A generalization of a theorem due to Cartwright (Q1420702)

For certain functional differential systems \[ x'(t)= A(x_t)+ B\phi(C(x_t),\quad t\in\mathbb R\tag{\(*\)} \] it is shown that any almost periodic (a.p.) solution is already quasi-periodic. Here \(E:= C([-h,0],\mathbb R^n)\), \(A: E\to\mathbb R^n\) and \(C: E\to \mathbb R^s\) are linear continuous, \(B\) is a real \(n\times r\) matrix, \(\phi: CS\to \mathbb R^r\) satisfies a Lipschitz condition on \(CS\), \(S\) open \(\subset E\), \(x_t(s):= x(t+ s)\), \(-h\leq s\leq 0\). Assumptions are: There is a positive \(r\) and a \(j\in \mathbb N\) such that the characteristic equation of \((*)\) has \(j\) roots \(z\) with \(\Re z>-r\) and no roots with \(\Re z= -r\), furthermore the Lipschitz constant of \(\phi\) is less than an explicitly given bound. These assumptions are fulfilled if \(\phi= 0\). In the proof the reduction theory of \textit{R. A. Smith} [Differ. Integral Equ. 5, No. 1, 213--240 (1992; Zbl 0754.34070)] is used, which gives an explicit one-one correspondence between (bounded) solutions of \((*)\) and solutions of the ordinary differential system \(y'= g(y)\) with Lipschitzian \(g\), for which \textit{M. L. Cartwright's} result [Proc. Lond. Math. Soc. (3) 17, 355--380 (1967; Zbl 0155.41901)] applies. Here \(f: \mathbb R\to\mathbb R^n\) quasi-periodic means \(f\) is Bohr-a.p. and there is a finite \(\Omega\subset\mathbb R\) such that the Bohr-spectrum of \(f\subset\{\sum_{\omega\in\Omega} r_\omega\omega: r_\omega\) rational\} (the usual definition allows only integer \(r_\omega\)).