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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" at http://math.GillesGnacadja.info/files/EquilExists.html. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs.

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    $\begingroup$ Shouldn't you require $f$ to be more than continuous (e.g. Lipschitz)? Currently $f$ doesn't (uniquely) define a (semi-)flow, for example when $f(x) = \sqrt{x}$ in a neighborhood of $x \ge 0$. $\endgroup$
    – Jaap Eldering
    Jun 21, 2011 at 13:33
  • $\begingroup$ Thanks Jaap, for catching and illustrating this insufficiency. I changed the statement. Now I explicitly require the existence of the semiflow. In my intended application, the map $f$ is a polynomial describing the kinetics of a chemical reaction network and time runs from zero to infinity. So I believe it would be too strong to require (global) Lipschitz continuity and too weak to require local Lipschitz continuity. Thanks again. $\endgroup$
    – Gilles Gnacadja
    Jun 22, 2011 at 1:37
  • $\begingroup$ A colleague showed me an article that essentially has the result: "The Brouwer Fixed Point Theorem Applied to Rumour Transmission", dx.doi.org/10.1016/j.aml.2006.02.007. The article is dated 2005/2006. There have to be earlier references. $\endgroup$
    – Gilles Gnacadja
    Aug 11, 2012 at 22:48

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This article from 1962 gives an earlier proof: "Axiomatic treatment of chemical reaction systems", dx.doi.org/10.1063/1.1732783.

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    $\begingroup$ Thank you, user48647, for this note. For others who might be interested, the specific part of interest in the paper is Part II, Section B (page 1579). $\endgroup$
    – Gilles Gnacadja
    Mar 25, 2014 at 3:28

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