Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 136048, 19 pages
doi:10.1155/2012/136048
Research Article
Dynamics of Fuzzy BAM Neural Networks with
Distributed Delays and Diffusion
Qianhong Zhang,1 Lihui Yang,2 and Jingzhong Liu3
1
Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics,
Guizhou College of Finance and Economics, Guiyang, Guizhou 550004, China
2
Department of Mathematics, Hunan City University, Yiyang, Hunan 413000, China
3
Department of Computer and Information Science, Hunan Institute of Technology, Hengyang,
Hunan 421002, China
Correspondence should be addressed to Qianhong Zhang, zqianhong68@163.com
Received 23 May 2011; Accepted 17 November 2011
Academic Editor: Carlos J. S. Alves
Copyright q 2012 Qianhong Zhang 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.
Constructing a new Lyapunov functional and employing inequality technique, the existence,
uniqueness, and global exponential stability of the periodic oscillatory solution are investigated
for a class of fuzzy bidirectional associative memory BAM neural networks with distributed
delays and diffusion. We obtained some sufficient conditions ensuring the existence, uniqueness,
and global exponential stability of the periodic solution. The results remove the usual assumption
that the activation functions are differentiable. An example is provided to show the effectiveness
of our results.
1. Introduction
The bidirectional associative memory BAM neural network was first introduced by Kosko
1, 2. These models generalize the single-layer autoassociative Hebbian correlator to a
two layer pattern-matched heteroassociative circuits. BAM neural network is composed of
neurons arranged in two-layers, the X-layer and the Y-layer. The BAM neural network has
been used in many fields such as image processing, pattern recognition, and automatic
control 3. Recently, the stability and the periodic oscillatory solutions of BAM neural
networks have been studied see, e.g., 1–25. In 2002, Cao and Wang 7 derived some
sufficient conditions for the global exponential stability and existence of periodic oscillatory
solution of BAM neural networks with delays. Some authors 16, 19, 25 studied the BAM
neural networks with distributed delays, which are more appropriate when neural networks
have a multitude of parallel pathways with a variety of axon sizes and lengths.
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Journal of Applied Mathematics
However, strictly speaking, diffusion effects cannot be avoided in the neural networks
when electrons are moving in asymmetric electromagnetic fields. So we must consider that
activations vary in space as well as in time. Song et al. 25 have considered the stability
of BAM neural networks with diffusion effects, which are expressed by partial differential
equations. In this paper, we would like to integrate fuzzy operations into BAM neural
networks. Speaking of fuzzy operations, T. Yang and L. B. Yang 26 first introduced fuzzy
cellular neural networks FCNNs combining those operations with cellular neural networks.
So far, researchers have founded that FCNNs are useful in image processing, and some
results have been reported on stability and periodicity of FCNNs 26–32. However, to the
best of our knowledge, few authors consider the global exponential stability and existence
of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion
terms. In 33, Li studied the global exponential stabilities of both the equilibrium point and
the periodic solution for a class of BAM fuzzy neural networks with delays and reactiondiffusion terms. Motivated by the above discussion, in this paper, by constructing a suitable
Lyapunov functional and employing inequality technique, we will derive some sufficient
conditions of the global exponential stability and existence of periodic solutions for fuzzy
BAM neural networks with distributed delays and diffusion terms.
2. System Description and Preliminaries
In this paper, we consider the globally exponentially stable and periodic fuzzy BAM neural
networks with distributed delays and diffusion terms described by partial differential
equations with delays:
t
l
m
∂
∂ui
∂ui t, x Kji t − sfj vj s, x ds
Dik
− ai ui t, x cji
∂t
∂xk
∂xk
−∞
j1
k1
t
t
m
m
Kji t − sfj vj s, x ds βji
Kji t − sfj vj s, x ds
αji
j1
m
−∞
Tji ωj j1
l
∂vj t, x ∂
∂t
∂x
k
k1
Hji ωj Ii t,
j1
−∞
j1
m
∗
Djk
∂vj
∂xk
t
n
− bj vj t, x dij
Nij t − sgi ui s, xds
2.1
−∞
i1
t
t
n
n
Nij t − sgi ui s, xds qij
Nij t − sgi ui s, xds
pij
i1
−∞
n
n
Sij ωi Lij ωi Jj t,
i1
i1
−∞
i1
where n and m correspond to the number of neurons in X-layer and Y -layer, respectively.
For i 1, 2, . . . , n, j 1, 2, . . . , m, x x1 , x2 , . . . , xl T ∈ Ω ⊂ Rl , Ω is a bounded compact
set with smooth boundary ∂Ω and mess Ω > 0 in space Rl . u u1 , u2 , . . . , un T ∈ Rn , v v1 , v2 , . . . , vm T ∈ Rm . ui t, x and vj t, x are the state of the ith neuron and the jth neurons
at time t and in space x, respectively. ai > 0, bj > 0, and they denote the rate with which
the ith neuron and jth neuron will reset its potential to the resting state in isolation when
Journal of Applied Mathematics
3
disconnected from the network and external inputs; cji and dij are constants, denoting the
connection weights. αji , βji , Tji , and Hji are elements of fuzzy feedback MIN template and
fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward
MAX template in X-layer, respectively; pij , qij , Sij , and Lij are elements of fuzzy feedback
MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and
fuzzy feed-forward MAX template in Y -layer, respectively; and denote the fuzzy AND
and fuzzy OR operations, respectively; ωj , ωi denote external input of the ith neurons in Xlayer and external input of the jth neurons in Y -layer, respectively; Ii t and Jj t denote
the external inputs on the ith neurons in X-layer and the jth neurons in Y -layer at time
t, respectively; I i : R → R and Jj : R → R are continuously periodic functions with
periodic ω., that is, Ii t ω Ii t, Jj t ω Jj t. Kji · and Nij · are delay kernels
functions. gi · and fj · are signal transmission functions of ith neurons and jth neurons at
∗
t, x, v ≥ 0 correspond to the
time t and in space x. Smooth functions Dik t, x, u ≥ 0 and Djk
transmission diffusion operators along the ith neurons and the jth neurons, respectively.
The boundary conditions and the initial conditions are given by
∂ui
:
∂n
∂vj
:
∂n
∂ui ∂ui
∂ui
,
,...,
∂x1 ∂x2
∂xl
∂vj
∂vj ∂vj
,
,...,
∂x1 ∂x2
∂xl
T
0,
i 1, 2, . . . , n,
T
2.2
0,
j 1, 2, . . . , m,
ui s, x φui s, x,
s ∈ −∞, 0, i 1, 2, . . . , n,
vj s, x ϕvj s, x,
s ∈ −∞, 0, i 1, 2, . . . , m,
2.3
where φui s, x and ϕvj s, x i 1, 2, . . . , n; j 1, 2, . . . , m. are continuous bounded functions
defined on −∞, 0 × Ω, respectively.
Throughout the paper, we give the following assumptions.
A1 The signal transmission functions fj ·, gi · i 1, 2, . . . , n, j 1, 2, . . . , m are
Lipschtiz continuous on R with Lipschtiz constants μj and νi , namely, for any
x, y ∈ R,
fj x − fj y ≤ μj x − y,
gi x − gi y ≤ νi x − y,
fj 0 gi 0 0.
2.4
A2 The delay kernels Kji , Nij : 0, ∞ → 0, ∞i 1, 2, . . . , n; j 1, 2, . . . , m are
nonnegative continuous functions that satisfy the following conditions:
∞
∞
i 0 Kji sds 0 Nij sds 1,
∞
∞
ii 0 sKji sds < ∞, 0 sNij sds < ∞,
iii there exists a positive constant η such that
∞
0
se Kji sds < ∞,
ηs
∞
0
seηs Nij sds < ∞.
2.5
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To be convenient, we introduce some notations. Let ui ui t, x, vj vj t, x. Let
∗
u∗ u∗1 , u∗2 , . . . , u∗n T and v∗ v1∗ , v2∗ , . . . , vm
be the equilibrium of system 2.1. We denote
∗
ut, x − u n ui − u∗ 2 dx
i
Ω i1
φu s, x − u∗ sup
1/2
,
1/2
n 2
∗
φui s, x − u dx
,
−∞≤s≤0
Ω i1
i
⎤1/2
⎡
m 2
vt, x − v∗ ⎣
vj − vj∗ dx⎦ ,
2.6
Ω j1
⎡
⎤1/2
m 2
φv s, x − v∗ sup ⎣
ϕvj s, x − vj∗ dx⎦ .
−∞≤s≤0
Ω j1
∗ T
of the delay
Definition 2.1. The equilibrium u∗ u∗1 , u∗2 , . . . , u∗n T , v∗ v1∗ , v2∗ , . . . , vm
fuzzy BAM neural networks 2.1 is said to be globally exponentially stable, if there exist
positive constants M ≥ 1, λ > 0 such that
ut, x − u∗ vt, x − v∗ ≤ M φu s, x − u∗ ϕv s, x − v∗ e−λt
2.7
for every solution ut, x, vt, x of the delay fuzzy BAM neural networks 2.1 with the initial
conditions 2.2 and 2.3 for all t > 0.
Definition 2.2. If ft : R → R is a continuous function, then the upper right derivative of ft
is defined as
D ft lim sup
h→0
1
ft h − ft .
h
2.8
Lemma 2.3 see 26. Suppose x and y are two states of system 2.1, then one has
n
n n
αij gj x − αij gj y ≤
αij gj x − gj y ,
j1
j1
j1
n
n n
βij gj x − βij gj y ≤
βij gj x − gj y .
j1
j1
j1
2.9
The remainder of this paper is organized as follows. In Section 3, we will study
global exponential stability of fuzzy BAM neural networks 2.1. In Section 4, we present
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5
the existence of periodic solution for fuzzy BAM neural networks 2.1. In Section 5, an
example will be given to illustrate effectiveness of our results obtained. We will give a general
conclusion in Section 6.
3. Global Exponential Stability
In this section, we will discuss the global exponential stability of fuzzy BAM neural networks
2.1 by constructing suitable functional.
Theorem 3.1. Under assumptions (A1) and (A2), if there exist δi > 0, δnj > 0 such that
⎡
⎤
m m
cji αji βji μj ⎦ νi δnj dij pij qij < 0,
δi ⎣−2ai j1
j1
δnj
n n
dij pij qij νi μj δi cji αji βji < 0,
−2bj i1
3.1
i1
where i 1, 2, . . . , n, j 1, 2, . . . , m then the equilibrium u∗ , v∗ of system 2.1 is the globally
exponentially stable.
Proof. By using 3.1, we can choose a small number λ > 0 such that
⎡
⎤
m m
∞
cji αji βji μj ⎦ 2νi δnj dij pij qij δi ⎣λ − 4ai 2
Nij seλs ds < 0,
j1
n
δnj λ − 4bj 2
dij pij qij νi 2μj
i1
0
j1
n
∞
i1
0
δi cji αji βji Kji seλs ds < 0.
3.2
It is well known that the bounded functions always guarantee the existence of an equilibrium
point for system 2.1. The uniqueness of the equilibrium for system 2.1 will follow from
the global exponential stability to be established below.
Suppose u1 t, x, . . . , un t, x, v1 t, x, . . . , vm t, xT is any solution of system 2.1.
Rewrite 2.1 as follows
l
∂ ui − u∗i
∂ ui − u∗i
∂
Dik
− ai ui − u∗i
∂t
∂x
∂x
k
k
k1
t
m
cji
Kji t − s fj vj − fj vj∗ ds
j1
−∞
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Journal of Applied Mathematics
m
t
αji
−∞
j1
m
m
Kji t − sfj vj ds − αji
t
βji
−∞
j1
j1
Kji t − sfj vj ds −
m
t
−∞
t
βji
−∞
j1
Kji t − sfj vj∗ ds
Kji t − sfj vj∗ ds,
3.3
∂ vj − vj∗
∂t
⎛
⎞
∗
l
∂ ⎜ ∗ ∂ vj − vj ⎟
⎝Djk
⎠ − bj vj − vj∗
∂xk
∂xk
k1
t
n
dij
Nij t − s gi ui − gi u∗i ds
−∞
i1
t
n
pij
−∞
i1
m
t
qij
−∞
j1
Nij t − sgi ui ds −
n
pij
i1
Nij t − sgi ui ds −
m
t
−∞
t
qij
j1
−∞
3.4
Nij t − sgi u∗i ds
Nij t − sgi u∗i ds.
Multiply both sides of 3.3 by ui − u∗i and integrate, we have
1 d
2 dt
Ω
ui −
2
u∗i dx
Ω
ui −
j1
⎡
⎤
t
m
Kji t − sfj vj ds⎦dx
ui − u∗i ⎣ αji
j1
−∞
⎡
⎤
t
m
Kji t − sfj vj∗ ds⎦dx
ui − u∗i ⎣ αji
Ω
−∞
j1
−∞
⎡
⎤
t
m
Kji t − sfj vj ds⎦dx
ui − u∗i ⎣ βji
Ω
−
⎡
⎤
t
m
Kji t − s fj vj − fj vj∗ ds⎦dx
ui − u∗i ⎣ cji
Ω
−
l
∂ ui − u∗i
2
∂
Dik
dx −
ai ui − u∗i dx
∂xk
∂xk
Ω
k1
Ω
u∗i
j1
−∞
⎤
⎡
t
m
Kji t − sfj vj∗ ds⎦dx.
ui − u∗i ⎣ βji
Ω
j1
−∞
3.5
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7
Applying the boundary condition 2.2 and the Gauss formula, we get
l
∂ ui − u∗i
∂
Dik
dx
ui −
∂xk
∂xk
k1
⎡
l ⎤
∗
l
∂
u
−
u
i
i
⎦dx
ui − u∗i ⎣ ∇ Dik
∂xk
Ω
k1
k1
l
l
∗
∂ u i − ui
∂ ui − u∗i
∗
Dik
∇·
ui − ui Dik
dx −
∇ ui − u∗i dx
∂xk
∂xk
Ω
Ω
k1
k1
2
l
∗
∗
l
∂ u i − ui
∂ u i − ui
ds −
Dik
dx
ui − u∗i Dik
∂xk
∂xk
∂Ω
k1 Ω
k1
2
l ∂ ui − u∗i
−
Dik
dx ≤ 0
∂xk
k1 Ω
Ω
u∗i
3.6
in which ∇ ∂/∂x1 , ∂/∂x2 , . . . , ∂/∂xl T is the gradient operator and
l
∂ ui − u∗i
Dik
∂xk
T
∂ ui − u∗i
∂ ui − u∗i
∂ ui − u∗i
Di1
, Di1
, . . . , Di1
.
∂xi
∂x2
∂xl
k1
3.7
From 3.5, 3.6, hypothesis A1, Lemma 2.3, and the Holder inequality, we have
2
dui − u∗i dt
≤ −2ai
Ω
2
ui − u∗i dx 2
Ω
ui − u∗i
⎤
⎡
t
m
Kji t − s fj vj − fj vj∗ ds⎦dx
× ⎣ cji
j1
Ω
Ω
−∞
⎤
⎡
t
t
m
m
∗ ⎣
∗
2 ui − ui
αji
Kji t − sfj vj ds − αji
Kji t − sfj vj ds⎦dx
j1
⎡
m
2 ui − u∗i ⎣ βji
j1
−∞
t
−∞
j1
m
Kji t − sfj vj ds − βji
j1
−∞
t
−∞
⎤
Kji t − sfj vj∗ ds⎦dx
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Journal of Applied Mathematics
⎡
⎤
m
t
∗ 2
∗ ⎣
∗
≤ −2ai
μj cji Kji t − svj − vj ds⎦dx
ui − ui dx 2
u i − ui
Ω
Ω
−∞
j1
⎡
⎤
t
m
2
ui − u∗i ⎣ μj αji βji Kji t − svj − vj∗ ds⎦dx
Ω
−∞
j1
m t
2
cji αji βji ≤ −2ai ui − u∗i 2
Kji t − sui − u∗i μj vj − vj∗ ds.
−∞
j1
3.8
Multiply both sides of 3.4 by vi − vi∗ , similarly, we get
2
dvj − vj∗ dt
≤ −2bj vj −
vj∗
2
n t
dij pij qij
2
Nij t − svj − vj∗ νi ui − u∗i ds.
−∞
i1
3.9
We construct a Lyapunov functional
⎡
∞
n
m 2
cji αji βji μj
V t δi ⎣ui − u∗i eλt 2
Kji s
Ω i1
0
j1
×
t
t−s
⎤
2
eλζs vj ζ, x − vj∗ dζ ds ⎦dx
∞
n m
2
dij pij qij νi
δnj vj − vj∗ eλt 2
Nij s
Ω j1
0
i1
×
t
e
t−s
λζs ui ζ, x −
3.10
2
u∗i dζ ds
dx.
Calculating the upper right derivative D V t of V t along the solution of 3.3 and 3.4,
from 3.8 and 3.9, we get
n
∂ui − u∗i λt
2
∗
e λeλt ui − u∗i D V t δi 2 ui − ui
∂t
Ω i1
2eλt
m cji αji βji μj
j1
×
∞
0
m 2
cji αji βji μj
Kji seλs vj t, x − vj∗ ds − 2eλt
j1
Journal of Applied Mathematics
×
∞
0
9
2
∗
Kji s × vj t − s, x − vj ds dx
m
∂vj − vj∗ 2
eλt λeλt vj − vj∗ δnj ⎣2vj − vj∗ ∂t
Ω j1
⎡
∞
n 2
dij pij qij νi
2e
Nij seλs ui t, x − u∗i ds
λt
0
i1
⎤
∞
n 2
∗
dij pij qij νi
−2e
Nij sui t − s, x − ui ds⎦dx
λt
0
i1
⎛
⎡
m n
2
λt
cji αji βji δi ⎣2e ⎝ − 2ai ui − u∗i 2
≤
Ω i1
j1
×
t
−∞
⎞
2
Kji t − sui − u∗i × μj vj − vj∗ ds⎠ λeλt ui − u∗i ∞
m 2
cji αji βji μj
2e
Kji seλs vj t, x − vj∗ ds
λt
0
j1
− 2eλt
∞
m cji αji βji μj
Kji s
0
j1
⎤
2
×vj t − s, x − vj∗ ds⎦dx
m
2
λt
− 2bj vj − vj∗
δnj 2e
Ω j1
n dij pij qij 2
i1
×
t
−∞
∗ ∗
Nij t − svj − vj νi ui − ui ds
n 2
dij pij qij νi
λeλt vj − vj∗ 2eλt
×
∞
i1
2
Nij seλs ui t, x − u∗ ds
i
0
− 2eλt
×
∞
0
n dij pij qij νi
i1
2
∗
Nij sui t − s, x − ui ds dx
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Journal of Applied Mathematics
⎧⎡
m n
⎨
2
2
cji αji βji ≤
eλt δi ⎣ − 4ai ui − u∗i λui − u∗i 4
⎩
Ω
i1
j1
×
t
−∞
∗
∗
Kji t − sui − ui μj vj − vj ds
∞
m 2
cji αji βji μj
Kji seλs vj t, x − vj∗ ds
2
0
j1
⎫
∞
m 2 ⎬
cji αji βji μj
Kji svj t − s, x − vj∗ ds dx
−2
⎭
0
j1
m
e
δnj
%
λt
Ω
n 2
2
dij pij qij − 4bj vj − vj∗ λvj − vj∗ 4
j1
i1
t
×
−∞
∗ ∗
Nij t − svj − vj νi ui − ui ds
∞
n 2
dij pij qij × νi
Nij seλs ui t, x − u∗i ds
2
0
i1
n dij pij qij νi
−2
i1
×
∞
2
Nij sui t − s, x − u∗ ds
0
i
&
dx.
3.11
Estimating the right of 3.11 by elemental inequality 2ab ≤ a2 b2 , we obtain that
⎧⎡
⎤
m n
⎨
2
2
cji αji βji × μj ui − u∗ 2 ⎦
eλt δi ⎣−4ai ui − u∗i λui − u∗i 2
D V ≤
i
⎩
Ω
i1
j1
⎫
∞
m 2 ⎬
cji αji βji μj
2
Kji seλs vj t, x − vj∗ ds dx
⎭
0
j1
Ω
e
λt
m
%
δnj
2
2
− 4bj vj − vj∗ λvj − vj∗ j1
n 2
dij pij qij νi vj − v∗ 2
j
i1
Journal of Applied Mathematics
11
&
∞
n λs ∗ 2
dij pij qij νi
2
Nij se ui t, x − ui ds dx
0
i1
⎧ ⎡
⎤
m n ⎨
cji αji βji μj ⎦
δ ⎣λ − 4ai 2
eλt
⎩ i
Ω
i1
j1
⎫
m
⎬
2
∞
Nij seλs ds ui − u∗i dx
2νi δnj dij pij qij ⎭
0
j1
Ω
e
λt
m
%
n dij pij qij νi
λ − 4bj 2
δnj
j1
i1
&
n
2
∞
λs
Kji se ds vj − vj∗ dx ≤ 0.
2μj δi cji αji βji
0
i1
3.12
Therefore,
V t ≤ V 0,
t ≥ 0.
3.13
Noting that
eλt min δi ut, x − u∗ vt, x − v∗ ≤ V t,
1≤i≤nm
t ≥ 0.
3.14
On the other hand, we have
⎡
∞
n
m ∗ 2
⎣
cji αji βji μj
δi ui 0, x − ui 2
Kji s
V 0 Ω i1
j1
×
m
Ω j1
0
−s
0
⎤
2
eλζs vj ζ, x − vj∗ dζ ds⎦dx
∞
n 2
dij pij qij νi
δnj vj 0, x − vj∗ 2
Nij s
×
0
−s
i1
2
eλζs ui ζ, x − u∗i dζ ds dx
n dij pij qij νi
≤ maxδi 2max δnj
1≤i≤n
⎡
1≤j≤m
⎣ max δnj
1≤j≤m
0
∞
Nij sse φu s, x − u∗ λs
0
i1
⎤
∞
m cji αji βji μj
Kji sseλs ⎦ϕv s, x − v∗ .
2maxδi 1≤i≤n
j1
0
3.15
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Journal of Applied Mathematics
Let
%
n dij pij qij νi
M max maxδi 2max δnj
1≤i≤n
1≤j≤m
max δnj
1≤j≤m
∞
Nij sseλs ,
0
i1
⎫
∞
m ⎬
cji αji βji μj
Kji sseλs ,
2maxδi ⎭
1≤i≤n
0
j1
3.16
then M ≥ 1 and
ut, x − u∗ vt, x − v∗ ≤ M φu s, x − u∗ ϕv s, x − v∗ e−λt .
3.17
This implies that the equilibrium point of system 2.1 is globally exponentially stable. The
proof is completed.
Corollary 3.2. Suppose (A1) and (A2) hold. Then, the equilibrium u∗ , v∗ of system 2.1 is the
globally exponentially stable, if the following conditions are satisfied:
−2ai m m cji αji βji μj νi
dij pij qij < 0,
j1
−2bj j1
n
n dij pij qij νi μj
cji αji βji < 0.
i1
3.18
i1
4. Periodic Oscillatory Solutions of Fuzzy BAM Neural Networks
In this section, we consider the existence and uniqueness of periodic oscillatory solutions for
system 2.1.
Theorem 4.1. Suppose that assumption (A1) and (A2) hold. Then, there exists exactly an ω-periodic
solution of system 2.1, with the initial values 2.2 and 2.3, and all other solutions converge
exponentially to it as t → ∞. If there exists δi ≥ 0, δnj > 0 such that
⎡
⎤
m m
cji αji βji μj ⎦ νi δnj dij pij qij < 0,
δi ⎣−2ai δnj
j1
j1
n n
dij pij qij νi μj δi cji αji βji < 0,
−2bj i1
4.1
i1
where i 1, 2, . . . , n, j 1, 2, . . . , m.
Proof. Let
'
(
T T
C Φ | Φ φu , ϕv φu1 , . . . , φun , ϕv1 , . . . , ϕvm , Φ : −∞, 0 × −∞, 0 −→ Rnm .
4.2
Journal of Applied Mathematics
13
For any Φ ∈ C, we define
⎡
⎤1/2
1/2
n m 2
2
φμi dx
ϕvi dx⎦ .
sup ⎣
Φ sup
−∞≤s≤0
Ω i1
−∞≤s≤0
4.3
Ω j1
Then, C is the Banach space of continuous functions.
For any φu , ϕv T , φu , ϕv T ∈ C, we denote the solutions of system 2.1 through
T
0, 0 , φu , ϕv T , and 0, 0T , φu , ϕv T as
T
u t, φu , x u1 t, φu , x , . . . , un t, φu , x ,
T
v t, ϕv , x v1 t, ϕv , x , . . . , vm t, ϕv , x ,
T
u t, φu , x u1 t, φu , x , . . . , un t, φu , x ,
T
v t, ϕv , x v1 t, ϕv , x , . . . , vm t, ϕv , x ,
4.4
respectively. And define that
ut φu , x u t θ, φu , x ,
vt ϕv , x v t θ, ϕv , x ,
θ ∈ −∞, 0, t ≥ 0,
4.5
then ut φu , x, vt ϕv , xT φu , x, vt ϕv , xT ∈ C for t ≥ 0. Let uix ui t, φu , x −
ui t, φu , x, vjx vj t, ϕv , x − vj t, ϕv , x. Therefore, it follows from system 2.1 that
l
∂uix ∂
∂uix
Dik
− ai uix
∂t
∂xk
∂xk
k1
t
m
*
) cji
Kji t − s fj vj s, ϕv , x − fj vj s, ϕv , x ds
−∞
j1
m
t
αji
−∞
j1
−
m
t
αji
j1
m
βji
j1
−
m
−∞
t
−∞
t
βji
j1
−∞
Kji t − sfj vj s, ϕv , x ds
Kji t − sfj vj s, ϕv , x ds
Kji t − sfj vj s, ϕv , x ds
Kji t − sfj vj s, ϕv , x ds,
14
Journal of Applied Mathematics
l
∂vjx ∂
∂t
∂x
k
k1
∂vjx
∂xk
∗
Djk
− bj vjx
t
n
*
) dij
Nij t − s gi ui s, φu , x − fj vj s, φu , x ds
−∞
i1
t
n
pij
Nij t − sgi ui s, φu , x ds
−∞
i1
−
t
n
pij
Nij t − sgi ui s, φu , x ds
−∞
i1
t
n
qij
Nij t − sgi ui s, φu , x ds
−∞
i1
−
n
qij
t
i1
Nij t − sgi ui s, φv , x ds.
−∞
4.6
Considering the following Lyapunov functional
V t n
Ω i1
δi |uix | e dx n
Ω i1
× μj
2 λt
∞
2δi
×
Ω j1
∞
0
Ω j1
2
δnj vjx eλt dx
m cji αji βji j1
Kji s
t
0
m
m
t−s
2δnj
Nij s
n
2
eλζs vj ζ, ϕv , x − vj ζ, ϕv , x dζ dsdx
dij pij qij νi
i1
t
t−s
2
eλζs ui ζ, φu , x − ui ζ, φu , x dζ dsdx.
4.7
By a minor modification of the proof of Theorem 3.1, we can easily obtain
u t, φu , x − u t, φu , x v t, ϕv , x − v t, ϕv , x ≤ M φu − φu ϕv − ϕv e−λt ,
4.8
Journal of Applied Mathematics
15
for all t ≥ 0. We can choose a positive integer N such that
Me−λNω ≤
1
4
4.9
and define a Poincare mapping C → C by
P
T φu , ϕv
T
uω φu , uω ϕv
.
4.10
It follows that
PN
φu , ϕv
T T
uNω φu , uNω ϕv
.
4.11
Letting t nω, we, from 4.8 to 4.11, obtain that
T T 1 T T N
− P N φu , ϕv
φu , ϕv
≤ φu , ϕv − φu , ϕv .
P
2
4.12
It implies that P N is a contraction mapping. Hence, there exist a unique equilibrium point
φu∗ , ϕ∗v T ∈ C such that
PN
φu∗ , ϕ∗v
T T
φu∗ , ϕ∗v .
4.13
Note that
T T T P P N φu∗ , ϕ∗v
P φu∗ , ϕ∗v
.
P N P φu∗ , ϕ∗v
4.14
Let ut, φu∗ , vt, ϕ∗v T be the solution of system 2.1 through 0, 0T , φu∗ , ϕ∗v T , then
ut ω, φu∗ , x, vt ω, ϕ∗v , xT is also the solution of system 2.1. It is clear that
T ∗ ∗ T
utω φu∗ , vtω ϕ∗v
ut φu , vt ϕv
4.15
for all t ≥ 0. It follows from 4.15 that
T ∗ ∗ T
u t ω, φu∗ , x , v t ω, ϕ∗v , x
u t, φu , x , v t, ϕv , x ,
t ≥ 0,
4.16
which shows that ut, φu∗ , x, vt, ϕ∗v , xT is exactly an ω-periodic solution of system 2.1
with the initial conditions 2.2 and 2.3 and other solutions of system 2.1 with the initial
conditions 2.2 and 2.3 converge exponentially to it as t → ∞.
16
Journal of Applied Mathematics
5. An Illustrative Example
In this section, we will give an example to illustrate feasible our result.
Example 5.1. Consider the following system
l
∂ui t, x ∂
∂ui
Dik
− ai ui t, x
∂t
∂xk
∂xk
k1
m
t
cji
−∞
j1
m
t
αji
t
βji
−∞
j1
m
m
j1
l
∂vj t, x ∂
∂t
∂x
k
k1
n
−∞
∂vj
∂xk
− bj vj t, x
Nij t − sgi ui s, xds
t
n
pij
Nij t − sgi ui s, xds
n
qij
i1
∗
Djk
t
i1
dij
i1
Kji t − sfj vj s, x ds
Tji ωj ∨ Hji ωj Ii ,
j1
Kji t − sfj vj s, x ds
−∞
j1
m
Kji t − sfj vj s, x ds
−∞
t
−∞
Nij t − sgi ui s, xds
n
n
Sij ωi ∨ Lij ωi Jj ,
i1
i1
5.1
where
Kji t Nij t te−t ,
i 1, 2, j 1, 2,
f1 r f2 r g1 r g2 r 1
|r 1| − |r − 1|.
2
5.2
Journal of Applied Mathematics
17
It is obvious that f·, g· satisfy assumption A1 and K·, N· satisfy assumption A2,
moreover μ1 μ2 ν1 ν2 1. Let
c11 1
,
2
c21 1,
1
c12 − ,
2
c22 d11 1
,
3
d21 1
,
2
1
d12 − ,
3
d22 2
,
3
α11 2
,
3
α21 1
,
2
1
α12 − ,
4
α22 1
,
3
β11 1
,
3
β21 1
,
2
1
β12 − ,
4
β22 5
,
3
p11 2
,
3
p21 1
,
4
1
p12 − ,
2
p22 2
,
3
q11 2
,
3
q21 1
,
2
1
q12 − ,
4
q22 1
,
3
b1 3,
b2 4.5.
a1 4,
a2 3.5,
2
,
3
5.3
By simply calculating, we can get
−2a1 2 2 cj1 αj1 βj1 μj ν1
d1j p1j q1j −1.75 < 0,
j1
−2a2 2
j1
2 cj2 αj2 βj2 μj ν2
d2j p2j q2j −0.75 < 0,
j1
−2b1 −2b2 j1
2
2
i1
i1
|di1 | pi1 qi1 νi μ1
2
i1
|c1i | |α1i | β1i −0.583 < 0,
5.4
2
|di2 | pi2 qi2 νi μ2
|c2i | |α2i | β2i −1.583 < 0.
i1
Since all the conditions of Corollary 3.2 are satisfied, therefore the system 5.1 has a unique
equilibrium point which is globally exponentially stable.
6. Conclusion
In this paper, we have studied the exponential stability of the equilibrium point and the
existence of periodic solutions for fuzzy BAM neural networks with distributed delays and
diffusion. Some sufficient conditions set up here are easily verified, and the conditions which
are only correlated with parameters of the system 2.1 are independent of time delays.
The results obtained in this paper remove the assumption about the activation functions
with differentiable and only require the activation functions are bounded and Lipschitz
continuous. Thus, it allows us even more flexibility in choosing activation functions.
18
Journal of Applied Mathematics
Acknowledgments
The authors would like to thank the editor and anonymous reviewers for their helpful
comments and valuable suggestions, which have greatly improved the quality of this paper.
This work is partially supported by the Scientific Research Foundation of Guizhou Science
and Technology Department2011J2096.
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