Numerical oscillation and non-oscillation for differential equation with piecewise continuous arguments of mixed type

The paper deals with the oscillation and non-oscillation of the Runge-Kutta methods for a differential equation with piecewise continuous arguments of mixed type. The conditions of the oscillation and non-oscillation for the Runge-Kutta method are obtained. It is proved that oscillation of the analytic solution is not preserved by the Runge-Kutta method under any conditions. The conditions under which the non-oscillation of analytic solutions is preserved by the Runge-Kutta method are obtained. Some numerical experiments are given.