A necessary and sufficient condition for the oscillation of neutral equations (Q920273)

A necessary and sufficient condition is obtained for oscillation of the solutions of a neutral delay differential equation of the form \[ (1)\quad d/dt[x(t)+px(t-\tau)]+qx(t-s)-hx(t-\nu)=0, \] where \(p,q,h,\tau,s,\nu =const>0\). The following result is proved: All solutions of equation (1) oscillate if and only if the corresponding characteristic equation \[ (2)\quad F(\lambda)=\lambda +p\lambda e^{- \tau \lambda}+qe^{-s\lambda}-he^{-\nu \lambda}=0 \] has no real roots.