# Strong Convergence Theorems for Quasi-Bregman Nonexpansive Mappings in Reflexive Banach Spaces

Authors: Mohammed Ali Alghamdi; Naseer Shahzad; Habtu Zegeye
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We study a strong convergence for a common fixed point of a
finite family of quasi-Bregman nonexpansive mappings in the framework of
real reflexive Banach spaces. As a consequence, convergence for a common
fixed point of a finite family of Bergman relatively nonexpansive mappings is
discussed. Furthermore, we apply our method to prove strong convergence theorems
of iterative algorithms for finding a common solution of a finite family
equilibrium problem and a common zero of a finite family of maximal monotone
mappings. Our theorems improve and unify most of the results that have
been proved for this important class of nonlinear mappings.


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## Body

## 1. Introduction

Throughout this paper,E is a real reflexive Banach space with the dual space E*. The norm and the dual pair between E* and E are denoted by ∥·∥ and 〈·,·〉, respectively. We also assume that f:E→(-∞,+∞] is a proper, lower semicontinuous, and convex function. Denote the domain of f by dom⁡f; that is, dom⁡f={x∈E:f(x)<∞}. Let x∈int⁡(dom⁡f). The subdifferential of f at x∈E is the convex set defined by
(1)∂f(x)={x*∈E*:f(x)+〈x*,y-x〉≤f(y),∀y∈E}.
TheFenchel conjugate of f is the function f*:E*→(-∞,+∞] defined by f*(y)=sup⁡{〈y,x〉-f(x):x∈E}.Letx∈Dom⁡(f) and y∈E. The right-hand derivative of f at x in the direction of y is defined by f∘(x,z)=lim⁡t→0+(f(x+ty)-f(x))/t. The function f is calledGâteaux differentiable at x if lim⁡t→0+(f(x+ty)-f(x))/t exists for any y and hence f∘(x,y) coincides with ∇f(x), the value of the gradient ∇f of f at x. The function f is said to beGâteaux differentiable if it is Gâteaux differentiable for any x∈int domf. Furthermore, f is said to beFréchet differentiableat x if this limit is attained uniformly in ∥y∥=1 and it is calleduniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x∈C and ∥y∥=1.Letf:E→(-∞,+∞] be a Gâteaux differentiable function. The function Df:dom⁡f×int⁡(dom⁡f)→[0,+∞) defined by
(2)Df(x,y)∶=f(x)-f(y)-〈∇f(y),x-y〉
is called theBregman distancewith respect to f [1].WhenE is a smooth Banach space, setting f(x)=∥x∥2 for all x∈E, we have ∇f(x)=2Jx, for all x∈E, where J is the normalized duality mapping from E into 2E*, and hence Df(x,y) reduces to Df(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2, for all x,y∈E, which is the Lyapunov function introduced by Alber [2]. If E=H, a Hilbert space, J is identity mapping and hence Df(x,y) becomes Df(x,y)=∥x-y∥2, for all x, y∈H.LetC be a nonempty and convex subset of int⁡(dom⁡f) and let T:C→int⁡(dom⁡f) be a mapping. T is said to benonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C, and T is said to bequasinonexpansive if F(T)≠∅ and ∥Tx-p∥≤∥x-p∥, for all x∈K and p∈F(T), where F(T) stands for the fixed point set of T; that is, F(T)={x∈C:Tx=x}. A point p∈C is said to be an asymptotic fixed point of T (see [3]) if C contains a sequence {xn} which converges weakly to p such that lim⁡n→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T is denoted by F^(T).A mappingT:C→int⁡(dom⁡f) with F(T)≠∅ is called(i)
quasi-Bregman nonexpansive [4] if
(3)Df(p,Tx)≤Df(p,x),∀x∈C,p∈F(T);(ii)
Bregman relatively nonexpansive [4] if
(4)Df(p,Tx)≤Df(p,x),∀x∈C,p∈F(T),F^(T)=F(T);(iii)
Bregman firmly nonexpansive [5] if, for all x,y∈C,
(5)〈∇f(Tx)-∇f(Ty),Tx-Ty〉≤〈∇f(x)-∇f(y),Tx-Ty〉or, equivalently,
(6)Df(Tx,Ty)+Df(Ty,Tx)+Df(Tx,x)+Df(Ty,y)≤Df(Tx,y)+Df(Ty,x).
Iterative methods for approximating fixed points of nonexpansive, quasinonexpansive mappings and their generalizations have been studied by various authors (see, e.g., [6–16] and the references therein) in Hilbert spaces. But extending this theory to Banach spaces encountered some difficulties because the useful examples of nonexpansive operators in Hilbert spaces are no longer nonexpansive in Banach spaces (e.g., the resolvent RA=(I+rA)-1, for r>0, of a monotone mapping A:C→2H and the metric projection PC onto a nonempty, closed, and convex subset C of H). To overcome these difficulties, Bregman [1] discovered techniques with the use of Bregman distance function Df(·,·) instead of norm in the process of designing and analyzing feasibility and optimization problems. This opened a growing area of research for solving variational inequalities and approximating solutions or fixed points of nonlinear mappings (see, e.g., [1, 17–21] and the references therein).The existence and approximation of fixed points of Bregman firmly nonexpansive mappings were studied in [5]. It is also known that if T is Bregman firmly nonexpansive and f isLegendre function which is bounded, uniformly Frêchet differentiable, andtotally convex on bounded subsets of E, then F(T)=F^(T) and F(T) is closed and convex (see [5]). It also follows that every Bregman firmly nonexpansive mapping is Bregman relatively nonexpansive and hence quasi-Bregman nonexpansive mapping.ABregman projection [1] of x∈int⁡(dom⁡f) onto the nonempty closed and convex set C⊂ domf is the unique vector PCf(x)∈C satisfying
(7)Df(PCf(x),x)=inf⁡{Df(y,x):y∈C}.Remark 1.
IfE is a smooth Banach space and f(x)=∥x∥2 for all x∈E, then the Bregman projection PCf(x) reduces to the generalized projection ΠC(x) (see, e.g., [2]) which is defined by
(8)ϕ(ΠC(x),x)=min⁡y∈Cϕ(y,x),
where ϕ(y,x)=∥y∥2-2〈y,Jx〉+∥x∥2.
Very recently, by using Bregman projection, Reich and Sabach [4] introduced an algorithm for finding a common zero of many finitely maximal monotone mappings Ai:E→E(i=1,2,…,N) satisfying F∶=⋂i=1NAi-1(0)≠∅ in a reflexive Banach space E as follows:
(9)x0∈E,chosenarbitrarily,yni=ResλniAif(xn+eni),Cni={z∈E:Df(z,yni)≤Df(z,xn+eni)},Cn∶=⋂i=1NCni,Qn={z∈E:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=PCn∩Qnf(x0),∀n≥0,
where ResrAf∶=(∇f+rA)-1(∇f). Under suitable conditions, they proved that if, for each i=1,2,…,N, we have that lim⁡inf⁡n→∞λni>0 and the sequences of errors {eni}⊂E satisfy lim⁡inf⁡n→∞eni=0, then the sequence {xn} converges strongly to PFf(x0).
In [22], Reich and Sabach proposed an algorithm for finding a common fixed point of many finitely Bregman firmly nonexpansive mappings Ti:C→C(i=1,2,…,N) satisfying ⋂i=1NF(Ti)≠∅ in a reflexive Banach space E as follows:
(10)x0∈E,chosenarbitrarily,Q0i=E,yni=Ti(xn+eni),Qn+1i={z∈Qni:〈∇f(xn+eni)-∇f(yni),z-yni〉≤0},Qn=⋂i=1NQni,xn+1=PQn+1f(x0),∀n≥0.
They proved that, under suitable conditions, the sequence {xn} generated by (10), converges strongly to ⋂i=1NF(Ti) and applied it to the solution of convex feasibility and equilibrium problems.Remark 2.
But it is worth mentioning that the iteration processes (9) and (10) seem difficult in the sense that, at each stage of iteration, the set(s) Cn and/or Qn are/is computed and the next iterate is taken as the Bregman projection of x0 onto the intersection of Cn and/or Qn. This seems difficult to do in applications.In this paper, we investigate an iterative scheme for finding a common fixed point of a finite family of quasi-Bregman nonexpansive mappings in reflexive Banach spaces. We prove strong convergence theorems for the sequences produced by the method. Furthermore, we apply our method to prove strong convergence theorems for finding a solution of a finite family of equilibrium problems and for finding a common zero of a finite family of maximal monotone mappings. Our results improve and generalize many known results in the current literature (see, e.g., [4, 23])


## 2. Preliminaries

Letf:E→(-∞,+∞] be a Gâteaux differentiable function. The function f is said to beessentially smooth if ∂f is both locally bounded and single-valued on its domain. It is calledessentially strictly convex, if (∂f)-1 is locally bounded on its domain and is strictly convex on every convex subset of dom⁡∂f. f is said to be aLegendre, if it is both essentially smooth and essentially strictly convex. When the subdifferential of f is single-valued, it coincides with the gradient ∂f=∇f (see [24]).We note that ifE is a reflexive Banach space, then we have the following.(i)
f is essentially smooth if and only if f* is essentially strictly convex (see [25, Theorem 5.4]).(ii)
(∂f)-1=∂f* (see [26]).(iii)
f is Legendre if and only if f* is Legendre (see [25, Corollary 5.5]).(iv)
Iff is Legendre, then ∇f is a bijection satisfying ∇f=(∇f*)-1, ran∇f=dom⁡∇f*=int domf*, and ran∇f*=dom⁡∇f=int domf (see [25, Theorem 5.10]).When E is a smooth and strictly convex Banach space, one important and interesting example of Legendre function is f(x)∶=(1/p)∥x∥p(1<p<∞). In this case the gradient ∇f of f coincides with the generalized duality mapping of E; that is, ∇f=Jp(1<p<∞). In particular, ∇f=I, the identity mapping in Hilbert spaces.LetE be a Banach space and let Br∶={z∈E:∥z∥≤r}, for all r>0 and SE={x∈E:∥x∥=1}. Then a function f:E→R is said to beuniformly convex on bounded subsets of E [27, page 203] if ρr(t)>0, for all r,t>0, where ρr:[0,∞)→[0,∞] is defined by
(11)ρr(t)∶=inf⁡x,y∈Br,∥x-y∥=t,α∈(0,1)(αf(x)+(1-α)f(y)-f(αx+(1-α)y))×(α(1-α))-1,
for all t≥0. The function ρr is called thegauge of uniform convexity off. The function f is also said to beuniformly smooth on bounded subsets of E [27, page 207] if lim⁡t→0+(σr(t)/t)=0, for all r>0, where σr:[0,∞)→[0,∞] is defined by
(12)σr(t)∶=sup⁡x∈Br,y∈SE,α∈(0,1)(αf(x+(1-α)ty)+(1-α)f(x-αty)-f(x))×(α(1-α))-1,
for all t≥0.In the sequel, we will need the following lemmas.Lemma 3 (see [28]).
LetE be a Banach space, let r>0 be a constant, and let f:E→R be a uniformly convex on bounded subsets of E. Then
(13)f(∑k=0nαkxk)≤∑k=0nαkf(xk)-αiαjρr(∥xi-yj∥),
for all i,j∈{0,1,2,…,n},  xk∈Br,αk∈(0,1), and k=0,1,2,…,n with ∑k=0n‍αk=1, where ρr is the gauge of uniform convexity of f.Lemma 4 (see [24]).
Letf:E→(-∞,+∞] be a proper, lower semicontinuous, and convex function; then f*:E*→(-∞,+∞] is a proper, weak* lower semicontinuous, and convex function. Thus, for all z∈E, we have
(14)Df(z,∇f*(∑i=1Nti∇f(xi)))≤∑i=1NtiDf(z,xi).Lemma 5 (see [29]).
Letf:E→R be a Gâteaux differentiable on int⁡(dom⁡f) such that ∇f* is bounded on bounded subsets of dom⁡f*. Let x*∈X and {xn}⊂E. If {Df(x,xn)} is bounded, so is the sequence {xn}.A functionf on E iscoercive [30] if the sublevel set of f is bounded; equivalently, lim⁡∥x∥→∞f(x)=∞. A function f on E is said to bestrongly coercive [27] if lim⁡∥x∥→∞f(x)/∥x∥=∞.Lemma 6 (see [27]).
LetE be a reflexive Banach space and let f:E→R be a continuous convex function which is strongly coercive. Then the following assertions are equivalent.(1)
f is bounded on bounded subsets and uniformly smooth on bounded subsets of E;(2)
f* is Fréchet differentiable and ∇f* is uniformly norm-to-norm continuous on bounded subsets of E*;(3)
dom⁡f=E*, f* is strongly coercive and uniformly convex on bounded subsets of E*.Lemma 7 (see [5]).
Letf:E→(-∞,+∞] be a Legendre function. Let C be a nonempty closed convex subset of int⁡(dom⁡f) and let T:C→C be a quasi-Bregman nonexpansive mapping. Then F(T) is closed and convex.Letf:E→(-∞,+∞] be a Gâteaux differentiable function. The modulus of total convexity of f at x∈dom⁡f is the function νf(x,·):[0,+∞)→[0,+∞] defined by
(15)νf(x,t)∶=inf⁡{Df(y,x):y∈dom⁡f,∥y-x∥=t}.
The function f is calledtotally convex at x if νf(x,t)>0, whenever t>0. The function f is called totally convex if it is totally convex at any point x∈int⁡(dom⁡f) and is said to be totally convex on bounded sets if νf(B,t)>0 for any nonempty bounded subset B of E and t>0, where the modulus of total convexity of the function f on the set B is the function νf:int domf×[0,+∞)→[0,+∞] defined by
(16)νf(B,t)∶=inf⁡{νf(x,t):x∈B∩dom⁡f}.
We know that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets (see [18, Theorem 2.10]). The following lemmas will be useful in the proof of our main results.Lemma 8 (see [31]).
The functionf:E→(-∞,+∞) is totally convex on bounded subsets of E if and only if, for any two sequences {xn} and {yn} in int⁡(dom⁡f) and dom⁡f, respectively, such that the first one is bounded,
(17)lim⁡n→∞Df(yn,xn)=0⟹lim⁡n→∞∥yn-xn∥=0.Lemma 9 (see [27, 32]).
Letf:E→R be a strongly coercive and uniformly convex on bounded subsets of E; then f* is bounded on bounded sets and uniformly Fréchet differentiable on bounded subsets of E*.Lemma 10 (see [33]).
Letf:E→(-∞,+∞] be uniformly Fréchet differentiable and bounded on bounded sets of E; then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E*.Lemma 11 (see [18]).
LetC be a nonempty, closed, and convex subset of E. Let f:E→R be a Gâteaux differentiable and totally convex function and let x∈E; then (i)
z=PCf(x) if and only if 〈∇f(x)-∇f(z),y-z〉≤0,∀y∈C.(ii)
Df(y,PCf(x))+Df(PCf(x),x)≤Df(y,x),∀y∈C.Letf:E→R be a Legendre and Gâteaux differentiable function. Following [2, 34], we make use of the function Vf:E×E*→[0,+∞) associated with f, which is defined by
(18)Vf(x,x*)=f(x)-〈x,x*〉+f*(x*),∀x∈E,x*∈E*.
Then Vf is nonnegative and
(19)Vf(x,x*)=Df(x,∇f*(x*))∀x∈E,x*∈E*.
Moreover, by the subdifferential inequality,
(20)Vf(x,x*)+〈y*,∇f*(x*)-x〉≤Vf(x,x*+y*),
where ∀x∈E and x*,y*∈E* (see [35]).Lemma 12 (see [36]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(21)an+1≤(1-αn)an+αnδn,n≥n0,
where {αn}⊂(0,1) and {δn}⊂R satisfying the following conditions: lim⁡n→∞αn=0, ∑n=1∞‍αn=∞ and limsup⁡n→∞δn≤0. Then, lim⁡n→∞an=0.Lemma 13 (see [37]).
Let{an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈N. Then there exists an increasing sequence {mk}⊂N such that mk→∞ and the following properties are satisfied by all (sufficiently large) numbers k∈N:
(22)amk≤amk+1,ak≤amk+1.
In fact, mk is the largest number n in the set {1,2,…,k} such that the condition an≤an+1 holds.


## 3. Main Results

We now prove the following theorem.Theorem 14.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Ti:C→C, for i=1,2,…,N, be a finite family of quasi-Bregman nonexpansive mappings such that F^(Ti)=F(Ti), for i=1,2,…,N. Assume that F∶=⋂i=1NF(Ti) is nonempty. For u,x0∈C let {xn} be a sequence generated by
(23)yn=PCf∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where {αn}⊂(0,1) and {βi}i=0N⊂[c,d]⊂(0,1) satisfying lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then, {xn} converges strongly to p=PFf(u).Proof.
Lemma7 ensures that each F(Ti), for i∈{1,2,…,N}, and F are closed and convex. Thus, PFf is well defined. Let p=PFf(u). Then, from (23), Lemmas 11 and 4, and property of Df we get that
(24)Df(p,yn)=Df(p,PCf∇f*(αn∇f(u)+(1-αn)∇f(xn)))≤Df(p,∇f*(αn∇f(u)+(1-αn)∇f(xn)))=αnDf(p,u)+(1-αn)Df(p,xn).
Moreover, from (23), (18), and (19) we get that
(25)Df(p,xn+1)=Df(p,∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)))=V(p,β0∇f(yn)+∑i=1Nβi∇f(Tiyn))=f(p)-〈p,β0∇f(yn)+∑i=1Nβi∇f(Tiyn)〉+f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)).
Since f is uniformly Fréchet differentiable function we have that f is uniformly smooth and hence by Theorem 3.5.5 of [27] we get that f* is uniformly convex. This, with Lemmas 3 and (24), gives that
(26)Df(p,xn+1)≤f(p)-β0〈p,∇f(yn)〉-∑i=1Nβi〈p,∇f(Tiyn)〉+β0f*(∇f(yn))+∑i=1Nβif*(∇f(Tiyn))-β0βiρr*(∥∇f(yn)-∇f(Tixn)∥)=β0V(p,∇f(yn))+∑i=1NβiV(p,∇f(Tyn))-β0βiρr*(∥∇f(yn)-∇f(Tixn)∥)=β0Df(p,yn)+∑i=1NβiDf(p,Tyn)-β0βiρr*(∥∇f(yn)-∇f(Tixn)∥)≤β0Df(p,yn)+∑i=1NβiDf(p,yn)-β0βiρr*(∥∇f(yn)-∇f(Tixn)∥)≤Df(p,yn)-β0βiρr*(∥∇f(yn)-∇f(Tiyn)∥)≤Df(p,yn),(27)≤αnDf(p,u)+(1-αn)Df(p,xn),
for each i∈{1,2,…,N}. Thus, by induction,
(28)Df(p,xn+1)≤max⁡{Df(p,u),Df(p,x0)},∀n≥0,
which implies that {xn} is bounded. Now, let zn=∇f*(αn∇f(u)+(1-αn)∇f(xn)). Then we note that yn=PCfzn. Since f is strongly coercive, uniformly convex, uniformly Fréchet differentiable, and bounded on bounded subsets of E by Lemmas 10, 9, and 6 we have that ∇f and ∇f* are bounded on bounded sets and hence {zn} and {yn} are bounded. In addition, using (19), (20), and property of Df we obtain that
(29)Df(p,yn)≤Df(p,zn)=V(p,∇f(zn))≤V(p,∇f(zn)-αn(∇f(u)-∇f(p)))+αn〈∇f(u)-∇f(p),zn-p〉=Df(p,∇f*(αn∇f(p)+(1-αn)∇f(xn)))+αn〈∇f(u)-∇f(p),zn-p〉≤αnDf(p,p)+(1-αn)Df(p,xn)+αn〈∇f(u)-∇f(p),zn-p〉=(1-αn)Df(p,xn)+αn〈∇f(u)-∇f(p),zn-p〉.
Furthermore, from (26) and (29) we have that
(30)Df(p,xn+1)≤(1-αn)Df(p,xn)+αn〈∇f(u)-∇f(p),zn-p〉-β0βiρr*(∥∇f(yn)-∇f(Tiyn)∥)(31)≤(1-αn)Df(p,xn)+αn〈∇f(u)-∇f(p),zn-p〉.
Now, we consider two cases.
Case
1. Suppose that there exists n0∈N such that {Df(p,xn)} is nonincreasing for all n≥n0. In this situation, {Df(p,xn)} is convergent. Then, from (30) we have that
(32)β0βiρr*(∥∇f(yn)-∇f(Tiyn)∥)⟶0,
which implies, by the property of ρr*, that
(33)∇f(yn)-∇f(Tiyn)⟶0,asn⟶∞.
Now, since f is a strongly coercive and uniformly convex on bounded subsets of E, f* is uniformly Fréchet differentiable on bounded subsets of E* (see [27, Proposition 3.6.2]) and f is Legendre; we have by Lemma 10 that
(34)yn-Tiyn⟶0,asn⟶∞,
for each i∈{1,2,…,N}.
Furthermore, Lemma11, property of Df, and the fact that αn→0, as n→∞, imply that
(35)Df(xn,yn)=Df(xn,PCfzn)≤Df(xn,zn)=Df(xn,∇f*(αn∇f(u)+(1-αn)∇f(xn)))≤αnDf(xn,u)+(1-αn)Df(xn,xn)≤αnDf(xn,u)+(1-αn)Df(xn,xn)⟶0asn⟶∞,
and hence by Lemma 8 we get that
(36)xn-yn⟶0,xn-zn⟶0,n⟶∞.
Since {zn} is bounded and E is reflexive, we choose a subsequence {znj} of {zn} such that znj⇀z and lim⁡sup⁡n→∞〈∇f(u)-∇f(p),zn-p〉=lim⁡j→∞〈∇f(u)-∇f(p),znj-p〉. Then, from (36) we get that
(37)ynj⇀z,asj⟶∞.
Thus, from (34) and the fact that F^(Ti)=F(Ti) we obtain that z∈F(Ti), for each i∈{1,2,…,N} and hence z∈⋂i=1NF(Ti).
Therefore, by Lemma11, we immediately obtain that limsup⁡n→∞〈∇f(u)-∇f(p),zn-p〉=lim⁡j→∞〈∇f(u)-∇f(p),znj-p〉=〈∇f(u)-∇f(p),z-p〉≤0. It follows from Lemma 12 and (31) that Df(p,xn)→0, as n→∞. Consequently, by Lemma 8 we obtain that xn→p.
Case
2. Suppose that there exists a subsequence {nj} of {n} such that
(38)Df(p,xnj)<Df(p,xnj+1),
for all j∈N. Then, by Lemma 13, there exists a nondecreasing sequence {mk}⊂N such that mk→∞, Df(p,xmk)≤Df(p,xmk+1) and Df(p,xk)≤Df(p,xmk+1), for all k∈N. Then from (30) and the fact that αn→0 we obtain that
(39)ρr*(∥∇f(ymk)-∇f(Tiymk)∥)⟶0,ask⟶∞,
for each i∈{1,2,…,N}. Thus, following the method of proof of Case 1, we obtain that ymk-Tiymk→0, xmk-ymk→0, xmk-zmk→0, as k→∞, and hence we obtain that
(40)limsup⁡k→∞〈∇f(u)-∇f(p),zmk-p〉≤0.
Then from (31) we have that
(41)Df(p,xmk+1)≤(1-αmk)Df(p,xmk)+αmk〈∇f(u)-∇f(p),zmk-p〉.
Now, since Df(p,xmk)≤Df(p,xmk+1), inequality (41) implies that
(42)αmkDf(p,xmk)≤Df(p,xmk)-Df(p,xmk+1)+αmk〈∇f(u)-∇f(p),zmk-p〉≤αmk〈∇f(u)-∇f(p),zmk-p〉.
In particular, since αmk>0, we get
(43)Df(p,xmk)≤〈∇f(u)-∇f(p),zmk-p〉.
Then, from (40) we obtain Df(p,xmk)→0, as k→∞. This together with (41) gives Df(p,xmk+1)→0, as k→∞. But Df(p,xk)≤Df(p,xmk+1), for all k∈N; thus we obtain that xk→p. Therefore, from the above two cases, we can conclude that {xn} converges strongly to p=PFf(u) and the proof is complete.If, in Theorem14, we consider a single quasi-Bregman nonexpansive mapping, we get the following corollary.Corollary 15.
Letf:E→R be a coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let T:C→C be a quasi-Bregman nonexpansive mapping such that F^(T)=F(T)≠∅. Let {xn} be a sequence generated by
(44)yn=PCf∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β∇f(yn)+(1-β)∇f(Tyn)),∀n≥0,
where {αn}⊂(0,1) and β∈(0,1) satisfying lim⁡n→∞αn=0 and ∑n=1∞‍=∞. Then {xn} converges strongly to p=PF(T)f(u).If, in Theorem14, we assume that each Ti, for i∈{1,2,…,N}, is Bregman relatively nonexpansive mapping, then we have that F^(Ti)=F(Ti) for each i∈{1,2,…,N}. Thus, we get the following corollary.Corollary 16.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Ti:C→C, for i=1,2,…,N, be a finite family of Bregman relatively nonexpansive mappings. Assume that F∶=⋂i=1NF(Ti) is nonempty. For u,x0∈C let {xn} be a sequence generated by
(45)yn=PCf∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where {αn}⊂(0,1) and {βi}i=0N⊂[c,d]⊂(0,1) satisfying lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then, {xn} converges strongly to p=PFf(u).If, in Theorem14, we assume that each Ti(i=1,2,…,N) is Bregman firmly nonexpansive, then we get the following corollary.Corollary 17.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Ti:C→C, for i=1,2,…,N, be a finite family of Bregman firmly nonexpansive mappings with ⋂i=1NF(Ti)≠∅. Let {xn} be a sequence generated by
(46)yn=PCf∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where {αn}⊂(0,1) and {βi}i=0N⊂[c,d]⊂(0,1) satisfying lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then {xn} converges strongly to p=PFf(u).


## 4. Applications

### 4.1. Equilibrium Problems

Letf:E→(-∞,+∞] be a coercive Legendre function. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f). Let H:C×C→R be a bifunction with the following conditions (see example [38]).(B1)
H(x,x)=0, for all x∈C;(B2)
H is monotone; that is, H(x,y)+H(y,x)≤0, for all x,y∈C;(B3)
for allx,y,z∈C, limsup⁡t→0+H(tz+(1-t)x,y)≤H(x,y);(B4)
for eachx∈C, H(x,·) is convex and lower semicontinuous.The equilibrium problem corresponding to H is to find x*∈C such that
(47)H(x*,y)≥0,∀y∈C.
The set of solutions of (47) is denoted by EP(H). Equilibrium problem is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, and so forth; see [20, 30, 38–43].The resolvent of a bifunctionH:C×C→R [39] is the operator ResHf:E→2C, defined by
(48)ResHf(x)={z∈C:H(z,y)+〈∇f(z)-∇f(x),y-z〉≥0,∀y∈C}.
We know the following lemma in [23].Lemma 18.
Letf:E→(-∞,+∞] be a coercive Legendre function. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f). If the bifunction H:C×C→R satisfies conditions (B1)–(B4), then(1)
ResHf is single-valued and dom(Resgf))=E;(2)
ResHf is a Bregman firmly nonexpansive mapping;(3)
EP(H) is a closed and convex subset of C and EP(H)=F(ResHf);(4)
For allx∈E and for all u∈F(ResHf), we have
(49)Df(u,ResHf(x))+Df(ResHf(x),x)≤Df(u,x).In addition, iff is uniformly Fréchet differentiable and bounded on bounded subsets of E, then by Reich and Sabach [5] we have that F(ResHf)=F^(ResHf). Thus, considering Ti=ResHif, for i=1,2,…,N, in Corollary 17, we get the following result.Theorem 19.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Hi:C→C, for i=1,2,…,N, be bifunctions which satisfy the conditions (B1)–(B4) and F∶=⋂i=1NEP(Hi) is nonempty. For x0∈C let {xn} be a sequence generated by
(50)yn=∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where Ti=ResHif, {αn}⊂(0,1), and {βi}i=0N⊂[c,d]⊂(0,1) satisfy lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then, {xn} converges strongly to p=PFf(u).


### 4.2. Zeroes of Maximal Monotone Operators

In this section we present an algorithm for finding a common zero of a finite family of maximal monotone mappings.LetA:E→2E* be a mapping with range R(A)={x*∈E*:x*∈Ax} and domain D(A)={x∈E:Ax≠∅}. Then, A is said to bemonotone if, for any x,y∈dom⁡(A), we have
(51)x*∈Ax,y*∈Ay⟹〈x-y,x*-y*〉≥0.
It is said to bemaximal monotone if A is monotone and the graph of A is not a proper subset of the graph of any other monotone mappings, where graph of A is given by G(A)∶={(x,x*)∈E×E*:x*∈Ax}. It is known that if A is maximal monotone, then the set A-1(0)={z∈E:0∈Az} is closed and convex.LetA:C⊆E→2E* be maximal monotone mapping and f is Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of E. The resolvent of A with respect to f is the mapping ResAf:E→2E defined by
(52)ResAf=(∇f+A)-1∘∇f.
It is known that ResAf is single-valued, Bregman firmly nonexpansive and F(ResAf)=A-1(0) (see [17]). Furthermore, the result by Reich and Sabach [5] implies that F(ResAf)=F^(ResAf). If, in Theorem 14, we assume that Ti=ResAif, then we obtain an algorithm for finding a common zero of a finite family of maximal monotone mappings.Theorem 20.
Letf:E→R be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Ai:C→2E*,i=1,2,…,N, be N maximal monotone mappings with F∶=⋂i=1NAi-1(0)≠∅. Then, for each x1,u∈E, the sequence {xn} is defined by
(53)yn=∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where Ti=ResHif, {αn}⊂(0,1), and {βi}i=0N⊂[c,d]⊂(0,1) satisfy lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then, {xn} converges strongly to p=PFf(u).For a common point of a solution of equilibrium problem and a zero of maximal monotone mapping we have the following corollary.Corollary 21.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let H:C→C be bifunction which satisfy the conditions (B1)–(B4) and let A:C→2E* be a maximal monotone mapping. Assume that F∶=EP(Hi)∩A-1(0) is nonempty. For x0∈C let {xn} be a sequence generated by
(54)yn=∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+β1∇f(ResHfyn)+β2∇f(ResAfyn)),∀n≥0,
where {αn}⊂(0,1) and {βi}i=12⊂[c,d]⊂(0,1) satisfying lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=02‍βi=1. Then, {xn} converges strongly to p=PFf(u).Remark 22.
Corollary17 and Corollary 4.3 improve Theorem 1 of Reich and Sabach [23] and Theorem 4.2 of Reich and Sabach [4], respectively, in the sense that at each stage the computation of Cn or Qn is not required.


## 4.1. Equilibrium Problems

Letf:E→(-∞,+∞] be a coercive Legendre function. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f). Let H:C×C→R be a bifunction with the following conditions (see example [38]).(B1)
H(x,x)=0, for all x∈C;(B2)
H is monotone; that is, H(x,y)+H(y,x)≤0, for all x,y∈C;(B3)
for allx,y,z∈C, limsup⁡t→0+H(tz+(1-t)x,y)≤H(x,y);(B4)
for eachx∈C, H(x,·) is convex and lower semicontinuous.The equilibrium problem corresponding to H is to find x*∈C such that
(47)H(x*,y)≥0,∀y∈C.
The set of solutions of (47) is denoted by EP(H). Equilibrium problem is a unified model of several problems, namely, variational inequality problem, complementary problem, saddle point problem, optimization problem, fixed point problem, and so forth; see [20, 30, 38–43].The resolvent of a bifunctionH:C×C→R [39] is the operator ResHf:E→2C, defined by
(48)ResHf(x)={z∈C:H(z,y)+〈∇f(z)-∇f(x),y-z〉≥0,∀y∈C}.
We know the following lemma in [23].Lemma 18.
Letf:E→(-∞,+∞] be a coercive Legendre function. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f). If the bifunction H:C×C→R satisfies conditions (B1)–(B4), then(1)
ResHf is single-valued and dom(Resgf))=E;(2)
ResHf is a Bregman firmly nonexpansive mapping;(3)
EP(H) is a closed and convex subset of C and EP(H)=F(ResHf);(4)
For allx∈E and for all u∈F(ResHf), we have
(49)Df(u,ResHf(x))+Df(ResHf(x),x)≤Df(u,x).In addition, iff is uniformly Fréchet differentiable and bounded on bounded subsets of E, then by Reich and Sabach [5] we have that F(ResHf)=F^(ResHf). Thus, considering Ti=ResHif, for i=1,2,…,N, in Corollary 17, we get the following result.Theorem 19.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Hi:C→C, for i=1,2,…,N, be bifunctions which satisfy the conditions (B1)–(B4) and F∶=⋂i=1NEP(Hi) is nonempty. For x0∈C let {xn} be a sequence generated by
(50)yn=∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where Ti=ResHif, {αn}⊂(0,1), and {βi}i=0N⊂[c,d]⊂(0,1) satisfy lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then, {xn} converges strongly to p=PFf(u).


## 4.2. Zeroes of Maximal Monotone Operators

In this section we present an algorithm for finding a common zero of a finite family of maximal monotone mappings.LetA:E→2E* be a mapping with range R(A)={x*∈E*:x*∈Ax} and domain D(A)={x∈E:Ax≠∅}. Then, A is said to bemonotone if, for any x,y∈dom⁡(A), we have
(51)x*∈Ax,y*∈Ay⟹〈x-y,x*-y*〉≥0.
It is said to bemaximal monotone if A is monotone and the graph of A is not a proper subset of the graph of any other monotone mappings, where graph of A is given by G(A)∶={(x,x*)∈E×E*:x*∈Ax}. It is known that if A is maximal monotone, then the set A-1(0)={z∈E:0∈Az} is closed and convex.LetA:C⊆E→2E* be maximal monotone mapping and f is Legendre function which is bounded, uniformly Fréchet differentiable on bounded subsets of E. The resolvent of A with respect to f is the mapping ResAf:E→2E defined by
(52)ResAf=(∇f+A)-1∘∇f.
It is known that ResAf is single-valued, Bregman firmly nonexpansive and F(ResAf)=A-1(0) (see [17]). Furthermore, the result by Reich and Sabach [5] implies that F(ResAf)=F^(ResAf). If, in Theorem 14, we assume that Ti=ResAif, then we obtain an algorithm for finding a common zero of a finite family of maximal monotone mappings.Theorem 20.
Letf:E→R be a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let Ai:C→2E*,i=1,2,…,N, be N maximal monotone mappings with F∶=⋂i=1NAi-1(0)≠∅. Then, for each x1,u∈E, the sequence {xn} is defined by
(53)yn=∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+∑i=1Nβi∇f(Tiyn)),∀n≥0,
where Ti=ResHif, {αn}⊂(0,1), and {βi}i=0N⊂[c,d]⊂(0,1) satisfy lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=0N‍βi=1. Then, {xn} converges strongly to p=PFf(u).For a common point of a solution of equilibrium problem and a zero of maximal monotone mapping we have the following corollary.Corollary 21.
Letf:E→R be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of int⁡(dom⁡f) and let H:C→C be bifunction which satisfy the conditions (B1)–(B4) and let A:C→2E* be a maximal monotone mapping. Assume that F∶=EP(Hi)∩A-1(0) is nonempty. For x0∈C let {xn} be a sequence generated by
(54)yn=∇f*(αn∇f(u)+(1-αn)∇f(xn)),xn+1=∇f*(β0∇f(yn)+β1∇f(ResHfyn)+β2∇f(ResAfyn)),∀n≥0,
where {αn}⊂(0,1) and {βi}i=12⊂[c,d]⊂(0,1) satisfying lim⁡n→∞αn=0, ∑n=1∞‍αn=∞, and ∑i=02‍βi=1. Then, {xn} converges strongly to p=PFf(u).Remark 22.
Corollary17 and Corollary 4.3 improve Theorem 1 of Reich and Sabach [23] and Theorem 4.2 of Reich and Sabach [4], respectively, in the sense that at each stage the computation of Cn or Qn is not required.



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*Source: 580686-2014-07-21.xml*