Remarks on recent fixed point theorems for compatible maps

S L Singh,S N Mishra,V Chadha

doi:10.1155/s0161171296001111

Abstract

Following Gerald Jungck′s formulation of compatibility of maps, several interesting fixed point theorems for such maps have been obtained, wherein at least one of the maps is continuous. The main purpose of this note is to avoid the continuity condition from some of the main results of Jungck et al.

Highlights

11) Recently several fixed point theorems for compatible maps have been obtained ([1]-[10] and references of [7]), and a few of them have been extended to biased maps
[6] most of the results, especially those for compatible maps, require continuity of at least one of the maps ([1]-[5], [7]10]) In this note, we present fixed point theorems for four maps, wherein the continuity condition is not needed Our results are variants of some interesting results, mainly, of Jungck motivation to waive the continuity condition referred to above comes from 12]-[ 14] and [6]
2. ((, 6)-CONTRACTION FOR FOUR MAPS Meir-Keeler type contractions for four maps on a metric space introduced in [1] have been further studied in [4]-[6], [10] and elsewhere These motivate the following, where (X, d) is a metric space

Summary

Introduction

Self-maps f and g of a metric space (X, d) are compatible iff limnd(fgzn, gfzn)= 0 whenever {:r,} is a sequence in X such that fzn, gz for some in X (cf [1]-[2], see [3]-[10])Compatible maps are natural generalizations of commuting and weakly commuting maps 11 (see for instance [1]-[10]) Very recently a less restrictive concept called "biased maps" has been studied by Jungck and Pathak [6] Let A and S be self-maps of a metric space (X, d) The pair {A, S) is S-biased iff whenever {x,} is a sequence in X and Axe, Sx E X, od(SAx,Sx) < ad(ASxn,Ax) if a liminf andif a limsup.They have shown that if the pair {A, S} is compatible, it is both S and A-biased (see [6], Remark11) Recently several fixed point theorems for compatible maps have been obtained ([1]-[10] and references of [7]), and a few of them have been extended to biased maps [6] most of the results, especially those for compatible maps, require continuity of at least one of the maps ([1]-[5], [7]10]) In this note, we present fixed point theorems for four maps, wherein the continuity condition is not needed Our results are variants of some interesting results, mainly, of Jungck motivation to waive the continuity condition referred to above comes from 12]-[ 14] and [6]2. ((, 6)-CONTRACTION FOR FOUR MAPS Meir-Keeler type contractions for four maps on a metric space introduced in [1] have been further studied in [4]-[6], [10] and elsewhere These motivate the following, where (X, d) is a metric spaceS L SINGH, V CI-tADHA AND S N MISHRADEFINITION 1. 11) Recently several fixed point theorems for compatible maps have been obtained ([1]-[10] and references of [7]), and a few of them have been extended to biased maps

Results

Conclusion