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A new approach to generalized metric spaces. (English) Zbl 1111.54025

In [Bull. Calcutta Math. Soc. 84, No. 4, 329–336 (1992; Zbl 0782.54037)], B. C. Dhage initiated the study of a generalized metric spaces, namely, D-metric spaces. In the present paper, the authors introduce an alternative, more robust generalization of metric spaces, namely, G-metric spaces, where the G-metric satisfies the following axioms:

(1) G(x,y,z)=c if x=y=z, (2) 0<G(x,x,z) whenever xy, (3) G(x,x,y)G(x,y,z) whenever zy, (4) G is a symmetric function of its three variables, and (5) G(x,y,z)G(x,a,a)+G(a,y,z).

In Section 2, some properties of G-metric spaces are studied. Section 3, entitled “The G-metric topology”, contains: Convergence and continuity in G-metric spaces; Completeness of G-metric spaces and compactness in G-metric spaces. In the last section, products of G-metric spaces are studied.


MSC:
54E35Metric spaces, metrizability
47H10Fixed point theorems for nonlinear operators on topological linear spaces
46B20Geometry and structure of normed linear spaces
54E50Complete metric spaces