STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEM AND BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES∗

Sheng ZHUJianhua HUANGInstitute of Mathematics and Computer Science,Fuzhou University,Fuzhou 350002,China E-mail:jxsrsheng@163.com;fjhjh57@163.com

STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEM AND BREGMAN TOTALLY QUASI-ASYMPTOTICALLY NONEXPANSIVE MAPPING IN BANACH SPACES∗

In this paper,we propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces.Moreover,we prove some strong convergence theorems under suitable control conditions.Finally,the application to zero point problem of maximal monotone operators is given by the result.

equilibrium problem;Bregman totally quasi-asymptotically nonexpansive mapping;Bregman distance;reflexive Banach space 2010 MR Subject Classification26A18;47H10;54C05

1　Introduction

In 1994,Blum and Oettli[1]firstly studied the equilibrium problem:

where g:C×C→R is a bifunction,the set of solutions of problem(1.1)is denoted by(EP). Since then,various equilibrium problems were investigated.It is well known that equilibrium problems and their generalizations were important tools for solving problems arising in the fields of linear or nonlinear programming,variational inequalities,complementary problems, optimization problems,fixed point problems and were widely applied to physics,structural analysis,management science and economics[2–3]etc.One of the most important and interesting topics in the theory of equilibria was to develop efficient and implementable algorithms for solving equilibrium problems and their generalizations[2–4]etc.Since the equilibrium problems have very close connections with both the fixed point problems and the variational inequalities problems,finding the common elements of these problems drew many people’s attention and became one of the hot topics in the related fields in the past few years[5–15]etc.But some methods that were proposed to the equilibrium problem is in the Hilbert spaces.

In 1967,Bregman[16]discovered an effective technique for using of the so-called Bregman distance function Df(.)in the process of designing and analyzing feasibility and optimization algorithms.This opened a growing area of research in which Bregman’s technique was 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 equilibria,for computing fixed point of nonlinear mapping and so on[6,12,13, 17,18]etc.

We know that the fixed point theory of nonexpansive mappings can be applied to solutions of diverse problems such as finding zeroes of monotone mappings and solutions to certain evolution equations and solving convex feasibility,variational inequality and equilibrium problems.There are,in fact,many papers that deal with methods for finding fixed points of nonexpasive and quasi-nonexpansivemappings in Hilbert,uniformly convex and uniformly smooth Banach spaces [1o,2o–29]etc.

When we try to extend this theory to general Banach spaces we encounter some difficulties, because many of the useful examples of nonexpansive mappings in Hilbert space are no longer nonexpansive in Banach spaces,an example of a Bregman nonexpansive mapping which is not a nonexpansive mapping in Hilbert space[3o],for example,the resolvent RA:=(I+A)−1of a maximal monotone mapping A:H→2Hand the metric projection PC.There are several ways to overcome these difficulties.One of them is to use the Bregman distance instead of the norm,Bregman(quasi-)nonexpansive mappings instead of the(quasi-)nonexpansive mappings and the Bregman projection instead of the metric projection.

In 2o14,Chang and Wang et al.[31]introduced the concept of Bregman totally quasiasymptotically nonexpansive mapping and obtained the strong convergence of common fixed point for Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces.

In this paper,our results extend the one of Chang,Wang and Chan[31]to propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces.Moreover,we prove some strong convergencetheorems under suitable control conditions. Finally,the application to zero point problem of maximal monotone operators is given by the result.

2　Preliminaries

Let E be a real Banach space,E∗be the dual space of E,f:E→(−∞,+∞]be a proper and real function,f∗:E∗→(−∞,+∞]be the Fenchel conjugate of f defined by

We denote by domf the domain of f,that is,the set{x∈E:f(x)<∞}for any x∈int(domf)and y∈E,we define the right-hand derivative of f at x in the direction y by

The function f is said to be Gˆateaux differentiable at x ifxists for any y.In this case,fo(x,y)coincides with∇f(x),the value of the gradient∇f of f at x.The function f is said to be Gˆateaux differentiable if it is Gˆateaux differentiable for any x∈int(domf).The function f is said to be Frˆechet differentiable at x,if this limit is attained uniformly in‖y‖=1. Finally,f is said to be uniformly Frˆechet differentiable on a subset C of E,if the above limit is attained uniformly for x∈C and‖y‖=1.

The Legendre function f is defined from a general Banach space E into(−∞,+∞][17]. According to[17],the function f is Legendre if it satisfies the following conditions.

(L1)The interior of the domain of f,int(domf),is nonempty;f is Gˆateaux differentiable on int(domf)and dom∇f=int(domf).

(L2)The interior of the domain of f∗,int(domf∗),is nonempty;f∗is Gˆateaux differentiable on int(domf∗)and dom∇f∗=int(domf∗).

Remark 2.1If E is a real reflexive Banach space,f is the Legendre function,then the following conclusions hold:

(a)f is the Legendre function if and only if f∗is the Legendre function;

(b)(∂f)−1=∂f∗;

(c)∇f=(∇f∗)−1,ran∇f=dom∇f∗=int(domf∗),ran∇f∗=dom∇f=int(domf);

(d)the function f and f∗are strictly convex on the interior of respective domains.

Definition 2.2(see[33])Let f:E→(−∞,+∞]be a convex and Gˆateaux differentiable function.The function Df:domf×int(domf)→[o,+∞)defined by

is called the Bregman distance with respect to f.

It should be noted that the Bregman distance is not a distance in the usual sense of the term.Clearly,Df(x,x)=o,but Df(x,y)=o may not imply x=y.In our case,when f is the Legendre function this indeed holds.In general,Dfis not symmetric and does not satisfy the triangle inequality.But by the definition,we know it has the following important properties:

(1)(the two point identity)for any x,y∈int(domf),

(2)(the three point identity)for any x∈domf and y,z∈int(domf),

(3)(the four point identity)for any y,w∈domf and x,z∈int(domf),

Definition 2.3(see[16])Let f:E→(−∞,+∞]be a convex and Gˆateaux differentiable function.The Bregman projection of x∈int(domf)onto the nonempty,closed and convex set C⊂domf is the necessarily unique vectorsatisfying the following:

Definition 2.4(see[34])Let f:E→(−∞,+∞]be a convex and Gˆateaux differentiable function.f is called

(a)totally convex at a point x∈int(domf),if its modulus of total convexity at x,vf: int(domf)×[o,+∞)→[o,+∞),defined by vf(x,t):=inf{Df(y,x):y∈domf,‖y−x‖=t} is positive whenever t>o;

(b)totally convex if it is totally convex at every point x∈int(domf);

(c)totally convex on bounded sets if vf(B,t)is positive for any nonempty bounded subset B of E and t>o,where the modulus of total convexity of the function f on the set B is the function vf:int(domf)×[o,+∞)→[o,+∞)defined by vf(B,t):=inf{vf(x,t):x∈B∩domf}.

Lemma 2.5(see[32])If x∈int(domf),then the following statements are equivalent:

(1)the function f is totally convex at x;

(3)for any ε>o,there exists δ=δ(ε)>o such that if y∈domf and Df(y,x)≤δ,then‖x−y‖≤ε.

Lemma 2.6(see[34])The function f is totally convex on bounded sets if and only if it is sequentially consistent,i.e.,for any two sequences and{xn}and{yn}in int(domf)and domf, respectively,and{xn}is bounded,then

Lemma 2.7(see[32])Let f:E→(−∞,+∞]be uniformly Frˆechet differentiable and bounded on bounded subsets of E.Then f is uniformly continuous on bounded subsets of E and∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E∗.

Lemma 2.8(see[35])Let f:E→(−∞,+∞]be a Gˆateaux differentiable and totally convex function.If xo∈E and the sequence{Df(xn,xo)}is bounded,then the sequence{xn} is also bounded.

Lemma 2.9(see[19])Let f:E→(−∞,+∞]be Gˆateaux differentiable function and totally convex on int(domf).Let x∈int(domf)and C⊂int(domf)be a nonempty,closed convex set.If z∈C,then the following statements are equivalent:

(1)z∈C is the Bregman projection of x onto C with respect to f,i.e.,

(2)the vector z is the unique solution of the variational inequality:

(3)the vector z is the unique solution of the inequality:

Definition 2.10Let C be a subset of E.Denoted by F(T)={x∈C:Tx=x}the set of fixed points of T,T:C→C be called:

(a)nonexpansive,if‖Tx−Ty‖≤‖x−y‖;

(b)quasi-nonexpansive,if F(T)/=Ø and‖Tx−p‖≤‖x−p‖,∀x∈C,p∈F(T);

(c)closed,if for any sequence{xn}⊂C with xn→x∈C and Txn→y∈C,then Tx=y;

(d)uniformly asymptotically regular on C if

(e)Bregman relatively nonexpansive if F(T)/=Ø,F(T)=ˆF(T)and Df(p,Tx)≤Df(p,x),∀x∈C,p∈F(T).A point p∈C is called an asymptotic fixed point of T if C contains asequence{xn}which converges weakly to p such that

We denoteFˆ(T) the set of asymptotic fixed points of T;

(f)Bregman quasi-nonexpansive if F(T)/=Ø and Df(p,Tx)≤Df(p,x),∀x∈C,p∈F(T);

(g)Bregman quasi-asymptotically nonexpansive if F(T)/=Ø and there exists a real sequence{kn}=1 such that

In 2o14,Chang and Wang et al.[31]introduced the concept of Bregman totally quasiasymptotically nonexpansive mapping.

(h)A mapping T:C→C is said to be Bregman totally quasi-asymptotically nonexpasive, if F(T)/=Ø and there exist nonnegative real sequences{vn},{µn}with vn→o,µn→o(as n→∞)and a strictly increasing continuous function ζ:R+→R+with ζ(o)=o such that

Remark 2.11According to the definitions,it is obvious that

(1)each Bregman relatively nonexpansive mapping is Bregman quasi-nonexpansive mapping;

(2)each Bregman quasi-asymptotically nonexpansive mapping is Bregman totally quasiasymptotically nonexpansive mapping,but the converse may be not true.If taking,ζ(t)=t, t≥o,vn=kn−1 andµn=o,then(2.1)can be rewritten as

This implies that each Bregman quasi-asymptotically nonexpansive mapping must be a Bregman total quasi-asymptoticallynonexpansive mapping,but the converse is not true.

Example 2.12Let E be a real reflexive Banach space,A:E→2E∗be a maximal monotone mapping,if A−1(o)/=Ø and f:E→(−∞,+∞]be a uniformly Frˆechet differentiable and bounded on bounded subsets of E,then the resolvent with respect to A,

is a closed and Bregman relatively nonexpansive mapping from E onto D(A),so is closed Bregman total quasi-asymptotically nonexpansive mapping.

Assumption 2.13Let C be a nonempty,closed convex subset of a uniformly convex and uniformly smooth Banach space E,and let g:C×C→R satisfy the following conditions:

(C1)g(x,x)=o,∀x∈C;

(C2)g is monotone,i.e.,g(x,y)+g(y,x)≤o,∀x,y∈C;

(C4)∀x∈C,g(x,.)is convex and lower semicontinuous.

The bifunction g:C×C→R satisfying above assumption conditions is existence.For example,let X be a Hilbert space,A:C→X be monotone semicontinuous mapping.Then g(x,y)=〈Ax,y−x〉satisfies assumption conditions(C1)–(C4),or let X be a Banach space, A:C→X∗be monotone semicontinuous mapping.Then g(x,y)=〈Ax,y−x〉also satisfies assumption conditions(C1)–(C4).

Definition 2.14(see[36])Let g:C×C→R be a bifunction.The resolvent of g is theE→2Cdefined by

Lemma 2.15(see[32])Let f:E→(−∞,+∞]be a coercive

Legendre function.Let C be a nonempty,closed and convex subset of int(domf).Assume that g:C×C→R satisfes Assumption 2.13.Then the following results hold:

(3) EP(g)is a closed and convex subset of C and EP(g)=F

Lemma 2.16(see[37])Let f:E→R be a Legendre function such that∇f∗is bounded on bounded subsets of int(domf).Let x∈E,if{Df(x,xn)}is bounded,then the sequence {xn}is bounded.

Lemma 2.17(see[31])Let E be a real reflexive Banach space and C be a nonempty, closed and convex subset of E and f:X→(−∞,+∞]be a Legendre function which is total convex on bounded subsets of E.Let T:C→C be a closed and Bregman totally quasi-asymptotically nonexpansive mapping with nonnegative real sequences{vn},{µn}and a strictly increasing continuous functions ζ:R+→R+such that vn→o,µn→o(as n→∞) and ζ(o)=o,then the fixed point set F(T)of T is a closed and convex subset of C.

Lemma 2.18(see[35])Let f:E→R be a Gˆateaux differentiable and totally convex function,xo∈E and 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,xo)≤

3　Main Results

Theorem 3.1Let E be a real reflexive Banach space and f:E→R be a coercive Legendre function which is bounded,uniformly Frˆechet differentiable and totally convex on bounded subsets of E.Let C be a nonempty,closed and convex subset of int(domf),T:C→C be a closed Bregman totally asymptotically quasi-nonexpansive mapping with nonnegative real sequences{vn},{µn}and a strictly increasing continuous functions ζ:R+→R+such that vn→o,µn→o(as n→∞)and ζ(o)=o.Let g:C×C→R be a bifunction satisfying conditions(C1)–(C4).Assume that T is uniformly asymptotically regular,Ω:=F(T)∩EP(g)/= Ø.Let{xn}be a sequence generated by

If Ω:=F(T)∩EP(g)is bounded,then the sequence{xn}strongly converges toProofWe split the proof into five steps.

Step 1We show that Ω is nonempty closed and convex subset of E,Dnis closed and convex.

It follows from Lemmas 2.15 and 2.17 that F(T)∩EP(g)is a closed and convex,so is Ω.

For any p,q∈Cn,t∈(o,1),putting w=tp+(1−t)q,we prove that w∈Cn.

Note that Df(p,un)≤Df(p,xn)+ξnand Df(q,un)≤Df(q,xn)+ξn.

Since Df(z,un)≤Df(z,xn)+ξn,by the definition of Df(.),we have

It clearly shows that Cnis closed and convex for any n≥1,so is Dn.Thus the sequence {xn}is well defined.Since unand it is from Lemma 2.15 that

where ξn=vnsupζ(Df(v,xn))+µn.

It shows that v∈Cnfor any n≥1,hence Ω⊂Cn,further,we have Ω⊂Dn. Step 2We prove that{xn}is bounded.

From xn+1≤Df(v,u)for any v∈Ω.

Hence,the sequence Df(xn,u)is bounded,and by Lemma 2.8 the sequence{xn}is also bounded.

Step 3We prove that{xn}is a Cauchy sequence.

Therefore,the sequence Df(xn,u)is increasing,and since it is also bounded,thenexists.By the construction of Dn,we have thatfor any positive integer m≥n.Therefore,

Letting in m,n→∞in(3.4),we can obtain that

It follows from Lemma 2.6 that

Step 4We show{xn}that converges to a point in Ω:=F(T)∩EP(g).

Since E be a reflexive Banach space and{xn}is a Cauchy sequence.

Without loss of generality,let

First,we prove that x∗∈F(T).

By taking in m=n+1 in(3.5),we can obtain that=o,by Lemma 2.6,

It follows from Df(xn+1,xn)=o and vn→o,µn→o(as n→∞),we have

Moreover,since Df(v,un)≤Df(v,xn)+ξnand f is lower semi-continuous function,we get that Df(v,xn)+ξnis bounded,so is Df(v,un).By Lemma 2.16,we obtain that{un}is bounded, hence

Since f is uniformly Frˆechet differentiable,it follows from Lemma 2.7 that

Since f is uniformly Frˆechet differentiable,it is also uniformly continuous,therefore,

Since{un}is bounded,∇f(un)is also bounded.It follows from(3.11)–(3.13)we have

In view of un=,by Lemma 2.15(4)we have

Note that Df(v,xn)is bounded,so is ζ(Df(v,xn))and vn→o,µn→o(as n→∞),it follows from(3.15)that

By Lemma 2.6 shows that

Taking into account‖xn−Tnxn‖≤‖xn−un‖+‖un−Tnxn‖.From(3.11)and(3.18),we can obtain that

Furthermore,we have

Since T is uniformly asymptotically regular and by(3.2o),we obtain that

Next,we prove that x∗∈EP(g).

Since f is uniformly Frˆechet differentiable,∇f is uniformly continuous on bounded sets.

Thus,by(3.18),we have

Since un=we have

We have from Assumption 2.13(C2)that

Letting n→∞,we have from(3.21)and Assumption 2.13(C4)that

For any y∈C and t∈(o,1],let yt=ty+(1−t)x∗∈C.Hence,g(yt,x∗)≤o and

and so g(yt,y)≥o.Moreover,we have

which shows that x∗∈EP(g).To sum up,we have

Step 5We show that{xn}strongly converges to

By Lemmas 2.15 and 2.17,we know F(T)∩EP(g)is a nonempty,closed and convex subset of E.Thus,theis well defined.

thus,by Lemmma 2.18,we have xn→Therefore,the sequence{xn}strongly converges toThis completes the proof.

4　Application

Application to zero point problem of maximal monotone operators.

In Example 2.12,we point out that if E is a real reflexive Banach space,A:E→2E∗is a maximal monotone mapping.If A−1(o)/=Ø and f:E→(−∞,+∞]is a uniformly Frˆechet differentiable and bounded on bounded subsets of E,then the resolvent with respect to A,

is a closed and Bregman relatively nonexpansive mapping from E onto E.Hence from Theorem 3.1 we have the following

Theorem 4.1Let E be a real reflexive Banach space and f:E→R be a coercive Legendre function which is bounded,uniformly Frˆechet differentiable and totally convex on bounded subsets of E.Let C be a nonempty,closed and convex subset of int(domf)and

C→C be a closed Bregman totally asymptotically quasi-nonexpansive mapping with nonnegative real sequences{vn},{µn}and a strictly increasing continuous functions ζ:R+→R+such that vn→o,µn→o(as n→∞)and ζ(o)=o.Let g:C×C→R be a bifunction satisfying conditions(C1)–(C4).Assume thatuniformly asymptotically regular and Ω:=F∩EP(g)/=Ø.Let{xn}be a sequence generated by

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∗January 18,2015;revised October 12,2015.This research was supported by the Province Natural Science Foundation of China(2014J01008).†

Jianhua HUANG.