Disconjugacy of a second order linear differential equation and periodic solutions

TL;DR

This paper introduces a new criterion for disconjugacy of second order linear differential equations that does not require smallness assumptions on coefficients, with applications to operator factorization and generalized Rolle's theorem.

Contribution

It presents a novel disconjugacy criterion that broadens applicability and compares it with existing criteria, enhancing understanding of differential equation properties.

Findings

New disconjugacy criterion without smallness assumptions

Comparison with classical criteria and detailed proofs

Applications to operator factorization and generalized Rolle's theorem

Abstract

The present paper is devoted to a new criterion for disconjugacy of a second order linear differential equation. Unlike most of the classical sufficient conditions for disconjugacy, our criterion does not involve assumptions on the smallness of the coefficients of the equation. We compare our criterion with several known criteria for disconjugacy, for which we provide detailed proofs, and discuss the applications of the property of disconjugacy to the problem of factorization of linear ordinary differential operators, and to the proof of the generalized Rolle's theorem. The paper is self-contained, and may serve as a brief introduction to theory of disconjugacy of a second order linear differential equation.