Existence, uniqueness and conditional stability of periodic solutions to evolution equations

Abstract

Using an ergodic approach, we investigate the condition for existence and uniqueness of periodic solutions to linear evolution equation u˙=A(t)u+f(t), t≥0, and to semi-linear evolution equations of the form u˙=A(t)u+g(u)(t), where the operator-valued function t↦A(t) and the vector-valued function f(t) are T-periodic, and Nemytskii's operator g is locally Lipschitz and maps T-periodic functions to T-periodic functions. We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above semi-linear equation in the case that the family (A(t))t≥0 generates an evolution family having an exponential dichotomy.