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Comparison and oscillation results for delay difference equations with oscillating coefficients. (English) Zbl 0840.39006

The authors consider the following delay difference equation \[ x_{n - 1} - x_n + \sum^m_{i = 1} p_i (n) x_{n - k_i (n)} = 0,\;n \in \mathbb{N}_0, \tag{1} \] and the corresponding inequality \[ x_{n - 1} - x_n + \sum^m_{i = 1} p_i (n)x_{n - k_i (n)} \leq 0,\;n \in \mathbb{N}_0, \tag{2} \] where \(k_i (n) : \mathbb{N}_0 \to \mathbb{N}_0\), \(p_i (n) : \mathbb{N}_0 \to \mathbb{R} : = (- \infty, \infty)\). The fundamental assumptions are: (i) there exist positive \(k_1\), \(k_m \in \mathbb{N}_0\) such that \(k_1 \geq k_1(n) \geq k_2(n) \geq \cdots \geq k_m (n) \geq k_m > 0\) for \(n \in \mathbb{N}_0\); and (ii) \(p_1 (n) + p_2 (n) + \cdots + p_s(n) \geq 0\) holds for \(n \in \mathbb{N}_0\) and \(s = 1,2, \dots, m\), and for every \(N \in \mathbb{N}_0\) there exists an \(N_1 \in \mathbb{N}_0\) with \(N \leq N_1\) such that \(p_i (n) \geq 0\) holds for \(n = N_1\), \(N_1 + 1, \dots, N_1 + k_1\) and \(i = 1,2, \dots, m\).
Some comparison results on oscillation properties between solutions of (1) and (2) are proved. Then, using them and some known oscillation theorems due to G. Ladas, Ch. G. Philos and Y. G. Sficas [J. Appl. Math. Simulation, 2, No. 2, 101-112 (1989; Zbl 0685.39004)] and J. Yan and C. Qian [J. Math. Anal. Appl., 165, No. 2, 346-360 (1992; Zbl 0818.39002)], a necessary and sufficient condition and some sufficient conditions for oscillation of Eq. (1) are given. Some misprints are in the paper.

MSC:

39A12 Discrete version of topics in analysis
26D15 Inequalities for sums, series and integrals
39A10 Additive difference equations
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