Figen Özpinar, Sermin Öztürk and Zeynep Fidan Koçak OSCILLATION FOR CERTAIN IMPULSIVE PARTIAL DIFFERENCE EQUATIONS

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1 DEMONSTRATIO MATHEMATICA Vol. XLVII No Figen Özpinar, Sermin Öztürk and Zeynep Fidan Koçak OSCILLATION FOR CERTAIN IMPULSIVE PARTIAL DIFFERENCE EQUATIONS Abstract. In this paper, we obtain some sufficient criteria for the oscillation of the solutions of linear impulsive partial difference equations with continuous variables. 1. Introduction The impulsive differential equations are adequate mathematical models of various processes and phenomena studied in physics, chemical technology, population dynamics, technics and economics. These equations provide natural mathematical description of processes which are subject to short-time perturbations during their evolution. Various monographs [, 4, 5, 6, 7, 9, 10], devoted to the impulsive differential equations were published in the recent years. At the same time, partial difference equations arise in applications, involving populations dynamics with spatial migrations, chemical reactions, mathematical physics and finite-difference schemes. Recently, there are many papers that devoted to the development of qualitative theory of partial difference equations with continuous variables, see e.g. [8, 11, 13]. Let 0 x 0 ă x 1 ă... ă x n ă x n1 ă..., be fixed points with lim nñ8 x n 8, and let for n P N x nr x n τ, where r is a fixed natural number, and τ ą 0 is a constant. Define J imp tx n u 8 n 1, R r0, 8q, J tpx, yq : x P J imp, y P Ru. In 009, Agarwal and Karakoc studied the oscillation of solutions of impulsive partial difference equations with continuous variables of the type p 1 zpx a, y bq p zpx a, yq p 3 zpx, y bq p 4 zpx, yq P px, yqzpx τ, y σq 0, px, yq P pr RqzJ, 010 Mathematics Subject Classification: 39A11, 34K11, 34C10. Key words and phrases: partial difference equation, impulsive PDEs, oscillation, continuous variables. DOI: /dema c Copyright by Faculty of Mathematics and Information Science, Download WarsawDate University 1/1/19 of 9:39 Technology AM

2 80 F. Özpinar, S. Öztürk, Z. F. Koçak zpxǹ, yq zpxń, yq L n zpxń, yq, px n, yq P J, where zpx, yq lim pq,sqñpx,yq qąx zpq, sq, zpx, yq lim pq,sqñpx,yq zpq, sq []. qăx In this paper, we shall consider the impulsive partial difference equations with continuous variables of the form (1) c 1 upx a, y bq c upx a, yq c 3 upx, y bq upx, yq F px, yqupx τ, yq Gpx, yqupx, y σq Hpx, yqupx τ, y σq 0, () upxǹ, yq upxń, yq L n upxń, yq, px n, yq P J, where upx, yq lim pq,sqñpx,yq qąx px, yq P pr RqzJ, upq, sq, upx, yq lim pq,sqñpx,yq upq, sq and qăx F, G, H P CpR R, R t0uq, a, b, τ, σ are positive constants, c i, i 1,, 3, 4, are nonnegative constants. Our results extend the oscillation results of Agarwal and Karakoc []. A function upx, yq in r τ, 8q r σ, 8q is said to be solution of (1) () if (i) for px, yq P pr RqzJ, u is continuous and satisfies (1), (ii) for px, yq P J, upx, yq and upx, yq exist, upx, yq upx, yq, and satisfy (). A solution upx, yq of (1) () is said to be eventually positive (or negative) if upx, yq ą 0 (or upx, yq ă 0) for all large x and y. It is said to be oscillatory if it is neither eventually positive nor eventually negative otherwise, it is called nonoscillatory.. Preparatory lemmas Throughout this paper, we shall assume that (A1) tl n u 8 n 1 is a sequence of positive real numbers such that ř 8 n 1 L n ă 8, (A) c, c 3 ą 0, (A3) τ ka θ, σ lb η, where k, l are nonnegative integers, θ P r0, aq and η P r0, bq. We set, for any px, yq P R R, P px, yq mintf ps, tq, x s x a, y t y bu 0 ă lim sup P px, yq ă 8, x,yñ8 Qpx, yq mintgps, tq, x s x a, y t y bu and and 0 ă lim sup Qpx, yq ă 8, x,yñ8

3 Oscillation for certain impulsive partial difference equations 81 Rpx, yq minthps, tq, x s x a, y t y bu E P λ ą 0 : E Q λ ą 0 : E R λ ą 0 : E λ ą 0 : and 0 ă lim sup Rpx, yq ă 8, x,yñ8 T px, yq mintp px, yq, Qpx, yq, Rpx, yqu, * p1 L m q λp px, yq ą 0, eventually, * p1 L m q λqpx, yq ą 0, eventually, * p1 L m q λrpx, yq ą 0, eventually, * p1 L m q λt px, yq ą 0, eventually. Here, the symbol ś x 0 ăx măs a m denotes the product of the members of the sequence ta m u over m such that x m P px 0, sq X J imp. If px 0, sq X J imp H, or x 0 s then we assume that ś x 0 ăx măs a m 1. Lemma 1. Assume that upx, yq is an eventually positive solution of (1) (). Then for px, yq P R R the function ż xa ż yb (3) ωpx, yq 1 ups, tqdtds x y x 0 ăx măs is an eventually positive solution of the partial difference inequality (4) c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq p1 L m qωpx, yq P px, yqωpx ka, yq Qpx, yqωpx, y lbq Proof. So that x 0 ăx măx is continuous for px, yq P R R, p5q Bω Bx ż yb y ż yb y x 0 ăx măx 1upx, yq 1Lm 1upxa, tq 1Lm 1 x x măxa Rpx, yqωpx ka, y lbq 0. 1Lm 1upx, tq j dt 1Lm 1upxa, tq upx, tq j dt,

4 8 F. Özpinar, S. Öztürk, Z. F. Koçak p6q ż Bω xa By x x 0 ăx măs 1rups, y bq ups, yqsds. Since upx, yq is an eventually positive solution of (1) (), (7) c 1 upx a, y bq c upx a, yq c 3 upx, y bq upx, yq ă 0, for px, yq P pr RqzJ. From (A) and (7) upx a, yq upx, yq ă 0 and upx, y bq upx, yq ă 0, eventually. Moreover since 0 ă ś x x măxa 1 1, we obtain (8) 1upx a, yq upx, yq ă 0. x x măxa Let px, yq P J and, say, x x n. From () and (8), we have upx n, yq 1 1 L n upxǹ, yq x n x măx na So, for all px, yq P pr Rq Bω Bx (9) 1 1 L n xǹ x măxǹ a 1upxn a, yq. 0 and Bω By 1upxǹ a, yq 0. Therefore ωpx τ, y σq ωpx pka θq, y plb ηq ωpx ka, y lbq. Since integrating (1), 0 ă ś 1 1, we find 0 c 1 ż xa x c 3 ż xa ż yb y ż yb x y ż xa ż yb x y ż xa ż yb x which yields (10) 0 c 1 ż xa y x ups a, t bqdtds c ż xa ups, t bqdtds ż xa F ps, tqups τ, tqdtds Hps, tqups τ, t σqdtds, ż yb y x 0 ăx măs x ż yb y ż yb x y ż xa ż yb x y ups a, tqdtds ups, tqdtds 1ups a, t bqdtds Gps, tqups, t σqdtds

5 c ż xa x c 3 ż xa Oscillation for certain impulsive partial difference equations 83 ż yb x y ż xa ż yb x y ż xa ż yb x y ż xa ż yb x y y x 0 ăx măs ż yb x 0 ăx măs F ps, tq Gps, tq Hps, tq 1Lm 1upsa, tqdtds 1Lm 1ups, tbqdtds c4 ż xa x 0 ăx măs x 0 ăx măs x 0 ăx măs 1Lm 1ups τ, tqdtds 1Lm 1ups, t σqdtds 1Lm 1ups τ, t σqdtds. x ż yb y ups, tqdtds From (9), (10) and using the definition of ω, P, Q, R, we obtain eventually c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq ωpx, yq P px, yqωpx ka, yq Qpx, yqωpx, y lbq Rpx, yqωpx ka, y lbq 0. Therefore, ωpx, yq is an eventually positive solution of inequality (4). 3. Main results Theorem 1. Assume that there exist X 0 0, Y 0 0 such that if k ą l ą 0, 1 #c (11) inf λpe,x X 0,y Y 0 λ c3 c k l l l k l if l ą k ą 0, 1Lm l x 0 ăx măx plj 1qa k 1Lm 1 1Lm λt x ia, y ib 1 1Lm λt x pl jqa, y lb j 1 + ą 1,

6 84 F. Özpinar, S. Öztürk, Z. F. Koçak 1 #c (1) inf λpe,x X 0,y Y 0 λ c3 c l k 3 k k l k and if k l ą 0, k 1Lm l x 0 ăx măx pk 1qa 1 #c (13) inf λpe,x X 0,y Y 0 λ c3 k k 1Lm 1 1 1Lm λt x ia, y ib 1Lm λt x ka, y pk jqb j 1 + ą 1 k k Then every solution of (1) () is oscillatory. 1 1 λt x ia, y ib j 1 + ą 1. Proof. Suppose to the contrary that upx, yq is nonoscillatory solution of (1) (). Without loss of generality we may assume that upx, yq is an eventually positive solution of (1) (). Let ωpx, yq be as in Lemma 1. Set Spλq λ ą 0 : c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq * λt px, yq ωpx, yq 0, eventually. Since Bω Bx ă 0 and Bω By ă 0, from (4) we obtain

7 Oscillation for certain impulsive partial difference equations 85 c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq T px, yq ωpx, yq 0, which implies that 1 P Spλq. That is Spλq is nonempty. If λ 0 P Spλq, we have eventually c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c 4 λ0 T px, yq ωpx, yq. Therefore ś λ0 T px, yq ą 0 eventually, i.e., λ 0 P E. Thus Spλq Ă E. Let µ P Spλq. Noting that Bω Bω Bx ă 0 and By ă 0 eventually, we obtain eventually (14) 1 ωpx a, y bq and thus, (15) If k ą l ą 0 then (16) ωpx, yq l x 0 ăx măxa c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq µt px, yq ωpx, yq. x 0 ăx măx l 1 ωpx, yq µt px a, y bq ωpx a, y bq. 1 1 µt px ia, y ibq ωpx la, y lbq. Since ωpx, yq ą 0 for all large x and y then from (4) c ωpx a, yq ωpx, yq, c 3 ωpx, y bq ωpx, yq,

8 86 F. Özpinar, S. Öztürk, Z. F. Koçak eventually. Hence k (17) c ωpx pi 1qa, yq and (18) k c4 c k ωpx, yq c k 4 l c 3 ωpx, ypi 1qbq Moreover, from (14) and k l c4 c l 3ωpx, yq c l 4 c ωpx a, yq 1Lm ωpx ia, yq 1Lm ωpx ka, yq ωpx, y ibq lωpx, y lbq. µt px, yq ωpx, yq (19) ωpx la, y lbq 1 c4 1Lm µt px pl1qa, y lbq ωpx pl1qa, y lbq c 1 c k l x 0 ăx măx la k l x 0 ăx măx plj 1qa 1Lm µt px pljqa, y lbq j eventually. Substituting (16) (19) into (4), we obtain c 1 ωpxa, y bqc ωpxa, yqc 3 ωpx, y bq c P px, yq c3 Qpx, yq c k l Rpx, yq l k c 1 c c 3 1Lm lωpx, yq ωpx ka, y lbq, 1Lm ωpx, yq 1Lm 1 ωpx, yq 1Lm 1

9 l k l Oscillation for certain impulsive partial difference equations 87 x 0 ăx măx plj 1qa eventually. Then 1Lm µt px ia, y ibq 1Lm µt px pljqa, y lbq j 1 ωpx, yq 0, c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c P px, yq c3 Qpx, yq c k l Rpx, yq l k l l k x 0 ăx măx plj 1qa eventually. Thus 1Lm l 1Lm 1 1Lm 1 1Lm µt px ia, y ibq 1Lm µt px pl jqa, y lbq j 1 j* ωpx, yq c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c T px, yq c3 c k l l k 1Lm l 0, 1Lm 1 1Lm 1

10 88 F. Özpinar, S. Öztürk, Z. F. Koçak sup l x X 0,y Y 0 k l x 0 ăx măx plj 1qa eventually. Therefore c λ 0 c k l l sup k l x X 0,y Y 0 k l x 0 ăx măx plj 1qa 1Lm µt px ia, y ibq 1Lm µt px pljqa, y lbq j 1 j* ωpx, yq 1Lm 1 c3 1Lm 1 1Lm µt px ia, y ibq 0, 1Lm l 1Lm µt px pl jqa, y lbq j 1 P Spλq. However, on the other hand, from (11) there exists an α 1 P p1, 8q such that 1 c inf λpe,x X 0,y Y 0 λ c3 c k l l l k l c 1 c c 3 k 1Lm l x 0 ăx măx plj 1qa Therefore, for λ µ, we obtain 1Lm 1 1 1Lm λt px ia, y ibq * 1Lm λt px pljqa, y lbq j 1 α 1 ą 1.

11 inf λpe,x X 0,y Y 0 c3 c k l l l k l Oscillation for certain impulsive partial difference equations 89 c k 1Lm l x 0 ăx măx plj 1qa The above inequality implies k c (0) c3 c k l l l k l 1 1Lm 1 1Lm µt px ia, y ibq * 1Lm µt px pl jqa, y lbq j 1 µα 1. l x 0 ăx măx plj 1qa 1 1 µt px ia, y ibq µt px pl jqa, y lbq j 1 λ 0 µα 1. It is clear that if λ 0 P Spλq then λ 1 0 λ 0 implies that λ 1 0 P Spλq. Therefore µα 1 P Spλq. Repeating the above arguments with µ replaced by µα 1, we can prove that there exists α P p1, 8q such that µα 1 α P Spλq. Continuing in this way, we obtain µ ś 8 α i P Spλq, where α i P p1, 8q. This implies that Spλq is unbounded. On the other hand lim sup T px, yq ą 0 and c 4 λt px, yq ą 0, x,yñ8

12 90 F. Özpinar, S. Öztürk, Z. F. Koçak eventually. So E is unbounded. Because of Spλq Ă E, then Spλq is also bounded. If l ą k ą 0 then from (15), we obtain k ωpx, yq (1) 1 1 c 4 k c4 µt px ia, y ibq ωpx ka, y kbq, () (3) and c c3 ωpx, yq ωpx, yq k (4) ωpx ka, y kbq 1 l k x 0 ăx măx pk 1qa c l k 3 By (1) (4), from (14), we have j ωpx ka, yq, lωpx, y lbq c 1 ωpxa, y bqc ωpxa, yqc 3 ωpx, y bq c P px, yq c3 Qpx, yq c l k 3 Rpx, yq k l k and hence k k 1Lm µt px ka, y pkjqbq j ωpx ka, y lbq. c 1 c c 3 x 0 ăx măx pk lqa 1Lm lωpx, yq 1Lm 1 ωpx, yq 1Lm 1 1Lm µt px ia, y ibq 1Lm ωpx, yq 1Lm µt px ka, y pkjqbq j 1 ωpx, yq 0

13 Oscillation for certain impulsive partial difference equations 91 c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c T px, yq c3 c l k 3 k k l k k 1Lm l x 0 ăx măx pk 1qa 1Lm 1 1Lm 1 1Lm µt px ia, y ibq 1Lm µt px ka, y pk jqbq j 1 j* ωpx, yq eventually. Thus c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c T px, yq c3 c l k 3 k sup k c 1 c c 3 x X 0,y Y 0 l k k 1Lm l x 0 ăx măx pk 1qa eventually. It is clear that 1Lm 1 1Lm 1 1Lm µt px ia, y ibq 0, 1Lm µt px ka, y pkjqbq j 1 j* ωpx, yq 0,

14 9 F. Özpinar, S. Öztürk, Z. F. Koçak c λ 1 0 c l k 3 k sup k k x X 0,y Y 0 l k x 0 ăx măx pk 1qa 1Lm 1 c3 1Lm 1 1Lm µt px ia, y ibq 1Lm l 1Lm µt px ka, y pk jqbq j 1 P Spλq. In the same way as in the proof of case k ą l ą 0, we can show that Spλq is unbounded. For k l ą 0, (16) (18) hold eventually. From (14), we obtain c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c P px, yq c3 Rpx, yq k Qpx, yq k k 1Lm k 1Lm 1 1Lm 1 1Lm µt px ia, y ibq j 1 j* ωpx, yq 0 then c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c T px, yq c3 k 1Lm k 1Lm 1

15 k k Oscillation for certain impulsive partial difference equations 93 1Lm 1 1Lm µt px ia, y ibq j 1 j* ωpx, yq 0, eventually. The above inequality implies c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c T px, yq c3 k Hence c λ 0 k sup k x X 0,y Y 0 sup k k x X 0,y Y 0 1Lm k 1Lm 1 1Lm 1 1Lm µt px ia, y ibq j 1 j* 1Lm 1 c3 k ωpx, yq 0. 1Lm k 1 1Lm µt px ia, y ibq j 1 P Spλq. In the same way as in the proof of case k ą l ą 0, we can show that Spλq is unbounded. This completes the proof. Corollary 1. Assume that if k ą l ą 0, (5) lim inf T px, yq q x,yñ8 ą c 4 L k1 c k c l 3 c k l L k 1 k1 k k j 1,

16 94 F. Özpinar, S. Öztürk, Z. F. Koçak if l ą k ą 0, (6) lim inf T px, yq q x,yñ8 and if k l ą 0, ą c 4 L l1 (7) lim inf T px, yq q x,yñ8 ą L k1 c k c l 3 c l k 3 L l 1 l1 c k c k 3 c 1 c c 3 k 1 k1 L l l k k j 1, j 1, where L ś 8 m 1. Then every solution of (1) () is oscillatory. Proof. Since c4 L k1 k max λ L λq k k 0ăλ L q q k 1 k1 for k ą l ą 0, from (5) 1 #c inf λpe,x X 0,y Y 0 λ 1 λ c3 c k l l l k l k l x 0 ăx măx plj 1qa # c c3 L L q c c3 L L L c k l 1 1 λt x ia, y ib λt x pl jqa, y lb j 1 + ck l c 1 c + k c 3 L λq L c1 c c 3 l L max λ L λq k

17 Oscillation for certain impulsive partial difference equations 95 q c c3 c k l L L L L q c c4 k c l 3 c k l L k1 L k 1 k1 k k q c4 L k1 j ą 1. k 1 k1 Therefore from Theorem 1, every solution of (1) () is oscillatory. In the same way, we can prove the case of l ą k ą 0 and k l ą 0. Theorem. Assume that there exist X 0 0, Y 0 0 such that (8) c k inf λpe P,x X 0,y Y 0 λ k Then every solution of (1) () is oscillatory. k k 1Lm λp px ia, yq j 1 ą 1. Proof. Assume that upx, yq is an eventually positive solution of (1) (). From Lemma 1, there exists an eventually positive solution ωpx, yq of (4). Then we have (9) c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq ωpx, yq P px, yqωpx ka, yq 0, eventually. Let S P pλq λ ą 0 : c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq c 4 p1 L m q λp px, yq ωpx, yq 0, eventually (. Since Bω Bx ă 0 and Bω By ă 0 then from (9), we obtain c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq P px, yq ωpx, yq 0, eventually. This implies that 1 P S P pλq. That is S P pλq is nonempty. If λ 0 P S P pλq then we have eventually 0 ă c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq λ0 P px, yq ωpx, yq.

18 96 F. Özpinar, S. Öztürk, Z. F. Koçak Hence ś λ0 P px, yq ą 0, i.e., λ 0 P E P. Thus S P pλq Ă E P. Since lim sup P px, yq ą 0 and λp px, yq ą 0 xñ8,yñ8 then E P is bounded. Since S p pλq Ă E P then S p pλq is also bounded. Let µ P S P pλq. Noting that Bω Bω Bx ă 0 and By ă 0 eventually, we have j c ωpxa, yq ă 1Lm µp px, yq ωpx, yq, (30) c k ωpx, yq ă k From (9) and (30), we obtain c 1 ωpxa, y bqc ωpxa, yqc 3 ωpx, y bq c k sup k x X 0,y Y 0 eventually. Hence k sup c k x X 0,y Y 0 1Lm µp px ia, yq j ωpxa, yq. 1Lm ωpx, yq µp px ia, yq j 1 P px, yqωpx, yq 0, µp px ia, yq j 1 λ 0 P S P pλq. On the other hand, from (8) there exists α 1 P p1, 8q such that c k k inf λp px ia, yq j 1 λpe P,x X 0,y Y 0 λ Therefore for λ µ, k λ 0 c k sup x X 0,y Y 0 α 1 ą 1. µp px ia, yq j 1 µα 1 P S p pλq. Repeating the above arguments with µ replaced by µα 1, we can prove that there exists α P p1, 8q such that µα 1 α P S P pλq. Continuing in this way, we obtain µ ś 8 α i P S P pλq, where α i P p1, 8q. This implies that S P pλq

19 Oscillation for certain impulsive partial difference equations 97 is unbounded. This contradicts the boundness of S P pλq. The proof is complete. Theorem 3. Assume that there exist X 0 0, Y 0 0 such that (31) c l l 3 inf λqpx, y ibq j 1 ą 1. λpe Q,x X 0,y Y 0 λ Then every solution of (1) () is oscillatory. The proof of Theorem 3 is similar to the proof of Theorem. Corollary. If (3) lim inf P px, yq p ą x,yñ8 or (33) lim inf Qpx, yq q ą x,yñ8 c4 L k1 c k c4 L l1 then every solution of (1) () is oscillatory. Proof. From (3), we obtain c k inf λpe P,x X 0,y Y 0 λ k inf 0ăλ L p c k c l 3 λ L λp k ck k k pk 1q k1 l l pl 1q l1 λp px ia, yq j 1 p c4 L k1 pk 1q k1 k k ą 1. So (8) holds. Similarly, if (33) holds then (31) holds. Thus, by Theorem and Theorem 3, every solution of (1) () is oscillatory. Theorem 4. Assume that there exist X 0 0, Y 0 0 such that if k ą l ą 0, c k l l (34) inf 1 λpe R,x X 0,y Y 0 λ l k l if l ą k ą 0, x 0 ăx măx plj 1qa λrx ia, y ib λrx pl jqa, y lb j 1 ą 1,

20 98 F. Özpinar, S. Öztürk, Z. F. Koçak c l k k 3ś λpe R,x X 0,y Y 0 (35) inf k l k and if k l ą 0, x 0 ăx măx pk 1qa 1 (36) inf λpe R,x X 0,y Y 0 λ k k λrx ia, y ib 1 λrx ka, y pk jqb j 1 ą 1 Then every solution of (1) () is oscillatory. 1 λrx ia, y ib j 1 ą 1. Proof. Assume that upx, yq is an eventually positive solution of (1) (). From Lemma 1, there exists an eventually positive solution ωpx, yq of (4). Then we have (37) c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq ωpx, yq Rpx, yqωpx ka, y lbq 0, eventually. Set S R pλq λ ą 0 : c 1 ωpx a, y bq c ωpx a, yq c 3 ωpx, y bq * Rpx, yq ωpx, yq 0, eventually. It is easy to prove that S R pλq Ă E R, S R is nonempty and E R is bounded. Let µ P S R pλq. Noting that Bω Bω Bx ă 0 and By ă 0, eventually, we have 1 ωpx, yq c 4 µrpx a, y bq ωpx a, y bq, for k ą l ą 0, x 0 ăx măx

21 (38) ωpx, yq and l c4 Oscillation for certain impulsive partial difference equations 99 l (39) ωpx la, y lbq 1 k l c k l x 0 ăx măx plj 1qa From (37) and (39), we obtain c 1 ωpxa, y bqc ωpxa, yqc 3 ωpx, y bq Hence, c k l l c k l l sup l x X 0,y Y 0 k l x 0 ăx măx plj 1qa sup x X 0,y Y 0 k l 1 1 µrpx ia, y ibq ωpx la, y lbq 1Lm µrpx pljqa, y lbq j ωpx ka, y lbq. l x 0 ăx măx plj 1qa 1 1Lm ωpx, yq µrpx ia, y ibq µrpx pl jqa, y lbq j 1 1 Rpx, yqωpx, yq 0. µrpx ia, y ibq µrpx pl jqa, y lbq j 1 On the other hand, from (34) there exists an α 1 P p1, 8q such that λ 0 P S R pλq.

22 100 F. Özpinar, S. Öztürk, Z. F. Koçak c k l inf λpe R,x X 0,y Y 0 λ l k l Hence, for λ µ λ 0 c k l l x 0 ăx măx plj 1qa l l k l µα 1 P S R pλq. 1 λrpx ia, y ibq λrpx pl jqa, y lbq j 1 x 0 ăx măx plj 1qa 1 µrpx ia, y ibq α 1 ą 1. µrpx pl jqa, y lbq j 1 Repeating the above arguments with µ replaced by µα 1, we get µα 1 α P S R pλq, where α P p1, 8q. Continuing in this way, we have µ ś 8 α i P S R pλq, where α i P p1, 8q. This implies that S R pλq is unbounded. This contradicts the boundness of S R pλq. The proof is complete. Corollary 3. Assume that if k ą l ą 0, (40) lim inf Rpx, yq r ą plq k1 c l k x,yñ8 if l ą k ą 0, (41) lim inf Rpx, yq r ą plq l1 c k l x,yñ8 3 and if k l ą 0, (4) lim inf x,yñ8 Rpx, yq r ą plq k1 c 1 c l c 3 L c 1 c k c 3 L c 1 c k c 3 L k k pk 1q k1, l l pl 1q l1, k k pk 1q k1, where L ś 8 m 1 p1 L mq. Then every solution of (1) () is oscillatory.

23 Oscillation for certain impulsive partial difference equations 101 Proof. For k ą l ą 0, from (40), we have c k l inf λpe R,x X 0,y Y 0 λ ą 1. l k l l x 0 ăx măx plj 1qa inf λpe R,x X 0,y Y 0 c k l Thus (34) is fulfilled. fulfilled, respectively. oscillatory. L λp L λrq k 1 λrx ia, y ib λrx pl jqa, y lb j 1 r c k l c4 L k1 c 1 c c 3 pk 1q k1 L k k Similarly, if (41) and (4) hold, (35) and (36) are Hence by Theorem 4, every solution of (1) () is References [1] R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 199. [] R. P. Agarwal, F. Karakoc, Oscillation of impulsive partial difference equations with continuous variables, Math. Comput. Modelling 50 (009), [3] R. P. Agarwal, P. J. Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic, Dordrecht, [4] D. D. Baĭnov, M. B. Dimitrova, A. B. Dishlies, Oscillation of bounded solutions of impulsive differential difference of second order, Appl. Math. Comput. 114 (000), [5] D. D. Baĭnov, E. Minchev, Forced oscillation of solutions of impulsive nonlinear parabolic differential difference equations, J. Korean Math. Soc. 35(4) (1998), [6] D. D. Baĭnov, E. Minchev, Oscillation of solutions of impulsive nonlinear parabolic differential difference equations, Internat. J. Theoret. Phys. 35(1) (1996), [7] D. D. Baĭnov, P. S. Simeonov. Impulsive Differential Equations Asymtotic Properties of the Solutions, World Scientific, Singapore, [8] B. T. Cui, Y. Liu, Oscillatory for partial difference equations with continuous variables, J. Comput. Appl. Math. 154 (003), [9] V. Lakshmikantham, D. D. Baĭnov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, [10] E. Minchev, Oscillation of solutions of impulsive nonlinear hyperbolic differential difference equations, Math. Balcanica 1(1 ) (1998), 15 4.

24 10 F. Özpinar, S. Öztürk, Z. F. Koçak [11] B. G. Zhang, Oscillation criteria of partial difference equations with continuous variables, Acta Math. Sinica 4(3) (1999), , (in Chinese). [1] B. G. Zhang, B. M. Liu, Oscillation criteria of certain nonlinear partial difference equations, Comput. Math. Appl. 38 (1999), [13] B. G. Zhang, B. M. Liu, Necessary and sufficient conditions for oscillation of partial difference equations with continuous variables, Comput. Math. Appl. 38 (1999), [14] B. G. Zhang, Y. Zhou, Qualitative Analysis of Delay Partial Difference Equations, Hindawi Publishing Corporation, New York, 007. [15] Y. Zhou, Existence of bounded and unbounded nonoscillatory solutions of nonlinear partial difference equations, J. Math. Anal. Appl. 33() (007), F. Özpinar BOLVADIN VOCATIONAL SCHOOL AFYON KOCATEPE UNIVERSITY AFYONKARAHISAR, TURKEY S. Öztürk DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND ARTS ANS CAMPUS AFYON KOCATEPE UNIVERSITY 0300 AFYONKARAHISAR, TURKEY Z. F. Koçak DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE AND LITERATURE MUGLA UNIVERSITY MUGLA, TURKEY Received February 9, 01.