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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 Am1,n Am,n1 Am,n Pm,n Amk ,nl 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) i1 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. 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