Theorem. Consider a continuous map and suppose that the autonomous dynamical system has a semiflow . Let . If is nonempty, compact, convex and forward-invariant, then contains an equilibrium of the dynamical system, i.e. a zero of the map .
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“.