Periodic Solutions Of Semilinear Delay Evolution Equation In Banach Space

In this thesis,by using Schauder fixed point theorem、Banach contraction mapping theory、Leray-Schauder fixed point theorem、monotone iterative method、analytic semigroup theory,we discuss the existence、uniqueness、existence and uniqueness、asymptotic stability of positive periodic solution、regularity of periodic solutions for abstract evolution equation with delay in a Banach space X u’(t)+Au(t)=f(t,u(t),u(t-τ)),t∈R where A:D(A)(?)X→X is a closed linear operator and—A generates a Cosemigroup T(t)(t≥0)on X,f:R × X × X→ X is a continuous function mapping and f(t,x,y)is ω-periodic in t.In addition,a kind of existence of periodic solutions for impulsive delay evolution equation is considered by using monotone iterative method for upper and lower solutions.The main results of this paper are as follows:1.Under the condition of nonlinear term f satisfies linear growth,the existence and uniqueness of periodic solutions for evolution equation with delay is considered by using Schauder fixed point theorem and Banach contraction mapping principle;2.The existence and uniqueness、asymptotic stability of positive periodic solution for evolution equations with delay in an order Banach space is discussed by using the monotone iterative technique and Leray-Schauder fixed point theorem;3.Under the case of A is sector operator,apply analytic semigroup theory and the regularity of periodic solution,we obtain the existence of classical solution;4.The existence of periodic solutions for impulsive delay evolution equation is discussed by using upper lower solutions method and monotone iterative technique when semigroup is compact and non-compact.