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

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 $g\left(X\right)\not\subset f\left(X\right)$ because $f\left(X\right)=\left\{2,6\right\}$ and $g\left(X\right)=\left[2,20\right]$. So, Example 1 in paper [1] 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.

## 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:10.1186/1687–1812–2012–221

## Acknowledgements

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

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Correspondence to Saurabh Manro.

### Competing interests

The authors declare that they have no competing interests.

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All authors read and approved the manuscript.

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Manro, S., Kumam, P. Erratum to ‘Common fixed point theorems for expansion mappings in various abstract spaces using concept of weak reciprocal continuity’. Fixed Point Theory Appl 2013, 8 (2013). https://doi.org/10.1186/1687-1812-2013-8