# Correction: A fixed point theorem for cyclic generalized contractions in metric spaces. Fixed Point Theory and Applications 2012, 2012:122

## Abstract

The purpose of this short note is to present some corrections and clarifications concerning the proof of the main result given in the above mentioned paper.

Concerning the text and the proof of the main result (Theorem 2.2), we would like to do the following corrections and clarifications:

(1) In Theorem 2.2, an additional hypothesis is needed, namely:

3. There exists ${y}_{1}\in Y$ such that

(2) On page 3, the fact that the sequence ${\left({c}_{n}\right)}_{n\in \mathbb{N}}$ is bounded follows now from the above mentioned hypothesis, using the following remark: by 3., for each $x\in Y$, we have that

Indeed, let $n\in {\mathbb{N}}^{\ast }$ and suppose that ${y}_{1}\in {A}_{1}$. Let $x\in {A}_{l}$ (where $l\in \left\{1,\dots ,m\right\}$). Let us choose ${u}_{2}\in {A}_{2},\phantom{\rule{0.2em}{0ex}}\dots \phantom{\rule{0.2em}{0ex}},{u}_{l-1}\in {A}_{l-1}$. Then

$\begin{array}{rcl}d\left(x,{f}^{n}\left(x\right)\right)& \le & d\left(x,{y}_{1}\right)+d\left({y}_{1},{f}^{n}\left({y}_{1}\right)\right)+d\left({f}^{n}\left({y}_{1}\right),{f}^{n}\left(x\right)\right)\\ \le & d\left(x,{y}_{1}\right)+d\left({y}_{1},{f}^{n}\left({y}_{1}\right)\right)+d\left({f}^{n}\left({y}_{1}\right),{f}^{n}\left({u}_{2}\right)\right)+\cdots +d\left({f}^{n}\left({u}_{l-1}\right),{f}^{n}\left(x\right)\right)\le \cdots \\ \le & d\left(x,{y}_{1}\right)+d\left({y}_{1},{f}^{n}\left({y}_{1}\right)\right)+d\left({y}_{1},{u}_{2}\right)+\cdots +d\left({u}_{l-1},x\right)<+\mathrm{\infty }.\end{array}$

(3) On page 5, to prove that the Picard iteration converges to ${x}^{\ast }$, we have to do as follows.

Let us show now that the Picard iteration converges to ${x}^{\ast }$ for any initial point ${x}_{1}$. We know that f has a unique fixed point (denoted by ${x}^{\ast }$) and the sequence ${\left({x}_{n}\right)}_{n\in \mathbb{N}}$ converges to a certain $y\in {\bigcap }_{i=1}^{m}{A}_{i}$. We will show that y is also a fixed point of f. For this purpose, we have

This shows that y is a fixed point of f and, thus, $y={x}^{\ast }$.

## Author information

Authors

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and permissions

Alghamdi, M.A., Petruşel, A. & Shahzad, N. Correction: A fixed point theorem for cyclic generalized contractions in metric spaces. Fixed Point Theory and Applications 2012, 2012:122. Fixed Point Theory Appl 2013, 39 (2013). https://doi.org/10.1186/1687-1812-2013-39

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2013-39