Nonexpansive periodic operators in l₁ with application to superhigh-frequency oscillations in a discontinuous dynamical system with time delay

We prove that the iterates of certain periodic nonexpansive operators in t₁ uniformly converge to zero in l_∞ norm. As a by-product we show that, for any solution x(t) of the equation x(t)= - sign(*(t - 1))+f(x(t)), t≥0, x| _[-1,0] ∈ C[-l,0] where f: ℝ→(-1, 1) is locally Lipschitz, the number of zeros of x(t) on any unit interval becomes finite after a period of time, with the single exception of the case f(0) = 0 and x(t) ≡ 0.