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On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. (English) Zbl 1218.47118

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 [Y.-S. Song and H.-J. Wang, Comput. Math. Appl. 55, No. 12, 2999–3002 (2008; Zbl 1142.47344)]. Our results provide an affirmative answer to B. Panyanak’s question [Comput. Math. Appl. 54, No. 6, 872–877 (2007; Zbl 1130.47050)] in a more general setting.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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