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Oscillation of second-order damped dynamic equations on time scales. (English) Zbl 1128.34022

The authors study the oscillation of the solutions of the nonlinear second order dynamic equation with damping

${\left(a\left(t\right){x}^{{\Delta }}\left(t\right)\right)}^{{\Delta }}+p\left(t\right){x}^{{{\Delta }}_{\sigma }}\left(t\right)+q\left(t\right)\left(f\circ {x}^{\sigma }\right)=0$

on a time scale $𝕋,$ that is, on a nonempty closed subset of the real numbers. (For the definition of (delta) derivative and other related notions on dynamic equations, the reader is referred to [M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications. Basel: Birkhäuser (2001; Zbl 0978.39001)]). By imposing appropriate conditions (too involved to be described here) to the maps $a,p,q$ and $f,$ the authors establish a series of results ensuring the oscillatory character of the above mentioned dynamic equation. It is worth mentioning that by using this general approach of time scales, the authors unify the study of differential and difference equations (when $𝕋=ℝ$ and $𝕋=ℕ,$ respectively), and extend and improve some known results existing already in the literature. Moreover, they obtain new results for the time scales $𝕋=hℕ$, $h>0,\phantom{\rule{0.166667em}{0ex}}𝕋={q}^{ℕ},$ $q>1,$ among others.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 39A12 Discrete version of topics in analysis