Improvements on the Leighton‐type oscillation criteria for impulsive differential equations and extension to non‐canonical case.

We investigate the oscillatory behavior of solutions of second‐order linear differential equations with impulses. These equations can be categorized into two types: canonical and non‐canonical. This study examines the canonical and non‐canonical impulsive differential equations in connection with the famous Leighton‐type oscillation theorem. The well‐known theorem states that every solution of (r(t)x′)′+p(t)x=0,t≥t0,$$ {\left(r(t){x}^{\prime}\right)}^{\prime }+p(t)x=0,\kern0.30em t\ge {t}_0, $$ where r$$ r $$ and p$$ p $$ are continuous with r(t)>0$$ r(t)>0 $$, is oscillatory if ∫t0∞dtr(t)=∞(canonical case),(*)$$ \kern3.4cm \int_{t_0}^{\infty}\frac{\mathrm{d}t}{r(t)}=\infty \kern0.30em \left(\mathrm{canonical}\ \mathrm{case}\right),\kern3.4cm \left(\ast \right) $$ and ∫t0∞p(t)dt=∞.(**)$$ \kern4.6cm \int_{t_0}^{\infty }p(t)\kern0.1em \mathrm{d}t=\infty .\kern4.3cm \left(\ast \ast \right) $$ This pioneering work of Leighton has received considerable attention since its inception, and hence, it has been extended to various types of equations, including delay differential equations, dynamic equations, and impulsive differential equations. There are also studies generalizing the Leighton oscillation theorem when the condition (∗)$$ \left(\ast \right) $$ fails, that is, when the equation is of non‐canonical type. Our work focuses on refining the Leighton oscillation theorem to treat the canonical and non‐canonical cases. By doing so, we correct, supplement, and enhance the current literature on the oscillation theory of differential equations with impulses. Examples are also given to illustrate the significance of the obtained results. [ABSTRACT FROM AUTHOR]