On Asymptotic and Oscillatory Behavior of Third-Order Delay Dynamic Equations

Abstract We establish conditions ensuring the oscillatory and asymptotic behavior of solutions to a linear third-order delay dynamic equation on time scales. Our approach combines Fubini-type result for triple integrals on time scales with an iterative construction of auxiliary functions. In addition, we derive two-sided bounds for Kneser-type solutions. The results extend the existing oscillation theory for dynamic equations and are new even in the continuous case. The effectiveness of our main results is illustrated by an application to a generalized Euler-type equation on time scales.