On first-order differential operators with Bohr-Neugebauer type property
We consider a differential equation u(t)-Bu(t)=f(t), where the functions u and f map the real line into a Banach space X and B: X X is a bounded linear operator. Assuming that any Stepanov-bounded solution u is Stepanov almost-periodic when f is Bochner almost-periodic, we establish that any Stepanov-bounded solution u is Bochner almost-periodic when f is Stepanov almost-periodic. Some examples are given in which the operator -B is shown to satisfy our assumption.
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