Open Access

Erratum to ‘Common fixed point theorems for expansion mappings in various abstract spaces using concept of weak reciprocal continuity’

Fixed Point Theory and Applications20132013:8

https://doi.org/10.1186/1687-1812-2013-8

Received: 18 December 2012

Accepted: 22 December 2012

Published: 9 January 2013

On critical examination of the results given in our paper [1], we notice one crucial error. We need to carry out the following correction.

Example 1 given in paper [1] is wrong as g ( X ) f ( X ) because f ( X ) = { 2 , 6 } and g ( X ) = [ 2 , 20 ] . So, Example 1 in paper [1] is replaced by the following example.

Example 1 Let ( X , G ) be a G-metric space, where X = [ 0 , 1 ] and
G ( x , y , z ) = ( | x y | + | y z | + | z x | )

for all x , y , z X .

Define f , g : X X by f ( x ) = x 2 and g ( x ) = x 6 for all x X .

Then, clearly, g ( X ) f ( X ) as f ( X ) = [ 0 , 1 2 ] and g ( X ) = [ 0 , 1 6 ] .

Moreover,
G ( f x , f y , f z ) = ( | f x f y | + | f y f z | + | f z f x | ) = 3 2 ( | x y | ) q [ 1 2 ( | x y | ) ] = q G ( g x , g y , g z )

for 1 < q 3 and hence, the condition (2.2) of Theorem 2 is satisfied. Also, f and g are two weakly reciprocally continuous self-maps by taking the sequence { x n = 1 n } . However, the maps are compatible. Thus, all the conditions of Theorem 2 are satisfied and x = 0 is the unique common fixed point of f and g.

Declarations

Acknowledgements

The authors are grateful to Prof. Ravindra K. Bisht for pointing out some of these errors.

Authors’ Affiliations

(1)
School of Mathematics and Computer Applications, Thapar University
(2)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)

References

  1. Manro S, Kumam P: Common fixed point theorems for expansion mappings in various abstract spaces using the concept of weak reciprocal continuity. Fixed Point Theory Appl. 2012., 2012: Article ID 221. doi:10.1186/1687–1812–2012–221Google Scholar

Copyright

© Manro and Kumam; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.