In the paper an operator method is suggested using the fixed points index theory of condensing multivalued maps for the investigation of the existence of mild periodic solutions of the differential inclusion \[ x'(t)ı Ax(t)+F(t,x(t)), \] where $A$ is a linear operator and $F$ is a multivalued map in a separable Banach space. Some applications to the existence of optimal periodic solutions of control systems in a Banach space are considered.