Skip to main content
Fixed Point Theory and Algorithms for Sciences and Engineering
Download PDF

Erratum to: An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces

Fixed Point Theory and Applications volume 2015, Article number: 203 (2015) Cite this article

The Original Article was published on 04 December 2014

Unfortunately, the original version of this article [1] contained an error. In Theorem 3.1 the condition ‘left amenable’ is required instead of the condition ‘left invariant’, because the authors used the Lemma 2.2 and in that lemma ‘X’ is amenable.

The correct Theorem 3.1 is correctly included in full in this erratum:

FormalPara Theorem 3.1

Let S be a semigroup. Let C be a nonempty compact convex subset of a real strictly convex and reflexive and smooth Banach space E. Suppose that \(\mathcal{S}=\{T_{s}:s\in S\}\) is a representation of S as nonexpansive mapping from C into itself such that \(\operatorname{Fix}(\mathcal {S})\neq\emptyset\). Let X be a left amenable subspace of \(B(S)\) such that \(1\in X\), and the function \(t\mapsto\langle T_{t}x,x^{*}\rangle\) is an element of X for each \(x\in C\) and \(x^{*}\in E^{*}\). Let \(\{\mu_{n}\}\) be a left regular sequence of means on X. Suppose that f is an α-contraction on C. Let \(\epsilon_{n}\) be a sequence in \((0, 1)\) such that \(\lim_{n} \epsilon_{n}=0\). Then there exists a unique sunny nonexpansive retraction P of C onto \(\operatorname{Fix}(\mathcal{S})\) and \(x \in C \) such that the following sequence \(\{z_{n}\}\) generated by

$$ z_{n}=\epsilon_{n} f(z_{n})+(1- \epsilon_{n})T_{\mu_{n}}z_{n}\quad( n \in\mathbb{N}), $$
(1)

strongly converges to Px.

References

  1. Hussain, N, Lashkarizadeh Bami, M, Soori, E: An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2014, 238 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Nawab Hussain

  2. Department of Mathematics, University of Isfahan, Isfahan, Iran

    Mahmood Lashkarizadeh Bami

  3. Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran

    Ebrahim Soori

Authors
  1. Nawab Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mahmood Lashkarizadeh Bami
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ebrahim Soori
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ebrahim Soori.

Additional information

The online version of the original article can be found under doi:10.1186/1687-1812-2014-238.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hussain, N., Lashkarizadeh Bami, M. & Soori, E. Erratum to: An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2015, 203 (2015). https://doi.org/10.1186/s13663-015-0450-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13663-015-0450-y