Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 716971, 14 pages
Research Article

Convergence of the Sequence of Successive Approximations to a Fixed Point

Department of Basic Sciences, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Received 29 September 2009; Accepted 21 December 2009

Academic Editor: Mohamed A. Khamsi

Copyright © 2010 Tomonari Suzuki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


If (X,d) is a complete metric space and T is a contraction on X, then the conclusion of the Banach-Caccioppoli contraction principle is that the sequence of successive approximations {Tnx} of T starting from any point xX converges to a unique fixed point. In this paper, using the concept of τ-distance, we obtain simple, sufficient, and necessary conditions of the above conclusion.