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R E S E A R C H

Open Access

Proximal point algorithms for zero points of

nonlinear operators

Yuan Qing

1

and Sun Young Cho

2*

*Correspondence:

ooly61@yahoo.co.kr

2Department of Mathematics,

Gyeongsang National University, Jinju, 660-701, Korea

Full list of author information is available at the end of the article

Abstract

A proximal point algorithm with double computational errors for treating zero points of accretive operators is investigated. Strong convergence theorems of zero points are established in a Banach space.

MSC: 47H05; 47H09; 47H10; 65J15

Keywords: accretive operator; fixed point; nonexpansive mapping; proximal point algorithm; zero point

1 Introduction

In this paper, we are concerned with the problem of finding zero points of an operator

A:E→E; that is, findingxdomAsuch that Ax. The domaindomAofAis

de-fined by the set {xE:Ax= }. Many important problems have reformulations which

require finding zero points, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and optimization problems; see [–] and the references therein. One of the most popular techniques for solving the inclusion prob-lem goes back to the work of Browder []. One of the basic ideas in the case of a Hilbert spaceH is reducing the above inclusion problem to a fixed point problem of the opera-torRA:H→Hdefined byRA= (I+A)–, which is called the classical resolvent ofA. If Ahas some monotonicity conditions, the classical resolvent ofAis with full domain and firmly nonexpansive. Rockafellar introduced the algorithmxn+=RAxnand call it the

prox-imal point algorithm; for more detail, see [] and the references therein. Regularization methods recently have been investigated for treating zero points of monotone operators; for [–] and the references therein. Methods for finding zero points of monotone map-pings in the framework of Hilbert spaces are based on the good properties of the resolvent

RA, but these properties are not available in the framework of Banach spaces.

In this paper, we investigate a proximal point algorithm with double computational er-rors based on regularization ideas in the framework of Banach spaces. The organization of this paper is as follows. In Section , we provide some necessary preliminaries. In tion , strong convergence of the algorithm is obtained in a general Banach space. In Sec-tion , an applicaSec-tion is provided to support the main results.

2 Preliminaries

In what follows, we always assume thatEis Banach space with the dualE∗. Recall that a closed convex subsetCofEis said to havenormal structureif for each bounded closed

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convex subsetK ofCwhich contains at least two points, there exists an elementxofK

which is not a diametral point of K,i.e.,sup{xy:yK}<d(K), whered(K) is the diameter ofK. It is well known that a closed convex subset of uniformly convex Banach space has the normal structure and a compact convex subset of a Banach space has the normal structure; for more details, see [] and the references therein.

LetUE={xE:x= }.Eis said to besmoothor said to be have aGâteaux differ-entiable normif the limitlimtx+tytx exists for each x,yUE. E is said to have a uniformly Gâteaux differentiable normif for eachyUE, the limit is attained uniformly

for allxUE.Eis said to beuniformly smoothor said to have auniformly Fréchet differ-entiable normif the limit is attained uniformly forx,yUE. Let·,·denote the pairing

betweenEandE∗. The normalized duality mappingJ:E→Eis defined by

J(x) =fE∗:x,f=x=f

for allxE. In the sequel, we usejto denote the single-valued normalized duality map-ping. It is known that if the norm ofEis uniformly Gâteaux differentiable, then the duality mappingJis single-valued and uniformly norm to weak∗continuous on each bounded subset ofE.

LetCbe a nonempty closed convex subset ofE. LetT:CCbe a mapping. In this pa-per, we useF(T) to denote the set of fixed points ofT. Recall thatTis said to becontractive

if there exists a constantα∈(, ) such that

TxTyαxy, ∀x,yC.

For such a case, we also callTanα-contraction.Tis said to benonexpansiveif

TxTyxy, ∀x,yC.

LetDbe a nonempty subset ofC. LetQ:CD.Qis said to be acontractionifQ=Q;

sunnyif for eachxCandt∈(, ), we haveQ(tx+ ( –t)Qx) =Qx;sunny nonexpansive retractionifQis sunny, nonexpansive, and contraction.K is said to be anonexpansive retractofCif there exists a nonexpansive retraction fromContoD.

The following result, which was established in [], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

LetEbe a smooth Banach space andCbe a nonempty subset ofE. LetQ:ECbe a retraction andjbe the normalized duality mapping onE. Then the following are equiva-lent:

() Qis sunny and nonexpansive;

() QxQyxy,j(QxQy),x,yE;

() xQx,j(yQx) ≤,∀xE,yC.

LetIdenote the identity operator onE. An operatorAE×Ewith domainD(A) ={zE:Az=∅}and rangeR(A) ={Az:zD(A)}is said to beaccretiveif for eachxiD(A)

andyiAxi,i= , , there existsj(x–x)∈J(x–x) such thaty–y,j(x–x) ≥. An

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nonexpansive single-valued mappingJr:R(I+rA)→D(A) byJr= (I+rA)–for eachr> ,

which is called theresolventofA.

In order to prove our main results, we also need the following lemmas.

Lemma .[] Let E be a Banach space,and A an m-accretive operator.Forλ> ,μ> ,

and xE,we have

Jλx=

μ λx+

 –μ λ

Jλx

,

where Jλ= (I+λA)–and Jμ= (I+μA)–.

Lemma .[] Let{an}be a sequence of nonnegative numbers satisfying the condition an+≤( –tn)an+tnbn+cn,∀n≥,where{tn}is a number sequence in(, )such that limn→∞tn= and

n=tn=∞,{bn}is a number sequence such thatlim supn→∞bn≤, and{cn}is a positive number sequence such that

n=cn<∞.Thenlimn→∞an= .

Lemma .[] Let{xn}and{yn}be bounded sequences in a Banach space E,and{βn}be a sequence in(, )with

 <lim inf

n→∞ βn≤lim supn→∞ βn< .

Suppose that xn+= ( –βn)yn+βnxn,∀n≥and lim sup

n→∞

yn+–ynxn+–xn ≤.

Thenlimn→∞ynxn= .

Lemma .[] Let E a real reflexive Banach space with the uniformly Gâteaux differ-entiable norm and the normal structure,and C be a nonempty closed convex subset of E.

Let S:CC be a nonexpansive mapping with a fixed point,and f :CC be a fixed contraction with the coefficientα∈(, ).Let{xt}be a sequence generated by the follow-ing xt=tfxt+ ( –t)Sxt,where t∈(, ).Then{xt}converges strongly as t→to a fixed point xof S,which is the unique solution in F(S)to the following variational inequality

f(x∗–x∗),j(x∗–p) ≥,∀pF(S).

3 Main results

Theorem . Let E be a real reflexive Banach space with the uniformly Gâteaux differen-tiable norm and A be an m-accretive operators in E.Assume that C:=D(A)is convex and has the normal structure.Let f:CC be a fixedα-contraction.Let{αn},{βn},{γn},and

{δn}be real number sequences in(, )such thatαn+βn+γn+δn= .Let QCbe the sunny nonexpansive retraction from E onto C and{xn}be a sequence generated in the following manner:

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(a) limn→∞αn= and

n=αn=∞;

(b)  <lim infn→∞γn≤lim supn→∞γn< ;

(c) ∞n=en<∞andn=δn<∞;

(d) rnμfor eachn≥andlimn→∞|rnrn+|= .

Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequalityfx) –x¯,j(px¯) ≤,∀pA–().

Proof FixingpA–(), we find that

x–pαf(x) –p+βJr(x+e) –p+γx–p+δQC(g) –p

ααx–p+αf(p) –p+βx–p+βe

+γx–p+δg–p

≤ –α( –α) x–p+αf(p) –p+e+δg–p. (.)

Next, we prove that

xnpM+

n

i=

ei+ n–

i=

δigi, (.)

whereM=max{x–p,f(–p)–αp}<∞. In view of (.), we find that (.) holds forn= . We assume that the result holds for somem. Notice that

xm+–pαmf(xm) –p+βnJrm(xm+em+) –p+γmxmp

+δmQC(gm) –p

αmαxmp+αmf(p) –p+βmxmp+βmem+

+γmxmp+δmQC(gm) –p

≤ –αm( –α)xmp+αm( –α)

f(p) –p

 –α

+em++δmgmp

M+

m+

i=

ei+ m

i=

δigi.

This shows that (.) holds. In view of the restriction (c), we find that the sequence{xn}is

bounded. Putyn=Jrn(xn+en+) andzn= xn+–γnxn

–γn . Now, we computezn+–zn. Note that zn+–zn=

αn+

 –γn+

f(xn+) +

βn+

 –γn+

yn++

δn+

 –γn+

QC(gn+)

αn  –γn

f(xn) –

βn

 –γn yn

δn

 –γn QC(gn)

= αn+  –γn+

f(xn+) –yn+ +yn++

δn+

 –γn+

QC(gn+) –yn+

αn  –γn

f(xn) –ynyn

δn

 –γn

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This yields

zn+–zn

αn+

 –γn+

f(xn+) –yn++yn+–yn+

αn

 –γn

f(xn) –yn

+ δn+  –γn+

QC(gn+) –yn++

δn

 –γn

QC(gn) –yn. (.)

Next, we estimateyn+–yn. In view of Lemma ., we find that

ynyn+

rn rn+

(xn+en+) +

 – rn

rn+

Jrn+(xn+en+) – (xn++en+)

= rn

rn+

(xn+en+) – (xn++en+) +

rn+–rn rn+

Jrn+(xn+en+) – (xn++en+)

xnxn++en++en++

M

μ (rn+–rn), (.) whereMis an appropriate constant such that

M≥sup

n≥

Jr

n+(xn+en) – (xn++en+).

Substituting (.) into (.), we arrive at

zn+–znxnxn+

αn+

 –γn+

f(xn+) –yn++en++en++

M

μ (rn+–rn) + αn

 –γn

f(xn) –yn

+ δn+  –γn+

QC(gn+) –yn++

δn

 –γn

QC(gn) –yn.

In view of the restrictions (a), (b), (c), and (d), we find that

lim sup n→∞

zn+–znxnxn+ ≤.

It follows from Lemma . thatlimn→∞znxn= . It follows from the restriction (b)

that

lim

n→∞xn+–xn= . (.)

Notice that

xnJrn(xn+en+)

xnxn++xn+–Jrn(xn+en+)

xnxn++αnf(xn) –Jrn(xn+en+)+γnxnJrn(xn+en+)

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It follows that

( –γn)xnJrn(xn+en+)

xnxn++αnf(xn) –Jrn(xn+en+)+δnQC(gn) –Jrn(xn+en+).

In view of the restrictions (a), (b), and (c), we find from (.) that

lim

n→∞xnJrn(xn+en+)= . (.)

Notice that

xnJrnxnxnJrn(xn+en+)+Jrn(xn+en+) –Jrnxn

xnJrn(xn+en+)+en+.

Since∞n=en<∞, we see from (.) that

lim

n→∞xnJrnxn= .

Take a fixed numberrsuch that>r> . In view of Lemma ., we obtain

JrnxnJrxn= Jr

r rn

xn+

 – r

rn

Jrnxn

Jrxn

 – r

rn

(Jrnxnxn)

Jrnxnxn. (.)

Note that

xnJrxnxnJrnxn+JrnxnJrxn ≤xnJrnxn.

This combines with (.), yielding

lim

n→∞xnJrxn= . (.)

Next, we claim thatlim supn→∞fx) –x¯,j(xnx¯) ≤, wherex¯=limt→zt, andztsolves

the fixed point equationzt=tf(zt) + ( –t)Jrzt,∀t∈(, ), from which it follows that

ztxn=( –t)(Jrztxn) +t

f(zt) –xn .

For anyt∈(, ), we see that

ztxn= ( –t)

Jrztxn,j(ztxn)

+tf(zt) –xn,j(ztxn)

= ( –t)JrztJrxn,j(ztxn)

+Jrxnxn,j(ztxn)

+tf(zt) –zt,j(ztxn)

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≤( –t)ztxn+Jrxnxnztxn

+tf(zt) –zt,j(ztxn)

+tztxn

ztxn+Jrxnxnztxn+t

f(zt) –zt,j(ztxn)

.

It follows that

ztf(zt),j(ztxn)

≤

tJrxnxnztxn, ∀t∈(, ).

By virtue of (.), we find that

lim sup n→∞

ztf(zt),j(ztxn)

≤. (.)

Sincezt → ¯x, ast→ and the fact thatjis strong to weak∗ uniformly continuous on

bounded subsets ofE, we see that

fx) –x¯,j(xnx¯)

ztf(zt),j(ztxn)

fx) –x¯,j(xnx¯)

fx) –x¯,j(xnzt)

+fx) –x¯,j(xnzt)

ztf(zt),j(ztxn)

fx) –x¯,j(xnx¯) –j(xnzt)+fx) –x¯+ztf(zt),J(xnzt)

fx) –x¯j(xnx¯) –j(xnzt)

+fx) –x¯+ztf(zt)xnzt →, ast→.

Hence, for any> , there existsλ>  such that∀t∈(,λ) the following inequality holds:

fx) –x¯,j(xnx¯)

ztf(zt),j(ztxn)

+.

This implies that

lim sup n→∞

fx) –x¯,j(xnx¯)

≤lim sup n→∞

ztf(zt),j(ztxn)

+.

Sinceis arbitrary and (.), one finds thatlim supn→∞fx) –x¯,j(xnx¯) ≤. This implies

that

lim sup n→∞

fx) –x¯,j(xn+–x¯)

≤. (.)

Finally, we prove thatxn→ ¯xasn→ ∞. Note that

xn+–x¯≤αn

f(xn) –x¯,j(xn+–x¯)

+βnynx¯xn+–x¯

+γnxnx¯xn+–x¯+δnQC(gn) –x¯xn+–x¯

αn

f(xn) –x¯,j(xn+–x¯)

+βn

ynx¯+xn+–x¯

+γn

xnx¯+xn+–x¯ +

δn

QC(gn) –x¯

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Note thatynx¯ ≤ xnx¯+en+. It follows that

xn+–x¯≤αn

f(xn) –x¯,j(xn+–x¯)

+βnynx¯

+γnxnx¯+δnQC(gn) –x¯

≤αn

f(xn) –x¯,j(xn+–x¯)

+ ( –αn)xnx¯+νn, (.)

where νn=en+(en++ xnx¯) +δnQC(gn) –x¯. In view of the restriction (c),

we find that ∞n=νn<∞. Letρn=max{f(xn) –x¯,j(xn+–x¯), }. Next, we show that

limn→∞ρn= . Indeed, from (.), for any give> , there exists a positive integern

such that

f(xn) –x¯,j(xn+–x¯)

<,nn.

This implies that ≤ρn<,nn. Since>  is arbitrary, we see thatlimn→∞ρn= .

In view of (.), we find that

xn+–x¯≤( –αn)xnx¯+ αnρn+νn.

In view of Lemma ., we find the desired conclusion immediately.

If the mappingf maps any element inCinto a fixed elementuandδn= , then we have

the following result.

Corollary . Let E be a real reflexive Banach space with the uniformly Gâteaux differen-tiable norm and A be an m-accretive operators in E.Assume that C:=D(A)is convex and has the normal structure.Let{αn},{βn},and{γn}be real number sequences in(, )such thatαn+βn+γn= .Let{xn}be a sequence generated in the following manner:

x∈C, xn+=αnu+βnJrn(xn+en+) +γnxn, ∀n≥,

where u is a fixed element in C,{en}is a sequence in E,{gn}is a bounded sequence in E,{rn} is a positive real numbers sequence,and Jrn= (I+rnA)–.Assume that A–()is not empty and the above control sequences satisfy the following restrictions:

(a) limn→∞αn= and

n=αn=∞;

(b)  <lim infn→∞γn≤lim supn→∞γn< ;

(c) ∞n=en<∞;

(d) rnμfor eachn≥andlimn→∞|rnrn+|= .

Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequality:ux¯,j(px¯) ≤,∀pA–().

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4 Applications

In this section, we give an application of Theorem . in the framework of Hilbert spaces. For a proper lower semicontinuous convex functionw:H→(–∞,∞], the

subdifferen-tial mapping∂wofwis defined by

∂w(x) =x∗∈H:w(x) +yx,x∗≤w(y),∀yH, ∀xH.

Rockafellar [] proved that∂wis a maximal monotone operator. It is easy to verify that ∈∂w(v) if and only ifw(v) =minxHg(x).

Theorem . Let w:H→(–∞, +∞]be a proper convex lower semicontinuous function such that(∂w)–()is not empty.Let f :HH be aκ-contraction and let{x

n}be a sequence in H in the following process:x∈H and

⎧ ⎨ ⎩

yn=arg minzH{w(z) +

zxnen+

rn }, xn+=αnf(xn) +βnyn+γnxn, ∀n≥,

where{en}is a sequence in H,{gn}is a bounded sequence in H,and{rn}is a positive real numbers sequence.Assume that the above control sequences satisfy the restrictions(a), (b), (d),andn=en<∞.Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequality:fx) –x¯,j(px¯) ≤,∀p∈(∂w)–().

Proof Sincew:H→(–∞,∞] is a proper convex and lower semicontinuous function, we

see that subdifferential∂wofwis maximal monotone. We note that

yn=arg min zH

w(z) +zxnen+

rn

is equivalent to ∈∂w(yn) +rn(ynxnen+). It follows that

xn+en+∈yn+rn∂w(yn).

Puttingδn=  in Theorem ., we draw the desired conclusion from Theorem .

imme-diately.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this manuscript. Both authors read and approved the final manuscript.

Author details

1Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China.2Department of Mathematics,

Gyeongsang National University, Jinju, 660-701, Korea.

Acknowledgements

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10.1186/1687-1812-2014-42

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