Equivalent conditions and the Meir-Keeler type theorems
type: Item
instance of: scholarly article
title: Equivalent conditions and the Meir-Keeler type theorems
zbMATH Open document ID: 0834.54025
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-04-01T00:00:00Z
review text: This is an important contribution to fixed point theory. The author shows among others that the contractive condition given in [\textit{G. Jungck}, \textit{K. B. Moon}, \textit{S. Park}, and \textit{B. E. Rhoades}, J. Math. Anal. Appl. 180, No. 1, 221-222 (1993; Zbl 0790.54055)]\ is equivalent to the contractive condition considered in [\textit{S. Sessa}, \textit{R. N. Mukherjee}, and \textit{T. Som}, Math. Jap. 31, 235-245 (1986; Zbl 0603.54044)]. Moreover, he gives some new Meir-Keeler type conditions (ensuring convergence of successive approximations) which are equivalent to those given by the reviewer [Čas. Pěstování Mat. 105, 341-344 (1980; Zbl 0446.54042)]\ (in this paper there are some misprints in the statement of the main theorem). From the point of view of possible applications in the theory of functional equations, especially interesting is Theorem 2 which, in particular, unifies the Boyd-Wong theorem and a result of the reviewer [Diss. Math. 127 (1975; Zbl 0318.39005)]\ which is not included in the Meir-Keeler theorem. Some results for locally contractive mappings and \(c\)-chainable metric spaces are also given.
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 811235
zbMATH Keywords: contractive condition
zbMATH Keywords: Meir-Keeler type conditions
zbMATH Keywords: locally contractive mappings
zbMATH Keywords: \(c\)-chainable metric spaces
author: J. Jachymski
reviewed by: Janusz Matkowski
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1299
OpenAlex ID: W2036451606
DOI: 10.1006/JMAA.1995.1299
description: scientific article; zbMATH DE number 811235