On Periodic Solutions To Nonlinear Differential Equations In Banach Spaces

Abstract

Let $ A $ denote the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a complex Banach space $ H $. In this work, an analog of the resolvent operator which is called quasi-resolvent operator and denoted by $ R_{\lambda} $ is defined for points of the spectrum,some equivalent conditions for compactness of the quasi-resolvent operators $ R_{\lambda} $ are given.Then using these, some theorems on existence of periodic solutions to the non-linear equations $\Phi(A)x=f(x)$ are given, where $ \Phi (A) $ is a polynomial of $ A $ with complex cofficients and $ f $ is a continuous mapping of $ H $ into itself.