The Poincaré-Miranda theorem and viability condition

The aim of this note is to discuss the relation between the assumptions of the Poincaré-Miranda theorem and the viability condition, first used by Nagumo to prove existence of a solution to ODEs under state constraints (viable solutions). An interesting consequence of this observation is an extension of the Poincaré-Miranda theorem to arbitrary convex compact sets in locally convex Hausdorff vector spaces (instead of a parallelotope in an Euclidean space). We also recall a very short proof of this extension based on a Ky Fan inequality. This proof is not new, but seems to have passed unnoticed in the literature devoted to the Poincaré-Miranda theorem. In fact, recent variations of this theorem in ℓ2 follow then in a simple straightforward way. The above extension also implies a generalization of the Lax intermediate value theorem to infinite dimensional spaces.