In this study, we derive new criteria that ensure the oscillation of solutions to noncanonical dynamic equations that are half-linear sublinear functional. These results not only resolve an open issue in numerous works in the literature but also emulate Ohriskatype and Hille-type criteria for canonical dynamic equations. We provide examples to demonstrate the accuracy, usefulness, and flexibility of the main results.

We study the Kneser-type oscillation criterion for a class of second-order half-linear functional dynamic equations on an arbitrary time scale utilizing the integral averaging approach and the Riccati transformation method. The results show an improvement in Kneser-type when compared to some known results. We provide some illustrative examples to demonstrate the significance of our main results.

We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type $(p,q)$-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the $(p,q)$-fractional setting.