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 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.
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