Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ-periodic evolutionary process {U(t,s)}t≥s on the space Pτ(X) of all τ-periodic continuous X-valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−ϵF(p,ϵ) under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0). The proof is based on a new decomposition of the space Pτ(X) by constructing a right inverse of L.