Dynamics of Fuzzy BAM Neural Networks with Distributed Delays and ...

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, [email protected] 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.

2

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


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

4


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


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, δn j > 0 such that ⎡

⎤ m m cji αji βji μj ⎦ νi δn j dij pij qij < 0, δi ⎣−2ai j1

j1

δn j

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 δn j dij pij qij δi ⎣λ − 4ai 2 Nij seλs ds < 0,

j1 n

δn j 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

−∞

6


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


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

8

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 eλζ s vj ζ, x − vj∗ dζ ds ⎦dx

∞ n m 2 dij pij qij νi δn j vj − vj∗ eλt 2 Nij s Ω j1

0

i1

×

t e t−s

λζ s

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∗ δn j ⎣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∗ δn j 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

10

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 δn j

%

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

% δn j

2 2 − 4bj vj − vj∗ λvj − vj∗

j1 n 2 dij pij qij νi vj − v∗ 2 j i1


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 δn j dij pij qij ⎭ 0 j1

Ω

e

λt

m

%

n dij pij qij νi λ − 4bj 2

δn j

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≤n m

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 eλζ s vj ζ, x − vj∗ dζ ds⎦dx

∞ n 2 dij pij qij νi δn j vj 0, x − vj∗ 2 Nij s ×

0

−s

i1

2 eλζ s ui ζ, x − u∗i dζ ds dx

n dij pij qij νi ≤ maxδi 2max δn j

1≤i≤n

⎡

1≤j≤m

⎣ max δn j 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

12


Let %

n dij pij qij νi M max maxδi 2max δn j 1≤i≤n

1≤j≤m

max δn j

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, δn j > 0 such that ⎡

⎤ m m cji αji βji μj ⎦ νi δn j dij pij qij < 0, δi ⎣−2ai δn j

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 −→ Rn m . 4.2


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 δn j vjx eλt dx

m cji αji βji j1

Kji s

t

0

m

m

t−s

2δn j

Nij s

n

2 eλζ s vj ζ, ϕv , x − vj ζ, ϕv , x dζ dsdx

dij pij qij νi

i1

t t−s

2 eλζ s 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


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 ut ω φu∗ , vt ω 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


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


−∞

j1 m


−∞

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


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


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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