Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces. (English) Zbl 1130.47050

Summary: Let \(K\) be a nonempty compact convex subset of a uniformly convex Banach space and let \(T:K\to\mathcal P(K)\) be a multivalued nonexpansive mapping. We prove that the sequences of Mann and Ishikawa iterates converge to a fixed point of \(T\). This generalizes former results proved by K.P.R.Sastry and G.V.R.Babu [Czech.Math.J.55, No.4, 817–826 (2005; Zbl 1081.47069)]. We also introduce both of the iterative processes in a new sense, and prove a convergence theorem of Mann iterates for a mapping defined on a noncompact domain.
Editor’s remark: An erratum to this paper has been posted by Y.-S. Song and H.-J. Wang in [Comput. Math. Appl. 55, No. 12, 2999–3002 (2008; Zbl 1142.47344)].


47J25 Iterative procedures involving nonlinear operators
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators
Full Text: DOI


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