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

Abstract

For the higher-order abstract differential equation $u^{\left(n\right)}\left(t\right)=Au\left(t\right)+f\left(t\right)$, $t\in ℝ$, we give a new definition of mild solutions. We then characterize the regular admissibility of a translation-invariant subspace $ℳ$ of $\text{BUC}\left(ℝ,E\right)$ with respect to the above-mentioned equation in terms of solvability of the operator equation $AX-X{𝒟}^n=C$. As applications, periodicity and almost periodicity of mild solutions are also proved.