Periodic solutions for nonautonomous first order delay differential systems via Hamiltonian systems

The existence of the nontrivial periodic solutions for the nonautonomous first order delay differential equation x′(t)=−[f(t,x(t−1))+f(t,x(t−2))+⋯+f(t,x(t−(2N−1)))]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x'(t)=-[f(t, x(t-1))+f(t, x(t-2))+\cdots+f(t, x(t-(2N-1)))]$\end{document} is investigated, where f∈C(R×R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in C({\mathbf{R}}\times{\mathbf{R}}, {\mathbf{R}})$\end{document} is 2N-periodic in t and odd in x, N is a positive integer. We prove several new existence results by some recent critical point theorems.