A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations

In this paper we present a new approach to the spectral theory of non-uniformly continuous functions and a new framework for the Loomis–Arendt–Batty–Vu theory. Our approach is direct and free of C0-semigroups, so the obtained results, that extend previous ones, can be applied to large classes of evolution equations and their solutions.