A comparison result for the oscillation of delay differential equations

Abstract

We obtain a comparison result for the oscillation of all solutions of the equation\[y˙(t)+∑i=1nqi(t)y(t−σi(t))=0\dot y(t) + \sum \limits _{i = 1}^n {{q_i}(t)y(t - {\sigma _i}(t)) = 0}\]in terms of the oscillation of all solutions of the equation\[x˙(t)+∑i=1npi(t)x(t−τi(t))=0\dot x(t) + \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i}(t)) = 0}\]under appropriate hypotheses on the asymptotic behavior of the quotientspi(t)/qi(t){p_i}(t)/{q_i}(t)andτi(t)/σi(t){\tau _i}(t)/{\sigma _i}(t)fori=1,2,…,ni = 1,2, \ldots ,n. An application to the oscillation of the nonautonomous delay-logistic equation is given.