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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(3): 523-558

Published online September 30, 2021

Copyright © Kyungpook Mathematical Journal.

### Convergence Theorem for Finding Common Fixed Points of N-generalized Bregman Nonspreading Mapping and Solutions of Equilibrium Problems in Banach Spaces

Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo*

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
e-mail : jollatanu@yahoo.co.uk

School of Mathematics, Statistics and Computer Science University of Kwazulu-Natal Durban, South Africa
e-mail : mewomoo@ukzn.ac.za

Received: May 4, 2019; Revised: July 28, 2020; Accepted: August 4, 2020

In this paper, we study some fixed point properties of n-generalized Bregman nonspreading mappings in re exive Banach space. We introduce a hybrid iterative scheme for finding a common solution for a countable family of equilibrium problems and fixed point problems in re exive Banach space. Further, we give some applications and numerical example to show the importance and demonstrate the performance of our algorithm. The results in this paper extend and generalize many related results in the literature.

Keywords: nonspreading mapping, Bregman distance, equilibrium problem, fixed point prolem, reflexive Banach space.

Let E be a real Banach space, and C be a nonempty, closed and convex subset of E. Let g:C×C be a bifunction, the Equilibrium Problem with respect to g denoted by EP(g) is define as finding a point zC such that

(z,y)0,yC.

The EP(g) was shown by Blum and Oettli [7] to cover several other optimization problems such as monotone inclusion problems, saddle point problems, minimization problems, variational inequality problems and Nash equilibria in non-cooperative games. In addition, there are many other important problems, for example, the complementarity problem and fixed point problems, which can be written in the form of EP(g) (1.1). Thus, the EP(g) is a unifying model for several problems arising in physics, engineering, science, optimization, economics etc.

In the last two decades, the existence of solutions of the EP(g) have been mentioned in many papers, see for instance [7, 11, 13, 26, 36, 39, 40 ], and several iterative methods have been proposed for solving EP(g) and related optimization problems, see for instance [1, 2, 4, 14, 15, 17, 18, 19, 28, 29, 32, 30, 31, 38, 41, 42] and reference therein. In solving the EP(g) (1.1) it is necessary to assume that the bifunction g satisfies the following assumptions:

• (A1) g(x,x) =0 for all x ∈ C;

• (A2) g is monotone, that is g(x,y)+g(y,x) ≤ 0 for all x,y ∈ C;

• (A3) For all x,y, z∈ C

limsupt0+g(tz+(1t)x,y)g(x,y);

• (A4) For all x ∈ C, g(x,⋅) is convex and lower semicontinuous.

Definition 1.1. Let f:E(,+] be a convex and Gâteaux differentiable function. The function Df:dom f×int(dom f)[0,+) defined by

Df(y,x)=f(y)f(x)f(x),yx

is called a Bregman distance with respect to f.

From the definition, we know that the following properties are satisfied (see [6]):

(i) The three points identity, for any xdom f and y,zint(dom f)

Df(x,y)+Df(y,z)Df(x,z)=f(z)f(y),xy;

(ii) Four point identity, for any x,wdom f and y,zint(dom f)

Df(x,y)Df(x,z)Df(w,y)+Df(w,z)=f(z)f(y),xw.

Definition 1.2. Let C be a nonempty closed convex subset of int(domf) and T:CC be a mapping. A point x ∈ C is called a fixed point of T if Tx =x. We denote the set of all fixed points of T by F(T). The mapping T:CC is called

(a) Bregman nonexpansive [33] if

Df(Tx,Ty)Df(x,y)x,yC;

Bregman nonspreading [23] if

Df(Tx,Ty)+Df(Ty,Tx)Df(Tx,y)+Df(Ty,x),  x,yC,

(c) (α,β,γ,δ)-generalized Bregman nonspreading [3, 16] if there exist α,β,γ,δ such that

αDf(Tx,Ty)+(1α)Df(x,Ty)+γ{Df(Ty,Tx)Df(Ty,x)}βDf(Tx,y)+(1β)Df(x,y)+δ{Df(y,Tx)Df(y,x)},x,yC.

for all x,yC.

Next, we introduce a n-generalized Bregman nonspreading mapping in Banach spaces.

Definition 1.3. Let f:E{+} be a convex and Gâteaux differentiable function and C be a nonempty closed convex subset of int(domf). A mapping T:CC is called a n-generalized Bregman nonspreading mapping if there exist αi,βi,γi,δi (i=1,2,,n) such that

k=1nαkDf(Tn+1kx,Ty)+(1k=1nαk)Df(x,Ty)+k=1nγkDf(Ty,Tn+1kx)Df(Ty,x)k=1nβkDf(Tn+1kx,y)+(1k=1nβk)Df(x,y)+k=1nδkDf(y,Tn+1kx)Df(y,x),

for all x,yC.

Remark 1.4. From Definition 1.3,

(a) when n=2, (1.4) becomes

α1Df(T2x,Ty)+α2Df(Tx,Ty)+(1α1α2)Df(x,Ty)+γ1(Df(Ty,T2x)Df(Ty,x))+γ2(Df(Ty,Tx)Df(Ty,x))β1Df(T2x,y)+β2Df(Tx,y)+(1β1β2)Df(x,y)+δ1(Df(y,T2x)Df(y,x))+δ2(Df(y,Tx)Df(y,x)),

which is called 2-generalized Bregman nonspreading in the sense of [44], where f(x)=12||x||2.

(b) When n=1, then (1.4) becomes

α1Df(Tx,Ty)+(1α1)Df(x,Ty)+γ1(Df(Ty,Tx)Df(Ty,x))    β1Df(Tx,y)+(1β1)Df(x,y)+δ1(Df(y,Tx)Df(y,x)),

which is the generalized Bregman nonspreading mapping in the sense of [3, 16]. Note that, the 2-generalized Bregman nonspreading mapping reduces to the generalized Bregman nonspreading mapping if α1=β1=γ1=δ1=0.

(c) The class of generalized Bregman nonspreading mapping reduces to Bregman nonspreading [23] if α1=β1=γ1=1 and δ1=0.

(d) The class of generalized Bregman nonspreading mapping reduces to Bregman nonexpansive [35] if α1=1 and β1=γ1=δ1=0.

We now present an example of Bregman nonspreading mapping which is not nonspreading in the usual Hilbert space setting.

Example 1.5. Let E= with the usual metric. Let f:E be defined by f(x)=x10 for all x and T:[0,0.85][0,0.85] be defined by Tx=x2. We first show that T is not nonspreading, i.e.,

||TxTy||2||xy||2+2xTx,yTyx,yC,

does not hold. Taking x = 0.5 and y = 0.85, then

||TxTy||2=(x2y2)2=[(0.5)2(0.85)2]2=0.22325625,

while

||xy||2+2xTx,yTy=(xy)2+2(xx2)(yy2)          =(0.50.85)2+2(0.50.52)(0.850.852)          =0.18625.

Hence, T is not nonspreading. Put

h(x,y)=Df(Tx,Ty)+Df(Ty,Tx)Df(Tx,y)Df(Ty,x).

By simple calculations, we obtain

Df(Tx,Ty)=x20+9y2010x2y18,Df(Ty,Tx)=y20+9x2010x18y2,Df(Tx,y)=x20+9y1010y2x9,Df(Ty,x)=y20+9x1010x9y2.

Then

h(x,y)=9y10(y101)+9x10(x101)10x2y9(y91)10x9y2(x91)  0,

for all x,y[0,0.85]. Thus T is Bregman nonspreading.

We further give an example of 2-generalized Bregman nonspreading mapping which is not necessarily 1-generalized Bregman nonspreading.

Example 1.6. Let E= and f(x)=x22 then the associated Bregman distance is given by

Df(x,y)=f(x)f(y)xy,f(y)  =12x212y2(xy)(y)  12(xy)2,x,y.

Define T:[0,2][0,2] by

Tx=0,ifx[0,2),1,ifx=2.

It is easy to see that F(T)={0}. Let

h(x,y)=α1Df(T2x,Ty)+α2Df(Tx,Ty)+(1α1α2)Df(x,Ty)  +γ1(Df(Ty,T2x)Df(Ty,x))+γ2(Df(Ty,Tx)Df(Ty,x))  β1Df(T2x,y)β2Df(Tx,y)(1β1β2)Df(x,y)  δ1(Df(y,T2x)Df(y,x))δ2(Df(y,Tx)Df(y,x)),

for all x,y[0,2]. We consider the following possible cases.

Case I: Suppose x=y=2, then Tx=Ty=1 and T2x=0. Thus

Df(Tx,Ty)=Df(Ty,Tx)=Df(x,y)=Df(y,x)=0,Df(x,Ty)=Df(Ty,x)=Df(Tx,y)=Df(y,Tx)=12,Df(T2x,Ty)=Df(Ty,T2x)=12,Df(T2x,y)=Df(y,T2x)=2.

Hence

h(x,y)=1212(α2+γ2+β2+δ2)2(β1+δ1).

Case II: Suppose x = 2 and y[0,2), then Tx = 1 and Ty = T2x = 0. Thus

Df(Tx,Ty)=Df(Ty,Tx)=12,Df(x,Ty)=Df(Ty,x)=2,Df(Tx,y)=Df(y,Tx)=12(y1)2,Df(x,y)=Df(y,x)=12(y2)2,Df(T2x,y)=Df(y,T2x)=y22,Df(T2x,Ty)=Df(Ty,T2x)=0.

Hence

h(x,y)=12(y24y)2(α1+γ1)32(α2+γ2)    2(y2)(β2+δ1)12(2y3)(β2+δ2).

Case III: Suppose x,y[0,2) then Tx=Ty=T2x=0. Thus

Df(Tx,Ty)=Df(Ty,Tx)=Df(T2x,Ty)=Df(Ty,T2x)=0,Df(x,y)=Df(y,x)=12(xy)2,Df(x,Ty)=Df(Ty,x)=x22,Df(Tx,y)=Df(y,Tx)=Df(T2x,y)=Df(y,T2x)=y22.

Hence

h(x,y)=(1α1α2)x22x22(γ1+γ2)y22(β1+β2)    12(1β1β2)(xy)2δ1xyx22δ2xyy22.

Choosing suitable choices of α1,α2,β1,β2,γ1,γ,δ1,δ2, for instance, α1=α2=β1=β2=γ1=γ2=1 and δ1=δ2=1, we see that h(x,y)0 for all the cases. Hence, T is 2-generalized Bregman nonspreading. However, in this case, T is not 1-generalized Bregman nonspreading (since α10,β10,γ10,δ10).

In 2010, by making use of the Bregman projection, Reich and Sabach [33] studied some approximation methods for finding common zeros of maximal monotone operators in reflexive Banach spaces. They also studied some approximation techniques for finding common solutions of finitely many Bregman nonexpansive operators, see [35]. In the same sense, Kassay et al. [20] studied the approximation of solutions of system of variational inequalities in reflexive Banach spaces. It is worth noting that extension of many theory from Hilbert space to general Banach space suffer some difficulties because many of the useful techniques employed in Hilbert space (for instance the inner product and the nonexpansiveness of resolvent operators) are no longer valid in Banach spaces setting.

Motivated by the works given in [21, 35, 46], we prove some properties of the n-generalized Bregman nonspreading mappings in reflexive Banach space. Further, we introduce a hybrid method for finding a common solution of countable family of equilibrium problem and finite family of fixed points of n-generalized Bregman nonspreading mapping in reflexive Banach space. We also discuss some applications and numerical example to demonstrate the applicability of our iterative algorithm and result. The method and results present in this paper generalized and unify many previously known related results, see for instance [21, 22, 35, 45, 46].

In this section, we recall some definitions and preliminary results which will be used in the sequel. We denote the strong convergence (resp. weak convergence) of a sequence {xn}E to a point x ∈ E by xnx (resp. xnx).

Let E be a real reflexive Banach space with the dual space E* and C a nonempty closed convex subset of E. Throughout this paper, we shall assume that the mapping f:E{+} is proper, convex and lower semi-continuous and also denote the domain of f by domf, where dom f={xE:f(x)<}. Let x∈ int(domf), the subdifferential of f at x is the convex set defined by

f(x)={x*E*:f(x)+x*,yxf(y), yE}

and the Frénchet conjugate of f is the function f*:E*(,+] defined by

f*(y*)=sup{y*,xf(x):xE}.

Let x ∈ int(domf), for any y ∈ E, the directional derivative of f at x is defined by

fo(x,y):=limh0f(x+hy)f(x)h.

If the limit in (2.1) exists as h0 for each y, then the function f is said to be Ga^teaux differentiable at x. In this case, the gradient of f at x is the linear function f(x), which is defined by f(x),y:=fo(x,y) for all y ∈ E. The function f is said to be Ga^teaux differentiable if it is Ga^teaux differentiable at each x ∈ int(dom f). When the limit as h0 in (2.1) is attained uniformly for any y ∈ E with ||y||=1, we say that f is Fréchet differentiable at x. It is well known that f is Ga^teaux (resp. Fréchet) differentiable at x ∈ int(dom f) if and only if the gradient ∇ f is norm-to-weak* (resp. norm-to-norm) continuous at x (see [6]).

Let E be a reflexive Banach space. The function f is called Legendre if and only if it satisfies the following two conditions:

(L1) f is Gâteaux differentiable, int(dom f) and dom f = int(dom f),

(L2) f* is Gâteaux differentiable, int(dom f*) and dom f* = int(dom f*).

Since E is reflexive, we know that (f)1=f*, this together with conditions (L1) and (L2) implies that

ranf=domf*=int(domf*),

and

ranf*=domf=int(domf).

The notion of Legendre function in infinite dimensional spaces was first introduced by Bauschke, Borwein and Combettes in [6]. By their definition, the conditions (L1) and (L2) also yield that f and f* are Gâteaux differentiable and strictly convex in the interior of their respective domains. It follows that f is Legendre if and only if f* is Legendre (see [6], Corollary 5.5, p. 634).

One important and interesting example of Legendre function is 1p||||p (1<p<) when E is a smooth and strictly convex Banach space. In this case, the gradient f of f coincide with the generalized duality mapping of E. More examples of Legendre functions can be found in [5, 6]. In the rest of this paper, we always assume that f:E{+} is a Legendre function.

Definition 2.1. Let f:E(,+] be a convex and Gâteaux differentiable function. The Bregman projection of xint(domf) onto the nonempty, closed and convex subset Cdomf is the necessarily unique vector ProjCf(x)C satisfying

Df(ProjCf(x),x)=inf{Df(y,x):yC}.

Remark 2.2.

1. If E is a Hilbert space and f(x)=12||x||2, then the Bregman projection ProjCf(x) is reduced to the metric projection of x onto C.

2. If E is smooth and strictly convex and f(x)=1p||x||p (1<p<), then the Bregman projection ProjCf(x) reduces to the generalized projection ΠC(x), which is defined by

Dp(ΠC(x),x):=inf{Dp(z,x):zC}.

It is known from [10] that z = ProjCf(x) if and only if

f(x)f(z),yz0  for all yC.

We also have

Df(y,ProjCf(x))+Df(ProjCf(x),x)Df(y,x)  for all xE, yC.

Similar to the metric projection in Hilbert space, the Bregman projection also has a variational characterization which is given below.

Lemma 2.3. [33] (Characterization of Bregman Projection)) Let f be totally convex on int(domf). Let C be a nonempty, closed and convex subset of int(domf) and x ∈ int(domf), if ω ∈ C, then the following conditions are equivalent:

(i)the vector ω is the Bregman projection of x onto C, with respect to f,

(ii) the vector ω is the unique solution of the variational inequality

f(x)f(z),zy0  yC,

(iii) the vector ω is the unique solution of the inequality

Df(y,z)+Df(z,x)Df(y,x)  yC.

Definition 2.4. Let f:E(,+] be a convex and Ga^teaux differentiable function. The function f is called:

(i) totally convex at x if its modulus of totally convexity at x ∈ int(domf), that is, the bifunction vf: int(domf)×[0,+)[0,+) defined by

vf(x,t):=inf{Df(y,x):ydomf,||yx||=t}

is positive for any t>0,

(ii) totally convex if it is totally convex at every point x ∈ int(dom f),

(iii) totally convex on bounded subset B of E, if vf(B,t) is positive for any nonempty bounded subset B, where the function vf: int(dom f)×[0,+)[0,+] is defined by

vf(B,t):=inf{vf(x,t):xBint(domf)},t>0.

(iv) cofinite if domf*=E*,

(v) coercive if lim||x||+(f(x)||x||)=+,

(vi) sequentially consistent if for any two sequences {xn} and {yn} in E such that {xn} is bounded,

limnDf(yn,xn)=0limn||ynxn||=0.

For further details and examples on totally convex functions see [8, 9, 10].

Lemma 2.5. ([9]) The function f:E is totally convex on bounded subsets if and only if it is sequentially consistent.

Lemma 2.6. ([34]) Let f:E be a Ga^teaux differentiable and totally convex function. If x0E and the sequence {Df(x0,xn)} is bounded, then the sequence {xn} is also bounded.

Lemma 2.7. ([10]) Let f:E(,+] be a convex function whose domain contains at-least two points. Then the following statements holds:

(i) f is sequentially consistent if and only if it is totally convex on bounded subsets.

(ii) If f is lower semicontinuous, then f is sequential consistent if and only if it is uniformly convex on bounded subsets.

(iii) If f is uniformly strictly convex on bounded subsets, then it is sequentially consistent and the converse implication holds when f is lower semicontinuous, Fréchet differentiable on its domain, and the Fréchet derivative f is uniformly continuous on bounded subsets.

Lemma 2.8. ([33]) If f:E is 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*.

Let f:E be a convex Legendre and Gâteaux differentiable function. The function Vf:E×E*[0,) associated with f defined by

Vf(x,x*)=f(x)x*,x+f*(x*),  xE,x*E*.

Then, Vf is non-negative and Vf(x,x*)=Df(x,f*(x*)) for all xE and x*E*. More so, by the subdifferential inequality,

Vf(x,x*)+y*,f*(x*)xVf(x,x*+y*)

for all xE and x*,y*E* (see [24]). In addition, if f:E(,+] is a proper lower semicontinuous function, then f*:E*(,+] is a proper weak* lower semicontinuous and convex function. Hence, Vf is convex in the second variable. Thus, for all z ∈ E

Df(z,f*( i=1Ntif(xi))) i=1NtiDf(z,xi),

where {xi}E and {ti}(0,1) with i=1Nti=1.

Let E be a Banach space and let Br:={zE:||z||r} for all r>0. Then, a function f:E is said to be uniformly convex on bounded subsets of E if ρr(t)>0 for all t≥ 0, where ρr:[0,+)[0,]is defined by

ρr(t)=infx,yBr,||xy||=t,α(0,1)αf(x)+(1α)f(y)f(αx+(1α)y)α(1α).

The function ρr is called the gauge of uniform convexity of f. More so, the function f:E(,+] is called totally coercive if

lim||x||+(f(x)||x||)=+.

Lemma 2.9. ([27]) Let r > 0 be a constant and let f:E be a continuous uniformly convex function on bounded subsets of E. Then

f( k=0αkxk) k=0αkf(xk)αiαjρr*(||xixj||),

for all i,j0, xkBr, αk(0,1) and k0 with k=0αk=1, where ρr* is the gauge of uniform convexity of f.

Let l be the Banach lattice of bounded real sequences with the supremum norm. It is well known that there exists a bounded linear functional µ on l such that the following three conditions hold:

(i) if {tn} in l and tn0 for every n, then μ({tn})0,

(ii) if tn=1 for every n, then μ({tn})=1,

(iii) μ({tn+1})=μ({tn}) for all {tn} in l.

Here, {tn+1} denotes the sequence (t2,t3,,tn,tn+1,) in l. Such a functional µ is called a Banach limit and the value of µ at {tn} in l is denoted by μntn. Therefore, condition (3) means μntn=μntn+1. If µ satisfies conditions (1) and (2), we call µ a mean on l (see, for example, [43] for more details).

Lemma 2.10. ([12]) Let C be a nonempty, closed and convex subset of a real reflexive Banach space E. Let f:E be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and local uniformly convex on E. Let T:CC be a mapping and {xn} be a bounded sequence of C and µ be a mean on l. Supposet that

μnDf(xn,Ty)μnDf(xn,y)yC.

Then, T has a fixed point in C.

Let T be a mapping from C into itself. A point x ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {xn} in C which converges weakly to p and limn||xnTxn||=0. We denote the set of all asymptotic fixed points of T by F^(T).

Recall that a mapping T:CC is said to be Bregman quasi-nonexpansive [27] if F(T) and

Df(p,Tx)Df(p,x)xC,pF(T).

A mapping T:CC is to be Bregman relatively nonexpansive [27] if the following conditions are satisfied:

(i) F(T) is nonempty;

(ii) Df(p,Tv)Df(p,v), pF(T), vC;

(iii) F^(T)=F(T).

Lemma 2.11. ([37]) Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and let f:E be a strictly convex and Gâteaux differentiable function. Let g:C×C be a bifunction satisfying conditions (A1)-(A4). For all λ>0 be any given number and x ∈ E, there exists z ∈ C such that

g(z,y)+1r(z)(x),yz0,  yC.

Define the resolvent mapping Tr:E2C as follows

Resλ,gf(x)={zC:g(z,y)+1rf(z)f(x),yz0,  yC},

then, Resλ,gf has the following properties:

(i) Resλ,gf is single-valued;

(ii) Resλ,gf is a firmly nonexpansive mapping, that is;

Resλ,gfzResλ,gfy,f(Resλ,gfz)f(Resλ,gfy)Resλ,gfzResλ,gfy,f(z)f(y)

z,yE;

(iii) F(Resλ,gf)=EP(g);

(iv) EP(g) is closed and convex.

It is easy to see that the resolvent operator satisfies the following inequality: for all r > 0, u ∈ EP(g) and x ∈ E, then

Df(x,Resλ,gfx)+Df(Resλ,gfx,u)Df(x,u).

In this section, we present the existence and some properties of fixed points of n-generalized Bregman nonspreading mapping in a reflexive Banach space. This result extend the corresponding results of [45] and [25] to reflexive Banach space.

Proposition 3.1. Let E be a real reflexive Banach space and f:E be a strictly convex and Gâteaux differentiable function. Let Cint(domf) be a nonempty, closed and convex set and T:CC be a n-generalized Bregman nonspreading mapping. Then, the following are equivalent

(i) F(T) is nonempty;

(ii) {Tmz} is bounded for some z ∈ C and m.

proof First we show that (i) implies (ii). Suppose F(T), then {Tmz}={z} for zF(T). So {Tmz} is bounded. Next, we show that (ii) implies (i). Let {Tmz} be bounded for some z ∈ C. Since T is n-Bregman generalized nonspreading, then there exist αi,βi,γi,δi for i=1,2,,n,

such that

k=1nαkDf(Tn+1kx,Ty)+(1k=1nαk)Df(x,Ty)+k=1nγk{Df(Ty,Tn+1kx)Df(Ty,x)}k=1nβkDf(Tn+1kx,y)+(1k=1nβk)Df(x,y)+k=1nδk{Df(y,Tn+1kx)Df(y,x)},

for all x,y ∈ C. Replacing x by Tm-1z in (3.1), we have that for any y,z ∈ C,

k=1nαkDf(Tn+1kTm1z,Ty)+(1k=1nαk)Df(Tm1z,Ty)+k=1nγk{Df(Ty,Tn+1kTm1z)Df(Ty,Tm1z)}k=1nβkDf(Tn+1kTm1z,y)+(1k=1nβk)Df(Tm1z,y)+k=1nδk{Df(y,Tn+1kTm1z)Df(y,Tm1z)}.

Since {Tmz} is bounded, we can apply Banach limit µ to both sides of (3.2), then we have

μm(k=1nαkDf(Tm+nkz,Ty)+(1k=1nαk)Df(Tm1z,Ty)+k=1nγk{Df(Ty,Tm+nkz)Df(Ty,Tm1z)})μm(k=1nβkDf(Tm+nkz,y)+(1k=1nβk)Df(Tm1z,y)+k=1nδk{Df(y,Tm+nkz)Df(y,Tm1z)}).

Thus, we obtain

k=1nαkμmDf(Tm+nkz,Ty)+(1k=1nαk)μmDf(Tm1z,Ty)+k=1nγk{μmDf(Ty,Tm+nkz)μmDf(Ty,Tm1z)}k=1nβkμmDf(Tm+nkz,y)+(1k=1nβk)μmDf(Tm1z,y)+k=1nδk{μmDf(y,Tm+nkz)μmDf(y,Tm1z)}.

Then

k=1nαkμmDf(Tmz,Ty)+(1k=1nαk)μmDf(Tmz,Ty)+k=1nγk{μmDf(Ty,Tmz)μmDf(Ty,Tmz)}k=1nβkμmDf(Tmz,y)+(1k=1nβk)μmDf(Tmz,y)+k=1nδk{μmDf(y,Tmz)μmDf(y,Tmz)}.

Hence

μmDf(Tmz,Ty)μmDf(Tmz,y).

Therefore by Lemma 2.10, T has a fixed point in C. This completes the proof.

The following results follow as direct consequences of Theorem 3.1.

Corollary 3.2. Let C be a nonempty, closed and convex subset of a smooth, strictly convex Banach space E, let p be a real number such that 1<p<+ and let f be a function defined by f(x)=1p||x||p and T:CC be a n-generalized Bregman nonspreading mapping. Then, the following assertions are equivalent:

(i) F(T) is nonempty;

(ii) {Tmz} is bounded for some z ∈ C.

Corollary 3.3. Let C be a nonempty bounded closed convex subset of a real reflexive Banach space E and f:E be a strictly convex and Gâteaux differentiable function. Let T:CC be a n-generalized Bregman nonspreading mapping. Then, T has a fixed point.

Remark 3.4. Corollary 3.2 is a generalization of the corresponding result in Theorem 3.2 of [45], where the equivalence between the two assertions was shown for p=2.

We now show another important property of the fixed points of n-generalized Bregman nonspreading mapping.

Proposition 3.5. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and f:E be a strictly convex and Gâteaux differentiable function. Let T:CC be a n-generalized Bregman nonspreading mapping such that F(T). Then F(T) is closed and convex.

Proof. Let uF(T), then putting u=xF(T) in (1.4), we have

k=1nαkDf(u,Ty)+(1k=1nαk)Df(u,Ty)+k=1nγk{Df(Ty,u)Df(Ty,u)}k=1nβkDf(u,y)+(1k=1nβk)Df(u,y)+k=1nδk{Df(y,u)Df(y,u)},

which implies that

Df(u,Ty)Df(u,y),  uF(T),yC.

This means that T is quasi-Bregman nonexpansive. Now let {xn}F(T) such that xnp. Then

Df(p,Tp)=limnDf(xn,Tp)Df(xn,p)=Df(p,p)=0.

Hence, pF(T). Therefore F(T) is closed.

Next, we show that F(T) is convex. For any x,yF(T) and λ(0,1), let z=λx+(1λ)y. Then

Df(z,Tz)=f(z)f(Tz)f(Tz),zTz    =f(z)f(Tz)f(Tz),λx+(1λ)yTz    =f(z)+λDf(x,Tz)+(1λ)Df(y,Tz)λf(x)(1λ)f(y)    f(z)+λDf(x,z)+(1λ)Df(y,z)λf(x)(1λ)f(y)    =f(z)f(z)f(z),λx+(1λ)yz    =f(z)f(z)f(z),zz    =0.

Hence, z = Tz. Therefore, F(T) is convex.

Using Corollary 3.3 and Proposition 3.5, we prove the following common fixed point theorem for a commutative family of n-generalized Bregman nonspreading mapping in a reflexive Banach space.

Theorem 3.6. Let f:E be a strictly convex and Gâteaux differentiable function, C be a nonempty bounded closed convex subset of a real reflexive Banach space E and let {Tα}αI be a commutative family of n-generalized Bregman nonspreading mappings from C into itself. Then {Tα}αI has a common fixed point.

Proof. By Theorem 3.5, we know that F(Tα) is a closed convex subset of C. Since E is reflexive and C is a bounded closed and convex subset, C is weakly compact. To show that αIF(Tα) is nonempty, it is sufficient to show that {F(Tα)}αI has a nonempty finite intersection property.

Now, let {T1,T2,,TN} be a commutative finite family of n-generalized Bregman nonspreading mapping from C into itself. We prove by induction that {T1,T2,,TN} has a common fixed point. To do this, we start by showing the case for N=2. By Corollary 3.3 and Theorem 3.5, F(T1) is nonempty, bounded, closed and convex. Let uF(T1), since T1T2=T2T1, then we have T1T2u=T2T1u=T2u. This implies that T2uF(T1). Hence, F(T1) is T2-invariant. Thus, the restriction of T2 to F(T1) is a n-generalized Bregman nonspreading self mapping. By Corollary (3.3), T2 has a fixed point in F(T1), that is, we have zF(T1) such that T2z=z. Hence, zF(T1)F(T2).

Suppose that for some N ≥ 2, Γ=k=1NF(Tk) is nonempty. Then Γ is a nonempty, bounded, closed and convex subset of C and the restriction of TN+1 to Γ is a n-generalized Bregman nonspreading self mapping. By Corollary 3.3, TN+1 has a fixed point in Γ. This implies that ΓF(TN+1) is nonempty. Hence, k=1N+1F(Tk) is nonempty. This completes the proof.

The following result will be used in the sequel.

Proposition 3.7. Let E be a real reflexive Banach space and let C be a nonempty, closed and convex subset of E. Let f:E be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T:CC be a n-generalized Bregman nonspreading mapping. Then, for any x,yC, αi,βi,γi,δi, for i=1,2,,n, we have

0k=1n(βkαk)(Df(Tn+1kx,Ty)Df(x,Ty))+Df(Ty,y)+f(Ty)f(y),k=1nβk(Tn+1kxx)+xTy+k=1nδk{Df(y,Tn+1kx)Df(y,x)}k=1nγk{Df(Ty,Tn+1kx)Df(Ty,x)}.

Proof. From the definition of n-generalized Bregman nonspreading mapping, we have

k=1nαkDf(Tn+1kx,Ty)+(1k=1nαk)Df(x,Ty)+k=1nγk{Df(Ty,Tn+1kx)Df(Ty,x)}k=1nβkDf(Tn+1kx,y)+(1k=1nβk)Df(x,y)+k=1nδk{Df(y,Tn+1kx)Df(y,x)},

for all x,yC. This implies that

0k=1nβkDf(Tn+1kx,y)+(1k=1nβk)Df(x,y)+k=1nδk{Df(y,Tn+1kx)Df(y,x)}k=1nαkDf(Tn+1kx,Ty)(1k=1nαk)Df(x,Ty)k=1nγk{Df(Ty,Tn+1kx)Df(Ty,x)}.

Hence, from the three points identity (1.2), we have

0k=1nβk(Df(Tn+1kx,Ty)+Df(Ty,y)+f(Ty)f(y),Tn+1kxTy)+(1k=1nβk)(Df(x,Ty)+Df(Ty,y)+f(Ty)f(y),xTy)k=1nαkDf(Tn+1kx,Ty)(1k=1nαk)Df(x,Ty)k=1nγk{Df(Ty,Tn+1kx)Df(Ty,x)}+k=1nδk{Df(y,Tn+1kx)Df(y,x)}.

Therefore

0k=1n(βkαk)(Df(Tn+1kx,Ty)Df(x,Ty))+Df(Ty,y)+f(Ty)f(y),k=1nβk(Tn+1kxx)+xTy+k=1nδk{Df(y,Tn+1kx)Df(y,x)}k=1nγk{Df(Ty,Tn+1kx)Df(Ty,x)}.

The following result is another important property which characterized the n-generalized Bregman nonspreading mapping.

Proposition 3.8. Let T:CC be a n-generalized Bregman nonspreading mapping. Suppose F(T), then T is Bregman relatively nonexpansive.

Proof. It is clear that

Df(p,Tx)Df(p,x)pF(T),xC.

We show that F^(T)=F(T). It is easy to see that F(T)F^(T). Now let pF^(T), that is, there exist a sequence {xn}C such that xnp and ||xnTxn||0. Since f is uniformly Frćhet differentiable on bounded subsets of E, then ∇f is uniformly continuous and thus

limn||f(xn)f(Txn)||=limn||f(xn)f(Txn)||=0.

Putting x = xn and y = q in Proposition 3.7, we have

0k=1n(βkαk)(Df(Tn+1kxn,Tq)Df(xn,Tq))+Df(Tq,q)+f(Tq)f(q),k=1nβk(Tn+1kxnxn)+xnTq+k=1nδk{Df(q,Tn+1kxn)Df(q,xn)}k=1nγk{Df(Tq,Tn+1kxn)Df(Tq,xn)}.

Observe that

Df(Tn+1kxn,Tq)Df(xn,Tq)=f(Tn+1kxn)f(Tq)            f(Tq),Tn+1kxnTq            f(xn)+f(Tq)+f(Tq),xnTq          =f(Tn+1kxn)f(xn)            +f(Tq),xnTq            f(Tq),Tn+1kxnTq          =f(Tn+1kxn)f(xn)            +f(Tq),xnTn+1kxn.

Similarly

Df(q,Tn+1kxn)Df(q,xn)=f(xn)f(Tn+1kxn)+f(xn),Tn+1kxnxn            +f(xn)f(Tn+1kxn),qxn,

and

Df(Tq,Tn+1kxn)Df(Tq,xn)=f(xn)f(Tn+1kxn)            +f(xn),Tn+1kxnxn            +f(xn)f(Tn+1kxn),Tqxn.

Substituting (3.10), (3.11) and (3.12) into (3.9), we have

0k=1n(βkαk)(f(Tn+1kxn)f(xn)+f(Tq),xnTn+1kxn)+Df(Tq,q)+f(Tq)f(q),k=1nβk(Tn+1kxnxn)+xnTq+k=1nδk{f(xn)f(Tn+1kxn)+f(xn),Tn+1kxnxn+f(xn)f(Tn+1kxn),qxn}k=1nγk{f(xn)f(Tn+1kxn)+f(xn),Tn+1kxnxn+f(xn)f(Tn+1kxn),Tqxn}.

Taking limit as n in (3.13) and using (3.8), we have

0 Df(Tq,q)+f(Tq)f(q),qTq.

Using the four points identity (1.3), we have

0Df(Tq,q)+Df(Tq,Tq)Df(Tq,q)Df(q,Tq)+Df(q,q)=Df(q,Tq).

Thus Df(q,Tq)0 and then Df(q,Tq)=0. Since f is strictly convex, we have q = Tq. Hence, qF(T). Therefore F^(T)F(T). This thus implies that F^(T)=F(T).

In this section, we introduce a hybrid algorithm for finding common solutions of countable family of equilibrium problem and finite fixed points of n-generalized Bregman nonspreading mapping in reflexive Banach space.

Let {αn,i:n,i,1i} be sequences of real numbers such that {αn,i}(0,1). We define the following Wn:CC mapping generated by Ti,i=1,2,,N and {αn,i}, where Ti:CC is a finite family of n-generalized Bregman nonspreading mappings.

Sn,0x=x,Sn,1x=f*[αn,1f(T1x)+(1αn,1)f(x)]Sn,2x=f*[αn,2f(T2Sn,1x)+(1αn,2)f(Sn,1x)]Sn,3x=f*[αn,3f(T3Sn,2x)+(1αn,3)f(Sn,2x)]  Sn,N1x=f*[αn,N1f(TN1Sn,N2x)+(1αn,N1)f(Sn,N2x)]Wn=Sn,N=f*[αn,Nf(TNSn,N1x)+(1αn,N)f(Sn,N1x)].

Using the above definition, we have the following lemma.

Proposition 4.1. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and let f:E be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let {Ti}i=1N be a finite famiy of n-generalized Bregman nonspreading mapping of C into itself such that i=1NF(Ti). Let {αn,i} be real sequence in (0,1) such that liminfnαn,i>0, i{1,2,,N}. Let Wn be a Bregman W-mapping generated by T1,T2,,TN in (4.1). Then

(i) i=1NF(Ti)=F(Wn),

(ii) Wn is Bregman quasi-nonexpansive,

(iii) If in addition, Ti is Bregman relatively nonexpansive mapping, for each i, then Wn is Bregman relatively nonexpansive.

Proof. Let xi=1NF(Ti). Then Tix=x, i=1,2,,N. From (4.1), we have that Sn,1x=x, Sn,2x=x,, Sn,Nx=x. Thus i=1NF(Ti)F(Wn). Conversely, let yF(Wn) and xi=1NF(Ti). Then

Df(x,y)=Df(x,Wny)    =Df(x,f*(αn,Nf(TNSn,N1y)+(1αn,N)f(Sn,N1y)))    =f(x)x,αn,Nf(TNSn,N1y)+(1αn,N)f(Sn,N1y)    +f*(αn,Nf(TNSn,N1y)+(1αn,N)f(Sn,N1y))    αn,N(f(x)x,f(TNSn,N1y)+f*(f(TNSn,N1y)))    +(1αn,N)(f(x)x,f(Sn,N1y)+f*(f(TNSn,N1y)))    αn,N(1αn,N)ρr*(||f(TNSn,N1y)f(Sn,N1y)||)    =αn,NDf(x,TNSn,N1y)+(1αn,N)Df(x,Sn,N1y)    αn,N(1αn,N)ρr*(||f(TNSn,N1y)f(Sn,N1y))    Df(x,Sn,N1y)    αn,N(1αn,N)ρr*(||f(TNSn,N1y)f(Sn,N1y)||)        Df(x,y)αn,1(1αn,1)ρr*(||f(T1y)f(y)||)    αn,2(1αn,2)ρr*(||f(T2Sn,1y)f(Sn,1y)||)    αn,N(1αn,N)ρr*(||f(TNSn,N1y)f(Sn,N1y)||).

This implies that

αn,1(1αn,1)ρr*(||f(T1y)f(y)||)    =αn,2(1αn,2)ρr*(||f(T2Sn,1y)f(Sn,1y)||)    ==αn,N(1αn,N)ρr*(||f(TNSn,N1y)f(Sn,N1y)||)=0.

Then by the property of ρr* from Lemma 2.9 and the norm-to-norm continuity of ∇ f*, we have

T1y=y,T2Sn,1y=Sn,1y,TNSn,N1=Sn,N1y.

It follows that

Df(y,Sn,1y)=Df(y,f*(αn,1f(T1y)+(1αn,1)f(y)))    αn,1Df(y,T1y)+(1αn,1)Df(y,y)=0.

Therefore yF(Sn,1) and consequently, yF(T1). Following similar argument, we have that yF(Ti) for i=1,2,,N and hence y i=1NF(Ti).

(ii) Let yF(Wn). Then

Df(y,Wnx)=Df(y,f*(αn,Nf(TNSn,N1x)+(1αn,N)f(Sn,N1x)))    αn,NDf(y,TNSn,N1x)+(1αn,N)Df(y,Sn,N1x)    αn,NDf(y,Sn,N1x)+(1αn,N)Df(y,Sn,N1x)    =Df(y,Sn,N1x)=Df(y,f*(αn,N1f(TN1Sn,N2x)    +(1αn,N1)f(Sn,N2x)))    αn,N1Df(y,TN1Sn,N2x)+(1αn,N1)Df(y,Sn,N2x)    Df(y,Sn,N2x)        Df(y,x).

(iii) Let {xn}C such that xnx¯ and ||Wnxnxn||0 as n. From (1.2), we have

Df(x¯,Wnxn)Df(x¯,xn)αn,1(1αn,1)ρr*(||f(T1xn)f(xn)||)      αn,2(1αn,2)ρr*(||f(T2Sn,1xn)f(Sn,1xn)||)      αn,N(1αn,N)ρr*(||f(TNSn,N1xn)f(Sn,N1xn)||).

Using three points identity (1.2), we obtain

Df(x¯,xn)Df(x¯,Wnxn)=x¯xn,f(Wnxn)f(xn)          Df(xn,Wnxn).

Since xnx¯ and limn||xnWnxn||=0, we obtain

|Df(x¯,xn)Df(z¯,Wnxn)|||x¯xn||||f(Wnxn)f(xn)||          Df(xn,Wnxn)0as  n.

Therefore from (4.3), we have

αn,1(1αn,1)ρr*(||f(T1xn)f(xn)||)+αn,2(1αn,2)ρr*(||f(T2Sn,1xn)  f(Sn,1xn)||)+  +αn,N(1αn,N)ρr*(||f(TNSn,N1xn)f(Sn,N1xn)||)  Df(x¯,xn)Df(x¯,xn).

Taking limit as n, using (4.5) and property of ρr*, yields

limn||f(T1xn)f(xn)||=limn||f(T2Sn,1xn)f(Sn,1xn)||=      =limn||f(TNSn,N1xn)f(Sn,N1xn)||=0.

By the norm-to-norm uniform continuity of ∇f on bounded subset of E*, it follows that

limn||T1xnxn||=limn||T2Sn,1xnSn,1xn||=      =limn||TNSn,N1xnSn,N1xn||=0.

We next prove that Sn,ixnxn0 for each i=1,2,,N1. From (4.6), we get

Dp(xn,Sn,1xn)=Df(xn,f*[αn,1f(T1xn)+(1αn,1)f(xn)])      αn,1Df(xn,T1xn)+(1αn,1)Df(xn,xn).

Taking limit as n and using (4.6), we have

limnDf(xn,Sn,1xn)=0,

hence

limn||Sn,1xnxn||=0.

Thus

||T2Sn,1xnxn||||T2Sn,1xnSn,1xn||+||Sn,1xnxn||0n.

Similarly, we have

Df(xn,Sn,2xn)=Df(xn,f*[αn,2f(T2Sn,1xn)+(1αn,2)f(Sn,1xn)])      αn,2Df(xn,T2Sn,1xn)+(1αn,2)Df(xn,Sn,1xn)

Taking limit as n, we have

limnDf(xn,Sn,2xn)=0,

and hence

limn||Sn,2xnxn||=0.

Following similar approach as above, we have

limn||Sn,3xnxn||=limn||Sn,4xnxn||==limn||Sn,N1xnxn||=0.

Therefore

limn||Sn,ixnxn||=0for each  i=1,2,,N1.

This together with the Bregman relative nonexpansiveness of each Ti for i=1,2,,N, implies that x¯F(Sn,i) for i=1,2,,N. Hence x¯F(Wn). This therefore implies that Wn is Bregman relatively nonexpansive.

We are now in position to introduce our iterative algorithm.

Theorem 4.2. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and f:E be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. For i=1,2,,N, let {αn,i}(0,1), Ti:CC be finite family of n-generalized Bregman nonspreading mappings and Wn:CC be a Bregman W-mapping generated by {αn,i} and T1,T2,,TN in (4.1). Let gj:C×C be bifunctions satisfying assumptions (A1)-(A4) and suppose Γ:= i=1NF(Ti) j=1EP(gi). Define the sequence {xn} by the following process

x0=xC,C0=Q0=C,zn=f*[βn,0f(xn)+ j=1βn,jf(Resλn,gjfxn)],yn=f*[δnf(xn)+(1δn)f(Wnzn)],Cn={zC:Df(z,yn)Df(z,xn)},Qn={zC:f(x)f(xn),xnz0},xn+1=ProjCnQnfx,

for all n ≥ 0, where {λn}(0,), {βn,j} and {δn} are sequences in [0,1) satisfying the following control conditions:

(i) j=0βn,j=1,  n{0};

(ii) There exists k such that liminfnβn,jβn,k>0, j{0};

(iii) 0δn<1, n and liminfnδn<1;

(iv) liminfnλn>0.

Then, the sequence {xn} converges strongly to ProjΓfx as n.

Proof. We divide the proof into several steps.

Step 1: We show that ΓCnQn and xn+1 is well defined.

It is clear that Cn and Qn are closed and convex. Then CnQn is closed and convex for n0. Obviously, ΓC0Q0. Suppose ΓCmQm for some m.

Let pΓ, then

Df(p,ym)=Df(p,f*[δmf(xm)+(1δm)f(Wmzm)])    =Vf(p,δmf(xm)+(1δm)f(Wmzm))    =f(p)p,δmf(xm)+(1δm)f(Wmzm)    +f*(δmf(xm)+(1δm)f(Wmzm))    δm[f(p)p,f(xm)+f*(xm)]    +(1δm)[f(p)p,f(Wmzm)+f*(Wmzm)]    δm(1δm)ρr*(||xmWmzm||)    δmDf(p,xm)+(1δm)Df(p,zm)δm(1δm)ρr*(||xmWnzm||)    =δnDf(p,xm)+(1δm)Df(p,f*[βm,0f(xm)    + j=1βm,jf(ResEP(g)fxm)])    δm(1δm)ρr*(||xmWmzm||).

Hence

Df(p,ym)δmDf(p,xm)+(1δm)[βm,0Df(p,xm)    + j=1βm,jDf(p,ResEP(g)fxm)    βm,0 j=1βm,jρr*(||xmResEP(g)fxm||)]    δm(1δm)ρr*(||xmWmzm||)    δmDf(p,xm)+(1δm)[βm,0Df(p,xm)+ j=1βm,jDf(p,xm)]    (1δm)βm,0 j=1βm,jρr*(||xmResEP(g)fxm||)    δn(1δm)ρr*(||xmWmzm||)    =Df(p,xm)(1δm)βm,0 j=1βm,jρr*(||xmResEP(g)fxm||)    δn(1δn)ρr*(||xmWmzm||)    Df(p,xm).

Hence pCm, which implies that ΓCm. Since xm+1=ProjCmQmfx, then f(x)f(xm+1),zxm+10  zCmQm. In particular, f(x)f(xm+1),pxm+10 pΓ. Thus pQm+1. This proves that ΓCm+1Qm+1. Therefore ΓCnQn  n0. Consequently, since CnQn is closed and convex, then xn+1=ProfCnQnfx is well-defined.

Step 2: We prove that {xn},{yn},{zn},{Resλn,gjfxn} and {Wnzn} are bounded.

Since ΓCnQn for every n0 and xn+1=ProjCnQnfx, then

Df(p,xn+1)Df(p,x) n0.

So {Df(p,xn)} is bounded and hence there exists a constant M>0 such that

Df(p,xn)M n{0}.

In view of Lemma 2.6, we conclude that the sequence {xn} is bounded. Similarly, the sequences {yn},{zn},{Resλn,gjfxn} and {Wnzn} are bounded.

Step 3: Next, we show that limn||xn+1xn||=0, limn||Resλn,gjfxnxn||=0 and limn||Wnznzn||=0.

Since xn+1CnQnQn and xn=ProjQnf(x), we have

Df(xn+1,ProjQnf(x))+Df(ProjQnf(x1),x)Df(xn+1,x).

Thus

Df(xn+1,xn)+Df(xn,x)Df(xn+1,x).

Therefore the sequence {Df(xn,x)} is non-decreasing and thus limnDf(xn,x) exists. Hence, it follows that limnDf(xn+1,xn)=0, and by Lemma 2.5, we have

limn||xn+1xn||=0.

Also, since xn+1Cn, we have

Df(xn+1,yn)Df(xn+1,xn).

This yields that limnDf(xn+1,yn)=0 and thus

limn||xn+1yn||=0.

Therefore from (4.11) and (4.12), we get

limn||ynxn||=0.

By the uniform continuity of f and ∇f on bounded subsets of E and E* respectively, we have

limn||f(yn)f(xn)||=0

and

limn||f(yn)f(xn)||*=0.

Furthermore,

Df(p,xn)Df(p,yn)=f(p)f(xn)pxn,f(xn)          f(p)+f(yn)+pyn,f(yn)          =f(yn)f(xn)+pyn,f(yn)pxn,f(xn)          =f(yn)f(xn)+xnyn,f(yn)          pxn,f(yn)f(xn).

Therefore from (4.12) - (4.14), we get

limn[Df(p,xn)Df(p,yn)]=0.

Note that from (4.8), we have

Df(p,yn)Df(p,xn)(1δn)βn,0 j=1βn,jρr*(||xnResλn,gjfxn||)    δn(1δn)ρr*(||xnWnzn||).

Using the property of ρr* and conditions (ii) and (iii) together with (4.15), we have

limn||xnResλn,gjfxn||=0

and

limn||xnWnzn||=0.

By the uniform continuity of ∇f on bounded subsets of E*, we have

limn||f(xn)f(Resλn,gjfxn)||=0.

Hence from (4.7), we get

limn||f(zn)f(xn)||=limn j=1βn,j||f(Resλn,gjfxn)f(xn)||=0.

Furthermore, since f is Fréchet differentiable on bounded subset of E, then ∇f* is uniformly continuous on bounded subsets of E*. Thus

limn||znxn||=0.

Therefore

limn||Wnznzn||=limn[||Wnznxn||+||xnzn||]=0.

Since {xn} is bounded, there exists a subsequence {xnk} of {xn} which converges weakly to qE. Since ||Wnznzn||0 and ||znxn||0 as n, then from Lemma 2.11 we have that qF(Wn). Hence q i=1NF(Ti).

Also from Lemma 2.11, we have for each j=1,2,

gj(Resλn,gjfxn,y)+1λnyResλn,gjfxn,f(Resλn,gjfxn)f(xn)0yC.

Hence

gj(Resλ nk ,gjfxnk,y)+1λ nk yResλ nk ,gjfxnk,f(Resλ nk ,gjfxnk)f(xnk)0yC.

From the assumption (A2), we have

1λnk||yResλnk,gjfxnk||||f(Resλnkgjfxnk)f(xnk)||  1λnkyResλnk,gjfxnk,f(Resλnk,gjfxnk)f(xnk)  gj(Resλnkgjfxnk,y)gj(y,Resλnk,gjfxnk)yC.

Taking the limit as k in the above inequality, from (A4) and condition (iv), we have xnkq,||f(Resλ nk,gjfxnk)f(xnk)||0, we have that gj(y,q)0 for all yC. For 0<t<1 and yC, define yt=ty+(1t)q. Noting that ytC, which yields gj(yt,q)0. It therefore follows from (A1) that

0=gj(yt,yt)tgj(yt,y)+(1t)gj(yt,q)tgj(yt,y).

That is gj(yt,y)0.

Let t0, from (A3), we obtain gj(q,y)0 for any yC, j=1,2,. This implies that q j=1EP(gj). Therefore qΓ:= i=1NF(Ti) j=1EP(gj).

Now since xn+1=ProjCnQnfx, we have

f(x)f(xn+1),xn+1z0zCnQn.

Since ΓCnQn, we have

f(x)f(xn+1),xn+1z0zΓ.

Taking the limit of the above inequality, we have

f(x)f(q),qz0zΓ.

Therefore q=ProjΓfx. This completes the proof.

### 5. Application to Zeros of Maximal Monotone Operators

Sabach [37] showed that under some properties of the function f, the solution set of the equilibrium problem is equivalent to the set of zeros of a maximal monotone operator, that is the points x* ∈ dom A such that

0*Ax*,

where A:E2E* is a maximal monotone operator. We denotes the set of zeros of A by A1(0*). An operator A:E2E* is said to be monotone if for any x,ydom A, we have

ξAxandμAyξμ,xy0.

A monotone operator A is said to be maximal if the graph of A, Gr(A):={(x,ξ):ξAx} is not contained in the graph of any other monotone operator. The problem of finding the zeros of monotone operators is very important due to its applications in differential equations, evolution equations, optimization and other related fields. Many algorithms have also been introduced to find its solutions in Hilbert and Banah spaces.

Let g:C×C be a bifunction and define the following operator Ag:E2E* in the following manner

Ag(x)={ξE*:g(x,y)ξ,yx   yC},xC,xC.

The following result was proved for the mapping Ag in [37].

Proposition 5.1. (Sabach [37]) Let C be a nonempty, closed and convex subset of a reflexive Banach space E and let f:E be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Assume that the bifunction g:C×C satisfies conditions (A1)-(A4), then:

(i) EP(g)=Ag1(0*);

(ii) Ag is maximal monotone operator;

(iii) Resgf=ResAgf.

Based on the above result, we propose the following which can be obtain from Theorem 4.2 for finding common fixed point of finite family of n-generalized Bregman nonspreading mapping and zeros of maximal monotone operators in reflexive Banach space.

Theorem 5.2. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and f:E be a coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. For i=1,2,,N, let {αn,i}(0,1), Ti:CC be finite family of n-generalized Bregman nonspreading mappings and Wn:CC be a Bregman W-mapping generated by {αn,i} and T1,T2,,TN in (4.1). Let gj:C×C be bifunctions satisfying assumptions (A1)-(A4), Agj:E2E* be as defined in (5.3) for j=1,2, and suppose Γ:= i=1NF(Ti) j=1Agj1(0*). Define the sequence {xn} by the following process

x0=xC,C0=Q0=C,zn=f*[βn,0f(xn)+ j=1βn,jf(ResAgjfxn)],yn=f*[δnf(xn)+(1δn)f(Wnzn)],Cn={zC:Df(z,yn)Df(z,xn)},Qn={zC:f(x)f(xn),xnz0},xn+1=ProjCnQnfx,

for all n ≥ 0, where {βn,j} and {δn} are sequences in [0,1) satisfying the following control conditions:

(i) j=0βn,j=1,  n{0};

(ii) There exists k such that liminfnβn,jβn,k>0, j{0};

(iii) 0δn<1, n and liminfnδn<1.

Then, the sequence {xn} converges strongly to ProjΓfx as n.

We give a numerical example to demonstrate the performance of our algorithm (4.7).

Example 6.1. Let E=, C=[10,10] and let f: be defined by f(x)=23x2. Let g:C×C be defined by g(x,y)=x(yx), x,yC and T:CC be defined by Tix=13ix, i=1,2,,N. It is easy to observe that f is coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subset of and f(x)=43x. Also since f*(x*)=sup{x*,xf(x):x}, then f*(z)=38z2 and f*(z)=34z. Further, Ti is 1-generalized Bregman nonspreading mapping and Resλn,gjfz=z23λnj.

Choose {αn,i}=1(n+i)2, {δn}=1(n+1)2, {λn}=12 and for each n{0}, and j0, let {βn,j} be defined by

βn,j=13j+1nn+1,n>j,1nn+1 k=1 n13kn=j,0n<j.

Observe that g satisfy Assumption (A1)-(A4) and Γ={0}. After simplification, the hybrid iterative scheme (4.7) reduces to the following: Given x0,

zn=34βn,043(xn)+ j=1βn,j2xn3(23j);yn=3443(n+1)2(xn)+11(n+1)243(Wnzn);Cn=0,2(xn2+yn2)3;Qn=0,xn;xn+1=ProjCnQnfx0,

where Wnzn is computed as follow:

Sn,0zn=zn,Sn,1zn=zn3(n+1)2+11(n+1)2zn;Sn,2zn=zn6(n+2)2+11(n+2)2Sn,1zn;  Wnzn=Sn,N=zn3N(n+N)2+11(n+N)2Sn,N1zn.

Finally, we select the following values

Case(i): N=10 and x0 = -1,

Case(ii): N=50 and x0 = 0.5,

Case(iii): N= 100 and x0 = 2.

Using Matlab 2016(b) and ϵ=106 as stopping criterion, we plot the graphs of error ||xn+1xn|| against number of iteration in each case. The computational results can be found in Figure 1.

Figure 1. Example 6.1, Top-Left: Case(i); Top-Right: Case(ii); Bottom: Case(iii).

The authors sincerely thank the reviewer for his careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

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