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Zbl 1111.54025
Mustafa, Zead; Sims, Brailey
A new approach to generalized metric spaces. (English)
[J] J. Nonlinear Convex Anal. 7, No. 2, 286-297 (2006). ISSN 1345-4773; ISSN 1880-5221/e

In [Bull. Calcutta Math. Soc. 84, No. 4, 329--336 (1992; Zbl 0782.54037)], {\it 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 $x\ne y$, (3) $G(x,x,y)\le G(x,y,z)$ whenever $z\ne y$, (4) $G$ is a symmetric function of its three variables, and (5) $G(x,y,z)\le G(x,a,a)+ G(a,y,z)$. 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.
[Valeriu Popa (Bacău)]

MSC 2000:: *54E35 Metric spaces, metrizability
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
46B20 Geometry and structure of normed spaces
54E50 Complete metric spaces

Keywords: metric space; generalized metric space; $D$-metric space; $G$-metric space

Citations: Zbl 0782.54037

Cited in: Zbl 1249.54084 Zbl 1204.54039 Zbl 1179.54067

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