Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 132790, 19 pages
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, 1200801347@stu.ynu.edu.cn
Received 14 June 2010; Accepted 16 August 2010
Academic Editor: Kok Lay Teo
Copyright q 2010 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:
tk ,
xΔ t At, xtBt, xt Ft, xt , t ∈ T, t /
−
Δxtk x tk − x tk Ik xtk , t tk , k ∈ N,
1.1
where T is an ω/2-periodic time scale and if t ∈ T, θ ∈ E, then t θ ∈ T, E is a subset
of R− −∞, 0 , At, xt diaga1 t, x1 t, a2 t, x2 t, . . . , an t, xn t, Bt, xt f1 t, xt , . . . , fn t, xt T , fi t, xt b1 t, x1 t, b2 t, x2 t, . . . , bn t, xn tT , Ft, xt −
fi t, x1t , x2t , . . . , xnt , xit θ xi t θ, t ∈ T, θ ∈ E, i 1, 2, . . . , n, and xtk , xtk represent the right and left limits of xtk in the sense of time scales, {tl } is a sequence of real
numbers such that 0 < t1 < t2 < · · · < tn → ∞ as l → ∞. There exists a positive integer q such
that tlq tl ω/2, Ikq u −Ik −u, l ∈ Z, u ∈ R. Without loss of generality, we also
2
Journal of Inequalities and Applications
assume that 0, ω/2T ∩ {tl : l ∈ N} {t1 , t2 , . . . , tq }. For each interval I of R, we denote that
IT I ∩ T, especially, we denote that T T ∩ 0, ∞.
System 1.1 includes many neural continuous and discrete time networks 1–9 . For
examples, the high-order Hopfield neural networks with impulses and delays see 8 :
⎡
xi t −ai xi t⎣bi xi t −
n
n
aij tgj xj t − bij tgj xj t − τj t
j1
−
n
n j1
1.2
⎤
bijl tgj xj t − τj t gl xl t − τl t Ii t⎦,
t/
tk ,
j1 l1
Δxi tk xi tk − xi t−k eik xi tk ,
i 1, 2, . . . , n, k 1, 2, . . . ,
1.3
the Cohen-Grossberg neural networks with bounded and unbounded delays see 9 :
⎡
x i t −ai xi t⎣bi xi t −
n
n
cij tfj xj t − cij tgj xj t − τij t
j1
−
n
dij thj
j1
j1
∞
1.4
⎤
Kij uxj t − u du Ii t⎦,
t/
tk ,
0
Δxi tk xi tk − xi t−k lik xi tk ,
i 1, 2, . . . , n, k 1, 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, antiperiodic trigonometric polynomials are important in the study of interpolation problems
22, 23 , and anti-periodic wavelets are discussed in 24 .
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 system 1.1 by using the method of coincidence degree
theory and Lyapunov functions.
The initial conditions associated with system 1.1 are of the form
x0 φ,
that is, xi θ φi θ,
θ ∈ E, i 1, 2, . . . , n.
1.6
Throughout this paper, we assume that
H1 ai t, u ∈ CT × R, R , ai t ω/2, −u ai t, u, and there exist positive constants
m
M
M
am
i , ai such that 0 < ai < ai t, u < ai for all t ∈ T, u ∈ R, i 1, 2, . . . , n;
Journal of Inequalities and Applications
3
H2 bi t, u ∈ CT × R, R, bi t ω/2, −u −bi t, u. There exist positive constants μi
and Lbi such that
∂bi t, u
≥ μi ,
∂u
|bi t, u − bi t, v| ≤ Lbi |u − v|,
1.7
bi t, 0 0,
for all t ∈ T, u, v ∈ R, i 1, 2, . . . , n;
H3 fi ∈ CT × Rn , R, fi t ω/2, −u −fi t, u, for i 1, 2, . . . , n. There exist positive
constants ci such that
n
xjt − yjt ,
fi t, x1t , . . . , xnt − fi t, y1t , . . . , ynt ≤ ci
1.8
j1
for all t, x1t , . . . , xnt , t, y1t , . . . , ynt ∈ T × Rn and fi t, 0, . . . , 0 0, i 1, 2, . . . , n;
H4 Iik ∈ CR, R and there exist positive constants LIik such that
1.9
|Iik u − Iik v| ≤ LIik |u − v|,
for all u, v ∈ R, k ∈ N, i 1, 2, . . . , n.
For convenience, we introduce the following notation:
hM max |ht|,
t∈0,ω
hm min |ht|,
T
t∈0,ω
T
h
2
ω
1/2
|ht|2 Δt
,
1.10
0
where h is an ω-periodic function.
The organization of this paper is as follows. In Section 2, we introduce some definitions
and lemmas. In Section 3, by using the method of coincidence degree theory, we obtain the
existence of the anti-periodic solutions of system 1.1. In Section 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 Sections 3 and 4.
The conclusions are drawn in Section 6.
2. Preliminaries
In this section, we will first recall some basic definitions and lemmas which can be found in
books 31, 32 .
Definition 2.1 see 31 . A time scale T is an arbitrary nonempty closed subset of real numbers
R. The forward and backward jump operators σ, ρ : T → T and the graininess μ : T → R
are defined, respectively, by
σt : inf{s ∈ T : s > t},
ρt : sup{s ∈ T : s < t},
μt σt − t.
2.1
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Journal of Inequalities and Applications
Definition 2.2 see 31 . A function f : T → R is called right-dense continuous provided it
is continuous at right-dense point of T and left-side limit exists finite at left-dense point of
T. The set of all right-dense continuous functions on T will be denoted by Crd Crd T Crd T, R. If f is continuous at each right-dense and left-dense point, then f is said to be a
continuous function on T, the set of continuous function will be denoted by CT.
Definition 2.3 see 31 . For x : T → R, one defines the delta derivative of xt, xΔ t to be
the number if it exists with the property that for a given ε > 0, there exists a neighborhood
U of t such that
xσt − xt − xΔ tσt − s
≤ ε|σt − s|,
2.2
for all s ∈ U.
Definition 2.4 see 31 . If F Δ t ft, then one defines the delta integral by
t
2.3
fsΔs Ft − Fa.
a
Definition 2.5 see 33 . For each t ∈ T, let N be a neighborhood of t. Then, one defines the
generalized derivative or dini derivative, D uΔ t to mean that, given ε > 0, there exists a
right neighborhood Nε ⊂ N of t such that
uσt − us
< D uΔ t ε,
μt, s
2.4
for each s ∈ Nε, s > t, where μt, s σt − s.
In case t is right-scattered and ut is continuous at t, this reduces to
D uΔ t uσt − ut
.
σt − t
2.5
Similar to 34 , we will give the definition of anti-periodic function on a time scale as
following.
Definition 2.6. Let T /
R be a periodic time scale with period p. One says that the function
f : T → R is ω/2-anti-periodic if there exists a natural number n such that ω/2 np,
ft ω/2 −ft for all t ∈ T and ω is the smallest number such that ft ω/2 −ft.
If T R, one says that f is ω/2-anti-periodic if ω/2 is the smallest positive number
such that ft ω/2 −ft for all t ∈ T.
Definition 2.7 see 31 . A function p : T → R is called regressive if 1 μtpt /
0 for all
t ∈ Tk , where μt σt − t is the graininess function. If p is regressive and right-dense
continuous function, then the generalized exponential function ep is defined by
ep t, s exp
t
s
ξμτ pτ Δτ ,
2.6
Journal of Inequalities and Applications
5
for s, t ∈ T, with the cylinder transformation
⎧
⎪
⎨ Log1 hz ,
h
ξh z ⎪
⎩z,
if h /
0,
2.7
if h 0.
Let p, q : T → R be two regressive functions, we define
p ⊕ q : p q μpq,
p : −
p
,
1 μp
qp⊕
p
q .
2.8
Then the generalized exponential function has the following properties.
Lemma 2.8 see 31, 32 . Assume that p, q : T → R are two regressive functions, then
i e0 t, s ≡ 1 and ep t, t ≡ 1;
ii ep σt, s 1 μtptep t, s;
iii ep t, σs ep t, s/1 μsps;
iv 1/ep t, s e p t, s;
v ep t, s 1/ep s, t e p s, t;
vi ep t, sep s, r ep t, r;
vii epΔ ·, s pep ·, s.
Lemma 2.9 see 31 . Assume that f, g : T → R are delta differentiable at t ∈ Tk . Then
fg
Δ
t f Δ tgt fσtg Δ t ftg Δ t f Δ tgσt.
2.9
The following lemmas can be found in 35, 36 , respectively.
Lemma 2.10. Let t1 , t2 ∈ 0, w T . If x : T → R is ω-periodic, then
xt ≤ xt1 ω
xΔ s Δs,
xt ≥ xt2 −
0
ω
xΔ s Δs.
2.10
0
Lemma 2.11. Let a, b ∈ T. For rd-continuous functions f, g : a, b → R, one has
b
a
ftgt Δt ≤
b
a
1/2 2
ft Δt
b
1/2
2
gt Δt
.
2.11
a
Definition 2.12. The anti-periodic solution x∗ t x1∗ t, x2∗ t, . . . , xn∗ tT of system 1.1 is
said to be globally exponentially stable if there exist positive constants and M M ≥ 1,
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Journal of Inequalities and Applications
for any solution xt x1 t, x2 t, . . . , xn tT of system 1.1 with the initial value φt φ1 t, φ2 t, . . . , φn tT ∈ CET , Rn , such that
n
xi t − xi∗ t ≤ M e t, αφ − x∗ ,
2.12
i1
where
n
φ − x∗ sup φi s − xi∗ s ,
α ∈ ET .
2.13
i1 s∈ET
The following continuation theorem of coincidence degree theory is crucial in the
arguments of our main results.
Lemma 2.13 see 37 . Let X, X be two Banach spaces, Ω ⊂ X be open bounded and symmetric
with 0 ∈ Ω. Suppose that L : DL ⊂ X → Y is a linear Fredholm operator of index zero with
∅ and N : Ω → Y is L-compact. Further, one also assumes that
DL ∩ Ω /
H Lx − Nx /
λ−Lx − N−x for all x ∈ DL ∩ ∂Ω, λ ∈ 0, 1 .
Then the equation Lx Nx has at least one solution on DL ∩ Ω.
3. Existence of Antiperiodic Solutions
In this section, by using Lemma 2.13, we will study the existence of at least one anti-periodic
solution of 1.1.
Theorem 3.1. Assume that H1 –H4 hold. Suppose further that
H5 E eij n×n is a nonsingular M matrix, where, for i, j 1, 2, . . . , n,
⎧
2q
2q
⎪
1 I
⎪
m
I
2 m M b
⎪
−
ω
a
a
L
−
ωa
L
−
L −
⎨ωam
i
i
i i
i
ik
μi k1 ik
k1
eij ⎪
1
⎪
⎪
ωam
aM
⎩−
i ωci ,
i
μi
1
ωam
aM
i ωci ,
i
μi
i j,
3.1
i/
j.
Then system 1.1 has at least one ω/2-anti-periodic solution.
Proof. Let Ck 0, ω; t1 , . . . , tq , tq1 , . . . , t2q T {x : 0, ω T → Rnm |xk t is a piecewise
continuous map with first-class discontinuity points in 0, ω T ∩{tk }, and at each discontinuity
point it is continuous on the left}. Take
X
ω ω
−xt, ∀t ∈ 0,
,
x ∈ C 0, ω; t1 , . . . , tq , tq1 , . . . , t2q T : x t 2
2 T
YX×R
n×q
3.2
Journal of Inequalities and Applications
7
are two Banach spaces with the norms
x X
n
|xi |0 ,
z Y x X y,
3.3
i1
respectively, where |xi |0 maxt∈0,ω T |xi t|, i 1, . . . , n,
Set
· is any norm of Rn×q .
x −→ xΔ , Δxt1 , . . . , Δx tq ,
L : Dom L ∩ X −→ Y,
3.4
where
Dom L ω ω
x ∈ C1 0, ω; t1 , . . . , t2q T : x t −xt, ∀t ∈ 0,
,
2
2 T
N : X −→ Y,
⎞
⎛ ⎞⎞
I11 x1 t1 I1q x1 tq
A1 t
⎟ ⎜
⎟
⎜
⎟⎟
⎜⎜
⎟ ⎜
⎟
⎜
⎟⎟
⎜⎜
..
..
,
.
.
.
,
Nx ⎜⎜ ... ⎟, ⎜
⎟
⎜
⎟⎟,
.
.
⎠ ⎝
⎠
⎝
⎠⎠
⎝⎝
An t
In1 xn t1 Inq xn tq
⎛⎛
⎞ ⎛
3.5
where
Ai t ai t, xi t bi t, xi t fi t, xt ,
i 1, 2, . . . , n.
3.6
It is easy to see that
Ker L {0},
Im L z f, C1 , . . . , Cq ∈ Y :
ω
fsΔs 0 Y.
3.7
0
Thus, dim KerL 0 codim ImL, and L is a linear Fredholm mapping of index zero.
Define the projectors P : X → Ker L and Q : Y → Y by
Px ω
xsΔs 0,
3.8
0
Qz Q f, C1 , . . . , Cq 1
ω
ω
fsΔs, 0, . . . , 0 ,
0
3.9
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Journal of Inequalities and Applications
respectively. It is not difficult to show that P and Q are continuous projectors such that
Im P Ker L,
Im L Ker Q ImI − Q.
3.10
−1
Further, let L−1
P L|Dom L∩Ker P and the generalized inverse KP LP is given by
KP z t
0
fsΔs t>tk
Ck −
1
2
ω/2
1
Ck ,
2 k1
q
fsΔs −
0
3.11
in which Cqi −Ci for all 1 ≤ i ≤ q.
Similar to the proof of Theorem 3.1 in 38 , it is not difficult to show that QNΩ,
KP I − QNΩ are relatively compact for any open bounded set Ω ⊂ X. Therefore, N is
L-compact on Ω for any open bounded set Ω ⊂ X.
Corresponding to the operator equation Lx − Nx λ−Lx − N−x, λ ∈ 0, 1 , we
have
xΔ t λ
1
Gt, x −
Gt, −x,
1λ
1λ
t ∈ T , t / tk ,
1
λ
Δxtk Ik xtk −
Ik −xtk ,
1λ
1λ
3.12
t tk , k ∈ N,
or
xiΔ t 1
λ
Gi t, x −
Gi t, −x,
1λ
1λ
1
λ
Iik xi tk −
Iik −xi tk ,
Δxi tk 1λ
1λ
tk ,
t ∈ T , t /
3.13
t tk , i 1, 2, . . . , n, k ∈ N,
where
Gi t, x ai t, xi t bi t, xi t fi t, xt ,
Gi t, −x ai t, −xi t bi t, −xi t fi t, −xt , i 1, 2, . . . , n.
3.14
Journal of Inequalities and Applications
Set t0 t0 0,
ω
0
9
t2q1 ω, in view of 3.13, H1 –H4 and Lemma 2.11, we obtain that
xiΔ t Δt 2q1
tk
k1
tk−1
ω
≤
0
$
≤
xiΔ t Δt 2q
|Δxi tk |
k1
1
λ
Gi t, x −
Gi t, −x Δt
1λ
1λ
2q
k1
1
λ
Iik xi tk −
Iik −xi tk 1λ
1λ
1
λ
1λ 1λ
%
ω
max{|Gi t, x|, |Gi t, −x|}Δt
0
% 2q
λ 1
max{|Iik xi tk |, |Iik −xi tk |}
1 λ 1 λ k1
$
ω
≤
&
max ai t, xi t bi t, xi t fi t, xt ,
0
'
ai t, −xi t bi t, −xi t fi t, −xt Δt
2q
3.15
max{|Iik xi tk |, |Iik −xi tk |}
k1
$
≤
ω
aM
i
max{|bi t, xi t − bi t, 0|, |bi t, −xi t − bi t, 0|}Δt
0
ω
&
max fi t, x1t , . . . , xnt − fi t, 0, . . . , 0 ,
0
'
fi t, −x1t , . . . , −xnt − fi t, 0, . . . , 0 Δt
2q
max{|Iik xi tk − Iik 0|, |Iik −xi tk − Iik 0|} k1
⎡
⎣Lbi
≤ aM
i
ω
|xi t|Δt 0
√
b
≤ aM
i Li ω xi
ω
ci
0
2
aM
i ci
n
⎤
xjt Δt⎦ j1
%
2q
|Iik 0|
k1
2q
LIik |xi |0 k1
2q
|Iik 0|
k1
2q
2q
n √
xj ω LI |xi | |Iik 0|,
0
ik
2
j1
k1
k1
where i 1, 2, . . . , n. Integrating 3.13 from 0 to ω, we have from H1 –H4 that
ω$
0
%
ai t, xi tbi t, xi t λai t, −xi tbi t, −xi t
−
Δt
1λ
1λ
10
Journal of Inequalities and Applications
%
ω$
ai t, xi tbi t, xi t λai t, xi tbi t, xi t
Δt
1λ
1λ
0
ω
ai t, xi tbi t, xi tΔt
0
ω
1
1λ
0
λ
1λ
ω
ai t, −xi tfi t, −xt Δt
0
2q
2q
1 λ Iik xi tk −
Iik −xi tk 1 λ k1
1 λ k1
ω
≤ aM
i
ai t, xi tfi t, xt Δt −
&
max fi t, x1t , . . . , xnt − fi t, 0, . . . , 0 ,
0
2q
'
fi t, −x1t , . . . , −xnt − fi t, 0, . . . , 0 Δt
max{|Iik xi tk − Iik 0|, |Iik −xi tk − Iik 0|} k1
≤ aM
i ci
2q
|Iik 0|
k1
2q
2q
n √
xj ω LI |xi | |Iik 0|,
0
ik
2
j1
k1
i 1, 2, . . . , n,
k1
3.16
by H2 , we obtain that
ω
ai t, xi txi tΔt ≤
0
2q
2q
n √
1 M 1 I
1
ai ci xj 2 ω Lik |xi |0 |Iik 0|,
μi
μi k1
μi k1
j1
3.17
where i 1, 2, . . . , n. From Lemma 2.10, for any ti1 , ti2 ∈ 0, ω T , i 1, 2, . . . , n, we have
ω
ai t, xi txi tΔt ≤
0
ω
ω
0
ai t, xi txi tΔt ≥
0
Dividing by
ω
0
(ω
0
ai t, xi txi ti1 Δt ai t, xi txi ti2 Δt −
ω
ai t, xi t
0
ω
0
ω
ai t, xi t
0
ω
0
xiΔ t Δt Δt,
3.18
xiΔ t Δt Δt.
3.19
ai t, xi tΔt on the both sides of 3.18 and 3.19, respectively, we obtain that
xi ti1 ≥ ( ω
0
xi ti2 ≤ ( ω
0
1
ai t, xi tΔt
1
ai t, xi tΔt
ω
ai t, xi txi tΔt −
ω
0
0
ω
ω
0
ai t, xi txi tΔt 0
xiΔ t Δt,
i 1, 2, . . . , n,
3.20
xiΔ t
Δt,
i 1, 2, . . . , n.
Journal of Inequalities and Applications
11
Let ti , ti ∈ 0, ω T , such that xi ti max xi t, xi ti mint∈0,ω T xi t, by the arbitrariness
t∈0,ω
T
of ti1 , ti2 in view of 3.15, 3.17, 3.20, we have
xi ti1 ≥ ( ω
0
ω
1
ai t, xi tΔt
ω
ai t, xi txi tΔt −
0
xiΔ t Δt
0
ω
1
ω
xiΔ t Δt
a
x
t,
tΔt
0
0
i
i
0
⎤
⎡
2q
2q
n
√
1 ⎣ 1 M
1
1
I
≥− m
a
ci xj 2 ω L |xi | |Iik 0|⎦
ωai μi i j1
μi k1 ik 0 μi k1
≥ − (ω
⎡
√
b
− ⎣aM
i Li
ai t, xi txi tΔt −
⎤
2q
2q
n √
xj ω LI |xi | |Iik 0|⎦,
ω xj 2 aM
0
i ci
ik
2
j1
xi ti2 ≤ ( ω
0
ω
1
ai t, xi tΔt
k1
ω
ai t, xi txi tΔt 0
0
ω
1
xiΔ t
k1
3.21
Δt
ω
xiΔ t Δt
a
x
t,
tΔt
0
0
i
i
0
⎤
⎡
2q
2q
n √
1 ⎣1 M 1
1
≤
a ci xj 2 ω LI |xi | |Iik 0|⎦
ωam
μi i j1
μi k1 ik 0 μi k1
i
≤ (ω
⎡
b
⎣aM
i Li
√
ai t, xi txi tΔt ⎤
2q
2q
n √
M
I
ω xj 2 ai ci xj 2 ω Lik |xi |0 |Iik 0|⎦
j1
k1
k1
where i 1, 2, . . . , n. Thus, we have from 3.21 that
|xi |0 max |xi t|
t∈0,ω
T
⎤
2q
2q
n √
1 ⎣1 M 1
1
I
≤
a ci xj 2 ω L |xi | |Iik 0|⎦
ωam
μi i j1
μi k1 ik 0 μi k1
i
⎡
⎡
b
⎣aM
i Li
√
ω xi
2
aM
i ci
3.22
⎤
2q
2q
n √
I
xj ω L |xi | |Iik 0|⎦,
0
ik
2
j1
k1
k1
where i 1, 2, . . . , n. In addition, we have that
xi
2
ω
0
1/2
|xi s|Δs
≤
√
ω max |xi t| t∈0,ω
T
√
ω|xi |0 ,
i 1, 2, . . . , n.
3.23
12
Journal of Inequalities and Applications
By 3.22, we obtain that,
⎡
ωam
i |xi |0
⎤
2q
2q
n √
1
1
1
M
I
≤ ⎣ ai ci xj 2 ω L |xi | |Iik 0|⎦
μi
μi k1 ik 0 μi k1
j1
⎡
√
b
⎣aM
ωam
i Li ω xi
i
⎤
2q
2q
n √
M
xj ω LI |xi | |Iik 0| ⎦
2 ai ci
0
ik
2
j1
⎡
n
1
xj
≤ ⎣ aM
ωc
i
μi i
j1
k1
k1
3.24
⎤
1 I
1
Lik |xi |0 |Iik 0|⎦
0
μi k1
μi k1
2q
2q
⎡
b
M
⎣aM
ωam
i Li ω|xi |0 ai ωci
i
n
j1
⎤
2q
2q
xj 0 LIik |xi |0 |Iik 0|⎦,
k1
k1
where i 1, 2, . . . , n. That is,
)
ωam
i
−
M b
ω2 am
i ai Li
−
ωam
i
*
$
%
2q
2q
1 I
1
I
m
M
Lik −
L |xi |0 −
ωai ai ωci xj
μi k1 ik
μi
k1
2q
2q
1
≤
|Iik 0| ωam
|Iik 0| Di ,
i
μi k1
k1
0
3.25
i 1, 2, . . . , n.
Denote that,
D D1 , D2 , . . . , Dn T .
|x|0 |x1 |0 , |x2 |0 , . . . , |xn |0 T ,
3.26
Then 3.25 can be rewritten in the matrix form
E|x|0 ≤ D.
3.27
From the conditions of Theorem 3.1, E is a nonsingular M matrix, therefore,
|x|0 ≤ E−1 D M1 , M2 , . . . , Mn T .
3.28
Let
M
n
Mi 1
Clearly, M is independent of λ .
3.29
i1
Take
Ω {x ∈ X : x
X
< M}.
3.30
Journal of Inequalities and Applications
13
It is clear that Ω satisfies all the requirements in Lemma 2.13 and conditionH is satisfied. In
view of all the discussions above, we conclude from Lemma 2.13 that system 1.1 has at least
one ω/2-anti-periodic solution. This completes the proof.
4. Global Exponential Stability of Antiperiodic Solution
Suppose that x∗ t x1∗ t, x2∗ t, . . . , xn∗ tT is an ω/2-anti-periodic solution of system 1.1.
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 that H1 –H5 hold. Suppose further that
H6 there exist positive constants Lai such that
|ai t, u − ai t, v| ≤ Lai |u − v|,
∀u, v ∈ R, i 1, 2, . . . , n;
4.1
H7 for all u, v ∈ R, i 1, 2, . . . , n, there exist positive constants Lab
i such that
ai t, ubi t, u − ai t, vbi t, v u − v ≤ 0,
|ai t, ubi t, u − ai t, vbi t, v| ≥ Lab
i |u − v|,
i 1, 2, . . . , n,
i 1, 2, . . . , n;
4.2
H8 there are ω-periodic functions ri t such that ri t supu∈R |fi t, u|, i 1, 2, . . . , n;
H9 there exists a positive constant such that
Ψi , t a M
L
r
1 μt −Lab
i
i i
n
1 μt − θ e t − θ, taM
i ci > 0,
i 1, 2, . . . , n;
4.3
j1
H10 impulsive operator Iik xi tk satisfy
Iik xi tk −γik xi tk ,
0 < γik < 2,
i 1, . . . , n, k ∈ N.
4.4
Then the ω/2-anti-periodic solution of system 1.1 is globally exponentially stable.
Proof. According to Theorem 3.1, we know that system 1.1 has an ω/2-anti-periodic
solution x∗ t x1∗ t, x2∗ t, . . . , x∗n tT with initial value x∗ s, s ∈ ET , suppose that
14
Journal of Inequalities and Applications
xt x1 t, x2 t, . . . , xn tT is an arbitrary solution of system 1.1 with initial value
φs, s ∈ ET . Then it follows from system 1.1 that
ai t, xi tbi t, xi t − ai t, xi∗ t bi t, xi∗ t ai t, xi tfi t, xt − ai t, xi∗ t fi t, xt∗ ,
tk ,
t ∈ T , t /
Δ xi tk − xi∗ tk −γik xi tk − xi∗ tk , t tk , k ∈ N, i 1, 2, . . . , n.
xi t − xi∗ t
Δ
4.5
In view of system 4.5, for t ∈ T , t / tk , k ∈ N, i 1, 2, . . . , n, we have
xi t − xi∗ tΔ ai t, xi tbi t, xi t − ai t, xi∗ t bi t, xi∗ t
ai t, xi tfi t, xt − ai t, xi∗ t fi t, xt∗ 4.6
ai t, xi tbi t, xi t − ai t, xi∗ t bi t, xi∗ t
ai t, xi t − ai t, xi∗ t fi t, xt ai t, xi∗ t fi t, xt − fi t, xt∗ .
Hence, we can obtain from H6 –H9 that
D xi t − xi∗ t
Δ
∗
a M
xi t − xi∗ t
≤ −Lab
i xi t − xi t Li ri
aM
i
n
ci xj t θ − xj∗ t θ
4.7
j1
n
a M
−Lab
xj t θ − xj∗ t θ ,
xi t − xi∗ t aM
i Li ri
i ci
j1
for i 1, 2, . . . , n, and we have from H10 that
xi tk − xi∗ tk 1 − γik xi tk − xi∗ tk ,
i 1, 2, . . . , n, k ∈ N.
4.8
For any α ∈ E, we construct the Lyapunov functional
V t V1 t V2 t,
V1 t n
e t, α xi t − xi∗ t ,
i1
V2 t n
n t
i1 j1
tθ
∗
1 μs − θ e s − θ, αaM
i ci xj s − xj s Δs.
4.9
Journal of Inequalities and Applications
15
tk , k ∈ N, calculating the delta derivative D V tΔ of V t along solutions of
For t ∈ T , t /
system 4.5, we can get
D V1 tΔ ≤
n
e t, α xi t − xi∗ t n
i1
≤
e σt, αD xi t − xi∗ t
Δ
i1
⎧
n ⎨
e t, α xi t − xi∗ t e σt, α
⎩
i1
⎤⎫
⎡
n
⎬
a M
∗
M
∗
4.10
⎦
a
× ⎣ −Lab
x
x
L
r
c
−
x
θ
−
x
θ
t
t
t
t
i
i
j
i
j
i
i i
i
⎭
j1
n a M
1 μt −Lab
e t, α xi t − xi∗ t
i Li ri
i1
n
n ∗
1 μt e t, αci
aM
i xj t θ − xj t θ ,
i1 j1
D V2 tΔ ≤
n
n ∗
1 μt − θ e t − θ, αaM
i ci xj t − xj t
i1 j1
−
n
n ∗
1 μt e t, αaM
i ci xj t θ − xj t θ .
4.11
i1 j1
By assumption H8 , it concludes that
D V tΔ D V1 tΔ D V2 tΔ
≤
a M
1 μt −Lab
e t, α xi t − xi∗ t
i Li ri
n i1
n
n ∗
1 μt − θ e t − θ, αaM
i ci xj t − xj t
i1 j1
≤
n .
a M
1 μt −Lab
i Li ri
i1
/
∗
1 μt − θ e t − θ, taM
i ci e t, α xi t − xi t
n
j1
≤ 0,
t ∈ T , t /
tk , k ∈ N.
4.12
16
Journal of Inequalities and Applications
Also,
V tk V1 tk V2 tk
n
e tk , α xi tk − xi∗ tk
i1
n n
tk
i1 j1
tk θ
∗
1 μs − θ e s − θ, αaM
i ci xj s − xj s Δs
4.13
n
≤
e tk , α xi tk − xi∗ tk i1
n
n tk
i1 j1
tk θ
V tk ,
∗
1 μs − θ e s − θ, αaM
i ci xj s − xj s Δs
k ∈ N.
It follows that V t ≤ V 0 for all t ∈ T .
On the other hand, we have
V 0 V1 0 V2 0
n
e 0, α xi 0 − xi∗ 0
i1
≤
n
n 0
i1 j1
θ
∗
1 μs − θ e s − θ, αaM
i ci xj s − xj s Δs
⎧
n ⎨
n
0
⎩
j1
θ
e 0, α i1
⎫
⎬
4.14
∗
1 μs − θ e s − θ, αaM
i ci Δs⎭sup xi s − xi s
s∈ET
n
≤ M sup φi s − xi∗ s ,
i1 s∈ET
where
⎧
⎨
⎛
M max sup⎝e 0, α 1≤i≤n ⎩α∈E
T
n
0
j1
θ
⎞⎫
⎬
⎠ .
1 μs − θ e s − θ, αaM
i cij Δs
⎭
4.15
It is obvious that
n
i1
e 0, α xi t − xi∗ t ≤ V t ≤ V 0 ≤ M sup
n
s∈ET i1
φi s − xi∗ s .
4.16
Journal of Inequalities and Applications
17
So we can finally get
n
xi t − xi∗ t ≤ M e 0, αsup
n
s∈ET i1
i1
φi s − xi∗ s M e 0, 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:
xΔ t At, xtBt, xt Ft, xt , t ∈ T, t / tk,
Δxtk x tk − x t−k Ik xtk , t tk , k ∈ Z,
5.1
where
2
2
At, u diag 10 arctan |u|, 11 arctan |u| ,
π
π
⎞
⎛ 2
c
x
tg
i
j
jt ⎟
⎜
u
⎟
⎜ j1
1
⎟,
, Ft, xt ⎜
Bt, u ⎜
2
⎟
100 u
⎠
⎝
ci tgj xjt
5.2
j1
gj 2×1 1
1000
sin u
sin u
,
ω 2π,
ci 2×1
1
1000
0, 2π
T
sin t
cos t
,
Ik 2×2
1
500
−u − u
−u − u
,
∩ {tk : k ∈ N} {t1 , t2 },
when T R, system 5.1 has at least one exponentially stable π-anti-periodic solution.
m
m
M
a
a
b
b
Proof. By calculation, we have am
1 10, a1 11, a2 11, a2 12, L1 L2 2/π, L1 L2 M
M
I
I
I
I
1/100, c1 1/1000, c2 1/1000, L11 L21 L12 L22 1/500, and μ1 μ2 1/100.
It is obvious that H1 –H4 , H6 –H8 , and H10 are satisfied. Furthermore, we can easily
calculate that
E≈
7.52 −12.74
−11.25 3.61
5.3
is a nonsingular M matrix, thus H5 is satisfied.
When T R, μt 0. Take 0.01, θ −1, we have that
Ψ1 , t ≈ −0.04 < 0,
Ψ2 , t ≈ −0.03 < 0.
5.4
18
Journal of Inequalities and Applications
Hence H10 holds. By Theorems 3.1 and 4.1, system 5.1 has 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 scale T R or Z and 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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