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Simple proofs of the Hadamard and Poincaré-Miranda theorems using the Brouwer fixed point theorem. (English) Zbl 1415.54025
This is a nice popular note explaining how one can prove the Hadamard and Poincaré-Miranda theorems from Brouwer’s fixed point theorem by elementary arguments. To be somewhat more explicit the most elementary form of Brouwer’s fixed point theorem states that a continuous self-map \(f\) of the closed ball of radius \(R\), \(B_R\), in \(\mathbb{R}^n\) must have a fixed point. The author’s starting point is Hadamard’s observation that this assumption implies that \(\left<x,x-f(x)\right>\ge0\) whenever \(\|x\|=R\). Hadamard had used this condition to conclude that \(\text{id} -f\) must have a zero in \(B_R\). The present author gives an elementary proof for Hadamard’s theorem in the following form: Let \(g:B_R\to\mathbb{R}^n\) be continuous with \(\left<g(x),x\right>\ge0\) whenever \(\|x\|=R\) then \(g\) has a zero in \(B_R\). As to Poincaré-Miranda he proves the following result: Denote by \(P\) the box \([-R_1,R_1]\times\dotsb\times[-R_n,R_n]\) and let \(g:P\to\mathbb{R}^n\) be continuous such that for \(i\in\{1,\dotsc,n\}\) we have that \(g_(x)\le0\) whenever \(x_i=-R_i\) and \(g_i(x)\ge0\) whenever \(x_i=R_i\) Then \(g\) has a zero in \(P\).

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
Full Text: DOI
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