Global behavior of a higher order nonlinear difference equation (Q1766720)

The authors consider the difference equation \[ x_{n+1} = f(x_n,x_{n-k}) \] for \(k>1\) fixed. The function \(f(u,v)\) is continuous with respect to both arguments on \((0,\infty)\), decreasing in \(u\) and increasing in \(v\), with \(f(\bar{x},v)/v\) non-increasing in \(v\), where \(\bar{x}\) is the unique positive equilibrium of the equation. Two kinds of global solutions are considered: oscillatory solutions and semi-cycles. Several results on existence of various global solutions are given, incorporating previous results concerning global behavior of equations that are special cases of the above.