Localization of exact solutions of second-order quasilinear systems (Q1778169)

Consider the periodic differential system \[ dx/dt =y, \;dy/dt =-q^2x+F(t)+ \mu f(t,x,y,\mu), \tag{*} \] where \(F\) and \(f\) are sufficiently smooth and \(2\pi\)-periodic in \(t\), and \(\mu\) is a small parameter. Let \(\tilde{p} (t)\) be an approximate \(2\pi\)-periodic solution of \((*)\). The author derives conditions such that \((*)\) has an exact periodic solution in a \(\delta\)-neighborhood of \(\tilde{p} (t)\) for \(| \mu| \leq \mu^*\). The main goal is to estimate \(\mu^*\) as a function of \(\delta\). The key tool is Banach's fixed-point theorem.