Periodic Solutions For Semilinear Evolution Equation In Ordered Banach Space

This paper discusses the existence ofω-periodic solutions for semilinear evolution equationsin an ordered Banach space E. where A is a closed liner operator in E,—A generates a C₀-semigroupT(t)(i≥0) on E and f(t,u,v) :R×E×E→E is a continuous mapping which is w-periodic in t.In this paper, the quasi-upper and lower solution method and the positive operators semigroups theory are applied to periodic solution problems for semilinear evolutions equations. The results on the existence ofω-periodic solution are obtained by using the fixed point theorem and the monotone iteration scheme. on the other hand, the existence and uniqueness result of periodic solutions for associated linear evolution equation is established, and the spectral radius of periodic resolvent operator is accurately estimated. With the aid of the estimation the existence and uniqueness results of periodic solutions are obtained by using the condition of order. The present results generalize, unify and extend the relevant conclusions in ordinary differential equations, partial differential equations and abstract ordinary differential equations.