Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces
© F. Gu and Q. Fu. 2009
Received: 19 November 2008
Accepted: 9 April 2009
Published: 15 April 2009
We establish strong convergence theorem for multi-step iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Our results extend and improve the recent ones announced by Plubtieng and Wangkeeree (2006), and many others.
From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense.
The purpose of this paper is to establish a strong convergence theorem for common fixed points of the multistep iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in a uniformly convex Banach space. The results presented in this paper extend and improve the corresponding ones announced by Plubtieng and Wangkeeree , and many others.
Definition 2.1 (see ).
Lemma 2.2 (see ).
Lemma 2.3 (see ).
3. Main Results
The proof is clear by Lemma 2.3.
so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then we have the following:
This completes the proof.
so that . Let the sequence be defined by (1.3) whenever , , satisfy the same assumptions as in Lemma 3.2 for each and the additional assumption that there exists such that for all . Then converges strongly to a common fixed point of the mappings .
Theorem 3.4 improves and extends the corresponding results of Plubtieng and Wangkeeree  in the following ways.
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