ACCRETIVE OPERATORS WITH COMPACT DOMAINS IN GENERAL BANACH SPACES

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ACCRETIVE OPERATORS WITH COMPACT DOMAINS IN GENERAL BANACH SPACES

HIROMICHI MIYAKE AND WATARU TAKAHASHI Received 2 July 2004

We prove strong convergence theorems of Mann’s type and Halpern’s type for resolvents of accretive operators with compact domains and apply these results to find fixed points of nonexpansive mappings in Banach spaces.

1. Introduction

Let Ebe a real Banach space, letC be a closed convex subset ofE, let T be a nonexpansive mapping ofCinto itself, that is,Tx−T y ≤ x−yfor eachx,y∈C, and let A⊂E×Ebe an accretive operator. Forr >0, we denote byJr the resolvent ofA, that is, Jr=(I+rA)⁻¹. The problem of finding a solutionu∈Esuch that 0∈Auhas been inves- tigated by many authors; for example, see [3,4,7,16,26]. We know the proximal point algorithm based on a notion of resolvents of accretive operators. This algorithm generates a sequence{xn}inEsuch thatx1=x∈Eand

xn+1=Jrnxn forn=1, 2,..., (1.1) where{rn}is a sequence in (0,∞). Rockafellar [18] studied the weak convergence of the sequence generated by (1.1) in a Hilbert space; see also the original works of Martinet [12,13].

On the other hand, Mann [11] introduced the following iterative scheme for finding a fixed point of a nonexpansive mappingTin a Banach space:x1=x∈Cand

xn+1=αnxn+1−αn

Txn forn=1, 2,..., (1.2) where{αn}is a sequence in [0, 1], and studied the weak convergence of the sequence generated by (1.2). Reich [17] also studied the following iterative scheme for finding a fixed point of a nonexpansive mappingT:x1=x∈Cand

xn+1=αnx+1−αn

Txn forn=1, 2,..., (1.3)

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where{αn}is a sequence in [0, 1]; see the original work of Halpern [6]. Wittmann [27]

showed that the sequence generated by (1.3) in a Hilbert space converges strongly to the point ofF(T), the set of fixed points ofT, which is the nearest toxif{αn}satisfies limn→∞αn=0,^∞_n₌1αn= ∞, and^∞_n₌1|αn+1−αn|<∞. Since then, many authors have studied the iterative schemes of Mann’s type and Halpern’s type for nonexpansive mappings and families of various mappings; for example, see [1,2,19,20,21,22,23,24,14, 15].

Motivated by two iterative schemes of Mann’s type and Halpern’s type, Kamimura and Takahashi [8,9] introduced the following iterative schemes for finding zero points of m-accretive operators in a uniformly convex Banach space:x1=x∈Eand

xn+1=αnx+1−αn

Jrnxn forn=1, 2,..., xn+1=αnxn+1−αn

Jrnxn forn=1, 2,..., (1.4) where {αn} is a sequence in [0, 1] and {rn} is a sequence in (0,∞). They studied the strong and weak convergence of the sequences generated by (1.4). Such iterative schemes for accretive operators with compact domains in a strictly convex Banach space have also been studied by Kohsaka and Takahashi [10].

In this paper, we first deal with the strong convergence of resolvents of accretive operators defined in compact sets of smooth Banach spaces. Next, we prove strong convergence theorems of Mann’s type and Halpern’s type for resolvents of accretive operators with compact domains. We apply these results to find fixed points of nonexpansive mappings with compact domains in Banach spaces.

2. Preliminaries

Through this paper, we denote byNthe set of positive integers. We also denote byEa real Banach space with topological dualE^∗and byJthe duality mapping ofE, that is, a multivalued mappingJofEintoE^∗such that for eachx∈E,

J(x)=

f ∈E^∗:f(x)= x²= f²

. (2.1)

A Banach spaceEis said to besmoothif the duality mappingJofEis single-valued. We know that ifEis smooth, thenJis norm to weak-star continuous. LetS(E) be the unit sphere ofE, that is,S(E)= {x∈E:x =1}. Then, the norm ofEis said to beuniformly Gˆateaux diﬀerentiableif for eachy∈S(E), the limit

limλ→0

x+λy − x

λ (2.2)

exists uniformly inx∈S(E). We know that ifEhas a uniformly Gâteaux differentiable norm, thenEis smooth. We also know that ifEhas a uniformly Gâteaux differentiable norm, then the duality mappingJ ofEis norm to weak-star uniformly continuous on each bounded subsets ofE. For more details, see [25].

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LetDbe a subset ofCand letPbe a retraction ofContoD, that is,Px=xfor each x∈D. ThenPis said to besunny[16] if for eachx∈Candt≥0 withPx+t(x−Px)∈C,

PPx+t(x−Px)=Px. (2.3)

A subsetD ofCis said to be asunny nonexpansive retract of Cif there exists a sunny nonexpansive retractionPofContoD. We know that ifEis smooth andPis a retraction ofContoD, thenPis sunny and nonexpansive if and only if for eachx∈Candz∈D,

x−Px,J(z−Px)≤0. (2.4)

For more details, see [25].

Let A⊂E×E be a multivalued operator. We denote by D(A) and A⁻¹0 the eﬀec- tive domain ofA, that is,D(A)= {x∈E:Ax = ∅}and the set of zeros ofA, that is, A⁻¹0= {x∈E: 0∈Ax}, respectively. An operatorA is said to beaccretiveif for each (x1,y1), (x2,y2)∈A, there exists j∈J(x1−x2) such that

y1−y2,j≥0. (2.5)

Such an operator was first studied by Kato and Browder, independently. We know that for each (x1,y1), (x2,y2)∈Aandr >0,

x1−x2≤x1−x2+ry1−y2. (2.6) LetCbe a closed convex subset ofEsuch thatC⊂ _r>0R(I+rA), whereI denotes the identity mapping ofEandR(I+rA) is the range ofI+rA, that is,R(I+rA)=

{(I+ rA)x:x∈D(A)}. Then, for eachr >0, we define a mappingJronCbyJr=(I+rA)⁻¹. Such a mappingJris called theresolventofA. We know that the resolventJrofAis single- valued. For eachr >0, we define the Yosida approximationAr ofAbyAr=r⁻¹(I−Jr).

We know that for eachx∈C, (Jrx,Arx)∈A. We also know that for eachx∈C∩D(A), Arx ≤inf{y:y∈Ax}. An accretive operator A is said to bem-accretiveifR(I+ rA)=Efor eachr >0 andAis also said to bemaximalif the graph ofAis not properly contained in the graph of any other accretive operator. We know from [5, page 181] that ifAis anm-accretive operator, thenAis maximal.

We need the following theorem [14], which is crucial in the proofs of main theorems.

Theorem2.1. LetCbe a compact convex subset of a smooth Banach spaceE, letSbe a com- mutative semigroup with identity, let᏿= {T(s) :s∈S}be a nonexpansive semigroup onC, and letF(᏿)be the set of common fixed points of᏿. ThenF(᏿)is a sunny nonexpansive retract ofC, and a sunny nonexpansive retraction ofContoF(᏿)is unique. In particular, if Tis a nonexpansive mapping ofCinto itself, thenF(T)is a sunny nonexpansive retract ofC and a sunny nonexpansive retraction ofContoF(T)is unique.

3. Main results

LetEbe a Banach space and letA⊂E×Ebe an accretive operator. In this section, we study the existence of a sunny nonexpansive retraction ontoA⁻¹0 and the convergence of resolvents ofA.

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Theorem3.1. LetCbe a compact convex subset of a smooth Banach spaceEand letA⊂ E×Ebe an accretive operator such thatD(A)⊂C⊂ _r>0R(I+rA). Then the setA⁻¹0is a nonempty sunny nonexpansive retract ofCand a sunny nonexpansive retractionPofConto A⁻¹0is unique. In this case, for eachx∈C,limt→∞Jtx=Px.

Proof. SinceC⊂R(I+rA) for eachr >0, the resolventJrofAis well defined onC. We know thatJr is a nonexpansive mapping ofCinto itself andA⁻¹0=F(Jr), whereF(Jr) denotes the set of fixed points ofJr. Then, byTheorem 2.1,A⁻¹0 is a sunny nonexpansive retract ofCand a sunny nonexpansive retractionPofContoA⁻¹0 is unique.

Next, we will show that for eachx∈C, lim_t_→∞Jtxexists and lim_t_→∞Jtx=Px. Letx∈C be fixed. SinceCis compact, there exist a sequence{tn}of positive real numbers andz∈C such that lim_n_→∞tn= ∞and{Jtnx}converges strongly toz. Then,zis contained inA⁻¹0.

Indeed, we have, for eachr >0,

JrJtnx−Jtnx=Jr−IJtnx=rArJtnx

≤rinfy:y∈AJtnx

≤rAtnx=rx−Jtnx tn

≤ r tn

x+Jtnx

(3.1)

and hence lim_n_→∞JrJtnx−Jtnx =0. Then, from

Jrz−z≤Jrz−JrJtnx+JrJtnx−Jtnx+Jtnx−z

≤2Jtnx−z+JrJtnx−Jtnx, (3.2) we haveJrz=z. This implies thatz∈F(Jr)=A⁻¹0.

Let{Jtnx}and{Jsnx}be subsequences of {Jtx}such that{Jtnx} and{Jsnx}converge strongly toyandzastn→ ∞andsn→ ∞, respectively. Fromz∈A⁻¹0, we have

0≤

Atnx−0,JJtnx−z

= 1 tn

I−Jtn

x,JJtnx−z (3.3)

and henceJtnx−x,J(Jtnx−z) ≤0. Thus, we have y−x,J(y−z) ≤0. Similarily, we havez−x,J(z−y) ≤0 and hencey=z∈A⁻¹0.

Letybe the limit lim_t_→∞Jtx. By a similar argument, we have

y−x,J(y−Px)≤0. (3.4)

Thus, sincePis a sunny nonexpansive retraction ofContoA⁻¹0, we have y−Px²=

y−Px,J(y−Px)

=

y−x,J(y−Px)+x−Px,J(y−Px)

≤

y−x,J(y−Px)≤0.

(3.5)

This implies thaty=Px. This completes the proof.

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Next, we prove a strong convergence theorem of Mann’s type for resolvents of anm- accretive operator in a Banach space.

Theorem3.2. LetCbe a compact convex subset of a smooth Banach spaceEand letA⊂ E×Ebe anm-accretive operator such thatD(A)⊂C. Letx1=x∈Cand define an iterative sequence{xn}by

xn+1=αnxn+1−αn

Jrnxn forn=1, 2,..., (3.6) where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfylimn→∞αn=0andlimn→∞rn= ∞. Then{xn} converges strongly to an element ofA⁻¹0.

Proof. Letu∈A⁻¹0. Since for eachn∈N,

xn+1−u≤αnxn−u+1−αnJrnxn−u

≤αnxn−u+1−αnxn−u

=xn−u,

(3.7)

the limit limn→∞xn−uexists.

Let{xnk}be a subsequence of{xn}such that{xnk}converges strongly tov∈C. Since for eachn∈N,

xn+1−Jrnxn=αnxn−Jrnxn (3.8) and lim_n_→∞αn=0, we have

nlim→∞xn+1−Jrnxn=0. (3.9) Then, Jr_nk−1xnk−1 converges strongly to v∈C. Since A is accretive, we have, for each (y,z)∈Aandn∈N,

z−Arnxn,Jy−Jrnxn

≥0. (3.10)

We also have

nlim→∞Arnxn=lim

n→∞r_n⁻¹xn−Jrnxn=0. (3.11) Thus, we have, for each (y,z)∈A,

z,J(y−v)≥0. (3.12)

We know that anm-accretive operatorAis maximal. For the sake of completeness, we will give the proof. LetB⊂E×Ebe an accretive operator such thatA⊂Band let (x,u)∈B. SinceAism-accretive, there existsy∈D(A) such thatx+u∈(I+A)y. Choosev∈Ay such thatx+u=y+v. SinceBis accretive, we have

x−y ≤ x−y+u−v =0 (3.13)

and hencex=y∈D(A) andu=v∈R(A). This implies that (x,u)∈A. So,Ais maximal.

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From (3.12) and the maximality ofA, we havev∈A⁻¹0. Thus, we have

nlim→∞xn−v=lim

k→∞

xnk−v=0. (3.14)

This completes the proof.

The following is a strong convergence theorem of Halpern’s type for resolvents of an accretive operator in a Banach space.

Theorem 3.3. Let C be a compact convex subset of a Banach spaceEwith a uniformly Gˆateaux diﬀerentiable norm and letA⊂E×Ebe an accretive operator such thatD(A)⊂ C⊂ _r>0R(I+rA). Letx1=x∈Cand define an iterative sequence{xn}by

xn+1=αnx+1−αn

Jrnxn forn=1, 2,..., (3.15) where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfy

∞ n=1

αn= ∞, lim

n→∞αn=0, lim

n→∞rn= ∞. (3.16)

Then{xn}converges strongly toPx, wherePdenotes a unique sunny nonexpansive retraction ofContoA⁻¹0.

Proof. We know fromTheorem 3.1that there exists a unique sunny nonexpansive retrac- tionPofContoA⁻¹0. Forx1=x∈C, we define{xn}by (3.15). First, we will show that

lim sup

n→∞

x−Px,JJrnxn−Px≤0. (3.17)

Let>0 and letzt=Jtxfor eacht >0. SinceAis accretive andt⁻¹(x−zt)∈Azt, we have Arnxn−t⁻¹x−zt

,JJrnxn−zt

≥0 (3.18)

and hence,

x−zt,JJrnxn−zt

≤tArnxn,JJrnxn−zt

. (3.19)

Then, from lim_n_→∞Arnxn=0, we have lim sup

n→∞

x−zt,JJrnxn−zt

≤0 (3.20)

for eacht >0. FromTheorem 3.1, we have lim_n_→∞zt=Px. Since the norm ofEis uniformly Gˆateaux diﬀerentiable, there existst0>0 such that for eacht > t0andn∈N,

Px−zt,JJrnxn−zt≤ 2, x−Px,JJrnxn−zt

−JJrnxn−Px≤

2. (3.21)

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Thus, we have, for eacht > t0andn∈N, x−zt,JJrnxn−zt

−

x−Px,JJrnxn−Px

≤x−zt,JJrnxn−zt

−

x−Px,JJrnxn−zt +x−Px,JJrnxn−zt

−

x−Px,JJrnxn−Px

=Px−zt,JJrnxn−zt +x−Px,JJrnxn−zt

−JJrnxn−Px

≤.

(3.22)

This implies that lim sup

n→∞

x−Px,JJrnxn−Px≤lim sup

n→∞

x−zt,JJrnxn−zt

+≤. (3.23) Since>0 is arbitrary, we have

lim sup

n→∞

x−Px,JJrnxn−Px≤0. (3.24)

Fromxn+1−Jrnxn=αn(x−Jrnxn) and limn→∞αn=0, we havexn+1−Jrnxn→0. Since the norm ofEis uniformly Gˆateaux diﬀerentiable, we also have

lim sup

n→∞

x−Px,Jxn+1−Px≤0. (3.25)

From (3.15) and [25, page 99], we have, for eachn∈N, 1−αn2Jrnxn−Px²−xn+1−Px²≥ −2αn

x−Px,Jxn+1−Px. (3.26) Hence, we have

xn+1−Px²≤

1−αnJrnxn−Px²+ 2αn

x−Px,Jxn+1−Px. (3.27) Let>0. Then, there existsm∈Nsuch that

x−Px,Jxn−Px≤

2 (3.28)

for eachn≥m. We have, for eachn≥m, xn+1−Px²≤

1−αnxn−Px²+ 1−

1−αn

≤

1−αn

1−αn−1xn−1−Px²+ 1−

1−αn−1

+

1− 1−αn

≤ 1−αn

1−αn−1xn−1−Px² +

1− 1−αn

1−αn−1

≤ⁿ

k=m

1−αkxm−Px²+

1−ⁿ

k=m

1−αk .

(3.29)

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Thus, we have lim sup

n→∞

xn−Px²≤ ∞ k=m

1−αkxm−Px²+

1− ∞ k=m

1−αk

. (3.30)

From^∞_n₌₁αn= ∞, we have^∞_n₌₁(1−αn)=0. So, we have lim sup

n→∞

xn−Px²≤. (3.31)

Since>0 is arbitrary, we have lim_n_→∞xn−Px²=0. This completes the proof.

4. Applications

Using convergence theorems inSection 3, we prove two convergence theorems for finding a fixed point of a nonexpansive mapping in a Banach space.

Theorem4.1. LetCbe a compact convex subset of a smooth Banach spaceEand letTbe a nonexpansive mapping ofCinto itself. Letx1=x∈Cand define an iterative sequence{xn} by

xn= 1

1 +rnx+ rn

1 +rnTxn forn=1, 2,..., (4.1) where{rn} ⊂(0,∞)satisfieslimn→∞rn= ∞. Then{xn}converges strongly toPx, whereP denotes a unique sunny nonexpansive retraction ofContoF(T).

Proof. We define a mappingAofCintoEbyA=I−T. Forr >0, we denote byJr the resolvent ofA. Then,Ais an accretive operator which satisfiesD(A)=C⊂ _r>0R(I+rA).

From (4.1), we have, for eachn∈N,

xn+rn(I−T)xn=x (4.2)

and hencexn=Jrnx. It follows fromTheorem 3.1that{xn}converges strongly toPx. This

completes the proof.

As in the proof ofTheorem 4.1, fromTheorem 3.3, we obtain the following convergence theorem for finding a fixed point of a nonexpansive mapping.

Theorem 4.2. Let C be a compact convex subset of a Banach spaceEwith a uniformly Gˆateaux diﬀerentiable norm and letTbe a nonexpansive mapping ofCinto itself. Letx1= x∈Cand define an iterative sequence{xn}by

un= 1

1 +rnxn+ rn

1 +rnTun, xn+1=αnx+1−αn

un,

(4.3)

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where{αn} ⊂[0, 1]and{rn} ⊂(0,∞)satisfy ∞

n=1

αn= ∞, _nlim_→∞αn=0, _nlim_→∞rn= ∞. (4.4) Then{xn}converges strongly toPx, wherePdenotes a unique sunny nonexpansive retraction ofContoF(T).

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Hiromichi Miyake: Department of Mathematical and Computing Sciences, Graduate School of Information Science and Engineering, Tokyo Institute of Technology, Okayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan

E-mail address:[email protected]

Wataru Takahashi: Department of Mathematical and Computing Sciences, Graduate School of Information Science and Engineering, Tokyo Institute of Technology, Okayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan

E-mail address:[email protected]

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