Periodicity and continuous dependence in iterative differential equations

In this work, we study the existence, uniqueness and continuous dependence of periodic solutions of the iterative differential equation x′(t)=∑m=1N∑l=1∞Cl,m(t)x[m](t)l+ddtgt,x[1](t),x[2](t),…,x[N](t)+h(t).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x^{\prime }(t)=\sum \limits _{m=1}^{N}\sum \limits _{l=1}^{\infty }C_{l,m}(t)\left( x^{[m]}(t)\right) ^{l}+\frac{d}{dt}g\left( t,x^{[1]}(t),x^{[2]} (t),\ldots ,x^{[N]}(t)\right) +h(t). \end{aligned}$$\end{document}Using Schauder’s fixed point theorem, we obtain the existence of periodic solution and by the contraction mapping principle we obtain the uniqueness. An example is given to illustrate this work. The results obtained here extend the work of Zhao and Feckan [14].