Oscillatory criteria for the second order linear functional - differential equations with locally integrable coefficients

TL;DR

This paper develops new oscillation criteria for second order linear functional differential equations with locally integrable coefficients, extending classical theorems and providing a global solvability criterion.

Contribution

It introduces generalized oscillation theorems and a global solvability criterion for equations with advanced and retarded arguments, using the Riccati equation method.

Findings

Established an interval oscillation criterion.

Generalized Berezanski and Braverman's oscillation theorem.

Derived a new global solvability criterion.

Abstract

The Riccati equation method is used to establish some oscillatory criteria for the second order linear functional - differential equations of multiple terms with locally integrable coefficients. An interval oscillation criterion for the second order linear functional - differential equations is proved. We have obtained a generalization of an oscillation theorem of L. Berezanski and E. Braverman, a generalization of the well known Fite's oscillation criterion and a new global solvability criterion for the second order linear functional - differential equations with advanced and retarded arguments.