Systems-Disconjugacy of a Fourth-Order Differential Equation with a Middle Term

TL;DR

This paper extends the theory of systems-conjugate points for a class of fourth-order differential equations by removing sign restrictions on the middle term, broadening the applicability of previous results.

Contribution

It generalizes existing results on systems-conjugate points to fourth-order equations with an indefinite middle term, expanding the theoretical framework.

Findings

Extended the concept of systems-conjugate points to more general equations

Proved new properties for equations with sign-indefinite middle terms

Broadened the applicability of previous conjugacy results

Abstract

Systems-conjugate points have been introduced and studied by John Barrett \cite{barr} in relation with the self-adjoint fourth order differential equation $$(r(x)y")"-(q(x)y')'= p(x)y,$$ where $r(x)>0$, $p(x)>0$ and $q\equiv0$. In this paper we extend some of his results to more general cases, when $q(x)$ is free of any sign restrictions.