Monotone iterative technique for second order delayed periodic problem in Banach spaces

In this paper, we deal with the existence of ω-periodic solutions for second-order functional differential equation with delay in E−u′′(t)=f(t,u(t),u(t−τ)),t∈R,where E   is an ordered Banach space, f:R×E×E→Ef:R×E×E→E is a continuous function which is ω-periodic in t and τ ≥ 0 is a constant. We first build a new maximum principle for the ω-periodic solutions of the corresponding linear equation with delay. With the aid of this maximum principle, under the assumption that the nonlinear function is quasi-monotonicity, we study the existence of the minimal and maximal periodic solutions for abstract delayed equation by combining perturbation method and monotone iterative technique of the lower and upper solutions.