Oscillation of third order differential equation with damping term

Abstract

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ⩽ 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.