0 Boundary Value Problems

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Global exponential stability and existence of periodic solutions for delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions

Weiyuan Zhang1*, Junmin Li2 and Minglai Chen2

Correspondence: ahzwy@163.com 1 Institute of Mathematics and Applied Mathematics, Xianyang NormalUniversity, Xianyang, 712000, China

Full list ofauthor information is available at the end ofthe article

Abstract

In this paper, both global exponential stability and periodic solutions are investigated for a class of delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions. By employing suitable Lyapunov functionals, sufficient conditions of the global exponential stability and the existence of periodic solutions are established for reaction-diffusion BAM neural networks with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature. A numerical example is given to show the effectiveness of the obtained results.

Keywords: neural networks; reaction-diffusion; mixed time delays; global exponential stability; Poincaré mapping; Lyapunov functional

ringer

1 Introduction

Neural networks (NNs) have been extensively studied in the past few years and have found many applications in different areas such as pattern recognition, associative memory, combinatorial optimization, etc. Delayed versions of NNs were also proved to be important for solving certain classes of motion-related optimization problems. Various results concerning the dynamical behavior of NNs with delays have been reported during the last decade (see, e.g., [1-7]). Recently, the authors in [1] and [2] considered the problem of exponential passivity analysis for uncertain NNs with time-varying delays and passivity-based controller design for Hopfield NNs, respectively.

Since NNs related to bidirectional associative memory (BAM) were proposed by Kosko [8], the BAM NNs have been one of the most interesting research topics and have attracted the attention of researchers. In the design and applications of networks, the stability of the designed BAM NNs is one of the most important issues (see, e.g., [9-12]). Many important results concerning mainly the existence and stability of equilibrium of BAM NNs have been obtained (see, e.g., [9-15]).

However, strictly speaking, diffusion effects cannot be avoided in the NNs when electrons are moving in asymmetric electromagnetic fields. So, we must consider that the activations vary in space as well as in time. In [16-34], the authors considered the stability of NNs with diffusion terms which were expressed by partial differential equations. In par© 2013 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

ticular, the existence and attractivity of periodic solutions for non-autonomous reaction-diffusion Cohen-Grossberg NNs with discrete time delays were investigated in [20]. The authors derived sufficient conditions on the stability and periodic solutions of delayed reaction-diffusion NNs (RDNNs) with Neumann boundary conditions in [21-25]. In these works, due to the divergence theorem employed, a negative integral term with gradient was removed in their deduction. Therefore, the stability criteria acquired by them do not contain diffusion terms; that is to say, the diffusion terms do not have any effect on their deduction and results. Meanwhile, some conditions dependent on the diffusion coefficients were given in [30, 32-34] to ensure the global exponential stability and periodicity of RDNNs with Dirichlet boundary conditions based on 2-norm.

To the best of our knowledge, there are few reports about global exponential stability and periodicity of RDNNs with mixed time delays and Dirichlet boundary conditions, which are very important in theories and applications and also are a very challenging problem. In this paper, by employing suitable Lyapunov functionals, we shall apply inequality techniques to establish global exponential stability criteria of the equilibrium and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions. The derived criteria extend and improve previous results in the literature [22, 29].

Throughout this paper, we need the following notations. Rn denotes the «-dimensional Euclidean space. We denote

\u(t,#)-«*|| = / ¿_\ui - uj\rdx,

Jq i=i

||^u (s, x)-uj|| = sup

-«><s<0

I T2\vui(s,x)-u*\rdx Jq i=i

f n r f n

^(t, x)-v*W = I Vj - v*:\r dx, H^v (s, x)-v*W = sup I x)

Jq j=i -TO<s<0 Jq j=i

'q j=1 r > 2.

- vj \ dx

Let u = u;(t,x), Vj = Vj(t,x).

The remainder of this paper is organized as follows. In Section 2, the basic notations, model description and assumptions are introduced. In Sections 3 and 4, criteria are proposed to determine global exponential stability, and periodic solutions are considered for reaction-diffusion recurrent neural networks with mixed time delays, respectively. An illustrative example is given to illustrate the effectiveness of the obtained results in Section 5. We also conclude this paper in Section 6.

2 Model description and preliminaries

In this paper, the RDNNs with mixed time delays are described as follows:

"ft = £ ik{Dik S)- pi{ui(t,

+ E (bf(vj(t, x))) + E (bjjjjt - Qji(t), x)))

j=i j=i

n - ft

+ m bji kji(t - sjjs,x)) ds + Ii(t),

f=É j - qj x»

+ (dijgi(ui(t,x))) + /

Ui(t - Tij(t),x)))

^2dij kij(t - s)gi(ui(s, x)) ds + /j(t).

i=l m /. t

The RDNNs model given in (1) can be regarded as RDNNs with two layers, where m is the number of neurons in the first layer and n is the number of neurons in the second layer. x = (x1,x2,...,xl)T e ^ c Rl, ^ is a compact set with smooth boundary 3&, and mes^ > 0 in the space Rl; u = (u1,U2,...,um)T e Rm, v = (v1, V2,..., vn)T e Rn. ui(t,x) and Vj(t,x) represent the state of the ith neuron in the first layer and the jth neuron in the second layer at time t and in the space x, respectively. bji, bp, j dij, dj and dj are known constants denoting the synaptic connection strengths between the neurons in the two layers, respectively; fj, fj, fj, gi, g and g denote the activation functions of the neurons and the signal propagation functions, respectively. Ii and Jj denote the external inputs on the ith neuron and jth neuron, respectively; pi and qj are differentiable real functions with positive derivatives defining the neuron charging time; jt) and dji(t) represent continuous time-varying delay and discrete delay, respectively; Dik > 0 and Djk > 0, i = 1,2,..., m, k = 1,2,..., l and j = 1,2,..., n, stand for the transmission diffusion coefficient along the ith neuron and jth neuron, respectively.

System (1) is supplemented with the following boundary conditions and initial values:

ui(t,x) = 0, Vj(t,x) = 0, t > 0,x e da,

ui(s, x) = pui (s, x), Vj (s, x) = pVj(s, x), (s, x) e (-œ,0] x a

(2) (3)

for any i = 1,2,..., m and j = 1,2,..., n, where n is the outer normal vector of d a, p = = (pu1,...,pum,pv1,...,pvn)T e C are bounded and continuous, where C = {p|p = ,p :

(-to,0] xrm\ R„ v

(-to,0]xr^ ^ R

}. It is the Banach space of continuous functions which maps (( œ,0]

(-œ (-œ,0]Z

into Rm+n with the topology of uniform convergence for the norm

\ Pv -œ<s<0

f m 1 r f n

/ (Pui\rdx + sup q>vj\rdx

Ja i=i -œ<s<0 Ja j=i

Remark 1 Some famous NN models became a special case of system (1). For example, when Dik = 0 and Djk = 0 (i = 1,2,...,m, k = 1,2,...,l), the special case of model (1) is the model which has been studied in [13-15]. When bji = 0 and dj = 0, i = 1,2,..., m, j = 1,2,..., n, system (1) became NNs with distributed delays and reaction-diffusion terms [18, 22, 29].

Throughout this paper, we assume that the following conditions are made.

(Al) The functions Tj(t), dji(t) are piecewise-continuous of class C1 on the closure of each continuity subinterval and satisfy

0 < Tij(t) < Tij, 0 < Gji(t) < Gji, Tij(t) < / <1, Gji(t) < / < 1, t = max {Tij}, 0 = max {Gji}

1<i<m,1<j<n 1<i<m,1<j<n

with some constants Tij > 0, Gji > 0, t > 0, G > 0 forall t > 0.

(A2) The functions pi (•) and q;-(•) are piecewise-continuous of class C1 on the closure of each continuity subinterval and satisfy

ai = inf pi(f )>0, pi(0) = 0, cj = inf qj(Z) > 0, q/(0) = 0.

(A3) The activation functions and the signal propagation functions are bounded and Lipschitz continuous, i.e., there exist positive constants Lj, Lf, if, Lg, Lg and igf such that for all n1, n2 e R,

(A4) The delay kernels Kji(s),Kij(s) : [0, to) ^ [0,to) (i = 1,2, ...,m, j =1,2,...,«) are real-valued non-negative continuous functions that satisfy the following conditions:

Let (u*, v*) = (u*, «2,..., «*, v*, v*,..., vn) be the equilibrium point of system (1).

Definition 1 The equilibrium point of system (1) is said to be globally exponentially stable if we can find r > 2 such that there exist constants a >0 and ¡3 > 1 such that

II u(t, x) - u* | + || v(t, x)- V* ||

for all t > 0.

Remark 2 It is well known that bounded activation functions always guarantee the existence of an equilibrium point for system (1).

(i) i0+œ Kji(s) ds = 1, i0+œKji(s) ds = 1;

(ii) /0+to sKji(s) ds < to, /0+tosKj(s) ds < to;

(iii) There exist a positive ¡x such that

< pe-2at(\\Vu(s,x)-u*\\ + ||^(s,x)-v*\\)

Lemma 1 [33] Let ^ be a cube \xi | < dl (l = 1,..., m), and let h(x) be a real-valued function belonging to C1(^) which vanishes on the boundary d^ of i.e., h(x)\da=o. Then

/ h2(x) dx < d2 / Jq JQ

dx. (5)

3 Global exponential stability

Now we are in a position to investigate the global exponential stability of system (1). By constructing a suitable Lyapunov functional, we arrive at the following conclusion.

Theorem 1 Let (A1)-(A4) be in force. If there exist wi > 0 (i = 1,2,..., n + m), r > 2, yij >0, Pji > 0 such that

(n n \

-rna^Dil - rnar + 2(r -1) £ ar + (r -1) £ arj™

j=1 j=1 '

+ £ wm+jm^\dij\r(Lg)r + |dij|r — my + |dij|ry^) <0

wm+j -rmc^Djl - rmj + 2(r -1) £ cr + (r -1) £ c^

\ i=1 i=1 /

+ mj(Lfy + br^-(iy + M^Ly) < 0, (6)

i=1 \ 1 /XG /

in which i = 1,2,..., m, j = 1,2,..., n, Lf, Lf, Lf, Lg, Lg and L^^ are Lipschitz constants, Di = min1<i<l{Dii/d|}, D* = min1<i<l {D*/d|}, then the equilibrium point (u*, v*) of system (1) is unique and globally exponentially stable.

Proof If (6) holds, we can always choose a positive number S >0 (may be very small) such that

(n n \

-rna^DJ - rna\ + 2(r -1) £ a[ + (r -1) £ a[¡¡^

j=1 j=1 '

+ £ Wm+jm^ |dij|r(Lg)r + |dij |r J— (Lg)r + |dij |r yj(Lf)j + 5 <0

,+j -rm^Dp - rmc[ + 2(r -1) £ c^ + (r -1) £ c^y^-1

\ i=1 i=1 /

+ mw^f|bji|r j + |bji|^— j + bj^Pï^y) + 5 <0, (7)

\ 1 ¡9 /

Let us consider the functions

Fi(x*) = wJ -rnarr1Dil - rna\ + 2(r -1) J^al

n r /.+TO

* „„ ^r-1

+ (r -1)> ar j7-1 kji(s)ds + 2x* nar

+ £wm+jmr( Idijlr(Lf)r + ld,jlr--(if)r

j=1 \ 1 - 1

e2xiSl<ij(s) dsj

j = wm+j

m m r /.+TO _

r-1D*l - rmcj + 2(r -1) C + (r -1) £ Cy-7-1 kj(s) ds

.■I v 0

+ 2y*mcrj-1

i=1 i=1

+ x lU..lr^f)r

ee i - -f o * \

+ Ibjil^-(ij)r + Ibji|rP^iLf)r e jSkji(s)ds),

1 _ le Jo /

wherex*,y* e [0, i = 1,2,...,m, j = 1,2,...,n.

From (8) and (A4), we derive Fi(0) <-S <0, Gj(0) <-S <0; Fi(x*) and Gj(y*) are continuous for x*,y* e [0, Moreover, Fi(x*) ^ as x* ^ and Gj(y*) ^ as y* ^ Thus there exist constants ei, Oj e [0, such that

Fi(si) = wA -rnari 1Dil - rnari + 2(r -1) ^a\

n _ r r \ n /

+ (r -1) J2aj1 / kji(s) ds + 2sinar_1\ Wm+j^l ld^Lf j=1 / j=1 ^ et - C\

+ ldij lr--(if)r + ldij lr y[j(Lf)r e2s'skij(s) ds) =0

1 _ l^r jo /

Gj(oj) = wm+j I -rmcrj 1D*l - rmc[ + 2(r -1) ^ cj \ i=1 m r /._ \ m ( + (r-1)£cj1 / kj(s)ds + 2oimj_1)+Y.w^i lbjilr{Lfj)r i=1 / i=1 ^

ee f - r t\

+ l bji l ^-(Lj)r + l bji lr Pji(Lj)r e2a'skji (s) d^ =0, (9)

1 _1 Jo /

By using a = min1<i<m,1<j<n{ei, a,}, obviously, we get

Fi(a) = wA -rnar 1Dil - rna\ + 2(r -1) ^ a[

n _ t\ n /

+ (r -1) £ ar j-1 / js) ds + 2anarM + £ wm+jmM \d^(,Lg)r

j=1 / j=1 ^

eT ~ - - C\

+ \dj\^--(Lg)r + \dij\ry[j(Lf)r e2askij(s)ds) < 0

1 - xt Jq /

Gj(a) = wm+j i -rmcr 1D**l - rmcr + 2(r -1) ^ cr

m r /.+TO _ \ m /

+ (r -1) £c^Yi^7-1 hj(s)ds + 2amcrJ_1\ winr( \bji\r{Lf^r i=1 / i=1 ^

eG f - -f f\

+ \bji\^-(L^)r + ^fLf)r e2askji(s)ds) < 0, (10)

1 - Xg Jq /

where i = 1,2,..., m, j = 1,2,..., n.

Suppose (u, v) = (u1, u2,..., un, v1, v2,..., vn)T is any solution of model (1). Rewrite model (1) as

d(ui - u*) ^ d / d(ui - u*)

dt k=1

^¿D ^^uix^)- (Pi(ui(t, x)) - Pi(u*))

vj(t, x)) -fj) + V jjjt - Gji(t), x)) -f (v*)))

+ 2.^,bji^/AVj(t,x)j "))) +

j=1 j=1

n t ( ( ) ( ))

J^bji kji(t -s)(jj(vj(s,x))-fj(v*)) ds, (11)

= £ £ /D*k ^) - (« - «-(•*))

+ P(dij(gi(ui(t,x)) -gi(u*))) + ^(dij(gi(ui(t- Tij(t),x)) -gi(u*)))

i=1 i=1

m t ( ( ) ( ))

+ Edfij / j-s)(gi(ui(s,x)) -gi(u*)) ds. (12)

Multiplying (11) by u - u* and integrating over a yield

1 d i (ui - u*)2 dx

2 dt i ij

[ V^( *) 9 /n 9(ui - u*)) ,

= > (ui - u*)— Dik—---)dx

Ja^! dxk\ dxk J

pi feu u - uî)2 dx + / J2 ju- u*)f (vi)-f^v^)dx

Ja Ja j=i

è f jui - uf)(/5(Vj(t - fy(t),x)) -f'^))) dx j=l ,/a

E JJ (*,(« - u*)/ œ kji(t - s)(k(vj(s, x)) -ffy*))) ds d

According to Green's formula and the Dirichlet boundary condition, we get

l . * \ l

' d xk ^

j^u - u*)^ D Ux = -

D J dx.

a \ dxk

Moreover from Lemma l, we have

dJ dx < -

a \ dxk

[ T^T (u - u,*)2 dx < -Dil | ui - u*|2. Ja k=l d2

From (11)-(15), (A2), (A3) and the Holder integral inequality, we obtain that

ui - u* | dx

< -2Dil I |ui - u*|2 dx -2ai I |ui - u*|2 dx a i a i

+ 2 / ¿(ju -u*|zj|vj - v*|)dx a j=l

+ 2 V / (\bji\|ui -u*|f (vj(t-Oji(t),x)) -/(v*)|)dx

+ 2 E /J (n/' kji(t - s)|ui - (vy(s, x)) -fj(vj)\)ds

Multiplying both sides of (l2) by Vj - v*, similarly, we also have d

, . | V; - j dx dt j j

< -2D*l I | Vj - v*|2 dx -2cj I | Vj - v*|2 dx j a j a j

+ 2 J (\dij\Lg|u - u* 11Vj - V*|) dx

+ 2^ / (\dj\£(u^t - Tjj(t),x)) -g(u*) 11 Vj - v*|) dx

+ 2 ¿/ [\dij \ / kij(t - s)g(ui(s, x)) - gi(u*)||Vj - V*| ds J a L «/-œ

Choose a Lyapunov functional as follows:

y (t)=/ mwi

Jq i=1

nar - 1\ui-u*\re2at

^ e9 ft

E jrnr n e2a f-j, x)) -jj\rdf

Tf 1 - ¡9 Jt - 9;:(t)

1 - ¡9 Jt-9}i(t)

r> + TO 01

" /» + TO ft

bji |rnrPi kji(s) e2a(s+t) ,x)) -jj(vf) \r d£ ds

j=1 J0 Jt-s

+ f ltw»

Jq j=1

m eT ^t

+ E \dij\rmr-- e2a feitutè,x)) -gi(uf) \d%

i=1 1 - ¡T Jt-Tij(t)

m f+TO _ f t

+ E \dij|rmrYj kij(s) e2a(s+t)\gi(ui(^,x))-gi(uf) \d%ds

i=1 J0 Jt-s

Its upper Dini-derivative along the solution to system (1) can be calculated as follows:

r-1 \ * \r 2at

mcr 1 \ Vj - vf \ e

D+V(t) <

Jq i=1

r-1 \ * \r-1 d|ui uf | 2at 2at r-1\ * \r

rnar: 1 \ui - uf \ -— e2at + 2ae2atnar 1 \ui - uf \

i I i H dt i I i i I

+ e2at^2 \bji\rnr-— f(v,(t,x)) -f(vf)\r

1 - ¡¡9

- £ |bji|rnr -i— (1 - 9ji(t)) e2a(t-j(t)) |^(vy(t - 9ji(t), x)) -j, \r

jl 1 - ¡9

" /» + TO

+ e2at £ |bji |rnr Pi e2askji(s) fjj, x)) -jj \r ds

j=1 j0

n f+TO

-e2at£|bji|rnrPi kji(s)fj(vj(t-s,x)) -fj(vf)\rds

f / ■ ■ m+j

+ / w, j=1 m

r-1\ * \r-1 d|vj v/ | 2at ^ 2at r-1 \ * \r

rmcr 1 \ vj - vf \ -— e + 2ae2atmcr 1 \ v,- - vf \

t \ > ¡\ dt ' 1 j j

+ e2at^2 \d,i\rmr--\gi(ui(t,x))-gi(uf)\r

i=1 1 - ¡T

- £ |dij\ rmr -—^— e2a(t-x'>^'})(1 - Tij)\gi(ui(t- Tij(t),x)) -gi(uf)\r

1 - ¡¡T

m /> + TO

+ e2at £ \dij\rmryï e2askij(s)gi(ui(t,x)) -gi(uf) \rds i=1 j0

m f+TO _

Ed\rmry' ! kij(s)\gi{ui(t-s,x)) -gi(uf)\ds

-e2atV\di,\rmr

Jq i=1

rnari 1|ui - u*|r 2e2atl -Dil|ui - u*|2 - ai|ui - u*|2

+ £ (bil^i-u^L |vj-v*|)

■ E(lbjil |ui - u* | f (vj(t - eji(t), x)) -f (v*) |)

■ £(lbji^-s^i-^^^s,x)) -jj(v*)^ d^

+ 2ae^2atna¡_1|ui - u*|r + e2at £ lbjilV--fjx)) -f (v*)|

1 - le

- e2at ^ lbjilrnr|j-(vj(t- eji,x)) - fj(v;)|r

+ e2at £ lbji lrnrpA e2askji(s) jvj(t,x)) -f^) |r ds

j=1 ^ n f

-e2at£lbjilV^ / A/i(s)^(vj(t-s,x)) -jj(v^)|rds j=1 Jo

+ wm+j

Jq ;=1

r — 1| * | r_ 2 2at i r\* /| * 12 | * 12

rmcj 11Vj — v*| e l —D*l|vj — v*| -cj|Vj — v*|

+ J2(ldijlLg|ui _ u*| |vj _ vj* D + J2(ldijl |ij(ui(t _ rij,x)) _k(u*) ^j- vj* D i=1 i=1

+ i(\dijlj kij(t-s)\ki(ui(s,x)) -iki(u*)||vj-v*^ dsj

+ 2ae2^tmcrJ_1|vj - v* |r + e2a^ ldjlrmr-^~ g(m(t,x))-¿(a*) (

i=1 1 - lr

- e2at^2 ldijlrmr¿(«(t- rij,x))-¿(u*) |r i=1

+ e2at £ ldijlrmrYrjjo e^kji^kiiuitx))-gi(u*) \ds

-e2atV ldij lrmry[A kij(s)|ki(ui(t-s, x)) -gi(u*) |r

i=1 Jo

From (18) and the Young inequality, we can conclude

dx. (18)

D+V(t) <

/ e2atY\

Jq i=1

- rnar xDil\ui - u*|r -rnaru - u*|r

+ (r-1)^ a^ui-^l

+ E n IbjiI7(Lfy |v- — vjf) + (г — a' |ui — U*|r j=l j=l n

+ E -jt — в-i (t), x))—Ir

+ (г — 1)it(a7ß-T—l kji(t — s)|ui — u*Irds^)

t E(|bjikji(t — s)f(v,(s,x)) — fj(v*)Ird^J +2аna; n в | ( ) ( )|

+ E IbjiIrnr ^ fj (v-(t, x)) —f (vj) |r -1 1 — -в

— LÑv j—t-ш x)— fj(v}) Ir

n /■ +c

^ I bji |rnrßi e2c"skji(s) f(v-(t,x)) — fj(v;) Irds

j=l Jo

n /■+c

— J2 IbjiIrnr ßji kii(s)f (v-(t —s, x)) —fj- (v;) I

j=l Jo

W e2aty> •/a j=l

t (r — 1) £c;|v- — v;|r t £(|dij|7m^Lg)r|ui — U*|r)

i=l i=l mm

+ (г — 1) {crj |v- — v* |0 t E IdijIrmrlgi(ui(t — tij(t),x)) — ¿(и*) | i=l i=l

+ (г — 1) ¿(cjY—7—1 f k-(t —s)|v- —v;|rds) i=l \ J-œ /

t ¿(Idi—IrmrY[jj' k— (t —s)|gi(ui(s,x)) — gi(u*)|rds^

ui ui |

—rmc7 lDíl|v— — v* |r — rmc71v— — v* |r

t 2amc7 11 v— — v*|r

■J2I~dij Irmr--|gi(ui(t, x)) — ¿(и?) |r

i=l 1 — -т

— Id— Irmr\gi(ui(t— tij(t),x)) —gi(u*)|r i=l

^ |d— |rmrY—\ e2ask— (s)\g^ui(t,x)) — gi(u*) \ds i=l Jo

m f+c _

— V|dij |rmr y— i kij(s)\gi{ui(t — s, x)) —gi(u*) \ ds

/ e2atJ2 a i=l

-rnar lDil - rnar + 2(r -1) ^ a\ + 2anar l

+ (r-1) ^(arj^ f* kM(t-s) ds^ j + ¿wm+m^ \dii\r(Lf)r

ex - - /"+œ - - \

+ \dij r--(Lf)r + \dij\r yj\ e2a%(s)(Lf)r ds

l - ¿t Jo /

+ e2atV]

|ui-u*| dx

- rmc^Djl - rmcr + 2(r -1) J^ cj

+ (r-1) jr(cr Y-7- J œ ~kij(t - s) d^ + 2amcrA + ]Twinr(\bji\r (Lf )r

+ \bji\r TZT^ L)r + jj l

(Lf )'

e2askji(s) ds

From (6), we can conclude

Vj-V*|rdx. (l9)

D+V(t) < 0 and so V(t) < V(0), t > 0.

V (0) = J^Wi

nar l|ui(0, x) - u*

^ e0 f0 J2\bji\rnr T—l f,{Vj(è, x))-fjj [dÇ jl 1 - J j)

n f+œ p 0

Tjn'-fiU kji(s) e2a(s+t jjfê,x)) -jj\d%ds

j=l 0 s

m eT 0 | ( ) ( )|

^ 1 - ¿T J - Tij (t)

m p+œ _ p 0

£ \dij\rmryj kij(s) e2a(s+ï),x)) -gi(u*) |rdÇ ds

:_-I 7 0 7-s

+W j=l

mcr 1|v;(0, x) - V*|

max {wi}

nar l|ui(0, x) - u*

^ e0 f C0

+ £ \b/i\rnr--(Lf)r j, x) - VÎfdÇ

jl 1 - ¿0 J-e,i

n - p+œ p 0

+ £\bji\rnr(Lk)rfir kji(s) e2a(s+ïj,x) - V*|rdÇds

j=l 0 s

mcr 11 Vj (0, x) - V*|

+ / max {Wm+j}

Ja j~^l<j<n

m - eT f 0

+ E\dij\r(Lg)rmr-- ^(Ç,x) - u*|rdÇ

i=i 1 - /t j-Tij

n - f+œ _ f 0

+ E\dij\rmr(Lgi)r yj h (s) e2a(s+Ç ) |ui(Ç, x) - u* |r dÇ ds

j=l 0 s

< < max {Wi} + max {Wm+j} max

I l<i<m l<j<n l<j<n

( ) +œ

J2\dij\rmr{Li;) ryj kij(s)se2as ds

+ max {Wm+j} max

l<j<n l<j<n

E\dij\^Lf) rmr -

+ < max {Wm+j} + max {Wi} max

l<j<n l<i<m l<i<m

+ max {Wi} max

l<i<m l<i<m

T.\~bii\rnr(Lgj

Pu(s,x) - u*|

n - n+œ

Pi^ se2ask-i(s) ds

1 Pv(s,x) - V* |r.

j=l , e00

1 - /0

Noting that

e2at( min w^ (| u(t,x) - u*|| + II v(t,x) - v*|) < V(t), t > 0.

\1<i<m+n /" 11 11 11

fi = ma^| max {Wi} + max {Wm+;-} max

I l<i<m l<j<n l<j<n

m ( ) +œ

J2\~dij fmr(Lf)r y- k

- (s)se2as ds

+ max {Wm+j} max

l<j<n l<j<n

E\d -m rmr -

max {Wm+; } + max {Wi} max

l<j<n l<i<m l<i<m

n - p +œ

J2\bji\rnr{Lfj)r fii se2askji(s) ds j=l 0

+ max {Wi} max

l<i<m l<i<m

£ \ bji \ rnrL"r

e00 1 - /0

/ min {Wi}.

l<i<m+n

Clearly, p > 1. It follows that

||u(t,x) - u*| + ||v(t,x) - v*| < Pe_2at(IPu(s,x) - u*| + |pv(s,x) - v*|),

for any t > 0, where p > 1 is a constant. This implies that the solution of (1) is globally exponentially stable. This completes the proof of Theorem 1. □

Remark 3 In this paper, the derived sufficient condition includes diffusion terms. Unfortunately, in the proof in the previous papers [21 - 24], a negative integral term with gradient is left out in their deduction. This leads to the fact that those criteria are irrelevant to the

diffusion term. Obviously, Lyapunov functional to construct is more general and our results expand the model in [22, 29].

When bji = 0 and dij = 0 (i = 1,2,..., m, j = 1,2,..., n), system (1) becomes the following BAM NNs with distributed delays and reaction-diffusion terms:

f=¿¿K|) - MM'm)+t (bfjfvj M»

+ V but KM - srn(Vj(s, x)i as +

J2bji Kji(t - s)fj(vj(s,x)) ds + Ii(t),

-=1 J-<x>

l d i 3V'\ m

£ \D>kij) - x» + £(^(u^,x)))

m /• t _ + £dij / kij(t - s)gi(ui(s,x)) ds + /j(t).

For (23), we get the following result.

Corollary 1 Let (A1)-(A4) be in force. If there exist wi > 0 (i = 1,2,..., n + m), r > 2, yj >0, ^ > 0 such that

(n n \

-rna!r1Dil - rnar + 2(r -1) £ ar + (r -1) £ arffi

j=1 j=1

+ £ Wm+jm\\aij\r{Lgy + Idij|ry^) < 0

(m m \

-rmcrr1D*l - rmcj + 2(r -1) £ c^ + (r -1) £ c^y^

i=1 i=1

+ £Win^\bji\r(Lf)r + \bji\rPji(fr) < 0, (24)

where i = 1,2,..., m, j = 1,2,..., n, Lf, Lf, L^, Lg, Lf and Lgf are Lipschitz constants. Then the equilibrium point (u*, v*) of system (1) is unique and globally exponentially stable.

4 Periodic solutions

In this section, we consider the stability criterion for periodic oscillatory solutions of system (1), in which external input Ii : R+ ^ R, i = 1,2,..., m, and /j: R+ ^ R, j = 1,2,..., n, are continuously periodic functions with period «, that is,

Ii(t + w)=Ii(t), /j(t + w)=/j(t), i = 1,2,...,m, j = 1,2, ...,n.

By constructing a Poincare mapping, the existence of a unique «-periodic solution and its stability are readily established.

Theorem 2 Let (A1)-(A4) be in force. There exists only one rn-periodic solution of system (1), and all other solutions converge exponentially to itast ^ if there exist constants wi > 0 (i = 1, 2, ..., n + m), r > 2, yij > 0, fiji > 0 (i = 1, 2, ..., m, j =1, 2, ..., n) such that

(n n \

-rnar-1Dil - rnar + 2(r -1) £ ar + (r -1) £ ar¿j-7-1

j=1 j=1 '

e f _ g \T W _ g \ r^

+ ¿wm+Jmr(| dj| r(if)r + | dj | r ^ (L)r + | dj| rKiT(if< 0

wm+j -rmc^Djl - rmc[ + 2(r -1) £ c^ + (r -1) £ c^y^

\ i=1 i=1 /

+ m WinY|bji|rj + ibjii^-^^-(Lf\r + |bji|r¿;i(Lf< 0, (25)

i=! \ 1 lx0 /

where i = 1,2,..., m and j = 1,2,..., n, Lfj, Lf, Lj, Lg, Lg and Lgf are Lipschitz constants in (A3).

Proof For any , e C, we denote the solutions of system (1) through (Q), ((Ipu))

and (Q), ( (£))" V *

u(t, pM, x) = ( u1(t, (fu, x),...,um(t, (fu, x)) T, v(t, fv, x) = ( V:(t, fv, x),..., Vn(t, fv, x)) T

u(t, fa, x) = (ux(t, fa, x),...,um(t, f«, x)) T, v(t, fv, x) = ( V:(t, fv, x),...,Vn(t, fv, x)) T,

respectively. Define

ut((u,x) = u(t + 0,(pu,x), 0 e (-ro,0],t > 0, vt(pv,x) = v(t + 0, pv,x), 0 e (-ro,0],t > 0.

Clearly, for any t > 0, (ut(>) e C. Now, we define

yi = ui(t, Pu, x) - ui(t, fu, x), Zj = vj (t, Pv, x) - vj (t, fv, x).

Thus, we can obtain from system (1) that

=£ g^D - (p^ui(t, Pu,Pi(ui(t, fu,x)))

+ E (bjiflijt' fv *)) fv, x})))

{vj(t - Oji(t), Vv,x)) -jj(vj(t - Gji(t), tv,x))))

n /- ft

+ E( bji / kji(t - s)(fj(vj(s, Vv,x))-fj(vj(s, tv,x))) ) ds,

j=1\ J-œ

jj = X] ¿(^-¡¡xj)- (q/(v/(t, Vv,x))- qjj *v,x)))

[ui(t, Vu,x)) -gi(ui(t, tu,x))))

U^t - Tij(t), Vu, x)) -gi(ui(t - Tij(t), tu, x))))

+ E [din kij (t - s)(gi(ui(s, Vu, x))- gi(ui(s, tu, x.

We consider the following Lyapunov functional:

= J2wi

Jq i=i

+ J2 Ibji lrnr(1 - /) fjj, Vv,x)) -jjj, tv,x)) |r dÇ

j=1 Jt-eji (t)

n />+œ /> t

+ £ lbjilrnr j kji(s) e2a(s+) j=1 JO Jt-s

X |/j(v/(f, Vv,x)) -fj, tv,x)) |rdÇ ds

+ f ¿w

Jq j=1

mjV/e2^

m />t + E |ijij|rmr (1- / ) l&(ui(Ç, Vu, x))- gi(ui(Ç, tu, x)) |rdÇ ds

i=1 A-Tj«

m f+œ _ f t

+ £ |dij|rmrYj kj(s) e2a(s+Ç) i=1 O t-s

X ^(«ité,Vu,x)) -gi(u,i(Ç, tu,x))|rdÇds

By a minor modification of the proof of Theorem 1, we can easily get

Il u(t, Vu, x)-u(t, tu, x) I + I v(t, Vv, x) - v(t, tv < Pe-2at(IlVu - tu y + llVv - tvII)

for t > 0,in which p > 1 is a constant. Now, we can choose a positive integer N such that pe-aNm < 1, f e-aNm < 1. (7)

Defining a Poincare mapping P: C ^ C by

pM = ("ff), ()

due to the periodicity of system, we have

pNÎV") = ( "N- № "A. (()

W(/ VVNffl^v)/'

Let i = N-, then from (26)-(29) we can derive that

^/M _p^/M < 1 /M _/M W/ < 2 VVfv)

which shows that PN is a contraction mapping. Therefore, there exists a unique fixed point d") e Cnamdy ^$) = Ç").

Since PN (P(pJ})) = P(PN (vi")) = P0), then pffi is also a fixed point of PN. Because of the uniqueness of a fixed point of PN, then Pi^u) = C").

U ' ^ ' f » ' / / , v / o \ <?VV/

m, (p*, x), v(t + m, p*, x)) is also a solution of system (1). Clearly,

/Ut+«(p*, x)\ = / Ut (ua(p*))\ = x)\

Vt+4 p*v, x)) \ vt(va(p*))/ V Vt(pt, x)J

for t > 0. Hence (u(t + m, p*,x), v(t + m, p*,x))T = (u(t, p*,x), v(t, p*,x))T for t > 0.

This shows that (u(t, pU,x), v(t, p*,x))T is exactly one m-periodic solution of system (1), and it is easy to see that all other solutions of system (1) converge exponentially to it as t ^ The proof is completed. □

5 Illustration example

In this section, a numerical example is given to illustrate the effectiveness of the obtained results.

Example 1 Consider the following system on ^ = {(x1,x2)T|0 <xk < v02n, k = 1,2} c R2:

%=£ ¿D S) - "(u^)+: m» M»

n n r t

+ £ (bj»At - °n(t), x))) + £ ~bH kji(t - sjvj (s, x)) ds + Ii(t),

j=1 j=1

d l d / d \ m d»=l ix-\Dik ¿)- ^j(»(t,+£ (^iHix)))

m m n t

+ £{dijgi{ui(t - j),x))) + £ ddij / kij(t - s)gi(ui(s,x)) ds + /,-(t),

• i '_i «/-TO

Figure 1 The surface of u1(x1, 0.5018, t) when x2 = 0.5018.

Hi = 0, Vj = 0, t > 0, x e 3Q,

Ui(s,x) = Vui(s,x), Vj(s,x) = pvj(s,x), (s,x) e (-œ,0] x Q,

where kji(t) = kij(t) = te-t, i,j,l = 1,2. f1(n) = f2(r,) = fM = Mn) = fM = fi(ri) = gM = gi(ri) =gi(n)=g2(n)=gi(n) =g2(n) = tanh(n), n = m = l = 2,ki = 2.5, 0ji(t) = rv(t) = 0.02 -0.01 sin(2^t), Lf = Lf = I* = Lg = Lg = Lg = 1, i,j = 1,2. Pi(Hi(t,x)) = Hi(t,x), qj(Vj(t,x)) = 2vj(t,x), D1= D2 = 1, D{ = D2 = 2, a1 = a2 = 1, c1 = c2 = 2, r = 2, = ¡ie = 0.2, d11 = 0.5, d12 = 1, d21 = 0.5, d22 = 0.2, du = -0.1, d12 = 0.2, d21 = 0.3, d22 = 0.5, du = 0.2, d12 = 0.6, d21 = 0.5, d22 = 0.8, bn = 0.3, bu = 0.6, b^ = -0.5, b22 = -0.8, bn = -1, bu = 0.5, fa = 0.3, b22 = 0.3, b11 = -0.1, b12 = 0.2, b21 = 0.5, b22 = 0.4. By simple calculation with w1 = w2 =

W3 = W4 = 1, ßll = ßl2 = ß2l = ß22 = 1 and yii = Yl2 = Y21 = Y22 = 1, we have

- rna^Dlkl - rnal + 2(r -1) £ al + (r -1) £ alß;-

j=l j=l

m' i //•> ) , 1 r , lf..lrY rtlg\

+ ¿m^|dy|r(Lg)r + |dlj|r —L-(L^)r + |dlj|rYlr(Lf)^ = -1.15 < O, (31)

22 - rnar2-1D2kl - rna2 + 2(r -1) £ a2 + (r -1) £ a2ßj2r

j=l j=l

l^-.lr(t-g)r, _1_tàr, Yr'

+ ¿m^|d2j|r(Lf)r + |d2j|r(4)" + \d2j|rY2j(L2= -2.38 < O, (32)

- rm^Dfri - rmcri + 2(r - !) c[ + (r -VJ2 cYiT-T

i=i i=l

+ E\bii\r(Lfi)r + \bii\ri--f + \bii\rpiiiL)^ = -19.1 < 0 (33)

- rmcr2-1D2Xi - rmcr2 + 2(r -1) ^2 + (r -1) J2 cr2Ya^

i=i i=i

+ E\b2i\r{LJ2)r + \b2i\r L)r + m'-fiiL)r) =-20.98 <0, (34) that is, (6) holds.

The simulation results are shown in Figures 1-8. When x2 = 0.5018, the states surfaces of m(x1,0.5018, t) are shown in Figures 1-2, while x1 = 0.5018, the states surfaces of «(0.5018,x2, t) are shown in Figures 3-4. When x2 = 0.5018, the states surfaces of v(xi,0.5018, t) are shown in Figures 5-6, while xi = 0.5018, the states surfaces of v(0.5018,x2, t) are shown in Figures 7-8, which illustrates that the system states in (30) converge to equilibrium solution. Therefore, it follows from Theorem 1 and the simulation study that (30) has one unique equilibrium solution which is globally exponentially stable.

Remark 4 Since -a + 2 E^L)2 (| b/112 + |b/1|2) = 0.25 > 0, the conditions of Corollary 3.2 in [22] and -ra.1 + (r - 1)E'=1 L (|j + |bA|) + ZiEj=1(|dy| + \dv |) = 3.3 > 0, under the conditions of Example 1, the conditions of Theorem 1 in [29] are not satisfied. However, by (31)-(34) and Theorem 1, we can derive that (30) has one unique equilibrium solution which is globally exponentially stable.

6 Conclusions

In this paper, by employing suitable Lyapunovfunctionals, Young's inequality and Holder's inequality techniques, global exponential stability criteria of the equilibrium point and periodic solutions for RDNNs with mixed time delays and Dirichlet boundary conditions have been derived, respectively. The derived criteria contain and extend some previous NNs in the literature. Hence, our results have an important significance in design as well as in applications of periodic oscillatory NNs with mixed time delays. An example has been given to show the effectiveness of the obtained results.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

WZ designed and performed all the steps of proof in this research and also wrote the paper.JL and MC participated in the design of the study and suggested many good ideas that made this paper possible and helped to draft the first manuscript. Allauthors read and approved the finalmanuscript.

Author details

1 Institute of Mathematics and Applied Mathematics, Xianyang NormalUniversity, Xianyang, 712000, China. 2Schoolof Science, Xidian University, Xi'an, Shaanxi 710071, China.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work is partially supported by the NationalNaturalScience Foundation of China under Grant No. 60974139, the SpecialResearch Project in Shaanxi Province Department of Education (2013JK0578) and Doctor Introduced project of Xianyang Normal University under Grant No. 12XSYK008.

Received: 18 October 2012 Accepted: 12 April 2013 Published: 26 April 2013 References

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doi:10.1186/1687-2770-2013-105

Cite this article as: Zhang et al.: Global exponential stability and existence of periodic solutions for delayed reaction-diffusion BAM neural networks with Dirichlet boundary conditions. BoundaryValue Problems 2013 2013:105.

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