Oscillation and Nonoscillation of Difference Equations with Several Delays
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Oscillation and Nonoscillation of Difference Equations with Several Delays Ba¸sak Karpuz
and Ioannis P. Stavroulakis
Abstract. Consider the delay difference equation Δx(n) +
m
pk (n)x(n − τk ) = 0
for n = 0, 1, . . . ,
k=1
where Δ is the forward difference operator, i.e., Δx(n) := x(n+1)−x(n), τk is a nonnegative integer and {pk (n)}∞ n=0 is a nonnegative sequence of reals for k = 1, 2, . . . , m. New oscillation and nonoscillation results, which essentially improve known results in the literature, are established. These results are extended to the more general difference equation Δx(n) +
m
pk (n)x(σk (n)) = 0
for n = 0, 1, . . . .
k=1
Examples illustrating the significance of the results are given. Mathematics Subject Classification. Primary 39A10; Secondary 39A21. Keywords. Oscillation, Nonoscillation, Difference equations.
1. Introduction In this paper, we study oscillation and nonoscillation of solutions of the delay difference equation Δx(n) +
m
pk (n)x(n − τk ) = 0
for n = 0, 1, . . . ,
(1)
k=1
where Δ is the forward difference operator, i.e., Δx(n) := x(n + 1) − x(n), for k = 1, 2, . . . , m, τk is a nonnegative integer and {pk (n)}∞ n=0 is a nonnegative sequence of reals. Without loss of generality, suppose that τ1 < τ2 < · · · < τm . By a solution of Eq. (1), we mean a sequence {x(n)}∞ n=−τm for which m x(n + 1) = x(n) − k=1 pk (n)x(n − τk ) holds for n = 0, 1, . . .. A solution 0123456789().: V,-vol
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B. Karpuz and I. P. Stavroulakis
MJOM
{x(n)}∞ n=−τm of Eq. (1) is said to be oscillatory if its terms are neither eventually positive nor eventually negative. Otherwise, the solution {x(n)}∞ n=−τm is called nonoscillatory. Let us quote from the literature some important results on the oscillation and nonoscillation of (1). In 1989, Erbe and Zhang [11] gave the first result based on both lower and upper limit conditions. The result reads as follows. Theorem A [11, Theorem 4.4]. Assume lim inf n→∞
m
pk (n) =: α > 0
and
lim sup n→∞
k=1
m
pk (n) > 1 − α.
k=1
Then, every solution of (1) is oscillatory. In the same paper, Erbe and Zhang [11] also gave the following result, which can be regarded as sharp in some sense. Theorem B [11, Theorem 4.1]. Assume m (τk + 1)τk +1
τkτk
k=1
lim inf pk (n) > 1. n→∞
Then, every solution of (1) is oscillatory. In 1998, Tang and Deng [24] improved Theorem B by giving the following pointwise result. Theorem C [24, Theorem 1]. Assume lim inf n→∞
m (τk + 1)τk +1 k=1
τkτk
pk (n) > 1.
Then, every solution of (1) is oscillatory. In 1999, Tang and Yu [25] replaced the coefficients with their arithmetic means, and improved Theorem C as follows. Theorem D [25, Corollary 4]. Assume τ +1 n+τ m k τk + 1 k lim inf pk (i) > 1. n→∞ τk i=n+1 k=1
Then, every solution of (1) is oscillatory. Finally, in 2001, Tang and Zhang [26] gave the following upper limit condition. Theorem E [26, Theorem 4.1]. Assume lim sup n→∞
m n+τ k
pk (i) > 1.
k=1 i=n
Then, every solution of (1) is oscillatory.
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Difference Equations
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