Means and convergence of semigroup orbits
Aleksandra Grzesik, Wieslawa J Kaczor, Instytut Matematyki, UMCS, 20-031 Lublin, Poland, E-mail: [email protected], Tadeusz Kuczumow, Instytut Matematyki, UMCS, 20-031 Lublin, Poland, E-mail: [email protected], Simeon Reich · Fixed Point Theory · 2020
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 x on the boundary of C such that each orbit of F converges in norm to x. We also establish analogous results for semigroups and mappings which are asymptotically nonexpansive in the intermediate sense.