Oscillation conditions for difference equations with a monotone or nonmonotone argument (Q1727314)

Summary: Consider the first-order delay difference equation with a constant argument \(\Delta x(n)+p(n)x(n-k)=0\), \(n=0,1,2,\dots\), and the delay difference equation with a variable argument \(\Delta x(n)+p(n)x(\tau(n))=0\), \(n=0,1,2,\dots\), where \(p(n)\) is a sequence of nonnegative real numbers, \(k\) is a positive integer, \(\Delta x(n)=x(n+1)-x(n)\), and \(\tau(n)\) is a sequence of integers such that \(\tau(n)\leq n-1\) for all \(n\geq0\) and \(\lim_{n\rightarrow\infty}\tau(n)=\infty\). A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.