Existence of ω-periodic solutions for second order delay differential equation in Banach spaces

Abstract

The propose of the paper is devoted to study the existence of ω-periodic solutions for second order delay differential equation in abstract Banach space. Firstly, we build a new maximum principle for the ω-periodic solutions of the corresponding linear equation. Secondly, with the help of this maximum principle, we study the existence of the minimal and maximal periodic solutions for our concerns problem by means of perturbation method and monotone iterative technique of the lower and upper solutions. In addition, an example is presented to show the application of our main results.