Existence, uniqueness and conditional stability of periodic solutions to evolution equations (Q499198)

The authors have investigate conditions for existence and uniqueness of periodic solutions to the linear evolution equation \[ u^{\prime} = A(t)u + f(t), t \geq 0, \] and to the semi-linear evolution equation of the form \[ u^{\prime} = A(t)u + g(u)(t), \] where the operator \(A(t)\) and the function \(f(t)\) are \(T\)-periodic, and the Nemytskii operator \(g\) is locally Lipschitz and maps \(T\)-periodic functions to \(T\)-periodic functions. They apply the results to study existence, uniqueness, and conditional stability of periodic solutions to the above semi-linear equation in the case that the family \((A(t)),t\geq 0\) generates an evolution family having an exponential dichotomy.