Nonexpansive Periodic Operators in l1 with Application to Superhigh-Frequency Oscillations in a Discontinuous Dynamical System with Time Delay

Recall that a map F: D/X X on a Banach space (X, & }&) is called ``nonexpansive'' if &F(a)&F(b)& &a&b& for all a, b # D. Here we shall be interested in X=l1(Z), the Banach space of biinfinite absolutely summable real sequences (an | n # Z) , with &a&1 := n # Z |an |. As usual, l1 (Z)= [a # l1(Z) : an 0 \n # Z], and we shall be interested in nonexpansive maps F: l1 (Z) l + 1 (Z). Our maps will also be ``integral-preserving'' (so &F(a)&1 =&a&1 for all a # l1 (Z)) and ``order-preserving'' (defined below), and F will, in a precise sense described later, be periodic of period N, where N is a positive integer.