Oscillation criteria for quasi-linear functional dynamic equations on time scales

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation

$$(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$$

on a time scale \(\mathbb{T}\) where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on \(\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}\) and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty \). We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.