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Erratum to: Generalized metrics and Caristi’s theorem

The Original Article was published on 15 May 2013

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,yX and all distinct points u,vX, each distinct from x and y:

  1. (i)

    d(x,y)=0x=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×NR by putting, for all m,nN 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, supposem,nN with m>n and suppose p,qN 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 |np|, qp or |qm| is even, then, since

d(n,m)1,

clearly (Q) holds. If all three numbers are odd, then, since mn=(mq)+(qp)+(pn), mn 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:XX 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 xX.

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 nN, 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 nN, n2. Then

2 n + 2 = 2 ( 2 n + 1 ) 2 n ( n + 1 ) = ( n + n ) ( n + 1 ) ( n + 1 ) ( n + 2 ) .

References

  1. Kirk WA, Shahzad N: Generalized metrics and Caristi’s theorem. Fixed Point Theory Appl. 2013., 2013: Article ID 129

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  2. Branciari A: A fixed point theorem of Banach-Caccioppoli type on a class of generalizedmetric spaces. Publ. Math. (Debr.) 2000, 57: 31–37.

    MathSciNet  Google Scholar 

  3. Jachymski J, Matkowski J, Świa̧tkowski T: Nonlinear contractions on semimetric spaces. J. Appl. Anal. 1995, 1(2):125–134.

    Article  MathSciNet  Google Scholar 

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Correspondence to Naseer Shahzad.

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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

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