H9there exists a positive constantsuch that
Ψi, t
1μt−Lab
i LairiM
n
j1
1μt−θet−θ, taiMci>0, i1,2, . . . , n;
4.3
H10impulsive operatorIikxitksatisfy
Iikxitk −γikxitk, 0< γik<2, i1, . . . , n, k∈N. 4.4
Then theω/2-anti-periodic solution of system1.1is globally exponentially stable.
Proof. According to Theorem 3.1, we know that system 1.1 has an ω/2-anti-periodic
xt x1t, x2t, . . . , xntT is an arbitrary solution of system 1.1 with initial value
φs, s∈ET. Then it follows from system1.1that
xit−x∗it Δ
ait, xitbit, xit−ai
t, x∗itbi
t, x∗itait, xitfit, xt
−ai
t, x∗itfit, x∗t, t∈T, t /tk,
Δxitk−x∗itk
−γik
xitk−x∗itk
, ttk, k∈N, i1,2, . . . , n.
4.5
In view of system4.5, fort∈T, t /tk, k∈N, i1,2, . . . , n, we have
xit−x∗itΔait, xitbit, xit−ai
t, x∗itbi
t, x∗it
ait, xitfit, xt−ai
t, x∗itfit, x∗t
ait, xitbit, xit−ai
t, x∗itbi
t, xi∗t
ait, xit−ai
t, xi∗tfit, xt ai
t, xi∗tfit, xt−fit, x∗t
. 4.6
Hence, we can obtain fromH6–H9that
Dxit−x∗itΔ≤ −Labi xit−x∗itLairiMxit−x∗it
aMi
n
j1
cixjtθ−x∗jtθ
−Labi LairiMxit−x∗itaMi ci n
j1
xjtθ−x∗jtθ,
4.7
fori1,2, . . . , n, and we have fromH10that
xi
tk−x∗itk1−γikxitk−x∗itk, i1,2, . . . , n, k∈N. 4.8
For anyα∈E, we construct the Lyapunov functional
Vt V1t V2t,
V1t n
i1
et, αxit−x∗it,
V2t n
i1 n
j1
t
tθ
1μs−θes−θ, αaMi cixjs−x∗jsΔs.
Fort ∈ T, t /tk, k ∈ N, calculating the delta derivativeDVtΔ ofVtalong solutions of
system4.5, we can get
DV1tΔ≤ n
i1
et, αxit−x∗it n
i1
eσt, αDxit−x∗it
Δ
≤n i1
⎧ ⎨
⎩et, αxit−x∗iteσt, α
× ⎡
⎣−Labi LairiMxit−xi∗taMi ci n
j1
xjtθ−x∗jtθ ⎤ ⎦
⎫ ⎬ ⎭
n
i1
1μt−Lab
i LairiM
et, αxit−x∗it
1μtet, αci n
i1 n
j1
aMi xjtθ−x∗jtθ,
4.10
DV2tΔ≤ n
i1 n
j1
1μt−θet−θ, αaMi cixjt−x∗jt
−n i1 n
j1
1μtet, αaMi cixjtθ−x∗jtθ.
4.11
By assumptionH8, it concludes that
DVtΔDV1tΔDV2tΔ
≤n i1
1μt−Labi LairiMet, αxit−x∗it
n
i1 n
j1
1μt−θet−θ, αaMi cixjt−xj∗t
≤n i1
.
1μt−Labi LairiM
n
j1
1μt−θet−θ, taMi ci /
et, αxit−x∗it
≤0, t∈T, t /tk, k∈N.
Also,
VtkV1
tkV2
tk
n
i1 e
tk, αxi
tk−xi∗tk
n
i1 n
j1
tk
tkθ
1μs−θes−θ, αaMi cixjs−xj∗sΔs
≤n i1
etk, αxitk−x∗itk
n
i1 n
j1
tk
tkθ
1μs−θes−θ, αaMi cixjs−x∗jsΔs
Vtk, k∈N.
4.13
It follows thatVt≤V0for allt∈T. On the other hand, we have
V0 V10 V20
n
i1
e0, αxi0−x∗i0
n
i1 n
j1
0
θ
1μs−θes−θ, αaMi cixjs−x∗jsΔs
≤n i1
⎧ ⎨
⎩e0, α n
j1
0
θ
1μs−θes−θ, αaMi ciΔs ⎫ ⎬
⎭sups∈ETxis−x∗is
≤Mn
i1
sup
s∈ET
φis−xi∗s,
4.14
where
M max
1≤i≤n ⎧ ⎨ ⎩supα∈ET
⎛
⎝e0, α n j1
0
θ
1μs−θes−θ, αaMi cijΔs ⎞ ⎠
⎫ ⎬
⎭. 4.15
It is obvious that
n
i1
e0, αxit−x∗it≤Vt≤V0≤Msup s∈ET
n
i1
φis−x∗
So we can finally get
n
i1
xit−x∗it≤Me0, αsup s∈ET
n
i1
φis−x∗isMe0, αφ−x∗. 4.17
Since M ≥ 1, from Definition 2.12, the ω/2-anti-periodic solution of system 1.1 is globally exponential stable. This completes the proof.
5. An Example
Example 5.1. Consider the following impulsive generalized neural networks:
xΔt At, xtBt, xt Ft, xt , t∈T, t /tk,
Δxtk x
tk−xt−kIkxtk, ttk, k∈Z,
5.1
where
At, u diag
10 2
π arctan|u|,11
2
πarctan|u|
,
Bt, u 1
100
u
u
, Ft, xt ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
2
j1 citgj
xjt
2
j1 citgj
xjt
⎞ ⎟ ⎟ ⎟ ⎟ ⎠,
gj
2×1
1 1000
sinu
sinu
, ci2×1 10001
sint
cost
, Ik2×2 5001
−u −u
−u −u
,
ω2π, 0,2π T∩ {tk:k∈N}{t1, t2},
5.2
whenTR, system5.1has at least one exponentially stableπ-anti-periodic solution.
Proof. By calculation, we haveam
1 10, aM1 11, am2 11, am2 12, La1 La2 2/π, Lb1 Lb2
1/100, cM1 1/1000, cM2 1/1000, LI
11 LI21 LI12 LI22 1/500, andμ1 μ2 1/100.
It is obvious thatH1–H4,H6–H8,andH10are satisfied. Furthermore, we can easily
calculate that
E≈ −7.52 −12.74
11.25 3.61
5.3
is a nonsingularMmatrix, thusH5is satisfied.
WhenTR, μt 0. Take0.01, θ−1, we have that
HenceH10holds. By Theorems3.1and4.1, system5.1has at least one exponentially stable π-anti-periodic solution. This completes the proof.
6. Conclusions
Using the time scales calculus theory, the coincidence degree theory, and the Lyapunov functional method, we obtain sufficient conditions for the existence and global exponential stability of anti-periodic solutions for a class of generalized neural networks with impulses and arbitrary delays. This class of generalized neural networks include many continuous or discrete time neural networks such as, Hopfield type neural networks, cellular neural networks, Cohen-Grossberg neural networks, and so on. To the best of our knowledge, the known results about the existence of anti-periodic solutions for neural networks are all done by a similar analytic method, and only good for neural networks without impulse. Our results obtained in this paper are completely new even if the time scaleTRorZand are of great significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural networks with delays and impulses .
Acknowledgment
This work is supported by the National Natural Sciences Foundation of China under Grant 10971183.
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