Continuation theorems for periodic systems and applications to problems with nonlinear time-dependent differential operators

In this paper we propose some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}' = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \\ \, x_{n}' = h(t,x_{1},\ldots,x_{n}), \\ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n, \end{cases} \end{equation*} providing a unified framework that improves and extends earlier contributions by Jean Mawhin and collaborators to second-order differential problems governed by nonlinear time-dependent differential operators of the form \begin{equation*} \begin{cases} \, (\phi(t,x'))'=f(t,x,x'), \\ \, x(0)=x(T),\quad x'(0)=x'(T). \end{cases} \end{equation*} The proof is based on the topological degree theory.