# Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings in Banach Spaces

Authors: Chin-Tzong Pang; Eskandar Naraghirad; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. By exhibiting an example, we first show that the class of Bregman relatively nonexpansive mappings embraces properly the class of Bregman strongly nonexpansive mappings which was investigated by Martín-Márques et al. (2013). We then prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator 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

LetE be a (real) Banach space with norm ∥·∥ and dual space E*. For any x in E, we denote the value of x* in E* at x by 〈x,x*〉. When {xn}n∈ℕ is a sequence in E, we denote the strong convergence of {xn}n∈ℕ to x∈E by xn→x and the weak convergence by xn⇀x. Let C be a nonempty subset of E. Let T:C→E be a map. We denote by F(T)={x∈C:Tx=x} the set offixed points of T. We call the map T(i)
nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y in C,(ii)
quasi-nonexpansive if F(T)≠∅ and ∥Tx-y∥≤∥x-y∥ for all x in C and y in F(T).The nonexpansivity plays an important role in the study of theMann iteration, given by
(1)xn+1=βnxn+(1-βn)Txn,
where the sequence {βn}n∈ℕ satisfies some appropriate conditions. Construction of fixed points of nonexpansive mappings via Mann’s algorithm [1] has been extensively investigated in the literature (see, e.g., [2] and the references therein).Letg:E→ℝ be a strictly convex and Gâteaux differentiable function on a Banach space E. TheBregman distance [3] (see also [4, 5]) corresponding to g is the function Dg:E×E→ℝ defined by
(2)Dg(x,y)=g(x)-g(y)-〈x-y,∇g(y)〉,∀x,y∈E.
It follows from the strict convexity of g that Dg(x,y)≥0 for all x, y in E. However, Dg might not be symmetric and Dg might not satisfy the triangular inequality.WhenE is a smooth Banach space, setting g(x)=∥x∥2 for all x in E, we have that ∇g(x)=2Jx for all x in E. Here J is the normalized duality mapping from E into E*. Hence, Dg(·,·) reduces to the usual map ϕ(·,·) as
(3)Dg(x,y)=ϕ(x,y):=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E.
If E is a Hilbert space, then Dg(x,y)=∥x-y∥2. For more details, we refer the readers to [6].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 is defined and denoted by
(4)ΠC(x)=argmin⁡y∈C⁡ϕ(y,x),
where ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2. Let C be a nonempty, closed, and convex subset of a smooth Banach space E; let T be a mapping from C into itself. A point p∈C is said to be anasymptotic fixed point [7] of T if there exists a sequence {xn}n∈ℕ in C which converges weakly to p and lim⁡n→∞∥xn-Txn∥=0. We denote the set of all asymptotic fixed points of T by F^(T). A point p∈C is called astrong asymptotic fixed point of T if there exists a sequence {xn}n∈ℕ in C which converges strongly to p and lim⁡n→∞∥xn-Txn∥=0. We denote the set of all strong asymptotic fixed points of T by F~(T).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 beBregman quasi-nonexpansive, if F(T)≠Ø and
(5)Dg(p,Tx)≤Dg(p,x),∀x∈C,p∈F(T).
A mapping T:C→C is said to beBregman 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 mappingT:C→C is said to beBregman weak 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 mappingT:C→C is said to beBregman strongly nonexpansive (BSNE) 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);(4)
for any bounded sequence{xn}n∈ℕ⊂C and any p∈F(T) we have
(6)lim⁡n→∞[Dg(p,xn)-Dg(p,Txn)]=0⟹lim⁡n→∞Dg(Txn,xn)=0.It is obvious that any Bregman strongly nonexpansive mapping is a Bregman relatively nonexpansive mapping, but the converse is not true in general. In the following, we show that there exists a Bregman relatively nonexpansive mapping which is not a Bregman strongly nonexpansive mapping.Example 1.
LetE=l2, where
(7)l2={σ=(σ1,σ2,…,σn,…):∑n=1∞‍∥σn∥2<∞},∥σ∥=(∑n=1∞‍∥σn∥2)1/2,∥σ∥=σ=(σ1,…)(∑n=1∞‍∥σn∥2)1/2,∀σ∈l2,〈σ,η〉=∑n=1∞‍σnηn,∀δ=(σ1,σ2,…,σn,…),〈σ,η〉=∑n=1∞‍σnηn,η=(η1,η2,…,ηn,…)∈l2.
Let {xn}n∈ℕ∪{0}⊂E be a sequence defined by
(8)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
(9)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
(10)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
(11)g(x)=1k∥x∥k,x∈E.
It is easy to show that ∇g(x)=Jk(x) for all x∈E, where
(12)Jk(x)={∥x∥k-1x*∈E*:〈x,x*〉=∥x∥∥x*∥,∥x*∥=∥x∥k-1}.
It is also obvious that
(13)Jk(λx)=λk-1Jk(x),∀x∈E,∀λ∈ℝ.
Now, we define a mapping T:E→E by
(14)T(x)={-nn+1x,ifx=xn;-x,ifx≠xn.
It is clear that F(T)={0} and for any n∈ℕ(15)Dg(0,Txn)=g(0)-g(Txn)-〈0-Txn,∇g(Txn)〉=-nk(n+1)kg(xn)-〈nn+1xn,-nk-1(n+1)k-1∇g(xn)〉=-nk(n+1)kg(xn)+nk(n+1)k〈xn,∇g(xn)〉=nk(n+1)k[-g(xn)+〈xn,∇g(xn)〉]=nk(n+1)kDg(0,xn)≤Dg(0,xn).
If x≠xn, then we have
(16)Dg(0,Tx)=g(0)-g(Tx)-〈0-Tx,∇g(Tx)〉=-g(x)-〈x,-∇g(x)〉=-g(x)-〈-x,∇g(x)〉=Dg(0,x).
Therefore, T is a Bregman quasi-nonexpansive mapping. Next, we claim that for any subsequence {xnj}j∈ℕ of {xn}n∈ℕ, lim⁡j→∞∥xnj-Txnj∥≠0. If not, then there exists a subsequence {xnj}j∈ℕ of {xn}n∈ℕ such that lim⁡j→∞∥xnj-Txnj∥=0. This implies that lim⁡j→∞∥xnj-Txnj∥=lim⁡j→∞∥xnj-(-nj/(nj+1))xnj∥=lim⁡j→∞∥xnj+(nj/(nj+1))xnj∥=0, which is impossible. Now, we claim that T is a Bregman relatively nonexpansive mapping. Indeed, for any sequence {zn}n∈ℕ⊂E such that zn⇀z0 and ∥zn-Tzn∥→0 as n→∞, since {xn}n∈ℕ is not a Cauchy sequence, there exists a sufficiently large number N∈ℕ such that zn≠xm, for any n,m>N. 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=lim⁡i→∞zni=xm and z0=lim⁡i→∞Tzni=Txm=(m/(m+1))xm which is impossible due to the fact that ∥xm-Txm∥≠0 for all m∈ℕ. This implies that Tzn=-zn for all n>N. It follows from ∥zn-Tzn∥→0 that 2zn→0 and hence zn→0, which implies that z0=0. Since z0∈F(T), we conclude that T is a Bregman relatively nonexpansive mapping. Finally, we show that T is not a Bregman strongly nonexpansive mapping. To this end, we consider the sequence {xn}n∈ℕ∪{0} defined by (8); then, we have
(17)Dg(0,xn)-Dg(0,Txn)=g(0)-g(xn)-〈0-xn,∇g(xn)〉-[g(0)-g(Txn)-〈0-Txn,∇g(Txn)〉]=-g(xn)+〈xn,∇g(xn)〉+g(Txn)-〈Txn,∇g(Txn)〉=-1k∥xn∥2+〈xn,∇g(xn)〉+nkk(n+1)k∥xn∥2-nk(n+1)k〈xn,∇g(xn)〉=-1k(1-nk(n+1)k)∥xn∥2+(1-nk(n+1)k)〈xn,∇g(xn)〉.
This implies that
(18)lim⁡n→∞Dg(0,xn)-Dg(0,Txn)=0.
On the other hand, we have
(19)∥xn-Txn∥=∥xn--nn+1xn∥=2n+1n+1∥xn∥
which implies that
(20)lim⁡n→∞∥xn-Txn∥≠0.
Therefore, T is not a Bregman strongly nonexpansive mapping.We refer the readers to see some other examples of Bregman relatively nonexpansive mappings in [8].A Banach spaceE is said to satisfy theOpial property [9] if for any weakly convergent sequence {xn}n∈ℕ in E with weak limit x, we have
(21)limsup⁡n→∞∥xn-x∥<limsup⁡n→∞∥xn-y∥
for all y in E with y≠x. It is well known that all Hilbert spaces, all finite dimensional Banach spaces, and the Banach spaces lp(1≤p<∞) satisfy the Opial property. Working with the Bregman distance Dg, the following Bregman Opial-like inequality holds for every Banach space E such that ∇g is weakly sequentially continuous:
(22)limsup⁡n→∞Dg(x,xn)<limsup⁡n→∞Dg(y,xn),
whenever xn⇀x≠y. See Lemma 3 for details. The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, quasinonexpansive mappings. The Bregman-Opial property suggests introducing the notions of Bregman nonexpansive-like mappings and developing fixed point theorems and convergence results for the Mann iterations for these mappings.LetE be a reflexive Banach space with the dual space E* and let A:E→2E* be a set-valued mapping. We define the domain and range of A by dom⁡A={x∈E:Ax≠Ø} 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 bemonotone [10] if 〈x-y,x*-y*〉≥0 whenever (x,x*),(y,y*)∈A. It is also said to bemaximal monotone [11] if its graph is not contained in the graph of any other monotone operators 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 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→dom⁡A be defined by
(23)ResrAg=(∇g+rA)-1∇g.
The mapping ResrAg is called the g-resolvent of A (see [12]). It is well known that A-1(0)=F(ResrAg) for each r>0 (for more details, see, e.g., [13]).Examples and some important properties of such operators are discussed in [14].In this paper, using Bregman functions, we study Mann type iterative algorithms for finding fixed points of Bregman relatively nonexpansive mappings in Banach spaces. We prove weak convergence theorems for the sequences produced by the methods. Some application of our results to the problem of finding a zero of a maximal monotone operator in a Banach space is presented. Our results improve and generalize many known results in the current literature; see, for example, [15].


## 2. Properties of Bregman Functions and Bregman Distances

LetE be a (real) Banach space, and let g:E→ℝ. For any x in E, thegradient∇g(x) is defined to be the linear functional in E* such that
(24)〈y,∇g(x)〉=lim⁡t→0g(x+ty)-g(x)t,∀y∈E.
The function g is said to beGâteaux differentiable at x if ∇g(x) is well defined, and g isGâteaux differentiable if it is Gâteaux differentiable everywhere on E. We call g Fréchet differentiable at x (see, e.g., [16, page 13] or [17, page 508]) if, for all ϵ>0, there exists δ>0 such that
(25)|g(y)-g(x)-〈y-x,∇g(x)〉|≤ϵ∥y-x∥whenever∥y-x∥≤δ.
The function g is said to beFréchet 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., [16, Proposition 1.1.10]). If g is also Fréchet differentiable, then ∇g is norm-to-norm continuous (see, [17, page 508]).LetE be a Banach space, >0, and Br:={z∈E:∥z∥≤r}. A function g:E→ℝ is said to be(i)
strongly coercive if
(26)lim⁡∥xn∥→+∞g(xn)∥xn∥=+∞;(ii)
locally bounded if g(Br) is bounded for all r>0;(iii)
locally uniformly smooth on E ([18, pages 207, 221]) if the function σr:[0,+∞)→[0,+∞], defined by
(27)σr(t)=sup⁡x∈Br,y∈SE,α∈(0,1)(αg(x+(1-α)ty)+(1-α)×g(x-αty)-g(x))(α(1-α))-1,
satisfies
(28)lim⁡t↓0σr(t)t=0,∀r>0;(iv)
locally uniformly convex on E (oruniformly convex on bounded subsets of E ([18, pages 203, 221])) if the gauge ρr:[0,+∞)→[0,+∞] ofuniform convexity of g, defined by
(29)ρr(t)=inf⁡x,y∈Br,∥x-y∥=t,α∈(0,1)(-g(αx+(1-α)y)αg(x)+(1-α)g(y)=inf⁡x,y∈Br,∥x-y∥=t,α∈(0,1)-g(αx+(1-α)y))(α(1-α))-1,
satisfies
(30)ρr(t)>0,∀r,t>0.For a locally uniformly convex mapg:E→ℝ, we have
(31)g(αx+(1-α)y)≤αg(x)+(1-α)g(y)-α(1-α)ρr(∥x-y∥),
for all x,y in Br and for all α in (0,1).LetE be a Banach space and g:E→ℝ a strictly convex and Gâteaux differentiable function. By (2), the Bregman distance satisfies [3]
(32)Dg(x,z)=Dg(x,y)+Dg(y,z)+〈x-y,∇g(y)-∇g(z)〉,∀x,y,z∈E.
In particular,
(33)Dg(x,y)=-Dg(y,x)+〈y-x,∇g(y)-∇g(x)〉,=-Dg(y,x)+∀x,y∈E.Lemma 2 (see [8, 16]).
LetE be a Banach space and g:E→ℝ a Gâteaux differentiable function which is locally uniformly convex on E. Let {xn}n∈ℕ and {yn}n∈ℕ be bounded sequences in E. Then the following assertions are equivalent:(1)
lim⁡n→∞Dg(xn,yn)=0;(2)
lim⁡n→∞∥xn-yn∥=0.In the following, derive an Opial-like inequality for the Bregman distance. For original Opial’s inequality, we refer the readers to Lemma 1 of [9].Lemma 3.
LetE be a Banach space and let g:E→ℝ be a strictly convex and Gâteaux differentiable function such that ∇g is weakly sequentially continuous. Suppose that {xn}n∈ℕ is a sequence in E such that xn⇀x for some x in E. Then
(34)limsup⁡n→∞Dg(x,xn)<limsup⁡n→∞Dg(y,xn),
for all y in the interior of dom⁡g with y≠x.Proof.
In view of the definition of Bregman distance (see (2)), we obtain
(35)Dg(x,xn)-Dg(y,xn)=g(x)-g(xn)-〈x-xn,∇g(xn)〉-[g(y)-g(xn)-〈y-xn,∇g(xn)〉]=g(x)-g(y)+〈x-y,∇g(x)〉-〈x-y,∇g(x)〉+〈y-x,∇g(xn)〉=g(x)-g(y)+〈x-y,∇g(x)〉+〈y-x,∇g(xn)-∇g(x)〉=-Dg(y,x)+〈y-x,∇g(xn)-∇g(x)〉.
Since xn⇀x as n→∞ and ∇g is weakly sequentially continuous, we deduce that
(36)limsup⁡n→∞Dg(x,xn)=limsup⁡n→∞[Dg(x,xn)-Dg(y,xn)+Dg(y,xn)]=limsup⁡n→∞[Dg(x,xn)-Dg(y,xn)]+limsup⁡n→∞Dg(y,xn)=-Dg(y,x)+limsup⁡n→∞Dg(y,xn).
Taking into account that Dg(y,x)>0 for y≠x, we obtain that
(37)limsup⁡n→∞Dg(x,xn)<limsup⁡n→∞Dg(y,xn),
which completes the proof.We call a functiong:E→(-∞,+∞]lower semicontinuous if {x∈E:g(x)≤r} is closed for all r in ℝ. For a lower semicontinuous convex function g:E→ℝ, thesubdifferential∂g of g is defined by
(38)∂g(x)={x*∈E*:g(x)+〈y-x,x*〉≤g(y),∀y∈E}
for all x in E. It is well known that ∂g⊂E×E* is maximal monotone [19, 20]. For any lower semicontinuous convex function g:E→(-∞,+∞], theconjugate functiong* of g is defined by
(39)g*(x*)=sup⁡x∈E{〈x,x*〉-g(x)},∀x*∈E*.
It is well known that
(40)g(x)+g*(x*)≥〈x,x*〉,∀(x,x*)∈E×E*,(x,x*)∈∂gisequivalenttog(x)+g*(x*)=〈x,x*〉.
We also know that if g:E→(-∞,+∞] is a proper lower semicontinuous convex function, then g*:E*→(-∞,+∞] is a proper weak* lower semicontinuous convex function. Here, saying g isproper, we mean that dom⁡g:={x∈E:g(x)<+∞}≠∅.The following definition is slightly different from that in Butnariu and Iusem [16].Definition 4 (see [17]).
LetE be a Banach space. A 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 in E and r>0.The following lemma follows from Butnariu and Iusem [16] and Zălinescu [18].Lemma 5.
LetE be a reflexive Banach space and g:E→ℝ 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 in E and r>0;(4)
domg*=E*,g* is Gâteaux differentiable and ∇g*=(∇g)-1.The following two results follow from [18, Proposition 3.6.4].Proposition 6.
LetE be a reflexive Banach space and let g:E→ℝ be a convex function which is locally bounded. The following assertions are equivalent:(1)
gis strongly coercive and locally uniformly convex onE;(2)
domg*=E*,g*is locally bounded and locally uniformly smooth onE;(3)
domg*=E*,g*is Fréchet differentiable and∇g*is uniformly norm-to-norm continuous on bounded subsets ofE*.Proposition 7.
LetE be a reflexive Banach space and g:E→ℝ a continuous convex function which is strongly coercive. The following assertions are equivalent:(1)
gis locally bounded and locally uniformly smooth onE;(2)
g*is Fréchet differentiable and∇g*is uniformly norm-to-norm continuous on bounded subsets ofE;(3)
domg*=E*,g*is strongly coercive and locally uniformly convex onE.Lemma 8 (see [17, 21]).
LetE be a reflexive Banach space, let g:E→ℝ be a strongly coercive Bregman function, and let V be the function defined by
(41)V(x,x*)=g(x)-〈x,x*〉+g*(x*),∀x∈E,∀x*∈E*.
The following assertions hold:(1)
Dg(x,∇g*(x*))=V(x,x*) for all x in E and x* in E*;(2)
V(x,x*)+〈∇g*(x*)-x,y*〉≤V(x,x*+y*) for all x in E and x*,y* in E*.It also follows from the definition thatV is convex in the second variable x* and
(42)V(x,∇g(y))=Dg(x,y).LetE be a Banach space and let C be a nonempty and convex subset of E. Let g:E→ℝ be a strictly convex and Gâteaux differentiable function. Then, we know from [22] that, for x in E and x0 in C, one has
(43)Dg(x0,x)=min⁡y∈CDg(y,x)iff〈y-x0,∇g(x)-∇g(x0)〉≤0,iff〈y-x0,∇g(x)-∇g(x0)〉∀y∈C.
Further, 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 in E, there exists a unique x0 in C such that
(44)Dg(x0,x)=min⁡y∈CDg(y,x).
TheBregman projectionprojCg from E onto C defined by projCg(x)=x0 has the following property:
(45)Dg(y,projCgx)+Dg(projCgx,x)≤Dg(y,x),∀y∈C,∀x∈E.
See [16] for details.LetE be a reflexive Banach space and let g:E→ℝ be a lower-semicontinuous, strictly convex, and Gâteaux differentiable function. Let C be a nonempty, closed, and convex subset of E and let {xn}n∈ℕ be a bounded sequence in E. For any x in E, we set
(46)Br(x,{xn}n∈ℕ)=limsup⁡n→∞Dg(x,xn).
TheBregman asymptotic radius of {xn}n∈ℕ relative to C is defined by
(47)Br(C,{xn}n∈ℕ)=inf⁡{Br(x,{xn}n∈ℕ):x∈C}.
TheBregman asymptotic center of {xn}n∈ℕ relative to C is the set
(48)BA(C,{xn}n∈ℕ)={x∈C:Br(x,{xn}n∈ℕ)=Br(C,{xn}n∈ℕ)}.Proposition 9.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E, and let g:E→ℝ be strictly convex, Gâteaux differentiable, and locally bounded on E. If {xn}n∈ℕ is a bounded sequence of C, then BA(C,{xn}n∈ℕ) is singleton.Proof.
In view of the definition of Bregman asymptotic radius, we may assume that{xn}n∈ℕ converges weakly to z in C. By Lemma 3, we conclude that BA(C,{xn}n∈ℕ)={z}.Lemma 10 (see [23]).
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g:E→ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, and locally bounded on E. Let T:C→E be a Bregman quasi-nonexpansive mapping. Then F(T) is closed and convex.


## 3. Weak Convergence Theorems for Bregman Relatively Nonexpansive Mappings

In this section, we prove weak convergence theorems concerning Bregman relatively nonexpansive mappings in a reflexive Banach space. We propose the following Bregman Mann’s type iteration.LetE be a reflexive Banach space and let g:E→ℝ be a strictly convex and Gâteaux differentiable function. Let C be a nonempty, closed, and convex subset of E. Let T:C→C be a Bregman relatively nonexpansive mapping. Let {xn}n∈ℕ be a sequence defined by
(49)xn+1=projCg(∇g*[γn∇g(xn)+(1-γn)∇g(Txn)]),
where {γn}n∈ℕ is an arbitrary sequence in [0,1].Lemma 11.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g:E→ℝ be a strictly convex and Gâteaux differentiable function. Let T:C→C be a Bregman quasi-nonexpansive mapping with a nonempty fixed point set F(T). Let {xn}n∈ℕ be a sequence defined by (49) such that {γn}n∈ℕ is an arbitrary sequence in [0,1]. Then the following assertions hold: (1)
Dg(z,xn+1)≤Dg(z,xn) for all z in F(T) and n=1,2,…;(2)
lim⁡n→∞Dg(z,xn) exists for any z in F(T).Proof.
Letz∈F(T). In view of (49), we have
(50)Dg(z,xn+1)=Dg(z,projCg(∇g*[γn∇g(xn)+(1-γn)∇g(Txn)]))=Dg(z,∇g*[γn∇g(xn)+(1-γn)∇g(Txn)])=V(z,∇γn∇g(xn)+(1-γn)∇g(Txn))≤γnV(z,∇g(xn))+(1-γn)V(z,∇g(Txn))=γnDg(z,xn)+(1-γn)Dg(z,Txn)≤γnDg(z,xn)+(1-γn)Dg(z,xn)=Dg(z,xn).
This implies that {Dg(z,xn)}n∈ℕ is a bounded and nonincreasing sequence for all z in F(T). Thus we have lim⁡n→∞Dg(z,xn) that exists for any z in F(T).Theorem 12.
LetC be a nonempty, closed, and convex subset of a reflexive Banach space E. Let g:E→ℝ be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on E. Let T:C→C be a Bregman relatively nonexpansive mapping. Let {γn}n∈ℕ be a sequence in [0,1] satisfying the control condition
(51)∑n=1∞‍γn(1-γn)=+∞.
Let {xn}n∈ℕ be a sequence generated by the algorithm (49). Then {xn}n∈ℕ converges weakly to a fixed point of T.Proof.
The boundedness of the sequence{xn}n∈ℕ follows from Lemma 11 and Definition 4. Since T is a Bregman quasi-nonexpansive mapping, for any q in F(T), we have
(52)Dg(q,Txn)≤Dg(q,xn),∀n∈ℕ.
This, together with Definition 4 and the boundedness of {xn}n∈ℕ, implies that {Txn}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., [16, Proposition 1.1.11] for more details). This implies that the sequences {∇g(xn)}n∈ℕ and {∇g(Txn)}n∈ℕ are bounded in E*.
In view of Proposition7, we have that dom⁡g*=E* and g* is strongly coercive and uniformly convex on bounded subsets of E*. Let s2=sup⁡{∥∇g(xn)∥,∥∇g(Txn)∥:n∈ℕ}<∞ and let ρs2*:E*→ℝ be the gauge of uniform convexity of the conjugate function g*.
Claim. For any p in F(T) and n in ℕ,
(53)Dg(p,xn+1)≤Dg(p,xn)-γn(1-γn)ρs2*×(∥∇g(xn)-∇g(Txn)∥).
Let p∈F(T). For each n in ℕ, it follows from the definition of Bregman distance (2), Lemma 8, (32), and (49) that
(54)Dg(p,xn+1)=g(p)-g(xn+1)-〈p-xn+1,∇g(xn+1)〉=g(p)+g*(∇g(xn+1))-〈xn+1,∇g(xn+1)〉-〈p,∇g(xn+1)〉+〈xn+1,∇g(xn+1)〉=g(p)+g*((1-γn)∇g(xn)+γn∇g(Txn))-〈p,(1-γn)∇g(xn)+γn∇g(Txn)〉≤(1-γn)g(p)+γng(p)+(1-γn)g*(∇g(xn))+γng*(∇g(Txn))-γn(1-γn)ρs2*×(∥∇g(xn)-∇g(Txn)∥)-(1-γn)〈p,∇g(xn)〉-γn〈p,∇g(Txn)〉=(1-γn)[g(p)+g*(∇g(xn))-〈p,∇g(xn)〉]+γn[g(p)+g*(∇g(Txn))-〈p,∇g(Txn)〉]-γn(1-γn)ρs2*(∥∇g(xn)-∇g(Txn)∥)=(1-γn)[g(p)-g(xn)+〈xn,∇g(xn)〉=(1-γn)-〈p,∇g(xn)〉]+γn[g(p)-g(Txn)+〈Txn,∇g(Txn)〉+γn-〈p,∇g(Txn)〉]-γn(1-γn)ρs2*(∥∇g(xn)-∇g(Txn)∥)=(1-γn)D(p,xn)+γnD(p,Txn)-γn(1-γn)ρs2*(∥∇g(xn)-∇g(Txn)∥)≤(1-γn)D(p,xn)+γnD(p,xn)-γn(1-γn)ρs2*(∥∇g(xn)-∇g(Txn)∥)=D(p,xn)-γn(1-γn)ρs2*(∥∇g(xn)-∇g(Txn)∥).
Thus we have
(55)γn(1-γn)ρs2*(∥∇g(xn)-∇g(Tnxn)∥)≤Dg(p,xn)-Dg(p,xn+1).
Since {Dg(p,xn)}n∈ℕ converges, together with the control condition (60), we have
(56)liminf⁡n→∞ρs2*(∥∇g(xn)-∇g(Txn)∥)=0.
Therefore, from the property of ρs2* we deduce that
(57)liminf⁡n→∞∥∇g(xn)-∇g(Txn)∥=0.
Since ∇g* is uniformly norm-to-norm continuous on bounded subsets of E* (see, e.g., [18]), we arrive at
(58)liminf⁡n→∞∥xn-Txn∥=0.
Since E is reflexive, then there exists a subsequence {xni}i∈ℕ of {xn}n∈ℕ such that xni⇀p∈C as i→∞. Since T is a Bregman relatively nonexpansive mapping, we deduce that p∈F(T). We claim that xn⇀p as n→∞. If not, then there exists a subsequence {xnj}j∈ℕ of {xn}n∈ℕ such that {xnj}j∈ℕ converges weakly to some q in C with p≠q. This implies that q∈F(T). By Lemma 11, lim⁡n→∞Dg(z,xn) exists for all z in F(T). By the Bregman Opial-like property of E, we obtain that
(59)lim⁡n→∞Dg(p,xn)=lim⁡i→∞Dg(p,xni)<lim⁡i→∞Dg(q,xni)=lim⁡n→∞Dg(q,xn)=lim⁡j→∞Dg(q,xnj)<lim⁡j→∞Dg(p,xnj)=lim⁡n→∞Dg(p,xn).
This is a contradiction. Thus we have p=q, and the desired assertion follows.Corollary 13.
LetE be a reflexive Banach space and let g:E→ℝ be a strongly coercive Bregman function which is locally bounded, locally uniformly convex, and locally uniformly smooth on E. Let T:E→E be a Bregman relatively nonexpansive mapping. Let {γn}n∈ℕ be a sequence in [0,1] satisfying the control condition
(60)∑n=1∞‍γn(1-γn)=+∞.
Let {xn}n∈ℕ be a sequence generated by
(61)xn+1=∇g*[γn∇g(xn)+(1-γn)∇g(Txn)],
where ∇g is the right-hand derivative of g. Then {xn}n∈ℕ converges weakly to a fixed point of T.


## 4. Applications (Approximating Zeros of Maximal Monotone Operators)

As an application of our main result, we include a concrete example in support of Theorem12. Using Theorem 12, we obtain the following strong convergence theorem for maximal monotone operators.Theorem 14.
LetE be a reflexive Banach space and g:E→ℝ a strongly coercive Bregman function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let A be a maximal monotone operator from E to E* such that Z:=A-1(0)≠Ø. Let r>0 and ResrAg=(∇g+rA)-1∇g be the g-resolvent of A. Let {γn}n∈ℕ be an arbitrary sequence in [0,1] which satisfies the control condition
(62)∑n=1∞‍γn(1-γn)=+∞.
Let {xn}n∈ℕ be a sequence generated by
(63)xn+1=∇g*[γn∇g(xn)+(1-γn)∇g(ResrAgxn)],
where ∇g is the right-hand derivative of g. Then the sequence {xn}n∈ℕ defined in (63) converges weakly to an element in Z as n→∞.Proof.
LettingT=ResrAg, in Theorem 12, from (49), we obtain (63). We need only to show that T satisfies all the conditions in Theorem 12. In view of [8, Lemma  3.2], we conclude that T is a Bregman relatively nonexpansive mapping. Thus, we obtain
(64)Dg(p,ResrAgv)≤Dg(p,v),∀v∈E,p∈F(ResrAg),F~(ResrAg)=F(ResrAg)=A-1(0),
where F~(ResrAg) is the set of all strong asymptotic fixed points of ResrAg. Therefore, in view of Theorem 12, we have the conclusions of Theorem 14. This completes the proof.



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*Source: 573075-2014-05-07.xml*