Oscillation properties of solutions of second order neutral dynamic equations of non-canonical type on time scales (Q2064116)

In this paper, oscillatory properties of a class of second-order sub-linear and super-linear neutral dynamic equations on time scales of the form \[ [r(t)(z^\Delta(t))^\nu]^\Delta+q(t)y^\mu(\tau(t))=0 \] are investigated via Krasnoselskii's fixed point theorem and several inequalities, where \[ z(t)=y(t)+p(t)y(m(t)), \] and that \(\nu,\mu\) are quotients of odd positive integers and \(-1<p_1\leq p(t)\leq0\). The main result of the paper is on the resriction: For the oscillation of solutions, the unboudedness is necessary but it is not necessary to say that the solutions are almost oscillatory. Some examples are given to emphasize the main results which are new in case of \(\mathbb{T}=\mathbb{R}\) and \(\mathbb{T}=\mathbb{Z}\).