Stability of nonautonomous systems on Fréchet spaces

The main objective of the present paper is to formulate sufficient conditions under which a nonautonomous dynamics acting on a arbitrary Fréchet space exhibits shadowing property and (partial) linearization. These conditions require that the linear part is hyperbolic (in the sense of the concept recently introduced by Aragão Costa) and that the nonlinear part is Lipschitz. Our results extend those previously known in the setting of nonautonomous dynamics acting on Banach space. We consider both the case of discrete and continuous time dynamics.