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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:

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.

Keywords

  • Point Theorem
  • Initial Point
  • Differential Geometry
  • Fixed Point Theorem
  • Point Theory

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, P.O. Box 4087, Jeddah, 21491, Saudi Arabia
(2)
Department of Mathematics, Babeş-Bolyai University, Kogǎlniceanu Street No. 1, Cluj-Napoca, 400084, Romania
(3)
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21859, Saudi Arabia

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.

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