# Approximating Common Fixed Points of Bregman Weakly Relatively Nonexpansive Mappings in Banach Spaces

Authors: Chin-Tzong Pang; Eskandar Naraghirad
Journal: Journal of Function Spaces
 (2014)

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## Abstract

Using Bregman functions, we introduce a new hybrid iterative scheme for finding common fixed points of an infinite family of Bregman weakly relatively nonexpansive mappings in Banach spaces. We prove a strong convergence theorem for the sequence produced by the method. No closedness assumption is imposed on a mappingT:C→C, where C is a closed and convex subset of a reflexive Banach space E. Furthermore, we apply our method to solve a system of equilibrium problems in reflexive Banach spaces. Some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature.


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## Body

## 1. Introduction

Throughout this paper, we denote the set of real numbers and the set of positive integers byℝ and ℕ, respectively. Let E be a Banach space with the norm ∥·∥ and the dual space E*. For any x∈E, we denote the value of x*∈E* at x by 〈x,x*〉. Let {xn}n∈ℕ be a sequence in E; we denote the strong convergence of {xn}n∈ℕ to x∈E as n→∞ by xn→x and the weak convergence by xn⇀x. The modulus δ of convexity of E is denoted by
(1)δ(ϵ)=inf{1-∥x+y∥2:∥x∥≤1,∥y∥≤1,∥x-y∥≥ϵ}
for every ϵ with 0≤ϵ≤2. A Banach space E is said to be uniformly convex if δ(ϵ)>0 for every ϵ>0. Let SE={x∈E:∥x∥=1}. The norm of E is said to be Gâteaux differentiable if, for each x,y∈SE, the limit
(2)limt→0∥x+ty∥-∥x∥t
exists. In this case, E is called smooth. If the limit (2) is attained uniformly for all x,y∈SE, then E is called uniformly smooth. The Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 whenever x,y∈SE and x≠y. It is well known that E is uniformly convex if and only if E* is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if E* is smooth; for more details, see [1–3].LetC be a nonempty subset of E. Let T:C→E be a mapping. We denote the set of fixed points of T by F(T); that is, F(T)={x∈C:Tx=x}. A mapping T:C→E is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. A mapping T:C→E is said to be quasinonexpansive if F(T)≠∅ and ∥Tx-y∥≤∥x-y∥ for all x∈C and y∈F(T). The mapping T is called closed, if for any sequence {xn}n∈ℕ⊂C with limn→∞xn=x0 and limn→∞Txn=y0, then we have Tx0=y0. Let T:C→C be a nonexpansive mapping. Recall that the Mann-type [4] iteration is given by the following formula:
(3)xn+1=γnTxn+(1-γn)xn,x1∈C.
Here, {γn}n∈ℕ is a sequence of real numbers in [0,1] satisfying some appropriate conditions. A more general iteration scheme is the Halpern [5] iteration given by
(4)u∈C,x1∈Cchosenarbitrarily,yn=(1-βn)xn+βnTxn,xn+1=αnu+(1-αn)yn,
where the sequences {βn}n∈ℕ and {αn}n∈ℕ satisfy some appropriate conditions. Numerous results have been proved on Mann's and Halpern's iterations for nonexpansive mappings in Hilbert and Banach spaces (see, e.g., [6–11]).LetE be a smooth, strictly convex, and reflexive Banach space and let J be the normalized duality mapping of E. Let C be a nonempty, closed, and convex subset of E. The generalized projection ΠC from E onto C [12] is defined and denoted by
(5)ΠC(x)=argminy∈Cϕ(y,x),
where ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2. Let C be a nonempty, closed, and convex subset of smooth Banach space E and let T be a mapping from C into itself. A point p∈C is said to be an asymptotic fixed point [13] of T if there exists a sequence {xn}n∈ℕ in C which converges weakly to p and limn→∞∥xn-Txn∥=0. We denote the set of all asymptotic fixed points of T by F^(T). A point p∈C is called a strong asymptotic fixed point of T if there exists a sequence {xn}n∈ℕ in C which converges strongly to p and limn→∞∥xn-Txn∥=0. We denote the set of all strong asymptotic fixed points of T by F~(T).Following Matsushita and Takahashi [14], a mapping T:C→C is said to be relatively nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
ϕ(u,Tx)≤ϕ(u,x), ∀u∈F(T),x∈C;(3)
F^(T)=F(T).The mappingT is called relatively weak quasinonexpansive [15, 16] if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
ϕ(u,Tx)≤ϕ(u,x), ∀u∈F(T),x∈C.In 2005, Matsushita and Takahashi [14] proved the following strong convergence theorem for relatively nonexpansive mappings in a Banach space.Theorem 1.
LetE be a uniformly convex and uniformly smooth Banach space, let C be a nonempty, closed, and convex subset of E, let T be a relatively nonexpansive mapping from C into itself, and let {αn}n∈ℕ be a sequence of real numbers such that 0≤αn<1 and limsupn→∞αn<1. Suppose that {xn}n∈ℕ∪{0} is given by
(6)x0=x∈C,yn=J-1(αnJxn+(1-αn)JTxn),Hn={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=ΠHn∩Wnx.
Then {xn}n∈ℕ converges strongly to ΠF(T)x.In 2010, Plubtieng and Ungchittrakool [17] proved the following strong convergence theorem for relatively nonexpansive mappings in a Banach space.Theorem 2.
LetE be a uniformly convex and uniformly smooth Banach space and let C^ and C be two nonempty, closed, and convex subsets of E such that C⊂C^. Let {Tn}n∈ℕ be a sequence of relatively nonexpansive mappings from C into E such that ∩n=1∞F(Tn) is nonempty and let {xn}n∈ℕ be a sequence defined as follows:
(7)x0∈C^,C1=C,x1=ΠC1x0,yn=J-1(αnJxn+(1-αn)JTnxn),Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)},xn+1=ΠCnx0,n∈ℕ,
where {αn}n∈ℕ⊂[0,1] satisfies either(a)
0≤αn<1 for all n∈ℕ and limsupn→∞αn<1 or(b)
liminfn→∞αn(1-αn)>0.
Suppose that for any bounded subsetB of C, there exists an increasing, continuous, and convex function hB:[0,+∞)→[0,+∞) such that hB(0)=0, and limk,l→∞sup{hB(∥Tkz-Tlz∥):z∈B}=0. Let T be a mapping from C into E defined by Tx=limn→∞Tnx for all x∈C and suppose that F(T)=∩n=1∞F(Tn)=∩n=1∞F~(Tn)=F~(T). Then {xn}n∈ℕ, {Tnxn}n∈ℕ, and {yn}n∈ℕ converge strongly to ΠF(T)x0.In 2010, Cai and Hu [15] proved the following strong convergence theorem for a finite family of closed relatively weak quasinonexpansive mappings in a Banach space.Theorem 3.
LetC be a nonempty, closed, and let convex subset of uniformly convex and uniformly smooth Banach space E and let {T1,T2,…,TN} be a finite family of closed relatively weak quasinonexpansive mappings from C into itself with F:=∩i=1NF(Ti)≠⌀. Assume that Ti is uniformly continuous for all i∈{1,2,…,N}. Let {xn}n∈ℕ be a sequence generated by the following algorithm
(8)x0=x∈Cchosenarbitrarily,C1=C,x1=ΠC1x0,zn=J-1[βnJxn-1+(1-βn)JTnxn],Tn=Tn(mod)N,yn=J-1[αnJx1+(1-αn)Jzn],Cn+1={z∈Cn:ϕ(z,yn)≤ϕ(z,xn)+αnϕ(z,x1)mmmm+(1-αn)[βnϕ(z,xn-1)+(1-βn)ϕ(z,Tnxn)]},xn+1=ΠCn+1x1,n∈ℕ∪{0}.
Let {αn}n∈ℕ∪{0} and {βn}n∈ℕ∪{0} be sequences in [0,1) such that limn→∞αn=0 and limn→∞βn=0. Then {xn}n∈ℕ converge strongly to ΠFx1 as n→∞.


### 1.1. Some Facts about Gradients

For any convex functiong:E→(-∞,+∞], we denote the domain of g by domg={x∈E:g(x)<∞}. For any x∈int domg and any y∈E, we denote by go(x,y) the right-hand derivative of g at x in the direction y; that is,
(9)go(x,y)=limt↓0g(x+ty)-g(x)t.
The function g is said to be Gâteaux differentiable at x if limt→0((g(x+ty)-g(x))/t) exists for any y. In this case, go(x,y) coincides with ∇g(x), the value of the gradient  ∇g of g at x. The function g is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function g is said to be Fréchet differentiable at x if this limit is attained uniformly in ∥y∥=1. The function g is Fréchet differentiable at x∈E (see, e.g., [18, p. 13] or [19, p. 508]) if for all ϵ>0, there exists δ>0 such that ∥y-x∥≤δ implies that
(10)|g(y)-g(x)-〈y-x,∇g(x)〉|≤ϵ∥y-x∥.
The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function g:E→ℝ is Gâteaux differentiable, then ∇g is norm-to-weak* continuous (see, e.g., [18, Proposition 1.1.10]). Also, it is known that if g is Fréchet differentiable, then ∇g is norm-to-norm continuous (see, [19, p. 508]). The function g is said to be strongly coercive if
(11)lim∥xn∥→∞g(xn)∥xn∥=∞.
It is also said to be bounded if g(U) is bounded for each bounded subset U of E. Finally, g is said to be uniformly Fréchet differentiable on a subset X of E if the limit (9) is attained uniformly for all x∈X and ∥y∥=1.


### 1.2. Some Facts about Legendre Functions

LetE be a reflexive Banach space. For any proper, lower semicontinuous, and convex function g:E→(-∞,+∞], the conjugate function  g* of g is defined by
(12)g*(x*)=supx∈E{〈x,x*〉-g(x)},∀x*∈E*.
It is well known that g(x)+g*(x*)≥〈x,x*〉 for all (x,x*)∈E×E*. It is also known that (x,x*)∈∂g is equivalent to
(13)g(x)+g*(x*)=〈x,x*〉.
Here, ∂g is the subdifferential of g [20, 21]. We also know that if g:E→(-∞,+∞] is a proper, lower semicontinuous, and convex function, then g*:E*→(-∞,+∞] is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.The functiong:E→(-∞,+∞] is called Legendre if it satisfies the following conditions:(i)
∂g is both locally bounded and single-valued on its domain;(ii)
(∂g)-1 is locally bounded on its domain and g is strictly convex on every convex subset of dom ∂g.For more details, we refer to [22].IfE is a reflexive Banach space and g:E→(-∞,+∞] is a Legendre function, then in view of [23];
(14)∇g*=(∇g)-1,ran∇g=domg*=intdomg*,ran∇g=int domg.Examples of Legendre functions are given in [22, 24]. One important and interesting Legendre function is (1/s)∥·∥s(1<s<∞), where the Banach space E is smooth and strictly convex and, in particular, a Hilbert space.


### 1.3. Some Facts about Bregman Distances

LetE be a Banach space and let E* be the dual space of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [25, 26] corresponding to g is the function Dg:E×E→ℝ defined by
(15)Dg(x,y)=g(x)-g(y)-〈x-y,∇g(y)〉,∀x,y∈E.
It is clear that Dg(x,y)≥0 for all x,y∈E. In that case when E is a smooth Banach space, setting g(x)=∥x∥2 for all x∈E, we obtain that ∇g(x)=2Jx for all x∈E and hence Dg(x,y)=ϕ(x,y) for all x,y∈E.LetE be a Banach space and let C be a nonempty, convex, and subset of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then, we know from [27, 28] that for x∈E and x0∈C,
(16)Dg(x0,x)=miny∈CDg(y,x)iff〈y-x0,∇g(x)-∇g(x0)〉≤0,〈y-x0,∇g(x)-∇g(x0)〉∀y∈C.
Furthermore, if C is a nonempty, closed, and convex subset of a reflexive Banach space E and g:E→ℝ is a strongly coercive Bregman function, then for each x∈E, there exists a unique x0∈C such that
(17)Dg(x0,x)=miny∈CDg(y,x).
The Bregman projectionprojCg from E onto C is defined by projCg(x)=x0 for all x∈E. It is also well known that projCg has the following property [27]:
(18)Dg(y,projCgx)+Dg(projCgx,x)≤Dg(y,x)
for all y∈C and x∈E (see [18] for more details). Let E be a reflexive Banach space, let g:E→ℝ be a strongly coercive Bregman function, and let Dg:E×E→ℝ be the Bregman distance corresponding to g. Then, g*:E*→ℝ is convex and Gâteaux differentiable [29]. Let Dg*:E*×E*→ℝ be the function defined by
(19)Dg*(x*,y*)=g*(x*)-g*(y*)-〈∇g*(y*),x*-y*〉
for x*,y*∈E*, where ∇g* is the gradient of g*. We know from [28] that
(20)Dg*(∇g(x),∇g(y))=Dg(y,x)
for all x,y∈E. We have from the definition of Dg* that
(21)Dg*(x*,y*)=Dg*(x*,z*)+Dg*(z*,y*)+〈∇g*(x*)-∇g*(z*),z*-y*〉,mmmmmmmmml∀x*,y*,z*∈E*.
In particular,
(22)Dg*(x*,y*)=-Dg*(y*,x*)+〈∇g*(y*)-∇g*(x*),y*-x*〉,mmmmmmmmmmml∀x*,y*∈E*.
Indeed, there exist x,y,z∈E such that ∇g(x)=x*, ∇g(y)=y* and ∇g(z)=z*. Therefore,
(23)Dg*(x*,y*)=Dg*(∇g(x),∇g(y))=Dg(y,x)=Dg(y,z)+Dg(z,x)+〈y-z,∇g(z)-∇g(x)〉=Dg*(∇g(z),∇g(y))+ϕ*(∇g(x),∇g(z))+〈∇g*(y*)-∇g*(z*),z*-x*〉=Dg*(z*,y*)+Dg*(x*,z*)+〈∇g*(x*)-∇g*(z*),z*-y*〉.


### 1.4. Some Facts about Uniformly Convex Functions

LetE be a Banach space and let Br:={z∈E:∥z∥≤r} for all r>0. Then a function g:E→ℝ is said to be uniformly convex on bounded subsets ofE ([29, pp. 203, 221]) if ρr(t)>0 for all r,t>0, where ρr:[0,+∞)→[0,∞] is defined by
(24)ρr(t)=infx,y∈Br,∥x-y∥=t,α∈(0,1)(-g(αx+(1-α)y))×(α(1-α))-1(αg(x)+(1-α)g(y)mmmmmmmmmml-g(αx+(1-α)y))mmmmmmmmmml×(α(1-α))-1)
for all t≥0. The function ρr is called the gauge of uniform convexity of g. The function g is also said to be uniformly smooth on bounded subsets ofE ([29, pp. 207, 221]) if limt↓0(σr(t)/t)=0 for all r>0, where σr:[0,+∞)→[0,∞] is defined by
(25)σr(t)=supx∈Br,y∈SE,α∈(0,1)(×(α(1-α))-1αg(x+(1-α)ty)mmmmmmmmmmm+(1-α)g(x-αty)-g(x))mmmmmmmmmmm×(α(1-α))-1)
for all t≥0.


### 1.5. Some Facts about Resolvents

LetE be a Banach space with the norm ∥·∥ and the dual space E*. Let A:E→2E* be a set-valued mapping. We define the domain and range of A by domA={x∈E:Ax≠⌀} and ranA=∪x∈EAx, respectively. The graph of A is denoted by G(A)={(x,x*)∈E×E*:x*∈Ax}. The mapping A⊂E×E* is said to be monotone [30, 31] if 〈x-y,x*-y*〉≥0 whenever (x,x*),(y,y*)∈A. It is also said to be maximal monotone [20] if its graph is not contained in the graph of any other monotone operator on E. If A⊂E×E* is maximal monotone, then we can show that the set A-10={z∈E:0∈Az} is closed and convex. Let E be a reflexive Banach space with the dual space E* and let g:E→(-∞,+∞] be a proper, lower semicontinuous, and convex function. Let A be a maximal monotone operator from E to E*. For any r>0, let the mapping ResrAg:E→domA be defined by
(26)ResrAg=(∇g+rA)-1∇g.
The mapping ResrAg is called the g-resolvent of A (see [32]). It is well known that A-1(0)=F(ResrAg) for each r>0 (for more details, see, e.g., [1, 33]).


### 1.6. Some Facts about Bregman Quasinonexpansive Mappings

LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g:E→(-∞,+∞] be a proper, lower semicontinuous, and convex function. Recall that a mapping T:C→C is said to be Bregman quasinonexpansive, if F(T)≠⌀ and
(27)Dg(p,Tx)≤Dg(p,x),∀x∈C,p∈F(T).
Nontrivial examples of such mappings are given in [34].A mappingT:C→C is said to be Bregman relatively nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
Dg(p,Tv)≤Dg(p,v), ∀p∈F(T),v∈C;(3)
F^(T)=F(T).A mapping T:C→C is said to be Bregman weakly relatively nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
Dg(p,Tv)≤Dg(p,v), ∀p∈F(T), v∈C;(3)
F~(T)=F(T).It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasinonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weakly relatively nonexpansive mapping, but the converse is not true in general. Indeed, for any mapping T:C→C, we have F(T)⊂F~(T)⊂F^(T). If T is Bregman relatively nonexpansive, then F(T)=F~(T)=F^(T). It is easy to verify that any closed mapping T:C→C is a Bregman weakly relatively nonexpansive mapping. To this end, let {xn}n∈ℕ be a sequence of C such that xn→x∈C and ∥xn-Txn∥→0 as n→∞. This implies that Txn→x∈C as n→∞. From the closedness of T, we conclude that x∈F(T). Below we show that there exists a Bregman weakly relatively nonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a closed mapping.Example 4.
LetE=l2, where
(28)l2={σ=(σ1,σ2,…,σn,…):∑n=1∞‍∥σn∥2<∞},∥σ∥=(∑n=1∞‍∥σn∥2)1/2,mmmmmmmmmmmmmmmmmmmm∀σ∈l2,〈σ,η〉=∑n=1∞‍σnηn,∀δ=(σ1,σ2,…,σn,…),0000000η=(η1,η2,…,ηn,…)∈l2.
Let {xn}n∈ℕ∪{0}⊂E be a sequence defined by
(29)x0=(1,0,0,0,…)x1=(1,1,0,0,0,…)x2=(1,0,1,0,0,0,…)x3=(1,0,0,1,0,0,0,…)⋯xn=(σn,1,σn,2,…,σn,k,…)⋯,
where
(30)σn,k={1ifk=1,n+1,0ifk≠1,k≠n+1,
for all n∈ℕ. It is clear that the sequence {xn}n∈ℕ converges weakly to x0. Indeed, for any Λ=(λ1,λ2,…,λn,…)∈l2=(l2)*, we have
(31)Λ(xn-x0)=〈xn-x0,Λ〉=∑k=2∞‍λkσn,k⟶0
as n→∞. It is also obvious that ∥xn-xm∥=2 for any n≠m with n,m sufficiently large. Thus, {xn}n∈ℕ is not a Cauchy sequence. Let k be an even number in ℕ and let g:E→ℝ be defined by
(32)g(x)=1k∥x∥k,x∈E.
It is easy to show that ∇g(x)=Jk(x) for all x∈E, where
(33)Jk(x)={x*∈E*:〈x,x*〉=∥x∥∥x*∥,∥x*∥=∥x∥k-1}.
It is also obvious that
(34)Jk(λx)=λk-1Jk(x),∀x∈E,λ∈ℝ.
Now, we define a mapping T:E→E by
(35)T(x)={nn+1x,ifx=xn;-x,ifx≠xn.
Then F(T)={0} and T is a Bregman weakly relatively nonexpansive mapping which is not a Bregman relatively nonexpansive mapping; see [35] for more details. Now, we prove that T is not a closed mapping. Indeed, let yn=(1+1/n)x2 for all n in ℕ. Then yn→x2 as n→∞, Tyn=-(1+1/n)x2→-x2 (since yn≠xm for all n,m∈ℕ), but Tx2=(2/3)x2≠-x2.An example of a Bregman quasinonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a Bregman weakly relatively nonexpansive mapping can be found in [35].In this paper, we investigate the problem of finding zeros of mappingsA:E→2E*; that is, find x∈domA such that
(36)0*∈Ax.
Recently, Sabach [36] proved the following two strong convergence theorems for the products of finitely many resolvents of maximal monotone operators in a reflexive Banach space.Theorem 5.
LetE be a reflexive Banach space and let Ai:E→2E*,i=1,2,…,N, be N maximal monotone operators such that Z:=∩i=1NAi-1(0*)≠⌀. Let g:E→ℝ be a Legendre function that is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let {xn}n∈ℕ be a sequence defined by the following iterative algorithm:
(37)x0∈Echosenarbitrarily,yn=ResλnNANg,…,Resλn1A1g(xn+en),Cn={z∈E:Dg(z,yn)≤Dg(z,xn+en)},Qn={z∈E:〈z-xn,∇g(x0)-∇g(xn)〉≤0},xn+1=projCn∩Qngx0andn∈ℕ∪{0}.
If, for each i=1,2,…,N, liminfn→∞λni>0, and the sequences of errors {eni}n∈ℕ⊂E satisfy liminfn→∞eni=0, then each such sequence {xn}n∈ℕ converges strongly to projZg(x0) as n→∞.Theorem 6.
LetE be a reflexive Banach space and let Ai:E→2E*,i=1,2,…,N, be N maximal monotone operators such that Z:=∩i=1NAi-1(0*)≠⌀. Let g:E→ℝ be a Legendre function that is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let {xn}n∈ℕ be a sequence defined by the following iterative algorithm:
(38)x0∈Echosenarbitrarily,H0=E,yn=ResλnNANg,…,Resλn1A1g(xn+en),Hn+1={z∈Hn:Dg(z,yn)≤Dg(z,xn+en)},xn+1=projHn+1gx0andn∈ℕ∪{0}.
If, for each i=1,2,…,N, liminfn→∞λni>0, and the sequences of errors {eni}n∈ℕ⊂E satisfy liminfn→∞eni=0, then each such sequence {xn}n∈ℕ converges strongly to projZg(x0) as n→∞.The approximation of fixed points of Bregman nonexpansive type mappings via Bregman distances has been studied in the last ten years and much intensively in the last five years. For some recent articles on the existence and the construction of fixed points for Bregman nonexpansive type mappings, we refer the readers to [36–40].But it is worth mentioning that, in all the above results for Bregman nonexpansive type mappings, the assumptionF^(T)=F(T) is imposed on the map T or the closedness of T is required. So, the following question arises naturally in a Banach space setting.Question 1.
Is it possible to obtain strong convergence of modified Mann's type schemes to a common fixed point of an infinite family of Bregman quasinonexpansive mappings{Tj}j∈ℕ without imposing the closedness assumption, the uniformly continuity assumption, or the assumption F^(Tj)=F(Tj) on the mapping Tj?In this paper, using Bregman functions, we introduce a new hybrid iterative scheme for finding common fixed points of an infinite family of Bregman weakly relatively nonexpansive mappings in Banach spaces. We prove a strong convergence theorem for the sequence produced by the method. No closedness assumption is imposed on a mappingT:C→C, where C is a closed and convex subset of a reflexive Banach space E. Consequently, the above question is answered in the affirmative in reflexive Banach space setting. Furthermore, we apply our method to solve a system of equilibrium problems in reflexive Banach spaces. Some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature; see, for example, [8, 11–14, 41–48].


## 1.1. Some Facts about Gradients

For any convex functiong:E→(-∞,+∞], we denote the domain of g by domg={x∈E:g(x)<∞}. For any x∈int domg and any y∈E, we denote by go(x,y) the right-hand derivative of g at x in the direction y; that is,
(9)go(x,y)=limt↓0g(x+ty)-g(x)t.
The function g is said to be Gâteaux differentiable at x if limt→0((g(x+ty)-g(x))/t) exists for any y. In this case, go(x,y) coincides with ∇g(x), the value of the gradient  ∇g of g at x. The function g is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function g is said to be Fréchet differentiable at x if this limit is attained uniformly in ∥y∥=1. The function g is Fréchet differentiable at x∈E (see, e.g., [18, p. 13] or [19, p. 508]) if for all ϵ>0, there exists δ>0 such that ∥y-x∥≤δ implies that
(10)|g(y)-g(x)-〈y-x,∇g(x)〉|≤ϵ∥y-x∥.
The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function g:E→ℝ is Gâteaux differentiable, then ∇g is norm-to-weak* continuous (see, e.g., [18, Proposition 1.1.10]). Also, it is known that if g is Fréchet differentiable, then ∇g is norm-to-norm continuous (see, [19, p. 508]). The function g is said to be strongly coercive if
(11)lim∥xn∥→∞g(xn)∥xn∥=∞.
It is also said to be bounded if g(U) is bounded for each bounded subset U of E. Finally, g is said to be uniformly Fréchet differentiable on a subset X of E if the limit (9) is attained uniformly for all x∈X and ∥y∥=1.


## 1.2. Some Facts about Legendre Functions

LetE be a reflexive Banach space. For any proper, lower semicontinuous, and convex function g:E→(-∞,+∞], the conjugate function  g* of g is defined by
(12)g*(x*)=supx∈E{〈x,x*〉-g(x)},∀x*∈E*.
It is well known that g(x)+g*(x*)≥〈x,x*〉 for all (x,x*)∈E×E*. It is also known that (x,x*)∈∂g is equivalent to
(13)g(x)+g*(x*)=〈x,x*〉.
Here, ∂g is the subdifferential of g [20, 21]. We also know that if g:E→(-∞,+∞] is a proper, lower semicontinuous, and convex function, then g*:E*→(-∞,+∞] is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.The functiong:E→(-∞,+∞] is called Legendre if it satisfies the following conditions:(i)
∂g is both locally bounded and single-valued on its domain;(ii)
(∂g)-1 is locally bounded on its domain and g is strictly convex on every convex subset of dom ∂g.For more details, we refer to [22].IfE is a reflexive Banach space and g:E→(-∞,+∞] is a Legendre function, then in view of [23];
(14)∇g*=(∇g)-1,ran∇g=domg*=intdomg*,ran∇g=int domg.Examples of Legendre functions are given in [22, 24]. One important and interesting Legendre function is (1/s)∥·∥s(1<s<∞), where the Banach space E is smooth and strictly convex and, in particular, a Hilbert space.


## 1.3. Some Facts about Bregman Distances

LetE be a Banach space and let E* be the dual space of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [25, 26] corresponding to g is the function Dg:E×E→ℝ defined by
(15)Dg(x,y)=g(x)-g(y)-〈x-y,∇g(y)〉,∀x,y∈E.
It is clear that Dg(x,y)≥0 for all x,y∈E. In that case when E is a smooth Banach space, setting g(x)=∥x∥2 for all x∈E, we obtain that ∇g(x)=2Jx for all x∈E and hence Dg(x,y)=ϕ(x,y) for all x,y∈E.LetE be a Banach space and let C be a nonempty, convex, and subset of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then, we know from [27, 28] that for x∈E and x0∈C,
(16)Dg(x0,x)=miny∈CDg(y,x)iff〈y-x0,∇g(x)-∇g(x0)〉≤0,〈y-x0,∇g(x)-∇g(x0)〉∀y∈C.
Furthermore, if C is a nonempty, closed, and convex subset of a reflexive Banach space E and g:E→ℝ is a strongly coercive Bregman function, then for each x∈E, there exists a unique x0∈C such that
(17)Dg(x0,x)=miny∈CDg(y,x).
The Bregman projectionprojCg from E onto C is defined by projCg(x)=x0 for all x∈E. It is also well known that projCg has the following property [27]:
(18)Dg(y,projCgx)+Dg(projCgx,x)≤Dg(y,x)
for all y∈C and x∈E (see [18] for more details). Let E be a reflexive Banach space, let g:E→ℝ be a strongly coercive Bregman function, and let Dg:E×E→ℝ be the Bregman distance corresponding to g. Then, g*:E*→ℝ is convex and Gâteaux differentiable [29]. Let Dg*:E*×E*→ℝ be the function defined by
(19)Dg*(x*,y*)=g*(x*)-g*(y*)-〈∇g*(y*),x*-y*〉
for x*,y*∈E*, where ∇g* is the gradient of g*. We know from [28] that
(20)Dg*(∇g(x),∇g(y))=Dg(y,x)
for all x,y∈E. We have from the definition of Dg* that
(21)Dg*(x*,y*)=Dg*(x*,z*)+Dg*(z*,y*)+〈∇g*(x*)-∇g*(z*),z*-y*〉,mmmmmmmmml∀x*,y*,z*∈E*.
In particular,
(22)Dg*(x*,y*)=-Dg*(y*,x*)+〈∇g*(y*)-∇g*(x*),y*-x*〉,mmmmmmmmmmml∀x*,y*∈E*.
Indeed, there exist x,y,z∈E such that ∇g(x)=x*, ∇g(y)=y* and ∇g(z)=z*. Therefore,
(23)Dg*(x*,y*)=Dg*(∇g(x),∇g(y))=Dg(y,x)=Dg(y,z)+Dg(z,x)+〈y-z,∇g(z)-∇g(x)〉=Dg*(∇g(z),∇g(y))+ϕ*(∇g(x),∇g(z))+〈∇g*(y*)-∇g*(z*),z*-x*〉=Dg*(z*,y*)+Dg*(x*,z*)+〈∇g*(x*)-∇g*(z*),z*-y*〉.


## 1.4. Some Facts about Uniformly Convex Functions

LetE be a Banach space and let Br:={z∈E:∥z∥≤r} for all r>0. Then a function g:E→ℝ is said to be uniformly convex on bounded subsets ofE ([29, pp. 203, 221]) if ρr(t)>0 for all r,t>0, where ρr:[0,+∞)→[0,∞] is defined by
(24)ρr(t)=infx,y∈Br,∥x-y∥=t,α∈(0,1)(-g(αx+(1-α)y))×(α(1-α))-1(αg(x)+(1-α)g(y)mmmmmmmmmml-g(αx+(1-α)y))mmmmmmmmmml×(α(1-α))-1)
for all t≥0. The function ρr is called the gauge of uniform convexity of g. The function g is also said to be uniformly smooth on bounded subsets ofE ([29, pp. 207, 221]) if limt↓0(σr(t)/t)=0 for all r>0, where σr:[0,+∞)→[0,∞] is defined by
(25)σr(t)=supx∈Br,y∈SE,α∈(0,1)(×(α(1-α))-1αg(x+(1-α)ty)mmmmmmmmmmm+(1-α)g(x-αty)-g(x))mmmmmmmmmmm×(α(1-α))-1)
for all t≥0.


## 1.5. Some Facts about Resolvents

LetE be a Banach space with the norm ∥·∥ and the dual space E*. Let A:E→2E* be a set-valued mapping. We define the domain and range of A by domA={x∈E:Ax≠⌀} and ranA=∪x∈EAx, respectively. The graph of A is denoted by G(A)={(x,x*)∈E×E*:x*∈Ax}. The mapping A⊂E×E* is said to be monotone [30, 31] if 〈x-y,x*-y*〉≥0 whenever (x,x*),(y,y*)∈A. It is also said to be maximal monotone [20] if its graph is not contained in the graph of any other monotone operator on E. If A⊂E×E* is maximal monotone, then we can show that the set A-10={z∈E:0∈Az} is closed and convex. Let E be a reflexive Banach space with the dual space E* and let g:E→(-∞,+∞] be a proper, lower semicontinuous, and convex function. Let A be a maximal monotone operator from E to E*. For any r>0, let the mapping ResrAg:E→domA be defined by
(26)ResrAg=(∇g+rA)-1∇g.
The mapping ResrAg is called the g-resolvent of A (see [32]). It is well known that A-1(0)=F(ResrAg) for each r>0 (for more details, see, e.g., [1, 33]).


## 1.6. Some Facts about Bregman Quasinonexpansive Mappings

LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g:E→(-∞,+∞] be a proper, lower semicontinuous, and convex function. Recall that a mapping T:C→C is said to be Bregman quasinonexpansive, if F(T)≠⌀ and
(27)Dg(p,Tx)≤Dg(p,x),∀x∈C,p∈F(T).
Nontrivial examples of such mappings are given in [34].A mappingT:C→C is said to be Bregman relatively nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
Dg(p,Tv)≤Dg(p,v), ∀p∈F(T),v∈C;(3)
F^(T)=F(T).A mapping T:C→C is said to be Bregman weakly relatively nonexpansive if the following conditions are satisfied:(1)
F(T) is nonempty;(2)
Dg(p,Tv)≤Dg(p,v), ∀p∈F(T), v∈C;(3)
F~(T)=F(T).It is clear that any Bregman relatively nonexpansive mapping is a Bregman quasinonexpansive mapping. It is also obvious that every Bregman relatively nonexpansive mapping is a Bregman weakly relatively nonexpansive mapping, but the converse is not true in general. Indeed, for any mapping T:C→C, we have F(T)⊂F~(T)⊂F^(T). If T is Bregman relatively nonexpansive, then F(T)=F~(T)=F^(T). It is easy to verify that any closed mapping T:C→C is a Bregman weakly relatively nonexpansive mapping. To this end, let {xn}n∈ℕ be a sequence of C such that xn→x∈C and ∥xn-Txn∥→0 as n→∞. This implies that Txn→x∈C as n→∞. From the closedness of T, we conclude that x∈F(T). Below we show that there exists a Bregman weakly relatively nonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a closed mapping.Example 4.
LetE=l2, where
(28)l2={σ=(σ1,σ2,…,σn,…):∑n=1∞‍∥σn∥2<∞},∥σ∥=(∑n=1∞‍∥σn∥2)1/2,mmmmmmmmmmmmmmmmmmmm∀σ∈l2,〈σ,η〉=∑n=1∞‍σnηn,∀δ=(σ1,σ2,…,σn,…),0000000η=(η1,η2,…,ηn,…)∈l2.
Let {xn}n∈ℕ∪{0}⊂E be a sequence defined by
(29)x0=(1,0,0,0,…)x1=(1,1,0,0,0,…)x2=(1,0,1,0,0,0,…)x3=(1,0,0,1,0,0,0,…)⋯xn=(σn,1,σn,2,…,σn,k,…)⋯,
where
(30)σn,k={1ifk=1,n+1,0ifk≠1,k≠n+1,
for all n∈ℕ. It is clear that the sequence {xn}n∈ℕ converges weakly to x0. Indeed, for any Λ=(λ1,λ2,…,λn,…)∈l2=(l2)*, we have
(31)Λ(xn-x0)=〈xn-x0,Λ〉=∑k=2∞‍λkσn,k⟶0
as n→∞. It is also obvious that ∥xn-xm∥=2 for any n≠m with n,m sufficiently large. Thus, {xn}n∈ℕ is not a Cauchy sequence. Let k be an even number in ℕ and let g:E→ℝ be defined by
(32)g(x)=1k∥x∥k,x∈E.
It is easy to show that ∇g(x)=Jk(x) for all x∈E, where
(33)Jk(x)={x*∈E*:〈x,x*〉=∥x∥∥x*∥,∥x*∥=∥x∥k-1}.
It is also obvious that
(34)Jk(λx)=λk-1Jk(x),∀x∈E,λ∈ℝ.
Now, we define a mapping T:E→E by
(35)T(x)={nn+1x,ifx=xn;-x,ifx≠xn.
Then F(T)={0} and T is a Bregman weakly relatively nonexpansive mapping which is not a Bregman relatively nonexpansive mapping; see [35] for more details. Now, we prove that T is not a closed mapping. Indeed, let yn=(1+1/n)x2 for all n in ℕ. Then yn→x2 as n→∞, Tyn=-(1+1/n)x2→-x2 (since yn≠xm for all n,m∈ℕ), but Tx2=(2/3)x2≠-x2.An example of a Bregman quasinonexpansive mapping which is neither a Bregman relatively nonexpansive mapping nor a Bregman weakly relatively nonexpansive mapping can be found in [35].In this paper, we investigate the problem of finding zeros of mappingsA:E→2E*; that is, find x∈domA such that
(36)0*∈Ax.
Recently, Sabach [36] proved the following two strong convergence theorems for the products of finitely many resolvents of maximal monotone operators in a reflexive Banach space.Theorem 5.
LetE be a reflexive Banach space and let Ai:E→2E*,i=1,2,…,N, be N maximal monotone operators such that Z:=∩i=1NAi-1(0*)≠⌀. Let g:E→ℝ be a Legendre function that is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let {xn}n∈ℕ be a sequence defined by the following iterative algorithm:
(37)x0∈Echosenarbitrarily,yn=ResλnNANg,…,Resλn1A1g(xn+en),Cn={z∈E:Dg(z,yn)≤Dg(z,xn+en)},Qn={z∈E:〈z-xn,∇g(x0)-∇g(xn)〉≤0},xn+1=projCn∩Qngx0andn∈ℕ∪{0}.
If, for each i=1,2,…,N, liminfn→∞λni>0, and the sequences of errors {eni}n∈ℕ⊂E satisfy liminfn→∞eni=0, then each such sequence {xn}n∈ℕ converges strongly to projZg(x0) as n→∞.Theorem 6.
LetE be a reflexive Banach space and let Ai:E→2E*,i=1,2,…,N, be N maximal monotone operators such that Z:=∩i=1NAi-1(0*)≠⌀. Let g:E→ℝ be a Legendre function that is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let {xn}n∈ℕ be a sequence defined by the following iterative algorithm:
(38)x0∈Echosenarbitrarily,H0=E,yn=ResλnNANg,…,Resλn1A1g(xn+en),Hn+1={z∈Hn:Dg(z,yn)≤Dg(z,xn+en)},xn+1=projHn+1gx0andn∈ℕ∪{0}.
If, for each i=1,2,…,N, liminfn→∞λni>0, and the sequences of errors {eni}n∈ℕ⊂E satisfy liminfn→∞eni=0, then each such sequence {xn}n∈ℕ converges strongly to projZg(x0) as n→∞.The approximation of fixed points of Bregman nonexpansive type mappings via Bregman distances has been studied in the last ten years and much intensively in the last five years. For some recent articles on the existence and the construction of fixed points for Bregman nonexpansive type mappings, we refer the readers to [36–40].But it is worth mentioning that, in all the above results for Bregman nonexpansive type mappings, the assumptionF^(T)=F(T) is imposed on the map T or the closedness of T is required. So, the following question arises naturally in a Banach space setting.Question 1.
Is it possible to obtain strong convergence of modified Mann's type schemes to a common fixed point of an infinite family of Bregman quasinonexpansive mappings{Tj}j∈ℕ without imposing the closedness assumption, the uniformly continuity assumption, or the assumption F^(Tj)=F(Tj) on the mapping Tj?In this paper, using Bregman functions, we introduce a new hybrid iterative scheme for finding common fixed points of an infinite family of Bregman weakly relatively nonexpansive mappings in Banach spaces. We prove a strong convergence theorem for the sequence produced by the method. No closedness assumption is imposed on a mappingT:C→C, where C is a closed and convex subset of a reflexive Banach space E. Consequently, the above question is answered in the affirmative in reflexive Banach space setting. Furthermore, we apply our method to solve a system of equilibrium problems in reflexive Banach spaces. Some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature; see, for example, [8, 11–14, 41–48].


## 2. Preliminaries

In this section, we begin by recalling some preliminaries and lemmas which will be used in the sequel.Definition 7 (see [19]).
LetE be a Banach space. The function g:E→ℝ is said to be a Bregman function if the following conditions are satisfied: (1)
g is continuous, strictly convex, and Gâteaux differentiable;(2)
the set{y∈E:Dg(x,y)≤r} is bounded for all x∈E and r>0.The following lemma follows from Butnariu and Iusem [18] and Zălinescu [29].Lemma 8.
LetE be a reflexive Banach space and let g:E→ℝ be a strongly coercive Bregman function. Then (1)
∇g:E→E* is one-to-one, onto and norm-to-weak* continuous;(2)
〈x-y,∇g(x)-∇g(y)〉=0 if and only if x=y;(3)
{x∈E:Dg(x,y)≤r} is bounded for all y∈E and r>0;(4)
domg*=E*,g* is Gâteaux differentiable and ∇g*=(∇g)-1.We know the following two results; see [29, Proposition 3.6.4].Theorem 9.
LetE be a reflexive Banach space and let g:E→ℝ be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:(1)
g is strongly coercive and uniformly convex on bounded subsets of E;(2)
domg*=E*,g* is bounded on bounded subsets and uniformly smooth on bounded subsets of E*;(3)
domg*=E*,g* is Fréchet differentiable and ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*.Theorem 10.
LetE be a reflexive Banach space and let g:E→ℝ be a continuous convex function which is strongly coercive. Then the following assertions are equivalent:(1)
g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;(2)
g* is Fréchet differentiable and ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*;(3)
domg*=E*,g* is strongly coercive and uniformly convex on bounded subsets of E*.LetE be a Banach space and let g:E→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [48] (see also [25, 26]) satisfies the three point identity that is
(39)Dg(x,z)=Dg(x,y)+Dg(y,z)+〈x-y,∇g(y)-∇g(z)〉,∀x,y,z∈E.
In particular, it can be easily seen that
(40)Dg(x,y)=-Dg(y,x)+〈y-x,∇g(y)-∇g(x)〉,∀x,y∈E.
Indeed, by letting z=x in (39) and taking into account that Dg(x,x)=0, we get the desired result.Lemma 11 (see [39]).
LetE be a Banach space and let g:E→ℝ be a Gâteaux differentiable function which is uniformly convex on bounded subsets of E. Let {xn}n∈ℕ and {yn}n∈ℕ be bounded sequences in E. Then the following assertions are equivalent:(1)
limn→∞Dg(xn,yn)=0;(2)
limn→∞∥xn-yn∥=0.The following result was first proved in [49] (see also [19, 38]).Lemma 12.
LetE be a reflexive Banach space let g:E→ℝ be a strongly coercive Bregman function and V the function defined by
(41)V(x,x*)=g(x)-〈x,x*〉+g*(x*),∀x∈E,∀x*∈E*.
Then the following assertions hold:(1)
Dg(x,∇g*(x*))=V(x,x*) for all x∈E and x*∈E*;(2)
V(x,x*)+〈∇g*(x*)-x,y*〉≤V(x,x*+y*) for all x∈E and x*,y*∈E*.The following lemma which is a generalization of Lemma 3.2 in [50] plays a key role in our results.Lemma 13 (see [17]).
LetC be a subset of a real Banach space E and let {Tn}n∈ℕ be a family of mappings from C into E. Suppose that for any bounded subset B of C, there exists a continuous increasing function hB:[0,∞)→[0,∞) such that hB(0)=0 and limk,l→∞θlk=0, where θlk:=sup{hB(∥Tkz-Tlz∥):z∈B}<∞, for all k,l∈ℕ. Then, for each x∈C, {Tnx}n∈ℕ converges strongly to some point of E. Moreover, let the mapping T be defined by
(42)Tx=limn→∞Tnx,∀x∈C.
Then, limsupn→∞{hB(∥Tnz-Tz∥):z∈B}=0.Lemma 14.
LetE be a Banach space and let g:E→ℝ be a convex function which is uniformly convex on bounded subsets of E. Let s>0 be a constant, Bs:={z∈E:∥z∥≤s}, Bs*:={z*∈E*:∥z*∥≤s}, let ρs be the gauge of uniform convexity of g, and let ρs* be the gauge of uniform convexity of g*, respectively. Then(i)
for anyx,y∈Bs and α∈(0,1)(43)g(αx+(1-α)y)≤αg(x)+(1-α)g(y)-α(1-α)ρs(∥x-y∥);(ii)
for anyx,y∈Bs(44)ρs(∥x-y∥)≤Dg(x,y);(iii)
if, in addition,g is bounded on bounded subsets and uniformly convex on bounded subsets of E, then, for any x∈E, y*,z*∈Bs and α∈(0,1)(45)V(x,αy*+(1-α)z*)≤αV(x,y*)+(1-α)V(x,z*)-α(1-α)ρs*(∥y*-z*∥);(iv)
if, in addition,g is bounded on bounded subsets, uniformly convex, and uniformly smooth on bounded subsets of E, then, for any x∈E, y*,z*∈Bs(46)ρs*(∥x*-y*∥)≤Dg*(x*,y*).Proof.
In view of (24), we get (i). Let us prove (ii). If x,y∈Bs and α∈(0,1), then we obtain
(47)g(αx+(1-α)y)-g(y)α≤g(x)-g(y)-(1-α)ρs(∥x-y∥).
Letting α→0 in the above inequality, we arrive at
(48)〈x-y,∇g(y)〉≤g(x)-g(y)-ρs(∥x-y∥).
This implies that
(49)ρs(∥x-y∥)≤Dg(x,y).
(iii) Let x∈E, y*,z*∈Bs, and α∈(0,1). Then
(50)V(x,αy*+(1-α)z*)=g(x)-〈x,αy*+(1-α)z*〉+g*(αy*+(1-α)z*)≤g(x)-α〈x,y*〉-(1-α)〈x,z*〉+αg*(y*)+(1-α)g*(z*)-α(1-α)ρs*(∥y*-z*∥)=αg(x)+(1-α)g(x)-α〈x,y*〉-(1-α)〈x,z*〉+αg*(y*)+(1-α)g*(z*)-α(1-α)ρs*(∥y*-z*∥)=α[g(x)-〈x,y*〉+g*(y*)]+(1-α)[g(x)-〈x,z*〉+g*(z*)]-α(1-α)ρs*(∥y*-z*∥)=αV(x,y*)+(1-α)V(x,z*)-α(1-α)ρs*(∥y*-z*∥).
(iv) Sinceg is uniformly smooth on bounded subsets of E, g* is uniformly convex on bounded subsets of E. Then, in view of (i), there exists a continuous, strictly increasing, and convex function ρs*:[0,∞)→[0,∞) such that
(51)g*(αx*+(1-α)y*)≤αg*(x*)+(1-α)g*(y*)-α(1-α)ρs*(∥x*-y*∥)
for all x*,y*∈Bs* and all α∈(0,1). If x*,y*∈Bs*, then we obtain
(52)g*(αx*+(1-α)y*)-g*(y*)α≤g*(x*)-g*(y*)-(1-α)ρs*(∥x*-y*∥).
Letting α→0 in the above inequality, we conclude that
(53)〈∇g*(y*),x*-y*〉≤g*(x*)-g*(y*)-ρs*(∥x*-y*∥).
This implies that
(54)ρs*(∥x*-y*∥)≤Dg*(x*,y*),
which completes the proof.Lemma 15 (see [35]).
LetE be a Banach space, let r>0 be a constant, and let g:E→ℝ be a continuous and convex function which is uniformly convex on bounded subsets of E. Then
(55)g(∑k=0∞‍αkxk)≤∑k=0∞‍αkg(xk)-αiαjρr(∥xi-xj∥)
for all i,j∈ℕ∪{0}, xk∈Br, αk∈(0,1), and k∈ℕ∪{0} with ∑k=0∞αk=1, where ρr is the gauge of uniform convexity of g.Lemma 16 (see [51]).
Let{an}n∈ℕ be a sequence of real numbers such that there exists a subsequence {ni}i∈ℕ of {n}n∈ℕ such that ani<ani+1 for all i∈ℕ. Then there exists a subsequence {mk}k∈ℕ⊂ℕ such that mk→∞ and the following properties are satisfied by all (sufficiently large) numbers k∈ℕ:
(56)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.Lemma 17 (see [52–54]).
Let{sn}n∈ℕ be a sequence of nonnegative real numbers satisfying the inequality
(57)sn+1≤(1-γn)sn+γnδn,∀n≥1,
where {γn}n∈ℕ and {δn}n∈ℕ satisfy the following conditions:(i)
{γn}n∈ℕ⊂[0,1] and ∑n=1∞‍γn=∞, or equivalently, Πn=1∞(1-γn)=0;(ii)
limsupn→∞δn≤0;(ii)'
∑n=1∞‍γnδn<∞.Then,limn→∞sn=0.


## 3. Strong Convergence Theorems

In this section, we prove a strong convergence theorem concerning the approximation of fixed point of Bregman weak relatively nonexpansive mappings in a reflexive Banach space. We start with the following simple lemma which has been proved in [33].Lemma 18.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E. Let T:C→C be a Bregman quasinonexpansive mapping. Then F(T) is closed and convex.Lemma 19.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let {Tn}n∈ℕ be an infinite family of Bregman quasinonexpansive mappings from C into itself such that F:=∩n=1∞F(Tn)≠⌀. Let the mapping T:C→C be defined by
(58)Tx=limn→∞Tnx.
Then, T is a Bregman quasinonexpansive mapping.Proof.
Letx∈C and p∈F(T) be fixed. Then we have that {Tnx}n∈ℕ is a bounded sequence in E. The function g is bounded on bounded subsets of E and, thus, ∇g is also bounded on bounded subsets of E* (see, e.g., [18, Proposition 1.1.11] for more details). This implies that the sequence {∇g(Tnx)}n∈ℕ is bounded in E*. Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain
(59)Dg(p,Tx)=g(p)-g(Tx)-〈x-Tx,∇g(Tx)〉=g(p)-g(limn→∞Tnx)-〈x-limn→∞Tnx,∇g(limn→∞Tnx)〉=limn→∞[g(p)-g(Tnx)-〈x-Tnx,∇g(Tnx)〉]=limn→∞Dg(p,Tnx)≤Dg(p,x).
Thus, T is a Bregman quasinonexpansive mapping, which completes the proof.Theorem 20.
LetE be a reflexive Banach space and let g:E→ℝ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, closed and convex subset of E and let {Tn}n∈ℕ be an infinite family of Bregman weak relatively nonexpansive mappings from C into itself such that F:=∩n=1∞F(Tn)≠⌀. Suppose in addition that T0=I, where I is the identity mapping on E. Let {xn}n∈ℕ∪{0} be a sequence generated by
(60)x0=x∈Cchosenarbitrarily,C0=C,yn=∇g*[αn∇g(xn)+(1-αn)∇g(Tnxn)],Cn+1={z∈Cn:Dg(z,yn)≤Dg(z,xn)},xn+1=projCn+1gxandn∈ℕ∪{0},
where ∇g is the gradient of g. Let {αn}n∈ℕ∪{0} be a sequence in [0,1) such that liminfn→∞αn(1-αn)>0.
Suppose that for any bounded subsetB of C, there exists an increasing, continuous, and convex function hB:[0,+∞)→[0,+∞) such that hB(0)=0, and limk,l→∞sup{hB(∥Tkz-Tlz∥):z∈B}=0. Let T be a mapping from C into E defined by Tx=limn→∞Tnx for all x∈C and suppose that F(T)=∩n=1∞F(Tn)=∩n=1∞F~(Tn)=F~(T). Then {xn}n∈ℕ, {Tnxn}n∈ℕ, and {yn}n∈ℕ converge strongly to projFgx0.Proof.
We divide the proof into several steps.
Step  1. We prove that Cn is closed and convex for each n∈ℕ∪{0}.
It is clear thatC0=C is closed and convex. Let Cm be closed and convex for some m∈ℕ. For z∈Cm, we see that
(61)Dg(z,ym)≤Dg(z,xm)
is equivalent to
(62)〈z,∇g(xm)-∇g(ym)〉≤g(ym)-g(xm)+〈xm,∇g(xm)〉-〈ym,∇g(ym)〉.
It could easily be seen that Cm+1 is closed and convex. Therefore, Cn is closed and convex for each n∈ℕ∪{0}.
Step  2. We claim that F⊂Cn for all n∈ℕ∪{0}.
It is obvious thatF⊂C0=C. Assume now that F⊂Cm for some m∈ℕ. Employing Lemma 12, for any w∈F⊂Cm, we obtain
(63)Dg(w,ym)=Dg(w,∇g*[αm∇g(xm)+(1-αm)∇g(Tmxm)])=V(w,αm∇g(xm)+(1-αm)∇g(Tmxm))=g(w)-〈w,αm∇g(xm)+(1-αm∇g(Tmxm))〉+g*(αm∇g(xm)+(1-αm)∇g(Tmxm))≤αmg(w)+(1-αm)g(w)+αmg*(∇g(xm))+(1-αm)g*(∇g(Tmxm))=αmV(w,∇g(xm))+(1-αm)V(w,∇g(Tmxm))=αmDg(w,xm)+(1-αm)Dg(w,Tmxm)≤αmDg(w,xm)+(1-αm)Dg(w,xm)=Dg(w,xm).
This proves that w∈Cm+1 and hence F⊂Cn for all n∈ℕ∪{0}.
Step  3. We prove that {xn}n∈ℕ, {yn}n∈ℕ, {zn}n∈ℕ, and {Tnxn}n∈ℕ are bounded sequences in C.
In view of (18), we conclude that
(64)Dg(xn,x)=Dg(projCngx,x)≤Dg(w,x)-Dg(w,xn)≤Dg(w,x),∀w∈F⊂Cn,n∈ℕ∪{0}.
This implies that the sequence {Dg(xn,x)}n∈ℕ is bounded and hence there exists M1>0 such that
(65)Dg(xn,x)≤M1,∀n∈ℕ.
In view of Lemma 8(3), we conclude that the sequence {xn}n∈ℕ is bounded. Since {Tn}n∈ℕ is an infinite family of Bregman weak relatively nonexpansive mappings from C into itself, we have for any q∈F that
(66)Dg(q,Tnxn)≤Dg(q,xn),∀n∈ℕ.
This, together with Definition 7 and the boundedness of {xn}n∈ℕ, implies that the sequence {Tnxn}n∈ℕ is bounded.
Step  4. We show that xn→u for some u∈F, where u=projFgx.
From Step 3 it follows that{xn}n∈ℕ is bounded. By the construction of Cn, we conclude that Cm⊂Cn and xm=projCmgx∈Cm⊂Cn for any positive integer m≥n. This, together with (18), implies that
(67)Dg(xm,xn)=Dg(xm,projCngx)≤Dg(xm,x)-Dg(projCngx,x)=Dg(xm,x)-Dg(xn,x).
In view of (16), we conclude that
(68)Dg(xn,x)=Dg(projCngx,x)≤Dg(w,x)-Dg(w,xn)≤Dg(w,x),∀w∈F⊂Cn,n∈ℕ∪{0}.
It follows from (68) that the sequence {Dg(xn,x)}n∈ℕ is bounded and hence there exists M2>0 such that
(69)Dg(xn,x)≤M2,∀n∈ℕ.
In view of (67), we conclude that
(70)Dg(xn,x)≤Dg(xn,x)+Dg(xm,xn)≤Dg(xm,x),∀m≥n.
This proves that {Dg(xn,x)}n∈ℕ is an increasing sequence in ℝ and hence by (69) the limit limn→∞Dg(xn,x) exists. Letting m,n→∞ in (67), we deduce that Dg(xm,xn)→0. In view of Lemma 11, we obtain that ∥xm-xn∥→0 as m,n→∞. This means that {xn}n∈ℕ is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we conclude that there exists v∈C such that
(71)limn→∞∥xn-v∥=0.
Now, we show that v∈F. In view of (67), we obtain
(72)limn→∞Dg(xn+1,xn)=0.
Since xn+1∈Cn+1, we conclude that
(73)Dg(xn+1,yn)≤Dg(xn+1,xn).
This, together with (72), implies that
(74)limn→∞Dg(xn+1,yn)=0.
It follows from Lemma 11, (72), and (74) that
(75)limn→∞∥xn+1-xn∥=0,limn→∞∥xn+1-yn∥=0.
In view of (71), we get
(76)limn→∞∥yn-u∥=0.
From (71) and (76), it follows that
(77)limn→∞∥xn-yn∥=0.
Since ∇g is uniformly norm-to-norm continuous on any bounded subset of E, we obtain
(78)limn→∞∥∇g(xn)-∇g(yn)∥=0.
Applying Lemma 11, we derive that
(79)limn→∞Dg(yn,xn)=0.
It follows from the three point identity (see (39)) that
(80)|Dg(w,xn)-Dg(w,yn)|=|Dg(w,yn)+Dg(yn,xn)+〈w-yn,∇g(yn)-∇g(xn)〉-Dg(w,yn)|=|Dg(yn,xn)-〈w-yn,∇g(yn)-∇g(xn)〉|≤Dg(yn,xn)+∥w-yn∥∥∇g(yn)-∇g(xn)∥⟶0
as n→∞.
The functiong is bounded on bounded subsets of E and, thus, ∇g is also bounded on bounded subsets of E* (see, e.g., [18, Proposition 1.1.11] for more details). This implies that the sequences {∇g(xn)}n∈ℕ, {∇g(yn)}n∈ℕ, and {∇g(Tnxn):n∈ℕ∪{0}} are bounded in E*.
In view of Theorem10(3), we know that domg*=E* and g* is strongly coercive and uniformly convex on bounded subsets. Let s1=sup{∥∇g(xn)∥,∥∇g(Tnxn)∥:n∈ℕ∪{0}} and ρs1*:E*→ℝ be the gauge of uniform convexity of the conjugate function g*. We prove that for any w∈F,
(81)Dg(w,yn)≤Dg(w,xn)-αn(1-αn)ρs1*×(∥∇g(xn)-∇g(Tnxn)∥).
Let us show (81). For any given w∈F(T), in view of the definition of the Bregman distance (see (13)), (9), Lemma 14, we obtain
(82)Dg(w,yn)=Dg(w,∇g*[αn∇g(xn)+(1-αn)∇g(Tnxn)])=V(w,αn∇g(xn)+(1-αn)∇g(Tnxn))=g(w)-〈w,αn∇g(xn)+(1-αn)∇g(Tnxn)〉+g*(αn∇g(xn)+(1-αn)∇g(Tnxn))≤αng(w)+(1-αn)g(w)-αn〈w,∇g(xn)〉-(1-αn)〈w,∇g(Tnxn)〉+αng*(∇g(xn))+(1-αn)g*(∇g(Tnxn))-αn(1-αn)ρs1*(∥∇g(xn)-∇g(Tnxn)∥)=αnV(w,∇g(xn))+(1-αn)V(w,∇g(Tnxn))-αn(1-αn)ρs1*(∥∇g(xn)-∇g(Tnxn)∥)=αnDg(w,xn)+(1-αn)Dg(w,Tnxn)-αn(1-αn)ρs1*(∥∇g(xn)-∇g(Tnxn)∥)≤αnDg(w,xn)+(1-αn)Dg(w,xn)-αn(1-αn)ρs1*(∥∇g(xn)-∇g(Tnxn)∥)=Dg(w,xn)-αn(1-αn)ρs1*(∥∇g(xn)-∇g(Tnxn)∥).
In view of (80), we obtain
(83)Dg(w,xn)-Dg(w,yn)⟶0asn⟶∞.
In view of (81) and (83), we conclude that
(84)αn(1-αn)ρs1*(∥∇g(xn)-∇g(Tnxn)∥)≤Dg(w,xn)-Dg(w,yn)⟶0
as n→∞. From the assumption liminfn→∞αn(1-αn)>0, we get
(85)limn→∞ρs1*(∥∇g(xn)-∇g(Tnxn)∥)=0.
Therefore, from the property of ρs1* we deduce that
(86)limn→∞∥∇g(xn)-∇g(Tnxn)∥=0.
Since ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*, we arrive at
(87)limn→∞∥xn-Tnxn∥=0.
From the boundedness of {xn}n∈ℕ, it follows that there exists a bounded subset B of C such that {xn}n∈ℕ⊂B. Let Tx=limn→∞Tnx for all x∈C. In view of Lemma 19, T is a Bregman quasinonexpansive mapping. On the other hand, we have
(88)12∥xn-Txn∥≤12∥xn-Tnxn∥+12∥Tnxn-Txn∥,∀n∈ℕ.
Since hB:[0,+∞)→[0,+∞) is an increasing, continuous, and convex function, we have
(89)hB(12∥xn-Txn∥)≤hB(12∥xn-Tnxn∥)+hB(12∥Tnxn-Txn∥)≤12hB(∥xn-Tnxn∥)+12sup{hB(∥Tnz-Tz∥):z∈B}.
Exploiting Lemma 13 and (87), we obtain
(90)limn→∞hB(∥xn-Txn∥)=0.
By the properties of hB, we conclude that
(91)limn→∞∥xn-Txn∥=0.
This, together with Lemma 19 and (71), implies that v∈F(T)=∩n=1∞F(Tn)=F.
Finally, we show thatv=projFgx. From xn=projCngx, we conclude that
(92)〈z-xn,∇g(xn)-∇g(x)〉≥0,∀z∈Cn.
Since F⊂Cn for each n∈ℕ, we obtain
(93)〈z-xn,∇g(xn)-∇g(x)〉≥0,∀z∈F.
Letting n→∞ in (93), we deduce that
(94)〈z-v,∇g(u)-∇g(x)〉≥0,∀z∈F.
In view of (16), we have v=projFgx, which completes the proof.Remark 21.
Theorem20 improves Theorems 1, 2, and 3 in the following aspects.(1)
For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, and strongly coercive Bregman function which is bounded on bounded subsets, and uniformly convex and uniformly smooth on bounded subsets.(2)
For the mappings, we extend the mapping from a relatively nonexpansive mapping to a countable family of Bregman weak relatively nonexpansive mappings. We remove the assumptionF^(T)=F(T) on the mapping T and extend the result to a countable family of Bregman weak relatively nonexpansive mappings, where F^(T) is the set of asymptotic fixed points of the mapping T.(3)
For the algorithm, we remove the setWn in Theorem 1.(4)
Theorem20 extends and improves Theorem 3.1 in [17]. We note that the proof of Theorem 3.3 (lines 24-25) in [17] is not valid in our discussion.(5)
We note also that the main result of the paper cannot be deduced from the results of [35].We end this section with the following simple example in order to support Theorem20.Example 22.
LetE, {xn}n∈ℕ∪{0} and g be as in Example 4. We define a countable family of mappings Tj:E→E by
(95)Tj(x)={nn+1x,ifx=xn;-jj+1x,ifx≠xn,
for all j≥1 and n≥0. It is clear that F(Tj)={0} for all j≥1. Choose j∈ℕ; then, for any n∈ℕ,
(96)Dg(0,Tjxn)=g(0)-g(Tjxn)-〈0-Tjxn,∇g(Tjxn)〉=-nk(n+1)kg(xn)+nk(n+1)k〈xn,∇g(xn)〉=nk(n+1)k[-g(xn)+〈xn,∇g(xn)〉]=nk(n+1)k[Dg(0,xn)]≤Dg(0,xn).
If x≠xn, then we have
(97)Dg(0,Tjx)=g(0)-g(Tjx)-〈0-Tjx,∇g(Tjx)〉=-jk(j+1)kg(x)-jk(j+1)k〈x,-∇g(x)〉=jk(j+1)k[-g(x)-〈-x,∇g(x)〉]≤Dg(0,x).
Therefore, Tj is a Bregman quasinonexpansive mapping. Next, we claim that Tj is a Bregman weak relatively nonexpansive mapping. Indeed, for any sequence {zn}n∈ℕ⊂E such that zn→z0 and ∥zn-Tjzn∥→0 as n→∞, there exists a sufficiently large number N0∈ℕ such that zn≠xm, for any n,m>N0. If we suppose that there exists m≤N such that zn=xm for infinitely many n∈ℕ, then a subsequence {xni}i∈ℕ would satisfy zni=xm, so z0=limi→∞zni=xm and z0=limi→∞Tjzni=Tjxm=(m/(m+1))xm which is impossible. This implies that Tjzn=-(j/(j+1))zn for all n>N0. It follows from ∥zn-Tjzn∥→0 that ((2j+1)/(j+1))zn→0 and hence zn→z0=0. Since z0∈F(Tj), we conclude that Tj is a Bregman weak relatively nonexpansive mapping. It is clear that ∩j=1∞F~(Tj)=∩j=1∞F(Tj)={0}. Thus {Tj}j∈ℕ is a countable family of Bregman weak relatively nonexpansive mappings. Next, we show that {Tj}j∈ℕ is not a countable family of Bregman relatively nonexpansive mappings. In fact, though xn⇀x0 and
(98)∥xn-Tjxn∥=∥xn-nn+1xn∥=1n+1∥xn∥⟶0
as n→∞,x0∉F(Tj) for all j∈ℕ. Therefore, F^(Tj)≠F(Tj) for all j∈ℕ. This implies that ∩j=1∞F^(Tj)≠∩j=1∞F(Tj). Let Tx=limj→∞Tjx for all x∈E. It is easy to see that
(99)T(x)={nn+1x,ifx=xn;-x,ifx≠xn.
In view of Example 4, we obtain that T is a Bregman weak relatively nonexpansive mapping with F(T)={0}=F~(T). Let B be a bounded subset of E. Then there exists r>0 such that B⊂Br={z∈E:∥z∥≤r}. Let ρr be the gauge of uniform convexity of g. Then, in view of Lemma 14, we obtain
(100)ρr(∥x-y∥)≤Dg(x,y),∀x,y∈B.
On the other hand, for any k,l∈ℕ, we have
(101)sup{∥Tkz-Tlz∥:z∈B}=sup{∥-kk+1z--ll+1z∥:z∈B}=|k-l|(k+1)(l+1)sup{∥z∥:z∈B}.
This implies that
(102)limsupk,l→∞{∥Tkz-Tlz∥:z∈B}=0.
Furthermore, we have
(103)limsupk,l→∞{ρr(∥Tkz-Tlz∥):z∈B}=0.
It is clear that, for any j∈ℕ, Tj is not continuous. Finally, it is obvious that the family {Tj}j∈ℕ satisfies all the aspects of the hypothesis of Theorem 20.


## 4. Equilibrium Problems

LetC be a nonempty, closed, and convex of a reflexive Banach space E. Let f:C×C→ℝ be a bifunction. Consider the following equilibrium problem [55]. Find p∈C such that
(104)f(p,y)≥0,∀y∈C.
For solving the equilibrium problem, let us assume that f:C×C→ℝ satisfies the following conditions:(A1)
f(x,x)=0 for all x∈C;(A2)
f is monotone; that is, f(x,y)+f(y,x)≤0 for all x,y∈C;(A3)
for eachy∈C, the function x↦f(x,y) is upper semicontinuous;(A4)
for eachx∈C, the function y↦f(x,y) is convex and lower semicontinuous.The set of solutions of problem (104) is denoted by EP(f).Letg:E→ℝ be a Legendre function. The resolvent of a bifunction f:C×C→ℝ [36] is the operator Resfg:E→2E*, defined by
(105)Resfg(x)={z∈C:f(z,y)+〈y-z,∇g(z)-∇g(x)〉≥≥0∀y∈C}
for all x∈E. We also define the mapping Af:E→2E* in the following way:
(106)Af(x)={{ξ∈E*:f(x,y)≥〈ξ,y-x〉∀y∈C},x∈C,⌀,x∉C.Lemma 23 (see [36, 56]).
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let f:C×C→ℝ be a bifunction satisfying (A1)–(A4) and EP(f)≠⌀. Then, the following statements hold:(1)
dom(Resfg)=E;(2)
Resfg is single-valued;(3)
Resfg is a Bregman firmly nonexpansive mapping [57]; that is, for all x,y∈E,
(107)〈Resfg(x)-Resfg(y),∇g(Resfg(x))-∇g(Resfg(y))〉≤〈Resfg(x)-Resfg(y),∇g(x)-∇g(y)〉;(4)
the set of fixed points ofResfg is the solution of the corresponding equilibrium problem; that is, F(Resfg)=EP(f);(5)
EP(f) is a closed and convex subset of C;(6)
Dg(q,Resfgx)+Dg(Resfgx,x)≤Dg(q,x),∀q∈F(Resfg).Lemma 24 (see [36]).
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let f:C×C→ℝ be a bifunction satisfying (A1)–(A4) and EP(f)≠⌀. Then, the following statements hold:(1)
EP(f)=Af-1(0*);(2)
Af is a maximal monotone operator;(3)
Resfg=ResAfg.In this section, we propose a new Halpern-type iterative scheme for finding common zeros of an infinite family of maximal monotone operators and prove the following strong convergence theorem in a Banach space.Theorem 25.
LetE be a reflexive Banach space and let g:E→ℝ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. For any j∈ℕ, let Aj:E→2E* be a maximal monotone operator such that Z:=∩j=1∞Aj-1(0*)≠⌀. Let {αn}n∈ℕ, {βn,j}n∈ℕ,j∈ℕ∪{0} be sequences in [0,1] satisfying the following control conditions:(a)
limn→∞αn=0;(b)
∑n=1∞‍αn=∞;(c)
βn,0+∑j=1∞‍βn,j=1,∀n∈ℕ;(d)
liminfn→∞βn,0βn,j>0,∀j∈ℕ.
Let{xn}n∈ℕ be a sequence generated by
(108)u∈E,x1∈Echosenarbitrarily,yn=∇g*[βn,0∇g(xn)+∑j=1∞‍βn,j∇g(ResrAjg(xn))],xn+1=∇g*[αn∇g(u)+(1-αn)∇g(yn)]andn∈ℕ,
where ∇g is the gradient of g and r>0 is a constant. Then the sequence {xn}n∈ℕ defined in (108) converges strongly to projZgu as n→∞.Proof.
We divide the proof into several steps.
Letz=projZgu. For every j∈ℕ, we denote by Tj the resolvent Resrfjg. Therefore,
(109)yn=∇g*[βn,0∇g(xn)+∑j=1∞‍βn,j∇g(Tjxn)].
Step  1. We prove that {xn}n∈ℕ,{yn}n∈ℕ, and {Tjxn:n∈ℕ,j∈ℕ} are bounded sequences in E. We first show that {xn}n∈ℕ is bounded. Let p∈Z be fixed. In view of Lemma 12 and (108), we have
(110)Dg(p,yn)=Dg(p,∇g*[βn,0∇g(xn)+∑j=1∞‍βn,j∇g(Tjxn)])=V(p,βn,0∇g(xn)+∑j=1∞‍βn,j∇g(Tjxn))≤βn,0V(p,∇g(xn))+∑j=1∞‍βn,jV(p,∇g(Tjxn))=βn,0Dg(p,xn)+∑j=1∞‍βn,jDg(p,Tjxn)≤βn,0Dg(p,xn)+∑j=1∞‍βn,jDg(p,xn)=Dg(p,xn).
This implies that
(111)Dg(p,xn+1)=Dg(p,∇g*[αn∇g(u)+(1-αn)∇g(yn)])=V(p,αn∇g(u)+(1-αn)∇g(yn))≤αnV(p,∇g(u))+(1-αn)V(p,∇g(yn))=αnDg(p,u)+(1-αn)Dg(p,yn)≤αnDg(p,u)+(1-αn)Dg(p,yn)≤αnDg(p,u)+(1-αn)Dg(p,xn)≤max{Dg(p,u),Dg(p,xn)}.
By induction, we obtain
(112)Dg(p,xn+1)≤max{Dg(p,u),Dg(p,x1)}
for all n∈ℕ. It follows from (112) that the sequence {Dg(xn,x)}n∈ℕ is bounded and hence there exists M3>0 such that
(113)Dg(xn,x)≤M3,∀n∈ℕ.
In view of Lemma 11(3), we get that the sequence {xn}n∈ℕ is bounded. Since {Tj}j∈ℕ is an infinite family of Bregman relatively nonexpansive mappings from E into itself, we conclude that
(114)Dg(p,Tjxn)≤Dg(p,xn),∀j,n∈ℕ.
This, together with Definition 7 and the boundedness of {xn}n∈ℕ, implies that the sequence {Tjxn:j,n∈ℕ} is bounded. The function g is bounded on bounded subsets of E and therefore ∇g is also bounded on bounded subsets of E* (see, e.g., [18, Proposition 1.1.11] for more details). This, together with Step 1, implies that the sequences {∇g(xn)}n∈ℕ, {∇g(yn)}n∈ℕ, and {∇g(Tjxn):j,n∈ℕ} are bounded in E*. In view of Theorem 10(3), we obtain that dom g*=E* and g* is strongly coercive and uniformly convex on bounded subsets of E. Let s2=sup{∥∇g(xn)∥,∥∇g(Tjxn)∥:j,n∈ℕ} and ρs2*:E*→ℝ be the gauge of uniform convexity of the conjugate function g*.
Step  2. We prove that for any j,n∈ℕ(115)Dg(z,yn)≤Dg(z,xn)-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥).
Let us show (115). For each n∈ℕ, in view of the definition of Bregman distance (see (15)), Lemmas 13, 14, 15, and (110), we obtain
(116)Dg(z,yn)=g(z)-g(yn)-〈z-yn,∇g(yn)〉=g(z)+g*(∇g(yn))-〈yn,∇g(yn)〉-〈z,∇g(yn)〉+〈yn,∇g(yn)〉=g(z)+g*(βn,0∇g(xn)+∑j=1∞‍βn,j∇g(Tjxn))-〈z,βn,j∇g(xn)+∑j=1∞‍βn,j∇g(Tjxn)〉≤βn,0g(z)+∑j=1∞‍βn,jg(z)+βn,0g*(∇g(xn))+∑j=1∞‍βn,jg*(∇g(Tjxn))-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥)-βn,0〈z,∇g(xn)〉-∑j=1∞‍βn,j〈z,∇g(Tjxn)〉=βn,0[g(z)+g*(∇g(xn))-〈z,∇g(xn)〉]+∑j=1∞‍βn,j[g(z)+g*(∇g(Tjxn))-〈z,∇g(Tjxn)〉]-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥)=βn,0[g(z)-g(xn)+〈xn,∇g(xn)〉-〈z,∇g(xn)〉]+∑j=1∞‍βn,j[g(z)-g(Tjxn)+〈Tjxn,∇g(Tjxn)〉+∑j=1∞‍βn,j-〈z,∇g(Tjxn)〉]-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥)=βn,0Dg(z,xn)+∑j=1∞‍βn,jDg(z,Tjxn)-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥)≤βn,0Dg(z,xn)+∑j=1∞‍βn,jDg(z,xn)-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥)=Dg(z,xn)-βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥).
It then follows from Lemma 12 and (115) that
(117)Dg(z,xn+1)=Dg(z,∇g*[αn∇g(u)+(1-αn)∇g(yn)])=Dg(z,∇g*[αn∇g(u)+(1-αn)∇g(yn)])=V(z,αn∇g(u)+(1-αn)∇g(yn))≤αnV(z,∇g(u))+(1-αn)V(z,∇g(yn))=αnDg(z,u)+(1-αn)Dg(z,yn)≤αnDg(z,u)+(1-αn)Dg(z,yn)≤αnDg(z,u)+(1-αn)[Dg(z,xn)-βn,0βn,jρs2*+×(1-αn)×∥∇g(xn)-∇g(Tjxn)∥].
Let M4:=sup{|Dg(z,u)-Dg(z,xn)|+βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥):j,n∈ℕ}. Then we get from (117) that
(118)βn,0βn,jρs2*(∥∇g(xn)-∇g(Tjxn)∥)≤Dg(z,xn)-Dg(z,xn+1)+αnM4,∀j∈ℕ.
In view of Lemma 12 and (115), we obtain
(119)Dg(z,xn+1)=Dg(z,∇g*[αn∇g(u)+(1-αn)∇g(yn)])=V(z,αn∇g(u)+(1-αn)∇g(yn))≤V(z,αn∇g(u)+(1-αn)∇g(yn)-αn(∇g(u)-∇g(z)))-〈∇g*[αn∇g(u)+(1-αn)∇g(yn)]-z,-αn(∇g(u)-∇g(z))〉=V(z,αn∇g(z)+(1-αn)∇g(yn))+αn〈xn+1-z,∇g(u)-∇g(z)〉=Dg(z,∇g*[αn∇g(z)+(1-αn)∇g(yn)])+αn〈xn+1-z,∇g(u)-∇g(z)〉≤αnDg(z,z)+(1-αn)Dg(z,yn)+αn〈xn+1-z,∇g(u)-∇g(z)〉=(1-αn)Dg(z,xn)+αn〈xn+1-z,∇g(u)-∇g(z)〉.
We will show that Dg(z,xn)→0 as n→∞ by considering two possible cases on the sequence {Dg(z,xn)}n∈ℕ.
Case  1. If {Dg(z,xn)}n∈ℕ is eventually decreasing, then there exists n0∈ℕ such that {Dg(z,xn)}n=n0∞ is decreasing and hence {Dg(z,xn)}n∈ℕ is convergent. Thus, we have Dg(z,xn)-Dg(z,xn+1)→0 as n→∞. This, together with condition (c) and (118), implies that
(120)limn→∞ρs2*(∥∇g(xn)-∇g(Tjxn)∥)=0.
Therefore, from the property of ρs2*, we deduce that
(121)limn→∞∥∇g(xn)-∇g(Tjxn)∥=0.
Since ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*, we arrive at
(122)limn→∞∥xn-Tjxn∥=0,∀j∈ℕ.
Since, for any j∈ℕ, Tj is a Bregman relatively nonexpansive mapping, there exists a subsequence {xni}i∈ℕ of {xn}n∈ℕ converging weakly to some y∈Z such that
(123)limsupn→∞〈xn+1-z,∇g(u)-∇g(z)〉=limi→∞〈xni+1-z,∇g(u)-∇g(z)〉.
This, together with (13), implies that
(124)limsupn→∞〈xn+1-z,∇g(u)-∇g(z)〉=〈y-z,∇g(u)-∇g(z)〉≤0.
In view of Lemma 11 and (122), we obtain that
(125)limn→∞Dg(Tjxn,xn)=0.
This implies that
(126)Dg(Tjxn,yn)≤βn,0Dg(Tjxn,xn)+∑j=1∞‍βn,jDg(Tjxn,Tjxn)=βn,0Dg(Tjxn,xn)⟶0
as n→∞. Observe also that
(127)Dg(yn,xn+1)≤αnDg(yn,u)+(1-αn)Dg(yn,yn)=αnDg(yn,u)⟶0
as n→∞. In view of Lemma 11 and (119), (126), and (127), we conclude that
(128)limn→∞∥yn-Tjxn∥=limn→∞∥yn-xn+1∥=0,limn→∞∥xn+1-xn∥=limn→∞∥yn-xn∥=0.
Moreover, from (124) and (128), we deduce that
(129)limsupn→∞〈xn-z,∇g(u)-∇g(z)〉=limsupn→∞〈xn+1-z,∇g(u)-∇g(z)〉≤0.
Thus we have the desired result by Lemma 17.
Case  2. If {Dg(z,xn)}n∈ℕ is not eventually decreasing, then there exists a subsequence {ni}i∈ℕ of {n}n∈ℕ such that
(130)Dg(z,xni)<Dg(z,xni+1)
for all i∈ℕ. Applying Lemma 16, we can find a nondecreasing sequence {mk}k∈ℕ⊂ℕ such that mk→∞,
(131)Dg(z,xmk)<Dg(z,xmk+1),Dg(z,xk)≤Dg(z,xmk+1)
for all k∈ℕ. This, together with (118), implies that
(132)βmk(1-βmk)ρs2*(∥∇g(xmk)-∇g(Tjxmk)∥)≤Dg(z,xmk)-Dg(z,xmk+1)+αmkM4≤αmkM4
for all k∈ℕ. Then, by conditions (a) and (c), we get
(133)limk→∞ρs2*(∥∇g(xmk)-∇g(Tjxmk)∥)=0.
By the same argument, as in Case 1, we arrive at
(134)limsupk→∞〈xmk+1-z,∇g(u)-∇g(z)〉=limsupk→∞〈xmk-z,∇g(u)-∇g(z)〉≤0.
It follows from (119) that
(135)Dg(z,xmk+1)≤(1-αmk)Dg(z,xmk)+αmk〈xmk+1-z,∇g(u)-∇g(z)〉.
Since Dg(z,xmk)≤Dg(z,xmk+1), it follows that
(136)αmkDg(z,xmk)≤Dg(z,xmk)-Dg(z,xmk+1)+αmk〈xmk+1-z,∇g(u)-∇g(z)〉≤αmk〈xmk+1-z,∇g(u)-∇g(z)〉.
In particular, since αmk>0, we obtain
(137)Dg(z,xmk)≤〈xmk+1-z,∇g(u)-∇g(z)〉.
In view of (135), we deduce that
(138)limk→∞Dg(z,xmk)=0.
This, together with (136), implies that
(139)limk→∞Dg(z,xmk+1)=0.
On the other hand, we have Dg(z,xk)≤Dg(z,xmk+1) for all k∈ℕ which implies that xk→z as k→∞. Thus, we have xn→z as n→∞.In the following, we propose a new Halpern-type iterative scheme for finding common solutions of a system of equilibrium problems in a reflexive Banach space and obtain a strong convergence theorem.Theorem 26.
LetE be a reflexive Banach space and let g:E→ℝ be a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let {Cj}j∈ℕ be an infinite family of nonempty, closed, and convex subsets of E. For any j∈ℕ, let fj:Cj×Cj→ℝ be a bifunction that satisfies conditions (A1)–(A4) such that Z:=∩j=1∞EP(fj)≠⌀, where EP(fj) is the set of solutions to the equilibrium problem (104). Let {αn}n∈ℕ, {βn,j}n∈ℕ,j∈ℕ∪{0} be sequences in [0,1] satisfying the following control conditions:(a)
limn→∞αn=0;(b)
∑n=1∞‍αn=∞;(c)
βn,0+∑j=1∞‍βn,j=1,∀n∈ℕ;(d)
liminfn→∞βn,0βn,j>0,∀j∈ℕ.
Let{xn}n∈ℕ be a sequence generated by
(140)u∈E,x1∈Echosenarbitrarily,yn=∇g*[βn,0∇g(xn)+∑j=1∞‍βn,j∇g(Resrfjg(xn))],xn+1=∇g*[αn∇g(u)+(1-αn)∇g(yn)],n∈ℕ,
where ∇g is the gradient of g and r>0 is a constant. Then, the sequence {xn}n∈ℕ defined in (140) converges strongly to projZgu as n→∞.Remark 27.
(1) We propose a new type of Halpern iterative scheme for finding common zeros of an infinite family of maximal monotone operators in a reflexive Banach space. This scheme has an advantage that we do not use any projection which creates some difficulties in a practical calculation of the iterative sequence.
(2) In Theorem25, we present a strong convergence theorem for an infinite family of maximal monotone operators with a new algorithm and new control conditions. We remove the sets Cn and Qn in Theorems 5 and 6.
(3) Theorem20 improves and extends the corresponding results of [45, 46] from one maximal monotone operator in Hilbert spaces to more general an infinite family of maximal monotone operators in Banach spaces.
(4) Theorem20 improves and extends the corresponding result of [47] from two maximal monotone operators in Hilbert spaces to more general an infinite family of maximal monotone operators in Banach spaces.
(5) Theorems25 and 26 improve and generalize Theorems 5 and 6, respectively.


## 5. Applications

In this section, we consider the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space.Theorem 28.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let {gj}j∈ℕ be an infinite family of continuously Fréchet differentiable and convex functions on E such that the gradient of gj, ∇gj is continuous and monotone for each j∈ℕ. Assume that Ω:=∩j=1∞argminy∈Egj(y)={z∈E:gj(z)=∩j=1∞miny∈Cgj(y)}≠⌀. Let {αn}n∈ℕ, {βn,j}n∈ℕ,j∈ℕ∪{0} be sequences in [0,1] satisfying the following control conditions:(a)
limn→∞αn=0;(b)
∑n=1∞‍αn=∞;(c)
βn,0+∑j=1∞‍βn,j=1,∀n∈ℕ;(d)
liminfn→∞βn,0βn,j>0,∀j∈ℕ.Let {xn}n∈ℕ be a sequence generated by
(141)u∈E,x1∈Cchosenarbitrarily,un,j∈Csuchthat〈y-un,j,∇gj(un,j)〉+〈y-un,j,∇g(un,j)-∇g(xn)〉≥0,∀y∈C,yn=∇g*[βn,0∇g(xn)+∑j=1∞‍βn,j∇g(un,j)],xn+1=∇g*[αn∇g(u)+(1-αn)∇g(yn)]andn∈ℕ.
Then the sequence {xn}n∈ℕ defined in (141) converges strongly to projΩgu as n→∞.



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*Source: 743279-2014-02-03.xml*