On quasi-periodic solutions of differential equations with piecewise constant argument (Q5961160)

It is shown that NEWLINE\[NEWLINEx'(t)= ax(t)+ \sum^N_{j=-N} a_j x([t+ j])+ f(t)\tag{\(*\)}NEWLINE\]NEWLINE has a unique quasiperiodic (qp) solution \(x\in QP(\omega_1,\dots, \omega_r)\) whenever \(f\in QP(\omega_1,\dots, \omega_r):= \{g:\mathbb{R}\to \mathbb{R}: g\) qp with (rationally independent) frequencies \(\omega_1,\dots, \omega_r\) and absolutely convergent Fourier series\}, provided suitably defined eigenvalues \(\mu\) satisfy \(|\mu|\neq 1\). It is stated that even for periodic \(f\) there need not exist a periodic solution. The result is extended to small nonlinear qp perturbations of \((*)\).