We prove a generalized maximum principle for subsolutions of boundary value problems, with mixed type unilateral conditions, associated to a degenerate parabolic second-order operator in divergence form.
The aim of this contribution is to study the role of the coefficient r in the qualitative theory of the equation (r(t)Φ(y ∆))∆+p(t)Φ(y σ ) = 0, where Φ(u) = |u| α−1 sgn u with α > 1. We discuss sign and smoothness conditions posed on r, (non)availability of some transformations, and mainly we show how the behavior of r, along with the behavior of the graininess of the time scale, affect some comparison results and (non)oscillation criteria. At the same time we provide a survey of recent results acquired by sophisticated modifications of the Riccati type technique, which are supplemented by some new observations.
In this paper we study asymptotic properties of the third order trinomial delay differential equation $$ y'''(t)-p(t)y'(t)+g(t)y(\tau (t))= 0 $$ by transforming this equation to the binomial canonical equation. The results obtained essentially improve known results in the literature. On the other hand, the set of comparison principles obtained permits to extend immediately asymptotic criteria from ordinary to delay equations.