On oscillation of nonlinear delay differential equations

Necessary, sufficient, and necessary and sufficient conditions are obtained for all solutions of the nonlinear differential equation \[ \frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}f\left ( {y\left ( {t - {\tau _j}} \right )} \right ) = 0,\qquad t \ge 0,} \qquad \left ( * \right )\] to be oscillatory. These conditions are expressed in terms of the characteristic equation of the corresponding linear “variational” equation \[ \frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}y\left ( {t - {\tau _j}} \right ) = 0, \qquad t \ge 0. \qquad \left ( { * * } \right )} \] Our results show that for a certain class of nonlinear functions $f,\left ( * \right )$ oscillates if and only if $\left ( { * * } \right )$ oscillates. As an application of our results, we obtain simple sufficient and necessary and sufficient conditions for the oscillation of several nonlinear delay differential equations which appear in applications.