Oscillation of second-order damped dynamic equations on time scales

Abstract

The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. This way results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained. In this paper, by employing the Riccati transformation technique we will establish some oscillation criteria for second-order linear and nonlinear dynamic equations with damping terms on a time scale T . Our results in the special case when T = R and T = N extend and improve some well-known oscillation results for second-order linear and nonlinear differential and difference equations and are essentially new on the time scales T = h N , h > 0 , T = q N for q > 1 , T = N 2 , etc. Some examples are considered to illustrate our main results.