On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces

We prove strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued maps. We also construct an iteration scheme which removes a restrictive condition in Song and Wang results [Y. Song, H. Wang, Erratum to “Mann and Ishikawa iterative processes for multiva...

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Bibliographic Details
Published in:Nonlinear analysis 2009-08, Vol.71 (3), p.838-844
Main Authors: Shahzad, Naseer, Zegeye, Habtu
Format: Article
Language:English
Subjects:
Banach space
Fixed point
Nonexpansive multi-valued map
Quasi-nonexpansive multimap
Strong convergence
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Summary:We prove strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued maps. We also construct an iteration scheme which removes a restrictive condition in Song and Wang results [Y. Song, H. Wang, Erratum to “Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces” [Comput. Math. Appl. 54 (2007) 872–877], Comput. Math. Appl. 55 (2008) 2999–3002]. Our results provide an affirmative answer to Panyanak’s question [Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl., 54 (2007), 872–877], in a more general setting.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2008.10.112