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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 $$P$$.

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