# Convergence Theorems for Right Bregman Strongly Nonexpansive Mappings in Reflexive Banach Spaces

Authors: H. Zegeye; N. Shahzad
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We prove a strong convergence theorem for a common fixed point of a finite family of right Bregman strongly nonexpansive mappings in the framework of real reflexive Banach spaces.  Furthermore, we apply our method to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our theorems complement some recent results that have been proved for this important class of nonlinear mappings.


---

## Body

## 1. Introduction

In this paper, without other specifications, letE be a real reflexive Banach space and E* as its dual, let R be the set of real numbers, and let C be a nonempty, closed, and convex subset E. Let f:E→(-∞,∞] be a proper convex and lower semicontinuous function. Denote the domain of f by dom⁡f; that is, dom⁡f={x∈E:f(x)<∞}. TheFenchel conjugate of f is the function f*:E*→(-∞,+∞] defined by f*(y)=sup⁡{〈y,x〉-f(x):x∈E}. f is calledcofinite if dom⁡f*=E*. For any x∈int⁡(dom⁡f) and y∈E, the right-hand derivative of f at x in the direction of y is defined by f0(x,y):=lim⁡t→0+(f(x+ty)-f(x))/t.The functionf is calledGâteaux differentiable at x if lim⁡t→0+(f(x+ty)-f(x))/t exists for any y. In this case, f0(x,y) coincides with ∇f(x), the value of the gradient ∇f of f at x. The function f is calledGâteaux differentiable if it is Gâteaux differentiable for any x∈ int(domf). The function f is said to beFréchet differentiable atx if this limit isattained uniformly in∥y∥=1 and f is said to beuniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x∈C and ∥y∥=1.The functionf is said to bebounded if it maps bounded subsets of E into bounded sets. We note that if f:E→R is uniformly Fréchet differentiable and bounded, then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E* (Proposition 2.1, [1]) and f* is uniformly Fréchet on bounded subsets of E* (see [2]) and hence ∇f* is uniformly continuous on bounded subsets of E* from the strong topology of E* to the strong topology of E.Letf:E→(-∞,+∞] be a Gâteaux differentiable function. The function Df:dom⁡f×int⁡(dom⁡f)→[0,+∞) defined by
(1)Df(x,y):=f(x)-f(y)-〈∇f(y),x-y〉
is called theBregman distance with respect tof [3].ABregman projection [4] of x∈int⁡(dom⁡f) onto the nonempty closed and convex set C⊂int⁡(dom⁡f) is the unique vector PCf(x)∈C satisfying
(2)Df(PCf(x),x)=inf⁡{Df(y,x):y∈C}.Remark 1.
IfE is a smooth and strictly convex Banach space and f(x)=∥x∥2 for all x∈E, then we have that ∇f(x)=2Jx for all x∈E, where J is the normalized duality mapping from E into 2E*, and hence(i)
Df(x,y) reduces to ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2, for all x,y∈E, which is the Lyapunov function introduced by Alber [5] and(ii)
PCf(x) reduces to the generalized projection ΠC(x) (see, e.g., [5]) which is defined by
(3)ϕ(ΠC(x),x)=min⁡y∈Cϕ(y,x).If E=H, a Hilbert space, J is the identity mapping and hence the Bregman distance becomes Df(x,y)=∥x-y∥2, for x,y∈H, and the Bregman projection PCf(x) reduces to the metric projection of H onto C,PC(x).LetT:C→C be a nonlinear mapping. Denote by F(T)={x∈C:Tx=x} the set of fixed points of T. A mapping T is said to benonexpansive if ∥Tx-Ty∥≤∥x-y∥, for all x,y∈C, and T is calledquasinonexpansive if ∥Tx-p∥≤∥x-p∥, for all x∈C and p∈F(T). A point p∈C is called anasymptotic fixed point of T (see [6]) if C contains a sequence {xn} which converges weakly to p such that lim⁡n→∞∥xn-Txn∥=0. We denote by F^(T) the set of asymptotic fixed points of T.A mappingT:C→int⁡(dom⁡f) is called(i)
left quasi-Bregman nonexpansive [7] if F(T)≠∅ and
(4)Df(p,Tx)≤Df(p,x),∀x∈C,p∈F(T);(ii)
left Bregman relatively nonexpansive [7] if F(T)≠∅ and
(5)hhhhhhhlhDf(p,Tx)≤Df(p,x),∀x∈C,p∈F(T),F^(T)=F(T);(iii)
left Bregman strongly nonexpansive (see [8, 9]), with respect to nonempty F^(T), if
(6)Df(p,Tx)≤Df(p,x),∀x∈C,p∈F^(T);
and, if, whenever {xn}⊂C is bounded, p∈F^(T) and
(7)lim⁡n→∞(Df(p,xn)-Df(p,Txn))=0,
it follows that
(8)lim⁡n→∞Df(Txn,xn)=0;(iv)
left Bregman firmly nonexpansive [10] if F(T)≠∅ and for all x,y∈C,
(9)〈∇f(Tx)-∇f(Ty),Tx-Ty〉≤〈∇f(x)-∇f(y),Tx-Ty〉,
or, equivalently,
(10)Df(Tx,Ty)+Df(Ty,Tx)+Df(Tx,x)+Df(Ty,y)≤Df(Tx,y)+Df(Ty,x).If T is left Bregman firmly nonexpansive and f is Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E, then it is known in [10] that F(T)=F^(T) and F(T) is closed and convex (see [10]). It follows that every left Bregman firmly nonexpansive mapping is Bregman strongly nonexpansive with respect to a nonempty set F(T)=F^(T).Existence and approximation of fixed points of nonexpansive and quasinonexpansive mappings have been intensively studied for almost fifty years or so by various authors (see e.g., [11–24] and the references therein) in Hilbert spaces. But most of the methods failed to give the same conclusion in Banach spaces more general than Hilbert spaces. One of the reasons is that a nonexpansive mapping in Hilbert spaces may not be nonexpansive in Banach spaces (e.g., the resolvent RA=(I+A)-1 of a maximal monotone mapping A:H→2H and the metric projection PK onto a nonempty, closed, and convex subset C of H).To overcome this problem, researchers use the distance functionDf(·,·) introduced by Bregman [4] instead of norm which opened a growing area of research in designing and analyzing iterative techniques for solving variational inequalities, approximating equilibria, computing fixed points of nonlinear mappings, and approximating solutions of convex feasibility problems (see, e.g., [4, 25–28] and the references therein).In [29], Reich and Sabach proposed the following algorithm for finding a common fixed point of finitely many left Bregman firmly nonexpansive self-mappings Ti(i=1,2,…,N) on E satisfying ∩i=1NF(Ti)≠∅. For x1∈E let the sequence {xn} be defined by
(11)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≥1.
They proved that, under some suitable conditions, the sequence {xn} generated by (11) converges strongly to a point in ∩i=1NF(Ti) and applied it to the solution of convex feasibility and equilibrium problems.Very recently, by using Bregman projection, Reich and Sabach [9] proposed an algorithm for finding a common fixed point of finitely many left Bregman strongly nonexpansive mappings Ti:C→C(i=1,2,…,N) satisfying ∩i=1NF(Ti)≠∅ in a reflexive Banach space E as follows:
(12)x0∈E,chosenarbitrarily,yni=Ti(xn+eni),Cni={z∈E:Df(z,yni)≤Df(z,xn+eni)},Cn=⋂i=1NCni,Qni={z∈E:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=PCn∩Qnf(x0),∀n≥0.
Under some suitable conditions, they proved that the sequence {xn} generated by (12) converges strongly to a point in ∩i=1NF(Ti) and applied it to the solution of convex feasibility and equilibrium problems.The above results naturally bring us to the following:a natural question arises whether we can establish analogous results for right Bregman strongly nonexpansive mappings or not.A mappingT:C→int⁡(dom⁡f) is called(i)
right quasi-Bregman nonexpansive [30] if F(T)≠∅ and
(13)Df(Tx,p)≤Df(x,p),∀x∈C,p∈F(T);(ii)
right Bregman relatively nonexpansive [30] if F(T)≠∅ and
(14)hhhhhhlhhDf(Tx,p)≤Df(x,p),∀x∈C,p∈F(T),F^(T)=F(T);(iii)
right Bregman strongly nonexpansive (see [8, 9]), with respect to nonempty F^(T), if
(15)Df(Tx,p)≤Df(x,p),∀x∈C,p∈F^(T);
and if, whenever {xn}⊂C is bounded, p∈F^(T) and
(16)lim⁡n→∞(Df(xn,p)-Df(Txn,p))=0,
it follows that
(17)lim⁡n→∞Df(xn,Txn)=0;(iv)
right Bregman firmly nonexpansive [10] if F(T)≠∅ and for all x,y∈C,
(18)〈∇f(Tx)-∇f(Ty),Tx-Ty〉≤〈∇f(Tx)-∇f(Ty),x-y〉,
or, equivalently,
(19)Df(Tx,Ty)+Df(Ty,Tx)+Df(x,Tx)+Df(y,Ty)≤Df(x,Ty)+Df(y,Tx).Remark 2.
It is shown in [10] that if T is right Bregman firmly nonexpansive, then F^(T)=F(T) and hence it is right Bregman relatively nonexpansive mapping provided that the Legendre function f is uniformly Fréchet differentiable and bounded on bounded sets of E.The class of right Bregman firmly nonexpansive mappings associated with the Bregman distance induced by a convex function was introduced and studied by Martin-Marques et al. [30]. Examples of right Bregman firmly nonexpansive mappings are given in [30]. If C is a nonempty and closed subset of int⁡(dom⁡f), where f is a Legendre and Fréchet differentiable function, and T:C→int⁡(dom⁡f) is a right Bregman strongly nonexpansive mapping, it is proved that F(T) is closed (see [30]). In addition, they have shown that this class of mappings is closed under composition and convex combination and proved weak convergence of the Picard iterative method to a fixed point of a mapping under suitable conditions (see [31]). However, Picard iteration process has onlyweak convergence.In this paper, it is our purpose to introduce an iterative scheme which converges strongly to a common fixed point of a finite family of right Bregman strongly nonexpansive mappings. As a consequence, we use our results to approximate a common zero of a finite family of maximal monotone mappings and a solution of a finite family of convex feasibility problems in reflexive real Banach spaces. Our results complements the recent results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on C⊆E.


## 2. Preliminaries

Letf:E→(-∞,+∞] be a convex and Gâteaux differentiable function. Themodulus of total convexity of f at x∈dom⁡f is the function νf(x,·):[0;+∞)→[0;+∞] defined by
(20)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 calledtotally convex if it is totally convex at any point x∈int⁡(dom⁡f) and is said to betotally 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⁡(dom⁡f)×[0,+∞)→[0,+∞] defined by
(21)νf(B,t):=inf⁡{Vf(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 [27], Theorem 2.10).The functionf is calledessentially smooth, if ∂f is both locally bounded and single-valued on its domain and it is calledessentially strictly convex, if (∂f)-1 is locally bounded on its domain and f is strictly convex on every convex subset of dom⁡∂f. f is said to beLegendre, if it is both essentially smooth and essentially strictly convex. Since E is reflexive, we know that (∂f)-1=∂f* (see [34]), f is essentially smooth if and only if f* is essentially strictly convex (see [35], Theorem 5.4), and f is Legendre if and only if f* is Legendre (see [35], Corollary 5.5); if f is Legendre, then ∇f is a bijection satisfying ∇f=(∇f*)-1, ran ∇f=dom⁡∇f*=intdomf*, and ran ∇f*=dom⁡f=intdomf (see [35], Theorem 5.10). From now on, we assume that the convex function f:E→(-∞,+∞] is Legendre.IfE is a smooth and strictly convex Banach space, then an important and interesting 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.In the sequel, we shall use the following lemmas.Lemma 3 (see [31]).
Letf:E→R be a bounded, uniformly Fréchet differentiable, and totally convex on bounded subsets of E. For each i=1,…,N, let Ti:K⊂E→K be a right Bregman strongly nonexpansive mapping with respect to F^(Ti)=F(Ti), and let T:=TN∘TN-1∘⋯∘T1. If F:=∩i=1NF(Ti) is nonempty, then T is also right Bregman strongly nonexpansive and F(T)=∩i=1NF(Ti).Lemma 4 (see [30]).
Letf:E→R be a Fréchet differentiable function. Let C be a nonempty closed convex subset of int⁡(dom⁡f) and let T:C→int⁡(dom⁡f) be a right quasi-Bregman nonexpansive mapping. Then F(T) is closed.Lemma 5 (see [36]).
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 and
(22)lim⁡n→∞Df(yn,xn)=0⟹lim⁡n→∞∥yn-xn∥=0.Lemma 6 (see [27]).
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.Lemma 7 (see [37]).
Iff:E→(-∞,+∞] is 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
(23)Df(z,∇f*(∑i=1N‍ti∇f(xi)))≤∑i=1N‍tiDf(z,xi).Lemma 8 (see [31]).
Letf:E→R be admissible and totally bounded at a point x∈int⁡(dom⁡f). Let {xn}⊂dom⁡(f). If {Df(xn,x)} is bounded, then so is the sequence {xn}.Letf:E→R be a Gâteaux differentiable function. Following [3, 5], we make use of the function Vf:E×E*→[0,+∞) associated with f, which is defined by
(24)Vf(x,x*)=f(x)-〈x,x*〉+f*(x*),∀x∈E,x*∈E*.Then,Vf is nonnegative and
(25)Vf(x,x*)=Df(x,∇f*(x*))∀x∈E,x*∈E*.Moreover, by the subdifferential inequality,(26)Vf(x,x*)+〈y*,∇f*(x*)-x〉≤Vf(x,x*+y*),for allx∈E and x*,y*∈E* (see [38]).Lemma 9 (see [39]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(27)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 10 (see [40]).
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:
(28)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

Theorem 11.
Letf:E→R be a cofinite 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 right Bregman strongly nonexpansive mappings such that F(Ti)=F^(Ti), for each i∈{1,2,…,N}. Assume that F:=∩i=1NF(Ti) is nonempty. For u,x1∈C, let {xn} be a sequence generated by
(29)xn+1=αnu+(1-αn)T(xn),n=1,2,…,
where T=TN∘TN-1∘⋯∘T1, {αn}⊂(0,1) satisfy lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞. Then, {xn} converges strongly to a point p in F.Proof.
Note that from Lemma3 we have F=F(T)=∩i=1NF(Ti) and T is right Bregman strongly nonexpansive mapping. Let p∈F. Then, using (29), the convexity of f, and property of T we get that
(30)Df(xn+1,p)=Df(αnu+(1-αn)Txn,p)≤αnDf(u,p)+(1-αn)Df(Txn,p)≤αnDf(u,p)+(1-αn)Df(xn,p).
Thus, by induction we obtain that
(31)Df(xn+1,p)≤max⁡{Df(u,p),Df(x1,p)}∀n≥1,
which implies that {Df(xn,p)} and hence Df(Txn,p) are bounded. Thus, from Lemma 8 we get that {xn} and {Txn} are bounded. Now, let yn=∇f(xn). Then, iteration process (29) becomes
(32)yn+1=∇f(αn∇f*(∇f(u))+(1-αn)∇f*T*(yn)),KKKKKKKKKKKKKKKKKKKKKn=1,2,…,
where T*:=∇fT∇f*, a conjugate of T. Since ∇f and ∇f* are uniformly continuous on bounded subsets of int⁡(dom⁡f) and int⁡(dom⁡f*), respectively, we get that {yn} and {T*yn} are bounded and by Section 6 of Martin-Marquez et al.  [31] we have that T* is left Bergman strongly nonexpansive with respect to ∇f(F(T)). In addition, by Proposition 3.3 of [30] we have that ∇f(F(T))=F(T*)=F(T*)^:=F′ is closed and convex. Let p′=PF′f*(∇f(u)). Now, from (32), (25), (26), and Lemma 6 we obtain that
(33)Df*(p′,yn+1)=Df*(p′,∇f(αn∇f*(∇f(u))+(1-αn)∇f*(T*yn)))=Vf*(p′,αn∇f*(∇f(u))+(1-αn)∇f*(T*yn))≤Vf*((∇f*(∇f(u))-∇f*(p′))p′,αn∇f*(∇f(u))+(1-αn)∇f*(T*yn)KKKKKK-αn(∇f*(∇f(u))-∇f*(p′)))+〈αn(∇f*(∇f(u))-∇f*(p′)),yn+1-p′〉=Vf*(p′,αn∇f*(p′)+(1-αn)∇f*(T*yn))+αn〈∇f*(∇f(u))-∇f*(p′),yn+1-p′〉≤αnVf*(p′,∇f*(p′))+(1-αn)Vf*(p′,∇f*(T*yn))+αn〈∇f*(∇f(u))-∇f*(p′),yn+1-p′〉=(1-αn)Df*(p′,T*yn)+αn〈p′∇f*(∇f(u))KKKKKKKKKKKKKiKKKKK-∇f*(p′),yn+1-p′〉≤(1-αn)Df*(p′,yn)+αn〈(p′)∇f*(∇f(u))KKKKKKKKKKKKKiiKKK-∇f*(p′),yn+1-p′(∇f(u))〉.
Now, we consider two cases.
Case  1. Suppose that there exists n0∈N such that {Df*(p′,yn)} is decreasing for all n≥n0. Then, we get that {Df*(p′,yn)} is convergent and hence
(34)Df*(p′,yn)-Df*(p′,yn+1)⟶0asn⟶∞.
In addition, from (32) and Lemma 7 we have that
(35)Df(p′,yn+1)=Df*(p′,∇f(αn∇f*(∇f(u))+(1-αn)∇f*(T*yn)))≤αnDf*(p′,∇f(u))+(1-αn)Df*(p′,T*yn).
Following from (35), (34), and the fact that αn→0, as n→∞, we get that
(36)Df*(p′,yn)-Df*(p′,T*yn)=Df*(p′,yn)-Df*(p′,yn+1)+Df*(p′,yn+1)-Df*(p′,T*yn)≤Df*(p′,yn)-Df*(p′,yn+1)+αn(Df*(p′,u)-Df*(p′,T*yn))⟶0KIKIKKIKIKIKIKIKIKIKIKIKIKIasn⟶∞.
This with the fact that T* is left Bregman strongly nonexpansive implies that
(37)lim⁡n→∞Df*(T*yn,yn)=0.
Then, by Lemma 5 we obtain that
(38)lim⁡n→∞∥T*yn-yn∥=0.
Now, since E* is reflexive and {yn+1} is bounded, there exists a subsequence {ynk+1} of {yn+1} such that
(39)ynk+1⇀y∈E*,(40)limsup⁡n→∞〈∇f*(∇f(u))-∇f*(p′),yn+1-p′〉=limsup⁡k→∞〈∇f*(∇f(u))-∇f*(p′),ynk+1-p′〉.
Thus, from (39), (38), the fact that T* is left Bregman strongly nonexpansive mapping with F(T*)^=F(T*), and Lemma 6 we get that y∈F(T*)=F′ and
(41)limsup⁡n→∞〈∇f*(∇f(u))-∇f*(p′),yn+1-p′〉=limsup⁡k→∞〈∇f*(∇f(u))-∇f*(p′),ynk+1-p′〉=〈∇f*(∇f(u))-∇f*(p′),y-p′〉≤0.
Therefore, it follows from (33), (41), and Lemma 9 that Df(p′,yn)→0 as n→∞. Consequently, by Lemma 5 we obtain that yn→p′=PF'f*(∇f(u)) and hence xn=∇f*(yn)→∇f*(p′)=p∈F.
Case  2. Suppose that there exists a subsequence {ni} of {n} such that
(42)Df*(p′,yni)<Df*(p′,yni+1),
for all i∈N. Then, by Lemma 10, there exist a nondecreasing sequence {mk}⊂N such that mk→∞ and
(43)Df*(p′,ymk)≤Df*(p′,ymk+1),Df*(p′,yk)≤Df*(p′,ymk+1),
for all k∈N. Thus, we get that
(44)Df*(p′,ymk)-Df*(p′,T*ymk)≤Df*(p′,ymk)-Df*(p′,ymk+1)+Df*(p′,ymk+1)-Df*(p′,T*ymk)≤Df*(p′,ymk)-Df*(p′,ymk+1)+αmk(Df*(p′,∇f(u))-Df*(p′,T*ymk))⟶0.
This implies that Df*(T*ynk,ynk)→0 as k→∞. Now, following the method in Case 1 we obtain that
(45)limsup⁡k→∞〈∇f*(∇f(u))-∇f*(p′),ymk+1-p′〉≤0.
Now, from (33) we have that
(46)Df*(p′,ymk+1)≤(1-αmk)Df*(p′,ymk)+αmk〈∇f*(∇f(u))-∇f*(p′),ymk+1-p′〉.
But (43) and (46) imply that
(47)αmkDf*(p′,ymk)≤Df*(p′,ymk)-Df*(p′,ymk+1)+αmk〈∇f*(∇f(u))-∇f*(p′),ymk+1-p′〉≤αmk〈∇f*(∇f(u))-∇f*(p′),ymk+1-p′〉,
and noting that αmk>0, we get that
(48)Df*(p′,ymk)≤〈∇f*(∇f(u))-∇f*(p′),ymk+1-p′〉.
Thus, using (45) we get that Df*(p′,ymk)→0 and hence from (46) we have that Df*(p′,ymk+1)→0 as k→∞. But Df*(p′,yk)≤Df*(p′,ymk+1), for all k∈N, implies that Df*(p′,yk)→0 and hence by Lemma 5 we obtain that yk→p′ and xk=∇f*(yk)→p=∇f*(p′)∈F. Therefore, from the above two cases, we can conclude that {xn} converges strongly to p∈F and the proof is complete.Remark 12.
We note that the sequence{xn} in Theorem 11 converges strongly to a point p∈F such that p=∇f*(p′), where p′=PF′(∇f(u)).If, in Theorem11, we consider a single right Bregman strongly nonexpansive mapping, we get the following corollary.Corollary 13.
LetE be a reflexive Banach space and let f:E→R be a cofinite 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 right Bregman strongly nonexpansive mapping such that F^(T)=F(T)≠∅. For u,x1∈C let {xn} be a sequence generated by
(49)xn+1=αnu+(1-αn)Txn,n=1,2,…,
where {αn}⊂(0,1) satisfy lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞. Then, {xn} converges strongly to some p in F(T).If, in Theorem11, we assume that each Ti,(i=1,2,…,N) is right Bregman firmly nonexpansive, then we have that T=TN∘TN-1∘⋯∘T1 is right Bregman firmly nonexpansive with F^(T)=F(T)=∩i=1NF(Ti) (see [10]) and hence it is right Bregman strongly nonexpansive mapping. Thus, we have the following.Corollary 14.
Letf:E→R be a cofinite 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 right Bregman firmly nonexpansive mappings with F:=∩i=1NF(Ti)≠∅. For u,x1∈C let {xn} be a sequence generated by
(50)xn+1=αnu+(1-αn)Txn,n=1,2,…,
where T=TN∘TN-1∘⋯∘T1, {αn}⊂(0,1) satisfying lim⁡n→∞αn=0 and ∑n=1∞‍αn=∞. Then, {xn} converges strongly to p in F.


## 4. Applications

### 4.1. Zeroes of Maximal Mappings

LetA:E→2E* be a maximal monotone mapping. Recently, many authors studied zero points of monotone mappings using different methods (see e.g., [13, 25, 28, 30, 31, 38]). In this section we use Halpern’s type scheme to find common zeros of a finite family of maximal monotone set-valued mappings.Definition 15 (see [31]).
Letf:E→(-∞,+∞] be an admissible function and let A:E→2E* be a set-valued mapping such that int⁡(dom⁡f)∩dom⁡A≠∅. The conjugate resolvent of A with respect to f, or the conjugate ∇f-resolvent, is the operator CResAf:E*→2E* defined by
(51)CResAf:=(I+A∘∇f*)-1.Remark 16.
If, in addition,A is monotone and fint⁡(dom⁡f) is strictly convex, then it is shown in [31] that CResAf is right Bregman firmly nonexpansive and ∇f*(F(CResAf))=int⁡(dom⁡f)∩A-1(0*). Moreover, we know that if f is Legendre, bounded, and uniformly continuous on bounded subsets of E, then, for every right Bregman firmly nonexpansive operator T, F^(T)=F(T)  (see [10]). Thus, under these assumptions on A and f, the operator CResAf is right Bregman strongly nonexpansive mapping.We shall need the following lemma.Lemma 17 (see [30]).
Letf:E→R be a strictly convex, cofinite, and admissible function, and let A:E→2E* be a set-valued monotone mapping. Then A is maximal monotone if and only if dom⁡(CResAf)=E*.Theorem 18.
Letf:E→R be a cofinite function such that f* is uniformly Fréchet differentiable and totally convex on bounded subsets of E*. Let Ai:E→2E*, i=1,2,…,N, be maximal monotone mappings such that F:=∩i=1NAi-1(0*)≠∅. For each w,z1∈E*, consider the sequence {zn} generated iteratively by
(52)zn+1=αnw+(1-αn)T(zn),n=1,2,…,
where T=CResANf∘CResN-1f∘⋯∘CResA1f, {αn}⊂(0,1) satisfy lim⁡n→∞αn=0, and ∑n=1∞‍αn=∞. Then, {zn} converges strongly to p′ in F(T)=∩i=1NF(CResAif), and ∇f*(p′)=p∈F.Proof.
From Lemma17 we know that each Ti=CResAif, i=1,2,…,N is a mapping from E* into itself, since
(53)⋂i=1NF(CResAif)=⋂i=1N∇f(Ai-1(0*))=∇f(F)≠∅.
Remark 16 guarantees that each Ti,i=1,2,…,N is right Bregman strongly nonexpansive mapping with respect to F(Ti)=F^(Ti). Now the result follows immediately from Theorem 11 applied to E*.


### 4.2. Convex Feasibility Problems

The convex feasibility problem (CFP) is finding an elementx*∈∩i=1NKi, where Ki for i=1,2,…,N, are nonempty, closed, and convex subsets of E. Let K⊂int⁡(dom⁡f). Theright Bregman projection [30] onto K is the operator PKf:int⁡(dom⁡f)→K defined by
(54)PKf(x):=arg limy∈K⁡{Df(x,y)}={z∈K:Df(x,z)≤Df(x,y)∀y∈K}.
If f:E→R is Legendre and uniformly continuous on bounded subsets of E and f is weakly sequentially continuous, then the right Bregman projection PKif is right Bregman strongly nonexpansive mapping with F(PKif)=F^(PKif) (see [30]). Therefore, if we take Ti=PKif for each i∈{1,2,…,N}, then we get an algorithm for solving convex feasibility problems. More precisely, we have the following result.Theorem 19.
Letf:E→R be a cofinite function which is bounded, uniformly continuous, and totally convex on bounded subsets of E. Assume that ∇f is weakly sequentially continuous. Let Ki,i=1,…,N be N nonempty, closed, and convex subsets of E such that F:=∩i=1NKi≠∅. For each x1∈K, let the sequence {xn} be generated iteratively by
(55)xn+1=αnu+(1-αn)T(zn),n=1,2,…,
where T=PKNf∘PKN-1f∘⋯∘PK1f, {αn}⊂(0,1) satisfy lim⁡n→∞αn=0, and ∑n=1∞‍αn=∞. Then, {xn} converges strongly to p in F.Remark 20.
Theorem11 complements the results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on C, where C is nonempty, closed, and convex subset of E.


## 4.1. Zeroes of Maximal Mappings

LetA:E→2E* be a maximal monotone mapping. Recently, many authors studied zero points of monotone mappings using different methods (see e.g., [13, 25, 28, 30, 31, 38]). In this section we use Halpern’s type scheme to find common zeros of a finite family of maximal monotone set-valued mappings.Definition 15 (see [31]).
Letf:E→(-∞,+∞] be an admissible function and let A:E→2E* be a set-valued mapping such that int⁡(dom⁡f)∩dom⁡A≠∅. The conjugate resolvent of A with respect to f, or the conjugate ∇f-resolvent, is the operator CResAf:E*→2E* defined by
(51)CResAf:=(I+A∘∇f*)-1.Remark 16.
If, in addition,A is monotone and fint⁡(dom⁡f) is strictly convex, then it is shown in [31] that CResAf is right Bregman firmly nonexpansive and ∇f*(F(CResAf))=int⁡(dom⁡f)∩A-1(0*). Moreover, we know that if f is Legendre, bounded, and uniformly continuous on bounded subsets of E, then, for every right Bregman firmly nonexpansive operator T, F^(T)=F(T)  (see [10]). Thus, under these assumptions on A and f, the operator CResAf is right Bregman strongly nonexpansive mapping.We shall need the following lemma.Lemma 17 (see [30]).
Letf:E→R be a strictly convex, cofinite, and admissible function, and let A:E→2E* be a set-valued monotone mapping. Then A is maximal monotone if and only if dom⁡(CResAf)=E*.Theorem 18.
Letf:E→R be a cofinite function such that f* is uniformly Fréchet differentiable and totally convex on bounded subsets of E*. Let Ai:E→2E*, i=1,2,…,N, be maximal monotone mappings such that F:=∩i=1NAi-1(0*)≠∅. For each w,z1∈E*, consider the sequence {zn} generated iteratively by
(52)zn+1=αnw+(1-αn)T(zn),n=1,2,…,
where T=CResANf∘CResN-1f∘⋯∘CResA1f, {αn}⊂(0,1) satisfy lim⁡n→∞αn=0, and ∑n=1∞‍αn=∞. Then, {zn} converges strongly to p′ in F(T)=∩i=1NF(CResAif), and ∇f*(p′)=p∈F.Proof.
From Lemma17 we know that each Ti=CResAif, i=1,2,…,N is a mapping from E* into itself, since
(53)⋂i=1NF(CResAif)=⋂i=1N∇f(Ai-1(0*))=∇f(F)≠∅.
Remark 16 guarantees that each Ti,i=1,2,…,N is right Bregman strongly nonexpansive mapping with respect to F(Ti)=F^(Ti). Now the result follows immediately from Theorem 11 applied to E*.


## 4.2. Convex Feasibility Problems

The convex feasibility problem (CFP) is finding an elementx*∈∩i=1NKi, where Ki for i=1,2,…,N, are nonempty, closed, and convex subsets of E. Let K⊂int⁡(dom⁡f). Theright Bregman projection [30] onto K is the operator PKf:int⁡(dom⁡f)→K defined by
(54)PKf(x):=arg limy∈K⁡{Df(x,y)}={z∈K:Df(x,z)≤Df(x,y)∀y∈K}.
If f:E→R is Legendre and uniformly continuous on bounded subsets of E and f is weakly sequentially continuous, then the right Bregman projection PKif is right Bregman strongly nonexpansive mapping with F(PKif)=F^(PKif) (see [30]). Therefore, if we take Ti=PKif for each i∈{1,2,…,N}, then we get an algorithm for solving convex feasibility problems. More precisely, we have the following result.Theorem 19.
Letf:E→R be a cofinite function which is bounded, uniformly continuous, and totally convex on bounded subsets of E. Assume that ∇f is weakly sequentially continuous. Let Ki,i=1,…,N be N nonempty, closed, and convex subsets of E such that F:=∩i=1NKi≠∅. For each x1∈K, let the sequence {xn} be generated iteratively by
(55)xn+1=αnu+(1-αn)T(zn),n=1,2,…,
where T=PKNf∘PKN-1f∘⋯∘PK1f, {αn}⊂(0,1) satisfy lim⁡n→∞αn=0, and ∑n=1∞‍αn=∞. Then, {xn} converges strongly to p in F.Remark 20.
Theorem11 complements the results due to Reich and Sabach [9], Suantai et al. [32], and Zhang and Cheng [33] in the sense that our scheme is applicable for right Bregman strongly nonexpansive self-mappings on C, where C is nonempty, closed, and convex subset of E.



---

*Source: 584395-2014-05-27.xml*