Periodic solutions for neutral evolution equations with delays

The aim is to study the periodic solution problem for neutral evolution equation $$(u(t)-G(t,u(t-\xi)))'+Au(t)=F(t,u(t),u(t-\tau)),\ \ \ \ t\in\R$$in Banach space $X$, where $A:D(A)\subset X\rightarrow X$ is a closed linear operator, and $-A$ generates a compact analytic operator semigroup $T(t)(t\geq0)$. With the aid of the analytic operator semigroup theories and some fixed point theorems, we obtain the existence and uniqueness of periodic mild solution for neutral evolution equations. The regularity of periodic mild solution for evolution equation with delay is studied, and some the existence results of the classical and strong solutions are obtained. In the end, we give an example to illustrate the applicability of abstract results. Our works greatly improve and generalize the relevant results of existing literatures.