Oscillatory Behaviour of Higher-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Abstract

In this paper, new sufficient conditions are investigated for the oscillation of solutions of higher order dynamic equations
\begin{equation*}
\bigl[r(t)(z^{\Delta n-1}(t))^\alpha\bigr]^\Delta+q(t)f(x(\delta(t)))=0
\quad\text{for}\ t\in[t_{0},\infty)_{\T},
\end{equation*}
where $z(t):=x(t)+p(t)x(\tau(t))$ and $\alpha\geq1$ is a quotient of odd positive integers. Under less restrictive assumptions for the neutral coefficient, we present new comparison theorems and Generalized Riccati technique.