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  • Erratum
  • Open Access

Erratum to: Generalized metrics and Caristi’s theorem

Fixed Point Theory and Applications20142014:177

https://doi.org/10.1186/1687-1812-2014-177

  • Received: 1 August 2014
  • Accepted: 1 August 2014
  • Published:

The original article was published in Fixed Point Theory and Applications 2013 2013:129

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 i = 1 d ( p i , p i + 1 ) < , then { p n } 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 [2].

Definition 1 Let X be a nonempty set and d : X × X [ 0 , ) a mapping such that for all x , y X and all distinct points u , v X , each distinct from x and y:
  1. (i)

    d ( x , y ) = 0 x = y ,

     
  2. (ii)

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

     
  3. (iii)

    d ( x , y ) 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 : = N , and define the function d : N × N R by putting, for all m , n N with m > n :
d ( n , n ) : = 0 ; d ( m , n ) = d ( n , m ) : = 1 2 n if  m = n + 1 ; d ( m , n ) = d ( n , m ) : = 1 if  m n  is even ; d ( m , n ) = d ( n , m ) : = i = n m d ( i , i + 1 ) if  m n  is odd .
To see that ( X , d ) is a generalized metric space, suppose m , n N with m > n and suppose p , q N are distinct with each distinct from m andn. Also we assume q > p . We now show that
d ( n , m ) d ( n , p ) + d ( p , q ) + d ( q , m ) .
(Q)
If one of the three numbers | n p | , q p or | q m | is even, then, since
d ( n , m ) 1 ,
clearly (Q) holds. If all three numbers are odd, then, since m n = ( m q ) + ( q p ) + ( p n ) , m n is odd and
d ( n , m ) = i = n m d ( i , i + 1 ) .
In this instance there are four cases to consider:
  1. (i)

    n < p < q < m ,

     
  2. (ii)

    p < n < q < m ,

     
  3. (iii)

    n < p < m < q ,

     
  4. (iv)

    p < n < m < q .

     
If (i) holds then
d ( n , m ) = i = n m d ( i , i + 1 ) = i = n p d ( i , i + 1 ) + i = p q d ( i , i + 1 ) + i = q m d ( i , i + 1 ) = d ( n , p ) + d ( p , q ) + d ( q , m ) .
In the other three cases
d ( n , m ) < d ( n , p ) + d ( p , q ) + d ( q , m ) .

Therefore ( X , d ) is a generalized metric space. Now suppose { n k } is a Cauchy sequence in ( X , d ) . Then if n i 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, i = 1 d ( i , i + 1 ) < .

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 X in such a space should always have a fixed pointif there exists a lower semicontinuous function φ : X R + such that
d ( x , f ( x ) ) φ ( x ) φ ( f ( x ) ) for each  x X .

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, let f ( n ) = n + 1 for n N , and define φ : N R + by setting φ ( n ) = 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:
1 2 n = d ( n , f ( n ) ) φ ( n ) φ ( f ( n ) ) = 2 n 2 n + 1 .
To see this, observe that
1 2 n 2 n 2 n + 1 = 2 n ( n + 1 ) .
This is equivalent to the assertion that
2 n + 1 n ( n + 1 ) .
(C)
The proof is by induction. Clearly (C) holds if n = 1 or n = 2 . Assume (C) holds for some n N , n 2 . Then
2 n + 2 = 2 ( 2 n + 1 ) 2 n ( n + 1 ) = ( n + n ) ( n + 1 ) ( n + 1 ) ( n + 2 ) .

Notes

Authors’ Affiliations

(1)
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
(2)
Department of Mathematics, King Abdulaziz University, P.O. Box 80293, Jeddah, 21959, Saudi Arabia

References

  1. Kirk WA, Shahzad N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013., 2013: Article ID 129Google Scholar
  2. 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
  3. Jachymski J, Matkowski J, Świa̧tkowski T: Nonlinear contractions on semimetric spaces. J. Appl. Anal. 1995, 1(2):125–134.View ArticleMathSciNetGoogle Scholar

Copyright

© Kirk and Shahzad; licensee Springer. 2014

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.

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