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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(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,zX.

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

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<q3 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.

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

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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.

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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

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

Keywords

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