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.