A new proof of Poincaré-Miranda theorem based on the classification of one-dimensional manifolds

TL;DR

Addresses the Poincaré-Miranda problem for maps $f:I^{n}\to \mathbb{R}^{n}$ satisfying the opposite-face sign boundary condition: for each $i$, $f_i(x_1,\dots,a_i,\dots,x_n) f_i(x_1,\dots,b_i,\dots,x_n) < 0$. The authors use Sard's theorem together with the classification of one-dimensional manifolds to show that, for small regular values $q$, the preimage $f^{-1}(q)$ has odd cardinality, guaranteeing zeros in $I^{n}$, and in the smooth case an odd number of zeros. They then employ a perturbation argument to rule out the possibility of no zero, obtaining a contradiction and proving the theorem. The method provides a bridge between Sard-style transversality and fixed-point type results, with potential Poincaré-Hopf-like corollaries for vector fields on cubes.

Abstract

This note gives a new elementary proof of Poincaré-Miranda theorem based on Sard's theorem and the simple classification of one-dimensional manifolds.

Figures (1)

• Figure 1: The image of $f_{1}^{-1}(q_{1})$.

Theorems & Definitions (8)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Corollary 2.4
• Theorem 2.5
• proof
• proof