Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings
© J.-Z. Xiao and X.-H. Zhu. 2008
Received: 8 August 2007
Accepted: 26 November 2007
Published: 12 December 2007
A family of commuting nonexpansive self-mappings, one of which is weakly contractive, are studied. Some convergence theorems are established for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point. The error estimates of these iterations are also given.
1. Introduction and Preliminaries
As an important extension of the class of contractive mappings, the class of weakly contractive mappings was introduced by Alber and Guerre-Delabriere . In Hilbert and Banach spaces, Alber et al. [1–4] and Rhoades  established convergence theorems on iteration of fixed point for weakly contractive single mapping.
We will establish some convergence theorems for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point and to give their error estimates.
We define iterations which will be needed in the sequel.
We will make use of following result in the proof of Theorem 2.4.
Lemma 1.1 (see ).
2. Main Result
By (1.1), (1.2), and (2.11), we deduce
From (2.16) and (2.17), we obtain the error estimate (2.1). This completes the proof.
If in Theorem 2.1, where is the identity mapping of , then we conclude that the sequence converges to the unique common fixed point of weakly contractive mapping , with the error estimate , where . Thus, our Theorem 2.1 is a generalization of the corresponding theorem of Rhoades .
From (2.21)–(2.23), we obtain (2.18). This completes the proof.
Hence, the estimate (2.25) holds. This completes the proof.
This work is supported by the National Natural Science Foundation of China (10671094).
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