R E S E A R C H
Open Access
Proximal point algorithms for zero points of
nonlinear operators
Yuan Qing
1and 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, findingx∈domAsuch that ∈Ax. The domaindomAofAis
de-fined by the set {x∈E: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
convex subsetK ofCwhich contains at least two points, there exists an elementxofK
which is not a diametral point of K,i.e.,sup{x–y:y∈K}<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={x∈E:x= }.Eis said to besmoothor said to be have aGâteaux differ-entiable normif the limitlimt→x+tyt–x exists for each x,y∈UE. E is said to have a uniformly Gâteaux differentiable normif for eachy∈UE, the limit is attained uniformly
for allx∈UE.Eis said to beuniformly smoothor said to have auniformly Fréchet differ-entiable normif the limit is attained uniformly forx,y∈UE. Let·,·denote the pairing
betweenEandE∗. The normalized duality mappingJ:E→E∗is defined by
J(x) =f ∈E∗:x,f=x=f
for allx∈E. 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:C→Cbe 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
Tx–Ty ≤αx–y, ∀x,y∈C.
For such a case, we also callTanα-contraction.Tis said to benonexpansiveif
Tx–Ty ≤ x–y, ∀x,y∈C.
LetDbe a nonempty subset ofC. LetQ:C→D.Qis said to be acontractionifQ=Q;
sunnyif for eachx∈Candt∈(, ), 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:E→Cbe a retraction andjbe the normalized duality mapping onE. Then the following are equiva-lent:
() Qis sunny and nonexpansive;
() Qx–Qy≤ x–y,j(Qx–Qy),∀x,y∈E;
() x–Qx,j(y–Qx) ≤,∀x∈E,y∈C.
LetIdenote the identity operator onE. An operatorA⊂E×Ewith domainD(A) ={z∈ E:Az=∅}and rangeR(A) ={Az:z∈D(A)}is said to beaccretiveif for eachxi∈D(A)
andyi∈Axi,i= , , there existsj(x–x)∈J(x–x) such thaty–y,j(x–x) ≥. An
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 x∈E,we have
Jλx=Jμ
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+–yn–xn+–xn ≤.
Thenlimn→∞yn–xn= .
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:C→C be a nonexpansive mapping with a fixed point,and f :C→C 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 x∗of S,which is the unique solution in F(S)to the following variational inequality
f(x∗–x∗),j(x∗–p) ≥,∀p∈F(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:C→C 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:
(a) limn→∞αn= and
∞
n=αn=∞;
(b) <lim infn→∞γn≤lim supn→∞γn< ;
(c) ∞n=en<∞and∞n=δn<∞;
(d) rn≥μfor eachn≥andlimn→∞|rn–rn+|= .
Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequalityf(¯x) –x¯,j(p–x¯) ≤,∀p∈A–().
Proof Fixingp∈A–(), we find that
x–p ≤αf(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
xn–p ≤M+
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+–p ≤αmf(xm) –p+βnJrm(xm+em+) –p+γmxm–p
+δmQC(gm) –p
≤αmαxm–p+αmf(p) –p+βmxm–p+βmem+
+γmxm–p+δmQC(gm) –p
≤ –αm( –α)xm–p+αm( –α)
f(p) –p
–α
+em++δmgm–p
≤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) –yn –yn–
δn
–γn
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
yn–yn+
≤ 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+)
≤ xn–xn++en++en++
M
μ (rn+–rn), (.) whereMis an appropriate constant such that
M≥sup
n≥
Jr
n+(xn+en) – (xn++en+).
Substituting (.) into (.), we arrive at
zn+–zn–xn–xn+
≤ 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+–zn–xn–xn+ ≤.
It follows from Lemma . thatlimn→∞zn–xn= . It follows from the restriction (b)
that
lim
n→∞xn+–xn= . (.)
Notice that
xn–Jrn(xn+en+)
≤ xn–xn++xn+–Jrn(xn+en+)
≤ xn–xn++αnf(xn) –Jrn(xn+en+)+γnxn–Jrn(xn+en+)
It follows that
( –γn)xn–Jrn(xn+en+)
≤ xn–xn++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→∞xn–Jrn(xn+en+)= . (.)
Notice that
xn–Jrnxn ≤xn–Jrn(xn+en+)+Jrn(xn+en+) –Jrnxn
≤xn–Jrn(xn+en+)+en+.
Since∞n=en<∞, we see from (.) that
lim
n→∞xn–Jrnxn= .
Take a fixed numberrsuch that>r> . In view of Lemma ., we obtain
Jrnxn–Jrxn= Jr
r rn
xn+
– r
rn
Jrnxn
–Jrxn
≤
– r
rn
(Jrnxn–xn)
≤ Jrnxn–xn. (.)
Note that
xn–Jrxn ≤ xn–Jrnxn+Jrnxn–Jrxn ≤xn–Jrnxn.
This combines with (.), yielding
lim
n→∞xn–Jrxn= . (.)
Next, we claim thatlim supn→∞f(¯x) –x¯,j(xn–x¯) ≤, wherex¯=limt→zt, andztsolves
the fixed point equationzt=tf(zt) + ( –t)Jrzt,∀t∈(, ), from which it follows that
zt–xn=( –t)(Jrzt–xn) +t
f(zt) –xn .
For anyt∈(, ), we see that
zt–xn= ( –t)
Jrzt–xn,j(zt–xn)
+tf(zt) –xn,j(zt–xn)
= ( –t)Jrzt–Jrxn,j(zt–xn)
+Jrxn–xn,j(zt–xn)
+tf(zt) –zt,j(zt–xn)
≤( –t)zt–xn+Jrxn–xnzt–xn
+tf(zt) –zt,j(zt–xn)
+tzt–xn
≤ zt–xn+Jrxn–xnzt–xn+t
f(zt) –zt,j(zt–xn)
.
It follows that
zt–f(zt),j(zt–xn)
≤
tJrxn–xnzt–xn, ∀t∈(, ).
By virtue of (.), we find that
lim sup n→∞
zt–f(zt),j(zt–xn)
≤. (.)
Sincezt → ¯x, ast→ and the fact thatjis strong to weak∗ uniformly continuous on
bounded subsets ofE, we see that
f(¯x) –x¯,j(xn–x¯)
–zt–f(zt),j(zt–xn)
≤f(¯x) –x¯,j(xn–x¯)
–f(¯x) –x¯,j(xn–zt)
+f(¯x) –x¯,j(xn–zt)
–zt–f(zt),j(zt–xn)
≤f(¯x) –x¯,j(xn–x¯) –j(xn–zt)+f(¯x) –x¯+zt–f(zt),J(xn–zt)
≤f(¯x) –x¯j(xn–x¯) –j(xn–zt)
+f(¯x) –x¯+zt–f(zt)xn–zt →, ast→.
Hence, for any> , there existsλ> such that∀t∈(,λ) the following inequality holds:
f(¯x) –x¯,j(xn–x¯)
≤zt–f(zt),j(zt–xn)
+.
This implies that
lim sup n→∞
f(¯x) –x¯,j(xn–x¯)
≤lim sup n→∞
zt–f(zt),j(zt–xn)
+.
Sinceis arbitrary and (.), one finds thatlim supn→∞f(¯x) –x¯,j(xn–x¯) ≤. This implies
that
lim sup n→∞
f(¯x) –x¯,j(xn+–x¯)
≤. (.)
Finally, we prove thatxn→ ¯xasn→ ∞. Note that
xn+–x¯≤αn
f(xn) –x¯,j(xn+–x¯)
+βnyn–x¯xn+–x¯
+γnxn–x¯xn+–x¯+δnQC(gn) –x¯xn+–x¯
≤αn
f(xn) –x¯,j(xn+–x¯)
+βn
yn–x¯+xn+–x¯
+γn
xn–x¯+xn+–x¯ +
δn
QC(gn) –x¯
Note thatyn–x¯ ≤ xn–x¯+en+. It follows that
xn+–x¯≤αn
f(xn) –x¯,j(xn+–x¯)
+βnyn–x¯
+γnxn–x¯+δnQC(gn) –x¯
≤αn
f(xn) –x¯,j(xn+–x¯)
+ ( –αn)xn–x¯+νn, (.)
where νn=en+(en++ xn–x¯) +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¯)
<, ∀n≥n.
This implies that ≤ρn<,∀n≥n. Since> is arbitrary, we see thatlimn→∞ρn= .
In view of (.), we find that
xn+–x¯≤( –αn)xn–x¯+ 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) rn≥μfor eachn≥andlimn→∞|rn–rn+|= .
Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequality:u–x¯,j(p–x¯) ≤,∀p∈A–().
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) +y–x,x∗≤w(y),∀y∈H, ∀x∈H.
Rockafellar [] proved that∂wis a maximal monotone operator. It is easy to verify that ∈∂w(v) if and only ifw(v) =minx∈Hg(x).
Theorem . Let w:H→(–∞, +∞]be a proper convex lower semicontinuous function such that(∂w)–()is not empty.Let f :H→H be aκ-contraction and let{x
n}be a sequence in H in the following process:x∈H and
⎧ ⎨ ⎩
yn=arg minz∈H{w(z) +
z–xn–en+
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),and∞n=en<∞.Then the sequence{xn}converges strongly tox¯,which is the unique solution to the following variational inequality:f(¯x) –x¯,j(p–x¯) ≤,∀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 z∈H
w(z) +z–xn–en+
rn
is equivalent to ∈∂w(yn) +rn(yn–xn–en+). 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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