Equilibria Exist in Compact Convex Forward-Invariant Sets

Theorem. Consider a continuous map f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} and suppose that the autonomous dynamical system \dot{x} = f(x) has a semiflow \varphi : {\mathbb{R}}_{\geq{0}} \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}. Let K \subseteq {\mathbb{R}}^{n}. If K is nonempty, compact, convex and forward-invariant, then K contains an equilibrium of the dynamical system, i.e. a zero of the map f.

According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in Equilibria Exist in Compact Convex Forward-Invariant Sets. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs. Please contribute here or on MathOverflow at Equilibria Exist in Compact Convex Forward-Invariant Sets.

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2 Responses to Equilibria Exist in Compact Convex Forward-Invariant Sets

  1. Pete Donnell says:

    Hi Gilles, I also struggled to find references for this when writing up my thesis a few years ago. Strange when it’s such a fundamental result!

    You might like to take a look at “A fixed point theorem for bounded dynamical systems” by Richeson and Wiseman in the Illinois Journal of Mathematics (issue 46(2) pp. 491-495), which I think contains a proof. The authors mention other proofs in an addendum to their paper, in issue 48(3) pp. 1079-1080 of the same journal.

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