On \(n\)th-order differential operators with Bohr-Neugebauer type property (Q1100612)

A bounded linear operator B in a Banach space X is considered. Furthermore the differential operator \(d^ n/dt^ n-B\) has Bohr- Neugebauer property i.e. for any almost periodic X-valued function f(t) and any bounded (on \(J=^{def}\infty <t<\infty)\) solution of the equation \(d^ n/dt^ nu(t)-Bu(t)=f(t)\) on J, \(u^{(n-1)},...,u',u\) are all almost periodic. The author's main result is then that, for any Stepanov almost periodic function g(t) and any Stepanov-bounded solution of th differential equation \(d^ n/dt^ nu(t)-Bu(t)=g(t)\) on J, \(u^{(n-1)},...,u',u\) are all almost periodic.