We investigate the semi-linear, non-autonomous, first-order abstract differential equation x′(t)=A(t)x(t)+f(t,x(t),φ[α(t,x(t))]),t∈R. obtain results on existence and uniqueness of (ω,c)-periodic (second-kind periodic) mild solutions, assuming that A(t) satisfies so-called Acquistapace–Terreni conditions homogeneous associated problem has an integrable dichotomy. A new composition theorem furthe...

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 of BUC(R,E) with respect to the above-mentioned equation in terms of solvability of the operator equation AX −X n = C. As applications, periodicity and almost periodicity of mild solutio...

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are o...