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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:K𝒫(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.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H04Set-valued operators
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)