 Erratum
 Open access
 Published:
Erratum to: Generalized metrics and Caristi’s theorem
Fixed Point Theory and Applications volume 2014, Article number: 177 (2014)
The assertion in [1] that Caristi’s theorem holds in generalized metric spaces isbased, among other things, on the false assertion that if\{{p}_{n}\} is a sequence in a generalized metric space(X,d), and if \{{p}_{n}\} satisfies {\sum}_{i=1}^{\mathrm{\infty}}d({p}_{i},{p}_{i+1})<\mathrm{\infty}, then \{{p}_{n}\} is a Cauchy sequence. In Example 1 below wegive a counterexample 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 [2].
Definition 1 Let X be a nonempty set and d:X\times X\to [0,\mathrm{\infty}) a mapping such that for all x,y\in X and all distinct points u,v\in X, each distinct from x and y:

(i)
d(x,y)=0\iff x=y,

(ii)
d(x,y)=d(y,x),

(iii)
d(x,y)\le d(x,u)+d(u,v)+d(v,y) (quadrilateral inequality).
Then X is called a generalized metric space.
The following example is a modification of Example 1 of [3].
Example 1 Let X:=\mathbb{N}, and define the function d:\mathbb{N}\times \mathbb{N}\to \mathbb{R} by putting, for all m,n\in \mathbb{N} with m>n:
To see that (X,d) is a generalized metric space, supposem,n\in \mathbb{N} with m>n and suppose p,q\in \mathbb{N} are distinct with each distinct from m andn. Also we assume q>p. We now show that
If one of the three numbers np, qp or qm is even, then, since
clearly (Q) holds. If all three numbers are odd, then, since mn=(mq)+(qp)+(pn), mn is odd and
In this instance there are four cases to consider:

(i)
n<p<q<m,

(ii)
p<n<q<m,

(iii)
n<p<m<q,

(iv)
p<n<m<q.
If (i) holds then
In the other three cases
Therefore (X,d) is a generalized metric space. Now suppose\{{n}_{k}\} is a Cauchy sequence in (X,d). Then if {n}_{i}\ne {n}_{k} and d({n}_{i},{n}_{k})<1, {n}_{i}{n}_{k} must be odd. However, if \{{n}_{k}\} is infinite, {n}_{i}{n}_{k} cannot be odd for all sufficiently large i, k. (Suppose {n}_{i}>{n}_{j}>{n}_{k}. If {n}_{i}{n}_{j} and {n}_{j}{n}_{k} are odd, then {n}_{i}{n}_{k} is even.) Thus any Cauchy sequence in(X,d) must eventually be constant. It follows that(X,d) is complete and that \{n\} is not a Cauchy sequence in (X,d). However, {\sum}_{i=1}^{\mathrm{\infty}}d(i,i+1)<\mathrm{\infty}.
Theorem 2 of [1] asserts that the analog of Caristi’s theorem holds in acomplete generalized metric space (X,d). Thus a mapping f:X\to X in such a space should always have a fixed pointif there exists a lower semicontinuous function \phi :X\to {\mathbb{R}}^{+} such that
The following example shows this is not true in the space described inExample 1.
Example 2 Let (X,d) be the space of Example 1, letf(n)=n+1 for n\in \mathbb{N}, and define \phi :\mathbb{N}\to {\mathbb{R}}^{+} by setting \phi (n)=\frac{2}{n}. Obviously f has no fixed points and,because the space is discrete, φ is continuous. On the other hand, f satisfies Caristi’s condition:
To see this, observe that
This is equivalent to the assertion that
The proof is by induction. Clearly (C) holds if n=1 or n=2. Assume (C) holds for some n\in \mathbb{N}, n\ge 2. Then
References
Kirk WA, Shahzad N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013., 2013: Article ID 129
Branciari A: A fixed point theorem of BanachCaccioppoli type on a class of generalizedmetric spaces. Publ. Math. (Debr.) 2000, 57: 31–37.
Jachymski J, Matkowski J, Świa̧tkowski T: Nonlinear contractions on semimetric spaces. J. Appl. Anal. 1995, 1(2):125–134.
Author information
Authors and Affiliations
Corresponding author
Additional information
The online version of the original article can be found at 10.1186/168718122013129
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kirk, W.A., Shahzad, N. Erratum to: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl 2014, 177 (2014). https://doi.org/10.1186/168718122014177
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168718122014177