Oscillation Criteria for Delay Neutral Difference Equations with Positive and Negative Coefficients. Contents

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1 Bol. Soc. Paran. Mat. (3s.) v. 21 1/2 (2003): c SPM Oscillation Criteria for Delay Neutral Difference Equations with Positive and Negative Coefficients Chuan-Jun Tian and Sui Sun Cheng abstract: This paper is concerned with a class of neutral type difference equations with positive and negative coefficients of the form (x n r nx n m ) + p nx n k q nx n l = 0, n = 0, 1, 2,..., where m, k and l are nonnegative integers, and {p n}, {q n} as well as {r n} are nonnegative real sequences. Novel oscillation criteria for this equation are derived. Contents 1 Introduction 1 2 Oscillation Criteria 2 3 Examples Introduction Let x n be the state variable of a stage dependent process where n belongs to a set of consecutive integers. In case the change x n = x n+1 x n depends on the past changes as well as the past state sizes, we may then write x n = F ( x, x n 2,...; x n, x,...). In this paper, we consider a special case in the form of a neutral difference equation with positive and negative coefficients where we assume that (x n r n x n m ) + p n x n k q n x n l = 0, n = 0, 1, 2,..., (1) (i) m, k and l are three integers such that 0 l < k and m > 0, (ii) {p n } n=0, {q n } n=0 and {r n } n=0 are nonnegative sequences, and (iii) the sequence {h n } defined by is nonnegative and has a positive subsequence. h n = p n q n k+l, n k l, (2) 1991 Mathematics Subject Classification: 39A10 The first author is partially supported by the National Natural Science Foundation of China and partially by Natural Science Fund of ShenZhen. 1 Typeset by B S P M style. c Soc. Paran. Mat.

2 2 Chuan-Jun Tian and Sui Sun Cheng By a solution of (1), we mean a real sequence {x n } which is defined for n µ and satisfies (1) for n 0, where µ = max {m, k}. Let Ω = { µ, µ + 1,, 1, 0}. It is easy to see that for any given sequence {φ n } defined on Ω, there exists a solution {x n } of (1) with x n = φ n for n Ω. A solution {x n } of (1) is said to be eventually positive (or eventually negative) if x n > 0 (or x n < 0) for all large n. It is said to be oscillatory if it is neither eventually positive nor eventually negative. Among the existing studies, the oscillation of Eq. (1) has been discussed by several authors, see for example [1-7]. In particular, Ladas [1] and Qian and Ladas [2] considered the case where r n 0. Chen and Zhang [3] and Zhang and Wang [4] considered the case where r n r with 0 r < 1. Recently, Li and Cheng [5,6] and Tang et al. [7] considered the general case in which one of their main results is obtained under assumptions such as r n + k+l q s = 1. (3) In this paper, we are concerned with oscillation criteria for (1) which do not make use of (3). Our approach is new and is based on the monotonicity of an associated sequence of a solution as well as a functional inequality satisfied by it. As a result, our results can be used to show oscillation when the previous results cannot. In the sequel, empty sum will be taken to be zero as usual. 2. Oscillation Criteria For any integer t {0, 1,, k l}, let R n (t) = r n + In particular, if t = 0, then (4) is just and if t = k l, then (4) is t n t+k l 1 q s + p s. (4) n+k l 1 R n (0) = r n + p s, R n (k l) = r n + k+l q s. Let { m, if qn 0 and t = k l = max {m, k}, otherwise

3 Oscillation Criteria for Delay Neutral Difference... 3 and Obviously, m β(t). { m, if qn 0 and t = k l β(t) = min {m, l + 1}, otherwise Lemma 2.1. Suppose there exists an integer t {0, 1,, k l} such that R n (t) = r n + t n t+k l 1 q s + p s 1, (5) for all large n. Suppose further that {x n } is an eventually positive solution of the functional difference inequality Then the sequence {z n } defined by (x n r n x n m ) + p n x n k q n x n l 0. (6) z n = x n r n x n m t n+k l t 1 q s x s l p s x s k (7) for all large n will satisfy z n 0 and z n > 0 eventually. Proof. Let N 1 > µ such that x n > 0 for n N 1 µ. Then, from (6) and (7), we have z n (p n t+k l q n t ) x n t l = h n t+k l x n t l 0, n N 1. (8) Hence z n is nonincreasing for n N 1. If {z n } is not eventually positive, then in view of our assumption on {h n }, z n < 0 for all large n. Thus there exist an integer N 2 N 1 and a constant c > 0 such that z n < c for n N 2. Therefore, from (7), we have x n c + r n x n m + t n+k l t 1 q s x s l + p s x s k, n N 2. (9) If the solution {x n } is unbounded, then lim sup x n =. Hence there exists a sequence of integers {n i } i=1 such that n i N 2 + for i = 1, 2, 3,..., lim i n i =, lim i x ni = and x ni = max {x n N 2 n n i } for i = 1, 2, 3,.... From (5), (9) and the above equalities, we have x ni c + r ni x ni m + which is a contradiction. n i 1 i t n i+k l t 1 q s x s l + p s x s k c + x ni < x ni, i.

4 4 Chuan-Jun Tian and Sui Sun Cheng If {x n } is bounded, then there exists a constant θ 0 such that lim sup x n = θ <. Choose a sequence of integers {n i } i=1 such that lim i n i = and lim i x ni = θ. It is easy to see that there exists a sequence of integers {λ i } i=1 such that n i λ i n i β(t) and x λi = max {x n n i n n i β(t)}. Then lim i λ i = and lim sup x λi θ. Hence, from (5) and (9), we have x ni c + r ni x ni m + n i 1 i t n i+k l t 1 q s x s l + p s x s k c + x λi. i Taking the superior limit as i on both sides, we obtain θ c + θ < θ, which is also a contradiction. The proof is complete. Lemma 2.2. Suppose there exists an integer t {0, 1,, k l} such that R n (t) = r n + t n t+k l 1 q s + p s 1 (10) for all large n. Suppose further that the following second-order difference inequality 2 y n + 1 α (t) h n t+k ly n 0 (11) does not have any eventually positive solution. Then for any eventually positive solution {x n } of (6), the sequence {z n } defined by (7) satisfies z n < 0 and z n 0 for all large n. Proof. We will only consider the case where the conditions q n 0 and t = k l do not hold simultaneously, since the other case can be dealt with in a similar fashion. Then = max {m, k}. From (6) and (7), (8) holds for all large n. Hence z n 0 for all large n. Suppose to the contrary that {z n } is eventually positive, then there is an integer T > such that x n > 0, z n > 0 and z n 0 for any n T. Let M = min { x T, x T +1,, x T } > 0. Then, in view of (7) and (10), we have x T = z T + r T x T m + M ( r T + T 1 s=t t T 1 s=t t T +k l t 1 q s x s l + p s x s k s=t ) T +k l t 1 q s + p s M. s=t Hence x T +1 M ( r T +1 + T s=t +1 l ) T +k l t q s + p s M. s=t +1

5 Oscillation Criteria for Delay Neutral Difference... 5 In general, we can obtain x n M ( r n + t By induction, we can further obtain Hence ) n+k l t 1 q s + p s M, T n T +. x n M, T + (s 1) n T + s, s = 1, 2,.... x n M for any n T. (12) In view of the monotonicity of {z n }, we may set lim z n = c. If c = 0, there exists an integer T 1 > T such that z n < M/2 holds for n > T 1. Hence, from (12), we have x n 1 n+ 1 z s, T 1 n T 1 +. s=t 1 If c > 0, then z n c for n T. In view of (7), (10) and (12), we have x n c + r n x n m + In general, we get t n+k l t 1 q s x s l + p s x s k c + M, n T 1. x n sc + M, n T 1 + (s 1), s = 1, 2,.... Hence lim x n =. Thus there exists an integer T 2 > T 1 such that x n 1 n+ 1 z s, T 2 n T 2 +. s=t 2 In general, there exists an integer N T 2 such that x n 1 n+ 1 z s, N n N +. (13) For N + n N + + β(t), in view of the monotonicity of {z n }, from (7),

6 6 Chuan-Jun Tian and Sui Sun Cheng (10) and (13), x n = z n + r n x n m + z n + = By induction, we obtain x n 1 n+ 1 z s, for j = 1, 2,.... Hence ( r n + t t n+k l t 1 q s x s l + p s x s k n+k l t 1 q s + n+ 1 1 z s + 1 z s n+ 1 1 z s. p s ) 1 z s N + + (j 1) β(t) n N + + jβ(t), x n 1 n+ 1 z s, n N. (14) Let y n = z s. Then y n > 0 for all large n, and y n = z n and 2 y n = z n for n N. From (8) and (14), in view of t + l, we have z n h n t+k l x n t l h n t+k l and so 2 y n + 1 h n t+k ly n 0, n t l+ 1 z s h n t+k l n N +, i.e., {y n } is an eventually positive solution of (11). This is contrary to our assumption. The proof is complete. Theorem 1 Assume there exist two integers t 1, t {0, 1,, k l} such that and R n (t 1 ) = r n + R n (t) = r n + t 1 q s + t n t 1+k l 1 z s, p s 1 (15) n t+k l 1 q s + p s 1 (16) for all large n. Further assume that the functional inequality (11) does not have any eventually positive solution. Then every solution of (1) oscillates.

7 Oscillation Criteria for Delay Neutral Difference... 7 Indeed, from Lemma 2.1 and (15), for every eventually positive solution {x n } of (1), the sequence {z n } defined by (7) is eventually positive. But in view of (11), (16) and Lemma 2.2, the sequence {z n } is eventually negative, thus we obtain a contradiction. Thus (1) cannot have any eventually positive, nor any eventually negative, solutions. We remark that there are many sufficient conditions which will guarantee the nonexistence of eventually positive solutions of (11). For instance, the following result is taken from [7]: If {d n } n=0 is a nonnegative sequence and then the following functional inequality lim inf n d s > 1 4, (17) 2 y n + d n y n 0, n = 0, 1, 2,... (18) does not have any eventually positive solutions. Consequently, if (15) and (16) hold for t 1, t and all large n and if lim inf n (p s+k l q s ) > 4, (19) then every solution of (1) oscillates. Corollary 2.1. Assume there exists an integer t {1,, k l 1} such that (16) holds for all large n, and lim inf n (p s+k l q s ) > max{m, k}, 4 holds. Further assume that {q n / (p n+k l q n )} and {p n / (p n q n k+l )} are nondecreasing and there exist two nonnegative constants δ 1 and δ 2 such that r n k (p n q n k+l ) δ 1 (p n m q n k+l m ), (20) q n k (p n q n k+l ) δ 2 (p n l q n k ), (21) p n (p n+t+l q n+t k+2l ) δ 2 (p n q n k+l ), (22) for all large n and δ 1 + (k l) δ 2 = 1. Then every solution of Eq. (1) oscillates. Proof. Suppose to the contrary that Eq.(1) has an eventually positive solution {x n } and let {z n } be defined by (7). From Lemma 2.2, we have z n < 0 and z n 0

8 8 Chuan-Jun Tian and Sui Sun Cheng eventually. In view of (20), (21) and (22), z n = h n t+k l x n t l = h n t+k l (z n t l + r n t l x n t l m ) Thus ( n t l 1 h n t+k l 2t l n 2t+k 2l 1 q s x s l + t l h n t+k l z n t l δ 1 h n t+k l m x n t l m n t l 1 h n t+k l 2t l p s x s k ) n 2t+k 2l 1 q s h s+k l x s l h n t+k l h s+k l t l h n t+k l z n t l δ 1 h n t+k l m x n t l m q n t lh n t+k l h n t+k 2l p n 2t+k 2lh n t+k l h n 2t+k 2l n t l 1 2t l ( z s+t ) n 2t+k 2l 1 t l ( z s+t k+l ) n t l 1 h n t+k l z n t l + δ 1 z n m + δ 2 n 2t+k 2l 1 +δ 2 t l z s+t k+l 2t l z s+t p s h s x s k h s = h n t+k l z n t l + δ 1 z n m + δ 2 (z n l z n t l ) + δ 2 (z n t l z n k ) h n t+k l z n k + δ 1 z n m + δ 2 z n l δ 2 z n k = (h n t+k l + δ 2 ) z n k + δ 1 z n m + δ 2 z n l. (z n δ 1 z n m ) + (h n t+k l + δ 2 ) z n k δ 2 z n l 0, which implies { z n } is an eventually positive solution of the inequality (x n δ 1 x n m ) + (h n t+k l + δ 2 ) x n k δ 2 x n l 0. (23) In view of Lemma and δ 1 + (k l) δ 2 = 1, (23) has no eventually positive solution, thus we will obtain a contraction. In fact, if {x n } is an eventually positive solution of (23) and let r n = δ 1, p n = h n+k l + δ 2 and q n = δ 2, then R n (k l) = r n + k+l q s = δ 1 + (k l)δ 2 = 1 1, n+k l 1 n+k l 1 R n (0) = r n + p s = δ 1 + (k l)δ 2 + h s+k l 1,

9 Oscillation Criteria for Delay Neutral Difference... 9 and from the known condition, lim inf n (p s+k l q s ) > max{m, k} 4 = α(0) 4. Hence from Lemma 2.1, z n < 0 defined by (7) is negative. But from Lemma 2.2, z n > 0. We obtain a contradiction. The proof is complete. Corollary 2.2. Assume that (16) holds for t = 0 and all large n, and lim inf n (p s+k l q s ) > max{m, k} 4 holds. Further assume that {p n / (p n q n k+l )} is nondecreasing and there exist two nonnegative constants δ 1 and δ 2 such that δ 3 p n q n k+l eventually and for all large n, r n k (p n q n k+l ) δ 1 (p n m q n k+l m ), (24) p n (p n+l q n k+2l ) δ 3 (p n q n k+l ), (25) and δ 1 + (k l) δ 3 = 1 and h n < δ 3 eventually. Then every solution of Eq.(1) oscillates. Indeed, by arguments similar to those in the proof of Corollary 2.1, we obtain, z n (δ 3 h n+k l ) z n l + δ 1 z n m δ 3 z n k. Hence { z n } is an eventually positive solution of the inequality (x n δ 1 x n m ) + δ 3 x n k (δ 3 h n+k l ) x n l 0, which yields a contradiction by Lemma 2.1 and 2.2. Similarly, we can obtain the following result. Corollary 2.3. Assume that (16) holds eventually and (19) holds for t = k l. Further assume that {q n / (p n+k l q n )} is nondecreasing and there exist two nonnegative constants δ 1 and δ 2 such that for all large n, r n k (p n q n k+l ) δ 1 (p n m q n k+l m ), (26) q n k (p n q n k+l ) δ 2 (p n l q n k ), (27) and δ 1 + (k l) δ 2 = 1. Then every solution of Eq.(1) oscillates. Theorem 2 Suppose there exists an integer t {0, 1,, k l} such that (3) holds eventually. Suppose further that holds and that lim inf n (p s+k l q s ) > max(m, t + l) 4 r n k (p n q n k+l ) p n m q n k+l m (28) for all large n. Then every solution of Eq.(1) is oscillatory.

10 10 Chuan-Jun Tian and Sui Sun Cheng Proof. Suppose to the contrary that {x n } is an eventually positive solution of (1). Then, in view of Lemma 2.1, z n > 0 and z n 0 for all large n. In view of (1), we have Hence z n = h n t+k l x n t l = h n t+k l (z n t l + r n t l x n t l m ) ( n t l 1 h n t+k l 2t l n 2t+k 2l 1 q s x s l + t l p s x s k ) z n + h n t+k l z n t l + r n t l h n t+k l x n t l m 0 (29) for all large n. From (1), we have z n m + h n t+k l m x n t l m = 0. In view of (29) and the above equality, we obtain (z n z n m )+h n t+k l z n t l {h n t+k l m r n t l h n t+k l } x n t l m 0, i.e. {z n } is an eventually positive solution of the recurrence relation (z n z n m ) + h n t+k l z n t l 0, which yields a contradiction by Lemma 2.1 and 2.2. The proof is complete. If the assumption (28) is not satisfied, the following result is available. Theorem 3 Suppose there exists an integer t {0, 1,, k l} such that (3) holds eventually. Further assume that there exists a constant c [0, 1) such that r n k (p n q n k+l ) c (p n m q n k+l m ). (30) Then every solution of Eq.(1) is oscillatory provided that there exists a constant c [0, c) such that the following recurrence relation u n + c 1 c h n t+k lu n t l m 0, n = 0, 1, 2,... (31) does not have an eventually positive solution. Proof. Suppose to the contrary that {x n } is an eventually positive solution of (1). Then, from Lemma 2.1, we have z n > 0 and z n 0 for all large n. As in the proof of Theorem 2, we get (z n cz n m ) + h n t+k l z n t l 0,

11 Oscillation Criteria for Delay Neutral Difference for all large n. Let u n = z n cz n m. Similar to the proof of Lemma 2.1, we have u n > 0 and u n 0 for all large n. Hence there exists an integer N > 0 such that z n > 0 and z n 0, and u n > 0 and u n 0 for n N. Thus z n = u n + cz n m = u n + c (u n m + cz n 2m ) = = u n + cu n m + + c i u n im + cz n (i+1)m ( c + c c i) u n m = c ( 1 c i+1) 1 c u n m for n (i + 1) m + N. Hence (31) holds for all large n, which is contrary to the hypothesis that (31) has no eventually positive solution. The proof is complete. 3. Examples In this section, we give two examples to illustrate our results. Example 3.1. Consider the neutral difference equation (x n r n x n m ) + p n x n k q n x n l = 0, n = 1, 2,, (32) where k = l + 1 and m > 0, l 0, and r n = 1 c, p n = c and q n = c (n + 1) d for n {0, 1, 2, }, where c (0, 1) and d is a real constant. Let t 1 = 1 and t = 0. Then it is easy to see that for any positive integer n, and R n (t 1 ) = R n (1) = r n + q = 1 n d < 1 R n (t) = R n (0) = r n + p n = 1 1. Since h n = p n q = n d satisfies ( ) ( lim inf n h s+k l = lim inf n +1 s d ) = {, d (, 2) 1, d = 2. Hence by Theorem 1, every solution of (32) oscillates when d (, 2) or d = 2 and max{m, k} < 4. But it seems difficult to obtain the same conclusion from the results in [5-7]. Example 3.2. Consider the neutral difference equation (32), where m = l = 1, k = 3 and r = 0.5, and Take t 1 = 2 and t = 0, then p n = (n + 1) 2, n = 0, 1, 2,..., q n = 0.25 (n + 3) 2, n = 0, 1, 2,.... R n (t 1 ) = R n (2) = r n + q n 2 + q = 1 (n + 1) 2 (n + 2) 2 < 1

12 12 Chuan-Jun Tian and Sui Sun Cheng and R n (t) = R n (0) = r n + p n + p n+1 = 1 + (n + 1) 2 + (n + 2) 2 > 1 for all large n, and h n = p n q n 2 = 2 (n + 1) 2. Thus ( ) ( ) lim inf n h s+k l = 2 lim inf n s 2 = 2 > max(m, k) =. 4 Hence by Theorem 1, every solution of (32) oscillates. But it seems difficult to obtain the same conclusion from the results in [5-7]. References 1. G. Ladas, Oscillation of difference equations with positive and negative coefficients, Rocky Mountain J. Math., 20 (1990), C. X. Qian and G. Ladas, Oscillatory behavior of difference equations with positive and negative coefficients, Le Math., XLIV (1989), M. P. Chen and B. G. Zhang, Oscillation and comparison theorems of difference equations with positive and negative coefficients, Bull. Institute Math. Acad. Sinica 22(4) (1994), B. G. Zhang and H. Wang, The existence of oscillatory and nonoscillatory solutions of neutral difference equations, Chinese J. Math. 24(4) (1996), W. T. Li and S. S. Cheng, On a neutral difference equation with positive and negative coefficients, Southeast Asia Bull. Math., 22 (1998), W. T. Li and S. S. Cheng, An oscillation theorem for a neutral difference equation with positive and negative coefficients, Tamkang J. Math., 30(1) (1999), X. H. Tang, J. S. Yu and D. H. Peng, Oscillation and nonoscillation of neutral difference equations with positive and negative coefficients, Comput. Math. Applic., 39 (2000), Chuan-Jun Tien College of Information Engineering Shenzhen University Shenzhen P. R. China Sui Sun Cheng Department of Mathematics Hsinchu, Taiwan R. O. China