(1) a.j År..O for all n, je N,a.j--O if jÅrn,

SOME REMARKS ON THE MANN ITERATIVE PROCESS

FOR NONEXPANSIVE MAPPINGS

By

Ken-ichi MiyAzAKi, Takeo KAwATANi and Suga MiyAuRA (Received Oct. 30, 1981)

J. B. Baillon proved in [1] the following ergodic theorem for nonexpansive mappings in Hilbert space.

Let X be a real Hilbert space, D be a convex bounded closed subset of X and T be a n't

nonexpansive mapping fromDinto D. Then the mean 2 Tjxi!n for xieD, with TO=I j--o

identity mapping, converges weakly to a fixed point of T.

Since then, various extensions of the theorem have been succeeded. The first direction of the extension is the enlargement of Hilbert spaces X to Banach spaces or to complete metric spaces and the weakening of the limitations of the uniform convexity of X and the convexity, boundedness of D [2], [5]. The second direction of the extension is the gen- eralization of the nonexpansive mapping T[9], [8]. The third direction of that is the

. n-1study of the more general averaging method of Cesbro means 2 Tjxiln [6], [3].

j=o

By making fu11 use of the interesting equations, S. Ishikawa was successfu1 to delete the assumptions of uniform convexity of X and boundedness of D in the above Baillon's type theorem.

Generally speaking, when considering the weakening of nonexpansiveness of T, the uniform convexity of X and compactness, convexity of D can not be deleted as seen in the proofs of Wong [9] and Singh [8] with their crucial usage of these conditions.

In this paper we shall show some applications of the Mann iterative process to the n-1Ceshro means of ergodic theory 2 Tjxiln of nonexpansive mapping T from the subset D

of a Banach space X into X &i'tOh various conditions. Although most content of this paper is an extension of [5] and [8], but the following specific lemma and theorems are interested in themselves.

Now let us modify the Mann's iterative process and let us define the quasi-normal Mann process. Let N denote the set of positive integers, and suppose that A=[a.j].,j.N is an infinite real matrix satisfying

(1) a.j År..O for all n, je N,

a.j--O if jÅrn,

n

.2anj=1 forall neN J=1

and

]im a.j =O for all J' EN.

n-co

Let D denote a subset of a Banach space X, and let Tbe a mapping from D into X.

Throughout this paper, suppose that for each xiGD the sequence {x.}..Ai can be defined inductively by the following form and x,GD for all neN

n(2) xn=anixi+jZ.,2anjTxj"i, n=2, 3,•••.

We here note that if D is a convex set then this condition x. GD for all nEN is necessarily satisfied.

Generalizing the normal Mann process in [4], let us propose the following

DEFiNmoN 1. Let A=[a.j].,j.rv be a matrix satisfying (1). Suppose in addition that there exists an infinite subset Ni ={n(i)lieN} ofN such that n(i)Åqn(i+1) and

(3) ' an(i+i),J' '---- (1-an(i+i),n(i+i))an(i),J'

are satis.fied for all i andjEN. Let us denote by M'(xi, A, T) the sequence {x.}..N deLfined by (2) depending on the .iciED,A and T, M'(xi, A, T) is said to be the quasi- normal Mann process.

By this definition, it is noted that for the quasi-normal Mann process M'(xi, A, T) there exists a subsequence {x.(i)}.(i).N, of {x.} = M'(xi, A, T) satisfying

(4) Xn(i+i) =(1 -"' an(i+i),n(i+i))Xn(o+an(i+i),n(i+i)TXn(i) for all iGN.

Now, we shall show some important examples of the quasi-normal Mann process for later use.

If xi is any point of D, then the constructing process of the Cesatro mean n-1

S.xi==l Tjxiln, nGN, with TO=I identity operator, is equal to the picking out process J=O

of the subsequence {x2.}..N from the sequence {x.} of the following specific quasi-normal Mann process M'(xi, A, T). Using'the notations in Definition 1 suppose that

(5) A== [anj]n,jeN, Ni == {2m Ime N} cN, aii=1,

( 1 if 1' --- 2m - 2

a2m-i,j '-- i o if 1. s2m --2 for m= 2, 3, •••,

and

f 11(m+1) if J' -- 1, 2k (k=1, 2,•••, m)

a2M'j"--ny

io otherwise fOrall mEN.

DEFiNiTioN 2. This quasi-normal Mann process is said to be the Cesdro-Mann process and is denoted by M2(xi, A, T)•

We here note that the relations (4) in this case turn out

m+1 1

(6) X2(m+i)=m+2 x2.+m+2 Tx2., forall meN.

n

We next consider the general averaging process mentioned in [3] B.xi=: E) b.jTJ"ixi J=1

with the matrix B=[b.j].,j.N such that b.jÅr=O for all n, J'eN, b.j---O for all jÅrn and

;Ib b.j --- 1. Then the sequence {B.xi}..N is equal to the subsequence {x2.}..N of the quasi- normal Mann process {x.}..N=M'(xi, A, T) with A== [a.j].,j.N such thatJ=1

aii=1

{1 if j---2m-2

a2m-1,J =i o if j vE 2m .--2 for each m == 2, 3,•-

and

i bm+i,i if j -•- 1

a2M'j -'--iobM'i'k'i io'fthl'r=w= 2iske (k= 1, 2,''', m) for each meN.

With the view of studying the Cesbro-Mann process, we shall show the specific form of the Lemma 1 in [5] directly.

LEMMA. Let {zk}k.N be a sequence in a Banach space X. Then for any positive integers K and M, we have

(7) ,2K_{., Mz+k}k = M+MK-1 IM5K z.- M+IK-1 2.1k(z,.,- Mi)iiiSi1 z,)l.

In the special case of X=R, the set of real numbers, and zk=1for all k, (7) turns out

(s) ,2K

i

PRooF. The equation (7) is easily shown as follows

Åí., MZtk = th ,g., '- (k'1)(M+Mklf.k(M+k-1) .,

= iili I-,Åí.,(k-1)z,+,Åí.,-k(MM++kk-1) z,l

== th (-21 kzk+i+:l -k-(MM++kk- 1)-7k+ K(MM++KK-1)- z.l

== IiS, ( IÅq(MM++KK-L) z.--2.lk(,,.,m Mi+kk-1-.,))

== M+MK-i (M5K zK- M+iK-i År'2:.ik(zk.,- Mi+kii z,)l.

Now, using this Lemma, we shall show the key theorem for our main result.

THEoREM 1. Let D be a subset of a Banach space X and let T he a nonexpansive mapping from D into X, i.e. suppose thatfor any x, yED, H Tx- Tyll S. Ilx--yil is satisLfied.

Let xi eD and {x.}..N be the sequence defined by (2) of the Cesdro-Mann process M2(xi, A, T). if all the x.eD,nGN, and {x2.}..N is bounded, then x2.-- Tx2. converges to

zero as m--År co.

PROOF. Putting y.,=x2., fo,r, eacb mE IV, (6) is Written in the form

(9) .v.+i= mM + +S y.+-m-l]F -Ty. for all mc- N.

Thus we have

m+1 1

ym+i -' T.ym+i ==

m+2 Ym+m+2 TYm'Tym+i m+1

= m+h-2-- (Ym -- TY m) +(Ty tn - Tym+i), and

II Ty.- Ty.+ijl i-S lly,. -'- ym+iIi

m+1 1

= ym- m+2 y.- m+2 Ty.

1

= m+2 ilY,n-Tymll•

Thus we get

J 'liyh+i ;Ty.+i il l-S'll y.- Ty., ll , '

so lim lly. -• Ty.ll =r exists.

m-co

We shall show r==O. In fact, if rÅrO then for any positive e there exists a positive integer P such that

(10) r;Sl ll y. --- Ty.U ;S r+s for all mÅr P.

Let L=2iep,. Ilx2mll, then since, ), p+lk+2 =co there exists h positive integer K such

that

(m r:;l p+ik+2 ;SL+iÅq",$, p+Z+2'

we here note that from (11) and from log P+ pK+ 3+2 Åq:IIII p+ik ., 2 , K satjsfies

(12) KÅq(P+3)e(L"i)1'-(P+2).

Now setting zk = Typ+k-.vp+k, from (9) we get

P+k+1

Zk+1- p+k+2 Zk

= Ty ..,., -- ( Pp ++ 2 I: S yp.k+ p+Z+2 Ty p. ,) -- fl I: 2lS (Ty p.k --yp.,)

= Ty p+k+1 - Ty p+k •

Therefore, by virtue of the nonexpansiveness of Tand from (9) and (10) we have

P+k+1

Zk+i- p+k+2 Zk IS ilYp+k.i•-- vp.kll

(13) Sp+Z+2 lly.., --- Ty ..,ll

Åq r+8 m+m.

== P+k+2'

Further, using (9) and the Lemma replacing M by P+2 we have

YP+K+1-YP+1

K

=: E (yp.k.1--yp.k)

k=1 ' '

=: J?Ki).1 p+ili+2 (Typ.k-yp.k)

- P'.K.," I..".., z.- ..k., :lk(zk+i- ;I:2I:g zk)l•

Hence, from IIzKII )r and (B), and applying (8) of the Lemma and finally from (11), (12) we obtain

L ). IIy...., ---yp., ll

iP'..","(-..K.'-.-2-r-'p.k.r21lhp:'k6+2k)

== P'p"+ 2" ( p.KK+2 ---p Frk.r:l pi+2m)r--pT+e 2 21 p+kk+:, i;r,Åí, p+Z+2-- pK+62-

ÅrL+ 1 -- e(2e(L'i)lr - 1) .

Since sÅrO is arbitrarily small, this is the contradiction which completes the proof.

Now we shall show the main theorem.

THEoREM 2. Let D be a closed subset ofa Banach space X and Tbe a nonexpansive mapping from D into a compact subset of X. Let {x.}..N =M2(xi, A, T) he the Cesdro- Mann process, namely suppose that the matrix A=[a,j].,J•.N satisLfies the assumption (5) andfor each xi eD the sequence {x.}..N is deLfined by (2) and x.•eD for all 'nEN. Then n-1T has a fixed point in D. Further, the Cesdro mean 2 Tjxiln, which is realized as a j=osubsequence of the Cesdro-Mann process M2(xi, A, T), converges to a .fixed point of T.

PRooF. Since T(D) is compact, so co({xi}UT(D)) is compact. Using the same notations in the proof of Theorem 1, this means that the sequence {y.}..NcUo (.{xi} U T(D)) is bounded. Thus there exists a subsequence {y.(i)}i.N of {y.}..N and anfGD such that [ly.(i)---fll converges to zero as i.co, since D is closed. From the boundedness of {ym}meN, Theorem 1 is applicable, so we have

ll Tf-f ll = : ll Tf- Ty.(i) + Ty.(i) -- y.(i) + y.(i) --f 11 l:;l 2llf'-- J'tn(i)il + ll TYmÅqi) - Ym(i)Il ,

the last rjght hand side converges to zero as i. oo. This implies f is a fixed point of T.

Now, from (9) we have

llym+i --fN == -ll;i: -l ym + ".-1; ii+ Tym -- ( lll It 21 f+ ., IP 2 Tf)

.Åq. M.++2i Uy.-fll+.e2 IITy.-Tfll

S- lly.-fll ' for all `inelv,

which yields

II y. -f ll :il ll y.(i) -fU for all mÅr m(i) .

By the construction of the Cesaro-Mann process {x.}..N==M'.(.xi, A, T) mentioned in (5), this means that

m-1

y. == x2m ---' 2 Tjxilm converges to fas m --År co, j---o

which completes the proof of the second part of the theorem.

Next we shall try to weaken the limitation of the mapping Tin the Theorem 2. Then the assumptions of the uniform convexity of the Banach space X and the existence of fixed points are necessarily requested for proving the theorem. First remember the following

DEFiNmoN 3 ([8]). Let D be a subset of a Banach space, and let Tbe a mapping from D into X. iffor any x and y in D we have

11 Tx - Ty II S- max {llx ---- yH, (IIx- Tx II + lly -•-• TyH)12, (filx- Ty 11 + Ily- Tx ll )12} ,

Tis said to be a generalized contraction.

There are other various generalizations of nonexpansive mapping. About these definitions and their comparisons, we may refer to [7].

As our final result we shall prove the following

THEoREM 3. Let D be a closed subset of a uniformly convex Banach space X, and let Tbe a generalized contraction from D into a compact subset of X. Assume that the fixed points F=={feD: Tf =f} i,s nonempty. Then for each xiGD the Cesaro mean m-1S.xi == 2 Tjxilm converges to a fixed point qf T as m.oo where the sequenee {S.xi}..N j=ois egual to the subsequence {x2.}..N of the Cesaro-Mann process {x.}..N= M2(xi, A, T).

When using the same notations in Theorems 1 and 2, the proof of the lim 11y.- Ty.Il == O

m--oois done along the same line of the proof of Theorem 1.1 in [8]. Therefore we only sketch it.

SKETcH oF PRooF. In the first place, by the definition of the generalized contraction

we can prove that •

Next, since {x.}..N is the Ceshro-Mann process, using the same notations in Theorem 1 we have (9) with y.=x2., mGN. Further by the uniform convexity of X we notice that for any 6: OÅqsS.2

Åq1 5) S(e): == inf {1 - ll (x + y)/2 11: llx ll iS l, IIyll Sl and llx -- yll )- s}

is positive and monotonically nondecreasing.

With these preparations, using (14), (9) and (15) we can proceed the analogous argu- ment with Z replaced by 11(m+2)(meN) in the proof of Theorem 1,1 in [8], but omit here the precise one. Then for anyfeF and mEN we have

(!6) Ilf"--ym+ill :Åq= ,tr, Ii- 7• +22 6(llyj-- TyjllAlf-xill)lllf-xill.

Now from this inequality it is shown that y.-- Ty. converges to zero as m-ÅÄoo. In fact, suppose that {lly.- Ty.ll} does not converge to zero. Since from (14) and (16) IIy.- Ty.Il S- IIy. --f+f-- Ty.11

;-S 2jl y. -f Il

;!ll2ilxi--fll forany feE

there exists a subsequence O'.(i)}i.N of {y.}..N such that {lly.(i)-Ty.,(i)II} converges to some positive ct as i---År co. Thus for any positive number e there exists a positive integer K such that

I1 - -il-6(ct12 llf- xi II)]Kll f- )ci II Åqe

and

lly.(i)-•Ty.(i)ilÅrct12 forany iÅrK.

From this and (16) and the monotonousness of 6 we have for any integer mÅrm(2K+1)

llf- y.Il ;ll llf--- .y.(2K+i)ll

:; j.Kn.".', Ii - -l} 6(ll y.(j) - Ty.o)Il1Ilf- xi Il)} llf- xi ll

;s I1 •-- g 6(ct12ll f--- x, ll)lKllf-•- x, ll Åq Åí,

which shows that {y.}..N converges tofas m.co. Thus the inequalities

ll ym(i) - Tym(i)U i!il ll .vm(i) mf ll + IIf'-' Tym(i)11

E{. 2 ll .ym(i) -"-f ll

imply that lly.(i)--Ty.(i)ll converges to zero as i.oo. This contradiction shows that y.--- Ty. converges to zero as m. co.

The rest of the assertions of the theorem is proved as same as the corresponding part of Theorem 2.

Acknowledgement

The first and the third authors would like to express their gratitude to the Scientific Research Fund of the Ministry of Education for the partial financial support.

References

[ 1 ] J. B. Baillon, Un th6orem de type ergodique pour les contractions non lin(Saires dans un espace de Hilbert, C. R, Acad. Sci. Paris A 280 (1975), 1511-1514.

[2] J.B. Baillon, R.E. Bruck and S, Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J, Math. 4 (1978), 1-9.

[3] H. Br6zis and F.E. Browder, Nonlinear ergodictheorems, Bull, Amer, Math Soc. 82 (1976), 959- 961.

[4] W.G. Dotson Jr,, On the Mann iterative process, Trans, Amer. Math. Soc. 149 (1970), 65-73.

[5] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc.

Amer. Math. Soc. 59 (1976), 65-71.

[6] S. Reich, Fixed point iterations ofnonexpansive mappings, Paci,J, Math, 60 (1975), 195-198.

[7] B.E.Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. .

Math. Soc. 226 (1977), 257-290.

[8] K.L. Singh, Sequence ofiterates ofgeneralized contractions, Fund. Math. 105 (1980), 115-126.

, [9] C.S. Wong, Approximation to fixed pointsof generalized nonexpansive mappings, Proc. Amer.

Math. Soc. 54 (1976), 93-97.

Department of Mathematics Kyusha Institute of Technology, Kitakyushu Technical College and

Department of Control Engineering Kyushu Insti,tute of Technology