A fixed point theorem for generalized metric spaces
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.
scientific article; zbMATH DE number 894077