## 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.

Zbl 0963.54031
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