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Contractions over generalized metric spaces. (English) Zbl 1173.54311

Summary: A generalized metric space (g.m.s) is defined as a metric space in which the triangle inequality is replaced by the ‘quadrilateral inequality’, \(d(x, y) \leq d(x, a) + d(a, b) + d(b, y)\) for all pairwise distinct points \(x, y, a, b\) of \(X\). \((X, d)\) becomes a topological space when we define a subset \(A\) of \(X\) to be open if to each \(a\) in \(A\) there corresponds a positive number \(r_a\) such that \(b \in A\) whenever \(d(a, b) < r_a\). Cauchyness and convergence of sequences are defined exactly as in metric spaces and a g.m.s \((X, d)\) is called complete if every Cauchy sequence in \((X, d)\) converges to a point of \(X\). A. Branciari [Publ. Math. 57, No. 1–2, 31–37 (2000; Zbl 0963.54031)] has published a paper purporting to generalize Banach’s contraction principle in metric spaces to g.m.s. In this paper, we present a correct version and proof of the generalization.

MSC:

54E40 Special maps on metric spaces
54H25 Fixed-point and coincidence theorems (topological aspects)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 0963.54031
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