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
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 (H1) Functions ()
j
g
u(1,2,,j)n are globally
Lipschitz continuous with the Lipschitz constant 0
j
L,
that is,
Man-made neural networks have been widely used in the
fields of pattern recognition, image processing, associa-
tion, 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 insta-
bility. Thus, it is very important to investigate the dy-
namics of delay neural networks.
g
12
()( )
jj 12j
uguLuu
, for all
12
,uu R.

(H2) Functions ()
ij t

are nonnega-
tive, bounded and continuously differentiable defined on
R
(,1,2,, )ij n

and 1
ij


, where ()
ij t

 express the derivative of
()
ij t

.
(H3) There exist constant 0
j

, 0
j

, such
that
x
R

, ()
j
jj
gx x


, 1, 2,,.
jn
The existence of periodic oscillatory solutions of neu-
ral 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 so-
lutions. 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.
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. Prelimi-
naries 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 contrac-
tive operator.
2. Preliminaries
We consider the following model for neutral type
neural networks with periodic coefficient:
1
()() ()()((()))(),
n
iiiijjjiji
j
x
ta tx tbtgxttIt 

 


(1)
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
1
()inf{0|there is a finite number of
subsets,such that ,
and diam()}
E
ii
i
A
where n is the number of neurons in the network, ()
i
x
t
() corresponds to the state of the ith unit at
time t, denotes the neuron firing rate, repre-
sents the neutral delayed connection weight.
1, 2,,i
()
i
at
i
A
AA
A
A








n
()
ij
bt
denotes the (Kuratoskii) measure of noncompactness,
where diam( )
i
A
denotes the diameter of set i
A
. Let X,
Y be two Banach spaces and be a bounded open
subset of X. A continuous and bounded map

:
112 2
() ((),(), ,())
T
nn
gxg xgxgx:
nn
RR
is the activation function. 12 ()((),(),,())
T
n
I
tItIt Itis
the

-periodic external input to the neuron. thi
NY
is called k-set contractive if for any bounded set
A

we have X
((NA)) ()
Y
k A



, where k is a constant. In
addition, for a Fredholm operator with in-
dex zero, according to [7], we may define
:LXY
Throughout this paper, we assume that:
*Corresponding author.

W. X. ZHANG ET AL.
50
()sup{0|()(()), for all
bounded subset }.
XY
lLrr ALA
AX
  

Lemma 2.1 Let be a Fredholm operator
with index zero, and be a fixed point. Suppose
that is a k-set contractive with
:LX Y
aY
:NY()kl
L

,
where
X
 is bounded, open, and symmetric about
. Furthermore, we also assume that
0
1) LxNx r

 , for x, (0,1)

;
2) [( ,]0, for ) ,QNxQr x][QN Q
ker ,xL ]
()xrx
where [, is a bilinear form on
YX

and Q is the project of Y onto .
Co ker()L
Then there is a x, such that . Lx Nxr
In order to use Lemma (2.1) for studying Equation
(1.1), we set
1
{ ()((),,())(,):
()(),1,,},
n
n
ii
YCxtxtxt CRR
xtxt in


 
 


with the norm defined by [0, ]
1 0max( )
n
ti
i
x
xt
 

,
and
11
1
{ ()((),,())(,):
()(),1,,},
n
n
ii
XCxtxtxtCRR
xtxt in


 
 


with the norm 00 max{ , }
x
xx. Th en 1 ,CC


are all
Banach spaces. Let 1
:LC C


 defined by
1
(, , )
T
dx
n
dt
Lxx x
 ; 1 :NC C


,
11 11
1
1
1
() ()()((()))
() ()()( (()))
n
jjjj
j
n
nnnnjj jnj
j
atxtbtgx tt
x
N
xatxtb tg xtt





 




  











(2.1)
It is easy to see from [8] that L is a Fredholm operator
with index zero. Clearly, Equation (1.1) has a

-peri-
odic solution if and only if for some
LxNx r
1
x
C

, where .
1
Lemma 2.2 [9] The differential operator L is a Fred-
holm operator with index zero, and satisfies .
:()((),rIt It , ())
T
n
It
() 1lL 
Lemma 2.3 If ,
here
1
max{,1,, }1
n
ij j
j
kbLi



n
[0,]
max
ij t
bb



C( )
ij t, :N


 is a k-contractive
map.
Proof. Let
A
 be a bounded subset and let


1()
C
A


. Then, for any 0

, there is a finite family of
subsets Ai satisfying 1ii
A
A

 with diam( )
i
A



.
Now let
1
1
(, ,,,)()().
n
iin iijj
j
Vtx yyatxgy




Since 1 are uniformly continuous on
any compact subset of R
(, ,,,)
ii n
Vtx yy
1n
R

, A and i
A
are precom-
pact in 0
C

, it follows that there is a finite family of sub-
sets ij
A
of i
A
such that i1jij
A
A


( )),,(
()), , (
i n
i n
tu t
tu t

with
11
11
(),(( )))
(,(()))|,
in
in
Vtt utt
Vt utt
|,(
, ()
ii
ii
x
ut









for any ,ij
x
uA. Therefore we have

0
[0,]
11
[0,]
11
sup|(, (), ((()))
(,(),(()))|
sup|(,(),((()))
(,(),(
iin in
t
ii in
iin in
t
ii in
Nx Nu
Vtxt xtt
Vtut utt
Vtxt xtt
Vtxt ut












11
11
()), ,
()), , (
()), ,
()), , (
i
i n
i
i n
t tx
tu t
t tx
tu t 










11
(()),,
()), , (
)))( (
))((
i
i n
j j
j ij
t t
tu t
t gut
u tt

[0, ]
11
1
1
()))|
sup|(,(),(()))
(,(),(()))|
(((( )))
(( ))
(
iin in
t
ii in
n
ijjjijij
j
n
ij jjij
j
t
Vtxt uu tt
Vtut utt
bgxt t
bL x tt
kk













 




1) .  










As

is arbitrary small, it is easy to see that
01
C

((
C
NA)) ()kA



.
0 1
C

 Lemma 2.4 Let C, 

, and ()1t


, then
0
(())vt C


()tt
, where is the inverse function of ()vt


.
Throughout this paper, we assume that 1
ij C

 ,
() 1
ij t


 (,1,, )ij n

. So that (
ij t)

has a unique
inverse, and we set ij to represent the inverse of
function
()tv
().
ij
tt


. Meanwhile, we denote
0
:(1/ )()hhsds

  and

1/ 2
20
:()hhsds

,
3. Main Results
Set
21
max :
1(())
ij
ij ij
pt
vt 
R











,
12
12
2
1
diag(,,,), (), ,
diag(,,,), (),
.
ll lm
nijnnijijj
mmm
nijnn
n
m
iijji
j
ij
A
aaa B bbbp
CaaaHh
hb I












Theorem 3.1 Assume that (H1), (H2) hold, further-
more, assume that (H4)
A
BC

is non-singular M-ma-
trix, then Equation (1.1) has at least one positive

-
periodic solution.
Proof. We consider the operator Equation
, (0,1)LxNx r



 (3.1)

W. X. ZHANG ET AL. 51
Corresponding to Equation (3.1), we have That is,
1
()() ()()((()))()
n
iiiijjjij
j
i
x
ta tx tbtgxttI t 


 




22 2
1
2
1
n
mm
iii ijjijj
j
n
m
ij ji
j
x
axb px
bI









(3.5)
(3.2)
Suppose that 1 (( ),,( ))T
n
x
txt
(0,1)
is a solution of system
(3.2) for a parameter


i
. Multiplying the
Equation of system (3.2) by
thi
x
and integrating over
[0,]

gives
Then we may rewrite Equation (3.5) as
.YCXH

 (3.6)
Substituting (3.6) into (3.4), we obtain

2
2
1
0
11
0
1/2 1/2
2
2
00
1
()()( (()))()
()(())()
()(())
n
l
iiiij jjiji
j
nn
mm
iijjjij ijji
jj
n
m
ij jijij
j
axxtbtgxtt Itdt
x
tbxttb Itd
bxtdtxttdt
b




 








t














() .
A
BC XBHH

 (3.7)


It follows from Lemma 2.2 of paper [10] that
1
(, ,)
T
n
X
Hh h
 
 . That is
2, 1,,,
ii
x
hi n

 (3.8)
where 1
1
(,,) ()()
T
n
H
hh ABCBHH
 
 

Substituting (3.8) into (3.6), we obtain

1/2
2
0
1
1/2 1/2
22
00
()
()().
n
m
ij ji
j
ii
xt dt
xt dtIt dt






 2, 1,,.
ii
x
hi n

 (3.9)
It is easy to see that there exist two positive constants
(1,2
i
Ni )

such that
And according to Lemma 2.4, we have
1
00
, .
2
x
Nx N
 (3.10)
2
0
2
0
2
(())
()
1(())
.
jij
j
ij ij
ij j
x
ttd
xs ds
vs
px













t
Let 1
1
00
{: ,xC xNxN

 
2
}, and define a
bounded bilinear form [,]

 on 1 CC


 by
0
[,]()()yx ytxtdt

.
Thus
22
2
11
nn
lm m
iiijjijjijj i
jj
axb pxbI 




Also we define by .
Obviously,
:Coker(Qy L)dt
0()yyt


 (3.3)
11
{|ker} {|, or }
x
xL xxNxN

 .
Without loss of generality, we may assume that
1
x
N

. Thus
Then we may rewrite Equation (3.3) as
,
A
XBYH (3.4)
where 122 (,,)
T
n
Xx x, 122 (,,)
T
n
Yx x .
Multiplying the Equation of system (3.2) by thii
x

and integrating over [0, ]

gives

2
2
0
1
1
0
1
1/2 1/2
22
00
() ()
() ,
()((()))()
()(())
(),
()
()()
ii
n
ii
ijj jiji
j
n
mm
iiijjjij
j
in
m
ij ji
j
m
ii i
atxt
x
xt dt
btg xttIt
axtbxtt
x
tdt
bIt
a xtdtxtdt






























 



22
1
1111 1111
11
11
11
[() ,][(),]
[(0)][(
[(0)][(
nn
jj jj
jj
nn
nnjjnnnjj
jj
QNxQrxQNxQrx N
aNb gIaNb gI
aNbgIaNbgI




0)]
0)]
n


 









 





(3.11)
If 1
11
max 0
n
ij ji
j
in i
bI
Na










, then
1
11
(0)
nn
iijjiijj
jj
aNbIbgI 


i

,
(3.12)
So from (3.12) we get



1/2 1/2
2
2
00
1
1/2
2
0
1
1/2 1/2
22
00
()(())
()
()().
n
m
ij jijij
j
n
m
ij ji
j
ii
bxtdtxttdt
bxtdt
x tdtI tdt
















[()(),][() (),]0
iii iii
QNx QrxQNx Qrx

 
1, ,in


Therefore, by using Lemma 2.1, we obtain that Equa-
tion (1.1) has at least one positive

-periodic solution.
The proof is complete.

W. X. ZHANG ET AL.
52
4. Acknowledgements
This work is supported by the Natural Sciences Founda-
tion of China under Grant No. 60963025.
REFERENCES
[1] X. Y. Liu and J. D. Cao, “On Periodic Solution of Neural
Networks via Differential Inclusions,” Neural Networks,
Vol. 22, No. 4, 2009，pp. 329-334.
doi：10.1016/j.neunet.2008.11.003
[2] J. Zhang and Z. J. Gui, “Periodic Solutions of
Nonautonomous Cellular Neural Networks with Impulses
and Delays,” Nonlinear Analysis: Real World Applica-
tions, Vol. 10, No. 3, 2009，pp. 1891-1903.
doi：10.1016/j.nonrwa.2008.02.029
[3] K. Gopalsamy and I. Leung, “Delay Induced Periodicity
in a Neural Netlet of Excitation and Inhibition,” Physica
D: Nonlinear Phenomena, Vol. 89, No. 3-4, 1996，pp.
395-426. doi:10.1016/0167-2789(95)00203-0
[4] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation
and Multistability in a System of Two Coupled Neurons
with Multiple Delays,” SIAM Journal on Applied Math-
ematics, Vol. 61, No. 2, 2000，pp. 673-700.
doi:0.1137/S0036139998344015
[5] J. Wu, T. Faria and Y. S. Huang, “Synchronization and
Stable Phase-locking in a Network of Neurons with
Memory,” Mathematical and Computer Modeling, Vol.
30, No. 1-2, 1999，pp. 117-138.
doi:0.1016/S0895-7177(99)00120-X
[6] Y. Li and L. Lu, “Global Exponential Stability and Exis-
tence of Periodic Solution of Hopfield- type Neural Net-
works with Impulses,” Physics Letters A, Vol. 333, No.
1-2, 2004，pp. 62-71. doi:0.1016/j.physleta.2004.09.083
[7] Z. J. Gui, S. J. Lu and W. G. Ge, “Existence of Periodic
Solution of Two Competition Dynamics System with
Neutral Delay and Several Deviating Arguments,” Chi-
nese Journal of Engrg. Mathematics, Vol. 22, 2005，pp.
703-711.
[8] R. E. Gaines and J. L. Mawhin, Coincidence Degree and
Nonlinear Differential Equations, Springer-Verlag, Berlin,
1977.
[9] Z. Liu and Y. Mao, “Existence Theorem for Periodic
Solutions of Higher Order Nonlinear Differential Equa-
tions,” Journal of Mathematical Analysis and Appications,
Vol. 216, No. 2, 1997，pp. 481–490.
doi:.1006/jmaa.1997.5669
[10] Z. Liu and L. Liao, “Existence and Global Expo-
nential Stability of Periodic Solution of Cellular
Neural Networks with Time-varying Delays,”
Journal of Mathematical Analysis and Applications,
Vol. 290, No. 1, 2004，pp. 247–262.
doi:10.1016/j.jmaa.2003.09.052