Open Access

Erratum to "Some Fixed Point Theorems of Integral Type Contraction in Cone Metric Space"

Fixed Point Theory and Applications20112011:346059

https://doi.org/10.1155/2011/346059

Received: 20 December 2010

Accepted: 5 January 2011

Published: 11 January 2011

The original article was published in Fixed Point Theory and Applications 2010 2010:189684

We regret making following mistake in the above-mentioned paper [1]. We would like to correct it and explain some notations.

(1) In [1] we introduced a new concept of integral type contraction in cone metric spaces and generalized Brancieri and Meir-Keeler theorems in such spaces. [1, Theorem  2.9] is an extension of Brancieri's theorem, and [1, Theorem  3.2] is an extension of Brancieri and Meir-Keeler's results. We asserted the following in [1, Theorem  2.9].

(i)"Let be a complete cone metric space and be a normal cone. Suppose is a non-vanishing map and a sub-additive cone integrable on each such that for each , . If is a map such that for all
(1)

for some , then has a unique fixed point in ."

Also, we asserted in [1, Theorem  3.2] the following.

(ii)"Let be a complete regular cone metric space and be a mapping on . Assume that there exists a function from into itself satisfying the following:

(B1) and for all .

(B2) is nondecreasing and continuous function. Moreover, its inverse is continuous.

(B3)For all , there exists such that for all
(2)

(B4)For all

(3)

Then has a unique fixed point."

After this theorem, we asserted the following in [1, Remark  3.3] that:

(iii)"If is a non-vanishing map and a sub-additive cone integrable on each such that for each , and , then is satisfies in all conditions of [1, Theorem  3.2]. Equivalently [1, Theorem  2.9] is concluded from [1, Theorem  3.2]."

Note that, in (B2) of [1, Theorem  3.2] and [1, Remark  3.3], we have emphasized that the map must have the continuous inverse, but unfortunately this assumption has been forgotten mistakenly in [1, Theorem  2.9]. Note that this assumption is a necessary condition to prove [1, Theorem  2.9].

(2) To prove [1, Theorem  3.2] and [1, Theorem  2.9], it is sufficient that satisfy the following: for each sequence
(4)

On the other hand, (4) is equivalent to continuity of at zero.

(3) In [2] the authors gave a counterexample on [1, Theorem  2.9] only for our misprint that we have asserted it in the above as you have seen. They also gave a comment for us at the end of their paper to correct such misprint and emphasized that must have the continuous inverse. As you have seen, we have asserted and emphasized such note in (B2) of [1, Theorem  3.2] and [1, Remark  3.3] before the authors in [2] mentioned it.

Nevertheless, we do apologize to the readers for this mistake.

Notes

Declarations

Acknowledgment

The third author would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Teheran, Iran, for supporting this research (Grant no. 89470126).

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University
(2)
Department of Mathematics, Faculty of Science, Imam Khomeini International University
(3)
School of Mathematics, Institute for Research in Fundamental Sciences

References

  1. Khojasteh F, Goodarzi Z, Razani A: Some fixed point theorems of integral type contraction in cone metric spaces. Fixed Point Theory and Applications 2010, 2010:-13.Google Scholar
  2. Arandelović ID, Kečkić DJ: A counterexample on a theorem by Khojasteh, Goodarzi, and Razani. Fixed Point Theory and Applications 2010, 2010:-6.Google Scholar

Copyright

© Farshid Khojasteh et al. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.