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Generalised metric spaces and mappings with fixed point. (English) Zbl 0782.54037
A generalized metric on a set $$X$$ is a mapping $$D:X \times X \times X \to \mathbb{R}$$, satisfying three axioms which are quite analogous to the axioms of a usual metric. (The author also notes that his definition of a generalized metric is alike S. Gähler’s concept of a 2-metric [see S. Gähler, Math. Nachr. 26, 115-148 (1963; Zbl 0117.160)]. The principal interest of the author is the existence problem of a fixed point for a mapping of generalized metric spaces. Some theorems, quite analogous to the classic ones, are obtained in this direction. In particular, it is proved, that if a self-map $$T$$ of a complete bounded generalized metric space $$(X,D)$$ satisfies the condition $$D(Tx,Ty,Tz) \leq q \cdot D(x,y,z)$$ for all $$x,y,z \in X$$ and some $$q \in(0,1)$$, then $$T$$ has a unique fixed point.
Reviewer: A.Šostak (Riga)

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
##### Keywords:
generalized metric; 2-metric