Periodic waveform relaxation solutions of nonlinear dynamic equations. (Q1855979)

Consider the periodic system \[ \frac{dx}{dt}=f(x,t)\tag{*} \] in \(\mathbb{R}^n\), where \(f\) is \(T\)-periodic in \(t\). Let \(F(x,y,y)\) be a splitting of \(f\), that is \(f(x,t) \equiv F(x,x,t)\). To compute a \(T\)-periodic solution of (*), the author uses the iteration schema \[ \frac{dx^{k+1}}{dt}= F\bigl (x^t(t),x^{k+1},t\bigr),\;x^{k+1} (0)=x^{k+1}(T),\tag{**} \] known as waveform iteration. The author provides a sufficient condition for the convergence of (**). The conditions obtained are known in the theory of waveform iteration.