DOI: 10.24193/fpt-ro.2020.2.36

Authors: A. Grzesik, W. Kaczor, T. Kuczumow and S. Reich

Abstract: In this paper we prove the following general theorem. Let (E,‖·‖_E) be a uniformly convex Banach space, and let C be a bounded, closed and convex subset of E. Assume that C has nonempty interior and is locally uniformly rotund. Let ℱ be a commutative nonexpansive semigroup acting on C. If ℱ has no fixed point in the interior of C, then there exists a unique point on the boundary of C such that each orbit of ℱ converges in norm to . We also establish analogous results for semigroups and mappings which are asymptotically nonexpansive in the intermediate sense.

Key Words and Phrases: Asymptotically nonexpansive in the intermediate sense, fixed point, iterates, locally uniformly rotund set, nonexpansive mapping, semigroup of mappings, uniform convexity.

2010 Mathematics Subject Classification: 41A65, 47H10, 47H20.

Published on-line: July 1st, 2020.

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