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Some notes on Ishikawa iteration for multi-valued mappings. (English) Zbl 1218.47120
Summary: N. Shahzad and H. Zegeye [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 3–4, A, 838–844 (2009; Zbl 1218.47118)] introduced several Ishikawa iterative schemes for fixed points of multi-valued mappings in Banach spaces and proved some strong convergence theorems for them. In their proofs, it seems reasonable and simpler to prove for the iteration {x n } to be a Cauchy sequence. In the present paper, we thus modify and improve the proofs of the main results given by Shahzad and Zegeye. Two concrete examples also are given.
MSC:
47J25Iterative procedures (nonlinear operator equations)
47H04Set-valued operators
47H09Mappings defined by “shrinking” properties