Asymptotic integration of second-order differential equations: Levinson-Weyl theory, Poincaré-Perron property, Lyapunov type numbers, and dichotomy

Abstract

We discuss asymptotic behavior of solutions to a class of second order semi-linear ordinary differential equations. This equation exhibits dichotomic behavior in the sense that some solutions are small and some are large at infinity. The evolution of solutions to semi-linear equation mimics the dynamical pattern of the associated linear equation. Under flexible hypotheses, asymptotic representations for asymptotically small and asymptotically large solutions and their derivatives are obtained in terms of test functions introduced in the paper. Behavior of bounded solutions to the equation under study is also discussed. The main tools are direct analysis and Schauder-Tikhonov fixed point theorem.