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 λ is defined for points of the spectrum,some equivalent conditions for compactness of the quasi-resolvent operators R λ are given.Then using these, some theorems on existence of periodic solutions to the non-linear equations Φ(A)x = f (x) are given, where Φ(A) is a polynomial of A with complex cofficients and f is a continuous mapping of H into itself.
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