Small solutions of the damped half-linear oscillator with step function coefficients

摘要

In this paper we consider the damped half-linear oscillator x 00|x 0 | n?1 + c(t)|x 0 | n?1 x 0 + a(t)|x| n?1 x = 0, n ∈ R +. We give a sufficient condition guaranteeing the existence of a small solution, that is a non-trivial solution which tends to 0 as t tends to infinity, in the case when both damping and elasticity coefficients are step functions. With our main theorem we not just generalize the corresponding theorem for the linear case n = 1, but we even sharpen Hatvani’s theorem concerning the undamped half-linear differential equation.