A Convergence Criterion for the Solutions of Nonlinear Difference Equations and Dynamical Systems

A general sufficient condition for the convergence of subsequences of solutions of non-autonomous, nonlinear difference equations and systems is obtained. For higher order equations the delay sizes and patterns play essential roles in determining which subsequences of solutions converge. For systems the specific manner in which the equations are related is important and lead to different criteria. Applications to discrete dynamical systems, including some that model populations of certain species are discussed.