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# 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

• Accepted: 22 December 2012
• Published:

## Keywords

• Point Theorem
• Differential Geometry
• Fixed Point Theorem
• Computational Biology
• Common Fixed Point

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

Example 1 given in paper  is wrong as $g\left(X\right)\not\subset f\left(X\right)$ because $f\left(X\right)=\left\{2,6\right\}$ and $g\left(X\right)=\left[2,20\right]$. So, Example 1 in paper  is replaced by the following example.

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

for all $x,y,z\in X$.

Define $f,g:X\to X$ by $f\left(x\right)=\frac{x}{2}$ and $g\left(x\right)=\frac{x}{6}$ for all $x\in X$.

Then, clearly, $g\left(X\right)\subset f\left(X\right)$as $f\left(X\right)=\left[0,\frac{1}{2}\right]$ and $g\left(X\right)=\left[0,\frac{1}{6}\right]$.

Moreover,
$\begin{array}{rcl}G\left(fx,fy,fz\right)& =& \left(|fx-fy|+|fy-fz|+|fz-fx|\right)\\ =& \frac{3}{2}\left(|x-y|\right)\ge q\left[\frac{1}{2}\left(|x-y|\right)\right]=qG\left(gx,gy,gz\right)\end{array}$

for $1 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 $\left\{{x}_{n}=\frac{1}{n}\right\}$. 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, Patiala, Punjab, India
(2)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung Khru, Bangkok, 10140, Thailand

## References 