Fixed points and best approximation in Menger convex metric spaces. (English) Zbl 1109.47047

Beg, Ismat; Abbas, Mujahid

Arch. Math., Brno 41, No. 4, 389-397 (2005).

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.

Reviewer: Ondřej Došlý (Brno)

MSC:

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

Keywords:

uniformly convex metric space; best approximation; nonexpansive map