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

Shahzad, Naseer; Zegeye, Habtu

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 3-4, 838-844 (2009).

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.

Keywords:

quasi-nonexpansive multimap; nonexpansive multi-valued map; fixed point; strong convergence; Banach space

Citations:

Zbl 1142.47344; Zbl 1130.47050

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