Classification of nonoscillatory solutions of higher order nonlinear difference equations (Q2731131)

The paper deals with the equation NEWLINE\[NEWLINE\Delta^m \bigl(x(n)+ px(n-\tau) \bigr)+f \biggl(n,x \bigl(n- \sigma_1(n) \bigr), \dots,x \bigl(n-\sigma_v (n)\bigr) \biggr)=0NEWLINE\]NEWLINE where \(m\geq 2\), \(p\geq 0\), \(p\neq 1\), \(\tau>0\), \(\sigma_i \in \mathbb{N}_0\), \(n-\sigma_i(n) \to\infty\) for \(i=1,\dots,v\), and \(f\) continuous. The nonoscillatory solutions \(x(n)\) are classified with respect to their asymptotic behaviour for \(n\to\infty\). It is proved that, for even \(m\), they can belong to one of three types and, for odd \(m\), to one of five types. For \(p>0\) and \(f\) superlinear or sublinear in five theorems there are given necessary and sufficient conditions for \(x\) belonging to a concrete type.