Existence and asymptotic stability of periodic solution for evolution equations with delays

In this paper, we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a Hilbert space Hu′(t)+Au(t)=F(t,u(t),u(t−τ1),…,u(t−τn)),t∈R, where A:D(A)⊂H→HA:D(A)⊂H→H is a positive definite selfadjoint operator, F:R×Hn+1→HF:R×Hn+1→H is a nonlinear mapping which is ω-periodic in t  , and τ1,τ2,…,τnτ1,τ2,…,τn are positive constants. We present essential conditions on the nonlinearity F to guarantee that the equation has ω-periodic solutions or an asymptotically stable ω-periodic solution. The discussion is based on analytic semigroups theory and an integral inequality with delays.