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Some fixed point theorems for nonconvex spaces. (English) Zbl 0924.47041

Let \(E\) be a topological vector space over the scalar field \(\mathbb{R}\) or \(\mathbb{C}\). Let \(\tau\) and \(\tau_w\) denote the original and weak topologies of \(E\). For a nonempty subset \(X\subset E\), \((X,\tau_w)\) (resp. \((X,\tau)\)) denotes \(X\) with the relative topology \(\tau_w\) (resp. \(\tau\)). The main result is the following proposition.
Proposition 2.1. Let \(X\) be a nonempty compact (in topology \(\tau\)) subset of \(E\). If \((X,\tau)\) is Hausdorff, then \((X,\tau_w)= (X,\tau)\).
The authors derive fixed point theorems of Ky Fan, Kim, Kaczyński, Kelly, Namioka from this proposition and prove a fixed point theorem for multivalued functions.

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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