Asymptotic stability of two dimensional systems of linear difference equations and of second order half-linear differential equations with step function coefficients

摘要

We give a sufficient condition guaranteeing asymptotic stability with respect to $x$ for the zero solution of the half-linear differential equation [x''|x'|^{n-1} + q(t)|x|^{n-1}x=0, qquad 1le n in mathbb{R},] with step function coefficient $q$. The geometric method of the proof can be applied also to two dimensional systems of linear non-autonomous difference equations. The application gives a new simple proof for a sharpened version of 'A. Elbert's asymptotic stability theorems for such difference equations and linear second order differential equations with step function coefficients.