The Poincaré–Miranda theorem and viability condition

Abstract

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 2 follow then in a simple straightforward way. The above extension also implies a generalization of the Lax intermediate value theorem to infinite dimensional spaces.

A fixed point index approach to Krasnosel’skiĭ-Precup fixed point theorem in cones and applications

“…Poincaré-Miranda theorem is the n-dimensional version of Bolzano intermediate value theorem and it is well-known its equivalence to Brouwer fixed point theorem (see [22] for historical details). It has received a renewed interest in recent years, see for instance [7,10,11,23,30] and the references therein for some new versions, generalizations and applications of this result.…”

A fixed point index approach to Krasnosel'skii-Precup fixed point theorem in cones and applications

“…Poincaré-Miranda theorem is the n-dimensional version of Bolzano intermediate value theorem and it is well-known its equivalence to Brouwer fixed point theorem (see [18] for historical details). It has received a renewed interest in recent years, see for instance [6,9,10,19,26] and the references therein for some new versions, generalizations and applications of this result.…”

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

“…Our aim is to show how existence results can be obtained as a direct application of the Brouwer fixed point theorem, in the framework of the theory of positively invariant sets. The search of zeros of a vector field f , or, equivalently, equilibria for the differential system ẏ = f (y), (2.1) is a classical and well investigated topic in the area of Nonlinear Analysis, including important applications to different disciplines, like control theory [6,7,22].…”