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

Abstract

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