Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

TL;DR

This paper extends the Brouwer fixed point theorem to establish the existence of equilibrium points and periodic solutions in differential systems on various domains, including convex and non-convex shapes, under specific boundary conditions.

Contribution

It introduces a general approach for arbitrary bound sets and applies it to convex and star-shaped domains, addressing a recent open question.

Findings

Existence of equilibrium points and periodic solutions in general domains.

Application of Brouwer fixed point theorem to non-convex domains.

Resolution of a recent open question in the field.

Abstract

We show the direct applicability of the Brouwer fixed point theorem for the existence of equilibrium points and periodic solutions for differential systems on general domains satisfying geometric conditions at the boundary. We develop a general approach for arbitrary bound sets and present applications to the case of convex and star-shaped domains. We also provide an answer to a question raised in a recent paper of Cid and Mawhin.