On first-order differential operators with Bohr-Neugebauer type property (Q583457)

Let X be a Banach space; \(f\in L^ p_{loc}({\mathbb{R}},X)\) with \(p\in [1,+\infty [\) is said \(S^ p\)-bounded if \(| f|_{S^ p}\equiv \sup \{(\int^{t+1}_{t}| f(s)|^ p ds)^{1/p}:\quad t\in {\mathbb{R}}\}<+\infty.\) If f: \({\mathbb{R}}\to X\) is a continuously differentiable \(S^ 1\)-bounded function and \(f'\) is an \(S^ p\)-bounded function, then, if \(p=1\), f is bounded and, if \(p\in]1,+\infty [\), f is bounded and uniformly continuous. Let B: \(X\to X\) be a bounded linear operator such that any \(S^ 1\)-bounded u with \(u'-Bu=f\) is \(S^ 1\)-almost periodic for any (uniformly) almost-periodic X-valued function f. Then any \(S^ 1\)-bounded u such that \(u'-Bu=g\) is (uniformly) almost-periodic for any \(S^ 1\)-almost periodic continuous X-valued function g.