Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 716971, 14 pages
doi:10.1155/2010/716971

Convergence of the sequence of successive approximations to a fixed point

Abstract

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 x∈X 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.