Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 386706, 14 pages
doi:10.1155/2012/386706
Research Article
Stability and Existence of Periodic Solutions for
Cellular Neural Networks with State Dependent
Delays on Time Scales
Yaping Ren1 and Yongkun Li2
1
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan,
Kunming 650221, China
2
Department of Mathematics, Yunnan University, Yunnan, Kunming 650091, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 12 April 2012; Accepted 27 August 2012
Academic Editor: Josef Diblı́k
Copyright q 2012 Y. Ren and Y. Li. 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.
We study delayed cellular neural networks on time scales. Without assuming the boundedness of
the activation functions, we establish the exponential stability and existence of periodic solutions.
The results in this paper are completely new even in case of the time scale T R or Z and improve
some of the previously known results.
1. Introduction
Consider the following cellular neural networks with state-dependent delays on time scales:
xiΔ t −ci txiσ t n
j1
aij tfj xt n
bij tfj xj ξij t, xt Ii t,
t ∈ T,
1.1
j1
where i 1, 2, . . . , n, T is an ω-periodic time scale which has the subspace topology inherited
from the standard topology on R, xiσ t xi σt, σt will be defined in the next section,
xt x1 t, x2 t, . . . , xn t, n corresponds to the number of units in the neural network,
xi t corresponds to the state of the ith unit at time t, fj xj t denotes the output of the
jth unit on ith unit at time t, bij denotes the strength of the jth unit on the ith unit at time
ξij t, xt, Ii denotes the external bias on the ith unit at time t, t − ξij t, xt corresponds to
the transmission delay along the axon of the jth unit, ci represents the rate with which the
ith unit will reset its potential to the resting state in isolation when disconnected from the
network and external inputs.
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Discrete Dynamics in Nature and Society
It is well known that the cellular neural networks have been successfully applied to
signal processing, pattern recognition, optimization, and associative memories, especially
in image processing and solving nonlinear algebraic equations. They have been widely
studied both in theory and applications 1–3. Many results for the existence of their periodic
solutions and the exponential convergence properties for cellular neural networks have been
reported in the literatures. See, for instance, 4–17 and references cited therein.
In fact, continuous and discrete systems are very important in implementation and
applications. It is well known that the theory of time scales has received a lot of attention
which was introduced by Stefan Hilger in order to unify continuous and discrete analysis.
Therefore, it is meaningful to study dynamic systems on time scales which can unify
differential and difference systems see 18–28.
When T R, ξij t, · ≡ t − τij t, 1.1 reduces to
xi t −ci txi t n
n
aij tfj xj t bij tfj xj t − τij t Ii t,
j1
1.2
j1
where i 1, 2, . . . , n. By using Mawhin’s continuation theorem and Liapunov functions, the
authors 6, 14 obtained the existence and stability of periodic solutions of 1.2, respectively.
Furthermore, 1.1 also covers discrete system for when T Z, ξij n, · ≡ n − τij n;
see 15
xi n 1 xi ne−ci nh θi h
m
aij nfj xj n
j1
θi h
m
bij nfj xj n − τij n θi hIi n,
1.3
j1
where θi h 1−e−ci nh /ci n, i 1, 2, . . . , m. In 15, the author firstly obtained the discretetime analogue of 1.3 by the semidiscretization technique 29, 30, and then some sufficient
conditions for the existence and global asymptotical stability of periodic solutions of 1.3
were established by using Mawhin’s continuation theorem and Liapunov functions.
However, in 5, 13–15, the activation functions fj , j 1, 2, . . . , n are assumed to be
bounded. Our main purpose of this paper is to establish the stability and existence of periodic
solutions of 1.1 without assuming the boundedness of the activation functions.
For the sake of convenience, we denote
ci max |ci t|,
t∈0,ωT
ci min |ci t|,
bij max bij t,
t∈0,ωT
t∈0,ωT
I i max |Ii t|,
t∈0,ωT
aij max aij t,
t∈0,ωT
i, j 1, 2, . . . , n.
1.4
Throughout this paper, we assume that
H1 ξij ·, · ∈ CT × Rn , 0, ∞T is ω-periodic with respect to its first argument and
satisfies ξij t, x ≤ τ for all t, x ∈ T × Rn , ci ∈ CT, 0, ∞, aij , bij , Ii ∈ CT, R, i,
j 1, 2, . . . , n are ω-periodic functions;
Discrete Dynamics in Nature and Society
3
H2 fj ∈ CR, R, and there exists a positive number Mj such that |fj x − fj y| Mj |x − y| for all x, y ∈ R, j 1, 2, . . . , n.
The initial conditions of system 1.1 are of the following form:
xi s φi s,
i 1, 2, . . . , n, s ∈ −τ, 0T ,
τ
max
max τij t,
1in,1jn t∈0,ωT
1.5
where φi ∈ C−τ, 0T , R, i 1, 2, . . . , n.
The organization of this paper is as follows. In Section 2, we introduce some lemmas
and definitions and state some preliminary results needed in later sections, which will be
used in latter sections. In Section 3, we will study the existence of periodic solutions of
system 1.1 by using the method of coincidence degree. In Section 4, we will derive sufficient
conditions to ensure that the periodic solutions of 1.1 are globally exponentially stable. 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 introduce some notations and definitions and state some preliminary
results.
Let T be a nonempty closed subset time scale of 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
A point t ∈ T is called left dense if t > inf T and ρt t, left scattered if ρt < t,
right dense if t < sup T and σt t, and right scattered if σt > t. If T has a left-scattered
maximum m, then Tk T \ {m}; otherwise Tk T. If T has a right-scattered minimum m,
then Tk T \ {m}; otherwise Tk T.
A function f : T → R is right dense continuous provided that it is continuous at rightdense point in T, and its left-side limits exist at left-dense points in T. If f is continuous at
each right-dense point and each left-dense point, then f is said to be continuous function on
T. The set of continuous functions f : T → R will be denoted by CT.
For y : T → R and t ∈ Tk , we define the delta derivative of yt, yΔ 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
yσt − ys − yΔ tσt − s < ε|σt − s|
∀s ∈ U.
2.2
If y is continuous, then y is right-dense continuous, and if y is delta differentiable at t,
then y is continuous at t.
Let y be right dense continuous. If Y Δ t yt, then we define the delta integral by
t
ysΔs Y t − Y a.
a
Definition 2.1 see 31. We say that a time scale T is periodic if there exists p > 0 such that if
t ∈ T, then t ± p ∈ T. For T /
R, the smallest positive p is called the period of the time scale.
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Discrete Dynamics in Nature and Society
Definition 2.2 see 31. Let T /
R be a periodic time scale with period p. We say that the
function f : T → R is periodic with period ω if there exists a natural number n such that
ω np, ft ω ft for all t ∈ T, and ω is the smallest number such that ft ω ft. If
T R, we say that f is periodic with period ω > 0 if ω is the smallest positive number such
that ft ω ft for all t ∈ T.
If T is ω periodic, then σt ω σt ω and μt is an ω-periodic function.
Definition 2.3 see 32. A function p : T → R is said to be regressive provided that 1 μtpt / 0 for all t ∈ Tk , where μt σt − t is the graininess function. The set of all
regressive rd-continuous functions f : T → R is denoted by R while the set R is given by
{f ∈ R : 1 μtft > 0} for all t ∈ T. Let p ∈ R. The exponential function is defined by
ep t, s exp
t
ξμτ
pτ Δτ ,
2.3
s
where ξhz is the so-called cylinder transformation.
Let p, q : T → R be two regressive functions, and we define
p ⊕ q : p q μpq,
p : −
p
,
1 μp
p q : p ⊕ q .
2.4
Then the generalized exponential function has the following properties.
Lemma 2.4 see 32. Assume that p, q : T → R are two regressive functions, and then
1 ep σt, s 1 μtptep t, s;
2 ep t, s 1/ep s, t e
p s, t;
3 ep t, sep s, r ep t, r.
Lemma 2.5 see 32. 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.
2.5
Lemma 2.6 see 32. If a, b ∈ T, α, β ∈ R, and f, g ∈ CT, R, then
b
ftΔt α a gtΔt;
b
2 if ft ≥ 0 for all a ≤ t < b, then a ftΔt ≥ 0;
1
b
a
αft βgtΔt α
b
a
3 if |ft| ≤ gt on a, b : {t ∈ T : a ≤ t < b}, then |
b
a
ftΔt| ≤
b
a
gtΔt.
Lemma 2.7 see 33. Let t1 , t2 ∈ 0, ωT . If x : T → R is ω periodic, then
ω
Δ xt xt1 x sΔs,
0
ω
Δ xt xt2 −
x sΔs.
0
2.6
Discrete Dynamics in Nature and Society
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Definition 2.8. The periodic solution x∗ t x1∗ t, x2∗ t, . . . , xn∗ tT of system 1.1 is said
to be exponentially stable if there exists a positive constant α with α ∈ R such that
for every δ ∈ −τ, 0T , and there exists N Nδ 1 such that the solution xt x1 t, x2 t, . . . , xn tT of system 1.1 through δ, xδ satisfies
n xi t − x∗ t N φδ − x∗ δe
α t, δ,
i
t ∈ T : 0, ∞T ,
2.7
i1
where φδ − x∗ δ n
i1
maxδ∈−τ,0T |φi δ − xi∗ δ|.
In order to show that there exists at least one ω-periodic solution of system 1.1, we
need the following concepts and result which are cited from 34.
Let X, Y be Banach spaces, L : Dom L ⊂ X → dim Y be a linear mapping, and N :
X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index
zero if dim Ker L co dim Im L < ∞ and Im L is closed in Y. If L is a Fredholm mapping
of index zero and there exists continuous projector P : X → X and Q : Y → Y such that
Im P Ker L, Ker Q ImI − Q, it follows that mapping L|Dom L∩Ker P : I − P X → Im L is
invertible. We denote the inverse of that mapping by KP . If Ω is an open bounded subset of X,
the mapping N will be called L compact on Ω if QNΩ is bounded and KP I −QN : Ω → X
is compact.
Lemma 2.9 see 34. Let X, Y be two Banach spaces and 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 equation Lx Nx has at least one solution on DL ∩ Ω.
Definition 2.10 see 35. A real matrix P pij n×n is said to be a nonsingular M matrix if
j, and all successive principal minors of P are positive.
pij ≤ 0, i, j 1, 2, . . . , n, i /
For A aij m×n , B bij m×n ∈ Rmn , A ≥ BA > B means that each pair of
corresponding elements of A and B such that aij ≥ bij aij > bij .
Lemma 2.11 see 35. Assume that P is a nonsingular M matrix and P y ≤ h, then y ≤ P −1 h.
3. Existence of Periodic Solutions
In this section, by Lemma 2.9, we will study the existence of at least one periodic solution of
system 1.1.
Theorem 3.1. Suppose that (H1)-(H2) holds, E eij n×n is a nonsingular M matrix, where
ri dii
eij z
dij
i j,
i/
j,
3.1
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Discrete Dynamics in Nature and Society
and ri ci 1 − ωci /1 ci ω, dij −aij bij Mj , i, j 1, 2, . . . , n, then system 1.1 has at least
one ω-periodic solution.
Proof. Let
X Y x x1 , x2 , . . . , xn T ∈ CT, R : xt ω xt, t ∈ T
3.2
with the norm defined by x max1≤i≤n |xi |0 , where |xi |0 maxt∈0,ωT |xi t|, then X and Y
are Banach spaces.
Set
Lx xΔ t,
P x Qx 1
ω
ω
xtΔt,
x ∈ X,
3.3
0
and N : X → Y
Nxi t −ci txiσ t n
j1
n
aij tfj xj t bij tfj xj ξij t, xt Ii t,
3.4
j1
where i 1, 2, . . . , n. Obviously, Ker L Rn , Im L {x ∈ X :
and
ω
0
xsΔs 0} is closed in Y
dim Ker L n codim Im L.
3.5
Hence, L is a Fredholm mapping of index zero. Furthermore, similar to the proof of
Theorem 3.4 in 21, one can easily show that N is L compact on Ω with any open bounded
set Ω ⊂ X. Corresponding to the operator equation Lx − Nx λ−Lx − N−x, λ ∈ 0, 1, we
have
xiΔ t
⎤
⎡
n
n
1 ⎣
σ
−ci txi t aij tfj xj t bij tfj xj ξij t, xt Ii t⎦
1λ
j1
j1
⎤
⎡
n
n
λ ⎣
σ
−
ci txi t aij tfj −xj t bij tfj −xj ξij t, xt Ii t⎦,
1λ
j1
j1
3.6
where i 1, 2, . . . , n.
Discrete Dynamics in Nature and Society
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Suppose that x x1 , x2 , . . . , xn T is a solution of system 3.6 for some λ ∈ 0, 1. In
view of 3.6 and H2, we have
ω
ω
ω
Δ ci txσ tΔt |Ii t|Δt
xi tΔt ≤
i
0
0
0
⎡
⎤
ω n
n
1
⎣ aij fj xj t Δt
⎦
b
f
xt
x
ξ
t,
ij j
j ij
1 λ 0 j1
j1
⎡
⎤
ω n
n
λ
Δt
⎣ aij fj −xj t ⎦
−x
ξ
b
f
xt
t,
ij j
j ij
1 λ 0 j1
j1
ω
ci xσ tΔt i
0
⎡
ω n
⎣ aij Mj xj t fj 0
0
j1
n
bij
3.7
⎤
Mj xj ξij t, xt fj 0 I i ⎦Δt
j1
ci ω|xi |0 ω
n aij bij
j1
ci ω|xi |0 ω
n j1
⎤
⎡
n Mj xj 0 ω⎣
aij bij fj 0 I i ⎦
j1
aij bij Mj xj 0 ωαi ,
i 1, 2, . . . , n,
where αi nj1 aij bij |fj 0| I i .
Integrating both sides of 3.6 from 0 to ω, we obtain that
ω
0
ci txiσ tΔt
1
1λ
−
⎤
⎡
ω n
n
⎣ aij tfj xj t bij tfj xj ξij t, xt Ii t⎦Δt
0
λ
1λ
j1
j1
⎤
⎡
ω n
n
⎣ aij tfj −xj t bij tfj −xj ξij t, xt Ii t⎦Δt,
0
j1
j1
3.8
where i 1, 2, . . . , n. Then we have from 3.7 that
⎤
⎡
ω ω
n n ci txσ tΔt ⎣ aij tfj xj t bij tfj xj ξij t, xt |Ii t|⎦Δt
i
0
0
ω
j1
n j1
aij bij
j1
Mj xj 0 ωαi ,
3.9
i 1, 2, . . . , n.
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Discrete Dynamics in Nature and Society
From Lemma 2.7, for any ti1 , ti2 ∈ 0, ωT , i 1, 2, . . . , n, we have
ω
ω
0
0
ci txiσ tΔt ci txiσ tΔt
Dividing by
ω
0
ω
0
ω
0
ω
ω Δ ci txi ti1 Δt ci t
xi tΔt Δt,
ci txi ti2 Δt −
ω
0
0
0
ω Δ ci t
xi tΔt Δt,
3.10
i 1, 2, . . . , n.
0
ci tΔt on the two sides of the inequalities above, we obtain that
ω
ω
1
Δ −xi ti1 − ω
ci txiσ tΔt xi tΔt,
c
0
0 i tΔt 0
ω
ω
1
Δ xi ti2 ω
ci txiσ tΔt xi tΔt, i 1, 2, . . . , n.
c
0
0 i tΔt 0
3.11
3.12
Let |xi ti0 | maxt∈0,ωT |xi t|, i 1, 2, . . . , n. If for some i 1, 2, . . . , n, xi ti0 0, we
choose ti2 ti0 . Hence maxt∈0,ωT |xi t| xi ti2 . From 3.12, we have
ci |xi |0 ci
ω
1
ω
ci txiσ tΔt
ω
Δ xi tΔt
c
0
0 i tΔt 0
ω
ω
1
ci txiσ tΔt ci xiΔ tΔt.
ω 0
0
3.13
If for some i 1, 2, . . . , n, xi ti0 0, we choose ti1 ti0 . Hence maxt∈0,ωT |xi t| −xi ti1 . From
3.11, we also have 3.13.
By using 3.7 and 3.9 and 3.13, for i 1, 2, . . . , n, we obtain that
ci |xi |0 n j1
n aij bij Mj xj 0 αi ci ω|xi |0 ci ω
aij bij Mj xj 0 ci ωαi ,
3.14
j1
so,
n ci 1 − ωci aij bij Mj xj 0 αi ,
|xi |0 1 ci ω
j1
3.15
namely,
ri |xi |0 −
n
j1
dij xj 0 αi ,
i 1, 2, . . . , n.
3.16
Denote the following:
|x|0 |x1 |0 , |x2 |0 , . . . , |xn |0 T ,
C α1 , α2 , . . . , αn T .
3.17
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Thus 3.16 is rewritten in the matrix form
E|x|0 C.
3.18
From the conditions of Theorem 3.1, E is a nonsingular M matrix, therefore
|x|0 E−1 C : B1 , B2 , . . . , Bn T ,
3.19
that is, |xi |0 Bi , i 1, 2, . . . , n.
Take A max1≤i≤n Bi 1 and
Ω {x ∈ X : x < A}.
3.20
It is clear that Ω satisfies all the requirements in Lemma 2.9, and condition H is satisfied. In
view of all the discussions above, we conclude from Lemma 2.9 that system 1.1 has at least
one ω-periodic solution. This completes the proof.
From Theorem 3.1, when T R, ξij t, · ≡ t − τij t, we have the following.
Corollary 3.2. Under assumptions of Theorem 3.1, system 1.2 has at least one ω-periodic solution.
When T Z, ξij n, · ≡ n − τij n, from Theorem 3.1, we have the following.
Corollary 3.3. Under assumptions of Theorem 3.1, system 1.3 has at least one ω-periodic solution.
4. Global Exponential Stability of the Periodic Solution
In this section, we will establish some results for the global exponential stability of the
periodic solution of 1.1. The following lemma is essential.
Lemma 4.1 see 36. Let p : T → R be rd-continuous and regressive. Suppose that f : T → R is
rd-continuous, δ ∈ T, and y0 ∈ R. Then y is the unique solution of the initial value problem
yΔ t ptyσ t ht,
yδ y0 ,
4.1
if and only if
yt e
p t, δy0 t
e
p t, shsΔs.
4.2
δ
Theorem 4.2. Assume H1 , H2 , and H4 hold. Furthermore, suppose that
> 0, where M
max1≤i≤n {n aij Mj H5 for t ∈ 0, ωT , i 1, 2, . . . , n, ci t − M
j1
n
b
M
}
ij
j
j1
holds. Then the ω-periodic solution of system 1.1 is globally exponentially stable.
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Discrete Dynamics in Nature and Society
Proof. From Theorem 3.1, we see that system 1.1 has at least one ω-periodic solution x∗ t x1∗ t, x2∗ t, . . . , xn∗ tT . Let xt x1 t, x2 t, . . . , xn tT be an arbitrary solution of system
1.1. Let yt xt − x∗ t, and then we obtain that
yiΔ t − ci tyiσ t n
aij t f xj t − f xj∗ t
j1
n
bij t fj xj ξij t, xt − fj xj∗ ξij t, xt ,
4.3
i 1, 2, . . . , n
j1
with initial values given by
yi s ϕi s − xi∗ s,
s ∈ −τ, 0T , i 1, 2, . . . , n,
4.4
where each ϕi ∈ C−τ, 0T , R is bounded.
Let ht h1 t, h2 t, . . . , hn tT and
hi t n
aij t f xj t − f xj∗ t
j1
n
bij t fj xj ξij t, xt − fj xj∗ ξij t, xt ,
4.5
i 1, 2, . . . , n.
j1
From Lemma 4.1, then
yi t e
ci t, δy0 t
e
ci t, shi sΔs,
δ
i 1, 2, . . . , n, δ ∈ −∞, 0T , t ≥ δ
4.6
is the unique solution of 4.3 with initial value yδ y0 . From 4.6, we have
t
yi t
1
≤ y0 |hi s|Δs,
e
ci t, δ
e
δ
s,
c
δ
i
i 1, 2, . . . , n, t ≥ δ
4.7
and, according to 4.5, we can obtain that
⎛
h ≤ ⎝
n
j1
aij Mj n
⎞
y,
bij Mj ⎠y ≤ M
i 1, 2, . . . , n, t ≥ δ.
4.8
j1
Thus, by 4.7, we have
t
y
y ≤ y0 M
Δs,
e
ci t, δ
e
ci s, δ
δ
i 1, 2, . . . , n, t ≥ δ.
4.9
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By virtue of Gronwall’s inequality 32, Corollary 6.7, we obtain
y
≤ y0 eM
t, δ,
e
ci t, δ
i 1, 2, . . . , n, t ≥ δ.
4.10
Hence we have
y ≤ y0 e t, δe
c t, δ
i
M
≤ y0 e
ci ⊕M
t, δ
y0 e−ci −M/1μc
t, δ
i
≤ y0 e−γ t, δ, t ≥ δ,
4.11
μci }, i 1, 2, . . . , n. We can rewrite 4.10 as follows
where γ min1≤i≤n {ci − M/1
n xi t − x∗ t ≤ φδ − x∗ δe−γ t, δ,
i
t ≥ δ.
4.12
i1
Therefore, the ω-periodic solution of system 1.1 is globally exponentially stable. This
completes the proof.
From Theorem 4.2, when T R, ξij t, · ≡ t − τij t, we have the following.
Corollary 4.3. Under assumptions of Theorem 4.2, all the ω-periodic solutions of system 1.2 are
globally exponentially stable.
When T Z, ξij n, · ≡ n − τij n, we have the following.
Corollary 4.4. Under assumptions of Theorem 4.2, all the ω-periodic solutions of system 1.3 are
globally exponentially stable.
Remark 4.5. Corollaries 3.2 and 4.3 improve the corresponding results obtained in 4–6, 13, 14
and are different from those in 8, 11, 12, 16, 17 and the references cited therein. Therefore,
the results of this paper are new, and they complement previously known results. Corollaries
3.3 and 4.4 improve the results obtained in 15.
5. An Example
Consider the following cellular neural networks with delays:
xiΔ t −ci txi t 2
j1
2
aij tfj xj t bij tfj xj t − τij t Ii t,
5.1
j1
where t ∈ T, i 1, 2, T is a 1-periodic time scale, c1 t 1/4 1/8 sin2πt, c2 t 1/3 1/6 cos2πt, a11 t sin2 πt, a12 t a21 t 1/4sin4 πt, a22 t | cosπt|,
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b11 t 2cos2 πt, b12 t b21 t 1/8cos6 πt, b22 t 1/2| sinπt|, f1 x1 1/112x 1/56 sin1/2x1 3, f2 x2 1/126x 1/42 cos2/3x2 − 11, I1 t sin6 πt,
2
6
I2 t 3 cos2πt, τ11 t ecos2πt , τ12 t 3sin πt , τ21 t lnsin4 πt, τ22 t 3/75cos πt .
By calculating, we have c1 3/8, c1 1/8, c2 1/2, c2 1/6, M1 1/56, M2 1/42,
a11 1, a12 a21 1/4, a22 1, b11 2, b12 b21 1/8, b22 1/2. It is not difficult to verify
that H1-H2 are satisfied.
Also by calculating, we have that
E eij 2 × 2
⎛ 1
1 ⎞
−
⎜ 28
112 ⎟
⎜
1 ⎟
⎠
⎝ 15
−
228 28
5.2
is a nonsingular M matrix. If we take ξ1 ξ2 1, we can obtain that
−ξ1 c1 2
7
< 0,
ξj a1j b1j Mj −
112
j1
−ξ2 c2 2
167
< 0. 5.3
ξj a2j b2j Mj −
1344
j1
The condition H3 is satisfied. By Theorems 3.1 and 4.2, we know that system 5.1 has at
least one 1-periodic solution, and this solution is globally exponentially stable.
6. Conclusion
In this paper, without assuming the boundedness of the activation functions, we establish
the stability and existence of periodic solutions of cellular neural networks with delays on
time scales. Our results obtained in this paper are completely new even in case of the time
scale T R or Z. Besides, our method used in this paper may be used to study other type
neural networks such as BAM neural networks, high-order Hopfield neural networks, and
Cohen-Grossberg neural networks.
Acknowledgments
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant 10971183, and this work was also supported by IRTSTYN.
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