Oscillatory behavior of solutions of third order semi-canonical dynamic equations on time scale

Abstract

This paper investigates the oscillatory behavior of nonlinear third-order dynamic equations on time scales. Our main approach is to transform the equation from its semi-canonical form into a more tractable canonical form. This transformation simplifies the analysis of oscillation behavior and allows us to derive new oscillation criteria. These criteria guarantee that all solutions to the equation oscillate. Our results extend and improve upon existing findings in the literature, particularly for the special cases where $ \mathbb{T} = \mathbb{R} $ and $ \mathbb{T} = \mathbb{Z} $. Additionally, we provide illustrative examples to demonstrate the practical application of the developed criteria.

Keywords

• third order
• oscillatory
• semi-canonical
• non-canonical
• canonical