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Nonlinear ergodic theorems for semigroups of nonexpansive mappings and left ideals
Nonlinear
Analym,
Theory,
Methods
& Applrcofrons,
Pergamon
Vol. 26, No. 8, pp. 1411-1427. 1996 Copyright b 1996 Else&r Scicncc Ltd Printed in Great Britain. All rights reserved 0362-546X/% $15.@0+0.00
0362-546X(94)00347-5
NONLINEAR ERGODIC NONEXPANSIVE ANTHONY
THEOREMS MAPPINGS
T. M. LAU,t$
KOJI NISHIURAO
iDepartment of Mathematical Sciences, University §Department of Information Sciences, Tokyo Institute (Received Key words andphrases: minimizer, uniformly
3 1 May
FOR SEMIGROUPS AND LEFT IDEALS
1994; received
and WATARU
OF
TAKAHASHIS
of Alberta, Edmonton, Alberta T6G-2G1, of Technology, Oh-okayama, Meguro-ku, for publication
Nonlinear ergodic theorems, nonexpansive convex space, fixed point theorems.
17 November mappings,
Canada; and Tokyo 152, Japan
1994) semigroup,
left ideal orbit,
1. INTRODUCTION
Let S be a semitopological semigroup, i.e. S is a semigroup with a Hausdorff topology such that for each s E S the mappings s + a . s and s + s . a from S to S are continuous. Let E be a uniformly convex Banach space and let S = 1T, : s E S 1 be a continuous representation of S as nonexpansive mappings on a closed convex subset C of E into C, i.e. T,,x = T,T,x for every a, b E S and x E C and the mapping (s, x) -+ T,(x) from S x C into C is continuous when S x C has the product topology. Let F(S) denote the set (x E C: T,x = x for all s E S] of common fixed points of S in C. Then as well known, F(S) (possibly empty) is a closed convex subset of C (see [l]). In this paper, we shall study the distance between left ideal orbits and elements in the fixed point set F(S). We shall prove (theorem 3.11) among other things that if E has a Frechet differentiable norm, then for any semitopological semigroup S and x E C, the set Q(x) = ncO(T,x: t E L), with the intersection taking over all closed left ideals L of S, contains at most one common fixed point of S (where Co A denotes the closed convex hull of A). This result is then applied to show (theorem 4.1) that if F(S) fl Q(x) # 0 for any x E C, then there exists a retraction P from C onto F(S) such that T,P = PT, = P for every t E S and P(x) E Co{ T,x: t E S ] for every x E C. Both theorems 3.11 and 4.1 were established by Lau and Takahashi in [2] when S has finite intersection property for closed left ideals. The first nonlinear ergodic theorem for nonexpansive mappings was established in 1975 by Baillon [3]: let C be a closed convex subset of a Hilbert space and let T be a nonexpansive mapping of C into itself. If the set F(T) of fixed points of T is nonempty, then for each x E C, the Cesaro means
converge weakly to some y E F(T). In this case, putting y = Px for each x E C, P is a nonexpansive retraction of C onto F(T) such that PT = TP = P and Px E co{ T’x: n = 1,2,. . .) for each x E C. In [4], Takahashi proved the existence of such a retraction for an amenable semigroup. This result is further extended to certain Banach spaces by Hirano and Takahashi in [5]. t This research
is supported
by NSERC-grant
A7679. 1411
A. T. M.
1412
LAU
ef 01
Our paper is organized as follows: in Section 2 we define some terminologies that we use; in Section 3 we study the distance between ideals determined by left orbits and the fixed point set; in Section 4 we apply our results in Sectian 3 to establish our main nonlinear ergodic theorems; finally in Section 5 we study an almost fixed point property determined by the minimal left ideals in the enveloping semigroup of a semigroup of nonexpansive mappings on a weak compact convex set and obtain a generalization of De Marr’s fixed point theorem [6]. 2. PRELIMINARIES
Throughout this paper, we assume that a Banach (ar Hilbert) space is real. Let E be a Banach space and let E * be its dual. Then, the value off E E * at x E E will be denoted by (x, f > or f(x). The dualiry mapping f of E is a muiltivalued operator I: E -+ E*, where f(x) = IKE E*: (x,f) = (lxl12 = j-(1’] (w h ic h IS nonempty by simple application of the Hahn-Banach theorem). Let B = (X E E: jjxl\ = 11 be the unit sphere of E. Then the norm of E is said to be Frtchet differentiable if for each x E B, the limit lim ” x + x-0
AYll
- II4
A
is attained uniformly for y E B. In this case, J is a single-valued and norm ID norm continuous mapping from E into E* (see [l, 71 for more details). Let S be a nonempty set and Iet X be a subspace of i”(S) (bounded real-valued functions on S) containing constants. By a submean on X we shall mean a real-valued function p on X satisfying the following properties: (1) Af + g) 5 ii(f) + kG3 for every f, g E X; (2) ~(osf) = arp(f) for every f E X and 1y 2 0; (3) forf,g E X,/l g implies p(f) 5 Ag); (4) flu(c) = c for every constant function c. A semitopological semigroup S is called left reversible (resp. right reversible) if S has finite intersection property for “closed” right (resp. left) ideals. S is called reversible if S is both left and right reversible. Let S be a semjropological semigroup and let C(S) denote the closed subalgebra of I”(S) consisting of bounded continuous functions. For each j E C(S) and a E S, let &f)(t) = /(of) and (~=f)(t) = f(ra)- Let RUC(S) denote allf E C(S) such that the mapping S + C(S) defined by s + rs f is continuous when C(S) has the norm topology. Then RUC(S) is a translation invariant subalgebra of C(S) containing constants. Further, RUC(S) is precisely the space of bounded left uniformly continuous functions on S when S is a group (see [S]). A submean ~1on RffC(S) is called invariant if fr(f, f > = p(r, j) = &ff for every f E RVC(S) and II E S. If S is a discrete semigroup, then RUC(S) has an invariant submean if and only if S is reversible. Also, if S is normal and C(S) has an invariant submean, then S is reversible. However, S need not be reversible when C(S) has an invariant submean in general (see [9j for details). 3. LEFT
IDEAL
ORBITS
AND
THE
FIXED
Unless otherwise specified, S denotes a semitopological continuous representation of S as nonexpansive mappmgs subset C of a Banach space E into C.
POINT
SET
semigroup and S = IT,: s E SI a from a nonempty closed convex
Nonlinear
x
ergodic
theorems
1413
Let E(S) denote the collection of closed left ideals in S. Assume that F(S) # 0. For each E C and L E C(S), define the real-valued function qx,L on F(S) by h,L(f)
= inf(IITx
- fl12: t E L1
and let 4x(f) = suPlqX,,(f):
L E W91.
Then
4Af) = sup - fl12 s infllT,x t as readily checked. LEMMA 3.1. Let C be a nonempty closed convex subset of a Banach space E. If F(S) # 0, then for each x E C, qx is a continuous real-valued function on F(S) such that 0 I q*(f) I I/x - fl12 for each f E F(S) and qx(fn) + cc)if /If,]/ -+ 00. Further, if F(S) is convex, then qwis a convex function on F(S). Proof. Since 0 I I/TX - fl12 = II T,X - ~,fll’ I IIx - fl12 for every f E F(S) and t E S, it follows readily that 0 I qx(f) I IIx - ~11~. Also, if f E F(S) and t E S, then IIT,x - f II I
Ilx - f II. Hence,II~41 5 IITX - f II + Ilf II 5 Ilx - f II + Ilf II, i.e. M = SUP{II~41: t E SI < ~0.
Let If,) be a sequence in F(S) such that 11f,lj + 03. Then we have for each t E S
IITX - fnl121 (IITX11- llfnll)2 = llfnl12- m~llllfnll + IIWII” 2 Ilfnll’ - 2MlfnII =
llfX(l
- G
>
and, hence, for each L E C(S),
4x,L(fn) 2 lW(1 So we have qx(fn) + 00. To see that qx is continuous,
- 6)
+ co.
let 1f,] be a sequence in F(S) converging to some f E F(S) and
M’ = sup((l7;x - f,J + 117;~ - fll: n = 1,2,...
and t E S).
Then, since
I/TX -fnl12 - IITX -f II25 (IITX -frill + IITX -f ll)lllTx -fall - Ilm -f III 5 M’llfn -f II, we have for each L E .52(S)
qx,r.(fn) 5 4X&(f) + M’lk -f II. Similarly,
we have
4&L(f) 5 qx,L(fn) + M’llfn -fll. So we obtain
Mfn) - qx(f)I 5 M’llfn - f II. This implies that qx is continuous
on F(S).
1414
A. T. M.
LAU
et al.
If F(S) is convex, for each f, g E F(S) and (Y,p L 0 with 01 + /? = 1, af + pg E F(S). Let E > 0. Then there exists L, E g(S) such that sup
inf(cYII T,x - flj2 + PII T,x - gjj2) < ,‘,“,f (DII T,X -
LEoC(S) fCL
0
fl12 + AlT,X - gl12)+ i.
Let u E L,. Then Su s L, and, hence, sup
inf(allT,x
LEC(S)fEL Moreover,
- fl12 + /3II7;x - gll’) 5 inf(oll/7;,x tes
- fll’
+
PIIT& - A21 + i.
there exist u, w E S such that
IIT& - fl12 < fp&
- fl12 + ;
and
IITwoux- fl12 < yT”Ux
- fl12 + ; .
Therefore, we obtain q,W
+ Pg)
=
SUP infllTx - bf + &II2 LLqS) teL
5 Ly, < fpTux
fy
TX - Al2 + PllTX - gl12) - Al2 + PIITUX- sl12)+ ;
5 4Tvvux - fl12 + PIITVVUX - gl12+ ; 5 4Tvux - Al2 + PIIGvux - gl12+ ;
=
ut~~II~x-~ll~+Pt~~lI~x-~l12+~ 1 2 (where L, = Su and L, = SW)
Since E > 0 is arbitrary,
we have
Nonlinear ergodic theorems THEOREM 3.2. Let C be a nonempty
1415
closed convex subset of a uniformly
convex Banach space
E. Assume that F(S) # 0. Then for any x E C, there exists a unique element h E F(S) such that
Proof. Since E is uniformly convex, the fixed point set F(S) in C is closed and convex (see [l]). Hence, it follows from lemma 3.1 and [lo] that there exists h E F(S) such that 4A4
= WqAf):f
E W)l.
To see that h is unique, let k E F(S). Then by [l 11, there exists a strictly increasing and convex function (depending on h and k) g: [0, co) + [0, 00) such that g(0) = 0 and IIT,x - (Ah + (1 - A)k)112 = IIA(T,x - h) + (1 - A)(T,x - k)/i2 I 1117;~ - h(l’ + (1 - A)IIT,x - k(l’ - A(1 - A)g(llh - kli) for each t E S and A with 0 5 ,J I 1. So we have for each i with 0 I A I 1, MO
5 Mh
+ (1 - AN) 5 &dh)
+ (1 - ~)q.r(k) - A(1 - Ak(lth - kill
and, hence a(h)
5 a(k)
- Q(llh
- kll).
It follows that qx@) 5 qx(k) - s(llh - kli)
as A -+ 1.
Since g is strictly increasing, it follows that if qx(h) = q,(k), then h = k. We call the unique element h E F(S) in theorem 3.2 the minimizer For each x E C, let Q(x) =
n CO(7;x:tEL) L EC(S)
= n a(&X: se.s
a
of qx in F(S).
t Es]
>
.
3.3. Let C be a nonempty closed convex subset of a Hilbert space H. Let S = (T, : s E S] be a continuous representation of S as nonexpansive mappings from C into C. Then for any x E C, any element in Q(x) n F(S) is the unique minimizer of qx in F(S). In particular, Q(x) fl F(S) contains at most one point. THEOREM
Proof. Let z E F(S) be the minimizer there exists u E S such that
sup inf(ll7;,x
s I
of qx in F(S) and y E Q(x) fl F(S). Then for some E > 0,
- zlIz + 2(&,x
- z, z - u> + II2 - vl12)
< inf(II TUx - alI2 + 2(T,,x - z, z - y> + II2 - ul12) + i .
t
Moreover,
there exist u, w E S such that II T,,x - zl12 < infll TUx - ~11~+ $ t
1416
A. T. M. LAU
et al.
and (Twvux - z, z - y> < inf(T,,,x
- z, .z -
t
Y>
+ i.
Therefore, we obtain 4Ay) = sup i:fll Lx
- YV
= slip inf(llT,,x s
- 211’ + 2(7;,x - z, z - Y> + llz - Yl12)
t
< inf(llT,,x t
- zl12 + 2(T,,x - z,z - Y> + Il.2 - YI12) + i
5 /Twuux - z112 + 2(T,,,x 5 IIT,,x - z/j2 + 2(T,,,x
< infllT,,x t
- z(12 + 2 sup inf(T,,x s
= qx(z) + 2 sup inf(7;,x s t
+ ,tZ - Yl12 + ;
- z, z - y> + llz - yll’ + i + i + i
t
I
Y>
- z,z - Y> + 112- ~1,~ + 5
- z112+ 2 inf(7;,,x
5 sup infll7;,x s
- z, z -
t
- z, z - y> + llz - yll’ + E
- z, z - y> + llz - yl12 + E.
This implies that 2 sup inf(T,,x s t
- z, z - y> > q*(y) - qX(z) - llz - yl12 - E 2 - Ilz - yl12 - &.
So, there exists a E S such that 2(7;,x
- z, z - y> > - llz - Yl12 - E
for every t E S. From y E cO( T,x: t E Sl, we have 20 - z, 2 - y> 2 -llz - Yl12 - E.
This inequality
implies llz - yl12 I E. S’mce E > 0 is arbitrary,
we have z = y.
Remark 3.4. In view of theorem 3.3 it is natural to ask the following
H
questions.
Problem 1. If E is a uniformly of qx in F(S)?
convex Banach space, x E C and y E Q(x) fl F(S), is y always
Problem 2. If E is a uniformly point for each x E C?
convex Banach space, does Q(x) fl F(S) contain at most one
the minimizer
Clearly, by theorem 3.2, an affirmative answer for problem 1 gives an affirmative answer to problem 2. We now proceed to give an affirmative answer for problem 2 when E has a Frechet differentiable norm.
Nonlinear
ergodic
1417
theorems
LEMMA 3.5. Let C be a nonempty closed convex subset of a Banach space E. Let x E C and f E F(S). Then inf)lT,x -fll = infsupl(T,x -fll. 5 s f Proof.
Let r = inf,l/T,x
- j-11 and E > 0. Then there exists a E S such that /IT,x - f/l < r + c.
So, for each t E S, we have
IILJ -fll 5 Ilm -A < 7 + 1 and, hence, inf supl)7;,x - fll 5 supl/7;,x -fll s
Since E > 0 is arbitrary,
t
I r + c.
f
we have inf sup/I 7;,x - f 11I r. s I
It is clear that inf, supI II 7;,x - fll L r. So we have inf sup(j?;,x - fll = r. s
I
n
LEMMA 3.6. Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Let x E C, f E F(S) and 0 < (Y I /3 < 1, Then for any E > 0, there exists a closed left ideal L of S such that llT,(l,T,x + (1 - A)f) - (AT,T,x + (1 - 1)f)ll < &
for every s E S, t E L and a! % A I fi. Proof.
Let r = inf, 1)T,x - f 11.By lemma 3.5, for any d > 0, there exists to E S such that sup11Trox - f I/ I r + d. I
Apply now lemma 1 in [2] and let L = St,.
n
Let E be a Banach space and let S be a semigroup. Let lx,: cxE S] be a subset of E and x,y E E. Then we write x, -+ x (a + wR) if for any E > 0, there exists cq, E S such that Ilx,,o - XII < E for every 01E S (see [12]). We also denote by [x,y] the set (Ax + (1 - A)y:O c: A 5 1). LEMMA 3.7. Let C be a nonempty closed convex subset of a Banach space E with a Frechet differentiable norm and let S be a semigroup. Let lx,: CYE S] be a bounded subset of C. Let z E n, co[xac,: 01 E S), y E C and {y,: cyE S] be a subset of C with y, E [y, x,] and
II-Y, - zI/ = minlllu If Y, + ~(CY+ mR), theny = z.
- 211:u E [y, x,1).
141x
Proof.
A. T. M.
LAU
et al.
Since the duality mapping J of E is single-valued,
for each CYE S, it follows from [13]
that (2.4- Y,,J(Y, for every u E [y, x,]. Putting
- z)> 2 0
u = x, , we have (x, - y,, J(y, - 4) L 0
for every CYE S. Since (x,: (Y E S) is bounded, there exists K > 0 such that (Ix, - y(l I K and IIY, - ~115 K f or every CYE S. Let E > 0 and choose 6 > 0 so small that 26K < E. Then since the norm of E is Frechet differentiable, there exists &, > 0 such that 6, < 6 and Mu)
- 4Y - z)l/ < fJ
for every u E E with 11~- (y - z)\\ < 6, _ Since y, -+ y(a: + coR), there exists 01~E S such that IlY CYCI” - Yll < 4 for every CYE S. So, for each QI E S, we have k,,
- Yaoo7 J(Yaao - z)> - (Xolcu”- Y> J(Y - z)>/
5 l(xuuo-
Y,,(), J(Y,,()
- 2)) - (X,,” -
YTJ(Y,ao
- t)>l
+ Ik” - Y9 J(Ycq - 4) - CL0 - Y, J(Y - z)>l = I(Y - Y,,,, J(Y,,o - z)>l + Ikx,,o - YYJ(Y,,” - z) - J(Y - z)>l 5 IlY - Yan”llllYcvq - zll + II&,”
- YllllJ(Ycmo - z) - J(Y - z)II
and, hence, xRYo - y, J(y - 2)) > xxmao - y,,,, maa
- z)> - E 2 -8.
From z E CO(X,,~ : a E SJ, we have (z - y,
J(Y
- z)> 2 -E,
that is IIY - z/I2 5 E. Since E > 0 is arbitrary,
we have y = z.
W
3.8. Let C be a nonempty closed convex subset of a uniformly convex Banach space E with a Frechet differentiable norm. Let x E C. Assume that F(S) # 0. Then for y E F(S) and Y 6 Q(x), k = inf(jT,x - y\J > 0. s LEMMA
Proof.
Supposing that k = 0, by lemma 3.5, inf sup/( 7& - y(I = k = 0. 5 f
Nonlinear ergodic theorems
1419
Let z E Q(x). For each t E S, let y, be the unique element in [y, 7;x] such that
llyt - z/l = min(llu - z/I: u E Iv, TXII. So, for any E > 0, there exists s, E S such that
SUPII I 7;sp - YII < ; and, hence, we have Il~rs, - Yll 5 !YB,, - Tts,,xll + I17;sox -Y/l
5 2117&X - YII < E
for every t E S, that is, yt --f y(t + 03~). So by lemma contradiction. So we have k > 0. W
3.7, we have y = z,. This is a
LEMMA 3.9. Let C be a nonempty
closed convex subset of a uniformly convex Banach space E with a Frtchet differentiable norm. Let x E C. Then for any y E F(S) and z E Q(x), there exists a closed left ideal L of S such that (T,x - y, J(y - z)> I 0 for every t E L. Proof. If x = y ory = z, lemma 3.9 is obvious. So, let x # y and y # z. For any t E S, define a unique element y, such that yt E [y, 7;x] and
lly, - zll = minlllu
- zll: 2.4E [y, TxlI.
Then since y # z, by lemma 3.7 we have yI ft y(t --) ~0~). So we obtain c > 0 such that for any t E S, there exists t’ E S with I]ytnt - yI[ 1 c. Setting Y,, = QGw
+ (1 - %lY,
Q,,rE [O, 11,
we also obtain c,, > 0 so small that Q,,~L co. In fact, since & is nonexpansive and y E F(S), we have
c 5 IIY,,,- YII = at, IIT,x - YII 5 a,,r/Ix - YII* So, put c,, = c//Ix - yll. Let k = inf, /]T,x - yl]. By lemma 3.5 and y, ++ y(t -+ mR), we have k > 0.
Now we choose
E
> 0 so small that
(R+E)(l -d($g)
IIT,w&
+ (1 -
ColY) -
(msT,ox + (1 - CdY)II < E
(*)
1420
A. T. M.
LAU
et al.
for every s, t E S. Fix t, E S with ljyr,tO - yjj L c. Then, since a,, t0 1 co, we have
coT,tox + (1 -
ColY =
=
(
(
1- 2
I 0>
1- 2
1 II>
Y + $
10
(a,,,,7;1,“x
Y + EY,‘,” I 0
+ (1 - a,,,,)y)
E [Y,Yt,r,l
and, hence, Ilco~,tox + (1 - co)r - zll 5 max(lly
- zll, IIY,,~, - zlll 5 IIY - zll = R.
By using (*), we obtain
IICoTSqt,X+ (1 -
ColY -
z/I < lIT,(co~;,,ox+ (1 5 llcoqr,X + (1 -
ColY) -
ColY -
ZII + &
zll + E
~R+E
for every s E S. On the other hand, since Ily - zJI = R < R + E and
IlcJ&p
+ (1 -
for every s E S, we have, by uniform
ColY -
YII = CollL,r”X - Yll 2 cok
convexity,
x + (1 - cob - z) + (y - z))
(is))-
that is
for every s E S. Putting
we have
IIU, + ol(y - u,) - ZII =
Il4Y
- z) - (a - l)(u, - z)ll
2 ally - z/I - (o( - l)llu, - ZII 2 OlllY- zll - (o( - l)llY - zll = IIY - zll for every s E S and CYL 1. So, by theorem 2.5 in [13], we have (u, + My - 24,) - y,
J(Y
- z)> 2 0
for every s E S and (Y L 1 and, hence, (u, - y, J(Y - z)> 5 0
1421
Nonlinear ergodic theorems
for every s E S. Therefore,
we obtain
- ?y, (T,T,t,x - Y,J(Y - z)> =: s T,T,,,,x
J(y - z)
i
>
= i (u, - y, J(Y - z)> 5 0 for every s E S. Let L = St, t,. LEMMA
n
3.10. Let C be a nonempty
closed convex subset of a uniformly
convex Banach space
E. Let x E C. If for any y, z E Q(x) n F(S),
inf inf SUP(T,X LEC(S) #EEJ(y-z) IEL
- Y,$)
5 0,
then Q(x) n F(S) has at most one point. Proof. Let y, z E Q(x) fl F(S). Then by convexity of Q(x) II F(S), we have (y + z)/2 E Q(x) II F(S). Let E > 0. By assumption, there exist L E G(S) and 4 E J((y + z)/2 - 2) such that
Tx-y+z 2 (f
’
4 SE >
for every t E L. Since y E cO(Tx: t E Lj, it follows that
and, hence, &Y - z, 4) = $lly - z/l2 5 E. Since E > 0 is arbitrary, Combining THEOREM
we have y = z. H
lemmas 3.9 and 3.10, we have the following
result.
3.11. Let C be a nonempty closed convex subset of a uniformly convex Banach space norm. Let x E C. Then Q(x) f’ F(S) contains at most one point.
E with a Frechet differentiable
4.ERGODICTHEOREMS
We are now ready to prove our main nonlinear
ergodic theorems.
4.1. Let C be a nonempty closed convex subset of a uniformly convex Banach space norm. Let S = 1q : s E S 1 be a continuous representation of a semitopological semigroup S as nonexpansive mappings from C into C. Assume that F(S) # 0. Then the following are equivalent: (1) for each x E C, the set Q(x) fl F(S) is nonempty; (2) there exists a retraction P of C onto F(S) such that P5T;= 7;P = P for every t E S and Px E a( 7;x: t E SJ for every x E C. THEOREM
E with a Frechet differentiable
1422
A. T. M. LAU
et al
Proof. (1) * (2). If for each x E C, the set Q(x) tl F(S) # 0, then by theorem 3.11, Q(x) fl F(S) contains exactly one point Px. Then, clearly, P is a retraction of C onto F(S) and Px E co1T,x: t E S ] for every x E C. Clearly, 7; P = P for every t E S. Also, if u E S and x E C, we have
and, hence,
Q(x) n W9 = QGW n F(S). This implies P7; = P for every t E S. (2) * (1). Let x E C. Then it is obvious that Px E F(S). Since Px = PT,x E co{7j;T,x: t E As] = co(7;,: t E S)
for every s E S, we have PXE
n~5(T,,x:t~S] sss
= Q(x).
n
4.2. Let C be a nonempty closed convex subset of a Hilbert space H and let S = (T, : s E S) be a continuous representation of a semitopological semigroup S as nonexpansive mappings from C into C. If for each x E C, the set Q(x) fl F(S) is nonempty, then there exists a nonexpansive retraction P of C onto F(S) such that PT, = T,P = P for every t E S and Px E Cii( T,x: t E S) for every x E C.
THEOREM
Proof. For each x E C, let Px be the unique element in Q(x) f’ F(S). Then, as in the proof of theorem 4.1 (1) * (2), P is a retraction of C onto F(S) such that PT, = T,P = P for every t E S and Px E Co{ T,x: t E Sj for every x E C. It remains to show that P is nonexpansive. Let y E C and 0 < A < 1. Then as in the proof of theorem 3.3 we have for any E > 0,
4x((l - J)Px + APY) = supinf)l7;,x s
- ((1 - A)Px + APy)l12
I
= sup inf(l &x - Px + A(Px - Py)l)* s
f
= supinf(l/T,,x s
- Px112 + 2A(7;,x - Px, Px - Py> + A2)lPx - Pyl12)
t
< q,(Px)
+ 2A supinf(T,,x s
Since Px is the minimizer
I
- Px, Px - Py) + A*IlPx - Pull2 + .z.
of qX, we have
2A sup inf( csx - Px, Px - Py) + A2(lPx - Pull2 + E s
f
> qw((l - A)Px + APy) - q,(Px)
L 0.
Nonlinear ergodic theorems
Since E > 0 is arbitrary,
1423
we have
21, sup inf( Ksx - Px, Px - Py) + A’llPx - Pyj12 L 0 s I and, hence, 2 sup inf(7;,x s I
- Px, Px - Py) 2 -IIIPx
- Pyl12.
Now, if A + 0, then sup inf( 7;,x - Px, Px - Py) L 0. s I Let E > 0. Then there exists u E S such that (T,,x - Px, Px - Py> > -& for every t E S. For such an element u E S, we also have sup inf(7;,T, y - PT, y, PT, y - Px) L 0 5 I and, hence, there exists v E S such that (T,,,Y
- PT,Y,PT,Y
- Px) > --E
for every t E S. Then, from PT, y = Py, we have (T,,,Y
for every t E S. Therefore,
- PY,PY - Px) > --E
we have
-2~ < (T,,,x = (T,“&
- Px, Px - Py) + (T,,, y - Py, Py - Px> - T,,, y - (Px - PY), px - PY)
= (T,“,X 5 IITuvux-
T,,,Y,PX T,,,rllllPx
-
- llpx - PYl12 - Prll - IlJ’x - Pyli* PY)
5 I/x - YIIIIPX - PYII - llpx - PYl12. Since E > 0 is arbitrary,
this implies IIPx - Pyll 5 IIx - yll.
We now proceed to find conditions
n
on S and E such that Q(x) f’ F(S) # @ for every x E C.
LEMMA 4.3 [14]. Let C be a nonempty closed convex subset of a Hilbert space H, let S be an index set, and let lx, : t E S 1 be a bounded set of H. Let X be a subspace of I”(S) containing
constants, and let p be a submean on X. Suppose that for each x E C, the real-valued function f on S defined by “f(t) = II4 - XII2 for all t E S belongs to X. If ex) = iut/Ix, - XII2 for all x E C
1424
A. T. M.
LAU
et al.
and r = inf(r(x): x E C), then there exists a unique element z E C such that r(z) = r. Further, the following inequality holds for every x E C.
r + /Iz - x(1* 5 r(x)
4.4. Let C be a nonempty closed convex subset of a Hilbert space H and let S be a semitopological semigroup such that RUC(S) has an invariant submean. Let S = (T,: s E S] be a continuous representation of S as nonexpansive mappings from C into C. Suppose that (T,x: s E S] is bounded for some x E C. Then the set Q(x) fI F(S) is nonempty.
THEOREM
Proof. First we observe that for any y E H, the functionf(t) = 11T,x - ~(1’ is in RUC(S) (see [ 151). Let ,D be an invariant submean and define a real-valued function g on H by
g(Y) = PrlITX
- Yl12
for each y E H.
If r = inf(g(yj: y E H), then by lemma 4.3 there exists a unique element z E H such that g(z) = r. Further, we know that
r + llz - ~11’5 &) For each s E S, let Q, be the metric projection Q, is nonexpansive and for each t E S,
for every y E H. of H onto Co] T,x: t E S). Then by Phelps [16],
II’T;,x- Q,zl/* = IiQJtsx - Qszll* 5 IIT,x - z/I’. So, we have
and, thus QSz = z. This implies that z E co(L&,x: t E 6s)
for all s E S
and, hence, z E n CO~T,,~: t E SI. se.5 On the other hand, by lemma 4.3,
llz - Al2 5 rUtIIT,x - Yl12- fir IITX - zl12
for every y E H.
So, putting y = cz for each s E S, we have
II2 - T,zl125 rUtIIT,x - Tszl12 - rUrIITX - zI12 = )utllT,,x- T,zl12- rutIIm - zI12 5 pt IIT,x - zl12- P,IITX - zl12= 0. Therefore,
we have Tsz = z for every s E S.
n
Nonlinear
ergodic
theorems
1425
PROPOSITION 4.5. Let S be a discrete reversible semigroup and let C be a nonempty weakly compact convex subset of a Banach space E. Let S = 1T, : s E S 1 be a representation of S as affine nonexpansive mappings from C into C. Then for each x E C, the set Q(x) II F(S) is nonempty. Proof. Let x E C. Clearly, Q(x) = nscS Co1T,,x: t E S) is weakly compact and convex. Also Q(x) is nonempty. Indeed, for each s E S, let K, = co(T,,x: t E SJ. Then (K,: s E S] are weakly closed subsets of C with finite intersection property: if sr , . . . , s, E S, choose t, E fly= r Ss,. Then TOx E f-J;=, KS,. Consequently n, Es KS = Q(x) is nonempty. Let a, s E S, y E Q(x) and let lyk] be a sequence in KS such that l/y, - y 11+ 0. Then T, yk E KS (by affiness and continuity of T,) and liT,yk - CYII 5 IIY, - ~11-, 0. I-I ence T, y E K,. Consequently, T, y E n, E s KS = Q(x). Hence, Q(x) is S-invariant. Now, since S is left reversible, by [ 171 the space WAP(S) of weakly almost periodic functions on S has a left invariant mean (see also [18]). Hence by [19], there exists a common fixed point in Q(x), i.e. Q(x) fl F(S) # 0. n 5,MINIMALLEFTIDEALS
Let (Z, 0) be a compact right topological semigroup, i.e. a smigroup and a compact Hausdorff topological space such that for each T E C the mapping y + y 0 r from Z into C is continuous. In this case, ZEmust contain minimal left ideals. Any minimal left ideal in C is closed and any two minimal left ideals of C are homeomorphic and algebraically isomorphic (see [20]). LEMMA 5.1. Let X be a nonempty weakly compact convex subset of a Banach space E. Let S = 1q : s E S) be a representation of a semigroup S as nonexpansive and weak-weak continuous mappings from X into X. Let C be the closure of S in the product space (X, weakly. Then E is a compact right topological semigroup consisting of nonexpansive mappings from X into X. Further, for any T E C, there exists a sequence 1T,) of convex combination of operators from S such that )IT,x - Txll -+ 0 for every x E X. Proof. It is easy to see that Z is a compact right topological semigroup. We now prove the last statement (which implies that each T E C is nonexpansive). Consider S G (E, 1).II>” with the product topology. Let @ = co S. Then each T E CDis nonexpansive. Hence each T E & is also nonexpansive. Since the weak topology of the locally convex space (E, 11.I]>” is the product space (E, weaky, it follows that C E Tweak = 6, and hence the last statement holds. n
C is called the enveloping semigroup of S. A subset X of a Banach space E is said to have normal structure if for any bounded (closed) convex subset W of X which contains more than one point, there exists x E W such that sup(]lx - yII: y E W) < diam( W), where diam( W) = sup1 IIx - yll: x, y E W) (see (211 for more details). THEOREM 5.2. Let X be a nonempty weakly compact convex subset of a Banach space E and X has normal structure. Let S = 1T,: s E S) be a representation of a semigroup as normnonexpansive and weakly continuous mappings from X into X and let C be the enveloping of S. Let I be a minimal left ideal of Z and let Y be a minimal S-invariant closed convex subset of X. Then there exists a nonempty weakly closed subset C of Y such that 1 is constant on C.
1426
A. T. M. LAU
et al.
Proof. Since I is a minimal left ideal of C and Z is a compact right topological semigroup (lemma 5.1), I = Ce for a minimal indempotent e of C and G = eCe is a maximal subgroup contained in I (see [20]). Since each T E G is a nonexpansive mapping from Y into Y (lemma 5.1), by Broskii-Milman theorem [22], there exists x E Y such that TX = x for every T E G. Now put C = Ix. Then C is weakly closed and S-invariant. Also, if y, , y2 E C, yl = T, ex, y, = T,ex, T, , T2 E C, then, since eT,e E G, we have (Te)y, = Te(T,ex) = TX for every T E Z and similarly (Te)y, = TX for every T E Z. The assertion is proved.
n
The following improves the main theorem in [23] for Banach spaces (see also [24]). 5.3. Let C and X be as in theorem, 5.2. Then there exist TO E E and x E X such that T,Tx = T,x for every T E C.
COROLLARY
Proof. Pick x E C and TO E I of the above theorem.
n
Remark 5.4. If S is commutative, then for any T E C and s E S, T, 0 T = T 0 T,, i.e. z = T,x is in fact a common fixed point for Z (and, hence, for S). Note that if X is norm compact, the weak and norm topology agree on X. Hence every nonexpansive mapping from X into X must be weakly continuous. Therfore, corollary 5.3 improves the well known fixed point theorem of De Marr [6] for commuting semigroups of nonexpansive mappings on compact convex sets. REFERENCES 1. BROWDER
F. E., Nonlinear
operators
and nonlinear
equations
of evolution
in Banach
spaces,
Proc. Sympos.
Pure Math. 18(2) (1976). 2. LAU A. T. & TAKAHASHI W., Weak convergence and non-linear ergodic theorems for reversible semigroups of nonexpansive mappings, Pacific J. Math. 126, 277-294 (1987). 3. BAILLON J. B., Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert, C. r. Acad. Sci. Paris St+. A-B 280, 1511-1514 (1975). 4. TAKAHASHI W., A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Am. math. Sot. 81, 253-256 (1981). 5. HIRANO N. & TAKAHASHI W., Nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Banach space, Pacific J. Math 112, 333-346 (1984). 6. MARR R. De, Common fixed points for commuting contraction mappings, Pacific J. Math. 13, 1139-1141 (1963). 7. DIESTEL J., Geometry of Banach spaces, Selected Topics, Lecture Notes in Mathematics 485. Springer, Berlin (1975). 8. HEWITT E. & ROSS K., Abstract Harmonic Analysis I. Springer, Berlin (1963). 9. LAU A. T. & TAKAHASHI W., Invariant means and fixed point properties for nonexpansive representations of topological semigroups, Top. Meth. Nonlinear Analysis (to appear). 10. BARBU V. & PRECUPANU Th., Convexity and Optimization in Banach Spaces. Editura Academiei R. S. R., Bucuresti (1978). 11. XU H. K., Inequalities in Banach spaces with applications, Nonlinear Analysis 16, 1127-l 138 (1991). 12. RODE G., An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert Space, J. math. Analysis
Applic. 85, 13. DEUTSCH
SIAMRev.
172-178
(1982).
F. R. & MASERICK
9,
516-530(1967).
P. H.,
Applications
of the Hahn-Banach
theorem
in approximation
theory,
Nonlinear 14. MIZOGUCHI semigroups 15. LAU A. (1985). 16. PHELPS 17. HSU R., 18. LAU A.
19. 20. 21. 22. 23. 24.
ergodic
1427
theorems
N. & TAKAHASHI W., On the existence of fixed points in Hilbert spaces, Nonlinear Analysis 14, 69-80 (1990). T., Semigroup of nonexpansive mappings on a Hilbert space,
and ergodic
retractions
for Lipschitzian
J. math. Analysis Applic. 105, 514-522
R. R., Convex sets and nearest points, Proc. Am. math. Sot. 8, 790-797 (1957). Topics on weakly almost periodic functions. Ph. D. Thesis, SUNY, Buffalo, 1985. T.. Amenability and fixed point property for semigroup of nonexpansive mapppings, in Fixed Point Theory and Applications (Edited by J. B. BAILLON and M. A. THERA). Longman Scientific and Technology, Essex (1991). LAU A. T., Some fixed point theormes and their applications to W*-algebras, in Fixed Point Theory and its Applications (Edited by S. SWAMIRATHAN), pp.l21-129. Academic Press, New York (1976). BERGLUND J. F., JUNGHENN H. Q. & MILNES P. Ana/ysis on Semigroups. John Wiley and Sons, New York (1989). GOEBEL K. & KIRK W. A., Topics in metric fixed point theory, Camb. Stud. Adv. Math. 28 (1990). BROSKII M. S. & MILMAN D. P., On the center of a convex set, Dokl. Akad. Nauk SSSR 59, 837-840. HOLMES R. D. & LAU A. T., Almost fixed points of semigroups of nonexpansive mappings, Studia Math. XLlll, 217-218 (1972). PETTIS B. J., On Nikaido’s proof of the invariant mean-valued theorem, Studia Math. 33, 193-196 (1969).
Analym,
Theory,
Methods
& Applrcofrons,
Pergamon
Vol. 26, No. 8, pp. 1411-1427. 1996 Copyright b 1996 Else&r Scicncc Ltd Printed in Great Britain. All rights reserved 0362-546X/% $15.@0+0.00
0362-546X(94)00347-5
NONLINEAR ERGODIC NONEXPANSIVE ANTHONY
THEOREMS MAPPINGS
T. M. LAU,t$
KOJI NISHIURAO
iDepartment of Mathematical Sciences, University §Department of Information Sciences, Tokyo Institute (Received Key words andphrases: minimizer, uniformly
3 1 May
FOR SEMIGROUPS AND LEFT IDEALS
1994; received
and WATARU
OF
TAKAHASHIS
of Alberta, Edmonton, Alberta T6G-2G1, of Technology, Oh-okayama, Meguro-ku, for publication
Nonlinear ergodic theorems, nonexpansive convex space, fixed point theorems.
17 November mappings,
Canada; and Tokyo 152, Japan
1994) semigroup,
left ideal orbit,
1. INTRODUCTION
Let S be a semitopological semigroup, i.e. S is a semigroup with a Hausdorff topology such that for each s E S the mappings s + a . s and s + s . a from S to S are continuous. Let E be a uniformly convex Banach space and let S = 1T, : s E S 1 be a continuous representation of S as nonexpansive mappings on a closed convex subset C of E into C, i.e. T,,x = T,T,x for every a, b E S and x E C and the mapping (s, x) -+ T,(x) from S x C into C is continuous when S x C has the product topology. Let F(S) denote the set (x E C: T,x = x for all s E S] of common fixed points of S in C. Then as well known, F(S) (possibly empty) is a closed convex subset of C (see [l]). In this paper, we shall study the distance between left ideal orbits and elements in the fixed point set F(S). We shall prove (theorem 3.11) among other things that if E has a Frechet differentiable norm, then for any semitopological semigroup S and x E C, the set Q(x) = ncO(T,x: t E L), with the intersection taking over all closed left ideals L of S, contains at most one common fixed point of S (where Co A denotes the closed convex hull of A). This result is then applied to show (theorem 4.1) that if F(S) fl Q(x) # 0 for any x E C, then there exists a retraction P from C onto F(S) such that T,P = PT, = P for every t E S and P(x) E Co{ T,x: t E S ] for every x E C. Both theorems 3.11 and 4.1 were established by Lau and Takahashi in [2] when S has finite intersection property for closed left ideals. The first nonlinear ergodic theorem for nonexpansive mappings was established in 1975 by Baillon [3]: let C be a closed convex subset of a Hilbert space and let T be a nonexpansive mapping of C into itself. If the set F(T) of fixed points of T is nonempty, then for each x E C, the Cesaro means
converge weakly to some y E F(T). In this case, putting y = Px for each x E C, P is a nonexpansive retraction of C onto F(T) such that PT = TP = P and Px E co{ T’x: n = 1,2,. . .) for each x E C. In [4], Takahashi proved the existence of such a retraction for an amenable semigroup. This result is further extended to certain Banach spaces by Hirano and Takahashi in [5]. t This research
is supported
by NSERC-grant
A7679. 1411
A. T. M.
1412
LAU
ef 01
Our paper is organized as follows: in Section 2 we define some terminologies that we use; in Section 3 we study the distance between ideals determined by left orbits and the fixed point set; in Section 4 we apply our results in Sectian 3 to establish our main nonlinear ergodic theorems; finally in Section 5 we study an almost fixed point property determined by the minimal left ideals in the enveloping semigroup of a semigroup of nonexpansive mappings on a weak compact convex set and obtain a generalization of De Marr’s fixed point theorem [6]. 2. PRELIMINARIES
Throughout this paper, we assume that a Banach (ar Hilbert) space is real. Let E be a Banach space and let E * be its dual. Then, the value off E E * at x E E will be denoted by (x, f > or f(x). The dualiry mapping f of E is a muiltivalued operator I: E -+ E*, where f(x) = IKE E*: (x,f) = (lxl12 = j-(1’] (w h ic h IS nonempty by simple application of the Hahn-Banach theorem). Let B = (X E E: jjxl\ = 11 be the unit sphere of E. Then the norm of E is said to be Frtchet differentiable if for each x E B, the limit lim ” x + x-0
AYll
- II4
A
is attained uniformly for y E B. In this case, J is a single-valued and norm ID norm continuous mapping from E into E* (see [l, 71 for more details). Let S be a nonempty set and Iet X be a subspace of i”(S) (bounded real-valued functions on S) containing constants. By a submean on X we shall mean a real-valued function p on X satisfying the following properties: (1) Af + g) 5 ii(f) + kG3 for every f, g E X; (2) ~(osf) = arp(f) for every f E X and 1y 2 0; (3) forf,g E X,/l g implies p(f) 5 Ag); (4) flu(c) = c for every constant function c. A semitopological semigroup S is called left reversible (resp. right reversible) if S has finite intersection property for “closed” right (resp. left) ideals. S is called reversible if S is both left and right reversible. Let S be a semjropological semigroup and let C(S) denote the closed subalgebra of I”(S) consisting of bounded continuous functions. For each j E C(S) and a E S, let &f)(t) = /(of) and (~=f)(t) = f(ra)- Let RUC(S) denote allf E C(S) such that the mapping S + C(S) defined by s + rs f is continuous when C(S) has the norm topology. Then RUC(S) is a translation invariant subalgebra of C(S) containing constants. Further, RUC(S) is precisely the space of bounded left uniformly continuous functions on S when S is a group (see [S]). A submean ~1on RffC(S) is called invariant if fr(f, f > = p(r, j) = &ff for every f E RVC(S) and II E S. If S is a discrete semigroup, then RUC(S) has an invariant submean if and only if S is reversible. Also, if S is normal and C(S) has an invariant submean, then S is reversible. However, S need not be reversible when C(S) has an invariant submean in general (see [9j for details). 3. LEFT
IDEAL
ORBITS
AND
THE
FIXED
Unless otherwise specified, S denotes a semitopological continuous representation of S as nonexpansive mappmgs subset C of a Banach space E into C.
POINT
SET
semigroup and S = IT,: s E SI a from a nonempty closed convex
Nonlinear
x
ergodic
theorems
1413
Let E(S) denote the collection of closed left ideals in S. Assume that F(S) # 0. For each E C and L E C(S), define the real-valued function qx,L on F(S) by h,L(f)
= inf(IITx
- fl12: t E L1
and let 4x(f) = suPlqX,,(f):
L E W91.
Then
4Af) = sup - fl12 s infllT,x t as readily checked. LEMMA 3.1. Let C be a nonempty closed convex subset of a Banach space E. If F(S) # 0, then for each x E C, qx is a continuous real-valued function on F(S) such that 0 I q*(f) I I/x - fl12 for each f E F(S) and qx(fn) + cc)if /If,]/ -+ 00. Further, if F(S) is convex, then qwis a convex function on F(S). Proof. Since 0 I I/TX - fl12 = II T,X - ~,fll’ I IIx - fl12 for every f E F(S) and t E S, it follows readily that 0 I qx(f) I IIx - ~11~. Also, if f E F(S) and t E S, then IIT,x - f II I
Ilx - f II. Hence,II~41 5 IITX - f II + Ilf II 5 Ilx - f II + Ilf II, i.e. M = SUP{II~41: t E SI < ~0.
Let If,) be a sequence in F(S) such that 11f,lj + 03. Then we have for each t E S
IITX - fnl121 (IITX11- llfnll)2 = llfnl12- m~llllfnll + IIWII” 2 Ilfnll’ - 2MlfnII =
llfX(l
- G
>
and, hence, for each L E C(S),
4x,L(fn) 2 lW(1 So we have qx(fn) + 00. To see that qx is continuous,
- 6)
+ co.
let 1f,] be a sequence in F(S) converging to some f E F(S) and
M’ = sup((l7;x - f,J + 117;~ - fll: n = 1,2,...
and t E S).
Then, since
I/TX -fnl12 - IITX -f II25 (IITX -frill + IITX -f ll)lllTx -fall - Ilm -f III 5 M’llfn -f II, we have for each L E .52(S)
qx,r.(fn) 5 4X&(f) + M’lk -f II. Similarly,
we have
4&L(f) 5 qx,L(fn) + M’llfn -fll. So we obtain
Mfn) - qx(f)I 5 M’llfn - f II. This implies that qx is continuous
on F(S).
1414
A. T. M.
LAU
et al.
If F(S) is convex, for each f, g E F(S) and (Y,p L 0 with 01 + /? = 1, af + pg E F(S). Let E > 0. Then there exists L, E g(S) such that sup
inf(cYII T,x - flj2 + PII T,x - gjj2) < ,‘,“,f (DII T,X -
LEoC(S) fCL
0
fl12 + AlT,X - gl12)+ i.
Let u E L,. Then Su s L, and, hence, sup
inf(allT,x
LEC(S)fEL Moreover,
- fl12 + /3II7;x - gll’) 5 inf(oll/7;,x tes
- fll’
+
PIIT& - A21 + i.
there exist u, w E S such that
IIT& - fl12 < fp&
- fl12 + ;
and
IITwoux- fl12 < yT”Ux
- fl12 + ; .
Therefore, we obtain q,W
+ Pg)
=
SUP infllTx - bf + &II2 LLqS) teL
5 Ly, < fpTux
fy
TX - Al2 + PllTX - gl12) - Al2 + PIITUX- sl12)+ ;
5 4Tvvux - fl12 + PIITVVUX - gl12+ ; 5 4Tvux - Al2 + PIIGvux - gl12+ ;
=
ut~~II~x-~ll~+Pt~~lI~x-~l12+~ 1 2 (where L, = Su and L, = SW)
Since E > 0 is arbitrary,
we have
Nonlinear ergodic theorems THEOREM 3.2. Let C be a nonempty
1415
closed convex subset of a uniformly
convex Banach space
E. Assume that F(S) # 0. Then for any x E C, there exists a unique element h E F(S) such that
Proof. Since E is uniformly convex, the fixed point set F(S) in C is closed and convex (see [l]). Hence, it follows from lemma 3.1 and [lo] that there exists h E F(S) such that 4A4
= WqAf):f
E W)l.
To see that h is unique, let k E F(S). Then by [l 11, there exists a strictly increasing and convex function (depending on h and k) g: [0, co) + [0, 00) such that g(0) = 0 and IIT,x - (Ah + (1 - A)k)112 = IIA(T,x - h) + (1 - A)(T,x - k)/i2 I 1117;~ - h(l’ + (1 - A)IIT,x - k(l’ - A(1 - A)g(llh - kli) for each t E S and A with 0 5 ,J I 1. So we have for each i with 0 I A I 1, MO
5 Mh
+ (1 - AN) 5 &dh)
+ (1 - ~)q.r(k) - A(1 - Ak(lth - kill
and, hence a(h)
5 a(k)
- Q(llh
- kll).
It follows that qx@) 5 qx(k) - s(llh - kli)
as A -+ 1.
Since g is strictly increasing, it follows that if qx(h) = q,(k), then h = k. We call the unique element h E F(S) in theorem 3.2 the minimizer For each x E C, let Q(x) =
n CO(7;x:tEL) L EC(S)
= n a(&X: se.s
a
of qx in F(S).
t Es]
>
.
3.3. Let C be a nonempty closed convex subset of a Hilbert space H. Let S = (T, : s E S] be a continuous representation of S as nonexpansive mappings from C into C. Then for any x E C, any element in Q(x) n F(S) is the unique minimizer of qx in F(S). In particular, Q(x) fl F(S) contains at most one point. THEOREM
Proof. Let z E F(S) be the minimizer there exists u E S such that
sup inf(ll7;,x
s I
of qx in F(S) and y E Q(x) fl F(S). Then for some E > 0,
- zlIz + 2(&,x
- z, z - u> + II2 - vl12)
< inf(II TUx - alI2 + 2(T,,x - z, z - y> + II2 - ul12) + i .
t
Moreover,
there exist u, w E S such that II T,,x - zl12 < infll TUx - ~11~+ $ t
1416
A. T. M. LAU
et al.
and (Twvux - z, z - y> < inf(T,,,x
- z, .z -
t
Y>
+ i.
Therefore, we obtain 4Ay) = sup i:fll Lx
- YV
= slip inf(llT,,x s
- 211’ + 2(7;,x - z, z - Y> + llz - Yl12)
t
< inf(llT,,x t
- zl12 + 2(T,,x - z,z - Y> + Il.2 - YI12) + i
5 /Twuux - z112 + 2(T,,,x 5 IIT,,x - z/j2 + 2(T,,,x
< infllT,,x t
- z(12 + 2 sup inf(T,,x s
= qx(z) + 2 sup inf(7;,x s t
+ ,tZ - Yl12 + ;
- z, z - y> + llz - yll’ + i + i + i
t
I
Y>
- z,z - Y> + 112- ~1,~ + 5
- z112+ 2 inf(7;,,x
5 sup infll7;,x s
- z, z -
t
- z, z - y> + llz - yll’ + E
- z, z - y> + llz - yl12 + E.
This implies that 2 sup inf(T,,x s t
- z, z - y> > q*(y) - qX(z) - llz - yl12 - E 2 - Ilz - yl12 - &.
So, there exists a E S such that 2(7;,x
- z, z - y> > - llz - Yl12 - E
for every t E S. From y E cO( T,x: t E Sl, we have 20 - z, 2 - y> 2 -llz - Yl12 - E.
This inequality
implies llz - yl12 I E. S’mce E > 0 is arbitrary,
we have z = y.
Remark 3.4. In view of theorem 3.3 it is natural to ask the following
H
questions.
Problem 1. If E is a uniformly of qx in F(S)?
convex Banach space, x E C and y E Q(x) fl F(S), is y always
Problem 2. If E is a uniformly point for each x E C?
convex Banach space, does Q(x) fl F(S) contain at most one
the minimizer
Clearly, by theorem 3.2, an affirmative answer for problem 1 gives an affirmative answer to problem 2. We now proceed to give an affirmative answer for problem 2 when E has a Frechet differentiable norm.
Nonlinear
ergodic
1417
theorems
LEMMA 3.5. Let C be a nonempty closed convex subset of a Banach space E. Let x E C and f E F(S). Then inf)lT,x -fll = infsupl(T,x -fll. 5 s f Proof.
Let r = inf,l/T,x
- j-11 and E > 0. Then there exists a E S such that /IT,x - f/l < r + c.
So, for each t E S, we have
IILJ -fll 5 Ilm -A < 7 + 1 and, hence, inf supl)7;,x - fll 5 supl/7;,x -fll s
Since E > 0 is arbitrary,
t
I r + c.
f
we have inf sup/I 7;,x - f 11I r. s I
It is clear that inf, supI II 7;,x - fll L r. So we have inf sup(j?;,x - fll = r. s
I
n
LEMMA 3.6. Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Let x E C, f E F(S) and 0 < (Y I /3 < 1, Then for any E > 0, there exists a closed left ideal L of S such that llT,(l,T,x + (1 - A)f) - (AT,T,x + (1 - 1)f)ll < &
for every s E S, t E L and a! % A I fi. Proof.
Let r = inf, 1)T,x - f 11.By lemma 3.5, for any d > 0, there exists to E S such that sup11Trox - f I/ I r + d. I
Apply now lemma 1 in [2] and let L = St,.
n
Let E be a Banach space and let S be a semigroup. Let lx,: cxE S] be a subset of E and x,y E E. Then we write x, -+ x (a + wR) if for any E > 0, there exists cq, E S such that Ilx,,o - XII < E for every 01E S (see [12]). We also denote by [x,y] the set (Ax + (1 - A)y:O c: A 5 1). LEMMA 3.7. Let C be a nonempty closed convex subset of a Banach space E with a Frechet differentiable norm and let S be a semigroup. Let lx,: CYE S] be a bounded subset of C. Let z E n, co[xac,: 01 E S), y E C and {y,: cyE S] be a subset of C with y, E [y, x,] and
II-Y, - zI/ = minlllu If Y, + ~(CY+ mR), theny = z.
- 211:u E [y, x,1).
141x
Proof.
A. T. M.
LAU
et al.
Since the duality mapping J of E is single-valued,
for each CYE S, it follows from [13]
that (2.4- Y,,J(Y, for every u E [y, x,]. Putting
- z)> 2 0
u = x, , we have (x, - y,, J(y, - 4) L 0
for every CYE S. Since (x,: (Y E S) is bounded, there exists K > 0 such that (Ix, - y(l I K and IIY, - ~115 K f or every CYE S. Let E > 0 and choose 6 > 0 so small that 26K < E. Then since the norm of E is Frechet differentiable, there exists &, > 0 such that 6, < 6 and Mu)
- 4Y - z)l/ < fJ
for every u E E with 11~- (y - z)\\ < 6, _ Since y, -+ y(a: + coR), there exists 01~E S such that IlY CYCI” - Yll < 4 for every CYE S. So, for each QI E S, we have k,,
- Yaoo7 J(Yaao - z)> - (Xolcu”- Y> J(Y - z)>/
5 l(xuuo-
Y,,(), J(Y,,()
- 2)) - (X,,” -
YTJ(Y,ao
- t)>l
+ Ik” - Y9 J(Ycq - 4) - CL0 - Y, J(Y - z)>l = I(Y - Y,,,, J(Y,,o - z)>l + Ikx,,o - YYJ(Y,,” - z) - J(Y - z)>l 5 IlY - Yan”llllYcvq - zll + II&,”
- YllllJ(Ycmo - z) - J(Y - z)II
and, hence, xRYo - y, J(y - 2)) > xxmao - y,,,, maa
- z)> - E 2 -8.
From z E CO(X,,~ : a E SJ, we have (z - y,
J(Y
- z)> 2 -E,
that is IIY - z/I2 5 E. Since E > 0 is arbitrary,
we have y = z.
W
3.8. Let C be a nonempty closed convex subset of a uniformly convex Banach space E with a Frechet differentiable norm. Let x E C. Assume that F(S) # 0. Then for y E F(S) and Y 6 Q(x), k = inf(jT,x - y\J > 0. s LEMMA
Proof.
Supposing that k = 0, by lemma 3.5, inf sup/( 7& - y(I = k = 0. 5 f
Nonlinear ergodic theorems
1419
Let z E Q(x). For each t E S, let y, be the unique element in [y, 7;x] such that
llyt - z/l = min(llu - z/I: u E Iv, TXII. So, for any E > 0, there exists s, E S such that
SUPII I 7;sp - YII < ; and, hence, we have Il~rs, - Yll 5 !YB,, - Tts,,xll + I17;sox -Y/l
5 2117&X - YII < E
for every t E S, that is, yt --f y(t + 03~). So by lemma contradiction. So we have k > 0. W
3.7, we have y = z,. This is a
LEMMA 3.9. Let C be a nonempty
closed convex subset of a uniformly convex Banach space E with a Frtchet differentiable norm. Let x E C. Then for any y E F(S) and z E Q(x), there exists a closed left ideal L of S such that (T,x - y, J(y - z)> I 0 for every t E L. Proof. If x = y ory = z, lemma 3.9 is obvious. So, let x # y and y # z. For any t E S, define a unique element y, such that yt E [y, 7;x] and
lly, - zll = minlllu
- zll: 2.4E [y, TxlI.
Then since y # z, by lemma 3.7 we have yI ft y(t --) ~0~). So we obtain c > 0 such that for any t E S, there exists t’ E S with I]ytnt - yI[ 1 c. Setting Y,, = QGw
+ (1 - %lY,
Q,,rE [O, 11,
we also obtain c,, > 0 so small that Q,,~L co. In fact, since & is nonexpansive and y E F(S), we have
c 5 IIY,,,- YII = at, IIT,x - YII 5 a,,r/Ix - YII* So, put c,, = c//Ix - yll. Let k = inf, /]T,x - yl]. By lemma 3.5 and y, ++ y(t -+ mR), we have k > 0.
Now we choose
E
> 0 so small that
(R+E)(l -d($g)
IIT,w&
+ (1 -
ColY) -
(msT,ox + (1 - CdY)II < E
(*)
1420
A. T. M.
LAU
et al.
for every s, t E S. Fix t, E S with ljyr,tO - yjj L c. Then, since a,, t0 1 co, we have
coT,tox + (1 -
ColY =
=
(
(
1- 2
I 0>
1- 2
1 II>
Y + $
10
(a,,,,7;1,“x
Y + EY,‘,” I 0
+ (1 - a,,,,)y)
E [Y,Yt,r,l
and, hence, Ilco~,tox + (1 - co)r - zll 5 max(lly
- zll, IIY,,~, - zlll 5 IIY - zll = R.
By using (*), we obtain
IICoTSqt,X+ (1 -
ColY -
z/I < lIT,(co~;,,ox+ (1 5 llcoqr,X + (1 -
ColY) -
ColY -
ZII + &
zll + E
~R+E
for every s E S. On the other hand, since Ily - zJI = R < R + E and
IlcJ&p
+ (1 -
for every s E S, we have, by uniform
ColY -
YII = CollL,r”X - Yll 2 cok
convexity,
x + (1 - cob - z) + (y - z))
(is))-
that is
for every s E S. Putting
we have
IIU, + ol(y - u,) - ZII =
Il4Y
- z) - (a - l)(u, - z)ll
2 ally - z/I - (o( - l)llu, - ZII 2 OlllY- zll - (o( - l)llY - zll = IIY - zll for every s E S and CYL 1. So, by theorem 2.5 in [13], we have (u, + My - 24,) - y,
J(Y
- z)> 2 0
for every s E S and (Y L 1 and, hence, (u, - y, J(Y - z)> 5 0
1421
Nonlinear ergodic theorems
for every s E S. Therefore,
we obtain
- ?y, (T,T,t,x - Y,J(Y - z)> =: s T,T,,,,x
J(y - z)
i
>
= i (u, - y, J(Y - z)> 5 0 for every s E S. Let L = St, t,. LEMMA
n
3.10. Let C be a nonempty
closed convex subset of a uniformly
convex Banach space
E. Let x E C. If for any y, z E Q(x) n F(S),
inf inf SUP(T,X LEC(S) #EEJ(y-z) IEL
- Y,$)
5 0,
then Q(x) n F(S) has at most one point. Proof. Let y, z E Q(x) fl F(S). Then by convexity of Q(x) II F(S), we have (y + z)/2 E Q(x) II F(S). Let E > 0. By assumption, there exist L E G(S) and 4 E J((y + z)/2 - 2) such that
Tx-y+z 2 (f
’
4 SE >
for every t E L. Since y E cO(Tx: t E Lj, it follows that
and, hence, &Y - z, 4) = $lly - z/l2 5 E. Since E > 0 is arbitrary, Combining THEOREM
we have y = z. H
lemmas 3.9 and 3.10, we have the following
result.
3.11. Let C be a nonempty closed convex subset of a uniformly convex Banach space norm. Let x E C. Then Q(x) f’ F(S) contains at most one point.
E with a Frechet differentiable
4.ERGODICTHEOREMS
We are now ready to prove our main nonlinear
ergodic theorems.
4.1. Let C be a nonempty closed convex subset of a uniformly convex Banach space norm. Let S = 1q : s E S 1 be a continuous representation of a semitopological semigroup S as nonexpansive mappings from C into C. Assume that F(S) # 0. Then the following are equivalent: (1) for each x E C, the set Q(x) fl F(S) is nonempty; (2) there exists a retraction P of C onto F(S) such that P5T;= 7;P = P for every t E S and Px E a( 7;x: t E SJ for every x E C. THEOREM
E with a Frechet differentiable
1422
A. T. M. LAU
et al
Proof. (1) * (2). If for each x E C, the set Q(x) tl F(S) # 0, then by theorem 3.11, Q(x) fl F(S) contains exactly one point Px. Then, clearly, P is a retraction of C onto F(S) and Px E co1T,x: t E S ] for every x E C. Clearly, 7; P = P for every t E S. Also, if u E S and x E C, we have
and, hence,
Q(x) n W9 = QGW n F(S). This implies P7; = P for every t E S. (2) * (1). Let x E C. Then it is obvious that Px E F(S). Since Px = PT,x E co{7j;T,x: t E As] = co(7;,: t E S)
for every s E S, we have PXE
n~5(T,,x:t~S] sss
= Q(x).
n
4.2. Let C be a nonempty closed convex subset of a Hilbert space H and let S = (T, : s E S) be a continuous representation of a semitopological semigroup S as nonexpansive mappings from C into C. If for each x E C, the set Q(x) fl F(S) is nonempty, then there exists a nonexpansive retraction P of C onto F(S) such that PT, = T,P = P for every t E S and Px E Cii( T,x: t E S) for every x E C.
THEOREM
Proof. For each x E C, let Px be the unique element in Q(x) f’ F(S). Then, as in the proof of theorem 4.1 (1) * (2), P is a retraction of C onto F(S) such that PT, = T,P = P for every t E S and Px E Co{ T,x: t E Sj for every x E C. It remains to show that P is nonexpansive. Let y E C and 0 < A < 1. Then as in the proof of theorem 3.3 we have for any E > 0,
4x((l - J)Px + APY) = supinf)l7;,x s
- ((1 - A)Px + APy)l12
I
= sup inf(l &x - Px + A(Px - Py)l)* s
f
= supinf(l/T,,x s
- Px112 + 2A(7;,x - Px, Px - Py> + A2)lPx - Pyl12)
t
< q,(Px)
+ 2A supinf(T,,x s
Since Px is the minimizer
I
- Px, Px - Py) + A*IlPx - Pull2 + .z.
of qX, we have
2A sup inf( csx - Px, Px - Py) + A2(lPx - Pull2 + E s
f
> qw((l - A)Px + APy) - q,(Px)
L 0.
Nonlinear ergodic theorems
Since E > 0 is arbitrary,
1423
we have
21, sup inf( Ksx - Px, Px - Py) + A’llPx - Pyj12 L 0 s I and, hence, 2 sup inf(7;,x s I
- Px, Px - Py) 2 -IIIPx
- Pyl12.
Now, if A + 0, then sup inf( 7;,x - Px, Px - Py) L 0. s I Let E > 0. Then there exists u E S such that (T,,x - Px, Px - Py> > -& for every t E S. For such an element u E S, we also have sup inf(7;,T, y - PT, y, PT, y - Px) L 0 5 I and, hence, there exists v E S such that (T,,,Y
- PT,Y,PT,Y
- Px) > --E
for every t E S. Then, from PT, y = Py, we have (T,,,Y
for every t E S. Therefore,
- PY,PY - Px) > --E
we have
-2~ < (T,,,x = (T,“&
- Px, Px - Py) + (T,,, y - Py, Py - Px> - T,,, y - (Px - PY), px - PY)
= (T,“,X 5 IITuvux-
T,,,Y,PX T,,,rllllPx
-
- llpx - PYl12 - Prll - IlJ’x - Pyli* PY)
5 I/x - YIIIIPX - PYII - llpx - PYl12. Since E > 0 is arbitrary,
this implies IIPx - Pyll 5 IIx - yll.
We now proceed to find conditions
n
on S and E such that Q(x) f’ F(S) # @ for every x E C.
LEMMA 4.3 [14]. Let C be a nonempty closed convex subset of a Hilbert space H, let S be an index set, and let lx, : t E S 1 be a bounded set of H. Let X be a subspace of I”(S) containing
constants, and let p be a submean on X. Suppose that for each x E C, the real-valued function f on S defined by “f(t) = II4 - XII2 for all t E S belongs to X. If ex) = iut/Ix, - XII2 for all x E C
1424
A. T. M.
LAU
et al.
and r = inf(r(x): x E C), then there exists a unique element z E C such that r(z) = r. Further, the following inequality holds for every x E C.
r + /Iz - x(1* 5 r(x)
4.4. Let C be a nonempty closed convex subset of a Hilbert space H and let S be a semitopological semigroup such that RUC(S) has an invariant submean. Let S = (T,: s E S] be a continuous representation of S as nonexpansive mappings from C into C. Suppose that (T,x: s E S] is bounded for some x E C. Then the set Q(x) fI F(S) is nonempty.
THEOREM
Proof. First we observe that for any y E H, the functionf(t) = 11T,x - ~(1’ is in RUC(S) (see [ 151). Let ,D be an invariant submean and define a real-valued function g on H by
g(Y) = PrlITX
- Yl12
for each y E H.
If r = inf(g(yj: y E H), then by lemma 4.3 there exists a unique element z E H such that g(z) = r. Further, we know that
r + llz - ~11’5 &) For each s E S, let Q, be the metric projection Q, is nonexpansive and for each t E S,
for every y E H. of H onto Co] T,x: t E S). Then by Phelps [16],
II’T;,x- Q,zl/* = IiQJtsx - Qszll* 5 IIT,x - z/I’. So, we have
and, thus QSz = z. This implies that z E co(L&,x: t E 6s)
for all s E S
and, hence, z E n CO~T,,~: t E SI. se.5 On the other hand, by lemma 4.3,
llz - Al2 5 rUtIIT,x - Yl12- fir IITX - zl12
for every y E H.
So, putting y = cz for each s E S, we have
II2 - T,zl125 rUtIIT,x - Tszl12 - rUrIITX - zI12 = )utllT,,x- T,zl12- rutIIm - zI12 5 pt IIT,x - zl12- P,IITX - zl12= 0. Therefore,
we have Tsz = z for every s E S.
n
Nonlinear
ergodic
theorems
1425
PROPOSITION 4.5. Let S be a discrete reversible semigroup and let C be a nonempty weakly compact convex subset of a Banach space E. Let S = 1T, : s E S 1 be a representation of S as affine nonexpansive mappings from C into C. Then for each x E C, the set Q(x) II F(S) is nonempty. Proof. Let x E C. Clearly, Q(x) = nscS Co1T,,x: t E S) is weakly compact and convex. Also Q(x) is nonempty. Indeed, for each s E S, let K, = co(T,,x: t E SJ. Then (K,: s E S] are weakly closed subsets of C with finite intersection property: if sr , . . . , s, E S, choose t, E fly= r Ss,. Then TOx E f-J;=, KS,. Consequently n, Es KS = Q(x) is nonempty. Let a, s E S, y E Q(x) and let lyk] be a sequence in KS such that l/y, - y 11+ 0. Then T, yk E KS (by affiness and continuity of T,) and liT,yk - CYII 5 IIY, - ~11-, 0. I-I ence T, y E K,. Consequently, T, y E n, E s KS = Q(x). Hence, Q(x) is S-invariant. Now, since S is left reversible, by [ 171 the space WAP(S) of weakly almost periodic functions on S has a left invariant mean (see also [18]). Hence by [19], there exists a common fixed point in Q(x), i.e. Q(x) fl F(S) # 0. n 5,MINIMALLEFTIDEALS
Let (Z, 0) be a compact right topological semigroup, i.e. a smigroup and a compact Hausdorff topological space such that for each T E C the mapping y + y 0 r from Z into C is continuous. In this case, ZEmust contain minimal left ideals. Any minimal left ideal in C is closed and any two minimal left ideals of C are homeomorphic and algebraically isomorphic (see [20]). LEMMA 5.1. Let X be a nonempty weakly compact convex subset of a Banach space E. Let S = 1q : s E S) be a representation of a semigroup S as nonexpansive and weak-weak continuous mappings from X into X. Let C be the closure of S in the product space (X, weakly. Then E is a compact right topological semigroup consisting of nonexpansive mappings from X into X. Further, for any T E C, there exists a sequence 1T,) of convex combination of operators from S such that )IT,x - Txll -+ 0 for every x E X. Proof. It is easy to see that Z is a compact right topological semigroup. We now prove the last statement (which implies that each T E C is nonexpansive). Consider S G (E, 1).II>” with the product topology. Let @ = co S. Then each T E CDis nonexpansive. Hence each T E & is also nonexpansive. Since the weak topology of the locally convex space (E, 11.I]>” is the product space (E, weaky, it follows that C E Tweak = 6, and hence the last statement holds. n
C is called the enveloping semigroup of S. A subset X of a Banach space E is said to have normal structure if for any bounded (closed) convex subset W of X which contains more than one point, there exists x E W such that sup(]lx - yII: y E W) < diam( W), where diam( W) = sup1 IIx - yll: x, y E W) (see (211 for more details). THEOREM 5.2. Let X be a nonempty weakly compact convex subset of a Banach space E and X has normal structure. Let S = 1T,: s E S) be a representation of a semigroup as normnonexpansive and weakly continuous mappings from X into X and let C be the enveloping of S. Let I be a minimal left ideal of Z and let Y be a minimal S-invariant closed convex subset of X. Then there exists a nonempty weakly closed subset C of Y such that 1 is constant on C.
1426
A. T. M. LAU
et al.
Proof. Since I is a minimal left ideal of C and Z is a compact right topological semigroup (lemma 5.1), I = Ce for a minimal indempotent e of C and G = eCe is a maximal subgroup contained in I (see [20]). Since each T E G is a nonexpansive mapping from Y into Y (lemma 5.1), by Broskii-Milman theorem [22], there exists x E Y such that TX = x for every T E G. Now put C = Ix. Then C is weakly closed and S-invariant. Also, if y, , y2 E C, yl = T, ex, y, = T,ex, T, , T2 E C, then, since eT,e E G, we have (Te)y, = Te(T,ex) = TX for every T E Z and similarly (Te)y, = TX for every T E Z. The assertion is proved.
n
The following improves the main theorem in [23] for Banach spaces (see also [24]). 5.3. Let C and X be as in theorem, 5.2. Then there exist TO E E and x E X such that T,Tx = T,x for every T E C.
COROLLARY
Proof. Pick x E C and TO E I of the above theorem.
n
Remark 5.4. If S is commutative, then for any T E C and s E S, T, 0 T = T 0 T,, i.e. z = T,x is in fact a common fixed point for Z (and, hence, for S). Note that if X is norm compact, the weak and norm topology agree on X. Hence every nonexpansive mapping from X into X must be weakly continuous. Therfore, corollary 5.3 improves the well known fixed point theorem of De Marr [6] for commuting semigroups of nonexpansive mappings on compact convex sets. REFERENCES 1. BROWDER
F. E., Nonlinear
operators
and nonlinear
equations
of evolution
in Banach
spaces,
Proc. Sympos.
Pure Math. 18(2) (1976). 2. LAU A. T. & TAKAHASHI W., Weak convergence and non-linear ergodic theorems for reversible semigroups of nonexpansive mappings, Pacific J. Math. 126, 277-294 (1987). 3. BAILLON J. B., Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert, C. r. Acad. Sci. Paris St+. A-B 280, 1511-1514 (1975). 4. TAKAHASHI W., A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Am. math. Sot. 81, 253-256 (1981). 5. HIRANO N. & TAKAHASHI W., Nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Banach space, Pacific J. Math 112, 333-346 (1984). 6. MARR R. De, Common fixed points for commuting contraction mappings, Pacific J. Math. 13, 1139-1141 (1963). 7. DIESTEL J., Geometry of Banach spaces, Selected Topics, Lecture Notes in Mathematics 485. Springer, Berlin (1975). 8. HEWITT E. & ROSS K., Abstract Harmonic Analysis I. Springer, Berlin (1963). 9. LAU A. T. & TAKAHASHI W., Invariant means and fixed point properties for nonexpansive representations of topological semigroups, Top. Meth. Nonlinear Analysis (to appear). 10. BARBU V. & PRECUPANU Th., Convexity and Optimization in Banach Spaces. Editura Academiei R. S. R., Bucuresti (1978). 11. XU H. K., Inequalities in Banach spaces with applications, Nonlinear Analysis 16, 1127-l 138 (1991). 12. RODE G., An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert Space, J. math. Analysis
Applic. 85, 13. DEUTSCH
SIAMRev.
172-178
(1982).
F. R. & MASERICK
9,
516-530(1967).
P. H.,
Applications
of the Hahn-Banach
theorem
in approximation
theory,
Nonlinear 14. MIZOGUCHI semigroups 15. LAU A. (1985). 16. PHELPS 17. HSU R., 18. LAU A.
19. 20. 21. 22. 23. 24.
ergodic
1427
theorems
N. & TAKAHASHI W., On the existence of fixed points in Hilbert spaces, Nonlinear Analysis 14, 69-80 (1990). T., Semigroup of nonexpansive mappings on a Hilbert space,
and ergodic
retractions
for Lipschitzian
J. math. Analysis Applic. 105, 514-522
R. R., Convex sets and nearest points, Proc. Am. math. Sot. 8, 790-797 (1957). Topics on weakly almost periodic functions. Ph. D. Thesis, SUNY, Buffalo, 1985. T.. Amenability and fixed point property for semigroup of nonexpansive mapppings, in Fixed Point Theory and Applications (Edited by J. B. BAILLON and M. A. THERA). Longman Scientific and Technology, Essex (1991). LAU A. T., Some fixed point theormes and their applications to W*-algebras, in Fixed Point Theory and its Applications (Edited by S. SWAMIRATHAN), pp.l21-129. Academic Press, New York (1976). BERGLUND J. F., JUNGHENN H. Q. & MILNES P. Ana/ysis on Semigroups. John Wiley and Sons, New York (1989). GOEBEL K. & KIRK W. A., Topics in metric fixed point theory, Camb. Stud. Adv. Math. 28 (1990). BROSKII M. S. & MILMAN D. P., On the center of a convex set, Dokl. Akad. Nauk SSSR 59, 837-840. HOLMES R. D. & LAU A. T., Almost fixed points of semigroups of nonexpansive mappings, Studia Math. XLlll, 217-218 (1972). PETTIS B. J., On Nikaido’s proof of the invariant mean-valued theorem, Studia Math. 33, 193-196 (1969).