Nonexpansive periodic operators in \(\ell_1\) with application to superhigh-frequency oscillations in a discontinuous dynamical system with time delay (Q5935824)

The map \(F: D\subseteq E\to E\) on Banach space \(E\) is called nonexpansive if \(\|F(a)- F(b)\|\leq\|a-b\|\) for all \(a,b\in D\). For the case \(E= \ell^+_1(\mathbb{Z})\) they give sufficient conditions for pointwise convergence \(\|F^k(x)\|_\infty\to 0\) as \(k\to\infty\) and also uniform convergence. The absence of infinite frequency oscillations in system \[ \dot x(t)= -\text{sign}(x(t- 1))+ f(x(t)),\quad t\geq 0,\quad x|_{[-1, 0]}= \varphi\in C([-1,0]), \] is established. Here \(\text{sign}(v)= 1\) for \(v> 0\) and \(\text{sign}(v)= -1\) for \(v< 0\) and \(\text{sign}(v)= 0\) for \(v=0\).