On a higher order neutral difference equation. (English) Zbl 0976.39004
Rassias, Themistocles M. (ed.), Mathematical analysis and applications. Palm Harbour, FL: Hadronic Press. 37-64 (2000).
Consider the neutral delay difference equation
\[
\Delta^m (x_n-p_nx_{n-\tau})+q_nx_{n-\sigma}=0,\quad n=0,1,\dots\tag{*}
\]
where \(m\) is an odd integer greater or equal to 3, \(\tau\) is a positive integer and \(\sigma\) is a nonnegative integer, the sequences \(\{p_n\}\) and \(\{q_n\}\) are real.
After a lot of preparatory lemmas on the properties of sequences sufficient conditions (some of them also necessary) in order that every solution to (*) be oscillatory as well as criteria for the existence of eventually positive solutions are established.
For the entire collection see [Zbl 0958.00011].
After a lot of preparatory lemmas on the properties of sequences sufficient conditions (some of them also necessary) in order that every solution to (*) be oscillatory as well as criteria for the existence of eventually positive solutions are established.
For the entire collection see [Zbl 0958.00011].
Reviewer: Dobiesław Bobrowski (Poznań)
MSC:
39A11 | Stability of difference equations (MSC2000) |