Open Journal of Applied Sciences, 2013, 3, 49-52 doi:10.4236/ojapps.2013.31B1010 Published Online April 2013 (http://www.scirp.org/journal/ojapps)
Periodic Solution for Neutral Type Neural Networks Wenxiang Zhang, Yan Yan, Zhanji Gui, Kaihua Wang* School of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan Email: *
[email protected] Received 2013
ABSTRACT The principle aim of this paper is to explore the existence of periodic solution of neural networks model with neutral delay. Sufficient and realistic conditions are obtained by means of an abstract continuous theorem of k-set contractive operator and some analysis technique. Keywords: Neutral-type Neural Networks; k-Set Contractive Operator; Periodic Solution
1. Introduction Man-made neural networks have been widely used in the fields of pattern recognition, image processing, association, optimal computation, and others. However, owing to the unavoidable finite switching speed of amplifiers, time delays in the electronic implementations of analog neural networks are inevitable, which may cause undesirable dynamic network behaviors such as oscillation and instability. Thus, it is very important to investigate the dynamics of delay neural networks. The existence of periodic oscillatory solutions of neural networks model has been studied by many researchers [1-6]. Some authors [3-5] used the well-known Hopf bifurcation theory to discuss the bifurcating periodic solutions. However, the usual Hopf bifurcation theory cannot be applied to non-autonomous system. In [6], Li and Lu applied the theory of coincidence degree to non-autonomous neural networks system and obtained some new criteria for the existence of periodic solutions. We consider the following model for neutral type neural networks with periodic coefficient: n
xi (t ) ai (t ) xi (t ) bij (t ) g j ( x j (t ij (t ))) I i (t ), (1) j 1
where n is the number of neurons in the network, xi (t ) ( i 1, 2, , n ) corresponds to the state of the ith unit at time t, ai (t ) denotes the neuron firing rate, bij (t ) represents the neutral delayed connection weight. g ( x) ( g1 ( x1 ), g 2 ( x2 ), , g n ( xn )) : R R T
n
n
is the activation function. I (t ) ( I1 (t ), I 2 (t ), , I n (t ))T is the -periodic external input to the ith neuron. Throughout this paper, we assume that: *
Corresponding author.
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(H1) Functions g j (u ) ( j 1, 2, , n) are globally Lipschitz continuous with the Lipschitz constant L j 0 , that is, g j (u1 ) g j (u2 ) L j u1 u2 , for all u1 , u2 R . (H2) Functions ij (t ) (i, j 1, 2, , n) are nonnegative, bounded and continuously differentiable defined on R and ij 1 , where ij (t ) express the derivative of ij (t ) . (H3) There exist constant j 0 , j 0 , such that x R , g j ( x) j x j , j 1, 2, , n . Remark 1.1 It is easy to verify that if condition (H1) is satisfied then condition (H3) holds. The organization of this paper is as follows. Preliminaries will be given in the next section. In section 3, we will study the existence of periodic solutions of system (1.1) by the abstract continuous theorem of k-set contractive operator.
2. Preliminaries In order to study Equation (1.1), we should make some preparations. Let E be a Banach space. For a bounded subset A E , let
E ( A) inf{ 0 | there is a finite number of subsets Ai A,such that A i 1 Ai , and diam( Ai ) }
denotes the (Kuratoskii) measure of noncompactness, where diam( Ai ) denotes the diameter of set Ai . Let X, Y be two Banach spaces and be a bounded open subset of X. A continuous and bounded map N : Y is called k-set contractive if for any bounded set A we have Y ( N ( A)) k X ( A) , where k is a constant. In addition, for a Fredholm operator L : X Y with index zero, according to [7], we may define OJAppS
W. X. ZHANG
50
l ( L) sup{r 0 | r X ( A) Y ( L( A)), for all
ET AL.
sets Aij of Ai such that Ai j 1 Aij with | Vi (t , xi (t ), u1 (t i1 (t )), , un (t in (t ))) Vi (t , ui (t ), u1 (t i1 (t )), , un (t in (t ))) | ,
bounded subset A X }. Lemma 2.1 Let L : X Y be a Fredholm operator with index zero, and a Y be a fixed point. Suppose that N : Y is a k-set contractive with k l ( L) , where X is bounded, open, and symmetric about 0 . Furthermore, we also assume that 1) Lx Nx r , for x , (0,1) ; 2) [QN ( x ) Qr , x ] [QN ( x ) Qr , x ] 0, for x ker L , where [, ] is a bilinear form on Y X and Q is the project of Y onto Co ker( L) . Then there is a x , such that Lx Nx r . In order to use Lemma (2.1) for studying Equation (1.1), we set
for any x, u Aij . Therefore we have Nx Nu 0 sup | Vi (t , xi (t ), x1 (t i1 (t )), , xn (t in (t ))) t[0, ]
Vi (t , ui (t ), u1 (t i1 (t )), , un (t in (t ))) | sup | Vi (t , xi (t ), x1 (t i1 (t )), , xn (t in (t ))) t[0, ]
Vi (t , xi (t ), u1 (t i1 (t )), , un (t in (t ))) | sup | Vi (t , xi (t ), u1 (t i1 (t )), , un (t in (t ))) t[0, ]
Vi (t , ui (t ), u1 (t i1 (t )), , un (t in (t ))) |
Y C {x(t ) ( x1 (t ), , xn (t )) C ( R, R ) : n
n
xi (t ) xi (t ), i 1, , n},
bij g j ( x j (t ij (t ))) g j (u j (t ij (t ))) j 1
with the norm defined by x 0 i 1 max t[0, ] xi (t ) , n
n
bij L j x j (t ij (t )) u j (t ij (t ))
and
j 1
X C1 {x(t ) ( x1 (t ), , xn (t )) C1 ( R, R n ) : xi (t ) xi (t ), i 1, , n},
with the norm x max{ x 0 , x 0 } .Then C , C are all 1
Banach spaces. Let L : C C defined by 1
Lx
dx dt
( x1 , , xn )T ; N : C1 C ,
n x1 a1 (t ) x1 (t ) j 1 b1 j (t ) g j ( x j (t 1 j (t ))) (2.1) N n xn a (t ) x (t ) j 1 bnj (t ) g j ( x j (t nj (t ))) n n
It is easy to see from [8] that L is a Fredholm operator with index zero. Clearly, Equation (1.1) has a -periodic solution if and only if Lx Nx r for some x C1 , where r : I (t ) ( I1 (t ), , I n (t ))T . Lemma 2.2 [9] The differential operator L is a Fredholm operator with index zero, and satisfies l ( L) 1 . n Lemma 2.3 If k max{ j 1 bij L j , i 1, , n} 1 , here bij maxt[0,] bij (t ) , N : C is a k-contractive map. Proof. Let A be a bounded subset and let C1 ( A) . Then, for any 0 , there is a finite family of
subsets Ai satisfying A i 1 Ai with diam( Ai ) . Now let n
Vi (t , xi , y1 , , yn ) ai (t ) xi g j ( y j ). j 1
Since Vi (t , xi , y1 , , yn ) are uniformly continuous on any compact subset of R R n 1 , A and Ai are precompact in C0 , it follows that there is a finite family of subCopyright © 2013 SciRes.
k (k 1) .
As is arbitrary small, it is easy to see that C ( N ( A)) k C ( A) . Lemma 2.4 Let C0 , C1 , and (t ) 1 , then 0 (v(t )) C , where v(t ) is the inverse function of t (t ) . Throughout this paper, we assume that ij C1 , ij (t ) 1 (i, j 1, , n) . So that ij (t ) has a unique inverse, and we set vij (t ) to represent the inverse of function t ij (t ). . Meanwhile, we denote 0
1
h : (1/ ) h( s )ds and h 2 : 0
0
1/ 2
h( s ) ds
,
3. Main Results Set 1 : t R , pij2 max 1 ij (vij (t )) A diag(a1l , a2l , , anl ), B (bij ) n n , bij bijm j pij , C diag(a1m , a2m , , anm ), H (hij ) n n , n
hi bijm j I i 2 . j 1
Theorem 3.1 Assume that (H1), (H2) hold, furthermore, assume that (H4) A BC is non-singular M-matrix, then Equation (1.1) has at least one positive periodic solution. Proof. We consider the operator Equation Lx Nx r , (0,1)
(3.1) OJAppS
W. X. ZHANG
Corresponding to Equation (3.1), we have x i ( t ) a i ( t ) x i ( t )
ET AL.
That is,
b i j ( t ) g j ( x j ( t i j ( t ) ) ) I i ( t )
n
j 1
51
x i
(3.2)
2
a im x i
a
l i
xi
2
2
n x i ( t ) bij ( t ) g j ( x j ( t ij ( t ))) I i ( t ) dt j 1
0
0
x (t b x ( t ) dt x ( t ) dt I ( t ) dt . n
b j 1
m ij
j
n
0
1/ 2
2
j
j
0
ij
( t )) dt
1/ 2
i
hi , i 1, , n,
(3.8)
2
hi , i 1, , n.
(3.9)
(3.10)
x j ( s )
Let { x C : x 0 N 1 , x 0 N 2 }, and define a bounded bilinear form [, ] on C C1 by
2
ds
1 ij ( v ij ( s ))
0
[ y, x] y (t ) x(t )dt . 0
.
2
Also we define Q : y Co ker( L) by y y (t )dt . 0
ail xi
n
n
bijm j pij x j
2
2
j 1
bijm j I i j 1
(3.3)
2
Then we may rewrite Equation (3.3) as AX BY H ,
(3.4)
where X ( x1 2 , , xn 2 ) , Y ( x1 2 , , xn 2 ) . T
T
Multiplying the ith Equation of system (3.2) by xi and integrating over [0, ] gives 2
2
x 0 N1 , x 0 N 2 .
Thus
2
xi
1
p ij x j
x i
(3.7)
It follows from Lemma 2.2 of paper [10] that X H (h1 , , hn )T . That is
2
x j ( t ij ( t )) d t
0
(3.6)
It is easy to see that there exist two positive constants N i (i 1, 2) such that
And according to Lemma 2.4, we have
2
Substituting (3.6) into (3.4), we obtain
xi
i
0
Ii
j
(3.5)
2
Y CX H .
1/ 2
2
b ijm
b ijm j p ij x j
where H (h1 , , hn )T ( A BC ) 1 ( BH H ) Substituting (3.8) into (3.6), we obtain
1/ 2
2
j 1
Then we may rewrite Equation (3.5) as
1/ 2
2
i
0
2
0
x i ( t ) dt
m ij
j 1
n
( A BC ) X BH H .
n n x i ( t ) bijm j x j ( t ij ( t )) bijm j I i ( t ) d t j 1 j 1
j 1
T
Suppose that ( x1 (t ), , xn (t )) is a solution of system (3.2) for a parameter (0,1) . Multiplying the ith Equation of system (3.2) by xi and integrating over [0, ] gives
n
2
a i (t ) xi (t ) d t , x i ( t ) n b ij ( t ) g j ( x j ( t ij ( t ))) I i ( t ) j 1
0
0
n m m a i x i ( t ) b ij j 1 x i ( t ) n m b ij j I i ( t ) j 1
x (t ) d t b x ( t ) d t x ( t b x ( t ) d t x ( t ) d t I ( t ) d t . a im n
j 1 n
j 1
2
1/ 2
j
0
0
2
0
j
0
2
1/ 2
i
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2
j
1/ 2
i
0
2
i
1/ 2
[ a1 N 1 n b1 j g j (0) I 1 ][ a1 N 1 n b1 j g j ( 0) I 1 ] j 1 j 1 n n [ a n N 1 j 1 bnj g j (0) I n ][ a n N 1 j 1 bnj g j ( 0) I n ] N 1 m ax 1 i n
n
b j 1 ij ai
j
I i 0
n
n
j 1
j 1
, then
(3.12)
So from (3.12) we get
i
[ QN ( x ) Qr , x ] [ QN ( x ) Qr , x ] 2 N 1 2
ai N1 bij j I i bij g j (0) I i
1/ 2
2
Without loss of generality, we may assume that x N1 . Thus
If
i
1/ 2
m ij
dt
m ij
0
x i ( t )
0
{x | x ker L } {x | x N1 , or x N1} .
(3.11)
x j ( t ij ( t )) d t ,
j
Obviously,
ij
( t ))
2
1/ 2
dt
[QN ( xi ) Q(ri ), xi ] [QN ( xi ) Q(ri ), xi ] 0, i 1, , n
Therefore, by using Lemma 2.1, we obtain that Equation (1.1) has at least one positive -periodic solution. The proof is complete. OJAppS
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ET AL.
4. Acknowledgements This work is supported by the Natural Sciences Foundation of China under Grant No. 60963025.
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