トップページ - 横浜国立大学学術情報リポジトリ

全文

(1)A Generalization of Goebel and Kirk's Fixed. '. Point Theorem, ' ' '. ' '. '. By .. t.. '. Hisaharu UMEGAKI and Wataru TAKAHASHI*' '. (ReceivedMarch25,1974) '. '. '. 1. Introduction.. ' Let K be a subset of a Banach space X A transformation F:K--)K is said to be asymPtotically nonexPansive if for each x, yG K,. .･ ' 11 Ftx-Fiy[I ;;l lei[[x- yll ' such that lim ki=1. If ki :1 for where {ki} is a sequence of real numbers i=1, 2, 3, ･･･, F is said'to be nonexPansive. `Asymptotically nonexpansive' was defined by Goebel and Kirk [3] and it was proved by them that if K is a nonempty, closed, convex and bounded subset of a uniformly convex Banach. space,thenFhasafixedpoint. - ''. Our purposes here are to extend this result to a left amenable semigroup, by considering that {Ft:z = 1, 2, ･･･} is a semigroup and to give an elementary proof of the fixed point theorem [6], [8] for a left amenable semigroup of. nonexpansive mappings in a uniformly convex Banach space. Finally, we remark that this result can be proved for the case of a left reversible semi-. group... ･･',. 2. Preliminaries. ' '. '. '. '. '. '. ' ' ' ' , ' . Let 21 be an abstract semigroup and m(X) be the' space of all bounded real valued functions of X with the supremum norm. An element ptEm(X)"' '[the dual space of m(2)] is mean on X if ptt[e(t)]=11pt[[=1 where e(t) =1. tt. tt. for all tE X. A mean pt is left invariantif pt,[A,(t)]= pt,[f<t)] for all fEm(X>. and soEX, where .11,,(t)=f(sot) for all tEX. A semigroup that has a left. invariantmeaniscalledleftamenable;see[1]. ' Let .X be a left amenable semigroup, then sXAt.X ¥￠ for all s,tE2 (cf. [4], [5]). So if we define an order s;llt by tE s.YV {s}, .X is a quasi-directed. set'. Throughout the paper, we shall assume that X has the identity L abovtEaMnl)ilSel ;7G]'m(L.ge)1 illhebne i[leiehfatvgmenabie semzgroup wzth the order dofned. tt ' *) Present Address: Department of Information Sciences, Tokyo Institute of. Technology. . ･ t. t '. '.

(2) 2 . H. UMEGAKi an,d W. TAKAHAsHi ' '. '' '. tt Ln.S.lf< Slp. '. ' '. '. t) =< ptt[f( t)] ;:$ i?f ?.u,p f( t). '. ' foranyleftinvariantm'eanptonX. . . ･ PROoF. Let c be a real number satisfying infsupf(t)<c. Then there. t. exists an element sd E "X such that f(t) <c for all tl.l: so, tG .X. Since A,(t) =. i. s sSt. ll. f(sot) <c for all tG .X and pt is a left invariant mean on .S, we have ptt[f(t)] !･. =ptt[.ICk,(t)]$c. Therefore ptt[f(t)].<..ipt?.u,p.f<t).. Similarly, we obtain. ). supinff(t)Sptt[Kt)]･ ' '. A Banach space X is called unijbrmly convex if there exists an increas-. ing, positive function S:(O, 2].(O, 1] such that the inequalities,Ilx" :il d, 11yll. Sd and llx-yll >--e imply ,･. tt. jlx+y/211S[1-S(e/d)]d. ,. It is well known that a uniformly convex Banach space X is' reflexive and hence a bbunded closed convex subset of X'is weakly compact. Throughout the paper we denote by S(x, r) the spherical ball centered at x with radius. '. '. '. 3. Fixed point theorem.. '. tt. ･" ･1. THEoREM. Let K be a nonemPty, closed, convex and bounded subset of a unijCbrmly convex Banach sPace X, and let 2 be a left amenable semigroup of maPPings t on K satisytying the condition that there exists kGm(X) such that Ll'lll-XtYg/=',,le(j,)l.IX,-,;Ll ;b;,,aiE,,X}21S." a"d '?fs".ple(t);si･ Then th2re exists. 'LEMMA 2. Let K and .X satisf)7 the same assumPtions as in Theorem and let x, yE K. if ptt[Hty-xll]=Ofor some left invariantmean pt on X, then sx=x. for all sE .X., .,, PROOF.LetsEiX.Since' '' ' itet[listy-sxli] S. ,Ltt[k(s)llty-xll]. tt. = le(s),eet[l1ty- xli] = O 2ctt[l1sty-x11]= ,Ltt[11ty--xll] = O,. We. Obtai" llxpt s.lt = ,,,,[ ll'. x-sty +sty-sxll]. ;!ii,Ctt[UX-Styii]+ic.et["stly-sJu11]. = o+o == o.. ' Thereforesx==xforallsE.X. ' '. PROoF OF THEOREM. Let pu (iEK be fixed, and let the set R, consist of those numbers p for which there exists an element so E X such that. 1 1 .,L. j.

(3) A Generalization of Goebel and Kirk's Fixed Point Theorem 3 ' ' ' ' ' KA [,,A.stS(ty' P)] ¥ ip' g.et 'po =inf {p:p E!i R,} and for each E>O define'. ' ,' .Ce ==V[.,A-.,,S(tY, po+E)]･ Thus for each tt E>O, C,AK is nonem-pty and convex, and hence. ' C== ,9, (CeAK) iF ip･ -.. ' ' Notice that if xeC, infsuplltly-xli$po. In fact, lets>O. xeCimplies s sKt xGC-,i2AK Since C- ./2 ig the closure of C,i2, there exists zGC,i2= V [tl.,S(ty, po+e/2)] such that l[x-21I < e/2 and hence there exists so G 2 such. that ,. ' ･ . 2EAS(ty,p,-Fe/2). .' sos.t , t Therefore, IIx-----tyllS-l]x-21I+II2-tLyll ･･･ ' ''' '. '. '. '. <e/2+po+6/2== po+e ' 1. J. t.. tt t. for all tll so, tG ' X. These inequalities imply sup ]lx-tyll ;:;l po+s and hence. '. s'o$t. ''. 'iRf S,u.? IIxin tyll S- too+e. m. tt .. SinceE>Oarearbitrary,weobtain '. ' 11x-tyll ;El po･ infsup s sSt. mt ' ' ' ' Now, let xGC and suppose that ptt[]ltx-x]l]#O for all left invariant means pt on X. In fact if ptt[)ltx-xll]=O for some left invariant mean pt on X, by Lemma 2 we can prove the Theorem. By Lemma 1, we obtain e>O such thats<i?fs,u.plltx--xil and hence e<s,u.p11tx-xll for all seX. If po= O, by. '. '. '. '. '. ' ' .infsupllty-xll;-Spo - -. ,,, .ss$e･ ''. and Lemma 1, 2 we can prove the Theorem. So, assume pe>O and choose a>O so that [1-S(e/p,+a)](p,+cr)<p,. Since infsupk(t)$1, there exists . s,sSt. t'E.Xsuchthat ' ' '' (po+a/2) sup k(t) < po+cr .. t' 5t. So, we can select toEX such that lltox-x]]>e and (po+a/2)k(to)<po+a. Since i?f E2:.l,p "ty-xil S- po, there exists so E! jX such that s,,u=.e iity-x[l < po+c¥/2.. Let uoeX such that uo-). so and uo). toso, that is, zao= soui== tosou2 for some. ubu2EX.Then ' .. '. .;.

(4) 4 , ･ ' H.UMEGAKiandW.TAKAHAsHI ' '. '. '. '. iltox-uotyH=iltox-tosou2tNll. 51le(to)llx-s,u2tyil ･ < le(to)(po+a/2) < po+cr. -s. and llx-u,tLyll = llx-soztityll. , <po+cr/2<po+ev. i'. for all tE E. Thus by uniform convexity of N, if t). uo,. '. llx+ t,x/2- ty" ;S [1-6(e/po+ cr)](po+cr) < po. and this contradicts the definition of po. Hence we conclude po=O or ptt[IItx-xli]=O for some left invariant mean pt on X. Thus we obtain the-. Theorem. , - ''. ' REMARK.. We can prove the Theorem even if .X is a left reversible semigroup (i.e. for every pair of elements t, sEX, there exists a pair u, vE2 such that tu =sv). In fact, if for x,yGK, infsupllty-xll=O instead of s s-<.t. assuming ptt[11ly-xll]=O for some lett invariantmean pt on .X, we have that. iiilf S,U.P IISotY-Soxil == O and il?f fiil;L,p iisoty-Jtll =O for all s, E X. Therefore, sx= x.. for all sE.X;see' Lemma' 2. + '. '. References. t ... '. ･),. [1] DAy, M.M., Arnenable semigroup. Illinois J. Math. 1 (1957), 509-544.. [2] DuNFoRb, N. and J.T.ScHwARTz, Linearoperators,I. Interscience, New Yorki (1958).. [3] GoEBEL, K. qnd W.A. KiRK, A fixed point theorem for asymptotically non-expansive mappings. Proc. Amer. Math. Soc. 35 (1972), 171-174. [4] GRANiRER, E., On amenable semigroups with a finite dimensional set of in-･. . variantmeansI. IllinoisJ.Math.7(1963),'32-48.. [5] ,Atheoremonamenablesemigroups.Trans.Amer.Math.Soc.111 ' (1964),367-379.. [6] HoLMEs, R.D. and A.T. LAu, Nonexpansive actions of topological semigroups･ and fixed points. J. London Math. Soc. 5 (1972), 330-336. [7] TAKAHAsHi, W., Invariant.functibns for amenable semigroups of positive con-･ tractions on Li. K6dai Math. Sem. Rep. 23 (1971), 131-143.. [8] ,Onfixedpointsforreversiblesemigroupsofnonexpansivemappings. Sci. Rep. Yokohama National Univ. 19 (1972), 1-4.. DEPARTMENT OF INFORMATION SCIENCES, TOKYO INsTITUTE OF TECHNOLOGY AND DEPARTMENt OF MATHEMATICS, YOKOHAMA NATIONAL UNIVERSITY.. 1 J.

(5)