FILOMAT, vol.30, no.4, pp.1069-1076, 2016 (Peer-Reviewed Journal)
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.