Oscillation properties for advanced difference equations.

Öcalan, Özkan; Akin, Ömer

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This paper is concerned with the nonlinear advanced difference equation with constant coefficients $x_{n + 1} - x_{n} + \sum_{i = 1}^{m} p_{i} f_{i} (x_{n - k_{i}}) = 0, n = 0, 1, \dots$ where $p_{i} \in (- \infty, 0)$ and $k_{i} \in {\dots, - 2, - 1}$ for $i = 1, 2, \dots, m$ . We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients $x_{n + 1} - x_{n} + \sum_{i = 1}^{m} p_{i n} f_{i} (x_{n - k_{i}}) = 0, n = 0, 1, \dots$ where $p_{i n} \leq 0$ and $k_{i} \in {\dots, - 2, - 1}$ for $i = 1, 2, \dots, m$ . ...