Infinite Frequency Oscillations in a Class of Discontinuous Equations with a Variable Delay

This paper mainly investigates the oscillatory behaviors in a class of discontinuous equations with a variable delay. Under the restrictions of boundedness and monotonicity on the time-delay function, the type and distribution of zero points of the solution are firstly determined. And then, the non-existence of oscillatory solutions with infinite frequency is proven for the systems with two classes of discontinuous vector fields. In detail, by applying the inequality techniques on a backward shift operator, an exact lower bound of the time is provided to ensure the vanishing of infinite frequency oscillations after this time. The obtained results improve some previous works in the cases of the constant delay and constant vector fields.