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: 19 August 2014

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
(2)
Department of Mathematics, King Abdulaziz University

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.