What question did this study set out to answer?

This research aims to establish conditions for the oscillatory and asymptotic behavior of solutions to third-order delay dynamic equations.

May 20, 2026Open Access

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

Key Points

This research aims to establish conditions for the oscillatory and asymptotic behavior of solutions to third-order delay dynamic equations.
Developed conditions for oscillatory behavior using Fubini-type results and auxiliary functions
Derived two-sided bounds for Kneser-type solutions
Applied findings to a generalized Euler-type equation on time scales.
Conditions established ensure oscillatory behavior of solutions to dynamic equations
Results extend the current oscillation theory even in continuous cases
Illustrated effectiveness through application to Euler-type equations.

Abstract

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.

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