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f
(
0
)
=
0
A
∈
R
n
×
n
B
∈
R
n
×
n
and
represent the connection weight matrix and
n
×
n
n
×
n
P
∈
R
Q
∈
R
the delayed connection weight matrix, respectively;
and
(
tw
is a one-dimensional Brownian motion
defined on a complete probability space with a natural filtration;
)
are the diffusion coefficient matrix;
τ =
max{
i
τ
},
n
φ
(
t
)
∈
C
([
−
τ
,
],
R
)
is measurable and bounded random variable.
2 Preliminaries
A
) means that matrix
A
is symmetric posi-
tive definite (respectively, positive semi-definite).
A
>
0
≥
0
The notation
(respectively,
A
and
T
A
denote the transpose
2
,
n
+
+
and inverse of the matrix
A
. Let
C
(
R
×
R
;
R
)
denote the family of
R
n
which are continuously twice
differentiable in
x
and once differentiable in
t
. Let
+
V
(
x
,
t
)
×
R
all nonnegative functions
on
F
(
x
(
t
),
y
(
t
))
=
−
Cx
(
t
)
+
Af
(
x
(
t
))
+
Bf
(
y
(
t
))
G
(
x
(
t
),
y
(
t
))
=
Px
(
t
)
+
Qy
(
t
)
,
. Define an operator
n
n
+
LV
∈
C
(
R
×
R
×
R
;
R
)
by
1
T
LV
(
x
,
y
,
t
)
=
V
(
x
,
t
)
+
V
(
x
,
t
)
F
(
x
,
y
)
+
trace
{
G
(
x
,
y
)
V
(
x
,
t
)
G
(
x
,
y
)}.
t
x
xx
2
L
=
diag
(
1
l
,
"
,
l
)
Throughout this paper, we always assume that there exists
n
such that
f
(
u
)
−
f
(
v
)
i
i
0
≤
≤
l
,
u
,
v
∈
R
.
(2)
i
u
−
v
x
(
t
)
From (2), it follows that system (1) has a unique solution
[10]. Moreover,
F
(
x
(
t
),
y
(
t
))
G
(
x
(
t
),
y
(
t
))
(
x
,
y
)
and
are locally bounded in
.
Lemma 1. (
Lasalle-type theorem, [11]) Assume that
system (1) has a unique solution
)
x
(
t
F
(
x
(
t
),
y
(
t
))
G
(
x
(
t
),
y
(
t
))
(
x
,
y
)
. Moreover,
and
are locally bounded in
.
2
,
n
+
+
V
∈
C
(
R
×
R
;
R
)
W
≥
0
W
≥
0
Assume that
there are
and
such that
1
2
lim
inf
V
(
x
,
t
)
=
∞
,
|
x
|
→
∞
t
≥
0
LV
(
x
,
y
,
t
)
≤
−
W
(
x
)
+
W
(
y
),
W
(
x
)
=
W
(
x
)
−
W
(
x
)
≥
0
1
2
1
2
n
KerW
=
{
x
∈
R
:
W
(
x
)
=
0
Then
is nonempty and
lim
inf
x
−
y
=
0
a
.
s
.
(3)
t
→
∞
y
∈
KerW
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