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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)

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems