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 \[ y ˙ ( t ) + ∑ i = 1 n q i ( t ) y ( t − σ i ( t ) ) = 0 \dot y(t) + \sum \limits _{i = 1}^n {{q_i}(t)y(t - {\sigma _i}(t)) = 0} \] in terms of the oscillation of all solutions of the equation \[ x ˙ ( t ) + ∑ i = 1 n p i ( t ) x ( t − τ i ( t ) ) = 0 \dot x(t) + \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i}(t)) = 0} \] under appropriate hypotheses on the asymptotic behavior of the quotients p i ( t ) / q i ( t ) {p_i}(t)/{q_i}(t) and τ i ( t ) / σ i ( t ) {\tau _i}(t)/{\sigma _i}(t) for i = 1 , 2 , … , n i = 1,2, \ldots ,n . An application to the oscillation of the nonautonomous delay-logistic equation is given.