Erratum to: Generalized metrics and Caristi’s theorem
© Kirk and Shahzad; licensee Springer. 2014
Received: 1 August 2014
Accepted: 1 August 2014
Published: 19 August 2014
The original article was published in Fixed Point Theory and Applications 2013 2013:129
The assertion in  that Caristi’s theorem holds in generalized metric spaces isbased, among other things, on the false assertion that if is a sequence in a generalized metric space, and if satisfies , then is a Cauchy sequence. In Example 1 below wegive a counter-example to this assertion, and in Example 2 we show that, infact, Caristi’s theorem fails in such spaces. We apologize for anyinconvenience.
For convenience we give the definition of a generalized metric space. The conceptis due to Branciari .
Then X is called a generalized metric space.
The following example is a modification of Example 1 of .
Therefore is a generalized metric space. Now suppose is a Cauchy sequence in . Then if and , must be odd. However, if is infinite, cannot be odd for all sufficiently large i, k. (Suppose . If and are odd, then is even.) Thus any Cauchy sequence in must eventually be constant. It follows that is complete and that is not a Cauchy sequence in . However, .
The following example shows this is not true in the space described inExample 1.
- Kirk WA, Shahzad N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013., 2013: Article ID 129Google Scholar
- Branciari A: A fixed point theorem of Banach-Caccioppoli type on a class of generalizedmetric spaces. Publ. Math. (Debr.) 2000, 57: 31–37.MathSciNetGoogle Scholar
- Jachymski J, Matkowski J, Świa̧tkowski T: Nonlinear contractions on semimetric spaces. J. Appl. Anal. 1995, 1(2):125–134.View ArticleMathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), whichpermits unrestricted use, distribution, and reproduction in any medium, provided theoriginal work is properly credited.