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

TL;DR

The paper develops continuation theorems for periodic vector systems with nonlinear, time-dependent differential operators by leveraging the Leray–Schauder coincidence degree and a degree-reduction formula. It shows how to reduce the infinite-dimensional degree to a finite-dimensional Brouwer degree on $\ker L$, and derives two continuation results for periodic problems, including $L^1$-Carathéodory homotopies. These theoretical advances are then applied to second-order systems of the form $(\phi(t,x'))' = f(t,x,x')$, enabling existence results under more general and time-dependent operator settings, with special attention to mean curvature and Minkowski-type operators. The work broadens Mawhin’s and related results, offering a robust framework for nonlinear time-dependent differential operators in periodic problems.

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} \, (φ(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.

Theorems & Definitions (16)

• Theorem 1.1: Leray--Schauder continuation theorem
• Theorem 1.2: Mawhin's continuation theorem
• Theorem 1.3: Capietto--Mawhin--Zanolin continuation theorem
• Proposition 2.1: Reduction formula
• proof : Sketch of the proof
• Proposition 2.2: Product formula
• proof : Sketch of the proof
• Theorem 3.1
• proof
• Theorem 3.2