Erratum to: Generalized metrics and Caristi’s theorem
Fixed Point Theory and Applications volume 2014, Article number: 177 (2014)
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 .
Definition 1 Let X be a nonempty set and a mapping such that for all and all distinct points , each distinct from x and y:
Then X is called a generalized metric space.
The following example is a modification of Example 1 of .
Example 1 Let , and define the function by putting, for all with :
To see that is a generalized metric space, suppose with and suppose are distinct with each distinct from m andn. Also we assume . We now show that
If one of the three numbers , or is even, then, since
clearly (Q) holds. If all three numbers are odd, then, since , is odd and
In this instance there are four cases to consider:
If (i) holds then
In the other three cases
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, .
Theorem 2 of  asserts that the analog of Caristi’s theorem holds in acomplete generalized metric space . Thus a mapping in such a space should always have a fixed pointif there exists a lower semicontinuous function such that
The following example shows this is not true in the space described inExample 1.
Example 2 Let be the space of Example 1, let for , and define by setting . 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 or . Assume (C) holds for some , . Then
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 Banach-Caccioppoli 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.
The online version of the original article can be found at 10.1186/1687-1812-2013-129
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Kirk, W.A., Shahzad, N. Erratum to: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl 2014, 177 (2014). https://doi.org/10.1186/1687-1812-2014-177