Open Access

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

  • Nawab Hussain1,
  • Mahmood Lashkarizadeh Bami2 and
  • Ebrahim Soori3Email author
Fixed Point Theory and Applications20152015:203

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

Received: 27 October 2015

Accepted: 27 October 2015

Published: 10 November 2015

The original article was published in Fixed Point Theory and Applications 2014 2014:238

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:

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.

Notes

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, University of Isfahan, Isfahan, Iran
(3)
Department of Mathematics, Lorestan University, Khoramabad, Iran

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) MathSciNetView ArticleGoogle Scholar

Copyright

© Hussain et al. 2015