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 tau-periodic evolutionary process {U(t, s)}(t >= s) on the space P-tau(X) of all tau-periodic continuous X-valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp = -is an element of F(p, is an element of) under the condition that 1 is a normal eigenvalue of the monodromy operator U(tau, 0). The proof is based on a new decomposition of the space P-tau(X) by constructing a right inverse of L. (C) 2014 Elsevier Inc. All rights reserved.