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Journal of Applied Mathematics & Bioinformatics, vol. 1, no.2, 2011, 49-56
ISSN: 1792-6602 (print), 1792-6939 (online)
International Scientific Press, 2011
Oscillation of Neutral Delay Partial
Difference Equation
Guanghui Liu1
Abstract
In this paper, some sufficient conditions for oscillation of the neutral delay partial
equation :

1,2 ( Am, n  cAm , n  )   Pi (m, n) Am  ki , n li  0
i 1
are established. Our results as a special case when c  0 ,   1 , involve and
improve some well-known oscillation results.
Mathematics Subject Classification : 35B05
Keywords: Oscillation, Neutral, Partial difference
1
Introduction
It is well known that the partial difference equations appear in considerations of
random walk problems, molecular structure and chemical reactions problems
1
College of Science, Hunan Institute of Engineering, 88 East Fuxing Road,
Xiangtan 411104, Hunan, China, e-mail : [email protected]
Article Info:
Revised : July 25, 2011. Published online : November 30, 2011.
50
Oscillation of Neutral Delay Partial Difference Equation
[1-3].Oscillation and nonoscillation of solutions of delay partial difference
equations is receiving much attention [4-7].
In this paper, we consider the neutral delay partial difference equation

1,2 ( Am, n  cAm ,n  )   Pi (m, n) Am  ki , n li  0 ,
(1.1)
i 1
where
m, n  N 0  0,1, 2,
integers, the coefficients
and
, , ki , li (i  1, 2, , ) are
Pi (m, n)  N 02  0,1, 2,
2
nonnegative
is a sequences of
nonnegative real numbers, and 0  c  1 . We defined
1,2 ( Z m ,n )  Z m 1,n  Z m,n 1  Z m,n , Z m ,n  Am ,n  cAm ,n  .
A solution Am,n  of (1.1) is said to be eventually positive if Am, n  0 for all
large m and n . It is said to be oscillatory if it is neither eventually positive nor
eventually negative.
As a special case of Eq. (1.1), B.G.Zhang et al.[5] considered partial difference
equation
Am1,n  Am,n1  Am,n  Pm,n Amk ,nl  0 ,
(1.2)
And proved that: for all large m , n , and there exists  such that
Pm ,n   
(k  l ) k l
,
(k  l  1) k l 1
(1.3)
Then every solution of equation (1.2) oscillates.
2
Main Results
In this section, we give some oscillation for Eq.(1.1). In order to prove our main
results, we need the following auxiliary results.
Guanghui Liu
51
Lemma 1. Suppose that Am,n  is an eventually positive solution of equation
(1.1), then :
, and Z m ,n is monotone decreasing in m , n , that is
(i)  1,(
2 Z m , n) 0
Z m 1,n  Z m ,n , Z m,n 1  Z m ,n ;
(ii) Z m ,n  0 .
Proof. Since Am,n  is an eventually positive solution of (1.1), then there exists
enough M , N , when m M , n  N , such that
Am , n  0, Am  , n   0, Am  ki , n  li  0 , i  1, 2, ,  .
From (1.1), we obtain

1,2 ( Z m ,n )   Pi (m, n) Am  ki , n li  0 .
i 1
That is
Z m 1,n  Z m ,n 1  Z m ,n  0,Z m 1,n  Z m ,n , Z m,n 1  Z m ,n .
Next, we show that Z m,n is eventually positive in m, n . If Z m,n  0 , then there
exists d  0 , for all large M 1 , N 1 , when m  M 1 , n  N1 , such that Z m ,n  d .
Am,n  Am ,n  Am,n  cAm ,n  Z m,n  d , Am,n  d  Am ,n .
Therefore,
Am h ,n  h   d  Am( h 1) ,n ( h 1)  2d  Am( h  2) ,n ( h 2)    (h  1)d  Am ,n 
as h  , Am  h ,n  h   .Which contradiction to
positive solution. This completes the proof.
A 
m,n
is an eventually

52
Oscillation of Neutral Delay Partial Difference Equation
Theorem 1. Assume that
kˆ

lim inf (1  c) Pi (m, n) 
m , n 
where
i 0
lˆ
ˆ
ˆ
(kˆ  1) k 1 (lˆ  1)l 1
kˆ  min(k1 , k2 , , k ) , lˆ  min(l1 , l2 , , l ),
,
(2.1)
then every solution of
equation (1.1) oscillates.
Proof. Suppose to the contrary that the equation (1.1) has a nonoscillatory
solution Am,n  .Without loss of generality, we may assume that
A 
m,n
is an
eventually positive solution of equation (1.1), then from (1.1), we have

0  1,2 ( Z m, n )   Pi (m, n) Am  ki , n li
i 1

 1,2 ( Z m,n )   Pi (m, n)( Z m  ki ,n li  cAm  ki  , n li  )
i 1

 1,2 ( Z m,n )   Pi (m, n)( Z m  ki ,n li  cZ m  ki  ,n li  )
i 1
By Lemma 1, we have

 1,(
2 Z m , n)  (1  c ) Pi ( m, n) Z m  kˆ , n  lˆ  0,
i 1
That is

Z m 1,n  Z m ,n 1  Z m,n   (1  c) Pi (m, n) Z m  kˆ ,n lˆ ,
(2.2)
i 1
dividing the both sides of (2.2) by Z m ,n , we have
Z m 1,n
Z m ,n
Set rm ,n 
Z m ,n
Z m 1,n
, t m ,n 

Z m ,n
Z m ,n 1
Z m,n 1
Z m ,n

Z m  kˆ ,n lˆ
i 1
Z m,n
 1   (1  c) Pi (m, n)
,
(2.3)
. It is easy to see that rm ,n  1 , tm ,n  1 , for all large
m and n . rm,n , t m,n  are bounded. Let lim inf t m,n  b  1, then from (2.3),
m , n 
we have
Guanghui Liu
1
rm ,n

1
t m,n
53

Z m  kˆ ,n lˆ
i 1
Z m ,n
 1   (1  c) Pi (m, n)

 1   (1  c) Pi (m, n)rm  kˆ ,n rm  kˆ 1,n  rm 1,n t m lˆ ,n t m lˆ 1,n  t m 1,n (2.4)
i 1
from (2.4), we get
lim sup
m , n 
1
rm ,n
 lim sup
m , n 
1
tm, n


1 1
ˆ ˆ
  1  lim inf  (1  c) Pi (m, n)a k bl ,
m , n 
a b
i 1
or

lim inf  (1  c) Pi (m, n)  (1 
m , n 
i 1
1 1 1
(a  1)(b  1)
 ) kˆ lˆ 
 f (a, b).
ˆ
ˆ
a b a b
a k 1b l 1
Now it is easy to see that
max f (a, b) 
a 1,b 1
kˆ
lˆ
ˆ
ˆ
(kˆ  1) k 1 (lˆ  1) l 1
,
hence we obtain

lim inf  (1  c) Pi (m, n) 
m , n 
i 1
kˆ
lˆ
ˆ
ˆ
(kˆ  1) k 1 (lˆ  1) l 1
,
which is a contradiction to (2.1).this completes the proof.

According to theorem 1, if c  0,   1 , we obtain the following result:
Corollary 1. For all large m , n , there exists  such that
Pm,n   
kk
ll
.
(k  1) k 1 (l  1) l 1
Then every solution of equation oscillates.
Remark 1. From Corollary 1, compare (2.5) with (1.3), obviously
(2.5)
54
Oscillation of Neutral Delay Partial Difference Equation
kk
ll
(k  l ) k l
.

(k  1) k 1 (l  1) l 1 (k  l  1) k l 1
Theorem 2. Assume that
( i  1) i 1
lim inf  (1  c) Pi (m, n)2
 1,
m , n 
( i ) i
i 1

i
(2.6)
where  i  min(k i , li ), i  1,2,  , then every solution of equation (1.1) oscillates.
Proof. Suppose to the contrary that Am,n  is an eventually positive solution of
z m ,n
equation (1.1). Set s m ,n 
z m 1,n 1
. It is easy to see that s m,n  1 , for all large m
and n , s m ,n is bounded. Let lim inf s m ,n    1, then from (1.1) and by Lemma 1,
m , n 
we have

1,2 ( Z m ,n )   (1  c) Pi (m, n) Z m i ,n i  0 .
i 1
That is
2
s m ,n

2
s m,n
Z m 1,n
Z m,n

Z m ,n 1
Z m,n

Z m i ,n i
i 1
Z m,n
 1   (1  c) Pi (m, n)

,
 1   (1  c) Pi (m, n) sm i , n i sm i 1,n i 1  sm 1,n 1 .
(2.7)
i 1
Taking supremun limit on both sides of (2.7),we have
2


 1  lim inf  (1  c) Pi (m, n)  i ,
m , n 
i 1
which implies   2 and

lim inf  (1  c) Pi (m, n)
m , n 
i 1
  1
1.
 2
i
(2.8)
Guanghui Liu
55
Noticed that
min(
 2
(  1) 1
  1
.
)  2 i 
 2
i
i
i
i
i
Hence we have

lim inf  (1  c) Pi (m, n)2
m , n 
i
(i  1)i 1
i
i 1
i
1,
which contradicts (2.6). This completes the proof.

Corollary 2. Assume that
1

 ( lim inf  (1  c) Pi (m, n))  
,
i 1 m , n  i 1
2 (  1)  1


where  
1


(2.9)

( i ) , i  min(ki , li ) , i  1, 2, ,  . Then every solution of (1.1)
i1
oscillates.
Proof . In fact, from (2.8) we have
1

  1
  ( lim inf (1  c) Pi (m, n))  .
1  lim inf  (1  c) Pi (m, n)
m , n 
i 1 m , n 
 2
i 1

i
Hence
1

1   ( lim inf (1  c) Pi (m, n)) 2

i 1 m , n 

(  1) 1

.
That is

1
 ( lim inf (1  c) Pi (m, n)) 

i 1 m , n 
which contradicts (2.9).The proof is completed.

.
2  (  1)  1

56
Oscillation of Neutral Delay Partial Difference Equation
References
[1] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales,
Birkhäuser, Boston, 2003.
[2]
R. Courant, K. Friedrichs and H. Lewy, On the partial difference equations
of mathematical physics, Birkhäuser, Boston, 1967.
[3] X.P. Li, Partial difference equations used in the study of molecular orbits,
Acta Chimica Sinica, 40, (1982), 688-698.
[4] J.C. Strikwerda, Finite Difference Schemes and Partial Difference Equations,
Wadsworth, Belmont, CA,1989.
[5] B.G. Zhang and S.T. Liu, Oscillation of a class of delay partial difference
equation, Journal of difference Equation and its Application, 1, (1995),
215-226.
[6] B.G. Zhang and S.T. Liu, On the oscillation of two partial difference
equations, Journal of Mathematical Analysis and Applications, 206, (1997),
480-492.
[7] B.G. Zhang and S.T. Liu, Oscillation of partial difference equations,
Panamerican Math J ., 5, (1995), 61-70.
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