Electronic Journal of Differential Equations, Vol. 2007(2007), No. 99, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ALMOST PERIODIC SOLUTIONS FOR HIGHER-ORDER
HOPFIELD NEURAL NETWORKS WITHOUT BOUNDED
ACTIVATION FUNCTIONS
FUXING ZHANG, YA LI
Abstract. In this paper, we consider higher-order Hopfield neural networks
(HHNNs) with time-varying delays. Based on the fixed point theorem, Lyapunov functional method, differential inequality techniques, and without assuming the boundedness on the activation functions, we establish sufficient
conditions for the existence and local exponential stability of the almost periodic solutions. The results of this paper are new and they complement previously known results.
1. Introduction
Consider the following higher-order Hopfield neural networks (HHNNs), with
time-varying delays,
n
X
x0i (t) = −ci xi (t) +
aij (t)gj (xj (t − τij (t)))
j=1
+
n X
n
X
(1.1)
bijl (t)gj (xj (t − σijl (t)))gl (xl (t − νijl (t))) + Ii (t),
j=1 l=1
for i = 1, 2, . . . , n, where n corresponds to the number of units in a neural network,
xi (t) corresponds to the state vector of the ith unit at the time t, ci > 0 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, aij (t) and bijl (t)
are the first- and second-order connection weights of the neural network, τij (t) ≥ 0,
σijl (t) ≥ 0 and υijl (t) ≥ 0 correspond to the transmission delays, Ii (t) denote the
external inputs at time t, and gj is the activation function of signal transmission.
Due to the fact that high-order neural networks have stronger approximation
property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks have been the
object of intensive analysis by numerous authors in recent years. In particular,
there have been extensive results on the problem of the existence and stability of
2000 Mathematics Subject Classification. 34C25, 34K13.
Key words and phrases. High-order Hopfield neural networks; almost periodic solution;
exponential stability; time-varying delays.
c
2007
Texas State University - San Marcos.
Submitted March 11, 2007. Published July 13, 2007.
Supported by grant 10371034 from NNSF of China.
1
2
F. ZHANG, Y. LI
EJDE-2007/99
equilibrium points and periodic solutions of HHNNs (1.1) in the literature. We
refer readers to [1, 2, 7, 8] and the references cited therein. The assumption
(T0) for each j ∈ {1, 2, . . . , n}, gj : R → R is bounded, i.e., there exists a
constant Lj such that
|gj (u)| ≤ Lj ,
for all u ∈ R
(1.2)
has been considered as a fundamental condition for the existence and stability
of equilibrium points and periodic solutions solutions of HHNNs (1.1). To the
best of our knowledge, few authors have considered the problems of periodic and
almost periodic solutions of HHNNs (1.1) without the assumptions (T0). Thus, it
is worth while to investigate the existence and stability of almost periodic solutions
of HHNNs (1.1) in this case.
In this paper we shall study the existence and stability of almost periodic solutions for (1.1). By applying the fixed point theorem, Lyapunov functional method
and differential inequality techniques, we derive some new sufficient conditions ensuring the existence and local exponential stability of the almost periodic solution
of (1.1). These results are new and they complement previously known results.
In particular, an example is also provided to illustrate the effectiveness of the new
results.
Throughout this paper, for i, j, l = 1, 2, . . . , n, it will be assumed that Ii , aij ,
bijl , τij , σijl , νijl : R → R are almost periodic functions, and there exist constants
τ , aij , bijl and Ii such that
τ = max
max sup τij (t), max sup σijl (t), max sup νijl (t) ,
1≤i,j≤n t∈R
1≤i,j,l≤n t∈R
1≤i,j,l≤n t∈R
(1.3)
sup |bijl (t)| = bijl , sup |aij (t)| = aij , sup |Ii (t)| = Ii .
t∈R
t∈R
t∈R
We also assume that the following conditions hold:
(H1) For each j ∈ {1, 2, . . . , n}, there exists a nonnegative constant Lgj such that
gj (0) = 0, |gj (u) − gj (v)| ≤ Lgj |u − v|, for all u, v ∈ R.
(H2) Assume that there exist nonnegative constants L, q and δ such that
n
n X
n
X
X
Ii
g
L = max { }, δ = max {c−1
[
a
L
bijl Lgj Lgl ]} < 1,
+
ij
i
j
1≤i≤n ci
1≤i≤n
j=1
j=1
l=1
L
≤ 1,
1−δ
q = max
1≤i≤n
c−1
i
n
X
n
n
2L X X
< 1.
aij Lgj +
bijl Lgj Lgl
1 − δ j=1
j=1
l=1
For convenience, we introduce the following notation. We use x = (x1 , x2 , . . . , xn )T
in Rn to denote a column vector, in which the symbol (T ) denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x| =
(|x1 |, |x2 |, . . . , |xn |)T , and define kxk = max1≤i≤n |xi |. A vector x ≥ 0 means
that all entries of x are greater than or equal to zero. x > 0 is defined similarly.
For vectors x and y, x ≥ y (resp. x > y) means that x − y ≥ 0 (resp. x − y > 0).
For the rest of this paper, we set
{xj (t)} = (x1 (t), x2 (t), . . . , xn (t))T ,
B = {ϕ|ϕ = {ϕj (t)} = (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))T },
where ϕ is an almost periodic function on R. For all ϕ ∈ B, we define the induced
module kϕkB by kϕkB = supt∈R kϕ(t)k. Therefore B is a Banach space.
EJDE-2007/99
ALMOST PERIODIC SOLUTIONS
3
The initial conditions associated with system (1.1) are of the form
xi (s) = ϕi (s), s ∈ [−τ, 0], i = 1, 2, . . . , n,
T
(1.4)
n
where ϕ = (ϕ1 (t), ϕ2 (t), . . . , ϕn (t)) ∈ C([−τ, 0]; R ).
Definition. [3, 4] Let u(t) : R → Rn be continuous in t. u(t) is said to be almost
periodic on R if, for any ε > 0, the set T (u, ε) = {δ : |u(t + δ) − u(t)| < ε, for
all t ∈ R} is relatively dense, i.e., for ∀ε > 0, it is possible to find a real number
l = l(ε) > 0, for any interval with length l(ε), there exists a number δ = δ(ε) in
this interval such that |u(t + δ) − u(t)| < ε, for for all t ∈ R.
The remaining part of this paper is organized as follows. In Section 2, we shall
derive new sufficient conditions for the existence of almost periodic solutions of
(1.1). In Section 3, we present some new sufficient conditions for the local exponential stability of the almost periodic solution of (1.1). In Section 4, we shall
give some examples and remarks to illustrate our results obtained in the previous
sections.
2. Existence of Almost Periodic Solutions
Theorem 2.1. Let conditions (H1) and (H2) hold. Then, there exists a unique
δL
},
almost periodic solution to (1.1) in the region B ∗ = {ϕ|ϕ ∈ B, kϕ − ϕ0 kB ≤ 1−δ
where
Z
t
ϕ0 (t) =
exp(−cj (t − s))Ij (s)ds
−∞
t
=
Z
Z
t
exp(−c1 (t − s))I1 (s)ds,
−∞
Z t
exp(−c2 (t − s))I2 (s)ds,
−∞
T
exp(−cn (t − s))In (s)ds .
...,
−∞
Proof. For each ϕ ∈ B, we consider the almost periodic solution xϕ (t) to the
nonlinear almost periodic differential equations
x0i (t) = −ci xi (t) +
n
X
aij (t)gj (ϕj (t − τij (t)))
j=1
+
n X
n
X
(2.1)
bijl (t)gj (ϕj (t − σijl (t)))gl (ϕl (t − νijl (t))) + Ii (t),
j=1 l=1
for i = 1, 2, . . . , n. Then τij (t), bij (t) and Ii (t) are almost periodic functions.
According to [3, pp. 80-112] and [4, pp. 90-100], we know that the auxiliary system
(2.1) has exactly one almost periodic solution
ϕ
ϕ
T
xϕ (t) = (xϕ
1 (t), x2 (t), . . . , xn (t))
Z
n
t
hX
=
e−c1 (t−s)
a1j (s)gj ϕj (s − τ1j (s))
−∞
+
n X
n
X
j=1 l=1
j=1
i
b1jl (s)gj (ϕj (s − σ1jl (s)))gl (ϕl (s − ν1jl (s))) + I1 (s) ds,
4
F. ZHANG, Y. LI
t
Z
e−cn (t−s)
...,
n
hX
−∞
+
EJDE-2007/99
anj (s)gj (ϕj (s − τnj (s)))
j=1
n X
n
X
i T
bnjl (s)gj (ϕj (s − σnjl (s)))gl (ϕl (s − νnjl (s))) + In (s) ds .
j=1 l=1
Now, we define a mapping T : B → B by setting
T (ϕ)(t) = xϕ (t),
∀ϕ ∈ B.
it is easy to see that B ∗ is a closed convex
Since B ∗ = {ϕ ∈ B, kϕ − ϕ0 kB ≤
subset of B. According to the definition of the norm of Banach space B, we get
Z
t
kϕ0 kB = sup max
Ii (s) exp(−ci (t − s))ds
δL
1−δ },
t∈R 1≤i≤n
−∞
Ii
≤ sup max { }
t∈R 1≤i≤n ci
= max {
1≤i≤n
Ii
} = L.
ci
Therefore, for for all ϕ ∈ B ∗ , we have
kϕkB ≤ kϕ − ϕ0 kB + kϕ0 kB ≤
L
δL
+L=
≤ 1.
1−δ
1−δ
(2.2)
In view of (H1), we have
|gj (u)| ≤ Lgj |u| for all u ∈ R, j = 1, 2, . . . , n.
∗
(2.3)
∗
Now, we prove that the mapping T is a self-mapping from B to B . In fact, for
all ϕ ∈ B ∗ , from (2.2) and (2.3), we obtain
kT ϕ − ϕ0 kB
= sup max
t∈R 1≤i≤n
+
n
n X
X
Z
t
e−ci (t−s)
|
−∞
n
hX
aij (s)gj (ϕj (s − τij (s)))
j=1
i bijl (s)gj (ϕj (s − σijl (s)))gl (ϕl (s − νijl (s))) ds|
j=1 l=1
Z
≤ sup max {
t∈R 1≤i≤n
t
e−ci (t−s)
−∞
Z
t∈R 1≤i≤n
e−ci (t−s)
−∞
Z
t∈R 1≤i≤n
≤ max {c−1
i
1≤i≤n
δL
=
,
1−δ
e−ci (t−s)
−∞
n
hX
j=1
n
hX
aij Lgj
j=1
t
≤ sup max {
aij Lgj kϕkB +
j=1
t
≤ sup max {
n
hX
aij Lgj +
n
hX
j=1 l=1
j=1 l=1
n X
n
X
L
+
1 − δ j=1
aij Lgj +
j=1
n
n
XX
n X
n
X
n X
n
X
j=1 l=1
l=1
i
bijl Lgj Lgl kϕk2B ds}
bijl Lgj Lgl (
L 2i
) ds}
1−δ
i
L
bijl Lgj Lgl ds
}
1−δ
i L
bijl Lgj Lgl }
1−δ
EJDE-2007/99
ALMOST PERIODIC SOLUTIONS
5
Pn Pn
Pn
g
g g
where δ = max1≤i≤n {c−1
i [
j=1
l=1 bijl Lj Ll ]}. This implies that
j=1 aij Lj +
∗
T (ϕ)(t) ∈ B . Next, we prove that the mapping T is a contraction mapping on B ∗ .
In view of (2.2) and (H1), for all φ, ψ ∈ B ∗ , we have
|T (φ(t)) − T (ψ(t))|
T
= |(T (φ(t)) − T (ψ(t)))1 |, . . . , |(T (φ(t)) − T (ψ(t)))n |
n
Z t
hX
= |
e−c1 (t−s)
a1j (s)(gj (φj (s − τ1j (s))) − gj (ψj (s − τ1j (s))))
−∞
+
n X
n
X
j=1
b1jl (s) gj (φj (s − σ1jl (s)))gl (φl (s − ν1jl (s)))
j=1 l=1
i
− gj (ψj (s − σ1jl (s)))gl (ψl (s − ν1jl (s))) ds|, . . . ,
Z t
n
hX
|
e−cn (t−s)
anj (s)(gj (φj (s − τnj (s))) − gj (ψj (s − τnj (s))))
−∞
+
j=1
n
n X
X
bnjl (s) gj (φj (s − σnjl (s)))gl (φl (s − νnjl (s)))
j=1 l=1
i T
− gj (ψj (s − σnjl (s)))gl (ψl (s − νnjl (s))) ds|
n
Z t
hX
≤
e−c1 (t−s)
a1j Lgj sup |φj (t) − ψj (t)|
−∞
+
n X
n
X
j=1
t∈R
b1jl (|gj (φj (s − σ1jl (s)))gl (φl (s − ν1jl (s)))
j=1 l=1
− gj (ψj (s − σ1jl (s)))gl (φl (s − ν1jl (s)))|
+ |gj (ψj (s − σ1jl (s)))gl (φl (s − ν1jl (s)))
i
− gj (ψj (s − σ1jl (s)))gl (ψl (s − ν1jl (s)))|) ds,
Z t
n
hX
anj Lgj sup |φj (t) − ψj (t)|
...,
e−cn (t−s)
−∞
+
n X
n
X
t∈R
j=1
bnjl (|gj (φj (s − σnjl (s)))gl (φl (s − νnjl (s)))
j=1 l=1
− gj (ψj (s − σnjl (s)))gl (φl (s − νnjl (s)))|
+ |gj (ψj (s − σnjl (s)))gl (φl (s − νnjl (s)))
i T
− gj (ψj (s − σnjl (s)))gl (ψl (s − νnjl (s)))|) ds
n
Z t
X
≤
e−c1 (t−s) [
a1j Lgj sup |φj (t) − ψj (t)|
−∞
+
n X
n
X
j=1 l=1
j=1
t∈R
b1jl Lgj Lgl (sup |φl (t)| + sup |ψj (t)|)kφ − ψkB ]ds, . . . ,
t∈R
t∈R
6
F. ZHANG, Y. LI
Z
t
−∞
+
EJDE-2007/99
n
X
e−cn (t−s) [
anj Lgj sup |φj (t) − ψj (t)|
t∈R
j=1
n X
n
X
bnjl Lgj Lgl (sup |φl (t)| + sup |ψj (t)|)kφ − ψkB ]ds
j=1 l=1
n
X
≤ c−1
(
a1j Lgj +
1
j=1
n
X
−1
. . . , cn (
anj Lgj
j=1
t∈R
T
t∈R
n
n
2L X X
b1jl Lgj Lgl )kφ − ψkB ,
1 − δ j=1
l=1
n
n
T
2L X X
+
bnjl Lgj Lgl )kφ − ψkB ,
1 − δ j=1
l=1
which implies
kT (φ) − T (ψ)kB ≤ max
1≤i≤n
n
X
{c−1
i (
aij Lgj
j=1
n
n
2L X X
+
bijl Lgj Lgl )}kφ − ψkB
1 − δ j=1
l=1
= qkφ − ψkB .
Pn Pn
Pn
g
g g
2L
Note that q = max1≤i≤n {c−1
i (
j=1 aij Lj + 1−δ
j=1
l=1 bijl Lj Ll )} < 1; it is
clear that the mapping T is a contraction. Therefore the mapping T possesses a
unique fixed point Z ∗ ∈ B ∗ , T Z ∗ = Z ∗ . By (2.1), Z ∗ satisfies (1.1). So Z ∗ is an
almost periodic solution of (1.1) in B ∗ . The proof is complete.
3. Stability of the almost periodic solution
In this section, we establish some results for the stability of the almost periodic
solution of (1.1).
Theorem 3.1. Let
n
n
n X
X
X
L
g
a
L
bijl Lgj Lgl (1 + 2
)]} < 1.
max {c−1
[
+
ij
i
j
1≤i≤n
1−δ
j=1
j=1
l=1
Suppose that all the conditions of Theorem 2.1 are satisfied. Then (1.1) has exactly
one almost periodic solution Z ∗ (t) = (x∗1 (t), x∗2 (t), . . . , x∗n (t))T ∈ B ∗ . Moreover,
Z ∗ (t) is locally exponentially stable, the domain of the attraction of Z ∗ (t) is the set
G1 (Z ∗ ) = {ϕ|ϕ = (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))T ∈ C([−τ, 0]; Rn ), kϕ − ϕ∗ k1 < 1},
where ϕ∗ = {ϕ∗j (t)}, ϕ∗j (t) = x∗j (t), j = 1, 2, . . . , n, t ∈ [−τ, 0], and kϕ − ϕ∗ k1 =
sup−τ ≤s≤0 max1≤j≤n |ϕj (s) − ϕ∗j (s)|. Namely, there exist constants λ > 0 and
M > 1 such that for every solution Z(t) = {xj (t)} to system (1.1) with initial
value ϕ = {ϕj (t)} ∈ G1 (Z ∗ ), we have
|xi (t) − x∗i (t)| ≤ M kϕ − ϕ∗ k1 e−λt ,
∀t > 0, i = 1, 2, . . . , n.
Proof. From Theorem 2.1, system (1.1) has exactly one almost periodic solution
Z ∗ (t) = {x∗j (t)} ∈ B ∗ . Let Z(t) = {xj (t)} be an arbitrary solution of system (1.1)
with initial value ϕ = {ϕj (t)} ∈ G1 (Z ∗ ), let y(t) = {yj (t)} = {xj (t) − x∗j (t)} =
EJDE-2007/99
ALMOST PERIODIC SOLUTIONS
7
Z(t) − Z ∗ (t). Then
yi0 (t) = −ci yi (t) +
n
X
aij (t)(gj (xj (t − τij (t))) − gj (x∗j (t − τij (t))))
j=1
+
n X
n
X
bijl (t)(gj (xj (t − σijl (t)))gl (xl (t − νijl (t)))
(3.1)
j=1 l=1
− gj (x∗j (t − σijl (t)))gl (x∗l (t − νijl (t)))), i = 1, 2, . . . , n.
Pn Pn
Pn
g
g g
L
Since max1≤i≤n {c−1
i [
j=1
l=1 bijl Lj Ll (1 + 2 1−δ )]} < 1, we can
j=1 aij Lj +
easily get
n
n
n X
X
X
L
−ci +
aij Lgj +
bijl Lgj Lgl (1 + 2
) < 0, i = 1, 2, . . . , n,
(3.2)
1
−
δ
j=1
j=1
l=1
which implies that we can choose a positive constant λ such that
n
n X
n
X
X
L
aij Lgj eλτ +
bijl Lgj Lgl (e2λτ + 2eλτ
(λ − ci ) +
) < 0,
1
−
δ
j=1
j=1
(3.3)
l=1
for i = 1, 2, . . . , n. We consider the Lyapunov functional
Vi (t) = |yi (t)|eλt ,
i = 1, 2, . . . , n.
(3.4)
Calculating the upper right derivative of Vi (t) along the solution y(t) = {yj (t)} of
system (3.1) with the initial value ϕ̄ = ϕ − ϕ∗ , we have from (2.2), (2.3), (3.1) and
(H1) that
D+ (Vi (t))
≤ −ci |yi (t)|eλt +
n
hX
|aij (t)||gj (xj (t − τij (t))) − gj (x∗j (t − τij (t)))|
j=1
+
n X
n
X
|bijl (t)||gj (xj (t − σijl (t)))gl (xl (t − νijl (t)))
j=1 l=1
i
− gj (x∗j (t − σijl (t)))gl (x∗l (t − νijl (t)))| eλt + λ|yi (t)|eλt
≤ (λ − ci )|yi (t)|eλt +
n
X
|aij (t)|Lgj |yj (t − τij (t))|
j=1
+
n X
n
hX
|bijl (t)|(|gj (xj (t − σijl (t)))gl (xl (t − νijl (t)))
j=1 l=1
gj (x∗j (t −
σijl (t)))gl (xl (t − νijl (t)))| + |gj (x∗j (t − σijl (t)))gl (xl (t − νijl (t)))
i
− gj (x∗j (t − σijl (t)))gl (x∗l (t − νijl (t)))|) eλt
−
λt
≤ (λ − ci )|yi (t)|e
n
X
+[
aij Lgj |yj (t − τij (t))|
j=1
+
n X
n
X
bijl Lgj Lgl (|yj (t − σijl (t))||yl (t − νijl (t))
j=1 l=1
+ x∗l (t − νijl (t))| + |x∗j (t − σijl (t))||yl (t − νijl (t))|)]eλt
8
F. ZHANG, Y. LI
n
hX
≤ (λ − ci )|yi (t)|eλt +
EJDE-2007/99
aij Lgj |yj (t − τij (t))|
j=1
+
n X
n
X
bijl Lgj Lgl (|yj (t − σijl (t))||yl (t − νijl (t))|
j=1 l=1
i
L
L
+
|yl (t − νijl (t))|) eλt ,
1−δ
1−δ
where i = 1, 2, . . . , n. Set
+ |yj (t − σijl (t))|
kϕ − ϕ∗ k1 =
sup
(3.5)
max |ϕj (s) − ϕ∗j (s)| > 0.
−τ ≤s≤0 1≤j≤n
Since kϕ − ϕ∗ k1 < 1, we can choose a positive constant M > 1 such that
M kϕ − ϕ∗ k1 < 1,
(M kϕ − ϕ∗ k1 )2 < M kϕ − ϕ∗ k1 .
(3.6)
It follows from (3.4) that
Vi (t) = |yi (t)|eλt < M kϕ − ϕ∗ k1 ,
for all t ∈ [−τ, 0],
i = 1, 2, . . . , n.
Now we claim that
Vi (t) = |yi (t)|eλt < M kϕ − ϕ∗ k1 ,
for all t > 0, i = 1, 2, . . . , n.
Contrarily, there must exist an i ∈ {1, 2, . . . , n} and ti > 0 such that
Vi (ti ) = M kϕ − ϕ∗ k1
and Vj (t) < M kϕ − ϕ∗ k1 , ∀t ∈ [−τ, ti ),
for j = 1, 2, . . . , n. It follows that
Vi (ti ) − M kϕ − ϕ∗ k1 = 0 and Vj (t) − M kϕ − ϕ∗ k1 < 0, ∀t ∈ [−τ, ti ),
for j = 1, 2, . . . , n. This together with (3.5), yields
0 ≤ D+ (Vi (ti ) − M kϕ − ϕ∗ k1 )
= D+ (Vi (ti ))
≤ (λ − ci )|yi (ti )|eλti +
n
hX
aij Lgj |yj (ti − τij (ti ))|
j=1
+
n X
n
X
bijl Lgj Lgl (|yj (ti − σijl (ti ))||yl (ti − νijl (ti ))|
j=1 l=1
i
L
L
+
|yl (ti − νijl (ti ))|) eλti
1−δ
1−δ
n
X
+
aij Lgj |yj (ti − τij (ti ))|eλ(ti −τij (ti )) eλτij (ti )
+ |yj (ti − σijl (ti ))|
= (λ − ci )|yi (ti )|eλti
j=1
+
n X
n
X
bijl Lgj Lgl |yj (ti − σijl (ti ))|eλ(ti −σijl (ti )) |yl (ti − νijl (ti ))|
j=1 l=1
× eλ(ti −νijl (ti )) eλσijl (ti ) eλνijl (ti ) e−λti
+ |yj (ti − σijl (ti ))|eλ(ti −σijl (ti )) eλσijl (ti )
+
L
1−δ
L
|yl (ti − νijl (ti ))|eλ(ti −νijl (ti )) eλνijl (ti )
1−δ
(3.7)
EJDE-2007/99
ALMOST PERIODIC SOLUTIONS
≤ (λ − ci )M kϕ − ϕ∗ k1 +
n
X
9
aij Lgj eλτ M kϕ − ϕ∗ k1
j=1
+
n X
n
X
bijl Lgj Lgl ((M kϕ − ϕ∗ k1 )2 e2λτ e−λti
j=1 l=1
L
L
+
M kϕ − ϕ∗ k1 eλτ )
1−δ
1−δ
n
n X
n
h
X
X
aij Lgj eλτ +
bijl Lgj Lgl (e2λτ + 2eλτ
≤ (λ − ci ) +
+ M kϕ − ϕ∗ k1 eλτ
j=1
j=1 l=1
L i
) M kϕ − ϕ∗ k1 .
1−δ
Thus, we have
0 ≤ (λ − ci ) +
n
X
aij Lgj eλτ +
n X
n
X
bijl Lgj Lgl (e2λτ + 2eλτ
j=1 l=1
j=1
L
)
1−δ
which contradicts (3.3). Hence, (3.7) holds. It follows that
|yi (t)| < M kϕ − ϕ∗ k1 e−λt ,
t > 0, i = 1, 2, . . . , n.
This completes the proof.
4. An Example
In this section, we give an example to demonstrate the results obtained in previous sections.
Consider the following HHNNs with delays:
1
1
(sin t)g1 (x1 (t − sin2 t)) + (cos 3t)g2 (x2 (t − 7 sin2 t))
16
16
√
1
3
+ (cos t)g1 (x1 (t − 5 sin2 t))g2 (x2 (t − 2 sin2 t)) + sin( 2t),
8
4
1
1
x02 (t) = −x2 (t) + (sin 2t)g1 (x1 (t − cos2 t)) + (cos 8t)g2 (x2 (t − 5 sin2 t))
16
16
√
1
3
+ (cos 4t)g1 (x1 (t − sin2 t))g2 (x2 (t − 4 sin2 t)) + cos( 2t),
8
4
(4.1)
1
where g1 (x) = g2 (x) = |x|. Observe that c1 = c2 = Lg1 = Lg2 = 1, aij = 16
,
1
i, j = 1, 2, b112 = b212 = 8 , bijl = 0, i, j, l = 1, 2, ijl 6= 112, ijl 6= 212. Then
x01 (t) = −x1 (t) +
L=
3
,
4
2
2 X
2
X
X
1
g
δ = max {c−1
[
a
L
+
bijl Lgj Lgl ]} = < 1,
ij
i
j
1≤i≤2
4
j=1
j=1
l=1
2
X
q = max {c−1
i (
1≤i≤2
j=1
2
X
max {c−1
i [
1≤i≤2
aij Lgj +
j=1
aij Lgj +
2L
1−δ
2 X
2
X
j=1 l=1
2 X
2
X
bijl Lgj Lgl )} =
j=1 l=1
bijl Lgj Lgl (1 + 2
3
< 1,
8
L
1
)]} = < 1.
1−δ
2
Therefore, By Theorem 3.1, system (4.1) has a unique almost periodic solution
Z ∗ (t) in the region kϕ − ϕ0 kB ≤ 0.25. Moreover, Z ∗ (t) is locally exponentially
stable, the domain of the attraction of Z ∗ (t) is the set G1 (Z ∗ ).
10
F. ZHANG, Y. LI
EJDE-2007/99
We remark that (4.1) is a very simple form of HHNNs. Since g1 (x) = g2 (x) = |x|,
one can observe that the condition (T0) is not satisfied. Therefore, all the results in
[1, 2, 5, 6, 7, 8, 9] and the references cited therein can not be applicable to system
(4.1). This implies that the results of this paper are essentially new.
References
[1] Jinde Cao, Jinling Liang and James Lam; Exponential stability of high-order bidirectional
associative memory neural networks with time delays, Physica D: Nonlinear Phenomena, 199
(3-4) (2004) 425-436.
[2] A. Dembo, O. Farotimi, T. Kailath; High-order Absolutely Stable Neural Network,IEEE
Trans on Circuits and System, 8(1) (1991), 57-65 81, (1984) 3088-3092.
[3] A. M. Fink; Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377,
Springer, Berlin, 1974, pp. 80-112.
[4] C. Y. He; Almost periodic differential equation, Higher Education Publishing House, Beijing,
1992 pp. 90-100. [In Chinese]
[5] Haijun Jiang and Zhidong Teng; Boundedness and global stability for nonautonomous recurrent neural networks with distributed delays, Chaos, Solitons & Fractals, 30(1) (2006)
83-93.
[6] Chuandong Li, Xiaofeng Liao, Rong Zhang; Delay-dependent exponential stability analysis of
bi-directional associative memory neural networks with time delay: an LMI approach, Chaos,
Solitons & Fractals, 24(4) (2005) 1119-1134.
[7] P. G. Wang, H. R. Lian; Global exponential stability and periodic solutions of the high-order
Hopfield type neural networks with time-varying coefficient. Zeischrift fur Analysis und ihre
Anwendungen, 24(2) (2005) 419-429.
[8] Bingji Xu, Xinzhi Liu, Xiaoxin Liao; Global asymptotic stability of high-order Hopfield type
neural networks with time delays. Computers & Mathematics with Applications, 45 ( 2003)
1729-1737
[9] Haifeng Yang, Tianguang Chu and Cishen Zhang; Exponential stability of neural networks
with variable delays via LMI approach, Chaos, Solitons & Fractals, 30(1) (2006) 133-139.
Fuxing Zhang
Department of Mathematics, Shaoyang University, Shaoyang, Hunan, 422000, China
E-mail address: fuxingzhang2006@163.com
Ya Li
Editorial Department of Journal of Hunan University, Changsha 410082, China
E-mail address: yali88888@sohu.com