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We present existence and multiplicity theorems for periodic mild solutions to parabolic evolution equations. Their peculiarity is a link with the spectrum of the generator of the semigroup rather than with the spectrum of the linearized periodic BVP for the evolution equation. They provide a positive solution to the open problem risen by Becker [ 3 ], they extend some results of Castro and Lazer [ 5 ] from scalar to systems of parabolic equations, and they are new even for finite-dimensional ODEs.

Solving Becker’s Problem on Periodic Solutions of Parabolic Evolution Equations