Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces

Abstract

Let K be a nonempty compact convex subset of a uniformly convex Banach space, and T : K → P ( K ) 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 Sastry and Babu [K.P.R. Sastry, G.V.R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J. 55 (2005) 817–826]. 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.