Evolution Families of $(X,Y,\varphi)$-type and Periodic Solutions to Nonautonomous Evolution Equations

Abstract

Consider nonautonomous evolution equation $\dot{u} = A(t) u(t) + Bg(u)(t)$ in which the family of operators $(A(t))_{t \geq 0}$ generates the evolution family $(\mathcal{U}(t,s))_{t \geq s \geq 0}$ of $(X,Y,\varphi)$-type, i.e., $\|\mathcal{U}(t,0)x\|_{Y} \leq \varphi(t) \|x\|_{X}$, $t > 0$, for certain couple of Banach spaces $(X,Y)$ and real-valued, positive function $\varphi$ satisfying $\lim_{t \to \infty} \varphi(t) = 0$. Inspired by Serrin's technique, we develop a unified approach toward the problems on the existence of periodic solutions to above equation. As illustrations of our abstract results, we give applications to the existence and uniqueness of periodic solutions to Oseen–Navier–Stokes and damped wave equations, as well as the existence of local stable manifolds nearby the periodic solution to the damped wave equations.