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Necessary and sufficient conditions for oscillations of difference equations with several delays. (English) Zbl 0808.39004

Necessary and sufficient conditions for the oscillation of all solutions of the delay difference equation \(x_{n+1} - x_ n + \sum^ m_{i = 1} p_ i x_{n - k_ i} = 0\), \(n = 0, 1, \dots,\) are obtained. Using these results it is proved that the inequality \(y_{n + 1} - y_ n + \sum^ n_{k = 0} q_{n - k} y_ k \leq 0\), \(n = \ell\), \(\ell + 1, \dots\), cannot have a positive solution.

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
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