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 it-
erative technique of the lower and upper solutions. In addition, an example is
presented to show the application of our main results.