On the mild solutions of higher-order differential equations in Banach spaces (Q1773473)

A new definition of a mild solution for the higher-order abstract differential equation \[ u^{(n)}(t)=Au(t)+f(t), \quad t \in {\mathbb R}, \] is given, where \(A\) is a closed operator on a Banach space \(E\). The author characterizes the regular admissibility of a translation-invariant subspace \({\mathcal M}\) of the space \(\text{BUC}({\mathbb R}, E)\) with respect to the above equation. In this context, \(\text{BUC}({\mathbb R}, E)\) is the space of bounded uniformly continuous functions. The case when \(f\) is a periodic function or an almost-periodic function is also studied.