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.

Keywords: canonical equation, comparison theorem, property (B).

Mathematics Subject Classification: 34K11, 34C10.

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About the authors

Jozef Džurina (corresponding author)
https://orcid.org/0000-0002-6872-5695
Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Mathematics, Letná 9, 042 00 Košice, Slovakia

Blanka Baculíková
Technical University of Košice, Faculty of Electrical Engineering and Informatics, Department of Mathematics, Letná 9, 042 00 Košice, Slovakia

About the article

Communicated by Josef Diblík.

Received: 2026-01-18.
Revised: 2026-05-14.
Accepted: 2026-05-18.
Published online: 2026-06-02.