Strong periodic solutions for a class of abstract evolution equations

We study a class of abstract nonlinear evolution equations in a separable Hilbert space for which we prove existence of strong time periodic solutions, provided the right-hand side is periodic and C¹ in time, and small enough in the norm of the considered space. We prove also uniqueness and stability of the solutions. The results apply, in particular, in several models of hydrodynamics, such as magneto-micropolar and micropolar models, and classical magnetohydrodynamics and Navier-Stokes models with non-homogeneous boundary conditions. The existence part of the proof is based on a set of estimates for the family of finite-dimensional approximate solutions.