Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces

R E S E A R C H

Open Access

Strong convergence theorems for equilibrium

problems and weak Bregman relatively

nonexpansive mappings in Banach spaces

Ravi P Agarwal

_{, Jia-Wei Chen}

_{and Yeol Je Cho}

*_{Correspondence:}

[email protected]; [email protected]

3_{College of Sciences, Northwest}

A&F University, Yangling, Xianyang, Shaanxi 712100, China

4_{Department of Mathematics}

Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract

In this paper, a shrinking projection algorithm based on the prediction correction method for equilibrium problems and weak Bregman relatively nonexpansive mappings is introduced and investigated in Banach spaces, and then the strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field.

MSC: 26B25; 47H09; 47J05; 47J25

Keywords: equilibrium problem; weak Bregman relatively nonexpansive mapping; Bregman distance; Bregman projection; fixed point; shrinking projection algorithm; totally convex function; Legendre function

1 Introduction

In this paper, without other specifications, let R be the set of real numbers, C be a nonempty, closed and convex subset of a real reflexive Banach spaceEwith the dual space E∗. The norm and the dual pair betweenE∗andEare denoted by·and·,·, respectively. Letf :E→R∪ {+∞}be a proper convex and lower semicontinuous function. Denote the domain off bydomf,i.e.,domf ={x∈E:f(x) < +∞}. TheFenchel conjugateoff is the functionf∗:E∗→(–∞, +∞] defined byf∗(ξ) =sup{ξ,x–f(x) :x∈E}. LetT:C→C be a nonlinear mapping. Denote byF(T) ={x∈C:Tx=x}the set of fixed points ofT. A mappingTis said to benonexpansiveifTx–Ty ≤ x–yfor allx,y∈C.

In , Blum and Oettli [] firstly studied the equilibrium problem: findingx¯∈Csuch that

H(x,¯ y)≥, ∀y∈C, (.)

whereH:C×C→Ris functional. Denote the set of solutions of the problem (.) by EP(H). Since then, various equilibrium problems have been investigated. It is well known that equilibrium problems and their generalizations have been important tools for solving problems arising in the fields of linear or nonlinear programming, variational inequal-ities, complementary problems, optimization problems, fixed point problems and have been widely applied to physics, structural analysis, management science and economics etc.One of the most important and interesting topics in the theory of equilibria is to de-velop efficient and implementable algorithms for solving equilibrium problems and their

generalizations (see,e.g., [–] and the references therein). Since the equilibrium prob-lems have very close connections with both the fixed point probprob-lems and the variational inequalities problems, finding the common elements of these problems has drawn many people’s attention and has become one of the hot topics in the related fields in the past few years (see,e.g., [–] and the references therein).

In , Bregman [] discovered an elegant and effective technique for using of the so-called Bregman distance functionDf (see, Section , Definition .) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique has been applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equi-libria, for computing fixed points of nonlinear mappings and so on (see,e.g., [–] and the references therein). In , Butnariu and Resmerita [] presented Bregman-type iterative algorithms and studied the convergence of the Bregman-type iterative method of solving some nonlinear operator equations.

Recently, by using the Bregman projection, Reich and Sabach [] presented the follow-ing algorithms for findfollow-ing common zeroes of maximal monotone operatorsAi:E→E

∗

(i= , , . . . ,N) in a reflexive Banach spaceE, respectively:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E, yi

n=Res f

λin(xn+e

i n),

Ci

n={z∈E:Df(z,yin)≤Df(z,xn+ein)}, Cn=

N i=Cni,

Qn={z∈E:∇f(x) –∇f(xn),z–xn ≤}, xn+=projfCn∩Qnx, ∀n≥

and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E, ηi

n=ξni+λin(∇f(y

i

n) –∇f(xn)), ξni∈Aiyin, ω_ni =∇f∗(λi

nηni+∇f(xn)),

Ci_n={z∈E:Df(z,yin)≤Df(z,ωni)}, Cn=

N i=Cni,

Qn={z∈E:∇f(x) –∇f(xn),z–xn ≤}, xn+=projfCn∩Qnx, ∀n≥,

where {λi

Ti:C→C(i= , , . . . ,N) in a reflexive Banach spaceEif

N

i=F(Ti)=∅:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E, Qi

=E, i= , , . . . ,N,

yi_n=Ti(xn+ein),

Qi

n+={z∈Qin:∇f(xn+eni) –∇f(yin),z–yin ≤}, Qn=

N i=Qin,

xn+=projfQn+x, ∀n≥.

(.)

Under some suitable conditions, they proved that the sequence{xn}generated by (.) converges strongly toN_i=F(Ti) and applied the result to the solution of convex feasibility and equilibrium problems.

Very recently, Chenet al.[] introduced the concept of weak Bregman relatively non-expansive mappings in a reflexive Banach space and gave an example to illustrate the ex-istence of a weak Bregman relatively nonexpansive mapping and the difference between a weak Bregman relatively nonexpansive mapping and a Bregman relatively nonexpansive mapping. They also proved the strong convergence of the sequences generated by the con-structed algorithms with errors for finding a fixed point of weak Bregman relatively non-expansive mappings and Bregman relatively nonnon-expansive mappings under some suitable conditions.

This paper is devoted to investigating the shrinking projection algorithm based on the prediction correction method for finding a common element of solutions to the equilib-rium problem (.) and fixed points to weak Bregman relatively nonexpansive mappings in Banach spaces, and then the strong convergence of the sequence generated by the pro-posed algorithm is derived under some suitable assumptions.

2 Preliminaries

Let T :C→C be a nonlinear mapping. A point ω∈C is called anasymptotic fixed

point ofT (see []) if Ccontains a sequence{xn} which converges weakly toω such that limn→∞Txn –xn= . A point ω∈ C is called a strong asymptotic fixed point of T (see []) ifC contains a sequence{xn} which converges strongly toωsuch that limn→∞Txn–xn= . We denote the sets of asymptotic fixed points and strong asymp-totic fixed points ofTbyF(T) andˆ F(T), respectively.˜

Let{xn}be a sequence inE; we denote the strong convergence of{xn}tox∈Ebyxn→x. For anyx∈int(domf), theright-hand derivativeoff atxin the directiony∈Eis defined by

f(x,y) :=lim t

f(x+ty) –f(x)

t .

f is calledGâteaux differentiableatxif, for ally∈E,lim_tf(x+ty)–f(x)

The Legendre functionf :E→(–∞, +∞] is defined in []. From [], ifEis a reflex-ive Banach space, thenf is the Legendre function if and only if it satisfies the following conditions (L) and (L):

(L) The interior of the domain off,int(domf), is nonempty,f is Gâteaux differentiable onint(domf)anddomf=int(domf);

(L) The interior of the domain off∗,int(domf∗), is nonempty,f∗is Gâteaux differentiable onint(domf∗)anddomf∗=int(domf∗).

SinceEis reflexive, we know that (∇f)–_=∇f∗_{(see []). This, by (L) and (L), implies} the following equalities:

∇f=∇f∗–, ran∇f=dom∇f∗=intdomf∗

and

ran∇f∗=dom∇f =int(domf).

By Bauschkeet al.[, Theorem .], the conditions (L) and (L) also yield that the func-tionsf andf∗are strictly convex on the interior of their respective domains. From now on we assume that the convex functionf:E→(–∞, +∞] is Legendre.

We first recall some definitions and lemmas which are needed in our main results.

Assumption . LetCbe a nonempty, closed convex subset of a uniformly convex and

uniformly smooth Banach spaceE, and letH:C×C→Rsatisfy the following conditions (C)-(C):

(C) H(x,x) = for allx∈C;

(C) His monotone,i.e.,H(x,y) +H(y,x)≤for allx,y∈C; (C) for allx,y,z∈C,

lim sup t→+ H

tz+ ( –t)x,y≤H(x,y);

(C) for allx∈C,H(x,·)is convex and lower semicontinuous.

Definition .[, ] Letf :E→(–∞, +∞] be a Gâteaux differentiable and convex func-tion. The functionDf:domf×int(domf)→[, +∞) defined by

Df(y,x) :=f(y) –f(x) – ∇f(x),y–x

is calledthe Bregman distancewith respect tof.

Remark .[] The Bregman distance has the following properties:

() thethree point identity, for anyx∈domf andy,z∈int(domf),

Df(x,y) +Df(y,z) –Df(x,z) = ∇f(z) –∇f(y),x–y

;

() thefour point identity, for anyy,ω∈domf andx,z∈int(domf),

Df(y,x) –Df(y,z) –Df(ω,x) +Df(ω,z) = ∇f(z) –∇f(x),y–ω

Definition .[] Letf :E→(–∞, +∞] be a Gâteaux differentiable and convex func-tion. The Bregman projection ofx∈int(domf) onto the nonempty, closed and convex set C⊂domf is the necessarily unique vectorprojf_C(x)∈Csatisfying the following:

Df

projf_C(x),x=infDf(y,x) :y∈C

.

Remark .

() IfEis a Hilbert space andf(y) = _x_{for allx∈E, then the Bregman projection} projf_C(x)is reduced to the metric projection ofxontoC;

() IfEis a smooth Banach space andf(y) = x

_{for allx∈E, then the Bregman}

projectionprojf_C(x)is reduced to the generalized projectionC(x)(see [, ]),

which is defined by

φC(x),x

=min

y∈Cφ(y,x),

whereφ(y,x) =y– y,J(x)+x,Jis the normalized duality mapping fromE

toE∗.

Definition . [, , ] LetCbe a nonempty, closed and convex set ofdomf. The

operatorT:C→int(domf) withF(T)=∅is called:

() quasi-Bregman nonexpansiveif

Df(u,Tx)≤Df(u,x), ∀x∈C,u∈F(T);

() Bregman relatively nonexpansiveifF(T) =ˆ F(T)and

Df(u,Tx)≤Df(u,x), ∀x∈C,u∈F(T);

() Bregman firmly nonexpansiveif

∇f(Tx) –∇f(Ty),Tx–Ty≤ ∇f(x) –∇f(y),Tx–Ty, ∀x,y∈C,

or, equivalently,

Df(Tx,Ty)+Df(Ty,Tx)+Df(Tx,x)+Df(Ty,y)≤Df(Tx,y)+Df(Ty,x), ∀x,y∈C;

() a weak Bregman relatively nonexpansive mappingwithF(T)=∅ifF(T˜ ) =F(T)and

Df(u,Tx)≤Df(u,x), ∀x∈C,u∈F(T).

Definition .[] LetH:C×C→Rbe functional. TheresolventofHis the operator

Resf_H:E→C_{defined by}

Resf_H(x) =z∈C:H(z,y) + ∇f(z) –∇f(x),y–z≥,∀y∈C.

() totally convexatx∈int(domf)if its modulus of total convexity atx, that is, the functionνf :int(domf)×[, +∞)→[, +∞)defined by

νf(x,t) :=inf

Df(y,x) :y∈domf,y–x=t

,

is positive whenevert> ;

() totally convexif it is totally convex at every pointx∈int(domf);

() totally convex on bounded setsifνf(B,t)is positive for any nonempty bounded

subsetBofEandt> , where the modulus of total convexity of the functionf on the setBis the functionνf :int(domf)×[, +∞)→[, +∞)defined by

νf(B,t) :=inf

νf(x,t) :x∈B∩domf

.

Definition .[, ] The functionf :E→(–∞, +∞] is called:

() cofiniteifdomf∗=E∗;

() coerciveiflimx→+∞(f(x)/x) = +∞;

() sequentially consistentif for any two sequences{xn}and{yn}inEsuch that{xn}is

bounded,

lim

n→∞Df(yn,xn) =  ⇒ nlim→∞yn–xn= .

Lemma .[, Proposition .] If f :E→(–∞, +∞]is Fréchet differentiable and totally convex,then f is cofinite.

Lemma .[, Theorem .] Let f :E→(–∞, +∞]be a convex function whose domain contains at least two points.Then the following statements hold:

() f is sequentially consistent if and only if it is totally convex on bounded sets; () Iff is lower semicontinuous,thenf is sequentially consistent if and only if it is

uniformly convex on bounded sets;

() Iff is uniformly strictly convex on bounded sets,then it is sequentially consistent and the converse implication holds whenf is lower semicontinuous,Fréchet differentiable on its domain and the Fréchet derivative∇f is uniformly continuous on bounded sets.

Lemma .[, Proposition .] Let f :E→R be uniformly Fréchet differentiable and

bounded on bounded subsets of E.Then∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗.

Lemma .[, Lemma .] Let f :E→R be a Gâteaux differentiable and totally convex

function.If x∈E and the sequence{Df(xn,x)}is bounded,then the sequence{xn}is also bounded.

Lemma .[, Proposition .] Let f :E→R be a Gâteaux differentiable and totally

convex function,x∈E and let C be a nonempty,closed convex subset of E.Suppose that the sequence {xn}is bounded and any weak subsequential limit of {xn}belongs to C.If Df(xn,x)≤Df(projfC(x),x)for any n∈N,then{xn}∞n=converges strongly toproj

Lemma .[, Proposition .] Let f :E→(–∞, +∞]be the Legendre function.Let C be a nonempty,closed convex subset ofint(domf)and T:C→C be a quasi-Bregman nonexpansive mapping with respect to f.Then F(T)is closed and convex.

Lemma .[, Lemma .] Let f:E→(–∞, +∞]be Gâteaux differentiable and proper convex lower semicontinuous.Then,for all z∈E,

Df

z,∇f∗

_N

i=

ti∇f(xi)

≤

N

i=

tiDf(z,xi),

where{xi}Ni=⊂E and{ti}Ni=⊂(, )with

N i=ti= .

Lemma .[, Corollary .] Let f :E→(–∞, +∞]be Gâteaux differentiable and to-tally convex on int(domf).Let x∈int(domf)and C ⊂int(domf)be a nonempty,closed convex set.Ifxˆ∈C,then the following statements are equivalent:

() the vectorxˆis the Bregman projection ofxontoCwith respect tof; () the vectorxˆis the unique solution of the variational inequality:

∇f(x) –∇f(z),z–y≥, ∀y∈C;

() the vectorxˆis the unique solution of the inequality:

Df(y,z) +Df(z,x)≤Df(y,x), ∀y∈C.

Lemma .[, Lemmas  and ] Let f :E→(–∞, +∞]be a coercive Legendre function. Let C be a nonempty,closed and convex subset ofint(domf).Assume that H:C×C→R satisfies Assumption..Then the following results hold:

() Resf_His single-valued anddom(Resf_H) =E; () Resf_His Bregman firmly nonexpansive;

() EP(H)is a closed and convex subset ofCandEP(H) =F(Resf_H); () for allx∈Eand for allu∈F(Resf_H),

Df

u,Resf_H(x)+Df

Resf_H(x),x≤Df(u,x).

Lemma .[, Proposition ] Let f :E→R be a Legendre function such that∇f∗ is

bounded on bounded subsets ofint domf∗.Let x∈E.If{Df(x,xn)}is bounded,then the sequence{xn}is bounded too.

3 Main results

In this section, we will introduce a new shrinking projection algorithm based on the pre-diction correction method for finding a common element of solutions to the equilibrium problem (.) and fixed points to weak Bregman relatively nonexpansive mappings in Banach spaces, and then the strong convergence of the sequence generated by the pro-posed algorithm is proved under some suitable conditions.

Algorithm . Step : Select an arbitrary starting pointx∈C, letQ=CandC={z∈ C:Df(z,u)≤Df(z,x)}.

Step : Given the current iteratexn, calculate the next iterate as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=∇f∗(βn∇f(T(xn)) + ( –βn)∇f(xn)), yn=∇f∗(αn∇f(x) + ( –αn)∇f(zn)), un=ResfH(yn),

Cn={z∈Cn–∩Qn–:Df(z,un)≤αnDf(z,x) + ( –αn)Df(z,xn)}, Qn={z∈Cn–∩Qn–:∇f(x) –∇f(xn),z–xn ≤},

xn+=projfCn∩Qnx, ∀n≥.

(.)

Theorem . Let C be a nonempty,closed and convex subset of a real reflexive Banach

space E,f:E→R be a coercive Legendre function which is bounded,uniformly Fréchet dif-ferentiable and totally convex on a bounded subset of E,and∇f∗be bounded on bounded subsets of E∗. Let H :C ×C→R satisfy Assumption . and T :C →C be a weak Bregman relatively nonexpansive mapping such thatEP(H)∩F(T)=∅.Then the sequence

{xn} generated by Algorithm. converges strongly to the pointprojfEP(H)∩F(T)(x),where projf_EP(H)∩F(T)(x)is the Bregman projection of C ontoEP(H)∩F(T).

To prove Theorem ., we need the following lemmas.

Lemma . Assume thatEP(H)∩F(T)⊆Cn∩Qnfor all n≥.Then the sequence{xn}is bounded.

Proof Since∇f(x) –∇f(xn),v–xn ≤ for allv∈Qn, it follows from Lemma . that xn=projfQn(x) and so, byxn+=proj

f

Cn∩Qn(x)∈Qn, we have

Df(xn,x)≤Df(xn+,x). (.)

Letω∈EP(H)∩F(T). It follows from Lemma . that

Df

ω,projf_Qn(x)+Df

projf_Qn(x),x≤Df(ω,x)

and so

Df(xn,x)≤Df(ω,x) –Df(ω,xn)≤Df(ω,x).

Therefore,{Df(xn,x)}is bounded. Moreover,{xn} is bounded and so are{T(xn)}, {yn},

{zn}. This completes the proof.

Lemma . Assume thatEP(H)∩F(T)⊆Cn∩Qnfor all n≥.Then the sequence{xn}is a Cauchy sequence.

has

Df

xm,projfQn(x)

+Df

projf_Qn(x),x≤Df(xm,x)

and soDf(xm,xn)≤Df(xm,x) –Df(xn,x). Therefore, we have

lim

n→∞Df(xm,xn)≤nlim→∞

Df(xm,x) –Df(xn,x)

= . (.)

Sincef is totally convex on bounded subsets ofE, by Definition ., Lemma . and (.), we obtain

lim

n→∞xm–xn= . (.)

Thus{xn}is a Cauchy sequence and solimn→∞xn+–xn= . This completes the proof.

Lemma . Assume thatEP(H)∩F(T)⊆Cn∩Qnfor all n≥.Then the sequence{xn} converges strongly to a point inEP(H)∩F(T).

Proof From Lemma ., the sequence{xn}is a Cauchy sequence. Without loss of gener-ality, letxn→ ˆω∈C. Sincef is uniformly Fréchet differentiable on bounded subsets ofE,

it follows from Lemma . that∇f is norm-to-norm uniformly continuous on bounded

subsets ofE. Hence, by (.), we have

lim

n→∞∇f(xm) –∇f(xn)= 

and so

lim

n→∞∇f(xn+) –∇f(xn)= . (.)

Sincexn+∈Cn, we have

Df(xn+,un)≤αnDf(xn+,x) + ( –αn)Df(xn+,xn).

It follows fromlimn→∞αn=  andlimn→∞Df(xn+,xn) =  that{Df(xn+,un)}is bounded and

lim

n→∞Df(xn+,un) = .

By Lemma .,{un}is bounded. Hence,limn→∞xn+–un=  and so

lim

n→∞∇f(xn+) –∇f(un)= . (.)

Taking into accountxn–un ≤ xn–xn++xn+–un, we obtain

lim

and soun→ ˆωasn→ ∞. For anyω∈EP(H)∩F(T), from Lemma ., we get

Df(un,yn)

≤Df(ω,yn) –Df(ω,un)

=Df

ω,∇f∗αn∇f(x) + ( –αn)∇f(zn)

–Df(ω,un)

≤αnDf(ω,x) + ( –αn)Df

ω,∇f∗βn∇f

T(xn)

+ ( –βn)∇f(xn)

–Df(ω,un)

≤αnDf(ω,x) + ( –αn)Df(ω,xn) –Df(ω,un)

=αn

Df(ω,x) –Df(ω,xn)

+Df(ω,xn) –Df(ω,un).

By the three point identity of the Bregman distance, one has

Df(ω,xn) –Df(ω,un)

= –Df(xn,un) + ∇f(un) –∇f(xn),ω–xn

≤–f(xn) +f(un) + ∇f(un),xn–un

+ ∇f(un) –∇f(xn),ω–xn

.

Sincefis a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on a bounded subset ofE, it follows from Lemma . thatfis continuous onEand∇f is uniformly continuous on bounded subsets ofEfrom the strong topology ofEto the strong topology ofE∗. Therefore, we have

lim

n→∞Df(un,yn)

≤ lim n→∞

αn

Df(ω,x) –Df(ω,xn)

–f(xn) +f(un)

+ ∇f(un),xn–un

+ ∇f(un) –∇f(xn),ω–xn

≤ lim n→∞

αn

Df(ω,x) –Df(ω,xn)

–f(xn) +f(un)

+ lim

n→∞ ∇f(un),xn–un

+ ∇f(un) –∇f(xn),ω–xn

= ,

that is,limn→∞Df(un,yn) = . Furthermore, one haslimn→∞un–yn=  and thus

lim

n→∞∇f(un) –∇f(yn)= .

Sinceun→ ˆωasn→ ∞, we haveyn→ ˆωasn→ ∞. Further, in the light ofun=ResfH(yn) and Definition ., it follows that, for eachy∈C,

H(un,y) + ∇f(un) –∇f(yn),y–un

≥

and hence, combining this with Assumption .,

∇f(un) –∇f(yn),y–un

Consequently, one can conclude that

H(y,ω)ˆ ≤lim inf n→∞ H(y,un)

≤lim inf

n→∞ ∇f(un) –∇f(yn),y–un

≤lim inf

n→∞∇f(un) –∇f(yn)· y–un = .

For anyy∈Candt∈(, ], letyt=ty+ ( –t)ωˆ∈C. It follows from Assumption . that H(yt,ω)ˆ ≤ and

 =H(yt,yt)≤tH(yt,y) + ( –t)H(yt,ω)ˆ ≤tH(yt,y)

and soH(yt,y)≥. Moreover, one has

≤lim sup

t→+ H(yt,y) =lim sup_t→+ H

ty+ ( –t)ω,ˆ y≤H(ω,ˆ y), ∀y∈C,

which shows thatωˆ ∈EP(H).

Next, we prove thatωˆ∈F(T). Note that

_∇f(xn) –∇f(yn)

=∇f(xn) –∇f

∇f∗αn∇f(x) + ( –αn)∇f(zn)

=∇f(xn) –

αn∇f(x) + ( –αn)∇f(zn)

=αn

∇f(xn) –∇f(x)

+ ( –αn)

∇f(xn) –∇f(zn)

=αn

∇f(xn) –∇f(x)

+ ( –αn)

∇f(xn) –∇f

∇f∗βn∇f

T(xn)

+ ( –βn)∇f(xn)

=αn

∇f(xn) –∇f(x)

+ ( –αn)βn

∇f(xn) –∇f

T(xn)

≥( –αn)βn∇f(xn) –∇f

T(xn)–αn∇f(xn) –∇f(x).

This implies that

( –αn)βn∇f(xn) –∇f

T(xn)≤∇f(xn) –∇f(yn)+αn∇f(xn) –∇f(x). (.)

Lettingn→ ∞in (.), it follows fromlim inf_n→∞( –αn)βn>  andlimn→∞αn=  that

lim

n→∞∇f(xn) –∇f

T(xn)= .

Moreover, we have thatlimn→∞xn–T(xn)= . This together withxn→ ˆωimplies that

ˆ

ω∈ ˜F(T). In view ofF(T˜ ) =F(T), one hasωˆ ∈EP(H)∩F(T). Therefore, the sequence

{xn}generated by Algorithm . converges strongly to a pointωˆ inEP(H)∩F(T). This

completes the proof.

Proof of Theorem . From Lemmas . and ., it follows that EP(H)∩F(T) is a nonempty, closed and convex subset ofE. Clearly,CnandQnare closed and convex and soCn∩Qnare closed and convex for alln≥.

Now, we show thatEP(H)∩F(T)⊆Cn∩Qnfor alln≥. Takeω∈EP(H)∩F(T) arbi-trarily. Then

Df(ω,un) =Df

ω,Resf_H(yn)

≤Df(ω,yn) –Df

Resf_H(yn),yn

≤Df(ω,yn)

=Df

ω,∇f∗αn∇f(x) + ( –αn)∇f(zn)

≤αnDf(ω,x) + ( –αn)Df(ω,zn)

=αnDf(ω,x) + ( –αn)Df

ω,∇f∗βn∇f

T(xn)

+ ( –βn)∇f(xn)

≤αnDf(ω,x) + ( –αn)

βnDf

ω,T(xn)

+ ( –βn)Df(ω,xn)

≤αnDf(ω,x) + ( –αn)Df(ω,xn),

which implies thatω∈Cnand soEP(H)∩F(T)⊆Cnfor alln≥.

Next, we prove thatEP(H)∩F(T)⊆Qnfor all n≥. Obviously,EP(H)∩F(T)⊆Q (Q=C). Suppose thatEP(H)∩F(T)⊆Qkfor allk≥. In view ofxk+=projfCk∩Qk(x), it

follows from Lemma . that

∇f(x) –∇f(xk+),xk+–v

≥, ∀v∈Ck∩Qk.

Moreover, one has

∇f(x) –∇f(xk+),xk+–ω

≥, ∀ω∈EP(H)∩F(T)

and so, for eachω∈EP(H)∩F(T),

∇f(x) –∇f(xk+),ω–xk+

≤.

This implies thatEP(H)∩F(T)⊆Qk+. To sum up, we haveEP(H)∩F(T)⊆Qnand so

EP(H)∩F(T)⊆Cn∩Qn, ∀n≥.

This together with EP(H)∩F(T)=∅yields thatCn∩Qn is a nonempty, closed convex subset ofCfor alln≥. Thus{xn}is well defined and, from both Lemmas . and ., the sequence{xn}is a Cauchy sequence and converges strongly to a pointωˆ ofEP(H)∩F(T). Finally, we now prove thatω¯=projf_EP(H)∩F(T)(x). Sinceprojf_EP(H)∩F(T)(x)∈EP(H)∩F(T), it follows fromxn+=projf(Cn∩Qn)(x) that

Df(xn+,x)≤Df

projf_EP(H)∩F(T)(x),x.

Thus, by Lemma ., we havexn→projfEP(H)∩F(T)(x) asn→ ∞. Therefore, the sequence

Remark . () Iff(x) = _x_{for allx∈E, then the weak Bregman relatively} nonexpan-sive mapping is reduced to the weak relatively nonexpannonexpan-sive mapping defined by Suet al. [], that is,Tis called aweak relatively nonexpansive mappingif the following conditions are satisfied:

˜

F(T) =F(T)=∅, φ(u,Tx)≤φ(u,x), ∀x∈C,u∈F(T),

whereφ(y,x) =y_{– y,J(x)+x}_{for allx,y∈EandJis the normalized duality} map-ping fromEto E∗_;

() Iff(x) =_x_{for allx∈E, then Algorithm . is reduced to the following iterative} algorithm.

Algorithm . Step : Select an arbitrary starting pointx∈C, letQ=CandC={z∈ C:φ(z,u)≤φ(z,x)}.

Step : Given the current iteratexn, calculate the next iterate as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=J–(βnJ(T(xn)) + ( –βn)J(xn)),

yn=J–(αnJ(x) + ( –αn)J(zn)),

un=ResfH(yn),

Cn={z∈Cn–∩Qn–:φ(z,un)≤αnφ(z,x) + ( –αn)φ(z,xn)}, Qn={z∈Cn–∩Qn–:J(x) –J(xn),z–xn ≤},

xn+=Cn∩Qnx, ∀n≥.

(.)

() Particularly, ifEP(H) =C, then Algorithm . is reduced to the following iterative algorithm.

Algorithm . Step : Select an arbitrary starting pointx∈C, letQ=CandC={z∈ C:φ(z,u)≤φ(z,x)}.

Step : Given the current iteratexn, calculate the next iterate as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

zn=J–(βnJ(T(xn)) + ( –βn)J(xn)),

yn=J–(αnJ(x) + ( –αn)J(zn)),

Cn={z∈Cn–∩Qn–:φ(z,yn)≤αnφ(z,x) + ( –αn)φ(z,xn)}, Qn={z∈Cn–∩Qn–:J(x) –J(xn),z–xn ≤},

xn+=Cn∩Qnx, ∀n≥.

(.)

() IfTx=xfor allx∈C, then, by Algorithm ., we can get the following modified Mann iteration algorithm for the equilibrium problem (.).

Step : Given the current iteratexn, calculate the next iterate as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

yn=J–(αnJ(x) + ( –αn)J(xn)),

un=ResfH(yn),

Cn={z∈Cn–∩Qn–:φ(z,un)≤αnφ(z,x) + ( –αn)φ(z,xn)}, Qn={z∈Cn–∩Qn–:J(x) –J(xn),z–xn ≤},

xn+=Cn∩Qnx, ∀n≥.

(.)

Iff(x) =_xfor allx∈E, then, by Theorem . and Remark ., the following results hold.

Corollary . Let C be a nonempty,closed convex subset of a real reflexive Banach space E. Suppose that H:C×C→R satisfies Assumption.and T:C→C is a weak relatively nonexpansive mapping such thatEP(H)∩F(T)=∅.Then the sequence{xn}generated by Algorithm.converges strongly to the pointEP(H)∩F(T)(x),whereEP(H)∩F(T)(x)is the generalized projection of C ontoEP(H)∩F(T).

Corollary . Let C be a nonempty,closed convex subset of a real reflexive Banach space E. Let T:C→C be a weak relatively nonexpansive mapping such that F(T)=∅.Then the sequence{xn}generated by Algorithm.converges strongly to the pointfF(T)(x),where F(T)(x)is the generalized projection of C onto F(T).

Corollary . Let C be a nonempty,closed convex subset of a real reflexive Banach space E. Suppose that H:C×C→R satisfies Assumption.such thatEP(H)=∅.Then the se-quence{xn}generated by Algorithm.converges strongly to the pointEP(H)(x),where EP(H)(x)is the generalized projection of C ontoEP(H).

Remark .

() It is well known that any closed and firmly nonexpansive-type mapping (see [, ]) is a weak Bregman relatively nonexpansive mapping wheneverf(x) = _x_{for all} x∈E. Ifβn≡for alln≥andEis a uniformly convex and uniformly smooth

Banach space, then Corollary . improves [, Corollary .];

() Ifαn≡for alln≥andEis a uniformly convex and uniformly smooth Banach

space, then Corollary . is reduced to [, Theorem .];

() Ifβn≡ –βn for alln≥,βn∈[, ],f(x) = xfor allx∈EandEis a uniformly

convex and uniformly smooth Banach space, then Corollary . improves [, Theorem .].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results and they read and approved the final manuscript.

Author details

1_{Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA.}2_{Department of Mathematics,}

Faculty of Sciences, King Abdulaziz University, Jeddah, 21589, Saudi Arabia.3_{College of Sciences, Northwest A&F}

University, Yangling, Xianyang, Shaanxi 712100, China. 4_{Department of Mathematics Education and the RINS,}

Acknowledgements

The authors are indebted to the referees and the associate editor for their insightful and pertinent comments on an earlier version of the work. The second author (Jiawei Chen) was supported by the Natural Science Foundation of China and the Fundamental Research Fund for the Central Universities, the third author (Yeol Je Cho) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

Received: 20 September 2012 Accepted: 26 February 2013 Published: 20 March 2013 References

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doi:10.1186/1029-242X-2013-119