Massera type theorems for abstract non-autonomous evolution equations

We establish two fixed point theorems for affine maps in Banach spaces,  with weaker assumptions than those in the literature.  Then we establish some  Massera type results for  abstract linear evolution  equations without assuming the existence of  bounded solutions,  which is an indispensable condition in the classical Massera  theorem and in the earlier literature. As application, we present an existence result on periodic mild solutions to  abstract nonautonomous semilinear evolution equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/35/abstr.html