On Super-high Frequencies in Discontinuous 1st-Order Delay–Differential Equations

One considers a scalar 1st-order nonlinear differential equation with a delayed relay-output proportional feedback. One shows that, under a boundedness condition on the nonlinearity only, any solution of this equation has, after a finite time, a finite number of zeros on compact sets. An estimate of the time after which the super-high frequency disappears is provided. This improves some previous work by Shustin. As a consequence, using some work by Fridman et al., any solution of the system under study coincides, after a finite time, with one of the periodic solutions.