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“.

This entry was posted on Saturday, June 18th, 2011 at 15:38 and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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

Pete Donnell says:

28 August 2012 at 07:59

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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- Gilles Gnacadja says:
  
  28 August 2012 at 18:16
  
  Thank you, Pete. I will look for these two papers. A colleague recently told me about another paper that also has the result: “The Brouwer Fixed Point Theorem Applied to Rumour Transmission” by William Basener, Bernard P. Brooks, and David Ross; http://dx.doi.org/10.1016/j.aml.2006.02.007.
  
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