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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.

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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