A fixed point theorem for generalized metric spaces
type: Item
instance of: scholarly article
title: A fixed point theorem for generalized metric spaces
zbMATH Open document ID: 0857.54044
DOI: 10.1155/S0161171296000658
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1997-03-10T00:00:00Z
full work available at URL: https://eudml.org/doc/47598
review text: Summary: In [Bull. Calcutta Math. Soc. 84, 329-336 (1992; Zbl 0782.54037)] \textit{B. C. Dhage} defined a generalized metric space as follows: Let \(D:X\times X\times X\to\mathbb{R}\) with the properties (i) \(D(x,y,z)\geq 0\) for each \(x,y,z\in X\), with equality iff \(x=y=z\), (ii) \(D(x,y,z)= D(y,x,z)= D(x,z,y)=\dots\) (symmetry), (iii) \(D(x,y,z)\leq D(x,y,a)+ D(x,a,z)+ D(a,y,z)\) for each \(x,y,z\in X\). 2-metric spaces are defined by a function \(d:X\times X\times X\to\mathbb{R}\) with properties (ii) and (iii) above, and (i) replaced by (i') For each distinct pair \(x,y\in X\), there exists a \(z\in X\) such that \(d(x,y,z)\neq 0\) and \(d(x,y,z)= 0\) if any two of the triplet \(x,y,z\) are equal. A number of fixed point theorems have been proved for 2-metric spaces. However, \textit{C.-R.Hsiao} [A property of contractive type mappings in 2-metric spaces, Jñānābha 16, 223-239 (1986)] showed that all such theorems are trivial in the sense that the iterations of \(f\) are all colinear. The situation for \(D\)-metric spaces is quite different. Some specific examples of \(D\)-metric spaces appear in [Dhage, loc. cit.]. The purpose of this paper is to prove two general fixed point theorems for \(D\)-metric spaces.
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 894077
zbMATH Keywords: generalized metric space
zbMATH Keywords: 2-metric spaces
zbMATH Keywords: \(D\)-metric spaces
author: Billy E. Rhoades
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 894077