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Fixed points and best approximation in Menger convex metric spaces. (English) Zbl 1109.47047
The authors investigate the problem of the existence of fixed points of nonexpansive maps in a metric space. A typical result is the following statement: Let $$F$$ be a nonempty compact convex subset of a convex complete metric space having properties (A) and (B), then any nonexpansive self map $$T$$ on $$F$$ has a fixed point.
Recall that a metric space $$X$$ has property (A) if for every $$x, y\in X$$, the set $$B\big (x; (1-t)d(x,y)\big )\cap B\big (y; td(x,y)\big )=: m(x,y,t)$$ is a singleton set for $$t\in [0,1]$$, and it is said to have property (B) if $$d\big (m(x,y,t),$$ $$m(z, y,t)\big )\leq td(x,z)$$, where $$B(x,r):= \{y\in X,\, d(x,y)\leq r\}$$. Quasinonexpansive maps and the existence of approximate fixed points are treated as well.

##### MSC:
 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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