SOME ERGODIC THEOREMS FOR A FINITE FAMILY
OF NONEXPANSIVE MAPPINGS
By
Ken-ichi MIyAzAKi, Takeo KAwATANi and Suga MIyAuRA (Received Nov. 26, 1982)
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
Xn+i,3 = 11s {I+ Ti + T2 +Ti+ Ti T2+ T2 Ti + Ti+T?+T? T2 + Ti T2 Ti
+ T2 Ti + T3 Ti + T2 Ti T2 + Ti Ti + Tg}x.,3•
The Cesaro means of type (1) are given in [7] in the case of commuting submarkovian operators on Li.
REMARK2. If the family {Ti:i=1,2,•••,k} is cornmutative, then the assumption Alr•=i F(Ti)SÅë may be omitted as shown in [4] and the iteration formula (1) may be reduced to more simple one.
2. ProofofTheorem1. The proof of Theorem1 depends upon the following lemmas.
LEMMA 1. Let C be a closed subset ofa Banach space X and {Ti: i---= 1, 2,•••, k} be a finite family of nonexpansive mappings from C into a compact subset D of X. For a .fixed positive integer m let us deLfine the sequence {x.,.} by
(4) xn+i,m :aoxn,m+27=i2(ii,i2,•••,ii)ait,i2,•••,it TiiTi2'''TiiXn.m,
where (ii, i2,•••, ii) is any repeated permutation of l letters ,from (1, 2,•••, k) and ]iÅíÅqi,,i,,...,i,) stands for the summation running over atl such permutations (ii, i2,•••, ii) and ao, ai,,i,,.,.,i, are numbers satisfying OÅqaoÅq1, OÅqai,,i,,".,i,Åq1, ao+27=i 2(i.i,,•-,h) ai,,i,,...,i,=1 and xo,. =xEC is a given element. Suppose further that x.,.EC for all positive i,nteger n. Then the sequence {x.,.}.co.i converges to a point .y.GD such that (5) 27=i 2(i,,i,,-•,i,) ai,,i,,-•,i, Ti, Ti,'•• Ti,ym= 2T==i 2(i,,i,.-,i,) ai,,i,,-•,i,ym•
Before proving Lemma 1 we here cite Ishikawa's theorem (Theorem 1 in [6]).
LEMMA A (Ishikawa [6]). Let C be a closed subset of a Banach space X and let T be a nonexpansive mapping from C into a compact subset of X. Suppose that a point xi EC and a sequence {t.}IP.i satisfy the conditions: O.Åq. t.SbÅq1, 21P..i t.= co and x.eC for all positive integer n, where {x.}ge=i is deLfined by
(6) Xn+1 == (1-tn)Xn+tn TXn•
Then Thas afixed point in C and {x.} converges to a.fixed point of T.
PRooF oF LEMMA 1, From (4) we get
xn+i,m "aoxn,m+(1-ao)Z7=i 2(ii,i2,•••,h) a;i,i2,•-•,ii TiiTi2'''TiiXn,m
wjth al,,i,,•-,i,=ai,,i,,-,i,/(i--ao), 27=i2(i,,i,,•-,i,)ai,,i,,•-,i,=:i• Put T=2T=iE(i,,i,,-•,i,)
ali,i2,•t•,i, TitTi2''.Ti,• Then for any x, yec,
IlTx-Tyll
:-:S 2T= i Z) (i,,i,,•••.i,) a;,,i,,•••,i,ll Ti, Ti,••• Ti,x - Ti , Ti,••• Ti,y 1I
, i-:i{ 27T- i 2(i,,i,,•••,i,) al,,i,,•••,i, llx- .y ll - lix- .y ll •
Thus from the assumption x.,.eC and setting t.==1-ao for all n, LemmaA may be applicable to this T. Hence {x.,.}ge=i converges to a point y. in C as n.oo such that
2T=i 2(ii,iz,•t•,ii) al't,i2"••,ii TiiTi2''' TiiYm =Ym which implies (5).
In the case of strictly convex Banach space the following lemma was proved in [8].
LEMMA 2. Let X be a strictly convex Banach space and let yi, i-- 1, 2,•••, k, be any elements of X. Suppose that y :21.iai.vi with OÅqaiÅq1, i=1, 2,•••, k, 21.iai==1 and there exists at least an element yi such that yiiy. Then we have
(7) IlyllÅqmax{llyill:for all yi such that yisy}.
Now we shall prove the theorem 1 by making use of these LemmaS 1, 2.
PRooF oF THEoREM 1. We consider the sequence {x.,.} defined by more general form (4) taking the place of (1). Since {Ti:i=1, 2,•••, k} in Theorem 1 are self-mappings of
C, the assumptions x.,.eC for all n in Lemma 1 are satisfied. Thus by Lemma 1 the sequence {x.,.}ee.i converges to an element y,. in Alr•.iF(Ti) as n.oo which satisfies (5). Thisimplies
(8) y.=2rei 2(i,,i,,••,i,) al,,i,,•-,i, Ti,Ti,'''Ti,ym
with a;,,i2,•••,i, == ai,,i,,•••,iil(1 dao)•
Now by the assumption there exists an element weC: Tiw ==w for i-- 1, 2,•••, k, and by nonexpansiveness of Ti we have llTi,Ti,•••Ti,y•--wil :ll lly-wll for any (ii, i2,•••, ii) and
y e C. Thus, applying Lemma 2 to
y. - i,v =: 2T= i År: (i,,i,,•••,i,) al•,,i,,•••,i, (Ti, Ti,••• Ti,y. - w) ,
if there exists a term Ti,Ti,•••Ti,y.7Ey. then
ll y. •-• wIl Åqmax {II Ti, Ti,••• Ti,y.-wll : for Ti, Ti,''' Ti,ym#ym}
S ll.y.rvvll •
This is a contradiction which shows
y.=Ti,Ti,•••Ti,y. for ail (it, i2,•••, ii),
hence
y.= Tiy. for i= 1, 2,•••, k,
that is y. e A fr•=i F( Ti).
Since {y.}.co=icC and C is a compact subset of X, there is a subsequence {y.,}ee.i that converges to a point .v of C. Now since Ti,i=1,2,•••,k, are nonexpansive and
y., e A}• .. i F( Ti), we have
Il Tiy-y11 == H Tiy-y., +y., ---- yll S- 2il y., - .y II •
which implies .vEA5•=i F(Ti). From (8) the point .v satisfies
JI. !m. 27=Ji E)(i,,i,,-•,i,) al,,i,,-•,i, Ti, Ti,''' Ti,ym,•
