Periodic solutions to evolution equations: existence, conditional stability and admissibility of function spaces
Volume 116 / 2016
Abstract
We prove the existence and conditional stability of periodic solutions to semilinear evolution equations of the form $\dot{u}=A(t)u+g(t,u(t)),$ where the operator-valued function $t\mapsto A(t)$ is $1$-periodic, and the operator $g(t,x)$ is $1$-periodic with respect to $t$ for each fixed $x$ and satisfies the $\varphi$-Lipschitz condition $ \|g(t,x_1)-g(t,x_2)\|\leq \varphi(t)\|x_1-x_2\|$ for $\varphi(t)$ being a real and positive function which belongs to an admissible function space. We then apply the results to study the existence, uniqueness and conditional stability of periodic solutions to the above semilinear equation in the case that the family $(A(t))_{t\geq 0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.
