Three fixed points of ray avoiding mappings on a finite-dimensional space
If the reviewer understands the paper correctly, the author deals with the following situation: Let \(D\subset\mathbb{R}^n\) be a nonempty, compact convex set and call \(r:\mathbb{R}^n\to D\) the metric projection with respect to the Euclidean norm. Let \(h,k>0\). A continuous map \(A:D\to\mathbb{R}^n\) is said to be ray avoiding if the following conditions are satisfied: 1) For sufficiently large \(c>0\), we have that \(\tau Arx\not=x\) whenever \(0<\tau\leq 1-k/c\) and \(d(x,rx)=h/c\) where \(d\) is the metric associated to the max-norm. 2) Whenever \(V\) is relatively open in \(D\) and \(x\in V\) we have that \((1-k/c)x\in r^{-1}(V)\cap\{x|\;d(x,rx)<h/c\}\). Starting from the Brouwer degree the author defines a fixed point index for ray avoiding maps. He then uses this index to formulate conditions for a ray avoiding map \(A\) that imply the existence of three fixed points of \(A\).
scientific article; zbMATH DE number 1545778