# Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces

Authors: Jiawei Chen; Zhongping Wan; Liuyang Yuan; Yue Zheng
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007).


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## Body

## 1. Introduction

Throughout this paper, without other specifications, we denote byR the set of real numbers. Let E be 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. Let f:E→R∪{+∞} be proper convex and lower semicontinuous. The Fenchel conjugate of f is the function f*:E*→(-∞,+∞] defined by (1.1)f*(ξ)=sup⁡{〈ξ,x〉-f(x):x∈E}.
We denote by domf the domain of f, that is, domf={x∈E:f(x)<+∞}. Let C be a nonempty closed and convex subset of EandT:C→C a nonlinear mapping. Denote by F(T)={x∈C:Tx=x}, the set of fixed points of T. T is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C.In 1967, Brègman [1] discovered an elegant and effective technique for the using of the so-called Bregman distance function Df (see, Section 2, Definition 2.1) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman's technique is applied in various ways in order to design and analyze iterative algorithms for solving not only feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, for computing fixed points of nonlinear mappings, and so on (see, e.g., [1–25], and the references therein).Nakajo and Takahashi [26] introduced the following modification of the Mann iteration method for a nonexpansive mapping T:C→C in a Hilbert space H as follows:(1.2)x0∈C,yn=αnxn+(1-αn)Txn,C¯n={z∈C:‖z-yn‖≤‖z-xn‖},Q¯n={z∈C:〈xn-z,xn-x0〉≤0},xn+1=PC¯n∩Q¯nx0,∀n≥0,
where {αn}⊂[0,1] and PC¯ is the metric projection from H onto a closed and convex subset C of H. They proved that {xn} generated by (1.2) converges strongly to a fixed point of T under some suitable assumptions. Motivated by Nakajo and Takahashi [26], Matsushita and Takahashi [27] introduced the following modification of the Mann iteration method for a relatively nonexpansive mapping T:C→C in a Banach space E as follows:(1.3)x0∈C,yn=J-1(αnJ(xn)+(1-αn)J(Txn)),C¯n={z∈C:ϕ(z,yn)≤ϕ(z,xn)},Q¯n={z∈C:〈J(xn)-J(x0),xn-z〉≤0},xn+1=ΠC¯n∩Q¯nx0,∀n≥0,
where {αn}⊂[0,1], ϕ(y,x)=∥y∥2-2〈y,J(x)〉+∥x∥2 for all x,y∈E, J is the duality mapping of E and ΠC is the generalized projection (see, e.g., [2, 3, 28]) from E onto a closed and convex subset C of E. They also proved that {xn} generated by (1.3) converges strongly to a fixed point of T under some suitable assumptions. Martinez-Yanes and Xu [29] gave a Halpern-type iterative algorithm for a nonexpansive mapping T:C→C as follows: (1.4)x0∈C,yn=βnx0+(1-βn)Txn,C¯n={z∈C:‖z-yn‖2≤‖z-xn‖2+βn(‖x0‖2+2〈xn-x0,z〉)},Q¯n={z∈C:〈xn-z,xn-x0〉≤0},xn+1=PC¯n∩Q¯nx0,∀n≥0,
where {βn}⊂[0,1]. They derived that {xn} generated by (1.3) converges strongly to a fixed point of T under some suitable assumptions. Qin and Su [30] generalized the results of Martinez-Yanes and Xu [29] to a uniformly convex and uniformly smooth Banach space for a relatively nonexpansive mapping and proposed the following iterative algorithm:(1.5)x0∈C,yn=J-1(βnJ(x0)+(1-βn)J(Txn)),C¯n={z∈C:ϕ(z,yn)≤βnϕ(z,x0)+(1-βn)ϕ(z,xn)},Q¯n={z∈C:〈J(xn)-J(x0),xn-z〉≤0},xn+1=ΠC¯n∩Q¯nx0,∀n≥0,
where {βn}⊂[0,1], ΠC is the generalized projection (see, e.g., [2, 3, 28]) from E onto a closed and convex subset C of E. They also obtained that {xn} generated by (1.5) converges strongly to a fixed point of T under some suitable assumptions. In 2003, Butnariu et al. [13] studied several notions of convex analysis: uniformly convexity at a point, total convexity at a point, uniformly convexity on bounded sets, and sequential consistency, which are useful in establishing convergence properties for fixed point and optimization algorithms in infinite dimensional Banach spaces. They established connections between these concepts and used these relations in order to obtain improved convergence results concerning the outer Bregman projection algorithm for solving convex feasibility problems and the generalized proximal point algorithm for optimization in Banach spaces. In 2006, Butnariu and Resmerita [14] presented a Bregman-type iterative algorithms and studied the convergence of the Bregman-type iterative method of solving operator equations. Resmerita [19] investigated the existence of totally convex functions in Banach spaces and, further, established continuity and stability properties of Bregman projections. Very recently, by using Bregman projection, Reich and Sabach [21] presented the following algorithms for finding common zeroes of maximal monotone operators Ai:E→2E*(i=1,2,…,N) in reflexive Banach space E as follows: (1.6)x0∈E,yni=Resλnif(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=projCn∩Qnfx0,∀n≥0,(1.7)x0∈E,ηni=ξni+1λni(∇f(yni)-∇f(xn)),ξni∈Aiyni,ωni=∇f*(λniηni+∇f(xn)),Cni={z∈E:Df(z,yni)≤Df(z,ωni)},Cn=⋂i=1NCni,Qn={z∈E:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=projCn∩Qnfx0,∀n≥0.
Further, under some suitable conditions, they obtained two strong convergence theorems of maximal monotone operators in reflexive Banach spaces. Reich and Sabach [22] studied the convergence of two iterative algorithms for finitely many Bregman strongly nonexpansive operators in Banach spaces and obtained two strong convergence theorems for finitely many Bregman strongly nonexpansive operators under some assumptions. In [24], Reich and Sabach proposed the following algorithms for finding common fixed points of finitely many Bregman firmly nonexpansive operators Ti:C→C(i=1,2,…,N) in reflexive Banach space E as follows: if ⋂i=1NF(Ti)≠∅(1.8)x0∈E,Q0i=E,i=1,2,…,N,yni=Ti(xn+eni),Qn+1i={z∈Qni:〈∇f(xn+eni)-∇f(yni),z-yni〉≤0},Qn=⋂i=1NQni,xn+1=projQn+1fx0,∀n≥0.
Under some suitable conditions, they proved that the sequence {xn} generated by (1.8) converges strongly to ⋂i=1NF(Ti) and applied it to the solution of convex feasibility and equilibrium problems.Inspired and motivated by the works, we introduce the concept of weak Bregman relatively nonexpansive mappings in reflexive Banach space and give an example to illustrate the existence of weak Bregman relatively nonexpansive mapping and the difference between weak Bregman relatively nonexpansive mapping and Bregman relatively nonexpansive mapping. Secondly, by using the conception of the Bregman projection (see, e.g., [1, 13, 14]), we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. Moreover, the convergence rate of our algorithms is faster than that of Matsushita and Takahashi [27] and Qin and Su [30]. The main results in this paper develop, extend, and improve the corresponding results in the literature.


## 2. Preliminaries

LetC be a nonempty closed convex subset of a real reflexive Banach space E, and let T:C→C be a nonlinear mapping. A point ω∈C is called an asymptotic fixed point of T (see, e.g., [2, 3]) if C contains a sequence {xn} which converges weakly to ω such that limn→∞∥Txn-xn∥=0. A point ω∈C is called an strong asymptotic fixed point of T (see, e.g., [2, 3]) if C contains a sequence {xn} which converges strongly to ω such that limn→∞∥Txn-xn∥=0. We denote the sets of asymptotic fixed points and strong asymptotic fixed points of T by F̂(T) and F̃(T), respectively. When {xn} is a sequence in E, we denote strong convergence of {xn} to x∈E by xn→x. For any x∈int(domf) and y∈E, the right-hand derivative of f at x in the direction y defined by (2.1)f0(x,y):=lim⁡t↘0f(x+ty)-f(x)t.f is called Gâteaux differentiable at x if, for all y∈E,limt↘0(f(x+ty)-f(x))/t exists. In this case, f0(x,y) coincides with ∇f(x), the value of the gradient of f at x. f is called Gâteaux differentiable if it is Gâteaux differentiable for any x∈int(domf). f is called Fréchet differentiable at x if this limit is attained uniformly for ∥y∥=1. We say f is uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x∈C and ∥y∥=1.Legendre functionf:E→(-∞,+∞] is defined in [7]. From [7], if E is a reflexive Banach space, then f is Legendre if and only if it satisfies the following conditions (L1) and (L2):(L1)
the interior of the domain off, int(domf), is nonempty, f is Gâteaux differentiable on int(domf), and domf=int(domf),(L2)
the interior of the domain off*, int(domf*), is nonempty, f* is Gâteaux differentiable on int(domf*), and domf*=int(domf*).SinceE is reflexive, we know that (∂f)-1=∂f* (see, e.g., [31]). This, by (L1) and (L2), implies (2.2)∇f=(∇f*)-1,ran∇f=dom⁡∇f*=int⁡(dom⁡f*),ran∇f*=dom⁡∇f=int⁡(dom⁡f).
By Theorem 5.4 [7], conditions (L1) and (L2) also yield that the functions f and f* are strictly convex on the interior of their respective domains. From now on, we assume that the convex function f:E→(-∞,+∞] is Legendre.We first recall some definitions and lemmas which are needed in our main results.Definition 2.1 (see [1, 13]).
Letf:E→(-∞,+∞] be a Gâteaux differentiable and convex function. The function Df:domf×int(domf)→[0,+∞), defined by
(2.3)Df(y,x):=f(y)-f(x)-〈∇f(x),y-x〉
is called the Bregman distance with respect to f.Remark 2.2 (see [24]).
The Bregman distance has the following properties:(i)
the three point identity, for anyx∈domf and y,z∈int(domf),(2.4)Df(x,y)+Df(y,z)-Df(x,z)=〈∇f(z)-∇f(y),x-y〉,(ii)
the four point identity, for anyy,ω∈domf and x,z∈int(domf),(2.5)Df(y,x)-Df(y,z)-Df(ω,x)+Df(ω,z)=〈∇f(z)-∇f(x),y-ω〉.Definition 2.3 (see [1]).
Letf:E→(-∞,+∞] be a Gâteaux differentiable and convex function. The Bregman projection of x∈int(domf) onto the nonempty closed and convex set C⊂domf is the necessarily unique vector projCf(x)∈C satisfying
(2.6)Df(projCf(x),x)=inf⁡{Df(y,x):y∈C}.Remark 2.4.
(i) IfE is a Hilbert space and f(y)=(1/2)∥x∥2 for all x∈E, then the Bregman projection projCf(x) is reduced to the metric projection of x onto C.
(ii) IfE is a smooth Banach space and f(y)=(1/2)∥x∥2 for all x∈E, then the Bregman projection projCf(x) is reduced to the generalized projection ΠC(x) (see, e.g. [3]) which defined by
(2.7)ϕ(ΠC(x),x)=min⁡y∈Cϕ(y,x),
where ϕ(y,x)=∥y∥2-2〈y,J(x)〉+∥x∥2,J is the normalized duality mapping from E to 2E*.Definition 2.5 (see[12, 21]).
LetC be a nonempty closed and convex set of domf. The operator T:C→int(domf) with F(T)≠∅ is called: (i)
quasi-Bregman nonexpansive if(2.8)Df(u,Tx)≤Df(u,x),∀x∈C,u∈F(T);(ii)
Bregman relatively nonexpansive if
(2.9)Df(u,Tx)≤Df(u,x),∀x∈C,u∈F(T),
and F̂(T)=F(T),(iii)
Bregman firmly nonexpansive if
(2.10)〈∇f(Tx)-∇f(Ty),Tx-Ty〉≤〈∇f(x)-∇f(y),Tx-Ty〉,∀x,y∈C,
or equivalently
(2.11)Df(Tx,Ty)+Df(Ty,Tx)+Df(Tx,x)+Df(Ty,y)≤Df(Tx,y)+Df(Ty,x),∀x,y∈C.Definition 2.6.
LetC be a nonempty closed and convex set of domf. The operator T:C→int(domf) with F(T)≠∅ is called weak Bregman relatively nonexpansive if F̃(T)=F(T) and
(2.12)Df(u,Tx)≤Df(u,x),∀x∈C,u∈F(T).Remark 2.7.
It is easy to see that each nonexpansive mappingT is quasi-Bregman nonexpansive mapping with respect to f(x)=(1/2)∥x∥2 for all x∈E. Moreover, every relatively nonexpansive mapping T also is Bregman relatively nonexpansive mapping, where T is called relatively nonexpansive mapping (see, e.g., [32]) if the following conditions are satisfied:
(2.13)F̂(T)=F(T)≠∅,ϕ(u,Tx)≤ϕ(u,x),∀x∈C,u∈F(T).
Now, we give an example which is weak Bregman relatively nonexpansive mapping but not Bregman relatively nonexpansive mapping.Example 2.8.
LetE=l2,f(x)=(1/2)∥x∥2 for all x∈E, where
(2.14)l2={ξ=(ξ1,ξ2,…,ξn,…):∑n=1∞|ξn|2<∞},‖ξ‖=(∑n=1∞|ξn|2)1/2,∀ξ∈l2,
and for any ξ=(ξ1,ξ2,…,ξn,…),μ=(μ1,μ2,…,μn,…)∈E,〈ξ,μ〉=∑n=1∞ξnμn. It is well known that l2 is a Hilbert space. Let {xn}⊂E be a sequence defined by x0=(1,0,0,0,…), x1=(1,1,0,0,…),x2=(1,0,1,0,…),…,xn=(ξn,1,…,ξn,k,…),…, where
(2.15)ξn,k={1,ifk=1,n+1,0,otherwise,
for all n≥0.
Define a mappingT:E→E by
(2.16)T(x)={nxnn+1,ifx=xn(∃n≥1),-x,ifx≠xn(∀n≥1),
for all n≥0. It is easy to see that F(T)={0}, and so, {xn} converges weakly to x0. Indeed, for any g=(ζ1,ζ2…,ζk,…)∈E, we have
(2.17)g(xn-x0)=〈g,xn-x0〉=∑k=2∞ζkξn,k=ζn+1.
From ∑n=1∞|ζn|2<∞, it shows that limn→∞ζn+1=0. Moreover,
(2.18)lim⁡n→∞g(xn-x0)=lim⁡n→∞ζn+1=0.
Next, for any m≠n, one has ∥xn-xm∥=2≠0; that is, {xn} is not a Cauchy sequence. Owing to ∥Txn-xn∥=∥xn∥/(n+1), we obtain
(2.19)lim⁡n→∞‖Txn-xn‖=0.
Then, x0 is an asymptotic fixed point of T, but x0∉F(T)={0}. So, T is not Bregman relatively nonexpansive mapping.
For any strong convergent sequence{yn}⊂l2 such that yn→y0 and ∥Tyn-yn∥→0 as n→∞. Then, there exists a sufficiently large nature number M such that yn≠xm for any n,m>M. Thus, Tyn=-yn for n>M, which implies that 2yn→0 and yn→y0=0 as n→∞. That is, y0=0 is a strong asymptotic fixed point of T, and so, F̃(T)=F(T)={0}. Since
(2.20)Df(0,Tx)=f(0)-f(Tx)-〈∇f(Tx),0-Tx〉=-12‖Tx‖2+〈Tx,Tx〉=12‖Tx‖2≤12‖x‖2=f(0)-f(x)-〈∇f(x),0-x〉=Df(0,x),x∈E.
Therefore, T is a weak Bregman relatively nonexpansive mapping.Definition 2.9 (see [12]).
Letf:E→(-∞,+∞] be a convex and Gâteaux differentiable function. f is called: (i)
totally convex atx∈int(domf) if its modulus of total convexity at x; that is, the function νf:int(domf)×[0,+∞)→[0,+∞) defined by(2.21)νf(x,t):=inf⁡{Df(y,x):y∈dom⁡f,‖y-x‖=t}
is positive whenever t>0,(ii)
totally convex if, it is totally convex at every point x∈int(domf),(iii)
totally convex on bounded sets if νf(B,t) is positive 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(2.22)νf(B,t):=inf⁡{νf(x,t):x∈B∩dom⁡f}.Definition 2.10 (see [12, 21]).
The functionf:E→(-∞,+∞] is called:(i)
cofiniteif domf*=E*,(ii)
sequentially consistent if, for any two sequences {xn} and {yn} in E such that the first is bounded, and(2.23)lim⁡n→∞Df(yn,xn)=0⟹lim⁡n→∞‖yn-xn‖=0.Lemma 2.11 (see [21, Proposition 2.3]).
Iff:E→(-∞,+∞] is Fréchet differentiable and totally convex, then f is cofinite.Lemma 2.12 (see [14, Theorem 2.10]).
Letf:E→(-∞,+∞] be a convex function whose domain contains at least two points. Then, the following statements hold: (i)
f is sequentially consistent if and only if it is totally convex on bounded sets,(ii)
iff is lower semicontinuous, then f is sequentially consistent if and only if it is uniformly convex on bounded sets,(iii)
iff is uniformly strictly convex on bounded sets, then it is sequentially consistent and the converse implication holds when f is lower semicontinuous, Fréchet differentiable on its domain, and the Fréchet derivative f′ is uniformly continuous on bounded sets.Lemma 2.13 (see [20, Proposition 2.1]).
Letf:E→R be a uniformly Fréchet differentiable and bounded on bounded subsets of E. Then, ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E*.Lemma 2.14 (see [21, Lemma 3.1]).
Letf:E→R be a Gâteaux differentiable and totally convex function. If x0∈E and the sequence {Df(xn,x0)}n=1∞ is bounded, then the sequence {xn}n=1∞ is also bounded.Lemma 2.15 (see [21, Proposition 2.2]).
Letf:E→R be a Gâteaux differentiable and totally convex function, x0∈E, and let C be a nonempty closed convex subset of E. Suppose that the sequence {xn}n=1∞ is bounded and any weak subsequential limit of {xn}n=1∞ belongs to C. If Df(xn,x0)≤Df(projCf(x0),x0) for any n∈N, then {xn}n=1∞ converges strongly to projCf(x0).In [23], Reich and Sabach proved the following result.Lemma 2.16 (see[23, Lemma 15.5]).
Letf:E→(-∞,+∞] be a Legendre function. Let C be a nonempty closed convex subset of int(domf) and T:C→C a Bregman firmly nonexpansive mapping with respect to f. Then, F(T) is closed and convex.Motivated by Lemma2.16, we get the similar result for quasi-Bregman nonexpansive mapping.Proposition 2.17.
Letf:E→(-∞,+∞] be a Legendre function. Let C be a nonempty closed convex subset of int(domf) and T:C→C a quasi-Bregman nonexpansive mapping with respect to f. Then, F(T) is closed and convex.Proof.
Without loss of generality, setF(T) is nonempty. Firstly, we show that F(T) is closed. Let {xn}n=0∞ be a sequence in F(T) such that xn→x¯. By the definition of quasi-Bregman nonexpansive mapping, we have
(2.24)Df(xn,Tx¯)≤Df(xn,x¯),n≥0.
Since f:E→(-∞,+∞] is a Legendre function, f is continuous at x¯∈C⊂int(domf). Then, from the definition of Bregman distance,
(2.25)lim⁡n→∞Df(xn,Tx¯)=Df(x¯,Tx¯),lim⁡n→∞Df(xn,x¯)=Df(x¯,x¯)=0.
From (2.24) and (2.25), it follows that Df(x¯,Tx¯)=0, and so, from [7, Lemma 7.3(vi), page 642], Tx¯=x¯. Therefore, x¯∈F(T), and so, F(T) is closed.
We now show thatF(T) is convex. For any x,y∈F(T) and t∈(0,1), it yields that z=tx+(1-t)y∈C. From the definition of quasi-Bregman nonexpansive mapping, it follows that
(2.26)Df(z,Tz)=f(z)-f(Tz)-〈∇f(Tz),tx+(1-t)y-T(tx+(1-t)y)〉=f(z)+tDf(x,Tz)+(1-t)Df(y,Tz)-tf(x)-(1-t)f(y)≤f(z)+tDf(x,z)+(1-t)Df(y,z)-tf(x)-(1-t)f(y)=〈∇f(z),z-tx-(1-t)y〉=0.
Again, from [7, Lemma 7.3(vi), page 642], we get Tz=z. Therefore, F(T) is convex. This completes the proof.From the definitions of Bregman distance and the Fenchel conjugate off, we have the following result.Lemma 2.18.
Letf:E→(-∞,+∞] be a Gâteaux differentiable and proper convex lower semicontinuous. Then, for all z∈E,
(2.27)Df(z,∇f*(∑i=1Nti∇f(xi)))≤∑i=1NtiDf(z,xi),
where {xi}i=1N⊂E and {ti}i=1N⊂(0,1) with ∑i=1Nti=1.Lemma 2.19 (see [14, Corollary 4.4]).
Letf:E→(-∞,+∞] be a Gâteaux differentiable and totally convex on int(domf). Let x∈int(domf) and C⊂int(domf) a nonempty closed convex set. If x̂∈C, then the following statements are equivalent:(i)
the vectorx̂ is the Bregman projection of x onto C with respect to f,(ii)
the vectorx̂ is the unique solution of the variational inequality(2.28)〈∇f(x)-∇f(z),z-y〉≥0,∀y∈C,(iii)
the vectorx̂ is the unique solution of the inequality(2.29)Df(y,z)+Df(z,x)≤Df(y,x),∀y∈C.


## 3. Main Results

In this section, we introduce several modification of Mann-type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are proved under some suitable conditions.Theorem 3.1.
LetC be a nonempty closed convex subset of a real reflexive Banach space E and f:E→R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T:C→C be a weak Bregman relatively nonexpansive mapping such that F(T)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.1)x0∈C,Q0=C,zn=∇f*(βn∇f(T(xn+en))+(1-βn)∇f(xn+en)),yn=∇f*(αn∇f(xn+en)+(1-αn)∇f(zn)),Cn={z∈Cn-1∩Qn-1:Df(z,yn)≤Df(z,xn+en)},C0={z∈C:Df(z,y0)≤Df(z,x0)},Qn={z∈Cn-1∩Qn-1:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=projCn∩Qnfx0,∀n≥0,
where {αn},{βn}⊂[0,1] such that liminfn→∞(1-αn)βn>0, and {en} is an error sequence in E with en→0 as n→∞. Then, the sequences {xn} and {yn} converge strongly to the point projF(T)f(x0), where projF(T)f(x0) is the Bregman projection of C onto F(T).Proof.
By Proposition2.17, it follows that F(T) is a nonempty closed and convex subset of E. It is easy to verify that C0,C1,Q0, and Q1 are closed and convex. Suppose that Ck and Qk(k≥1) are closed and convex. Then, Ck∩Qk is closed and convex. For any z∈Ck+1,y∈Qk+1,
(3.2)Df(z,yk+1)≤Df(z,xk+1+ek+1)⟺f(z)-f(yk+1)-〈∇f(yk+1),z-yk+1〉≤f(z)-f(xk+1+ek+1)-〈∇f(xk+1+ek+1),z-(xk+1+ek+1)〉⟺〈∇f(xk+1+ek+1),z-(xk+1+ek+1)〉-〈∇f(yk+1),z-yk+1〉≤f(yk+1)-f(xk+1+ek+1)⟺〈∇f(xk+1+ek+1)-∇f(yk+1),z-yk+1〉≤f(yk+1)-f(xk+1+ek+1)-〈∇f(xk+1+ek+1),yk+1-(xk+1+ek+1)〉⟺〈∇f(xk+1+ek+1)-∇f(yk+1),z-yk+1〉≤Df(yk+1,xk+1+ek+1),〈∇f(x0)-∇f(xk),y-xk〉≤0,
which implies that Ck+1 and Qk+1 are closed and convex. As a consequence, Cn and Qn are closed and convex for all n≥0. Taking p∈F(T) arbitrarily,
(3.3)Df(p,yn)=Df(p,∇f*(αn∇f(xn+en)+(1-αn)∇f(zn)))≤αnDf(p,xn+en)+(1-αn)Df(p,zn)=αnDf(p,xn+en)+(1-αn)Df(p,∇f*(βn∇f(T(xn+en))+(1-βn)∇f(xn+en)))≤αnDf(p,xn+en)+(1-αn)[(1-βn)Df(p,xn+en)+βnDf(p,T(xn+en))]≤αnDf(p,xn+en)+(1-αn)[(1-βn)Df(p,xn+en)+βnDf(p,xn+en)]=Df(p,xn+en),
that is, p∈Cn, and so, F(T)⊂Cn for all n≥0. We now show that F(T)⊂Qn for all n≥0. Clearly, F(T)⊂Q0=C. Assume that F(T)⊂Qk for all k≥0. Note that xk+1=projCk∩Qkf(x0), and we have
(3.4)〈∇f(x0)-∇f(xk+1),xk+1-z〉≥0,z∈Ck∩Qk.
Therefore,
(3.5)〈∇f(x0)-∇f(xk+1),xk+1-p〉≥0,p∈F⊂Ck∩Qk,
which yields that p∈Qk+1. Then, F(T)⊂Qn for all n≥0. Consequently, F(T)⊂Cn∩Qn and Cn∩Qn is nonempty closed and convex for all n≥0. Moreover, {xn} is well defined.
Secondly, we show that{xn} is a Cauchy sequence and bounded. Since
(3.6)〈∇f(x0)-∇f(xn),z-xn〉≤0,∀z∈Qn,
it follows that xn=projQnf(x0). Therefore, by xn+1=projCn∩Qnf(x0)∈Qn,
(3.7)Df(xn,x0)≤Df(xn+1,x0).
Taking p∈F(T) arbitrarily. From Lemma 2.19, it yields that
(3.8)Df(p,projQnf(x0))+Df(projQnf(x0),x0)≤Df(p,x0).
Moreover, one has
(3.9)Df(xn,x0)≤Df(p,x0)-Df(p,xn)≤Df(p,x0).
Hence, {Df(xn,x0)} is bounded and so {xn}, {yn}, and {zn} are also bounded. From (3.7), it shows that limn→∞Df(xn,x0) exists. In the light of xm∈Qm-1⊂Qn for any m>n, by Lemma 2.19,
(3.10)Df(xm,projQnf(x0))+Df(projQnf(x0),x0)≤Df(xm,x0),
that is,
(3.11)Df(xm,xn)≤Df(xm,x0)-Df(xn,x0).
Consequently, one has
(3.12)lim⁡n→∞Df(xm,xn)=0.
Since f is totally convex on bounded subsets of E, by Lemma 2.12 and (3.12), we have
(3.13)lim⁡n→∞‖xm-xn‖=0.
Thus, {xn} is a Cauchy sequence, and so,
(3.14)lim⁡n→∞‖xn+1-xn‖=0.
Since en→0 as n→∞, one has
(3.15)lim⁡n→∞‖(xn+1+en+1)-(xn+en)‖=0,lim⁡n→∞‖xn+1-(xn+en)‖=0.
Let xn→ω¯∈C. Then, xn+en→ω¯.
Thirdly, we show that{xn} converges strongly to a point of F(T). Since f is uniformly Fréchet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-norm uniformly continuous on bounded subsets of E. So, by (3.15),
(3.16)lim⁡n→∞‖∇f(xn+1)-∇f(xn+en)‖=0.
It follows from xn+1∈Cn that
(3.17)Df(xn+1,yn)≤Df(xn+1,xn+en).
By the uniformly Fréchet differentiable of f on bounded subsets of E, f is also uniformly continuous on bounded subsets of E. Hence, from (3.12) and limn→∞en=0,
(3.18)lim⁡n→∞Df(xn+1,xn+en)=lim⁡n→∞(f(xn+1)-f(xn+en)-〈∇f(xn+en),xn+1-(xn+en)〉)=0.
As a consequence, limn→∞Df(xn+1,yn)=0 and so, limn→∞∥xn+1-yn∥=0. Moreover, one has
(3.19)lim⁡n→∞‖∇f(xn+1)-∇f(yn)‖=0.
Since ∥xn-yn∥≤∥xn-xn+1∥+∥xn+1-yn∥,∥xn-yn∥→0 and yn→ω¯ as n→∞. Noticing that
(3.20)‖∇f(xn+1)-∇f(yn)‖=‖∇f(xn+1)-(αn∇f(xn+en)+(1-αn)∇f(zn))‖≥(1-αn)‖∇f(xn+1)-∇f(zn)‖-αn‖∇f(xn+1)-∇f(xn+en)‖=(1-αn)‖∇f(xn+1)-(βn∇f(T(xn+en))+(1-βn)∇f(xn+en))‖-αn‖∇f(xn+1)-∇f(xn+en)‖≥-αn‖∇f(xn+1)-∇f(xn+en)‖+(1-αn)βn‖∇f(xn+1)-∇f(T(xn+en))‖-(1-αn)(1-βn)‖∇f(xn+1)-∇f(xn+en)‖.
Therefore,
(3.21)(1-αn)βn∥∇f(xn+1)-∇f(T(xn+en))∥≤∥∇f(xn+1)-∇f(yn)∥+αn∥∇f(xn+1)-∇f(xn+en)∥+(1-αn)(1-βn)∥∇f(xn+1)-∇f(xn+en)∥.
In view of liminfn→∞(1-αn)βn>0 and from both (3.16) and (3.19), one has
(3.22)lim⁡n→∞‖∇f(xn+1)-∇f(T(xn+en))‖=0.
Furthermore, we have
(3.23)lim⁡n→∞‖xn+1-T(xn+en)‖=0,
and so, by (3.14),
(3.24)lim⁡n→∞‖(xn+en)-T(xn+en)‖=0.
Since xn→ω¯ and en→0, we get ω¯∈F̃(T)=F(T).
Finally, we showω¯=projF(T)f(x0). Since projF(T)f(x0)∈F(T)⊂Cn∩Qn, it follows from xn+1=proj(Cn∩Qn)f(x0) that Df(xn+1,x0)≤Df(projF(T)f(x0),x0). By Lemma 2.15, xn→projF(T)f(x0) as n→∞. Therefore, {xn} and {yn} converge strongly to projF(T)f(x0). This completes the proof.Theorem 3.2.
LetC be a nonempty closed convex subset of a real reflexive Banach space E and f:E→R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T:C→C be a Bregman relatively nonexpansive mapping such that F(T)≠∅. Assume that {αn},{βn}⊂[0,1] such that liminfn→∞(1-αn)βn>0, and {en} is an error sequence in E with en→0 as n→∞. Then, the sequences {xn} and {yn} generated by (3.1) converge strongly to the point projF(T)f(x0), where projF(T)f(x0) is the Bregman projection of C onto F(T).Proof.
As in the proof of Theorem3.1, we know that the sequences {xn} and {yn} converge strongly to ω¯∈C, and so,
(3.25)lim⁡n→∞‖(xn+en)-T(xn+en)‖=0.
Then, for any subsequence {xnk} of {xn} converges weakly to ω¯,
(3.26)lim⁡k→∞‖(xnk+enk)-T(xnk+enk)‖=0.
Therefore, ω¯∈F̂(T)=F(T). By the similar proof of Theorem 3.2, the sequences {xn} and {yn} converge strongly to projF(T)f(x0). This completes the proof.Ifαn≡0,  en≡0, and f(x)=(1/2)∥x∥2 for all x∈E, n≥0, then from Remark 2.4 and Theorem 3.1, we have the following result.Corollary 3.3.
LetC be a nonempty closed convex subset of a real reflexive, smooth, and strictly convex Banach space E, and let T:C→C be a relatively nonexpansive mapping such that F(T)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.27)x0∈C,Q0=C,yn=J-1(βnJ(T(xn))+(1-βn)J(xn)),Cn={z∈Cn-1∩Qn-1:ϕ(z,yn)≤ϕ(z,xn)},C0={z∈C:ϕ(z,y0)≤ϕ(z,x0)},Qn={z∈Cn-1∩Qn-1:〈J(x0)-J(xn),z-xn〉≤0},xn+1=ΠCn∩Qnx0,∀n≥0,
where J is the duality mapping on E, {βn}⊂[0,1] such that liminfn→∞βn>0. Then, the sequences {xn} and {yn} converge strongly to the point ΠF(T)(x0), where ΠF(T)(x0) is the generalized projection (see, e.g., [2, 3, 28]) of C onto F(T).
In [27], Matsushita and Takahashi proved the following result.Theorem MT (see [27, Theorem 3.1]).
LetC be a nonempty closed convex subset of a real uniformly convex and uniformly smooth Banach space E, and let T:C→C be a relatively nonexpansive mapping such that F(T)≠∅. Assume that {αn} is a sequence of real numbers such that 0≤αn<1 and limsupn→∞αn<1. Then, the sequence {xn} generated by (1.3) converges strongly to the point ΠF(T)(x0), where ΠF(T)(x0) is the generalized projection (see, e.g., [2, 3, 28]) of C onto F(T).Remark 3.4.
Corollary3.3 extends Theorem MT [27] from uniformly convex and uniformly smooth Banach spaces to reflexive, smooth, and strictly convex Banach space.
Now, we investigate convergence theorems for Halpern-type iterative algorithms with errors.Theorem 3.5.
LetC be a nonempty closed convex subset of a real reflexive Banach space E and f:E→R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T:C→C be a weak Bregman relatively nonexpansive mapping such that F(T)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.28)x0∈C,Q0=C,zn=∇f*(βn∇f(x0)+(1-βn)∇f(T(xn+en))),yn=∇f*(αn∇f(zn)+(1-αn)∇f(xn+en)),Cn={z∈Cn-1∩Qn-1:Df(z,yn)≤(1-αnβn)Df(z,xn+en)+αnβnDf(z,x0)},C0={z∈C:Df(z,y0)≤Df(z,x0)},Qn={z∈Cn-1∩Qn-1:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=proj(Cn∩Qn)fx0,∀n≥0,
where {αn},{βn}⊂[0,1] such that liminfn→∞αn>0 and limn→∞βn=0, and {en} is an error sequence in E with en→0 as n→∞. Then, the sequences {xn} and {yn} converge strongly to the point projF(T)f(x0), where projF(T)f(x0) is the Bregman projection of C onto F(T).Proof.
By Proposition2.17, it follows that F(T) is a nonempty closed and convex subset of E. It is easy to see that Cn is closed and Qn is closed and convex for all n≥0. For any z∈Cn, n≥1,
(3.29)Df(z,yn)≤(1-αnβn)Df(z,xn+en)+αnβnDf(z,x0)⟺f(z)-f(yn)-〈∇f(yn),z-yn〉≤(1-αnβn)(f(z)-f(xn+en)-〈∇f(xn+en),z-xn-en〉)+αnβn(f(z)-f(x0)-〈∇f(x0),z-x0〉)⟺(1-αnβn)f(xn+en)+αnβnf(x0)-f(yn)≤〈∇f(yn),z-yn〉-(1-αnβn)〈∇f(xn+en),z-xn-en〉-αnβn〈∇f(x0),z-x0〉,
which implies that Cn is closed and convex for all n≥1. Since, for any z∈C0,
(3.30)Df(z,y0)≤Df(z,x0)⟺f(z)-f(y0)-〈∇f(y0),z-y0〉≤f(z)-f(x0)-〈∇f(x0),z-x0〉⟺〈∇f(x0),z-x0〉-〈∇f(y0),z-y0〉≤f(y0)-f(x0)⟺〈∇f(x0),z-x0〉-〈∇f(y0),z-x0+x0-y0〉≤f(y0)-f(x0)⟺〈∇f(x0)-∇f(y0),z-x0〉+Df(x0,y0)≤0,
which shows that C0 is closed and convex. As a consequence, Cn is closed and convex for all n≥0. Taking p∈F(T) arbitrarily, by Lemma 2.18,
(3.31)Df(p,yn)=Df(p,∇f*(αn∇f(zn)+(1-αn)∇f(xn+en)))≤αnDf(p,zn)+(1-αn)Df(p,xn+en)=(1-αn)Df(p,xn+en)+αnDf(p,∇f*(βn∇f(x0)+(1-βn)∇f(T(xn+en))))≤(1-αn)Df(p,xn+en)+αnβnDf(p,x0)+αn(1-βn)Df(p,T(xn+en))≤(1-αn)Df(p,xn+en)+αnβnDf(p,x0)+αn(1-βn)Df(p,xn+en)=(1-αnβn)Df(p,xn+en)+αnβnDf(p,x0),
that is, p∈Cn, and so, F(T)⊂Cn for all n≥0. As in the proof of Theorem 3.1, we get F(T)⊂Qn for all n≥0, {xn} is a Cauchy sequence, {xn},{yn}, and {zn} are also bounded, and thus,
(3.32)lim⁡n→∞Df(xn+1,xn+en)=0,lim⁡n→∞‖(xn+1+en+1)-(xn+en)‖=0,(3.33)lim⁡n→∞‖xn+1-(xn+en)‖=0.
Consequently, F(T)⊂Cn∩Qn and Cn∩Qn is nonempty closed and convex for all n≥0. Moreover, {xn} is well defined. Set xn→ω¯∈C.
Secondly, we show that{xn} converges strongly to a point of F(T). Since f is uniformly Fréchet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-norm uniformly continuous on bounded subsets of E. So, by (3.33),
(3.34)lim⁡n→∞‖∇f(xn+1)-∇f(xn+en)‖=0.
In view of xn+1=proj(Cn∩Qn)f(x0)∈Cn, we have
(3.35)Df(xn+1,yn)≤(1-αnβn)Df(xn+1,xn+en)+αnβnDf(xn+1,x0).
Due to limn→∞βn=0, from (3.32), one has
(3.36)lim⁡n→∞Df(xn+1,yn)=0.
Therefore, limn→∞∥xn+1-yn∥=0. Moreover, one has
(3.37)lim⁡n→∞‖∇f(xn+1)-∇f(yn)‖=0.
Since ∥xn-yn∥≤∥xn-xn+1∥+∥xn+1-yn∥, by (3.32) and (3.33),
(3.38)lim⁡n→∞‖xn-yn‖=0,
and thus, yn→ω¯ as n→∞. Noticing that
(3.39)‖∇f(xn+1)-∇f(yn)‖=‖∇f(xn+1)-(αn∇f(zn)+(1-αn)∇f(xn+en))‖≥αn‖∇f(xn+1)-∇f(zn)‖-(1-αn)‖∇f(xn+1)-∇f(xn+en)‖=αn‖∇f(xn+1)-(βn∇f(x0)+(1-βn)∇f(Tn(xn+en)))‖-(1-αn)‖∇f(xn+1)-∇f(xn+en)‖≥αn(1-βn)‖∇f(xn+1)-∇f(Tn(xn+en))‖-αnβn‖∇f(xn+1)-∇f(x0)‖-(1-αn)‖∇f(xn+1)-∇f(xn+en)‖.
That is,
(3.40)αn(1-βn)∥∇f(xn+1)-∇f(T(xn+en))∥≤‖∇f(xn+1)-∇f(yn)‖+αnβn‖∇f(xn+1)-∇f(x0)‖+(1-αn)‖∇f(xn+1)-∇f(xn+en)‖.
Together with liminfn→∞αn>0,limn→∞βn=0, and (3.37), this yields that
(3.41)lim⁡n→∞‖∇f(xn+1)-∇f(T(xn+en))‖=0.
Since f is uniformly Fréchet differentiable on bounded subsets of E, from Lemma 2.12, ∇f is norm-to-norm uniformly continuous on bounded subsets of E and so is ∇f*. Then, by (3.41), we get
(3.42)lim⁡n→∞‖xn+1-T(xn+en)‖=0.
From ∥(xn+en)-T(xn+en)∥≤∥(xn+en)-(xn+1+en+1)∥+∥(xn+1+en+1)-T(xn+en)∥, it follows that limn→∞∥(xn+en)-T(xn+1+en+1)∥=0. So, ω¯∈F̃(T)=F(T). By the same argument of Theorem 3.1, we know that {xn} and {yn} converge strongly to projF(T)f(x0). This completes the proof.Ifαn≡1 and en≡0 for all n≥0, then from Theorem 3.5, we have the following result.Corollary 3.6.
LetC be a nonempty closed convex subset of a real reflexive Banach space  E and  f:E→R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of  E, and let  T be a weak Bregman relatively nonexpansive mapping from  C into itself such that  F(T)≠∅. Define a sequence  {xn} in  C by the following algorithm:
(3.43)x0∈C,Q0=C,yn=∇f*(βn∇f(x0)+(1-βn)∇f(Txn)),Cn={z∈Cn-1∩Qn-1:Df(z,yn)≤(1-βn)Df(z,xn)+βnDf(z,x0)},C0={z∈C:Df(z,y0)≤Df(z,x0)},Qn={z∈Cn-1∩Qn-1:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=projCn∩Qnfx0,∀n≥0,
where {βn}⊂[0,1] such that limn→∞βn=0 and {en} is an error sequence in E with en→0 as n→∞. Then, the sequences {xn} and {yn} converges strongly to the point projF(T)f(x0), where projF(T)f(x0) is the Bregman projection of C onto F(T).Now, we develop a strong convergence theorem for a Bregman relatively nonexpansive mapping.Theorem 3.7.
LetC be a nonempty closed convex subset of a real reflexive Banach space E and f:E→R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T:C→C be a Bregman relatively nonexpansive mapping such that F(T)≠∅. Define a sequence {xn} in C by the following algorithm:
(3.44)x0∈C,Q0=C,zn=∇f*(βn∇f(x0)+(1-βn)∇f(T(xn+en))),yn=∇f*(αn∇f(zn)+(1-αn)∇f(xn+en)),Cn={z∈Cn-1∩Qn-1:Df(z,yn)≤(1-αnβn)Df(z,xn+en)+αnβnDf(z,x0)},C0={z∈C:Df(z,y0)≤Df(z,x0)},Qn={z∈Cn-1∩Qn-1:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=projCn∩Qnfx0,∀n≥0,
where {αn},{βn}⊂[0,1] such that liminfn→∞αn>0 and limn→∞βn=0 and {en} is an error sequence in E with en→0 as n→∞. Then, the sequences {xn} and {yn} converges strongly to the point projF(T)f(x0), where projF(T)f(x0) is the Bregman projection of C onto F(T).Proof.
The proof is similar to Theorem3.5 and so is omitted. This completes the proof.Ifαn≡1 and en≡0 for all n≥0, then from Theorem 3.7, we get the following corollary.Corollary 3.8.
LetE be a real reflexive Banach space and f:E→R a Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on bounded subset of E, and let T:E→E be a Bregman relatively nonexpansive mapping such that F(T)≠∅. Assume that {βn} is a real sequence in [0,1] such that limn→∞βn=0. Define a sequence {xn} by the following algorithm:
(3.45)x0∈C,Q0=C,yn=∇f*(βn∇f(x0)+(1-βn)∇f(Txn)),Cn={z∈Cn-1∩Qn-1:Df(z,yn)≤(1-αnβn)Df(z,xn)+αnβnDf(z,x0)},C0={z∈C:Df(z,y0)≤Df(z,x0)},Qn={z∈Cn-1∩Qn-1:〈∇f(x0)-∇f(xn),z-xn〉≤0},xn+1=projCn∩Qnfx0,∀n≥0.
Then, the sequences {xn} and {yn} converge strongly to the point projF(T)f(x0), where projF(T)f(x0) is the Bregman projection of C onto F(T).In [30], Qin and Su obtained the following.Theorem QS (see [30, Theorem 2.2]).
LetC be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T:C→C be a relatively nonexpansive mapping such that F(T)≠∅. Assume that {βn} is a real sequence in [0,1) such that limn→∞βn=0. Then, the sequence {xn} generated by (1.5) converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection (see, e.g., [2, 3]) from E onto F(T).Remark 3.9.
Corollary3.8 extends Theorems QS [30] from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces.


## 4. Conclusions

In this paper, we introduce a conception of weak Bregman relatively nonexpansive mapping in reflexive Banach space and give an example to illustrate the existence of weak Bregman relatively nonexpansive mapping and the difference between weak Bregman relatively nonexpansive mapping and Bregman relatively nonexpansive mapping which enlarge the Bregman operator theory. Secondly, by using projection techniques, we construct several modification of Mann-type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. Thirdly, strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. By further research, on the one hand, we may apply our algorithms to find zeros of finite families of maximal monotone operators, solutions of system of convex minization problems, solutions of system of variational inequalities, equilibrium, and equation operators (see, e.g., [24]). On the other hand, one may give some numerical experiments to verify the theoretical assertions and show how to compute the generalized projections. These topics will be done in the future.



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*Source: 420192-2011-07-28.xml*