R E S E A R C H
Open Access
An algorithm with general errors for the zero
point of monotone mappings in Banach
spaces
Yan Tang
*, Zhiqing Bao and Daojun Wen
Dedicated to Professor Shih-sen Chang on the occasion of his 80th birthday
*Correspondence:
[email protected] College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China
Abstract
In this paper, we introduce a proximal point iterative algorithm with general errors for monotone mappings in Banach spaces. We prove that the proposed algorithm converges strongly to a proximal point for monotone mappings. Our theorems in this paper improve and unify most of the results that have been proposed for this important class of nonlinear mappings.
MSC: 47H09; 47H10; 47L25
Keywords: monotone mappings; fixed point; viscosity approximation; resolvent mappings
1 Introduction
LetEbe a real Banach space with dualE∗. A normalized duality mappingJ:E→E∗is
defined by
Jx=f∗∈E∗:x,f∗=x=f∗,
where·,·denotes the generalized duality pairing betweenEandE∗. It is well known that Eis smooth if and only ifJ is single-valued and ifEis uniformly smooth thenJis uni-formly continuous on bounded subsets ofE. Moreover, ifEis reflexive and strictly convex Banach space with a strictly convex dual thenJ–is single-valued, one-to-one, surjective, uniformly continuous on bounded subsets and it is the duality mapping fromE∗intoE; hereJJ–=I
E∗andJJ–=IE.D(A) denotes the domain ofA.
A mappingA:D(A)⊂E→E∗is said to be monotone if for eachx,y∈D(A), the follow-ing inequality holds:
x–y,Ax–Ay ≥.
The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. The mappingAis said to be maximal monotone if it is not properly contained in any other monotone operator. In the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal
monotone mappings. A variety of problems, for example, convex optimization, linear pro-gramming, monotone inclusions, and elliptic differential equations can be formulated as finding a zero of maximal monotone operators. The proximal point algorithm is recog-nized as a powerful and successful algorithm in finding a solution of maximal monotone operators. In Hilbert spaces, many authors have studied monotone mappings by the vis-cosity approximation methods and obtained a series of good results, see [–] and the references therein.
LetCbe closed convex subset of Banach spaceE. A mappingA:C→Eis called accre-tive if there existsj(x–y)∈J(x–y) such that
j(x–y),Ax–Ay≥.
The mappingAis calledm-accretive if it is accretive andR(I+rA), the range of (I+rA), isEfor allr> ; and an accretive mappingAis said to satisfy the range condition if
D(A)⊆C⊆
r>
R(I+rA), (.)
for some nonempty, closed, and convex subsetCof a real Banach spaceE(see [, , ]). Denote the zero set ofAby
A–() =z∈D(A) : ∈Az.
The accretive mappings and monotone mappings have different natures in Banach spaces, these being more general than Hilbert spaces. For solving the original problem of finding a solution to the inclusion ∈Az, Rockafellar [] introduced the following algorithm:
zk+ek∈zk++ckAzk+, (.)
where{ek}is a sequence of errors. Rockafellar obtained the weak convergence of the
algo-rithm (.).
WhenAis maximal monotone mapping in Hilbert spaces, Xu [] proposed the follow-ing regularization for the proximal algorithm:
zk+=JcAk
λku+ ( –λk)zk+ek ,
whereJckA= (I+ckA)–is the resolvent ofA, converge strongly to a point inA–().
Recently Yao and Shahzad [] proved that sequences generated from the method of resolvent given by
zk+=γkzk+δkJcAk(zk) +λkek,
whereJA
ck= (I+ckA)–is the resolvent ofA, converge strongly to a point inA–().
On the other hand, with regard to a finite family ofm-accretive mappings, Zegeye and Shahzad [] proved that under appropriate conditions in Hilbert spaces, an iterative pro-cess of Halpern type defined by
whereSr=aI+aJr+aJr+· · ·+aNJrNwithJri= (I+rAi)–forai∈(, ),{i= , , , . . . ,N}
andNi=ai= , for{Ai,i= , , . . .N}accretive mappings, converge strongly to a point in
N
i=A–i (). Very recently, Kimura and Takahashi [] defined the mapping in Banach
space:
yi=J∗
αiJxi+ ( –αi)JSixi ,
where{Si}are sequences of nonexpansive mappings ofCinto itself. Then{Jyi}converges
weakly and{Jxi–JSixi}converges strongly to ifϕ(x,yi)≤ϕ(x,xi) for alli∈N.
Motivated and inspired by the above results, our concern now is the following: Is it pos-sible to construct a new sequence with general errors in Banach spaces which converges strongly to a zero of monotone operators? LetCbe a nonempty, closed, and convex subset of a uniformly smooth strictly convex real Banach spaceE. LetA:C→E∗be a continuous monotone mapping satisfying the range condition (.) withA–()=∅. It is our purpose in this paper to introduce an iterative scheme (two-step iterative scheme) which converges strongly to a zero of a monotone operators in Banach spaces. Our theorems presented in this paper improve and extend the corresponding results of Yao and Shahzad [], and Zegeye and Shahzad [] and some other results in this direction.
2 Preliminaries
LetCbe a nonempty, closed, and convex subset of a uniformly smooth strictly convex real Banach spaceEwith dualE∗. A monotone mappingAis said to satisfy the range condition if we have
D(A)⊆C⊆
r>
J–R(J+rA) (.)
for some nonempty, closed, and convex subsetCof a smooth, strictly convex, and reflexive Banach spacesE (see [, , ]). In the sequel, the resolvent of a monotone mapping A:C→E∗ shall be denoted bySA
r = (J+rA)–Jforr> . It is well known that the fixed
points of the operatorSAr are the zeros of the monotone mappingA, denoteF:=A–(). We recall that a Banach spaceEhas theKadee-Kleeproperty if for any sequence{xn} ⊂E
andx∈Ewithxnxandxn → x,xn→xasn→ ∞. We note that every uniformly
convex Banach space enjoys theKadee-Kleeproperty.
Lemma .[] Let C be a nonempty,closed,and convex subset of a uniformly smooth
strictly convex real Banach space E with dual E∗.A⊂E×E∗is a monotone mapping
sat-isfying(.).Let SA
r be the resolvent of A for r⊂(,∞).If{xn}is a bounded sequence of C
such that SArxnz,then z∈A–().
Lemma . Forλ,μ> ,the identity
SAλx=SAμ
μ λJx+
–μ λ
JSλAx
Proof By the definition of the operatorSA
r, the identity holds.
LetEbe a smooth Banach space with dualE∗. Let the Lyapunov functionϕ:E×E→R, introduced by Alber [], be defined by
ϕ(y,x) =y– y,Jx+x ∀x,y∈E.
Obviously, by the definition of the mappingJ,ϕ(·,·)≥.
Lemma .[] Let E be a smooth and strictly convex Banach space,and C be a nonempty,
closed,and convex subset of E.Let A⊂E×E∗ be a monotone mapping satisfying(.),
A–()be nonempty and SrAbe the resolvent of A for some r> .Then,for each r> ,we have
ϕp,SArx +ϕSArx,x ≤ϕ(p,x) ∀p∈A–(),x∈C.
LetEbe a reflexive, strictly convex, and smooth Banach space, and letCbe a nonempty, closed, and convex subset ofE. The generalized projection mapping, introduced by Alber [], is a mappingC:E→Cthat assigns an arbitrary pointx∈Eto the minimizer,x¯=
Cx, wherex¯is the solution to the minimization problem
ϕ(x,x) =¯ minϕ(y,x),y∈C.
Lemma .[] Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly convex real reflexive Banach space E.Let x∈E,then,∀y∈C,
ϕ(y,Cx) +ϕ(Cx,x)≤ϕ(y,x).
Lemma .[] Let C be a nonempty closed and convex subset of a real smooth Banach
space E.A mappingC:E→C is a generalized projection.Let x∈E,then x=Cx if and
only if
z–x,Jx–Jx ≤, ∀z∈C.
We make use of the functionV:E×E∗→Rdefined byV(x,x∗) =ϕ(x,J–x∗). That is V(x,x∗) =x– x,x∗+x∗for allx∈Eandx∗∈E∗.
Lemma .[] Let E be a reflexive,strictly convex,and smooth real Banach space with dual E∗.Then
Vx,x∗ + J–x∗–x,y∗≤Vx,x∗+y∗ ,
for all x∈E and x∗,y∗∈E∗.
Lemma .[] Let{an}be a sequence of nonnegative real numbers satisfying the following
relation:
where{θn}is a sequence in(, )and{σn}is a real sequence such that
(i) ∞n=θn=∞;
(ii) lim supn→∞σnθn ≤or∞n=σn<∞.
Thenlimn→∞an= .
Lemma . [] Let E be a uniformly convex, and smooth real Banach space with dual E∗,and let{xn}and{yn}be two sequences of E.If either{xn}or{yn}is bounded and
ϕ(xn,yn)→as n→ ∞,then xn–yn→as n→ ∞.
Lemma .[] Let E be a uniformly convex and smooth real Banach space and BR()
be a closed ball of E. Then there exists a continuous strictly increasing convex function
g: [,∞)→[,∞)with g() = such that
αx+αx+· · ·+αNxN≤ N
i=
αixi–αiαjg
xi–xj ,
forαi∈(, )such that
N
i=αi= and xi∈BR() :={x∈E:x ≤R}for some R> .
3 Main results
In this section, we introduce our algorithm and state our main result.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Banach space E which also enjoys the Kadee-Klee property.Let A:C→E∗be
a continuous monotone mapping satisfying(.),and u∈C be a constant.Assume that
F:=A–()=∅,for x,u∈C arbitrarily,let the sequence{x
n}be generated iteratively by
⎧ ⎨ ⎩
yn=C(J–(λnJxn+ ( –λn)JSAcxn)),
xn+=J–(αnJu+βnJxn+γnJScAyn+εnJen),
(.)
where c⊂(,∞),{en} ∈E is an error, λn∈[, ],{αn}, {βn}, {γn}, {εn} are sequences of
nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) lim supn→∞λn= ;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequence(.)converges strongly tox¯=Fu.
Proof From the Lemma . and Kamimuraet al.[], we see thatA–() is closed and
convex. ThusFuis well defined. First we prove that{xn}is bounded. Takep∈F:=A–(),
then from (.) and Lemmas ., ., and the property ofϕ, we have
ϕ(p,yn) =ϕ
p,C
J–λnJxn+ ( –λn)JSAcxn
≤ϕp,J–λnJxn+ ( –λn)JSAcxn
=p– p, ( –λn)JSAcxn+λnJxn
+( –λn)JScAxn+λnJxn
≤ p– ( –λn)
p,JSAcxn
– λnp,Jxn
+ ( –λn)JScAxn
≤( –λn)ϕ
p,ScAxn +λnϕ(p,xn)
≤( –λn)ϕ(p,xn) +λnϕ(p,xn) =ϕ(p,xn). (.)
Forn≥, by using the definition ofϕand Lemma . we have from (.)
ϕ(p,xn+) =ϕ
p,J–αnJu+βnJxn+γnJSAcyn+εnJen
=p– p,αnJu+βnJxn+γnJSAcyn+εnJen
+αnJu+βnJxn+γnJSAcyn+εnJen
≤ p– αnp,Ju– βnp,Jxn– γn
p,JSAcyn
– εnp,Jen+αnu
+βnxn+γnSAcyn
+εnen
=αnϕ(p,u) +βnϕ(p,xn) +γnϕ
p,ScAyn +εnϕ(p,en)
≤αnϕ(p,u) + (βn+γn)ϕ(p,xn) +εnϕ(p,en). (.)
Sincelim supn→∞en= , assume thatsupϕ(p,en)≤Mis a real nonnegative constant,
then by induction, we have
ϕ(p,xn+)≤max
ϕ(p,u),ϕ(p,x),M, ∀n≥,
which implies that{xn}is bounded. Therefore, we see that{ScAxn}, {ScAyn}, and{yn}are
bounded. Next we show thatϕ(xn,xn+)→.
Because the sequence{xn}is bounded, by the reflexivity ofE, there exists a subsequence
{xnj}of{xn}andp∈Csuch thatxnj→p∈Cweakly. Denotex∗=Cu, thenx∗∈Cis
the unique element that satisfiesinfx∈Cϕ(x,u) =ϕ(x∗,u). By using the weakly lower
semi-continuity of the norm onE, we get
ϕx∗,u ≤ϕ(p,u)≤lim inf
j→∞ϕ(xnj,u) ≤lim sup
j→∞ϕ(xnj,u)≤xinf∈Cϕ(x,u) =ϕ
x∗,u , (.)
which implies that
lim
j→∞ϕ(xnj,u) =ϕ
x∗,u =ϕ(p,u) =inf
x∈Cϕ(x,u). (.)
Thus, from (.) and Lemma ., we have
z–x∗,Jx∗–Ju≥, ∀z∈C, (.)
z–p,Jp–Ju ≥, ∀z∈C. (.)
Puttingz:=pin (.) andz:=x∗in (.), we get
p–x∗,Jx∗–Ju≥, ∀z∈C, (.)
Adding (.) and (.) we havex∗–p,Jp–Jx∗ ≥,i.e.x∗–p,Jx∗–Jp ≤. By using the fact thatJis monotone we getx∗–p,Jx∗–Jp= . Since strict convexity ofEimplies thatJis strictly monotone, we havep=x∗. Consequently,xnj→x∗weakly.
Furthermore, by the definition ofϕand (.), we have
lim
n→∞
xn– xn,Ju+u =x∗–
x∗,Ju+u,
which shows thatlimn→∞xn=x∗. In view of theKadee-Kleeproperty ofE, we have
xn–x∗ →. Thus,ϕ(xn,xn+)→.
Now we show that x∗ ∈ A–(). By using the definition of ϕ and Lemma . and Lemma ., we get
ϕ(p,xn+) =ϕ
p,J–αnJu+βnJxn+γnJSAcyn+εnJen
≤ p– p,αnJu+βnJxn+γnJSAcyn+εnJen
+αnJu+βnJxn+γnJSAcyn+εnJen
≤ p– αnp,Ju– βnp,Jxn– γn
p,JSAcyn
– εnp,Jen+αnu
+βnxn+γnSAcyn
+εnen–βnγngJxn–JSAcyn
=αnϕ(p,u) + ( –αn–εn)ϕ(p,xn) +εnϕ(p,en) –βnγngJxn–JScAyn .
Becausexn→x∗,ϕ(p,xn) is convergent, and noticing the conditions (i), (ii) and (iii), we get
gJxn–JScAyn →. (.)
The property of the functiongimplies that
Jxn–JSAcyn→, n→ ∞. (.)
Thus, sinceJ–is uniformly continuous on bounded sets, we obtain
xn–SAcyn→, n→ ∞. (.)
From Lemma ., the property ofϕ, and the condition (ii) we have
ϕ(xn,yn) =ϕ
xn,C
J–λnJxn+ ( –λn)JScAxn
≤ϕxn,J–
λnJxn+ ( –λn)JSAcxn
≤( –λn)ϕ
xn,ScAxn +λnϕ(xn,xn)→. (.)
Using Lemma ., we getxn–yn→, furthermoreyn→x∗, and from (.) we getyn–
SA
cyn→ and from Lemma ., we obtainx∗∈F=A–().
Now takex¯=Fu, thenx¯∈F, by the Lemma . and Lemma ., then we have
ϕ(x,¯ u) =V(x,¯ Ju)
=ϕ(x,¯ x) + ¯ u–x,¯ Ju–Jx¯
= u–x,¯ Ju–Jx¯ ≤. (.)
Sinceϕ(·,·)≥, so we haveϕ(x,¯ u) = . Furthermore, from (.) we have
ϕ(x,¯ xn+)≤(βn+γn)ϕ(x,¯ xn) +αnϕ(x,¯ u) +εnϕ(x,¯ en)
≤( –αn–εn)ϕ(x,¯ xn) + (αn+εn)
ϕ(x,¯ u) +ϕ(x,¯ en) . (.)
Notice the conditions (i) and (iii), from (.), by the Lemma . we haveϕ(x,¯ xn+)→
asn→ ∞. Consequently, from Lemma . we havexn→ ¯x. In fact, we getx∗=x. The¯
proof is complete.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Banach space E which also enjoys the Kadee-Klee property.Let A:C→E∗be
a continuous monotone mapping satisfying(.),and u∈C be a constant.Assume that
F:=A–()=∅,let x
nbe a sequence generated by x∈C
xn+=J–
αnJu+βnJxn+γnJSAcxn+εnJen , (.)
where{en} ∈E is an error,{αn},{βn},{γn},{εn}are sequences of nonnegative real numbers
in[, ]and
(i) αn+βn+γn+εn= ,n≥;
(ii) limn→∞αn= ,
∞
n=αn=∞;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequence converges strongly toFu.
Proof Puttingλn= in Theorem ., we obtain the result.
We note the method of the proof of Theorem . provides a convergence theorem for a finite family of continuous monotone mappings. In fact, we have the following theorem.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Banach space E which also enjoys the Kadee-Klee property.Let Ai:C→E∗for
i= , , . . . ,N be continuous monotone mappings satisfying(.),and u∈C be a constant. Assume that F:=Ni=A–
i ()=∅,for x,u∈C arbitrarily,let the sequence{xn}be
gener-ated iteratively by
⎧ ⎨ ⎩
yn=C(J–(λnJxn+ ( –λn)
N
i=μiJSAcixn)),
xn+=J–(αnJu+βnJxn+γnNi=σiJScAiyn+εnJen),
(.)
where c⊂(,∞),{en} ∈E is an error,λn∈[, ],λn∈[, ],{αn},{βn},{γn},{εn}are
se-quences of nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥,Ni=σi= ,Ni=μi= ;μi≥,σi≥;
(ii) lim supn→∞λn= ;limn→∞αn= ,
∞
n=αn=∞;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
If in Theorem . and Theorem . we putu≡, we have the following corollaries.
Corollary . Let C be a nonempty, closed, and convex subset of a uniformly smooth
strictly convex real Banach space E which also enjoys the Kadee-Klee property.Let A:C→
E∗be a continuous monotone mapping satisfying(.),and u∈C be a constant.Assume
that F:=A–()=∅,for x∈C arbitrarily,let the sequence{xn}be generated iteratively by
⎧ ⎨ ⎩
yn=C(J–(λnJxn+ ( –λn)JSAcxn)),
xn+=J–(βnJxn+γnJSAcyn+εnJen),
(.)
where c⊂(,∞),{en} ∈E is an errorλn∈[, ],{βn},{γn},{εn}be sequences of nonnegative
real numbers in[, ],and
(i) βn+γn+εn= ,n≥;
(ii) lim supn→∞λn= ;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequence converges strongly tox¯=F().
Corollary . Let C be a nonempty, closed, and convex subset of a uniformly smooth
strictly convex real Banach space E which also enjoys the Kadee-Klee property.Let A:C→
E∗be a continuous monotone mapping satisfying(.).Assume that F:=A–()=∅,let x
n
be a sequence generated by
xn+=J–
βnJxn+γnJScAxn+εnJen , (.)
x∈C,where c> and{en} ∈E is an error,{βn},{γn},{εn}are sequences of nonnegative
real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥;
(ii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequence converges strongly toF().
IfEis a Hilbert space, thenEis a uniformly convex and smooth real Banach space. In this case,J=I, andϕ(x,y) =x–y, and
C=PC, the projection mapping fromEonC. It
is well known thatPCandScAare nonexpansive. Furthermore, (.) reduces to (.). Thus
the following theorems hold.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Hilbert space E which also enjoys the Kadee-Klee property.Let A:C→E∗
be a continuous monotone mapping satisfying(.),and f :C→C be a contraction with
a contraction coefficientρ∈(, ).Assume that F:=A–()=∅,let the sequence{x
n}be
generated iteratively by
⎧ ⎨ ⎩
yn=PC(λnxn+ ( –λn)SAcxn),
xn+=αnf(xn) +βnxn+γnScAyn+εnen,
(.)
where c> and{en} ∈H is an error,SAc = (I+cA)–,and the sequencesλn∈[, ],{αn},
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) <lim infn→∞λn<lim supn→∞λn< ;
(iii) lim supn→∞εn= ,andlim supn→∞en= ,
then the sequence converges strongly tox¯=PFf(x).¯
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Hilbert space E which also enjoys the Kadee-Klee property.Let A:C→E∗be a
continuous monotone mapping satisfying(.),and f :C→C be a contraction with a
con-traction coefficientρ∈(, ).Assume that F:=A–()=∅,let x
nbe a sequence generated
by x∈C
xn+=αnf(xn) +βnxn+γnSAcxn+εnen, (.)
where c> and{en} ∈H is an error,ScA= (I+cA)–,and the sequences{αn},{βn},{γn},{εn}
are nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) lim supn→∞εn= ,andlim supn→∞en= ,
then the sequence converges strongly tox¯=PFf(x).¯
If in Theorem . and Theorem . we putf:≡uconstant, we have the following corol-laries.
Corollary . Let C be a nonempty, closed, and convex subset of a uniformly smooth
strictly convex real Hilbert space E which also enjoys the Kadee-Klee property.Let A:C→
E∗ be a continuous monotone mapping satisfying(.),and u∈C be a constant.Assume
that F:=A–()=∅,let the sequence{x
n}be generated iteratively by
⎧ ⎨ ⎩
yn=PC(λnxn+ ( –λn)SAcxn),
xn+=αnu+βnxn+γnSAcγnyn+εnen,
(.)
where c> and{en} ∈H is an error,SAc = (I+cA)–,and the sequencesλn∈[, ],{αn},
{βn},{γn},{εn}be nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) <lim infn→∞λn<lim supn→∞λn< ;
(iii) lim supn→∞εn= ,andlim supn→∞en= ,
then the sequence converges strongly tox¯=PFu.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Hilbert space E which also enjoys the Kadee-Klee property.Let A:C→E∗be
a continuous monotone mapping satisfying(.),and u∈C be a constant.Assume that
F:=A–()=∅,let x
nbe a sequence generated by x∈C
xn+=αnu+βnxn+γnScAxn+εnen, (.)
where c> and{en} ∈H is an error,ScA= (I+cA)–,and the sequences{αn},{βn},{γn},{εn}
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) lim supn→∞εn= ,andlim supn→∞en= ,
then the sequence converges strongly tox¯=PFu.
Remark . In fact, ifAis a maximal monotone mapping, then all the above results hold.
4 Applications
In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Banach spaces. Letgbe continuously Fréchet differen-tiable convex functionals such that the gradient ofg(∇g|C) are continuous and monotone. DenoteB:=∇g, then the following theorem holds.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Banach space E which also enjoys the Kadee-Klee property.Let g be
continu-ously Fréchet differentiable convex functionals such that the gradient of g,B:=∇g,is con-tinuous,monotone,and F:=arg miny∈Cg(y) :={z∈C:g(z) =miny∈Cg(y)} =∅.Let u∈C be a constant,for x,u∈C arbitrarily,and the sequence{xn}be generated iteratively by
⎧ ⎨ ⎩
yn=C(J–(λnJxn+ ( –λn)JSBcun)),
xn+=J–(αnJu+βnJxn+γnJScByn+εnJen),
(.)
where c⊂(,∞),{en} ∈E is an error, λn∈[, ],{αn}, {βn}, {γn}, {εn} are sequences of
nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) lim supn→∞λn= ;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequences{xn},{un}converge strongly tox¯=Fu.
In addition, we can extend it to the equilibrium problem for a bifunctionφ:C×C→
RwhereCis a nonempty, closed, and convex subset of a smoothly, strictly convex, and reflexive real Banach spaceEwith dualE∗. The problem is to findx∈Csuch thatφ(x,y)≥ for ally∈Cand the set of solutions if denoted byEP(φ). The mappingTrn:E→Cis
defined as follows: forx∈E,
Trn(x) :=
z∈C:φ(z,y) + rn
j(y–z),z–x≥,∀y∈C
.
It is proved in [] that{Trn}is single-valued and nonexpansive. Furthermore, we see that
F(Trn) =EP(φ) is closed and convex if the bifunctionφ satisfies (A)φ(x,x) = ,∀x∈C,
(A) φ(x,y) +φ(y,x)≤,∀x,y∈C, (A) for eachx,y,z∈C, limt→φ(tz+ ( –t)x,y)≤
φ(x,y), and (A) for eachx∈C,y→φ(x,y) is convex and lower semicontinuous.
The following theorems are connected with the problem of obtaining a common ele-ment of the sets of zeros of a monotone operator and an equilibrium problem.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Banach space E which also enjoys the Kadee-Klee property.Letφbe a bifunction
satisfying(.),and u∈C be a constant.Assume that F:=A–()∩EP(φ)=∅,for x,u∈C arbitrarily,let the sequence{xn}be generated iteratively by
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
φ(un,y) +rnj(y–un),un–xn ≥, ∀y∈C,
yn=C(J–(λnJun+ ( –λn)JSAcun)),
xn+=J–(αnJu+βnJxn+γnJScAyn+εnJen),
(.)
where c⊂(,∞),{en} ∈E is an error, λn∈[, ],{αn}, {βn}, {γn}, {εn} are sequences of
nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥,limn→∞αn= ,
∞
n=αn=∞;
(ii) lim supn→∞λn= ;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequences{xn},{un}converge strongly tox¯=Fu.
Theorem . Let C be a nonempty,closed,and convex subset of a uniformly smooth strictly
convex real Banach space E which also enjoys the Kadee-Klee property.Letφbe a
bifunc-tion from C→Rsatisfying(A)-(A),A:C→E∗be a continuous monotone mapping
sat-isfying(.),and u∈C be a constant.Assume that F:=A–()∩EP(φ)=∅,for x,u∈C arbitrarily,let the sequence{xn}be generated iteratively by
⎧ ⎨ ⎩
φ(un,y) +rnj(y–un),un–xn ≥, ∀y∈C,
xn+=J–(αnJu+βnJxn+γnJScAun+εnJen),
(.)
where c⊂(,∞),{en} ∈E is an error, λn∈[, ],{αn}, {βn}, {γn}, {εn} are sequences of
nonnegative real numbers in[, ]and
(i) αn+βn+γn+εn= ,n≥;
(ii) limn→∞αn= ,
∞
n=αn=∞;
(iii) lim supn→∞εn= ,lim supn→∞en= ,
then the sequences{xn},{un}converge strongly tox¯=Fu.
Remark . Our results extend and unify most of the results that have been proved for this important class of nonlinear operators. In particular, our theorems provide a conver-gence scheme to the proximal point of a monotone mapping which improves the results of Yao and Shahzad [] to Banach spaces, being more general than Hilbert spaces. We also note that our results complement the results of Zegeye and Shahzad [], which are convergence results for accretive mappings. At the same time, Theorem . extends the results of Tang [] and Zegeye and Shahzad [] which have not involved errors.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
Chongqing (CSTC, 2014jcyjA00026) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1400614).
Received: 3 May 2014 Accepted: 14 November 2014 Published:07 Dec 2014
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