Journal of Applied Mathematics and Physics, 2013, 1, 1-5
http://dx.doi.org/10.4236/jamp.2013.14001 Published Online October 2013 (http://www.scirp.org/journal/jamp)
Existence of Periodic Solutions for Neutral-Type Neural
Networks with Delays on Time Scales
Zhenkun Huang, Jinxiang Cai
School of Science, Jimei University, Xiamen, China
Received June 2013
ABSTRACT
In this paper, we employ a fixed point theorem due to Krasnosel’skii to attain the existence of periodic solutions for
neutral-type neural networks with delays on a periodic time scale. Some new sufficient conditions are established to
show that there exists a unique periodic solution by the contraction mapping principle.
Keywords: Neutral-Type; Neural Networks; On Time Scales; Periodic Solution
1. Introduction
Recently, scholars and researchers have paid more atten-
tion to the discussion of neural networks described by
neutral-type differential equations with delays (see [1-6]).
Meanwhile, difference equations or discrete-time analo-
gues of differential equations can preserve the conver-
gence dynamics of their continuous time counterparts in
some degree [7]. Due to their usage in applications, these
discrete-type neural networks with or without delays
have been discussed by [8,9] and references therein. It is
interesting to study that neural systems on time scales
can unify the continuous and discrete situations. The
theory of time scales initiated by S. Hilger [10,11] has
been incorporated to investigate neural networks [12,13]
and so on.
However, few works have considered for neutral-type
neural networks on time scales [14,15 ]. In this paper, we
consider the existence of periodic solutions for the neu-
tral networks with delays
1
1
()()()() ()
()(())
(), :{1,2,}
n
iii ijj
j
n
ijj j
j
i
x
tatxtc txtk
btgxt k
It in
 


 





(1)
For a review of dynamic equations o-time scales, we
direct the reader to the monographs [9,10] and begin with
a few definitions.
Definition 1.1. A time scale  is p-periodic if there
exists 0p and p such that if t

 then
tp. For , the smallest positive p is called
the period of.
Definition 1.2. Let  be a p-periodic time scale.
f:

 is periodic with period  if there exists a
natural number n such ,() ()np f tTf t

 for all
t

 and

is the smallest number such that
()()
f
tT ft

.
Without other statements, let  be a p-periodic time
scale such that 0

. We will show the existence of
periodic solutions for (1) where ,.kmpm
2. Preliminaries
Theorem 2.1. ([10,11]) Assume :

 is strictly
increasing and :()



 is a time scale. Let
.  
： If


t 
 and
 
t 

 exist for
,
k
t

 then one has



.



 
Theorem 2.2. ([10,11]) Assume :

 is strictly
increasing and :()



 is a time scale. If : f


is a rd-continuous and

is differentiable with rd-con-
tinuous derivative , then one gets
 




1, ,.
bb
aa
ftt tfs sab









A function : p

 is said to be regressive pro-
vided




10tpt 

 for all .
k
t The set of all
regressive rd-continuous function : f is devote
by R while the set R

is given by




:10, t.RfRtpt 


 
Let pR

. The exponential function is defined by
()
(,)exp(( ))
t
ps
ets p








 (2)
where


hz
 is called cylinder transformation.

Z. K. HUANG, J. X. CAI
2
Lemma 2.1.
Let ,pqR. One gets that
(i) 0(,)1 and e(,)1
p
ets tt;
(ii) ((),)(1()())(,)
pp
ets tptets

 ;
(iii) 1()
(,), where
(, )1()()
p
p
pt
ets p
etstpt



;
(iv) (, )1(,)(,)
ppp
etseste st

;
(v) (,)(,)(,)
pp p
etsesr etr;
(vi) (,)(,)(,)
pq pq
etsets ets

;
(vii) (, )(, )
(,)
ppq
q
ets ets
ets 
;
(viii) 1()
(, )(, )
pp
pt
es es







 ;
Finally, we state Krasnosel’skii fixed point theorem
which enables us to prov e the existence of period ic solu-
tions.
Theorem 2.3.
([16]) Let  be a closed convex nonempty subset of
a Banach space


B, . Suppose that
A
and B map
 into B such that (i) ,xy

 imply
xy; (ii)  is compact and continuous; (iii)
 is a contraction mapping; Then there exists z


with .zzz 
3. Existence of Periodic Solutions
Let 0, , TTkT be fixed and if

,
np for some n. By the notation


,ab, we
mean




,:.abtatb The intervals




,, ,ab ab and


,ab are defined similarly, Defined

(, ):()()
n
T
P
CR tTt   ,
where (, )
n
CR is the space of all real valued conti-
nuous functions. Then T
P is a Banach space when it is
endowed with norm

0,
sup( )
tT
x
xt

.
For each ,ij, we make basic assumption 1
()
H
:
j
aR

 is continuous, ()0
i
at and
()()
ii
at Tat for all tT.
()(),()()
ijij ii
ctTct ItTIt  and ()
ij
ct
 is con-
tinuous.
()
j
g
u is continuous, (0)0
j
g and
() ()
jj j
g
ugv Luv for some 0
j
L.
Lemma 3.1.
Assume that 1
()
H
holds. {()}
ii T
x
tP
  is a solu-
tion of (1) if and only if
1
1
1
()() () (1(,))
[()()
()(()()](, )
i
i
n
iijja
j
t
ij
tT
n
ijj jia
j
xtc txtkettT
rsx s k
btgxskIsets s








 



，
(3)
where 1
():() ()()
n
iijiij
j
rsc sascs
 


 


 and i

.
Proof. Let


()
iT
i
x
tP


is a solution of (1). It fol-
lows from (1) that
1
1
() ()() ()
()(())(),
n
iiiijj
j
n
ijj ji
j
xatxt ctxtk
btgxt kIt
 


 



(4)
where i

. Multiply both sides of (4) by (,0)
i
a
et
and integrate from tT

to t, one obt a i n that
1
1
[(,0)()][()( )
()( ())()](,0).
i
i
n
tt
ai ijj
tTtT j
n
ijj jia
j
es xsscsxsk
bsgxskIse ss










Divide both sides of above equation by (,0)
i
a
et , due
to ()()
ii
x
txtT

 and Lemma 2.1, we have


1
1
()1(,)[()()
()(()()(,).
i
i
n
t
ia ijj
tT j
n
ijj jia
j
x
tettT csxsk
bsgxsk Ise tss




 




(5)
It follows from integration by parts and the period icity
of ()
ij
c

and ()
j
x

, we get
1
1
1
(, )()()
() ()[1(,)]
()[(,)()].
i
i
i
n
t
aijj
tT j
n
ij ja
j
n
ts
jaij
tTj
etscsxsks
ctxtke ttT
x
sketscss


















(6)
Substitute (6) into (5) and simplify, we get (3). From
Lemma 2.1, we get the desired result and the proof is
complete.
Define the mapping : PP
TT
H by


1
1
1
()()
() ()1,
()( )()(( ))()
(,),
i
i
i
n
ij ja
j
n
t
ijijjj i
tT j
a
Ht
cttke ttT
rss kbsgskIs
etss





















(7)
where i

. Let ()()()()()(),
iii
H
tBtAt


where ,AB are given by
1
()()() (),
n
iijj
j
Bt cttk 



 (8)

Z. K. HUANG, J. X. CAI
3
and



1
()()1(,)()( )
() ( ())()(,),
i
i
t
ia ij
tT
n
ijj jia
j
A
tettT rssk
bsgskIse tss






 


 (9)
where ()
i
rs is defined in Lemma 3.1 and i. Next,
we will prove that
A
is compact and B is a contrac-
tion mapping in Lemma 3.2 and Lemma 3.3, respective-
ly.
Lemma 3.2. Assume that 1
()
H
holds. : PP
TT
A
defined by (9) is compact.
Proof. We first show that
A
maps PT into PT. It
follows from (9) that




1
1
()(+)1( ,)
()()
()( ())()(,),
i
i
ia
t
ij
tT
n
ijj jia
j
AtT etTt
ru Tuk
bugukIuetTuT u









 
 



(10)
and 1
1
() ()()()
=()()()().
n
iijiij
j
n
ij iiji
j
ruTcuTauT cuT
cuau curu







 






i.e., ()
i
ru is T-periodic. It follows from (2) and Theo-
rem 2.2 that (, )=(,)
ii
aa
etTuTetu

 and
(,)(,).
ii
aa
etTtettT

 
Thus (10) becomes



1
1
()(+)1(, )
()( )()(())
()(,) =()().
i
i
ia
n
t
ij ijjj
tT j
ia i
AtT ettT
ruukb uguk
Iuetu uAt








 
 



where i. That is, A: PP
TT
.
Secondly, we will show that is continuous. Let
, T P

 with , CC   and define

1
[0,]
[,]
[,] [0,]
:max1(, )
:max(,)
:max() :max()
:max
i
i
ia
tT
ia
utTt
ijij ii
ttTtt T
jj
j
ettT
etu
bbt rt



 





 













 


，
(11)
Given 0

 and take ()nM


 such that
.

  By making use of Lipschitz inequality of
1
()
H
, we get
1
()() (
),
n
t
iiii ijj
tTj
ij jjj
AA
bLdu M

 
 

 



where max{ }
i
M
M

and
1
().
n
iii iijj
j
M
TnbL  



Hence, it follows that
max ()()
ii
i
AAA A




 ,
That proves A is continuous.
Thirdly, we need to show
A
is compact. Consider
the sequence {}
nT
P   and assume that the sequence is
uniformly bounded. Let 0R be such ()nR


for
all n

. It is easy to estimate that




1
()
() ()
1
() ()
1
1
()() 1(,)
()()()(( ))
() (,)
(() )
[],
i
i
nia
n
tnn
ij ijjj
tT j
ia
n
tnn
iiijij jji
tT j
n
iiiij ji
j
At ettT
rss kbsgs k
Isetss
rsbLIs
nRbLRIT D



 
 






 
 








where sup{( )}.
ii
sR
I
Is

 Thus the sequence () {}
n
A  is
uniformly bounded. Now, it can be easily checked that
() ()
()
1
()
1
1
( )()()()( )()1()()
{[(()()())]()
()(())()},
nn
ii i
ii
nn
ij iiji
j
nn
ijj ji
j
AtatAtat t
ctat cttk
btgtk It









 

 



which leads to
()
1
()().
n
niiijji
j
A
tDanRbLRI 



 

For all n.That is ()
()
n
A
F 
 for some positive
constant
F
.Thus the sequence () {}
n
A  is uniformly
bounded and equi-continuous. The Arzela-Ascoli theo-
rem [16] implies that () {}
k
n
A  uniformly co nverg es to a
continuous T-periodic function

. Thus
A
is com-
pact.
Lemma 3.3. Let B is defined by (8) and assume that
2[0,]
1
(): sup(),
max 1.
ij ij
tT
n
ij
ij
Hct 


 



Then : PP
TT
B is a contraction.
Proof. Trivially, : PP
TT
B. For any , T
P


,
we have

Z. K. HUANG, J. X. CAI
4
[0, ]1
1
()()max(() ())
.
n
iiijj j
tT
j
n
ij
j
BBctk tk  



 



which leads to .BB

 
Hence, B de-
fines a contraction mapping with contraction constant

.
Theorem 3.1. Assume that 1
()
H
and 2
()
H
hold. If
all solutions {x (t)}P
iT
 of (1) satisfy with x(t)
iG

and
1
max [()]
n
iiiij jiii
ij
TnbLG ITG  
 

 



wher e 0G and sup{( )},
ii
sR
I
Is

 then (1) has a
T-periodic solution.
Proof. Define {:}.
T
PG    Lemma 3.1
implies : PP
TT
A and
A
is compact and conti-
nuous. Lemma 3.2 implies B is a contraction and
: PP
TT
B. Now, we need to show that if ,



,
we have

.
ii
A
BG  
Let ,

  with
,.G   It follows from (8) and (9) that

1
1
(()
())
[()]
t
ii ij
ii tT
n
ij jji
j
n
iiiij j
j
iii
AB rs
bLIss
TnbLG
ITG



 











All the conditions of Krasnosel’skii theorem are satis-
fied on the set . Thus there exists a fixed point z in
 such that .zAzBz By Lemma 3.1, this fixed
point is a solution of (1). Hence (1) has a T-periodic so-
lution.
Theorem 3.2. Assume that 1
()
H
and 2
()
H
hold.
Let ,,
iii

be given by ( 11). If
1
()1
n
iiiij j
j
Tn bL
 



holds, then (1) has a unique T-periodic solution.
Proof. Let the mapping
H
be given by (7). For
, ,
T
P

 we have

1
1
(
)
[()],
n
t
iii jj
ii tTj
ij jjj
n
iiiij j
j
HH
bL s
Tn bL



 


 

 



which leads to


1
max
[()].
ii
i
n
iiiij j
j
HHH H
Tn bL
 
 


 
 

That is,
H
defines a contraction mapping and there
exists a unique fixed point which is a T-periodic solution
of (1). This completes the proof.
4. Conclusion
Due to time scales calculus theory and the fixed point
theorem, we obtained some more generalized results to
ensure the existence of the periodic solutions for neutral-
type neural networks with delays. The conditions can be
easily checked in practice by simple algebraic methods.
The method in this paper can be applied to prove the ex-
istence of the periodic solutions of some other similar
systems such as neutral-type networks with leakage-
terms [17].
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