Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 510789, 14 pages
doi:10.1155/2011/510789
Research Article
Pseudo Almost-Periodic Solution of
Shunting Inhibitory Cellular Neural
Networks with Delay
Haihui Wu
Sunshine College, Fuzhou University, Fuzhou, Fujian, 350002, China
Correspondence should be addressed to Haihui Wu, whh3346@sina.com
Received 28 June 2010; Accepted 29 July 2010
Academic Editor: Ivanka Stamova
Copyright q 2011 Haihui Wu. 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.
Shunting inhibitory cellular neural networks are studied. Some sufficient criteria are obtained
for the existence and uniqueness of pseudo almost-periodic solution of this system. Our results
improve and generalize those of the previous studies. This is the first paper considering the pseudo
almost-periodic SICNNs. Furthermore, several methods are applied to establish sufficient criteria
for the globally exponential stability of this system. The approaches are based on constructing
suitable Lyapunov functionals and the well-known Banach contraction mapping principle.
1. Introduction
It is well known that the cellular neural networks CNNs are widely applied in signal
processing, image processing, pattern recognition, and so on. The theoretical and applied
studies of CNNs have been a new focus of studies worldwide see 1–12. Bouzerdoum
and Pinter in 1 have introduced a new class of CNNs, namely, the shunting inhibitory
CNNs SICNNs. Shunting neural networks have been extensively applied in psychophysics,
speech, perception, robotics, adaptive pattern recognition, vision, and image processing.
Recently, Chen and Cao 9 have studied the existence of almost-periodic solutions of the
following system of SICNNs:
ẋij t −aij xij t −
cijkl f xij t − τ xij t Lij t,
Ckl ∈Nr i,j 1.1
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Journal of Applied Mathematics
where Cij denotes the cell at the i, j position of the lattice, the r-neighborhood Nr i, j of Cij
is
Nr i, j Cijkl : max |k − l|, l − j ≤ r , 1 ≤ k ≤ m, 1 ≤ l ≤ n ,
1.2
xij is the activity of the cell Cij , Lij t is the external input to Cij , the constant aij > 0 represents
the passive decay rate of the cell activity, Cijkl ≥ 0 is the connection or coupling strength of
postsynaptic activity of the cell transmitted to the cell Cij , and the activation function fxkl is a positive continuous function representing the output or firing rate of the cell Cij . Since
studies on neural dynamic systems not only involve a discussion of stability properties, but
also involve many dynamic properties such as periodic oscillatory behavior, almost-periodic
oscillatory properties, chaos, and bifurcation. To the best of our knowledge, few authors
have studied almost-periodic solutions for SINNs with delays and variable coefficients, and
most of them discuss the stability, periodic oscillation in the case of constant coefficients.
In their paper, they investigated the existence and stability of periodic solutions of SINNs
with delays and variable coefficients. They considered the SICNNs with delays and variable
coefficients
ẋij t −aij txij t −
Bkl ∈Nr
−
i,j Bijkl tfij xkl txij t
Cijkl tgij xkl t − τkl xij t Lij t,
Ckl ∈Nr i,j 1.3
where, for each i 1, 2, . . . , n, j 1, 2, . . . , m, aij t, Bijkl t, Cijkl t,and Lij t are all continuous
ω− periodic functions and aij t > 0, Bijkl t ≥ 0, Cijkl t ≥ 0, and τij is a positive constant.
In this paper, we consider the following more general SICNNs:
ẋij t −aij txij t −
Bkl ∈Nr
−
Ckl ∈Nr i,j Bijkl tfij xkl txij t
i,j Cijkl tgij xkl t − τkl txij t Lij t.
1.4
By using the Lyapunov functional and contraction mapping, a set of criteria are established
for the globally exponential stability, the existence, and uniqueness of pseudo almost-periodic
solution for the SICNNs. This is the first paper considering the pseudo almost-periodic
solution of SICNNs. Since the nature is full of all kinds of tiny perturbations, either the
periodicity assumption or the almost-periodicity assumption is just approximation of some
degree of the natural perturbations. A well-known extension of almost periodicity is the
asymptotically almost periodicity, which was introduced by Frechet. In 1992, Zhang 13, 14
introduced a more general extension of the concept of asymptotically almost periodicity,
the so-called pseudo almost periodicity, which has been widely applied in the theory
of ODEs and PDEs. However, it is rarely applied in the theory of neural networks or
mathematical biology. This paper is expected to establish criteria that provide much flexibility
Journal of Applied Mathematics
3
in the designing and training of neural networks and to shed some new light on the
application of pseudo almost periodicity in neural networks, population dynamics, and the
theory of differential equations.
Throughout this paper, we will use the notations g M supt∈R gt, g L supt∈R gt,
where gt is a bounded continuous function on R.
In this paper, we always use i 1, 2, . . . , m; j 1, 2, . . . , n; unless otherwise stated.
In this paper, we always consider system 1.4 together with the following
assumptions.
A1 aij t are almost periodic on R with aij t > 0, and Lij t and τij t are pseudo almost
periodic on R with Lij > 0.
A2 τij t is bounded, continuous and differentiable with 0 ≤ τij ≤ τ, 1 − τ > 0 for t ∈ R,
where τ is constant.
A3 fij , gij ∈ CR, R are bounded, and continuous, and there exist positive numbers
μij , νij such that |fij x − fij y| ≤ μij |x − y|, |gij x − gij y| ≤ νij |x − y|, for all
x, y ∈ R.
2. Preliminaries and Basic Results of Pseudo Almost-Periodic Function
In this section, we explore the existence of pseudo almost-periodic solution of 1.4. First, we
would like to recall some basic notations and results of almost periodicity and pseudo almost
periodicity 15, 16 which will come into play later on.
Let Ω ⊂ Cn be close and let LR resp., LR × Ω denote the C∗ -algebra of bounded
continuous complex-valued functions on R respectively, R × Ω with supremum norm | · |∞
denoting the Euclidean norm in Cn ; that is, |x|∞ max1≤i≤n |xi |, x ∈ Cn .
Definition 2.1. A function g ∈ LR is called almost periodic if, for each > 0, there exists
an l > 0 such that every interval of length l contains a number τ with the property that
|gt τ − gt| < , t ∈ R.
Definition 2.2. A function g ∈ LR × Ω is called almost periodic in t ∈ R, uniformly in Z ∈ Ω,
if, for each > 0 and any compact set M of Ω, there exists an l > 0 such that every interval
of length l contains a number τ with the property that |gt τ, z − gt, z| < , t ∈ R, Z ∈ Ω.
The number τ is called an -translation number of g. Denote by APRAPR × Ω the set
of all such function.
Set
PAP0 R PAP0 R × Ω 1
ϕ ∈ LR : lim
t → ∞ 2t
1
ϕ ∈ LR × Ω : lim
t → ∞ 2t
t
−t
t
−t
ϕsds 0 ,
ϕs, Zds 0, uniformly in Z ∈ Ω .
2.1
Definition 2.3. A function f ∈ LRLR × Ω is called pseudo almost periodic pseudo
almost periodic in t ∈ R, uniformly in Z ∈ Ω, if f g ϕ, where g ∈ PARPAR × Ω and
ϕ ∈ PAP0 RPAP0 R × Ω. The function g and ϕ are called the almost periodic component
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Journal of Applied Mathematics
and the erigodic perturbation, respectively, of the function f. Denote by PAPRPAPR ×
Ω the set of all such functions f.
Define x∞ supt∈R |xt|∞ , x ∈ PAP. It is trivial to show that PAP is a Banach space
with · ∞ .
Let At aij t be a complex n × n matrix-valued function with elements entries
which are continuous on R. We consider the homogeneous linear ODE and nonhomogeneous
linear ODE as follows:
dx
Atx,
dt
dx
Atx ft,
dt
2.2
2.3
where x denotes an n-column vector.
Definition 2.4 see 15, 16. The homogeneous linear ODE 2.2 is said to admit an
exponential dichotomy if there exist a linear projection p i.e., p2 P on Cn and positive
constants k, α, β such that
XtP X −1 s ≤ ke−αt−s , t ≥ s,
XtI − P X −1 s ≤ ke−βs−t , t ≤ s,
2.4
where Xt is a fundamental matrix of 2.2 with X0 E; E is the n × n identity matrix.
Definition 2.5 see 15, 16. The matrix At is said to be row dominant if there exists a
number δ > 0 such that | Re aij t| ≥ nj1,j / i |aij t| δ for all t ∈ −∞, ∞ and i 1, 2, . . . , n.
Lemma 2.6 see 15, 16. If At is a bounded, continuous, and row-dominant n×n matrix function
on R, and there exists k ≤ n such that Re aij < 0 i 1, 2, . . . , k. Then 2.2 has a fundamental
matrix solution Xt satisfying
XtP X −1 s ≤ ke−δt−s , t ≥ s,
XtI − P X −1 s ≤ ke−δs−t , t ≤ s,
2.5
where K is a positive constant, and P diagEk , 0 with Ek being a k × k identity matrix.
For H h1 , h2 , . . . , hn ∈ LRn , suppose that Ht ∈ Ω for all t ∈ R. Define H × l : R →
Ω × R by H × lt h1 t, h2 t, . . . , hn t, t t ∈ R.
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5
Lemma 2.7 see 15, 16. Assume that the function ft, z ∈ PAPR × Ω is continuous in
z ∈ M uniformly in t ∈ R for all compact subsets M ⊂ Ω and F ∈ PAPRn such that FR ⊂ Ω.
Then f ◦ F × l ∈ PAPR.
It is obvious that if f satisfies a Lipschitz condition; that is, there is an L > 0 such that
f z , t − fz, t ≤ Lz − z
z , z ∈ M, t ∈ R ,
2.6
then f is continuous in z ∈ M uniformly in t ∈ R. Obviously, if ft ∈ PAPR is uniformly
continuous in t ∈ R and ϕ ∈ PAPR such that ϕR ⊂ Im f, then f ◦ ϕ ∈ PAPR.
Lemma 2.8 see 15, 16. Assume that At is an almost-periodic matrix function and ft ∈
PAPRn . If 2.2 satisfies an exponential dichotomy, then 2.3 has unique pseudo almost periodic
solution xt reading
xt t
−∞
XtP X −1 sfsds −
∞
XtE − P X −1 sfsds
2.7
t
and satisfying x ≤ K/α K/βf, where Xt is a fundamental matrix solution of 2.2.
Definition 2.9. System 1.4 is said to be globally exponentially stable GES, if for any two
solutions xt and yt of 1.4, there exist positive numbers M and ε such that
xt − yt ≤ Me−εt−t0 ϕ − ψ ,
p
p
2.8
t > t0 ,
where xt xt, ϕ and yt yt, ψ denoting the solution of 1.4 through t0 , ϕ and t0 , ψ
respectively. Here ε is called the Lyapunov exponent of 1.4.
3. Existence and Stability of Pseudo Almost-Periodic Solution
Theorem 3.1. Assume that (A1 )−(A3 ) hold and
A4 ⎫
⎬
r sup max
e− s aij ηdη ⎝
μij Bijhl s νij Cijhl s⎠ds < 1.
⎭
t∈R i,j ⎩ −∞
Bhl ∈Nr i,j Chl ∈Nr i,j ⎧
⎨ t
t
⎛
⎞
3.1
Then 1.4 has a unique pseudo almost-periodic solution, say x∗ t, satisfying x∗ ∞ ≤ L/1 − r.
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Journal of Applied Mathematics
Proof. For any ϕ ∈ PAPR, consider
ẋij −aij txij t −
Bhl ∈Nr ij −
Chl ∈Nr ij Bijhl tfij ϕhl t ϕij t
3.2
Cijhl tgij ϕhl t − τhl t ϕij t Lij t.
Since −aij t < 0, from Lemmas 2.6, 2.7, and 2.8, it follows that 3.2 has a unique pseudo
almost-periodic solution, which is given by
⎛
xϕ t ⎝
t
−∞
e−
t
s
⎡
aij ηdη ⎣
−
Bhl ∈Nr i,j
Bijhl sfij ϕhl s ϕij s
−
Chl ∈Nr
i,j ⎤
⎞
Cijhl sgij ϕhl s − τhl s ϕij s Lij s⎦ds⎠
.
mn×1
3.3
Define the mapping T : PAPR → PAPR by Tϕt xϕ t, ϕ ∈ PAPR.
Let B∗ {ϕ | ϕ ∈ PAPR, ϕ − ϕ0 ∞ ≤ γ/1 − γL}, where ϕ0 t t
t
t
t
t − t a ηdη
L11 sds, . . . , −∞ e− s aij ηdη Lij sds, . . . , −∞ e− s amn ηdη Lmn sdsT .
−∞ e s 11
Clearly, B∗ is closed and convex in B. Note that
t
t
− s aij ηdη
ϕ0 sup max e
L
sds
ij
∞
i,j
−∞
t∈R
t
−a−ij s−t ≤ sup max e
Lij ds
−∞
t∈R i,j
≤ sup max
t∈R
i,j
Lij
a−ij
max
i,j
Lij
a−ij
3.4
L.
Therefore, for any ϕ ∈ B∗ , we have
ϕ ≤ ϕ − ϕ0 ϕ0 ≤
γ
L
LL
.
1−γ
1−γ
3.5
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Now, we will show that T maps B∗ into itself. In fact, for any ϕ ∈ B∗ , by using L/1 − γ ≤ 1,
we have
⎧
⎡
⎨ t
t
Tϕ − ϕ0 sup max e− s aij ηdη ⎣−
Bijhl sfij ϕhl s ϕij s
∞
⎩
−∞
t∈R i,j
Bhl ∈Nr i,j −
Chl ∈Nr i,j
≤ sup max
t∈R
i,j
⎧
⎨ t
⎩
−∞
e−
t
s
⎡
aij ηdη ⎣
μij
Cijhl sgij
Bhl ∈Nr i,j ⎞
ϕhl s − τhl s ϕij s⎦ds⎠
⎤
Bijhl s ϕhl s ϕij s
⎤ ⎞
νij
Cijhl sϕhl s − τhl sϕij s⎦ds⎠
Chl ∈Nr i,j ⎧
⎞ ⎫
⎛
⎬ ⎨ t
t
2
≤ max
e− s aij ηdη ⎝μij
Bijhl s νij
Cijhl s⎠ds ϕ .
⎭
i,j ⎩ −∞
Bhl ∈Nr i,j Chl ∈Nr i,j γL
.
≤ γ ϕ ≤ γ ϕ ≤
1−γ
3.6
For any ϕ, φ ∈ B∗ , it follows from L/1 − γ ≤ 1 that
Tϕ − Tψ ∞
⎧
⎨ t
t
sup max e− s aij ηdη
t∈R i,j ⎩ −∞
⎡
×⎣
Bhl ∈Nr i,j ≤ sup max
t∈R
i,j
Cijhl sgij
ϕhl s − τhl s ϕij s − gij
Chl ∈Nr i,j ⎧
⎨ t
⎩
Bijhl sfij ϕhl s ϕij s − fij ψhl s ψij s
e−
−∞
⎡
×⎣
t
s
aij ηdη
Bhl ∈Nr i,j Bijhl s fij ϕhl s · ϕij s − ψij s
⎤ ⎞
ψhl s − τhl s ψij s⎦ds⎠
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Journal of Applied Mathematics
Cijhl s gij ϕhl s − τhl s fij ϕhl s − fij ψhl s · ψij s Chl ∈Nr i,j ⎫
⎬
×ϕij s − ψij s gij ϕhl s − τhl s − gij ψhl s − τhl s · ψij s ds
⎭
≤ max
i,j
⎧
⎨ t
⎩
−∞
e
−
⎡
t
s aij ηdη ⎣
μij
Bijhl s
νij
⎤
Chl ∈Nr i,j i,j 2L ϕ − ψ 2Lγ ϕ − ψ δϕ − ψ .
γ·
1−γ
1−γ
Cijhl s⎦
Bhl ∈Nr
⎫
⎬
ϕ ψ ϕ − ψ ds
∞
∞
⎭
3.7
Since δ < 1, T is a contraction mapping. Therefore, there exists a unique fixed point x∗ ∈ B∗
such that Tx∗ x∗ . That is, system 1.4 has a unique pseudo almost-periodic solution x∗ ∈ B∗
with x∗ − ϕ0 ≤ γ/1 − γL.
Now we go ahead with the GES of 1.4. The approaches involve constructing suitable
Lyapunov functions and application of a generalized Halanay’s delay differential inequality.
We will stop here to see our first criteria for the globally exponential stability of 1.4, which
is delay dependent.
Theorem 3.2. In addition to (A1 )−(A4 ), if one further assumes that
A5 ⎧
⎨
⎛
⎡
L
⎣μij
c min inf βij ⎝2aij t − 3βij
i,j i,j⎩
1−γ
Bhl ∈Nr i,j Bijhl t νij
Chl ∈Nr i,j ⎤⎞
Cijhl t⎦⎠
⎞⎫
ij
−1
⎪
⎬
C
ξ
t
ij
hl
L
L
ij
⎟
⎜
βhl ⎝μhl
νhl
Bhl t ⎠ > 0
⎪
−1
1 − γ Bij ∈N h,l
1−γ
⎭
Chl ∈Nr i,j 1 − τij ξij t
r
⎛
3.8
or
A6 ⎧
⎨
⎛
⎡
L
⎣μij
c inf βij ⎝aij t − βij
i,j⎩
1−γ
Bhl ∈Nr i,j Bijhl t
νij
⎤⎞
Cijhl t⎦⎠
Chl ∈Nr i,j ⎞⎫
ij
−1
⎪
⎬
C
ξ
t
ij
hl
L
L
ij
⎟
⎜
−βhl ⎝μhl
νhl
Bhl t ⎠ > 0.
⎪
−1
1 − γ Bij ∈N h,l
1−γ
⎭
Chl ∈Nr i,j 1 − τij ξij t
r
⎛
3.9
Then there exists a unique pseudo-almost periodic solution of system 1.4 and all other solutions
converge exponentially to the (pseudo) almost-periodic attractor.
Journal of Applied Mathematics
9
Proof. By Theorem 3.2, there exists a unique pseudo almost-periodic solution, namely, xt xt, ϕ. Let y yt, ϕ be any other solution of 1.4 through t0 , ϕ. Assume that A5 is
satisfied and consider the auxiliary functions Fij ε defined on 0, ∞ as follows:
⎧
⎨
Fij ε inf βij
t∈R ⎩
⎡
L ⎣
2aij t − ε − 3βij
μij
1−γ
Bhl ∈Nr i,j Bijhl t
νij
Chl ∈Nr i,j ⎤
Cijhl t⎦
⎞⎫
ij
−1
⎪
⎬
ξ
C
t
ij
hl
L
L
ij
⎟
⎜
νhl
Bhl t βhl ⎝μhl
⎠ .
⎪
−1
1 − γ Bij ∈N h,l
1−γ
⎭
Chl ∈Nr i,j 1 − τij ξij t
r
⎛
3.10
FromA1 −A3 , one can easily show that Fij ε is well defined and is continuous. From A5 ,
it follows that Fij 0 > 0, Fij ε → −∞ as ε → ∞ it follows that there exists an εij > 0 such
that Fij εij > 0. Let ε mini,j εij . Then we have Fij ε > 0, 1 ≤ i ≤ n, 1 ≤ j ≤ m.
Consider the Lyapunov functional defined by
⎡
2
1 ⎣
V t βij xij t − yij t eεt
2 i,j
Vij
Chl ∈Nr i,j L
1−γ
t
t−τhl t
⎤
−1 Chl ξhl
t 2 εsτ M hl ds⎦.
xhl s − yhl s e
−1 1 − τhl ξhl
t
3.11
Calculating the upper-right derivative of V t and using the inequality 2ab ≤ a2 b2 , one has
⎧
⎨1
2
2
V t ≤
βij
xij t − yij t εeεt − eεt aij t xij − yij t
⎩
2
i,j
eεt
Bhl ∈Nr i,j Bijhl txij t − yij t · fij xhl txij t − fij yhl t yij t
Cijhl txij t − yij t
Chl ∈Nr i,j · gij xhl t − τhl txij t − gij yhl t − τhl t yij t
−1 Cijhl ξhl
t 2
M
1 L
νij
−1 xhl t − yhl t eεtτhl 2 Chl ∈N i,j 1 − γ 1 − τhl ξhl t
eεt
r
−
⎫
⎬
M
L
1
Chl txhl t − τhl t − yhl t − τhl t2 eεt−τhl tτhl νij
⎭
2 Chl ∈N i,j 1 − γ ij
r
10
Journal of Applied Mathematics
≤
βij
i,j
⎧
⎨1
⎩2
2
2
xij t − yij t εeεt − eεt aij t xij − yij t
eεt
Bhl ∈Nr i,j eεt
Bijhl txij t − yij t · fij xhl txij t − fij yhl t yij t
Cijhl txij t − yij t
Chl ∈Nr i,j · gij xhl t − τhl txij t − gij yhl t − τhl t yij t
−1 Cijhl ξhl
t 2
M
1 L
νij
−1 xhl t − yhl t eεtτhl 2 Chl ∈N i,j 1 − γ 1 − τhl ξhl t
r
⎫
⎬
L
1 2
Cijhl t xhl t − τhl t − yhl t − τhl t eεt
−
νij
⎭
2 Chl ∈N i,j 1 − γ
r
≤e
εt
⎧
⎨ ε
2
− aij t xij t − yij t
βij
⎩
2
i,j
Bhl ∈Nr i,j "
Bijhl txij t − yij t fij xhl txij t − yij t
fij xhl t − fij yhl t yij t
Chl ∈Nr i,j Cijhl txij t − yij t
"
× gij xhl t − τhl txij t − yij t
gij xhl t − τhl t − gij yhl t − τhl t yij t
−1 Cijhl ξhl
t 2 M
1 L
νij
−1 xhl t − yhl t eετhl
2 Chl ∈N i,j 1 − γ 1 − τhl ξhl t
r
⎫
⎬
L
1 2
Cijhl t xhl t − τhl t − yhl t − τhl t
−
νij
⎭
2 Chl ∈N i,j 1 − γ
r
Journal of Applied Mathematics
⎧
⎨ ε
2
≤ eεt βij
− aij t xij t − yij t
⎩
2
i,j
Bhl ∈Nr i,j Bijhl tμij
L
1−γ
#
×
xij t − yij t
Cijhl tνij
Chl ∈Nr i,j
2
$
xij t − yij txhl t − yhl t
2
xij t − yij t
L
1−γ
#
×
11
xij t − yij t
×xhl t − τhl t − yhl t − τhl t
−1 Cijhl ξhl
t 2 M
L
1 νij
−1 xij t − yij t eετhl
2 Chl ∈N i,j 1 − γ 1 − τhl ξhl t
r
⎫
⎬
L
1 2
Cijhl t xhl t − τhl t − yhl t − τhl t
−
νij
⎭
2 Chl ∈N i,j 1 − γ
r
≤ eεt
βij
i,j
⎧
⎨ ε
⎩ 2
− aij t
Bhl ∈Nr i,j Bijhl tμij
%
×
xij t − yij t
L
1−γ
xij t − yij t
Cijhl tνij
Chl ∈Nr i,j
%
×
2
2
2 2
1
xij t − yij t xhl t − yhl t
2
L
1−γ
2
1
xij t − yij t
2
2 $
× xhl t − τhl t − yhl t − τhl t
xij t − yij t
2
−1 Cijhl ξhl
t 2 M
L
1 νij
−1 xij t − yij t eετhl
2 Chl ∈N i,j 1 − γ 1 − τhl ξhl t
r
⎫
2 ⎬
L
1 hl
C t xhl t − τhl t − yhl t − τhl t
−
νij
⎭
2 Chl ∈N i,j 1 − γ ij
r
&
12
Journal of Applied Mathematics
≤ eεt
⎧
⎨
ε
⎩
i,j
2
βij
− aij t
xij t − yij t
2
3
2
2
L Bijhl tμij
xij t − yij t
1
−
γ
Bhl ∈Nr i,j 2
L 3
βij
Cijhl tνij
xij t − yij t
2 Chl ∈N i,j
1−γ
r
2
L 1 ij
xij t − yij t
βhl Bhl tμhl
2 Bij ∈N h,l
1−γ
r
⎫
ij
−1
ξ
C
t
2 ετ M ⎬
ij
hl
L
1
νhl
xij t − yij t e hl
⎭
2 Cij ∈N h,l 1 − γ 1 − τ ξ−1 t
ij
r
ij
2
c0 ≤ − eεt xij t − yij t ≤ 0,
2
3.12
where c0 > 0 is defined by
⎧
⎨
c0 min inf βij
i,j t∈R ⎩
⎡
L ⎣
μij
2aij t − ε − 3βij
1−γ
⎤
Bijhl t νij
Cijhl t⎦
Bhl ∈Nr i,j Chl ∈Nr i,j ⎞⎫
ij
−1
⎪
⎬
C
ξ
t
ij
hl
L
L
ij
⎟
⎜
βhl ⎝μhl
νhl
Bhl t ⎠ > 0.
⎪
−1
1 − γ Bij ∈N h,l
1−γ
⎭
Chl ∈Nr i,j 1 − τij ξij t
r
⎛
3.13
From the above, we have V t ≤ V t0 , t ≥ t0 , and
n
2
1 εt
e min βhl
xij t − yij t ≤ V t,
h,l
2
i1
t ≥ t0 ,
⎡
−1 ⎤
2
Chl ξhl
t
L μ ετ μ
1 εt0 ⎣
V t0 ≤ e
τhl e hl sup
βij βij νij
−1 ⎦ϕ − ψ 2 .
2
1−γ
t∈t0 −τ,t0 1 − τhl ξhl t
i,j
i,j h,l
3.14
Thus, it follows that there exists a positive constant M > 1 such that
xt − yt ≤ Me−ε/2t−t0 ϕ − ψ ,
2
2
which implies that 1.4 is GES.
t ≥ t0 ,
3.15
Journal of Applied Mathematics
13
Now we assume that A6 is satisfied. By carrying out similar arguments as above, one
can easily show that there exists an ε > 0 such that
⎧
⎨
inf
βij
i,j⎩
⎡
L ⎣
μij
aij t − ε − βij
1−γ
Bhl ∈Nr i,j Bijhl t
νij
Chl ∈Nr i,j ⎤
Cijhl t⎦
⎞⎫
ij
−1
⎪
⎬
ξ
C
t
ij
hl
L
L
ij
⎜
⎟
−βhl ⎝μhl
νhl
Bhl t ⎠ > 0.
⎪
−1
1 − γ Bij ∈N h,l
1−γ
⎭
Chl ∈Nr i,j 1 − τij ξij t
r
⎛
3.16
Consider the Lyapunov function
⎧
⎨
V t βij xij t − yij teεt ⎩
i,j
×
t
t−τhl t
Chl ∈Nr i,j
νij
L
1−γ
⎫
−1 Cijhl ξhl
s εtτ μ ⎬
hl ds
.
−1 xhl s − yhl se
⎭
1 − τhl ξhl
s
3.17
Similar to the above arguments, calculating the upper-right derivative D V t produces
D V t ≤ −c2 eεt
xij t − yij t ≤ 0,
3.18
i,j where
⎧
⎨
⎡
L ⎣
μij
aij t − ε − βij
1−γ
⎤
Bijhl t νij
Cijhl t⎦
hl
C ∈Nr i,j i,j ⎛
⎞⎫
ij
−1
⎪
⎬
C
ξ
t
ij
hl
L
L
ij
⎟
⎜
−βhl ⎝μhl
νhl
Bhl t ⎠ > 0.
⎪
−1
1 − γ Bij ∈N h,l
1−γ
⎭
Chl ∈Nr i,j 1 − τij ξij t
r
3.19
c2 min inf βij
i,j t∈R ⎩
Bhl ∈Nr
Then we have
'
(
xij t − yij t ≤ V t ≤ V t0 ,
eεt min βhl
h,l
i,j t ≥ t0 .
3.20
Note that
⎡
V t0 ≤ eεt0 ⎣ βij βij
i,j i,j Cij ∈Nr h,l
μ
μ
τhl eετhl
−1 ⎤
Cijhl ξhl
t
sup
−1 ⎦ϕ − ψ 1 .
t∈t0 −τ,t0 1 − τhl ξhl t
3.21
14
Journal of Applied Mathematics
Then there exists a positive constant M > 1 such that
n xt − yt xi t − yi t ≤ Mϕ − ψ eεt−t0 ,
1
1
t ≥ t0 .
3.22
i1
The proof is complete.
References
1 A. Bouzerdoum and R. B. Pinter, “Shunting inhibitory cellular neural networks: derivation and
stability analysis,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 3, pp. 215–221, 1993.
2 Y. Liu, Z. You, and L. Cao, “On the almost periodic solution of generalized shunting inhibitory cellular
neural networks with continuously distributed delays,” Physics Letters A, vol. 360, no. 1, pp. 122–130,
2006.
3 T. Zhou, Y. Liu, and A. Chen, “Almost periodic solution for shunting inhibitory cellular neural
networks with time-varying delays and variable coefficients,” Neural Processing Letters, vol. 23, no.
3, pp. 243–255, 2006.
4 L. Chen and H. Zhao, “Global stability of almost periodic solution of shunting inhibitory cellular
neural networks with variable coefficients,” Chaos, Solitons & Fractals, vol. 35, no. 2, pp. 351–357, 2008.
5 A. Chen and J. Cao, “Almost periodic solution of shunting inhibitory CNNs with delays,” Physics
Letters A, vol. 298, no. 2-3, pp. 161–170, 2002.
6 Y. Li, C. Liu, and L. Zhu, “Global exponential stability of periodic solution for shunting inhibitory
CNNs with delays,” Physics Letters A, vol. 337, no. 1-2, pp. 46–54, 2005.
7 X. Huang and J. Cao, “Almost periodic solution of shunting inhibitory cellular neural networks with
time-varying delay,” Physics Letters A, vol. 314, no. 3, pp. 222–231, 2003.
8 B. Liu and L. Huang, “Existence and stability of almost periodic solutions for shunting inhibitory
cellular neural networks with time-varying delays,” Chaos, Solitons & Fractals, vol. 31, no. 1, pp. 211–
217, 2007.
9 A. Chen and J. Cao, “Almost periodic solution of shunting inhibitory CNNs with delays,” Physics
Letters A, vol. 298, no. 2-3, pp. 161–170, 2002.
10 Y. Xia, J. Cao, and Z. Huang, “Existence and exponential stability of almost periodic solution for
shunting inhibitory cellular neural networks with impulses,” Chaos, Solitons & Fractals, vol. 34, no. 5,
pp. 1599–1607, 2007.
11 Q. Zhou, B. Xiao, Y. Yu, and L. Peng, “Existence and exponential stability of almost periodic solutions
for shunting inhibitory cellular neural networks with continuously distributed delays,” Chaos, Solitons
& Fractals, vol. 34, no. 3, pp. 860–866, 2007.
12 B. Liu, “Almost periodic solutions for shunting inhibitory cellular neural networks without global
Lipschitz activation functions,” Journal of Computational and Applied Mathematics, vol. 203, no. 1, pp.
159–168, 2007.
13 C. Y. Zhang, Pseudo almost periodic functions and their applications, Ph.D. thesis, University of Western
Ontario, Ontario, Canada, 1992.
14 C. Y. Zhang, “Pseudo-almost-periodic solutions of some differential equations,” Journal of Mathematical Analysis and Applications, vol. 181, no. 1, pp. 62–76, 1994.
15 C. Y. He, Almost Periodic Differential Equations, Higher Education Publishing House, Beijing, China,
1992.
16 A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer,
Berlin, Germany, 1974.
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