Existence of Time Periodic Solutions to Parabolic Delayed Evolution Equation in Ordered Banach Space

The purpose of the article is to systematically research time -periodic problem to the general parabolic delayed evolution equation. First of all, by applying theory of operator semigroup, the Schauder fixed point theorem, and monotone iterative method, the existence and uniqueness results of -periodic mild solutions and positive mild solutions to the abstract evolution equation with delay are gained when nonlinear term satisfies some suitable ordered conditions. Next, we use the abstract results to get similar results of periodic problem to the parabolic evolution equation with finite delay, which improves and generalizes the related work of this fields. Ultimately, two examples to parabolic delayed evolution equation are presented.