This paper derives new oscillation criteria for a class of second-order non-canonical advanced dynamic equations of the form

$${\left(\zeta (\ell ){\varkappa }^{\mathrm{\Delta }}(\ell )\right)}^{\mathrm{\Delta }}+q(\ell )\varkappa (\mathrm{\wp }(\ell ))=0.$$

The derived results are based on establishing dynamic inequalities, which lead to novel monotonicity properties of the solutions. These properties are then used to derive new oscillatory conditions. This approach has been successfully applied to difference and differential equations due to the sharpness of its criteria. However, no analogous studies have adopted a similar methodology for dynamic equations on time scales. Furthermore, this study includes examples to illustrate the importance and sharpness 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.

This work presents new Kneser-type oscillation criteria for second-order quasilinear functional dynamic equations defined on arbitrary unbounded above time scales. Our approach employs the Riccati transformation technique in conjunction with the integral averaging method. The results show a significant improvement over recent Kneser-type oscillation criteria. We provided several illustrative examples to highlight the importance of our findings.