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OF NONEXPANSIVE MAPPINGS

TOMONARI SUZUKI

Received 14 September 2004 and in revised form 24 March 2005

Using the notion of Banach limits, we discuss the characterization of fixed points of non-expansive mappings in Banach spaces. Indeed, we prove that the two sets of fixed points of a nonexpansive mapping and some mapping generated by a Banach limit coincide. In our discussion, we may not assume the strict convexity of the Banach space.

1. Introduction

LetCbe a closed convex subset of a Banach spaceE. A mappingT onCis called a non-expansive mappingifTx−T y ≤ x−yfor allx,y∈C. We denote byF(T) the set of fixed points ofT. Kirk [17] proved thatF(T) is nonempty in the case thatCis weakly compact and has normal structure. See also [2,3,5,6,11] and others.

Convergence theorems to fixed points are also proved by many authors; see [1,7,8,

9,10,13,15,18,23,30] and others. Very recently, the author proved the convergence theorems for two nonexpansive mappings without the assumption of the strict convexity of the Banach space. To prove this, the author proved the following theorem, which plays an extremely important role in [26].

Theorem1.1 (see [26]). LetCbe a compact convex subset of a Banach spaceEand letT be a nonexpansive mapping onC. Thenz∈Cis a fixed point ofTif and only if

lim inf

n→∞

1n

n

i=1

Tizz

=0 (1.1)

holds.

The author also proved the following theorem. Using it, we give one nonexpansive retraction onto the set of all fixed points.

Theorem1.2 (see [27]). LetEbe a Banach space with the Opial property and letCbe a weakly compact convex subset ofE. LetTbe a nonexpansive mapping onC. Put

M(n,x)=n1

n

i=1

Tix (1.2)

Copyright©2005 Hindawi Publishing Corporation

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forn∈Nandx∈C. Then forz∈C, the following are equivalent: (i)zis a fixed point ofT;

(ii){M(n,z)}converges weakly toz;

(iii)there exists a subnet{M(νβ,z) :β∈D}of a sequence{M(n,z)}inCconverging weakly toz.

In this paper, using the notion of Banach limits, we generalize Theorems1.1and1.2. We remark that the proofs of our results are simpler than the proofs of Theorems1.1and

1.2. In our discussion, we may not assume the strict convexity of the Banach space.

2. Preliminaries

Throughout this paper, we denote byNthe set of all positive integers and byRthe set of all real numbers. For a subsetAofN, we define a functionIAfromNintoRby

IA(n)=

  

1 ifn∈A,

0 ifn∈A. (2.1)

LetEbe a Banach space. We denote byE∗ the dual ofE. We recall that Eis said to have theOpial property[21] if for each weakly convergent sequence{xn}inEwith weak limitx0, lim infnxn−x0<lim infnxn−xfor allx∈Ewithx=x0. All Hilbert spaces,

all finite-dimensional Banach spaces, andp(1≤p <∞) have the Opial property. A Ba-nach space with a duality mapping which is weakly sequentially continuous also has the Opial property; see [12]. We know that every separable Banach space can be equivalently renormed so that it has the Opial property; see [31]. Gossez and Lami Dozo [12] prove that every weakly compact convex subset of a Banach space with the Opial property has normal structure. See also [19,20,22,25] and others.

We denote bythe Banach space consisting of all bounded functions fromNintoR (i.e., all bounded real sequences) with supremum norm. We recall thatµ∈()is called amean ifµ =µ(IN)=1. It is equivalent to

inf

n∈Na(n)≤µ(a)supnNa(n) (2.2)

for alla∈∞. We also know that ifa(n)b(n) for allnN, thenµ(a)µ(b) holds. Sometimes, we denote byµn(a(n)) the valueµ(a).µ∈()is called aBanach limitif the following hold:

(i)µis a mean;

(ii)µ(a)=µn(a(n+ 1)) for alla∈∞. That is, puttingb(n)=a(n+ 1) forn∈N, we haveµ(a)(b).

It is obvious that

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for a Banach limitµ,a∈∞, andk∈N. We know that Banach limits exist; see [4]. We also know that

lim inf

n→∞ a(n)≤µ(a)lim supn→∞ a(n) (2.4)

for alla∈∞.

LetT be a nonexpansive mapping on a weakly compact convex subsetCof a Banach spaceE. Letµbe a Banach limit. Then we know that for x∈C, there exists a unique elementx0ofCsatisfying

µnfTnx=fx0

(2.5)

for allf ∈E∗; see [14,19]. Following Rod´e [24], we denote suchx0byTµx. We also know thatis a nonexpansive mapping onC.

We now prove the following lemmas, which are used inSection 3. Lemma2.1. Leta,b∈∞and letµbe a Banach limit. Then the following hold.

(i)If there existsn0Nsuch thata(n)≤b(n)for alln≥n0, thenµ(a)≤µ(b)holds.

(ii)If there existsn0Nsuch thata(n)=b(n)for alln≥n0, thenµ(a)(b)holds.

Proof. We first show (i). We note thata(n0+n)≤b(n0+n) for alln∈N. Sinceµis a

Banach limit, we have

µna(n)=µnan0+n

≤µnbn0+n

=µnb(n). (2.6)

It is obvious that (ii) follows from (i). This completes the proof. Lemma2.2. LetA1,A2,A3,...,Akbe subsets ofNand letµbe a Banach limit. Put

A=

k

j=1

Aj, α= k

j=1

µIAj

−k+ 1. (2.7)

Suppose thatα >0. Then,

µIA≥α (2.8)

holds and

n∈N:n≥n0

∩A=∅ (2.9)

holds for alln0N.

Proof. It is obvious thatn∈Aif and only if

k

j=1

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andn∈N\Aif and only if

k

j=1

IAj(n)≤k−1. (2.11)

Therefore we obtain

IA(n) k

j=1

IAj(n)−k+ 1 (2.12)

for alln∈N. Hence,

µIA≥µn

 k

j=1

IAj(n)−k+ 1

 

=k

j=1

µIAj

−k+ 1

=α >0.

(2.13)

We suppose that{n∈N:n≥n0} ∩A=∅for somen0N. ThenIA(n)=0 forn≥n0.

So, fromLemma 2.1, we obtain

0< α≤µIA=µ(0)=0. (2.14)

This is a contradiction. This completes the proof.

3. Main results

In this section, we prove our main results.

We first prove the following theorem, which plays an important role in this paper. Theorem3.1. LetEbe a Banach space and letCbe a weakly compact convex subset ofE. LetTbe a nonexpansive mapping onC. Letµbe a Banach limit. Suppose thatTµz=zfor somez∈C. Then there exist sequences{pn}inNand{fn}inE∗such that

pn+1> pn,

Tpnzzλ 1 3n+1,

fTpnz−z≤2 +1

3+1 for=1, 2,...,n−1,

fn=1, fn

Tpnzz=Tpnzz

(3.1)

for alln∈N, where

λ=lim sup

n→∞

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Before proving this theorem, we need some preliminaries. In the following lemmas and the proof ofTheorem 3.1, we put

A(f,ε)= n∈N:fTnz−z≤ε (3.3)

for f ∈E∗andε >0, and

B(ε)= n∈N:Tnz−z≥λ−ε (3.4)

forε >0.

Lemma3.2. For everyn∈N,

Tnzzλ (3.5)

holds.

Proof. Sinceµis a Banach limit, we haveµn(Tnz−z)≤λ. Fixm∈N. By the Hahn-Banach theorem, there exists f ∈E∗such that

f =1, fTmz−z=Tmzz. (3.6)

Forn∈N, we have

Tmzz=fTmzz

=fTmzTm+nz+fTm+nzz

≤ fTmzTm+nz+fTm+nzz

=TmzTm+nz+fTm+nzz

≤Tnzz+fTm+nzz.

(3.7)

Hence

Tmzz=µnTmzz

≤µnTnz−z+fTm+nz−z =µnTnz−z+µnfTm+nz−f(z)

=µnTnz−z+µnfTnz−f(z)

=µnTnz−z+fTµz−f(z)

=µnTnz−z

≤λ.

(3.8)

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Lemma3.3. Suppose thatm∈N,f ∈E∗, andδ >0satisfy that

f =1, fTmz−z=Tmzzλδ. (3.9)

Then

µIA(f,ε)

εε

+δ (3.10)

holds for allε >0.

Proof. Forn > m, byLemma 3.2, we have

fTnzz= fTnzTmz+fTmzz

≥ −fTnzTmz+ fTmzz

= −TnzTmz+Tmzz

≥ −Tn−mzz+Tmzz

≥ −λ+λ−δ= −δ.

(3.11)

On the other hand, by the definition ofA(f,ε),

fTnzz> ε (3.12)

for alln∈N\A(f,ε). Therefore, forn∈Nwithn > m, we have

fTnzz≥ −δIA

(f,ε)(n) +εIN\A(f,ε)(n)

= −δIA(f,ε)(n) +εIN\A(f,ε)(n) + (ε−ε)IA(f,ε)(n)

= −(δ+ε)IA(f,ε)(n) +εIN(n)

= −(δ+ε)IA(f,ε)(n) +ε.

(3.13)

ByLemma 2.1, we have

0= fTµz−f(z) =µnfTnz−f(z)

=µnfTnz−z

≥µn−(δ+ε)IA(f,ε)(n) +ε

= −(δ+ε)µIA(f,ε)+ε.

(3.14)

Hence, we obtain

µIA(f,ε)

εε

+δ . (3.15)

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Lemma3.4. µ(IB(ε))=1holds for allε >0.

Proof. We fixε >0 andη∈Rwith 1/2< η <1 and put

δ=ε(1−η)

2η . (3.16)

We note that 0< δ < ε/2. By the definition ofλ, there existsm∈Nsuch that

Tmzzλδ. (3.17)

Fix f ∈E∗with

f =1, fTmz−z=Tmzz. (3.18)

So, usingLemma 3.3, we have

µIA(f,ε/2)

ε/ε/2

2 +δ =η. (3.19)

Forn∈Nwithm+n∈A(f,ε/2), we have

TnzzTmzTm+nz

≥fTmzTm+nz

=fTmzz+fzTm+nz

=Tmzz+fzTm+nz

≥λ−δ−ε 2 ≥λ−ε,

(3.20)

and hencen∈B(ε). Therefore

IB(ε)(n)≥IA(f,ε/2)(m+n) (3.21)

for alln∈N. So we obtain

µIB(ε)

≥µnIA(f,ε/2)(m+n)

=µnIA(f,ε/2)(n)

≥η.

(3.22)

Sinceηis arbitrary, we obtain the desired result.

Proof ofTheorem 3.1. By the definition ofλ, there existsp1Nsuch that

Tp1z−z≥λ− 1

32. (3.23)

Fix f1∈E∗with

f1=1, f1

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ByLemma 3.3, we have

µIA(f1,(2/3)2)

22

22+ 1. (3.25)

We now define inductively sequences{pn}inNand{fn}inE∗. Suppose thatpk∈Nand

fk∈E∗are known. Since

µIB(1/3k+2)+

k

=1

µIA(f,(2/3)+1)−k

1 +

k

=1

2+1

2+1+ 1−k

1 +

k

=1

2+11

2+1 −k=1 + k

=1

−1 2+1

>12>0,

(3.26)

we have

m∈N:m≥pk+ 1∩B

1 3k+2

∩k

=1

A

f,

2 3

+1

=∅ (3.27)

byLemma 2.2. So we can choosepk+1Nsuch thatpk+1> pk,

Tpk+1z−z≥λ− 1

3k+2, f

Tpk+1z−z≤2

+1

3+1 (3.28)

for=1, 2,...,k. Fix fk+1∈E∗with

fk+1=1, fk+1

Tpk+1z−z=Tpk+1z−z. (3.29)

Note that

µIA(fk+1,(2/3)k+2)

2k+2

2k+2+ 1 (3.30)

byLemma 3.3. Hence we have defined{pn}and{fn}.

Now, we prove our main results.

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Proof. We first assume thatz∈Cis a fixed point ofT. Then, we have

fTµz=µnfTnz=µnf(z)=f(z) (3.31)

for all f ∈E∗, and henceTµz=z. Conversely, we assume thatTµz=z. ByTheorem 3.1, there exist sequences{pn}inNand{fn}inE∗satisfying the conclusion ofTheorem 3.1.

We putλ=lim supnTnzz. Since Cis weakly compact, there exists a subsequence

{pnk}of{pn}such that{Tpnkz}converges weakly to some pointu∈C. Ifnk> , then

fTpnkz−z≤2 +1

3+1. (3.32)

So we obtain

f(u−z)2

+1

3+1 (3.33)

for allN. Since

Tpzu=fTpzu ≥fTpz−u

=fTpz−z+f(z−u)

=Tpzz+f (z−u)

≥λ− 1 3+1

2+1

3+1

(3.34)

forN, we have

lim inf

→∞ T

pzuλ, (3.35)

and hence

lim inf

k→∞

Tpnkzzλlim inf →∞

Tpzu

lim inf

k→∞ T

pnkzu. (3.36)

By the Opial property ofE, we obtainz=u. We also have

lim inf

k→∞ T

pnkzTzlim inf k→∞ T

pnk1zzλ, (3.37)

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Theorem3.6. LetCbe a compact convex subset of a Banach spaceE. LetTbe a nonexpan-sive mapping onC. Letµbe a Banach limit. Thenz∈Cis a fixed point ofTif and only if Tµz=z.

Proof. From the proof ofTheorem 3.5, we know thatTz=zimplies thatTµz=z.

Con-versely, we assume thatTµz=z. ByTheorem 3.1, there exist sequences{pn}inNand{fn}

inE∗satisfying the conclusion ofTheorem 3.1. We putλ=lim supnTnzz. SinceCis

compact, there exists a subsequence{pnk}of{pn}such that{Tpnkz}converges strongly to some pointu∈C. As in the proof ofTheorem 3.5, we obtain lim infTpz−u ≥λ.

This implies thatλ=0, and hence{Tnz}converges toz. So we have

Tz=Tlim

n→∞T

nz=lim n→∞T

n+1z=z. (3.38)

This completes the proof.

Appendix

In some sense, Theorems3.5and3.6are generalizations of Theorems1.2and1.1, respec-tively. To show this, we use the notion of universal nets. We recall that a net{yβ:β∈D} in a topological spaceYis universal if for each subsetAofY, there existsβ0∈Dsatisfying

either of the following:

(i) yβ∈Afor allβ∈Dwithβ≥β0; or

(ii) yβ∈Y\Afor allβ∈Dwithβ≥β0.

For every net{yβ:β∈D}, by the axiom of choice, there exists a universal subnet{yβγ:

γ∈D}of{yβ:βD}. If f is a mapping fromY into a topological spaceZ and{yβ:

β∈D}is a universal net inY, then{f() :β∈D}is also a universal net inZ. IfY is

compact, then a universal net{yβ:β∈D}inYalways converges. See [16] and others for

details.

PropositionA.1. Let{νβ:β∈D}be a universal subnet of a sequence{n:n∈N}inN. Define a functionµfrom∞intoRby

µ(a)=lim

β∈D

1

νβ νβ

i=1

a(i) (A.1)

for alla∈∞. Thenµis a Banach limit.

Proof. We note thatµis well defined because{νβ:β∈D}is universal. It is obvious that

µis linear. Fora∈∞, we have

µ(a)=lim

β∈D

1

νβ νβ

i=1

a(i)lim

β∈D

1

νβ νβ

i=1

a =lim

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Similarly, we obtainµ(a)≥ −a. Henceµ ≤1. Sinceµ(IN)=1, we haveµ =µ(IN)= 1, that is,µis a mean on. We also have

µna(n+ 1)=lim β∈D

1

νβ νβ

i=1

a(i+ 1)

=lim

β∈D

 1

νβ νβ

i=1

a(i)−aν(1)

β +

aνβ+ 1

νβ

 

(a)

(A.3)

for alla∈∞. This completes the proof.

PropositionA.2. LetCbe a weakly compact convex subset of a Banach spaceEand letT be a nonexpansive mapping onC. DefineM(·,·)as inTheorem 1.2. Take a universal subnet {νβ:β∈D}of a sequence{n:n∈N}inNand define a mappingUonCby

Ux=weak-lim

β∈D M

νβ,x (A.4)

for allx∈C. Then there exists a Banach limitµsatisfyingTµx=Uxfor allx∈C.

Proof. Define a Banach limitµas inProposition A.1. Then forx∈Candf ∈E∗, we have

f(Ux)=lim

β∈Df

1

νβ νβ

i=1

Tix

=lim

β∈D

1

νβ νβ

i=1

fTix

=µnf(Tnx)

=fTµx.

(A.5)

Since f is arbitrary, we haveUx=Tµxfor allx∈C. This completes the proof.

UsingProposition A.2, we obtain the following proposition.

PropositionA.3. LetCbe a weakly compact convex subset of a Banach spaceEand letT be a nonexpansive mapping onC. Then ifz∈CsatisfiesTheorem 1.2(iii), then there exists a Banach limitµsatisfyingTµz=z.

Proof. DefineM(·,·) as inTheorem 1.2. Take a universal subnet{νβγ:γ∈D}of{νβ:

β∈D}. We note that{νβγ:γ∈D}is also a universal subnet of a sequence{n:n∈N}in N. We also note that{M(νβγ,z) :γ∈D}converges weakly tozbecause{M(νβ,z) :β∈D} does. Define a mappingUonCby

Ux=weak-lim

γ∈D M

νβγ,x

(A.6)

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In the case thatEis a Hilbert space, orEis a uniformly convex Banach space with a Fr´echet differentiable norm,itself is a nonexpansive retraction fromContoF(T); see

Baillon [1] and Bruck [8]. In general, this does not hold. We finally give an example. Example A.4(see [27,28]). Define a compact convex subsetCof (R2, ·

) by

C= x1,x2

: 0≤x21,−x2≤x1≤x2

. (A.7)

Define a nonexpansive mappingTonCby

Tx1,x2

=−x1,x1 (A.8)

for all (x1,x2)∈C. Then,F(T)= {(0, 0)}and

Tµx1,x2=0,x1 (A.9)

for (x1,x2)∈Cand a Banach limitµ. That is,is not a nonexpansive retraction fromC

ontoF(T).

References

[1] J. B. Baillon,Un th´eor`eme de type ergodique pour les contractions non lin´eaires dans un espace de Hilbert, C. R. Acad. Sci. Paris S´er. A-B280(1975), no. 22, A1511–A1514 (French). [2] ,Quelques aspects de la th´eorie des points fixes dans les espaces de Banach. II, S´eminaire

d’Analyse Fonctionnelle (1978–1979), `Ecole Polytechnique, Palaiseau, 1979, pp. 1–32, exp. no. 8.

[3] ,Quelques aspects de la th´eorie des points fixes dans les espaces de Banach. I, S´eminaire d’Analyse Fonctionnelle (1978–1979), `Ecole Polytechnique, Palaiseau, 1979, pp. 1–13, exp. no. 7.

[4] S. Banach,Th´eorie des op´erations lin´eaires, Monografie Mat., PWN, Warszawa, 1932.

[5] F. E. Browder,Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. U.S.A.53(1965), 1272–1276.

[6] ,Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A.54 (1965), 1041–1044.

[7] ,Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces, Arch. Rational Mech. Anal.24(1967), 82–90.

[8] R. E. Bruck,A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math.32(1979), no. 2-3, 107–116.

[9] M. Edelstein,A remark on a theorem of M. A. Krasnoselski, Amer. Math. Monthly73(1966), 509–510.

[10] M. Edelstein and R. C. O’Brien,Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc. (2)17(1978), no. 3, 547–554.

[11] D. G¨ohde,Zum Prinzip der kontraktiven Abbildung, Math. Nachr.30(1965), 251–258. [12] J.-P. Gossez and E. Lami Dozo,Some geometric properties related to the fixed point theory for

nonexpansive mappings, Pacific J. Math.40(1972), 565–573.

[13] B. Halpern,Fixed points of nonexpanding maps, Bull. Amer. Math. Soc.73(1967), 957–961. [14] N. Hirano, K. Kido, and W. Takahashi,Nonexpansive retractions and nonlinear ergodic theorems

in Banach spaces, Nonlinear Anal.12(1988), no. 11, 1269–1281.

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[16] J. L. Kelley,General Topology, D. Van Nostrand, New York, 1955.

[17] W. A. Kirk,A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly72(1965), 1004–1006.

[18] M. A. Krasnosel’ski˘ı,Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.)10(1955), no. 1(63), 123–127 (Russian).

[19] E. Lami Dozo,Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc.38(1973), 286–292.

[20] P. -K. Lin, K. K. Tan, and H. K. Xu,Demiclosedness principle and asymptotic behavior for asymp-totically nonexpansive mappings, Nonlinear Anal.24(1995), no. 6, 929–946.

[21] Z. Opial,Weak convergence of the sequence of successive approximations for nonexpansive map-pings, Bull. Amer. Math. Soc.73(1967), 591–597.

[22] S. Prus,Banach spaces with the uniform Opial property, Nonlinear Anal.18(1992), no. 8, 697– 704.

[23] S. Reich,Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl.67(1979), no. 2, 274–276.

[24] G. Rod´e,An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl.85(1982), no. 1, 172–178.

[25] B. Sims,A support map characterization of the Opial conditions, Miniconference on Linear Anal-ysis and Function Spaces (Canberra, 1984), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 9, The Australian National University, Canberra, 1985, pp. 259–264.

[26] T. Suzuki,Strong convergence theorem to common fixed points of two nonexpansive mappings in general Banach spaces, J. Nonlinear Convex Anal.3(2002), no. 3, 381–391.

[27] ,Common fixed points of two nonexpansive mappings in Banach spaces, Bull. Austral. Math. Soc.69(2004), no. 1, 1–18.

[28] T. Suzuki and W. Takahashi,Weak and strong convergence theorems for nonexpansive mappings in Banach spaces, Nonlinear Anal.47(2001), no. 4, 2805–2815.

[29] W. Takahashi,A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc.81(1981), no. 2, 253–256.

[30] R. Wittmann,Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel)58 (1992), no. 5, 486–491.

[31] D. van Dulst,Equivalent norms and the fixed point property for nonexpansive mappings, J. Lon-don Math. Soc. (2)25(1982), no. 1, 139–144.

Tomonari Suzuki: Department of Mathematics, Kyushu Institute of Technology, Sensui-cho, To-bata-ku, Kitakyushu 804-8550, Japan

References

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