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

Panyanak, Bancha

Comput. Math. Appl. 54, No. 6, 872-877 (2007).

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)].

MSC:

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

Keywords:

multivalued mappings; fixed points; Mann iterates; Ishikawa iterates; uniformly convex Banach spaces

Citations:

Zbl 1081.47069; Zbl 1142.47344

References:

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