Convergence of splitting algorithms for the sum of two accretive operators with applications

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R E S E A R C H

Open Access

Convergence of splitting algorithms for the

sum of two accretive operators with

applications

Xiaolong Qin

1

_{, Sun Young Cho}

2

_{and Lin Wang}

3*

*_{Correspondence:}

[email protected] 3_{College of Statistics and}

Mathematics, Yunnan University of Finance and Economics, Kunming, China

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

Abstract

We study a splitting algorithm for problems involving the sum of two accretive operators. We prove the strong convergence of the algorithm. Applications to variational inequality, ﬁxed point, equilibrium, and minimization problems are provided.

Keywords: accretive operator; ﬁxed point; nonexpansive mapping; resolvent; zero point

1 Introduction

Many important real world problems have reformulations which require finding zero points of some nonlinear operators, for instance, evolution equations, complementarity problems, mini-max problems, variational inequalities and optimization problems; see [– ] and the references therein. It is well known that minimizing a convex functionf can be reduced to finding zero points of the subdifferential mapping∂f. Forward-backward splitting algorithms were proposed by Lions and Mercier [], by Passty [], and, in a dual form for convex programming, by Han and Lou []. The algorithms, which pro-vide a range of approaches to solving large-scale optimization problems and variational inequalities, have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation, and this operator is decomposed as the sum of two nonlinear operators. This paper concerns a forward-backward splitting algorithm with computational errors designed to find zeros of the sum of two accretive operatorsAandB.

The paper is organized in the following way. In Section , we present the preliminaries that are needed in our work. In Section , we present a splitting algorithm for ﬁnding zeros of the sum of two accretive operatorsAandB. Convergence analysis of the algorithms is investigated. In Section , applications of our main results are provided.

2 Preliminaries

LetE be a real Banach space with the dual E∗. Given a continuous strictly increasing functionϕ:R+_→_R+_{, where}_R+_{denotes the set of nonnegative real numbers, such that}

ϕ() =  andlimr→∞ϕ(r) =∞, we associate with it a (possibly multivalued) generalized

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duality mapJϕ:E→E

∗

, deﬁned asJϕ(x) ={x∗∈E∗:x∗(x) =xϕ(x),x∗=ϕ(x)},

∀x∈E. In this paper, we use the generalized duality map associated with the gauge func-tionϕ(t) =tq–_for_q_{> . Let}_ρ

E: [,∞)→[,∞) be the modulus of smoothness of E byρE(t) =sup{x+y–_x–y– ,x∈UE,y ≤t}. A Banach spaceE is said to be uniformly smooth if ρE(t)

t → ast→. Letq> .Eis said to beq-uniformly smooth if there exists a ﬁxed constantc>  such thatρE(t)≤ctq. The modulus of convexity ofEis the function

δE() : (, ]→[, ] deﬁned byδE() =inf{ –x_+t:x=y= ,x–y ≥}. Recall that Eis said to be uniformly convex ifδE() >  for any∈(, ]. LetEbe a smooth Banach space, and letCbe a nonempty subset ofE. LetProj_C:E→Cbe a retraction andJbe the normalized duality mapping onE. Then the following are equivalent []: ()Proj_Cis sunny and nonexpansive; ()x–Proj_Cx,J(y–Proj_Cx) ≤,∀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 be accretive iff, fort>  and x,y∈D(A),x–y ≤ x–y+t(u–v),∀u∈Ax,v∈Ay. It follows from Kato [] thatA is accretive iff, forx,y∈D(A), there existsjq(x–x) such thatu–v,jq(x–y) ≥. An accretive operatorAis said to bem-accretive iffR(I+rA) =Efor allr> . In this paper, we useA–_{() to denote the set of zeros of}_{A. For an accretive operator}_{A, we can define a}

nonexpansive single-valued mappingJr:R(I+rA)→D(A) byJr= (I+rA)–for eachr> , which is called the resolvent ofA. Recall that a single-valued operatorA:C→Eis said to beα-inverse strongly accretive if there exists a constantα>  and somejq(x–y)∈Jq(x–y) such thatAx–Ay,jq(x–y) ≥αAx–Ayq_,_∀_x,_y_∈_C.

LetT:C→Cbe a mapping. Recall thatT is said to beκ-contractive iﬀ there exists a constantκ∈(, ) such thatTx–Ty ≤κx,y,∀x,y∈C.Tis said to be nonexpansive iﬀ

κ= .Tis said to beκ-strictly pseudocontractive iﬀ there exists a constantκ∈(, ) such that

Tx–Ty,jq(x–y)≤ x–yq–κ(x–Tx) – (y–Ty)q, ∀x,y∈C

for somejq(x–y)∈Jq(x–y).T is said to be pseudocontractive iﬀTx–Ty,jq(x–y) ≤

x–yq_,_∀_x,_y_∈_C_{for some}_j

q(x–y)∈Jq(x–y).

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

Lemma .[] Let E be a real q-uniformly smooth Banach space.Then the following inequality holds:x+yq_≤_xq_+q_y,_J_q(x_+y)_and_x_+yq_≤_xq_+q_y,_J_q(x)₊_Kq_yq_,

∀x,y∈E,where Kqis some ﬁxed positive constant.

Lemma .[] Let E be a real Banach space,and let C be a nonempty closed and convex subset of E.Let A:C→E be a single-valued operator,and let B:E→E_{be an m-accretive} operator.Then F(Ja(I–aA)) = (A+B)–_(),_{where Ja}_(I_–_aA)_{is the resolvent of B for a}_{> .}

Lemma .[] Let E be a real Banach space,and A be 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 real numbers such that an+≤( –

tn)an+bn+cn,where{cn}is a sequence of nonnegative real numbers,{tn} ⊂(, )and{bn} is a number sequence.Assume that∞_n_=tn=∞,lim sup_n_→∞bn

tn ≤,and

∞

n=cn<∞.

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Lemma .[] Let q> .Then the following inequality holds:ab≤a_qq +q–_q b

q

q–_,_for ar-bitrary positive real numbers a and b.

Lemma .[] Let C be a nonempty closed convex subset of a real uniformly smooth Banach space E.Let f :C→C be a contractive mapping,and let T:C→C be a non-expansive mapping.For each t∈(, ),let xt be the unique solution of the equation x= tf(x) + ( –t)Tx.Then{xt}converges strongly to a ﬁxed pointx¯=QF(T)f(x).¯

3 Main results

Theorem . Let E be a real q-uniformly smooth Banach space with the constant Kq.Let B:E→E _{be an m-accretive operator such that D(B)}_{is convex.}_{Let A}_:_D(B)_→_{E be an}

α-inverse strongly accretive operator.Assume that (A+B)–_()₌_∅_._{Let f} _:_D(B)_→_D(B)

be a ﬁxedκ-contraction.Let{rn},{αn},{βn},and{γn}be positive real number sequences, where{αn},{βn},and{γn}are in(, ).Let{xn}be a sequence generated in the following iterative process:

x∈C, xn+=αnf(xn) +βnJrn(xn–rnAxn+en) +γnfn, ∀n≥,

where Jrn = (I+rnB)–, {en}is a sequence in E,and{fn}is a bounded sequence in D(B).

Assume that the sequences{αn},{βn},{γn},{en},and{rn}satisfy the following restrictions:

() αn+βn+γn= ;

() limn→∞αn= ,

_∞

n=αn=∞;

() ∞_n_=|βn–βn–|<∞;

() lim inf_n_→∞rn> ,rn≤(q_Kα_q) 

q–_,∞

n=|rn–rn–|<∞;

() ∞_n_=en<∞,∞_n_=γn<∞.

Then the sequence{xn}converges strongly to x=Proj₍_A₊_B₎–_()f(x),whereProj₍_A₊_B₎–_()is the unique sunny nonexpansive retraction of C onto(A+B)–_().

Proof First, we show that{xn}is bounded. In view of Lemma ., we ﬁnd that

(I–rnA)x– (I–rnA)yq

≤ x–yq–qrnAx–Ay,Jq(x–y)+Kqr_nqAx–Ayq

≤ x–yq–qrnαAx–Ayq+KqrqnAx–Ayq

=x–yq–αq–Kqrq_n–rnAx–Ayq.

From restriction (), we ﬁnd thatI–rnAis nonexpansive. Fixingp∈(A+B)–(), we ﬁnd from Lemma . thatp=Jrn(xn–rnAxn)p. It follows from restriction () that

xn+–p

≤αnf(xn) –p+βnJrn(xn–rnAxn+en) –p+γnfn–p

≤αnκxn–p+αnf(p) –p

+βn(xn–rnAxn+en) – (I–rnA)p+γnfn–p

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≤max xn–p,f(p) –p  –κ

+en+γnfn–p

≤max xn––p,

f(p) –p  –κ

+en–+en

+γn–fn––p+γnfn–p ..

.

≤max x–p,

f(p) –p  –κ

+ n

i=

ei+ n

i=

γiM

≤max x–p,

f(p) –p  –κ

+

∞

i=

ei+

∞

i=

γiM<∞,

whereM=sup_n_≥_{fn–p}. This shows that{xn}is bounded. Setyn= (I–rnA)xn+en. It follows that

yn––yn ≤ xn–xn–+|rn–rn–|Axn–+en+en–.

Using Lemma ., we ﬁnd that

Jrn–yn––Jrnyn

=Jrn–

rn–

rn yn+

 –rn– rn

Jrnyn

–Jrn–yn–

≤rn–

rn (yn–yn–) +

 –rn– rn

(Jrnyn–yn–)

≤ yn–yn–+|

rn–rn–|

rn

Jrnyn–yn

≤ xn–xn–+|rn–rn–|

Axn–+

Jrnyn–yn

rn

+en+en–. (.)

On the other hand, we have

xn+–xn ≤αnκxn–xn–+|αn–αn–|f(xn–)

+γnfn–fn–+|γn–γn–|fn–

+βnJrn–yn––Jrnyn+|βn–βn–|Jrn–yn–. (.) Using (.) and (.), we ﬁnd

xn+–xn ≤

 –αn( –κ)

xn–xn–+|αn–αn–|f(xn–)

+γnfn–fn–+|γn–γn–|fn–

+|rn–rn–|

Axn–+

Jrnyn–yn rn

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Using restrictions (), (), and (), we obtain from Lemma . that

lim

n→∞xn+–xn= . (.)

On the other hand, we have

Jrnyn–xn ≤

xn+–xn

βn

+αn

βn

xn–f(xn)+γn

βn

xn–fn.

Using restrictions () and (), we obtain from (.) that

lim

n→∞Jrnyn–xn= . (.)

Since

Jrn(xn–rnAxn) –xn≤Jrn(xn–rnAxn) –Jrnyn+Jrnyn–xn

≤ en+Jrnyn–xn,

we ﬁnd from (.) and restriction () that

lim

n→∞Jrn(xn–rnAxn) –xn= . (.)

Without loss of generality, let us assume that there exists a real numberrsuch thatrn≥ r> . SinceBis accretive, we have

xn–Jr(I–rA)xn

r –

xn–Jrn(I–rnA)xn

rn

,Jq

Jr(I–rA)xn–Jrn(I–rnA)xn

≥.

It follows that

q

≤rn–r

rn

xn–Jrn(I–rnA)xn,Jq

≤xn–Jrn(I–rnA)xnJr(I–rA)xn–Jrn(I–rnA)xn

q–

.

This implies from (.) that

lim

n→∞Jr(xn–rAxn) –xn= . (.)

SinceJr(I–rA) is nonexpansive andf is contractive, we ﬁnd that the mappingtf + ( – t)Jr(I–rA) is contractive. Letxtbe the unique ﬁxed point of the mappingtf+ ( –t)Jr(I– rA), that is,xt=tf(xt) + ( –t)Jr(I–rA)xt,∀t∈(, ). Settingx=limt→xt, we havex= Proj₍_A₊_B₎–_()f(x), whereProj₍_A₊_B₎–_()is the unique sunny nonexpansive retraction fromC onto (A+B)–().

Next, we show that

lim sup n→∞

f(x) –x,Jq(xn–x)

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Since

xt–xnq≤tf(xt) –xn,Jq(xt–xn)

+ ( –t)Jr(I–rA)xt–xn,Jq(xt–xn)

≤tf(xt) –xt,Jq(xt–xn)+txt–xn,Jq(xt–xn)

+ ( –t)Jr(I–rA)xt–Jr(I–rA)xn,Jq(xt–xn)

+ ( –t)Jr(I–rA)xn–xn,Jq(xt–xn)

≤tf(xt) –xt,Jq(xt–xn)+xt–xnq

+Jr(I–rA)xn–xnxt–xnq–, we have

f(xt) –xt,Jq(xn–xt)≤

tJr(I–rA)xn–xnxt–xn q–_.

Fixtand letn→ ∞. It follows from (.) that

lim sup n→∞

f(xt) –xt,Jq(xn–xt)≤. (.)

Since the duality map Jq is single-valued and strong-weak∗ uniformly continuous on bounded sets of a Banach spaceEwith a uniformly Gâteaux diﬀerentiable norm, one has

f(xt) –xt,Jq(xn–xt)–f(x) –x,Jq(xn–x)

=f(x) –x,Jq(xn–x) –Jq(xn–xt)

+f(x) –x–f(xt) –xt,Jq(xn–xt)

≤f(x) –x,Jq(xn–x) –Jq(xn–xt)

+f(x) –x–f(xt) –xtxn–xtq–. Hence,∀> ,∃δ>  such thatt∈(,δ), one has

f(x) –x,Jq(xn–x)≤f(xt) –xt,Jq(xn–xt)+.

Using (.), we see that

lim sup n→∞

f(x) –x,Jq(xn–x)¯

≤. (.)

Using Lemma ., one has

xn+–xq≤αn

f(xn) –f(x),Jq(xn+–x)

+αn

f(x) –x,Jq(xn+–x)¯

+βnJrn(xn–rnAxn+en) –xxn+–x

q–

+γnfn–xxn+–xq–

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+αn

f(x) –x,Jq(xn+–x)

+enxn+–xq–

+γnfn–xxn+–xq–

≤ –αn( –κ)

qxn–x

q₊q– 

q xn+–x q

+αn

+enxn+–xq–

+γnfn–xxn+–xq–.

It follows that

xn+–xq≤

 –αn( –κ)xn–xq

+qαn

+qenxn+–xq–

+qγnfn–xxn+–xq–.

Using restrictions () and (), we see from (.) that{xn}converges strongly tox. This

completes the proof.

Remark . The framework of the space in Theorem . can be applicable toLp, where p> .

Corollary . Let E be a real q-uniformly smooth Banach space with the constant Kq.Let B:E→E_{be an m-accretive operator such that D(B)}_{is convex.}_{Assume that B}–_()₌_∅_.

Let f :D(B)→D(B)be a ﬁxedκ-contraction.Let{rn}and{αn}be positive real number sequences,where{αn}is in(, ).Let{xn}be a sequence generated in the following iterative process:

x∈C, xn+=αnf(xn) + ( –αn)Jrnxn, ∀n≥,

where Jrn= (I+rnB)–.Assume that the sequences{αn}and{rn}satisfy the following

restric-tions:

() limn→∞αn= ,

_∞

n=αn=∞;

() ∞_n_=|αn–αn–|<∞;

() lim infn→∞rn> ,∞n=|rn–rn–|<∞.

Then the sequence{xn}converges strongly to x=ProjB–_()f(x),whereProj_B–_()is the unique sunny nonexpansive retraction of C onto B–_().

Corollary . Let E be a real q-uniformly smooth Banach space with the constant Kq,and let C be a nonempty closed and convex subset of E.Let f :C→C be a ﬁxedκ-contraction. Let T:C→C be anα-strictly pseudocontractive mapping with a nonempty ﬁxed point set. Let{rn},{αn},{βn},and{γn}be positive number sequences,where{αn},{βn},and{γn}in (, ).Let{xn}be a sequence generated in the following process:

x∈C, xn+=αnf(xn) +βn( –rn)xn+rnβnTxn+γnfn, ∀n≥,

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() αn+βn+γn= ;

() limn→∞αn= ,

∞

n=αn=∞;

() ∞_n_=|βn–βn–|<∞;

() lim infn→∞rn> ,rn≤(q_Kα_q) 

q–_,∞

n=|rn–rn–|<∞;

() ∞_n_=γn<∞.

Then the sequence{xn}converges strongly to x=Proj_F₍_T₎f(x),whereProj_F₍_T₎is the unique sunny nonexpansive retraction of C onto F(T).

Proof PuttingA=I–T, we ﬁnd thatAisα-inverse strongly accretive andF(T) =A–_().

Notice that

xn+=αnf(xn) +βn( –rn)xn+rnβnTxn+γnfn

=αnf(xn) +βn

( –rn)xn+rnTxn+γnfn

=αnf(xn) +βn

xn–rn(I–T)xn

+γnfn

=αnf(xn) +βn(xn–rnAxn) +γnfn.

Using Theorem ., we ﬁnd the desired conclusion immediately.

4 Applications

In this section, we give some applications of our main results in the framework of Hilbert spaces.

From now on, we always assume thatCis a nonempty closed and convex subset of a real Hilbert spaceHandPCstands for the metric projection fromHontoC. LetA:C→Hbe a monotone operator. Recall that the classical variational inequality is to ﬁndx∈Csuch that

Ax,y–x ≥, ∀y∈C. (.)

The solution set of the variational inequality is denoted byVI(C,A). LetiCbe a function deﬁned by

iC(x) =

⎧ ⎨ ⎩

, x∈C,

∞, x∈/C.

It is easy to see thatiCis a proper lower and semicontinuous convex function onH, and the subdiﬀerential∂iC ofiCis maximal monotone. Deﬁne the resolventJr:= (I+r∂iC)–

of the subdiﬀerential operator∂iC. Lettingx=Jry, we ﬁnd that

y∈x+r∂iCx ⇐⇒ y∈x+rNCx

⇐⇒ y–x,v–x ≤, ∀v∈C

⇐⇒ x=PCy,

whereNCx:={e∈H:e,v–x,∀v∈C}.

PuttingB=∂iCin Theorem ., we ﬁnd thatJrn=PC. Hence, the following result can be

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Theorem . Let f :C→C be a ﬁxed κ-contraction.Let A:C→E be an α-inverse strongly monotone operator withVI(C,A)=∅.Let{rn}be a positive number sequence.Let

{αn},{βn},and{γn}be real number sequences in(, ).Let{xn}be a sequence generated in the following iterative process:

x∈C, xn+=αnf(xn) +βnPC(xn–rnAxn+en) +γnfn, ∀n≥,

where{en}is a sequence in H and{fn}is a bounded sequence in C.Assume that the se-quences{αn},{βn},{γn},{en},and{rn}satisfy the following restrictions:

() αn+βn+γn= ;

() lim_n_→∞αn= ,

∞

n=αn=∞;

() ∞_n_=|βn–βn–|<∞;

() lim inf_n_→∞rn> ,rn≤α,

∞

n=|rn–rn–|<∞;

() ∞_n_=en<∞,∞_n_=γn<∞.

Then the sequence{xn}converges strongly to x=PVI(C,A)f(x),where PVI(C,A) is the unique

metric projection from C ontoVI(C,A).

Next, we consider the problem of ﬁnding a solution of a Ky Fan inequality [], which is known as an equilibrium problem in the terminology of Blum and Oettli; see [] and the references therein.

LetFbe a bifunction ofC×CintoR, whereRdenotes the set of real numbers. Recall the following equilibrium problem:

Findx∈Csuch thatF(x,y)≥, ∀y∈C. (.)

The solution set of the problem is denoted byEP(F) in this section.

To study the equilibrium problem (.), we may assume thatF satisﬁes the following restrictions:

(A) F(x,x) = for allx∈C;

(A) Fis monotone,i.e.,F(x,y) +F(y,x)≤for allx,y∈C; (A) for eachx,y,z∈C,lim sup_t_↓_F(tz+ ( –t)x,y)≤F(x,y); (A) for eachx∈C,y→F(x,y)is convex and lower semicontinuous.

The following lemma can be found in [].

Lemma . Let F:C×C→Rbe a bifunction satisfying(A)-(A).Then,for any r>  and x∈H,there exists z∈C such that F(z,y) +_ry–z,z–x ≥,∀y∈C.Further,deﬁne

Trx= z∈C:F(z,y) +

ry–z,z–x ≥,∀y∈C

(.)

for all r> and x∈H.Then()Tris single-valued and ﬁrmly nonexpansive; ()F(Tr) = EP(F)is closed and convex.

Lemma . Let F be a bifunction from C×C toRwhich satisﬁes(A)-(A),and let AFbe a multivalued mapping of H into itself deﬁned by

AFx=

⎧ ⎨ ⎩

{z∈H:F(x,y)≥ y–x,z,∀y∈C}, x∈C,

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Then AF is a maximal monotone operator with the domain D(AF)⊂C,EP(F) =A–

F (), and Trx= (I+rAF)–_x,_∀_x_∈_H,_r_{> ,}_{where Tr}_{is deﬁned as in}_(.).

Based on Lemmas . and ., we ﬁnd from Theorem . the following immediately.

Theorem . Let F:C×C→Rbe a bifunction satisfying(A)-(A)such thatEP(F)is not empty.Let f :C→C be a ﬁxedκ-contraction.Let{rn}be a positive number sequence. Let{αn},{βn},and{γn}be real number sequences in(, ).Let{xn}be a sequence generated in the following iterative process:

x∈C, xn+=αnf(xn) +βnTrn(xn–rnAxn+en) +γnfn, ∀n≥,

where Jrn= (I+rnAF)–,{en}is a sequence in H and{fn}is a bounded sequence in C.Assume

that the sequences{αn},{βn},{γn},{en},and{rn}satisfy the following restrictions:

() αn+βn+γn= ;

() limn→∞αn= ,

∞

n=αn=∞;

() ∞_n_=|βn–βn–|<∞;

() lim infn→∞rn> ,rn≤α,∞n=|rn–rn–|<∞;

() ∞_n_=en<∞,

∞

n=γn<∞.

Then the sequence{xn}converges strongly to some point inEP(F).

For any matrixD∈Rm×n, we denote its transpose byDT_{and its operator norm by}_D₌ maxx∈Rn:x=Dx.

Consider the inclusion problem [] ∈A(x,x,y) +B(x,x,y), where A(x,x,y) =

(DT_y,_ET_{y, –Dx}

–Ex),B(x,x,y) =Tx×Tx× {b}andTandTare maximal

mono-tone mappings onRnandRn, respectively, andD∈Rm×n,E∈Rm×n,b∈Rm. Then,A andBare maximal monotone andAis Lipschitz continuous onRm+n+nwith the constant

η=DT₊_ET₊_D₊_E_.

The special case whereT=∂f,T=∂fyields the following convex program:

⎧ ⎨ ⎩

minimizef(x) +f(x)

subject toDx+Ex=b,

where f andf are closed proper convex functions on, respectively,Rn andRn. The

special case wheren=n,D= –E=Iandb=  yields the inclusion ∈Tx+Tx.

Finally, we consider ﬁnding minimizers of proper lower semicontinuous convex func-tions.

For a proper lower semicontinuous convex functionh:H→(–∞,∞], the subdiﬀeren-tial mapping∂hofhis deﬁned by

∂h(x) =x∗∈H:h(x) +y–x,x∗≤h(y),∀y∈H, ∀x∈H.

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Theorem . Let f :C→C be a ﬁxedκ-contraction.Let h:H→(–∞, +∞]be a proper convex lower semicontinuous function such that(∂h)–_()_{is not empty.}_Let_{_r

n}be a positive number sequence.Let{αn},{βn},and{γn}be real number sequences in(, ).Let{xn}be a sequence generated in the following iterative process:

x∈C, xn+=αnf(xn) +βnarg min z∈H h(z) +

z–xn–en rn

+γnfn, ∀n≥,

where{en}is a sequence in H and{fn}is a bounded sequence in C.Assume that the se-quences{αn},{βn},{γn},{en},and{rn}satisfy the following restrictions:

() αn+βn+γn= ;

() limn→∞αn= ,

∞

n=αn=∞;

() ∞_n_=|βn–βn–|<∞;

() lim infn→∞rn> ,rn≤α,∞n=|rn–rn–|<∞;

() ∞_n_=en<∞,

∞

n=γn<∞.

Then the sequence{xn}converges strongly to some minimizer of h.

Proof Sinceh:H→(–∞,∞] is a proper convex and lower semicontinuous function, we see that the subdiﬀerential∂hofhis maximal monotone. PuttingA=  andyn=Jrn(xn+

en), we see that

yn=arg min z∈H h(z) +

z–xn–en

rn

is equivalent to

∈∂h(yn) +  rn

(yn–xn–en).

It follows that

xn+en∈yn+rn∂h(yn).

By using Theorem ., we draw the desired conclusion immediately.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department of Mathematics, Hangzhou Normal University, Hangzhou, China.}2_{Department of Mathematics,}

Gyeongsang National University, Jinju, Korea.3_{College of Statistics and Mathematics, Yunnan University of Finance and} Economics, Kunming, China.

Acknowledgements

The authors are grateful to the anonymous referees for useful suggestions, which improved the contents of the article.

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