Asymptotic behavior of solutions for a class of difference equations
Let the functions \(F\), \(G\in C(\mathbb{R},\mathbb{R})\) and let \(F\) be increasing. If \(F(y)\geq G(y)\) \([F(y)\leq G(y)]\) for all \(y\in\mathbb{R}\), then each solution of the nonlinear difference equation \[ x_n-x_{n-1}=-F(x_n)+G(x_{n-k}),\quad n\geq n_0, \] with positive integer \(k\), tends to either a constant or \(-\infty\) \([+\infty]\) as \(n\to\infty\). Some illustrative examples are included.
scientific article; zbMATH DE number 1005602