Open Access

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

Fixed Point Theory and Applications20132013:39

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

Received: 31 January 2013

Accepted: 10 February 2013

Published: 26 February 2013

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 Y such that
d ( y 1 , f n ( y 1 ) ) < + for all  n N .
(2) On page 3, the fact that the sequence ( c n ) n N is bounded follows now from the above mentioned hypothesis, using the following remark: by 3., for each x Y , we have that
d ( x , f n ( x ) ) < + for all  n N .
Indeed, let n N and suppose that y 1 A 1 . Let x A l (where l { 1 , , m } ). Let us choose u 2 A 2 , , u l 1 A l 1 . Then
d ( x , f n ( x ) ) d ( x , y 1 ) + d ( y 1 , f n ( y 1 ) ) + d ( f n ( y 1 ) , f n ( x ) ) d ( x , y 1 ) + d ( y 1 , f n ( y 1 ) ) + d ( f n ( y 1 ) , f n ( u 2 ) ) + + d ( f n ( u l 1 ) , f n ( x ) ) d ( x , y 1 ) + d ( y 1 , f n ( y 1 ) ) + d ( y 1 , u 2 ) + + d ( u l 1 , x ) < + .

(3) On page 5, to prove that the Picard iteration converges to x , we have to do as follows.

Let us show now that the Picard iteration converges to x for any initial point x 1 . We know that f has a unique fixed point (denoted by x ) and the sequence ( x n ) n N converges to a certain y i = 1 m A i . We will show that y is also a fixed point of f. For this purpose, we have
d ( y , f ( y ) ) d ( y , x n + 1 ) + d ( f ( x n ) , f ( y ) ) d ( y , x n + 1 ) + d ( y , x n ) 0 as  n .

This shows that y is a fixed point of f and, thus, y = x .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University
(2)
Department of Mathematics, Babeş-Bolyai University
(3)
Department of Mathematics, King Abdulaziz University

Copyright

© Alghamdi et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.