Asymptotic behavior of almost-orbits of nonexpansive semigroups without convexity
Hirobumi Kiuchi, Wataru Takahashi · Kodai Mathematical Journal · 1992
We first prove a resul on the asymptotic behavior of almost-orbits of nonexpansive semigroups without convexity in a Hubert space.This is a generalization of results of Rode [7] and Takahashi [10].Further we prove a fixed point theorem for Lipschitzian semigroups without convexity.This is a generalization of results of Lau [3], Takahashi [8], [10] and Ishihara [2], 1. Introduction.Let H be a real Hubert space with norm || || and inner product ( , •) and let C be a nonempty subset of H.A mapping T: C->C is said to be Lipschitzian if there exists a nonnegative number k such that \\Tx-Ty\\^k\\x-y\\ for every x, ys t of 5 into itself are continuous.Then a family S~ {T s : seS} of mappings of C into itself is called a Lipschitzian semigroup on C if it satisfies the following:(1) τ st x=T s T t x for all s, tt=S and XGC; (2) for each xeC, the mapping s-^T s x is continuous on S;(3) for each seS, T s is a Lipschitzian mapping of C into itself with Lipschitz constant k g .A Lipschitzian semigroup cS-{T t : t^S} on C is said to be nonexpansive if k s -l for every s^S.Recently, Takahashi [10] proved a nonlinear ergodic theorem and a fixed point theorem for nonexpansive semigroups without convexity in a Hubert space.On the other hand, Miyadera-Kobayasi [4] introduced the notion of an almost-orbit of nonexpansive semigroups and established the weak and strong almost convergence of such an almost-orbit; see also [1], [11], [12].