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].