Oscillatory and nonoscillatory behavior of nonlinear delay dynamic equations on time scales

This paper is devoted to the qualitative analysis of first-order nonlinear delay dynamic equations on time scales. We establish sufficient conditions for oscillation and nonoscillation of solutions to dynamic equations of the form$ x^{\Delta}(t) + p(t)\big(x(\tau(t))\big)^{\alpha} = 0, \quad t \ge t_0, $where $ \alpha $ is a quotient of two odd integers and $ p $, $ \tau $ satisfy standard regularity and delay conditions on an arbitrary time scale $ \mathbb{T} $. Employing comparison principles, the time scales version of L'Hôpital's rule, and integral transformations, we derive new oscillation criteria that generalize several classical results from the differential and difference equation settings to the unified framework of time scales. Furthermore, we provide sufficient conditions ensuring the existence of eventually positive (nonoscillatory) solutions. Our results extend and complement the existing theory in the literature.