Some Ergodic Theorems for a Finite Family of Nonexpansive Mappings
Ken-ichi Miyazaki, Takeo Kawatani, Suga Miyaura · 1983
1. In [1], J. B, Baillon proved the first ergodic theorem for nonlinear mappings jn Hilbert space: Let C be a closed convex bounded subset of a Hilbert space X and Tbe a nonexpansive self-mapping of C, then for each xEC the Cesbro means ili-2:;6Tkx converge weakly to a fixed point of Tas n-År oo. Since then, various extensions and devel-opment of this theorem have been given in [2I,[3] and [5]. The purpose of this paper is to extend the ergodic theorem to the folloWing one in the case of finite family of non-expansive mappings under certain restrictions. THEoREM 1. Let C be a eonv'ex compact subset ofa strictly convex Banach space X and {Ti: i=1, 2,•••, k} (kl2) a family of nonexpansive self-mappings of C with a non-empty set of common fixed points. For any point xeC, nonnegative integer n and positive integer m, we define (1) Xn+i,m= ' p- ' (iÅíl; k) {Xn,m+27=i 2(ii,-,ii) TiiTi2'''TiiXn,m}, where (ii, i2,•••, ii) is any repeated permutation of l letters from (1, 2,•••, k), and 2(i,,...,i,) stands for the summation running over atl the permutations (ii,•••, it) and p(m, k) = (km'i-1)1(k-1), x,,.=x for m = 1, 2,•••, Then for a.fixed m the sequence {x.,.}ee..o converges to an element y. in Alr•=i F(Ti) as n.co, where F(T) is the set offixed points of T. And this y. satisLfies (2) ynt = p(hll,-kr) {Ym+27-i2(it,-•,ii)TiiTi2'''TiiYm}' Further there is a subsequence {y.,}ge.,i that converges to a common fixed point y of Ti, i= = 1, 2,•••, k which satisfies (3) Y=}.!II} p(ml,., k)'{Ym,+27='i 2(ii,•-,ii) TiiTi2"'TiiYmj} ' REMARK 1. We shall here give some examples of (1). The expressio,n of (1) for k=2, m = = 3 will be illustrated as follows