Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation

Abstract This paper examines the oscillatory behavior of a class of third-order semi-canonical nonlinear delay dynamic equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msup> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>a</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>ℓ</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mfenced open="[" close="]"> <m:msup> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>r</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>ℓ</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>ϰ</m:mi> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>ℓ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mfenced> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:msup> </m:mfenced> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>q</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>ℓ</m:mi> <m:mo stretchy="false">)</m:mo> <m:msup> <m:mi>ϰ</m:mi> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msup> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">℘</m:mi> <m:mo stretchy="false">(</m:mo> <m:mi>ℓ</m:mi> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">)</m:mo> <m:mo>=</m:mo> <m:mn>0.</m:mn> </m:math> $$ \begin{equation*}\left( a(\ell) \left[\left(r(\ell) \varkappa^{\Delta} (\ell) \right)^{\Delta} \right]^{\alpha} \right)^{\Delta} + q(\ell) \varkappa^{\beta}(\wp(\ell))=0. \end{equation*}$$ The core objective is to streamline the analysis of the non-oscillatory solutions by converting the semi-canonical operator to an equivalent canonical form. This transformation facilitates the derivation of novel oscillation results by establishing criteria for the non-existence of non-oscillatory solutions. The practical examples provided in this paper illustrate the novelty of our findings, which contribute to enhancing, generalizing, and extending the existing literature.