## 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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