Comparison theorems for property (B) of the third-order differential equations with deviating arguments

Abstract

The aim of this paper is to introduce a new comparison theorem (in both delayed and advanced cases) that allows us to investigate the properties of third-order differential equations with quasi-derivatives $(r_{1}(t)(r_{2}(t)y'(t))')'-p(t)y(\tau(t))=0$ using the following simpler differential equations $(r(t)(r(t)z'(t))')'-p(t)z(\tau(t))=0$ and $y'''(t)-q(t)y(\sigma(t))=0.$ The obtained comparison principles allow for the immediate transcription of the oscillatory results known for the simpler equations into studied equation with quasi-derivatives. The progress achieved will be illustrated through several examples.