Equilibria Exist in Compact Convex Forward-Invariant Sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:53:19Z http://mathoverflow.net/feeds/question/68174 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68174/equilibria-exist-in-compact-convex-forward-invariant-sets Equilibria Exist in Compact Convex Forward-Invariant Sets Gilles Gnacadja 2011-06-19T00:12:11Z 2011-06-22T01:31:45Z <blockquote> <strong>Theorem.</strong> 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 $f$. </blockquote> <p>According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in <i>"Equilibria Exist in Compact Convex Forward-Invariant Sets"</i> at <a href="http://math.GillesGnacadja.info/files/EquilExists.html" rel="nofollow">http://math.GillesGnacadja.info/files/EquilExists.html</a>. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs.</p>