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

TL;DR

This paper develops continuation theorems for periodic systems with nonlinear time-dependent operators, extending previous results and utilizing topological degree theory to analyze solutions of such differential problems.

Contribution

It introduces unified continuation theorems for periodic differential systems with nonlinear operators, broadening the scope of earlier Mawhin-type results.

Findings

Provides new continuation theorems for nonlinear periodic systems

Extends existing results to more general differential operators

Uses topological degree theory for proofs

Abstract

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.