Open Journal of Applied Sciences, 2013, 3, 62-64
doi:10.4236/ojapps.2013.31B1012 Published Online April 2013 (http://www.scirp.org/journal/ojapps)
Periodic Solution of Impulsive Lotka-Volterra Recurrent
Neural Networks with Delays
Yan Yan, Kaihua Wang, Zhanji Gui*
School of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan, China
Received 2013
ABSTRACT
In this paper, periodic solution of impulsive Lotka-Volterra recurrent neural networks with delays is studied. Using the
continuation theorem of coincidence degree theory and analysis techniques, we establish criteria for the existence of
periodic solution of impulsive Lotka-Volterra recurrent neural networks with delays.
Keywords: Lotka-Volterra; Delays; Periodic Solution; Impulsive
1. Introduction
In recent years, applications of theory differential equa-
tions in mathematical ecology have been developed rap-
idly. Various mathematical models have been proposed
in the study of population dynamics. The Lotka-Volterra
competition system is the most famous models for dy-
namics of population. Owing to its theoretical and prac-
tical significance, the Lotka-Volterra systems have been
studied extensively [1,2]. The Lotka-Volterra type neural
networks, derived from conventional membrane dynam-
ics of competing neurons, provide a mathematical basis
for understanding neural selection mechanisms. Recently,
periodic solutions of impulsive Lotka-Volterra recurrent
neural networks have be en reported.
It is well known that delays are important phenomenon
in neural networks [3]. Thus, studying the dynamic prop-
erties of neural networks with delays has interesting im-
plications in both theory and applications [4-7]. In this
paper, we will study the following impulsive Lotka-Volterra
recurrent neural networks system with delays:
1
()()[ ()()()()
( )(( ))],,
()(),1,2,,1,2, ,,
n
iiii ijj
j
ii ik
ik ikik
xtxthtxta txt
ctxtttt
x
tbxti nkn







 





(1)
where each ()
i
x
t denotes the activity of neuron, ()


ijn n
Aa
is real matrices, each of their elements denotes a
synaptic weight and represents the strength of the synaptic
connection from neuron j to neuron i, denotes ex-
ternal inputs. The variable delays
nn
()
i
ht
()
it

for
are nonnegative continuous functions satisfying
1, 2,,in
0()


it


for , where
0t0

is a constant. i, ,
i are all positive periodic continuous functions with
period .
()ht ()
ij
at
(ct)T0

2. Existence of Positive Periodic Solutions
Lemma 1 [8] Let X and Y be two Banach spaces. Con-
sider an operator equation Lx Nx

where :DomL

LX
[0,1] Y is a Fredholm operator of index zero and


is a parameter. Let P and Q denote two pro-
jectors such that and QY .
Assume that :PX KerL:/ImY L
:NY is L-compact on

, where

is open bounded in X. Furthermore, assume that
(a) For each (0,1),


Dom ,
x
L ,

Lx Nx

(b) For each Ker

 
x
L, , 0QNx
(c) deg{,,0} 0Ker

JQN
K
L, where
:Imer
J
QL is an isomorphism and repre-
sents the Brouwer degree. deg{*}
Then the equation

LxNx has at least one solution in
Dom L.
For the sake of convenience, we introduce the follow-
ing notation:
0[0, ]
[0, ]
1
1
1(), m
), (
:
with res
and (


in (),
max(1,2,,),
() is contin
pect to,...,)
(,) ) exsitat,...,
and()(),1,2,...,




uous
; (
;

,



















T
ii
tT
u
ii
tT
kk
uu gt
T
ggi n
Rxt
tt
PC JRt
xtxt kp
l
tdtg
t
xJ
xt
p
p
txt
t
*Corresponding author.

Y. YAN ET AL. 63
()ut , ()
g
t
()zt
where are T-perio dic fu n ctions.
is an T-periodic solution of (1) if and
only if an T -periodic solution of
Lemma 2i
( )}zt is ln{ i
1
()()exp{ ()}()exp{()}
( )exp{(( ))},,
() ln(1),1,2,,1,2, ,.
n
iiiij j
j
iii k
ik ik
zt htztatzt
ctztttt
x
tbink


 




 




 n
(2)
n
resent pa
Theorem Assume that
where 12 ((ln{()},ln{( )},,ln{
i
zt ztzln{)}( )})ztt
e reaNow we ardy to state and prove the main results
of the ppe r. 1


ij i
ac
ution. , then system (1) has at least one T-periodic sol
Proof. To complete the proof, we only need to search
for an appropriate open bounded subset verifying
requirements in Lemma 1.
Let
all the
12
((),(),,()),
{(,)| ()()

 
T
n
n
zztzt zt
2
},
, p
Z
zPCRRztT zt
YZR
that both Z
e norms
en it is standard to show th and Y are Banach
space when they are endowed with th
[0,]
||||sup|()|


ctT
zzt
and
12
22 2
11
|| (,,,) ||(||||||||).
pp p
zc czcc
Set Y as
j
prove tL is a Fredholm mapping of
inero.
Consider the operator equation
:DomLL
1
()()((),(),,()),

 p Lztztztzt
where Dom{|()(,)}.

 n LZzZztPCRR
At the same time, we denote :NZ Y as
1
1
()()((()exp{ ()}()exp{()}
()exp{(())}),(, ))

 



n
iiij
j
iii p
Nzth tztatzt
ctzttI I 
It is easily to hat
dex z
(0,1).Lz Nz

 (3)
al , we obtain Integra t i ng (3) ov er the interv[0,]T
0
1
01
ij j j
T
(4)
0()exp{
(1,2,,




ii
ct z
in
ln(1)exp{( )}
()exp{ ()}
(( ))},
).

 




pT
iiki
k
n
T
i
hTbz tdt
at ztdt
tt dt 
Then, we can deriv e
01
|()|2 ln(11,2,,).
 


p
Ti
ii
zt hTbn ),(


k
k
i
Since there exist ( )([0,],),n
i
ztPCT R,
i

[0, ]

iT

2,
12
[,

tt
(
, ,],

p t such that
)inf(), ()
[0, ][0, ]
sup(), (
1,,).


iii iii
tT tT
zinzztz t

For (4) we can see
0
1ln(1
iik
k
hT b


0
0
0
1
0
1
)exp{( )}
()exp{ ()}
()exp{ (
exp{()}
xp{ ()}
()exp{ ()}
ln(1)(1)exp{
pT
i
T
ii i
T
ii
p
i i
k
T
i i
iii
p
ii
ik i
k
z tdt
at ztdt
ct ztd
zdt
z dt
ctz dt
bac z











 





()},
ii
T 
which implies
0()
e
ii
T
at

())}
ln(1 )
i
T
ik
t t
b

 
1
1ln(1)
() ln:
1






 



p
iik
k
ii i
ii
hb
T
zA
ac

Thus, [0, ]

tT, we have
0
1
()( )ln(1)|()|2:.
pT
iiiik ii
k
ztzbztdt AhTM 


 

Similarly, according to (4), we have
0
1
01
0
0
1
0
()exp{}
n
T
ij
at Mdt
1
0
1
ln(1)exp{( )}
()exp{ }
())}
ln(1)exp{()}
()exp{()}
ln(1)








 









pT
iik i
k
n
T
ij
j
T
iii
pT
iki i
k
j
T
iii
p
ik
k
hTbz tdt
at Mdt
cttdt
bzdt
ctz dt
b



()
exp{ (tz
1()exp{ }
(1)exp{()},




n
ij
j
iii
at MT
czT


Y. YAN ET AL.
64
which implies, Software Technology (HrZD201101).
11
1ln(1)exp{ }
() ln1
:.











pn
iij
ik
kj
ii i
hba
T
zc
B






M
Thus, , we have
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22ln(1):.


  
 


pT
iiiiki
k
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iik
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zt zbztdt
BhTb N

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Now, we can derive
Obviously,
[4] Y. Zhang and K. T. Kok, “Global Convergence of
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Circuits and syetemsm, Vol. 52, 2005, pp. 2482-2489.
1
|()|max{||,||}:.
i
ztM NM
1
M
is independent of

. Then, there exists
such that Let a constant 0,Fmax{| |.
i
, then it is clear that
and N is
}zF
L-com
1,rM F
 satisfies
pact on
{ } r
condition Lemma1
:||zZ
(a)||c
z
of

. Let
:Im, ,,( ,00)
J
Qxd d, a odirect c
[5] J. Zhang and Z. J. Gui, “Periodic Solutions of
Nonautonomous Cellular Neural Networks with Impulses
and Delays,” Real World Applications, Vol. 1, No. 3,
2009, pp. 1891-1903.
doi:10.1016/j.nonrwa.2008.02.029
mes
By now we have proved that satisfies all the re-
qu nun theorem (Lemma
1), system (2) has at least one T-periodic solution
in
putation giv
deg{,,0} 0. JQN KerL [6] F. Y. Wei and S. H. Wang, “Almost Periodic Solution
and Global Stability for Cooperative L-V Diffusion Sys-
tem,” Journal of Mathematical Research and Exposition,
Vol. 30, 2010, pp. 1108-1116.

atioirements in Mawhin’s conti
. Hence
[7] X. Z. Meng and L. S. A. Chen, “Permanence and Global
Stability in an Impulsive Lotka-Volterra Species Com-
petitive System with both Discrete Delays and Continu-
ous Delays,” International Journal of Biomathematics,
Vol. 1, No. 2, 2008, pp. 179-196.
doi: 10.1142/S1793524508000151
12
()((), (),,())T
n
ztz tztztDom L. The
3. Acknowledgements
Th by the National Natural Science
Foo: 60963025); the Foundation of
the Office of Education of Hainan Province (No:
Hd f
proof
is completes.
is work is supported
undation of China (N[8] R. E. Gaines and J. L. Mawhin, “Coincidence Degree and
Nonlinear Deferential Equations,” Berlin: Springer Ver-
lag, 1977.
j2009-36) anthe Foundation oHainan College of