# Bregman Asymptotic Pointwise Nonexpansive Mappings in Banach Spaces

Authors: Chin-Tzong Pang; Eskandar Naraghirad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We first introduce a new class of mappings called Bregman asymptotic pointwise nonexpansive mappings and investigate the existence and the approximation of fixed points of such mappings defined on a nonempty, bounded, closed, and convex subsetC of a real Banach space E. Without using the original Opial property of a Banach space E, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes for Bregman asymptotic pointwise nonexpansive mappings
in a reflexive Banach space E. Our results are applicable in the function spaces Lp, where 1<p<∞ is a real number.


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## Body

## 1. Introduction

Throughout this paper, we denote the set of real numbers and the set of positive integers byℝ and ℕ, respectively. Let E be a Banach space with the norm ∥·∥ and the dual space E*. For any x∈E, we denote the value of x*∈E* at x by 〈x,x*〉. Let {xn}n∈ℕ be a sequence in E; we denote the strong convergence of {xn}n∈ℕ to x∈E as n→∞ by xn→x and the weak convergence by xn⇀x. The modulus δ of convexity of E is denoted by
(1)δ(ϵ)=inf{1-∥x+y∥2:∥x∥≤1,∥y∥≤1,∥x-y∥≥ϵ},
for every ϵ with 0≤ϵ≤2. A Banach space E is said to be uniformly convex if δ(ϵ)>0 for every ϵ>0. Let SE={x∈E:∥x∥=1}. The norm of E is said to be Gâteaux differentiable if for each x,y∈SE, the limit
(2)limt→0∥x+ty∥-∥x∥t
exists. In this case, E is called smooth. If the limit (2) is attained uniformly for all x,y∈SE, then E is called uniformly smooth. The Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 whenever x,y∈SE and x≠y. It is well known that E is uniformly convex if and only if E* is uniformly smooth. It is also known that if E is reflexive, then E is strictly convex if and only if E* is smooth; for more details, see [1, 2].LetC be a nonempty subset of E. Let T:C→E be a mapping. We denote the set of fixed points of T by F(T); that is, F(T)={x∈C:Tx=x}. A mapping T:C→E is said to be nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈C. A mapping T:C→E is said to be quasi-nonexpansive if F(T)≠∅ and ∥Tx-y∥≤∥x-y∥ for all x∈C and  y∈F(T). The nonexpansivity plays an important role in the study of Mann iteration [3] for finding fixed points of a mapping T:C→C. Recall that the Mann iteration is given by the following formula:
(3)xn+1=γnTxn+(1-γn)xn,x1∈C.
Here, {γn}n∈ℕ is a sequence of real numbers in [0,1] satisfying some appropriate conditions. A more general iteration is the Ishikawa iteration [4], given by
(4)yn=βnTxn+(1-βn)xn,xn+1=γnTyn+(1-γn)xn,
where the sequences {βn}n∈ℕ and {γn}n∈ℕ satisfy some appropriate conditions. When all βn=0, the Ishikawa iteration reduces to the classical Mann iteration. Construction of fixed points of nonexpansive mappings via Mann’s and Ishikawa’s algorithms [3] has been extensively investigated in the literature (see, e.g., [5] and the references therein). A powerful tool in deriving weak or strong convergence of iterative sequences is due to Opial [6]. A Banach space E is said to satisfy the Opial property [6] if for any weakly convergent sequence {xn}n∈ℕ in E with weak limit x, we have
(5)limsupn→∞∥xn-x∥<limsupn→∞∥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. However, not every Banach space satisfies the Opial property; see, for example, [7].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 [8] is defined and denoted by
(6)ΠC(x)=argminy∈Cϕ(y,x),
where ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2. Let C be a nonempty, closed, and convex subset of a smooth Banach space E, and let T be a mapping from C into itself.


### 1.1. Some Facts about Gradients

For any convex functiong:E→(-∞,+∞] we denote the domain of g by dom g={x∈E:g(x)<∞}. For any x∈intdomg and any y∈E, we denote by go(x,y) the right-hand derivative of g at x in the direction y; that is,
(7)go(x,y)=limt↓0g(x+ty)-g(x)t.
The function g is said to be Gâteaux differentiable at x if limt→0(g(x+ty)-g(x))/t exists for any y. In this case go(x,y) coincides with ∇g(x), the value of the gradient∇g of g at x. The function g is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function g is said to be Fréchet differentiable at x if this limit is attained uniformly in ∥y∥=1. The function g is Fréchet differentiable at x∈E (see, e.g., [9, page 13] or [10, page 508]) if for all ϵ>0, there exists δ>0 such that ∥y-x∥≤δ implies that
(8)|g(y)-g(x)-〈y-x,∇g(x)〉|≤ϵ∥y-x∥.
The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function g:E→ℝ is Gâteaux differentiable, then ∇g is norm-to-weak* continuous (see, e.g., [9, Proposition 1.1.10]). Also, it is known that if g is Fréchet differentiable, then ∇g is norm-to-norm continuous (see, [10, page 508]). The mapping ∇g is said to be weakly sequentially continuous if xn⇀x as n→∞ implies that ∇g(xn)⇀*∇g(x) as n→∞ (for more details, see [9, Theorem 3.2.4] or [10, page 508]). The function g is said to be strongly coercive if
(9)lim∥xn∥→∞g(xn)∥xn∥=∞.
It is also said to be bounded on bounded subsets ofE if g(U) is bounded for each bounded subset U of E. Finally, g is said to be uniformly Fréchet differentiable on a subset X of E if the limit (7) is attained uniformly for all x∈X and ∥y∥=1.LetE be a reflexive Banach space. For any proper, lower semicontinuous, and convex function g:E→(-∞,+∞], the conjugate functiong* of g is defined by
(10)g*(x*)=supx∈E{〈x,x*〉-g(x)},
for all x*∈E*. It is well known that g(x)+g*(x*)≥〈x,x*〉 for all (x,x*)∈E×E*. It is also known that (x,x*)∈∂g is equivalent to
(11)g(x)+g*(x*)=〈x,x*〉.
Here, ∂g is the subdifferential of g [11, 12]. We also know that if g:E→(-∞,+∞] is a proper, lower semicontinuous, and convex function, then g*:E*→(-∞,+∞] is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.


### 1.2. Some Facts about Bregman Distances

LetE be a Banach space and let E* be the dual space of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] corresponding to g is the function Dg:E×E→ℝ defined by
(12)Dg(x,y)=g(x)-g(y)-〈x-y,∇g(y)〉,∀x,y∈E.
It is clear that Dg(x,y)≥0 for all x,y∈E. In that case when E is a smooth Banach space, setting g(x)=∥x∥2 for all x∈E, we obtain that ∇g(x)=2Jx for all x∈E and hence Dg(x,y)=ϕ(x,y) for all x,y∈E.LetE be a Banach space and let C be a nonempty and convex subset of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then, we know from [15] that for x∈E and x0∈C, Dg(x0,x)=miny∈CDg(y,x) if and only if
(13)〈y-x0,∇g(x)-∇g(x0)〉≤0,∀y∈C.
Furthermore, if C is a nonempty, closed, and convex subset of a reflexive Banach space E and g:E→ℝ is a strongly coercive Bregman function, then for each x∈E, there exists a unique x0∈C such that
(14)Dg(x0,x)=miny∈CDg(y,x).
The Bregman projectionprojCg from E onto C is defined by projCg(x)=x0 for all x∈E. It is also well known that projCg has the following property:
(15)Dg(y,projCgx)+Dg(projCgx,x)≤Dg(y,x),
for all y∈C and x∈E (see [9] for more details).For any bounded subsetA of a reflexive Banach space E, we denote the Bregman diameter of A by
(16)Bdiam(A):=sup{Dg(x,y):x,y∈A}<∞.


### 1.3. Some Facts about Uniformly Convex Functions

LetE be a Banach space and let Bs:={z∈E:∥z∥≤s} for all s>0. Then a function g:E→ℝ is said to be uniformly convex on bounded subsets ofE ([16, Pages 203, 221]) if ρs(t)>0 for all s, t>0, where ρs:[0,+∞)→[0,∞] is defined by
(17)ρs(t)=infx,y∈Bs,∥x-y∥=t,α∈(0,1)(αg(x)+(1-α)g(y)=infx,y∈Bs,∥x-y∥=t,α∈(0,1)(-g(αx+(1-α)y)×(α(1-α))-1,
for all t≥0. The function ρs is called the gauge of uniform convexity of g. The function g is also said to be uniformly smooth on bounded subsets ofE ([16, Pages 207, 221]) if limt↓0(σs(t)/t)=0 for all s>0, where σs:[0,+∞)→[0,∞] is defined by
(18)σs(t)=supx∈Bs,y∈SE,α∈(0,1)(αg(x+(1-α)ty)=supx∈Bs,y∈SE,α∈(0,1)(+(1-α)g(x-αty)-g(x))×(α(1-α))-1,
for all t≥0. The function g is said to be uniformly convex if the function δg:[0,+∞)→[0,+∞], defined by
(19)δg(t):=sup{12g(x)+12g(y)-g(x+y2):∥y-x∥=t},
satisfies that limt↓0(σs(t)/t)=0.Remark 1.
LetE be a Banach space, let s>0 be a constant, and let g:E→ℝ be a convex function which is uniformly convex on bounded subsets. Then
(20)g(αx+(1-α)y)≤αg(x)+(1-α)g(y)-α(1-α)ρs(∥x-y∥),
for all x,y∈Bs:={z∈E:∥z∥≤s} and α∈(0,1), where ρs is the gauge of uniform convexity of g.Definition 2.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. A mapping T:C→E is said to be Bregman asymptotic pointwise nonexpansive if there exists a sequence of mappings θn:C→[0,∞) such that
(21)Dg(Tnx,Tny)≤θn(x)Dg(x,y),∀x,y∈C,limsupn→∞θn(x)≤1,∀x∈C.
Denoting an(x)=max{θn(x),1}, we note that without loss of generality we can assume that T is Bregman asymptotic pointwise nonexpansive if
(22)Dg(Tnx,Tny)≤an(x)Dg(x,y),∀x,y∈C,n∈ℕ,(23)limn→∞an(x)=1,an(x)≥1,∀x∈C,n∈ℕ.
Define bn(x)=an(x)-1. In view of (23), we obtain
(24)limn→∞bn(x)=0.
Next, we denote by ℬ𝒯(C) the class of all Bregman asymptotic pointwise nonexpansive mappings T:C→C.
Imposing some restrictions on the behavior ofan and bn, we can define the following subclass of Bregman asymptotic pointwise nonexpansive mappings.Definition 3.
LetC and ℬ𝒯(C) be as in Definition 2. We define ℬ𝒯r(C) as a class of all T∈ℬ𝒯(C) such that
(25)∑n=1∞‍bn(x)<∞,∀x∈C,anisaboundedfunction∑n=1∞‍bn(x)<∞∀x∈C,anisaforeveryninℕ.
Kirk and Xu [17] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings with respect to the norm of a Banach space E. Recently, Kozlowski [18] proved weak and strong convergence theorems for asymptotic pointwise nonexpansive mappings in a Banach space. To see some other related works, we refer the reader to [19, 20].In this paper, we first investigate the approximation of fixed points of a new class of Bregman asymptotic pointwise nonexpansive mappings defined on a nonempty, bounded, closed, and convex subsetC of a real Banach space E. Without using the Opial property of a Banach space E, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes. Our results improve and generalize many known results in the current literature; see, for example, [18, 21].


## 1.1. Some Facts about Gradients

For any convex functiong:E→(-∞,+∞] we denote the domain of g by dom g={x∈E:g(x)<∞}. For any x∈intdomg and any y∈E, we denote by go(x,y) the right-hand derivative of g at x in the direction y; that is,
(7)go(x,y)=limt↓0g(x+ty)-g(x)t.
The function g is said to be Gâteaux differentiable at x if limt→0(g(x+ty)-g(x))/t exists for any y. In this case go(x,y) coincides with ∇g(x), the value of the gradient∇g of g at x. The function g is said to be Gâteaux differentiable if it is Gâteaux differentiable everywhere. The function g is said to be Fréchet differentiable at x if this limit is attained uniformly in ∥y∥=1. The function g is Fréchet differentiable at x∈E (see, e.g., [9, page 13] or [10, page 508]) if for all ϵ>0, there exists δ>0 such that ∥y-x∥≤δ implies that
(8)|g(y)-g(x)-〈y-x,∇g(x)〉|≤ϵ∥y-x∥.
The function g is said to be Fréchet differentiable if it is Fréchet differentiable everywhere. It is well known that if a continuous convex function g:E→ℝ is Gâteaux differentiable, then ∇g is norm-to-weak* continuous (see, e.g., [9, Proposition 1.1.10]). Also, it is known that if g is Fréchet differentiable, then ∇g is norm-to-norm continuous (see, [10, page 508]). The mapping ∇g is said to be weakly sequentially continuous if xn⇀x as n→∞ implies that ∇g(xn)⇀*∇g(x) as n→∞ (for more details, see [9, Theorem 3.2.4] or [10, page 508]). The function g is said to be strongly coercive if
(9)lim∥xn∥→∞g(xn)∥xn∥=∞.
It is also said to be bounded on bounded subsets ofE if g(U) is bounded for each bounded subset U of E. Finally, g is said to be uniformly Fréchet differentiable on a subset X of E if the limit (7) is attained uniformly for all x∈X and ∥y∥=1.LetE be a reflexive Banach space. For any proper, lower semicontinuous, and convex function g:E→(-∞,+∞], the conjugate functiong* of g is defined by
(10)g*(x*)=supx∈E{〈x,x*〉-g(x)},
for all x*∈E*. It is well known that g(x)+g*(x*)≥〈x,x*〉 for all (x,x*)∈E×E*. It is also known that (x,x*)∈∂g is equivalent to
(11)g(x)+g*(x*)=〈x,x*〉.
Here, ∂g is the subdifferential of g [11, 12]. We also know that if g:E→(-∞,+∞] is a proper, lower semicontinuous, and convex function, then g*:E*→(-∞,+∞] is a proper, weak* lower semicontinuous, and convex function; see [2] for more details on convex analysis.


## 1.2. Some Facts about Bregman Distances

LetE be a Banach space and let E* be the dual space of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] corresponding to g is the function Dg:E×E→ℝ defined by
(12)Dg(x,y)=g(x)-g(y)-〈x-y,∇g(y)〉,∀x,y∈E.
It is clear that Dg(x,y)≥0 for all x,y∈E. In that case when E is a smooth Banach space, setting g(x)=∥x∥2 for all x∈E, we obtain that ∇g(x)=2Jx for all x∈E and hence Dg(x,y)=ϕ(x,y) for all x,y∈E.LetE be a Banach space and let C be a nonempty and convex subset of E. Let g:E→ℝ be a convex and Gâteaux differentiable function. Then, we know from [15] that for x∈E and x0∈C, Dg(x0,x)=miny∈CDg(y,x) if and only if
(13)〈y-x0,∇g(x)-∇g(x0)〉≤0,∀y∈C.
Furthermore, if C is a nonempty, closed, and convex subset of a reflexive Banach space E and g:E→ℝ is a strongly coercive Bregman function, then for each x∈E, there exists a unique x0∈C such that
(14)Dg(x0,x)=miny∈CDg(y,x).
The Bregman projectionprojCg from E onto C is defined by projCg(x)=x0 for all x∈E. It is also well known that projCg has the following property:
(15)Dg(y,projCgx)+Dg(projCgx,x)≤Dg(y,x),
for all y∈C and x∈E (see [9] for more details).For any bounded subsetA of a reflexive Banach space E, we denote the Bregman diameter of A by
(16)Bdiam(A):=sup{Dg(x,y):x,y∈A}<∞.


## 1.3. Some Facts about Uniformly Convex Functions

LetE be a Banach space and let Bs:={z∈E:∥z∥≤s} for all s>0. Then a function g:E→ℝ is said to be uniformly convex on bounded subsets ofE ([16, Pages 203, 221]) if ρs(t)>0 for all s, t>0, where ρs:[0,+∞)→[0,∞] is defined by
(17)ρs(t)=infx,y∈Bs,∥x-y∥=t,α∈(0,1)(αg(x)+(1-α)g(y)=infx,y∈Bs,∥x-y∥=t,α∈(0,1)(-g(αx+(1-α)y)×(α(1-α))-1,
for all t≥0. The function ρs is called the gauge of uniform convexity of g. The function g is also said to be uniformly smooth on bounded subsets ofE ([16, Pages 207, 221]) if limt↓0(σs(t)/t)=0 for all s>0, where σs:[0,+∞)→[0,∞] is defined by
(18)σs(t)=supx∈Bs,y∈SE,α∈(0,1)(αg(x+(1-α)ty)=supx∈Bs,y∈SE,α∈(0,1)(+(1-α)g(x-αty)-g(x))×(α(1-α))-1,
for all t≥0. The function g is said to be uniformly convex if the function δg:[0,+∞)→[0,+∞], defined by
(19)δg(t):=sup{12g(x)+12g(y)-g(x+y2):∥y-x∥=t},
satisfies that limt↓0(σs(t)/t)=0.Remark 1.
LetE be a Banach space, let s>0 be a constant, and let g:E→ℝ be a convex function which is uniformly convex on bounded subsets. Then
(20)g(αx+(1-α)y)≤αg(x)+(1-α)g(y)-α(1-α)ρs(∥x-y∥),
for all x,y∈Bs:={z∈E:∥z∥≤s} and α∈(0,1), where ρs is the gauge of uniform convexity of g.Definition 2.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. A mapping T:C→E is said to be Bregman asymptotic pointwise nonexpansive if there exists a sequence of mappings θn:C→[0,∞) such that
(21)Dg(Tnx,Tny)≤θn(x)Dg(x,y),∀x,y∈C,limsupn→∞θn(x)≤1,∀x∈C.
Denoting an(x)=max{θn(x),1}, we note that without loss of generality we can assume that T is Bregman asymptotic pointwise nonexpansive if
(22)Dg(Tnx,Tny)≤an(x)Dg(x,y),∀x,y∈C,n∈ℕ,(23)limn→∞an(x)=1,an(x)≥1,∀x∈C,n∈ℕ.
Define bn(x)=an(x)-1. In view of (23), we obtain
(24)limn→∞bn(x)=0.
Next, we denote by ℬ𝒯(C) the class of all Bregman asymptotic pointwise nonexpansive mappings T:C→C.
Imposing some restrictions on the behavior ofan and bn, we can define the following subclass of Bregman asymptotic pointwise nonexpansive mappings.Definition 3.
LetC and ℬ𝒯(C) be as in Definition 2. We define ℬ𝒯r(C) as a class of all T∈ℬ𝒯(C) such that
(25)∑n=1∞‍bn(x)<∞,∀x∈C,anisaboundedfunction∑n=1∞‍bn(x)<∞∀x∈C,anisaforeveryninℕ.
Kirk and Xu [17] studied the existence of fixed points of asymptotic pointwise nonexpansive mappings with respect to the norm of a Banach space E. Recently, Kozlowski [18] proved weak and strong convergence theorems for asymptotic pointwise nonexpansive mappings in a Banach space. To see some other related works, we refer the reader to [19, 20].In this paper, we first investigate the approximation of fixed points of a new class of Bregman asymptotic pointwise nonexpansive mappings defined on a nonempty, bounded, closed, and convex subsetC of a real Banach space E. Without using the Opial property of a Banach space E, we prove weak convergence theorems for the sequences produced by generalized Mann and Ishikawa iteration processes. Our results improve and generalize many known results in the current literature; see, for example, [18, 21].


## 2. Preliminaries

In this section, we begin by recalling some preliminaries and lemmas which will be used in the sequel.Definition 4 (see [10]).
LetE be a Banach space. The function g:E→ℝ is said to be a Bregman function if the following conditions are satisfied:(1)
gis continuous, strictly convex, and Gâteaux differentiable;(2)
the set{y∈E:Dg(x,y)≤r} is bounded for all x∈E and r>0.Lemma 5 (see [9, 16]).
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∈E and r>0;(4)
domg*=E*,g* is Gâteaux differentiable and ∇g*=(∇g)-1.We know the following two results; see [16, Proposition 3.6.4].Theorem 6.
LetE be a reflexive Banach space and g:E→ℝ a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:(1)
g is strongly coercive and uniformly convex on bounded subsets of E;(2)
domg*=E*,g* is bounded on bounded subsets and uniformly smooth on bounded subsets of E*;(3)
domg*=E*,g* is Fre´chet differentiable and ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*.Theorem 7.
LetE be a reflexive Banach space and g:E→ℝ a continuous convex function which is strongly coercive. Then the following assertions are equivalent:(1)
g is bounded on bounded subsets and uniformly smooth on bounded subsets of E;(2)
g* is Fre´chet differentiable and ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*;(3)
domg*=E*,g* is strongly coercive and uniformly convex on bounded subsets of E*.LetE be a Banach space and let g:E→ℝ be a convex and Gâteaux differentiable function. Then the Bregman distance [13, 14] does not satisfy the well known properties of a metric, but it does have the following important property, which is called the three point identity [22]:
(26)Dg(x,z)=Dg(x,y)+Dg(y,z)+〈x-y,∇g(y)-∇g(z)〉,∀x,y,z∈E.
In particular, it can easily be seen that
(27)Dg(x,y)=-Dg(y,x)+〈y-x,∇g(y)-∇g(x)〉,=-Dg(y,x)+∀x,y∈E.
Indeed, by letting z=x in (26) and taking into account that Dg(x,x)=0, we get the desired result.Lemma 8 (see [23]).
LetE be a Banach space and g:E→ℝ a Gâteaux differentiable function which is uniformly convex on bounded subsets of E. Let {xn}n∈ℕ and {yn}n∈ℕ be bounded sequences in E. Then the following assertions are equivalent:(1)
limn→∞Dg(xn,yn)=0;(2)
limn→∞∥xn-yn∥=0.Lemma 9 (see [10, 24]).
LetE be a reflexive Banach space, g:E→ℝ a strongly coercive Bregman function, and V the function defined by
(28)V(x,x*)=g(x)-〈x,x*〉+g*(x*),x∈E,x*∈E*.
Then the following assertions hold:(1)
Dg(x,∇g*(x*))=V(x,x*) for all x∈E and x*∈E*;(2)
V(x,x*)+〈∇g*(x*)-x,y*〉≤V(x,x*+y*) for all x∈E and x*,y*∈E*.LetC and D be nonempty subsets of a real Banach space E with D⊂C. A mapping RD:C→D is said to be sunny if
(29)RD(RDx+t(x-RDx))=RDx,
for each x∈E and t≥0. A mapping RD:C→D is said to be a retraction if RDx=x for each x∈C.Lemma 10 (see [25]).
Suppose{rk}k∈ℕ is a bounded sequence of real numbers and {dk,n}k,n∈ℕ is a doubly index sequence of real numbers which satisfy
(30)limsupk→∞limsupn→∞dk,n≤0,rk,n≤rk+dk,n,
for each k,n≥1. Then {rk}k∈ℕ converges to an r∈ℝ.LetE be a reflexive Banach space and let g:E→(-∞,+∞] be an admissible function, that is, a proper, lower-semicontinuous, 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
(31)Br({xn},x)=limsupn→∞Dg(xn,x).
The Bregman asymptotic radius of {xn}n∈ℕ relative to C is defined by
(32)Br({xn},C)=inf{Br({xn},x):x∈C}.
The Bregman asymptotic center of {xn}n∈ℕ relative to C is the set
(33)BA({xn},C)={x∈C:Br({xn},x)=Br({xn},C)}.The following Bregman Opial-like inequality has been proved in [26]. It is worth mentioning that the Bregman Opial-like inequality is different from the ordinary Opial inequality [6] and can be applied in uniformly convex Banach spaces.Lemma 11 (see [26]).
LetE be a Banach space and let g:E→(-∞,+∞] be a proper strictly convex function so that it is Gâteaux differentiable on intdomg. Suppose {xn}n∈ℕ is a sequence in domg such that xn⇀v for some v∈intdomg. Then
(34)limsupn→∞Dg(xn,v)<limsupn→∞Dg(xn,y),∀y∈intdomgwithy≠v.Theorem 12 (see [16]).
Letg:E→(-∞,+∞] be a function. Then the following assertions are equivalent:(1)
g is convex and lower semicontinuous;(2)
g is convex and weakly lower semicontinuous;(3)
epi(g) is convex and closed;(4)
epi(g) is convex and weakly closed,
whereepi(g)={(x,t)∈E×ℝ:g(x)≤t} denotes the epigraph of g.


## 3. Fixed Point Theorems and Demiclosedness Principle

Proposition 13.
LetE be a reflexive Banach space and let g:E→ℝ be a strongly admissible function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T∈ℬ𝒯r(C). Then T has a fixed point. Moreover, F(T) is closed and convex.Proof.
We first show thatF(T) is nonempty. Let x in C be fixed. We define a function f:C→[0,∞) by
(35)f(y)=limsupn→∞Dg(y,Tnx),y∈C.
In view of Remark 1, it is easy to see that f is convex. Since g is continuous, by Theorem 12 we conclude that the Bregman distance Dg is weakly lower-semicontinuous in the first argument. Since E is a uniformly convex Banach space and C is weakly compact, in view of [1] there exists a unique point z∈C such that
(36)f(z)=miny∈Cf(y).
We show that {Tnz}n∈ℕ is convergent in norm. To this end, let s1=sup{∥Tnz∥,∥z∥:n∈ℕ} and ρs1:E→ℝ be the gauge of uniform convexity of g. For any k, m∈ℕ, put uk,m=(1/2)Tkz+(1/2)Tmz. Then we have uk,m∈C. In view of Remark 1, we obtain
(37)Dg(uk,m,Tk+m+nx)=g(uk,m)-g(Tk+m+nx)-〈uk,m-Tk+m+nx,∇g(Tk+m+nx)〉=g(12Tkz+12Tmz)-g(Tk+m+nx)-〈12Tkz+12Tmz-Tk+m+nx,∇g(Tk+m+nx)〉≤12g(Tkz)+12g(Tmz)-14ρs1(∥Tkz-Tmz∥)-g(Tk+m+nx)-12〈Tkz-Tk+m+nx,∇g(Tk+m+nx)〉-12〈Tmz-Tk+m+nx,∇g(Tk+m+nx)〉=12[g(Tkz)-g(Tk+m+nx)-〈Tkz-Tk+m+nx,∇g(Tk+m+nx)〉]+12[g(Tmz)-g(Tk+m+nx)-〈Tmz-Tk+m+nx,∇g(Tk+m+nx)〉]-14ρs1(∥Tkz-Tmz∥)=12Dg(Tkz,Tk+m+nx)+12Dg(Tmz,Tk+m+nx)-14ρs1(∥Tkz-Tmz∥)≤12ak(z)Dg(z,Tm+nx)+12ak(z)Dg(z,Tk+nx)-14ρs1(∥Tkz-Tmz∥).
Applying to both sides of the above inequalities limsupn→∞ we obtain
(38)f(uk,m)≤12[ak(z)+am(z)]f(z)-14ρs1(∥Tkz-Tmz∥).
This, together with f(z)≤f(uk,m), implies that
(39)ρs1(∥Tkz-Tmz∥)≤4[12[ak(z)+am(z)]f(z)-f(uk,m)]≤4[12[ak(z)+am(z)]f(z)-f(z)].
Letting k, m→∞ in (39) we conclude that
(40)limk,m→∞ρs1(∥Tkz-Tmz∥)=0.
From the properties of ρs1, we deduce that limk,m→∞∥Tkz-Tmz∥=0. Thus, {Tkz}k∈ℕ is a norm-Cauchy sequence and hence convergent. Let
(41)v=limn→∞Tnz.
Since T is a Bregman asymptotic pointwise nonexpansive mapping, we have, for all n∈ℕ,
(42)Dg(Tn+1z,Tv)≤a1(Tnz)Dg(Tnz,v).
Letting n→∞ in (42), we conclude that Tv=v. This shows that F(T)≠∅.
Now, we show thatF(T) is closed. Let {pn}n∈ℕ be a sequence in F(T) such that pn→p as n→∞. Then we have that {pn}n∈ℕ is a bounded sequence in E. We claim that p∈F(T). Since g is continuous, we conclude that g(pn)→g(p) as n→∞. This implies that
(43)Dg(pn,p)=g(pn)-g(p)-〈pn-p,∇g(p)〉≤|g(pn)-g(p)|+∥pn-p∥∥∇g(p)∥⟶0≤|g(pn)-g(p)|+∥pn-p∥(n⟶∞).
In view of the definition of T, we obtain
(44)Dg(Tp,pn)≤an(p)Dg(p,pn)⟶0(n⟶∞).
This implies that
(45)0=limn→∞Dg(pn,Tp)=limn→∞[g(pn)-g(Tp)-〈pn-Tp,∇g(Tp)〉]=Dg(p,Tp).
It follows from Lemma 8 that Tp=p. Thus we have p∈F(T).
Let us show thatF(T) is convex. For any p, q∈F(T), t∈(0,1), and n∈ℕ, we set x=tp+(1-t)q and en(p,q)=max{an(p),an(q)}. We prove that x∈F(T). By the definition of Bregman distance (see (12)), we get
(46)Dg(x,Tnx)=g(x)-g(Tnx)-〈x-Tnx,∇g(Tnx)〉=g(x)-g(Tnx)-〈tp+(1-t)q-Tnx,∇g(Tnx)〉=g(x)-g(Tnx)-t〈p-Tnx,∇g(Tnx)〉-(1-t)〈q-Tnx,∇g(Tnx)〉+tg(p)+(1-t)g(q)-[tg(p)+(1-t)g(q)]=g(x)+t[g(p)-g(Tnx)=g(x)+t[-〈p-Tnx,∇g(Tnx)〉]+(1-t)[g(q)-g(Tnx)+(1-t)[-〈q-Tnx,∇g(Tnx)〉]=g(x)+tDg(p,Tnx)+(1-t)Dg(q,Tnx)-[tg(p)+(1-t)g(q)]≤g(x)+tan(p)Dg(p,x)+(1-t)an(q)Dg(q,x)-tg(p)-(1-t)g(q)≤g(x)+ten(p,q)Dg(p,x)+(1-t)en(p,q)Dg(q,x)-tg(p)-(1-t)g(q)=g(x)+en(p,q)×[t(g(p)-g(x)-〈p-x,∇g(x)〉)+(1-t)(g(q)-g(x)-〈q-x,∇g(x)〉)]-tg(p)-(1-t)g(q)=g(x)+en(p,q)×[-g(x)-〈t(p-x),∇g(x)〉-〈(1-t)(q-x),∇g(x)〉]+en(p,q)[tg(p)+(1-t)g(q)]-tg(p)-(1-t)g(q)=g(x)-en(p,q)g(x)-en(p,q)×[〈tp+(1-t)q-x,∇g(x)〉]+en(p,q)[tg(p)+(1-t)g(q)]-tg(p)-(1-t)g(q)=(en(p,q)-1)(-g(x))+(en(p,q)-1)tg(p)+(en(p,q)-1)(1-t)g(q)=(en(p,q)-1)×[-g(x)+tg(p)+(1-t)g(q)].
This implies that limn→∞Dg(x,Tnx)=0. Thus for each ϵ>0, there exists n0∈ℕ such that
(47)Dg(x,Tnx)≤ϵ,∀n≥n0.
This means that the sequence {Dg(x,Tnx)}n∈ℕ is bounded. In view of Definition 4, we conclude that the sequence {Tnx}n∈ℕ is bounded. Then, by Lemma 8, we obtain limn→∞∥x-Tnx∥=0. Thus we have Tn+1x→x; that is, T(Tnx)→x. On the other hand, in view of three-point identity (see (26)), we deduce that
(48)Dg(x,Tx)=Dg(x,Tn+1x)+Dg(Tn+1x,Tx)+〈x-Tn+1x,∇g(Tn+1x)-∇g(Tx)〉≤Dg(x,Tn+1x)+a1(Tnx)Dg(Tnx,x)+∥x-Tn+1x∥∥∇g(Tn+1x)-∇g(Tx)∥.
Letting n→∞ in the above inequalities we deduce that Dg(x,Tx)=0 and hence by Lemma 8 we conclude that Tx=x, which completes the proof.Lemma 14.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T∈ℬ𝒯r(C). If {xn}n∈ℕ is a sequence of C such that limn→∞∥Txn-xn∥=0, then, for any m∈ℕ, limn→∞∥Tmxn-xn∥=0.Proof.
In view of (25), there exists a finite constant M1>0 such that
(49)∑j=1m-1‍sup{aj(x):x∈C}≤M1.
It follows from three-point identity (see (26)) that
(50)Dg(Tmxn,xn)=Dg(Tmxn,Tm-1xn)+Dg(Tm-1xn,xn)+〈Tmxn-Tm-1xn,∇g(Tm-1xn)-∇g(xn)Tm-1〉≤am-1(xn)Dg(Txn,xn)+Dg(Tm-1xn,Tm-2xn)+Dg(Tm-2xn,xn)+〈Tm-1xn-Tm-2xn,∇g(Tm-2xn)-∇g(xn)Tm-1〉+M2∥Tmxn-Tm-1xn∥⋮≤(∑j=1m-1‍aj(xn)+1)Dg(Txn,xn)+2M2∑j=1m-1‍∥Tmxn-Tm-1xn∥,
where M2:=sup{∥∇g(Tjxn)∥:j=1,2,…,mandn∈ℕ}. This, together with Lemma 8, implies that limn→∞∥Tmxn-xn∥=0. This completes the proof.Theorem 15.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex and uniformly smooth on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T∈ℬ𝒯r(C). If {xn}n∈ℕ converges weakly to z and limn→∞∥Txn-xn∥=0, then Tz=z. That is, I-T is demiclosed at zero, where I is the identity mapping on E.Proof.
Let the functionϕ:E→[0,∞) be defined by
(51)ϕ(x)=limsupn→∞Dg(xn,x)(x∈E).
For any m,n∈ℕ with m>2, in view of three-point identity (see (26)), we obtain
(52)Dg(Tmxn,x)=D(Tmxn,Tm-1xn)+Dg(Tm-1xn,x)+〈Tmxn-Tm-1xn,∇g(Tm-1xn)-∇g(x)Tm-1〉≤am-1(xn)Dg(Txn,xn)+D(Tm-1xn,Tm-2xn)+Dg(Tm-2xn,x)+〈Tm-1xn-Tm-2xn,∇g(Tm-2xn)〉-∇g(xn)Tm-1〉+M2′∥Tmxn-Tm-1xn∥⋮≤(∑j=1m-1‍aj(x)+1)Dg(Txn,xn)+Dg(Txn,x)+M2′∑j=1m-1‍∥Tmxn-Tm-1xn∥,
where M2′:=sup{∥∇g(Tjxn)∥:j=1,2,…,mandn∈ℕ}. This, together with Lemma 8, implies that
(53)limsupn→∞Dg(Tmxn,x)≤limsupn→∞Dg(xn,x)=ϕ(x).
In view of (25), there exists a finite constant M3>0 such that
(54)∑j=1m-1‍sup{aj(x):x∈C}≤M3,
where M4:=sup{∥∇g(Tm-1xn)∥:m=1,2,…,mandn∈ℕ}. This, together with Lemma 8, implies that limn→∞∥Tmxn-xn∥=0. Employing Lemma 8, we conclude that
(55)ϕ(x)=limsupn→∞Dg(xn,x)≤limsupn→∞Dg(xn,Tmxn)+limsupn→∞Dg(Tmxn,x)+limsupn→∞M2′∥xn-Tmxn∥≤limsupn→∞Dg(Tmxn,x)≤ϕ(x).
This means that
(56)ϕ(x)=limsupn→∞Dg(Tmxn,x).
Since T is a Bregman asymptotic pointwise nonexpansive mapping, it follows that
(57)ϕ(Tmx)≤am(xn)ϕ(x),∀x∈C.
In view of (57), we deduce that
(58)limsupm→∞ϕ(Tmz)≤ϕ(z).
By the Bregman Opial-like inequality ((34)) we obtain that for any x≠z(59)ϕ(z)=limsupn→∞Dg(xn,z)<limsupn→∞Dg(xn,x)=ϕ(x).
This shows that ϕ(z)=inf{ϕ(x):x∈C}. Thus we have
(60)ϕ(Tmz)=ϕ(z).
Put s2=sup{∥∇g(z)∥,∥∇g(Tmz)∥:m∈ℕ}, zm=∇g*((1/2)∇g(z)+(1/2)∇g(Tmz)), and um=projCg(zm) for all m∈ℕ. Then we have um∈C. In view of Remark 1, we obtain a continuous strictly increasing convex function ρs2*:[0,+∞)→[0,+∞) with ρs2*(0)=0 such that
(61)Dg(xn,um)=Dg(xn,projCg(zm))≤Dg(xn,zm)=g(xn)-g(zm)-〈xn-zm,∇g(zm)〉=g(xn)+g*(zm)-〈zm,∇g(zm)〉-〈xn,∇g(zm)〉+〈zm,∇g(zm)〉=g(xn)+g*(12∇g(z)+12∇g(Tmz))-〈xn,12∇g(z)+12∇g(Tmz)〉≤12g(xn)+12g(xn)+12g*(∇g(xn))+12g*(∇g(Tmz))-14ρs2*(∥∇g(z)-∇g(Tmz)∥)-12〈xn,∇g(xn)〉-12〈xn,∇g(Tmz)〉=12[g(xn)+g*(∇g(z))-〈xn,∇g(xn)〉]+12[g(xn)+g*(∇g(Tmz))-〈xn,∇g(Tmz)〉]-14ρs2*(∥∇g(z)-∇g(Tmz)∥)=12[g(xn)-g(z)+〈xn,∇g(z)〉-〈xn,∇g(z)〉]+12[g(xn)-g(Tmz)+〈Tmz,∇g(Tmz)〉-〈xn,∇g(Tmz)〉]-14ρs2*(∥∇g(z)-∇g(Tmz)∥)=12Dg(xn,z)+12Dg(xn,Tmz)-14ρs2*(∥∇g(z)-∇g(Tmz)∥).
Applying to both sides limsupn→∞ and remembering that ϕ(z)=inf{ϕ(x):x∈C} we obtain
(62)ϕ(z)≤12ϕ(z)+12ϕ(Tmz)-14ρs2*(∥∇g(z)-∇g(Tmz)∥).
This implies that
(63)ρs2*(∥∇g(z)-∇g(Tmz)∥)≤2ϕ(Tmz)-2ϕ(z).
Letting m→∞ in (63) we conclude that
(64)limm→∞ρs2*(∥∇g(z)-∇g(Tmz)∥)=0.
From the properties of ρs2*, we deduce that limm→∞∥∇g(z)-∇g(Tmz)∥. Since ∇g* is uniformly norm-to-norm continuous on bounded subsets of E*, we arrive at limm→∞∥z-Tmz∥=0. Thus we have Tm+1z→z; that is, T(Tmz)→z. On the other hand, in view of three-point identity (see (26)), we deduce that
(65)Dg(z,Tz)=Dg(z,Tm+1z)+Dg(Tm+1z,Tz)+〈z-Tm+1z,∇g(Tm+1z)-∇g(Tz)〉≤Dg(z,Tm+1z)+a1(Tmz)Dg(Tmz,z)+∥z-Tm+1z∥∥∇g(Tm+1z)-∇g(Tz)∥.
Letting m→∞ in the above inequalities we deduce that Dg(z,Tz)=0 and hence by Lemma 8 we conclude that Tz=z, which completes the proof.


## 4. Weak Convergence Theorems of Generalized Mann Iteration Process

In this section, we prove weak convergence theorems of generalized Mann iteration process in a reflexive Banach space.Definition 16.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E and let T∈ℬ𝒯r(C). Let {nk}k∈ℕ be an increasing sequence in ℕ and let {γk}k∈ℕ⊂(0,1) be such that liminfk→γk(1-γk)>0. The generalized Mann iteration process generated by the mapping T, the sequence {γk}k∈ℕ, and the sequence {nk}k∈ℕ, denoted by gM(T,{γk}k∈ℕ,{nk}k∈ℕ), is defined by the following iterative formula:
(66)xk+1=γkTnk(xk)+(1-γk)xk,wherex1∈Cischosenarbitrarily.Definition 17.
We say that a generalized Mann iteration processgM(T,{γk}k∈ℕ,{nk}k∈ℕ) is well defined if
(67)limsupk→∞ank(xk)=1.Remark 18.
Observe that by the definition of Bregman asymptotic pointwise nonexpansive,limk→∞ank(x)=1 for every x∈C. Hence we can always select a subsequence {ank}k∈ℕ of {an}n∈ℕ such that (67) holds. In other words, by a suitable choice of {nk}k∈ℕ we can always make gM(T,{γk}k∈ℕ,{nk}k∈ℕ) well defined.We will prove a series of lemmas necessary for the proof of the generalized Mann process convergence theorem.Lemma 19.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {γk}k∈ℕ⊂(0,1) such that liminfk→∞γk(1-γk)>0. Let w∈F(T) and let gM(T,{γk}k∈ℕ,{nk}k∈ℕ) be a generalized Mann process. Then there exists λ∈ℝ such that limk→∞D(xk,w)=λ.Proof.
Letw∈F(T) be arbitrary chosen. In view of (66), we obtain
(68)Dg(xk+1,w)≤γkDg(Tnk(xk),w)+(1-γk)Dg(xk,w)≤γkDg(Tnk(xk),Tnk(w))+(1-γk)Dg(xk,w)≤γk(1+bnk(xk))Dg(xk,w)+(1-γk)Dg(xk,w)≤γkbnk(xk)Dg(xk,w)+Dg(xk,w)≤bnk(xk)Bdiam(C)+Dg(xk,w).
This implies that for every n∈ℕ,
(69)Dg(xk+n,w)≤Dg(xk,w)+Bdiam(C)∑i=kk+n-1‍bni(xi).
Put rp=Dg(xp,w) for every p∈ℕ and dk,n=Bdiam(C)∑i=kk+n-1‍bni(xi). Since T∈ℬ𝒯r(C), we obtain that limsupk→∞limsupn→∞dk,n=0. In view of Lemma 11, there exists λ∈ℝ such that limk→∞Dg(xk,w)=λ. This completes the proof.Lemma 20.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {γk}k∈ℕ⊂(0,1) such that liminfk→∞γk(1-γk)>0 and let gM(T,{γk}k∈ℕ,{nk}k∈ℕ) be a generalized Mann process. Then
(70)limk→∞Dg(Tnk(xk),xk)=0,limk→∞Dg(xk+1,xk)=0.Proof.
In view of Proposition13, we conclude that F(T)≠∅. Let w∈F(T) be fixed. It follows from Lemma 19 that there exists λ∈ℝ such that limk→∞Dg(xk,w)=λ. Let s3=sup{∥Tnk(xk)∥,∥xk∥:k∈ℕ} and let ρs3:E→ℝ be the gauge of uniform convexity of g. By the definition of T, we obtain
(71)Dg(xk+1,w)=Dg(γkTnk(xk)+(1-γk)xk,w)≤γkDg(Tnk(xk),w)+(1-γk)Dg(xk,w)=γkDg(Tnk(xk),Tnk(w))+(1-γk)Dg(xk,w)≤γkank(xk)Dg(xk,w)+(1-γk)Dg(xk,w)-γk(1-γk)ρs3(∥Tnk(xk)-xk∥)=γk(ank(xk)-1)Dg(xk,w)+Dg(xk,w)-γk(1-γk)ρs3(∥Tnk(xk)-xk∥).
This implies that
(72)γk(1-γk)ρs3(∥Tnk(xk)-xk∥)≤(ank(xk)-1)Dg(xk,w)+Dg(xk,w)-Dg(xk+1,w).
Letting k→∞ in (72) we conclude that
(73)limk→∞ρs3(∥Tnk(xk)-xk∥)=0.
From the properties of ρs3, we deduce that limk→∞∥Tnk(xk)-xk∥=0. Employing Lemma 8, we conclude that
(74)limk→∞Dg(Tnk(xk),xk)=0,limk→∞Dg(xk+1,xk)=0.
This completes the proof.In the next lemma, we prove that under suitable assumption the sequence{xk}k∈ℕ becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to recall the following notions.Definition 21.
A strictly increasing sequence{ni}i∈ℕ⊂ℕ is called quasiperiodic if the sequence {ni+1-ni}i∈ℕ is bounded or equivalently if there exists a number p∈ℕ such that any block of consecutive natural numbers must contain a term of the sequence {ni}i∈ℕ. The smallest of such numbers will be called a quasi period of {ni}i∈ℕ.Lemma 22.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {γk}k∈ℕ⊂(0,1) such that liminfk→∞γk(1-γk)>0. Let {nk}k∈ℕ⊂ℕ be such that the generalized Mann process gM(T,{γk}k∈ℕ,{nk}k∈ℕ) is well defined. If, in addition, the set of indices 𝒥={j∈ℕ:nj+1=1+nj} is quasi-periodic, then {xk}k∈ℕ is an approximate fixed point sequence; that is,
(75)limk→∞∥T(xk)-xk∥=0.Proof.
In view of (66), we have
(76)xk+1-xk=γk(Tnk(xk)-xk).
This, together with Lemmas 8 and 20, implies that
(77)limk→∞∥xk+1-xk∥=0.
In view of Lemma 8, we conclude that
(78)limm→∞Dg(xk+1,xk)=0.
Let p∈ℕ be a quasi-period of 𝒥. We first prove that ∥xk-T(xk)∥→0 as k→∞ through 𝒥. Since nk+1=nk+1 for such k, we obtain
(79)Dg(Tnk+1(xk+1),Tnk+1(xk))=Dg(Tnk+1(xk+1),Tnk+1(xk))≤ank+1(xk+1)Dg(xk+1,xk),Dg(TTnk(xk),T(xk))≤M5Dg(Tnk(xk),xk),
where M5=sup{a1(x):x∈C}<∞. This implies that
(80)limk→∞Dg(Tnk+1(xk+1),Tnk+1(xk))=0,limk→∞Dg(TTnk(xk),T(xk))=0.
It follows from Lemma 8 that
(81)limk→∞∥Tnk+1(xk+1)-Tnk+1(xk)∥=0,limk→∞∥TTnk(xk)-T(xk)∥=0.
On the other hand, we have
(82)∥xk-T(xk)∥≤∥xk-xk+1∥+∥xk+1-Tnk+1(xk+1)∥+∥Tnk+1(xk+1)-Tnk+1(xk)∥+∥TTnk(xk)-T(xk)∥.
Thus, we obtain ∥xk-T(xk)∥→0 as k→∞ through 𝒥. In view of Lemma 8, we conclude that Dg(xk,T(xk))→0 as k→∞ through 𝒥. Now, let ϵ>0 be fixed. It follows from Dg(xk,T(xk))→0 as k→∞ through 𝒥 that there exists N0∈ℕ such that
(83)Dg(xk,T(xk))<ϵ3,∀k≥N0.
Since 𝒥 is quasi-periodic, for any k∈ℕ there exists jk∈𝒥 such that |k-jk|≤p. Assume that k-p≤jk≤k (the proof of the other case is identical). Since T is a Bregman M6-Lipschitzian mapping where M6=sup{a1(x):x∈C}, there exists 0<δ<ϵ/3 such that
(84)Dg(Tx,Ty)<ϵ3ifDg(x,y)<δ.
In view of (66) and (83), there exists N1∈ℕ such that
(85)∥xk+1-xk∥<δp,∀k≥N1.
We also obtain
(86)∥xk-xjk∥≤∥xk-xk-1∥+∥xk-1-xk-2∥+⋯+∥xjk-1-xjk∥≤pδp=δ.
This implies that
(87)limk→∞Dg(xk,xjk)=0,limk→∞Dg(T(xk),T(xjk))=0.
If follows from (77)-(78) that
(88)∥xk-Txk∥≤∥xk-xjk∥+∥xjk-T(xjk)∥+∥T(xjk)-T(xk)∥≤δ+ϵ3+ϵ3<ϵ.
This completes the proof.Theorem 23.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {γk}k∈ℕ⊂(0,1) such that liminfk→∞γk(1-γk)>0. Let {nk}k∈ℕ⊂ℕ be such that the generalized Mann process gM(T,{γk}k∈ℕ,{nk}k∈ℕ) is well defined. If, in addition, the set of indices 𝒥={j∈ℕ:kj+1=1+kj} is quasi-periodic, then the sequence {xk}k∈ℕ generated by gM(T,{γk}k∈ℕ,{nk}k∈ℕ) converges weakly to a fixed point of T.Proof.
In view of Lemma22, we obtain
(89)limk→∞∥Txk-xk∥=0.
Let y,z∈C be weak cluster points of the sequence {xk}k∈ℕ. Then there exist subsequences {yk}k∈ℕ and {zk}k∈ℕ of {xk}k∈ℕ such that yk⇀y and zk⇀z as k→∞. In view of (89) and Theorem 15, we conclude that Ty=y and Tz=z. It follows from Lemma 19 that there exist real numbers λ1 and λ2 such that
(90)limk→∞Dg(yk,y)=λ1,limk→∞Dg(zk,z)=λ2.
We claim that y=z. Assume on the contrary that y≠z. By the Bregman Opial-like property we obtain
(91)λ1=limsupk→∞Dg(yk,y)<limsupk→∞Dg(yk,z)=limsupk→∞Dg(zk,z)<limsupk→∞Dg(zk,y)=limsupk→∞Dg(yk,y)=λ1.
This is a contradiction and hence there exists w∈C such that xk⇀w as k→∞. Since C is weakly sequentially compact, such a weak cluster point w is unique. In view of Theorem 15, we conclude that T(w)=w, which completes the proof.


## 5. Weak Convergence of Generalized Ishikawa Iteration Process

The two-step Ishikawa iteration process is a generalization of the one-step Mann iteration process. The Ishikawa iteration process provides more flexibility in defining the algorithm parameters which is important from numerical implementation perspective.Definition 24.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ be an increasing sequence. Let {βk}k∈ℕ⊂(0,1) and {γk}k∈ℕ⊂(0,1) be sequences of real numbers such that liminfk→∞βk(1-βk)>0 and liminfk→∞γk(1-γk)>0. The generalized Ishikawa iteration process generated by the mapping T, the sequences {βk}k∈ℕ⊂(0,1), and {γk}k∈ℕ⊂(0,1), and the sequence {nk}k∈ℕ denoted by gI(T,{βk}k∈ℕ,{γk}k∈ℕ,{nk}k∈ℕ) is defined by the following iterative scheme:
(92)x1∈Cchosenarbitrarily,yk=βkTnk(xk)+(1-βk)xk,xk+1=γkTnk(yk)+(1-γk)xk.Definition 25.
We say that a generalized Ishikawa iteration processgI(T,{βk}k∈ℕ,{γk}k∈ℕ,{nk}k∈ℕ) is well defined if
(93)limsupk→∞ank(xk)=limsupk→∞ank(yk)=1.Lemma 26.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {βk}k∈ℕ⊂(0,1) and {γk}k∈ℕ⊂(0,1) be sequences of real numbers such that liminfk→∞βk(1-βk)>0 and liminfk→∞γk(1-γk)>0. Let w∈F(T) and gI(T,{βk}k∈ℕ,{γk}k∈ℕ,{nk}k∈ℕ) be a generalized Ishikawa iteration process. Then there exists θ∈ℝ such that limk→∞Dg(xk,w)=θ.Proof.
LetM7>1 be fixed. Since limk→∞ank(yk)=1, there exists k0∈ℕ such that for any k>k0, ank(yk)≤M7. Let s4=sup{∥Tnk(xk)∥,∥xk∥:k∈ℕ} and let ρs4:E→ℝ be the gauge of uniform convexity of g. By the definition of T and in view of (93), we obtain
(94)Dg(xk+1,w)=Dg(γkTnk(yk)+(1-γk)xk,w)≤γkDg(Tnk(yk),w)+(1-γk)Dg(xk,w)=γkDg(Tnk(yk),Tnk(w))+(1-γk)Dg(xk,w)≤γkank(yk)Dg(yk,w)+(1-γk)Dg(xk,w)≤γkank(yk)[βkDg(Tnk(xk),w)≤γkank(yk)+(1-βk)Dg(xk,w)≤γkank(yk)-ρs4(∥Tnk(xk)-xk∥)]+(1-γk)Dg(xk,w)=γkank(w)[βkDg(Tnk(xk),Tnk(w))=γkank(w)+(1-βk)Dg(xk,w)=γkank(w)-ρs4(∥Tnk(xk)-xk∥)]+(1-γk)Dg(xk,w)≤γkank(yk)[βkank(xk)Dg(xk,w)≤γkank((yk)+(1-βk)Dg(xk,w)≤γkank((yk)-ρs4(∥Tnk(xk)-xk∥)]+(1-γk)Dg(xk,w)=γkank(yk)[βk(ank(xk)-1)Dg(xk,w)=γkank(yk)[+βkDg(xk,w)=γkank(yk)[+(1-βk)Dg(xk,w)=γkank(yk)[-ρs4(∥Tnk(xk)-xk∥)]+(1-γk)Dg(xk,w)=γkβkank(yk)(ank(xk)-1)Dg(xk,w)+γkank(yk)Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥)+(1-γk)Dg(xk,w)≤γkank(yk)(ank(xk)-1)Dg(xk,w)+γkank(yk)Dg(xk,w)+(1-γk)Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥)=γkank(yk)ank(xk)Dg(xk,w)+(1-γk)Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥)=γkank(yk)ank(xk)Dg(xk,w)-Dg(xk,w)+γkDg(xk,w)+(1-γk)Dg(xk,w)-γkβkρs4(∥Tnk(xk)-xk∥)≤[ank(yk)ank(xk)-1]Dg(xk,w)+Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥)=ank(yk)ank(xk)Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥)=ank(yk)[ank(xk)-1]Dg(xk,w)+[ank(xk)-1]Dg(xk,w)+Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥)≤M7bnk(xk)Bdiam(C)+Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥).
This implies that for every n∈ℕ,
(95)Dg(xk+n,w)≤Dg(xk,w)+M7Bdiam(C)∑i=kk+n-1‍bni(xi).
Put rp=Dg(xp,w) for every p∈ℕ and dk,n=M7Bdiam(C)∑i=kk+n-1‍bni(xi). Since T∈ℬ𝒯r(C), we obtain that limsupk→∞limsupn→∞dk,n=0. In view of Lemma 11, there exists θ∈ℝ such that limk→∞Dg(xk,w)=θ. Let M8>1 be fixed. Since limk→∞ank(xk)=1, there exists k0∈ℕ such that for any k>k0, ank(xk)≤M8. Therefore, by the same argument, as in the proof of Lemma 19, we conclude that for k>k0 and n>1(96)Dg(xk+n,w)≤Dg(xk,w)+M7Bdiam(C)∑i=kk+n-1‍bni(xi)≤Dg(xk,w)+M7Bdiam(C)∑i=kk+n-1‍bni(xi).
By the same manner as in the proof of Lemma 19, we deduce that there exists θ∈ℝ such that limk→∞Dg(xk,w)=θ, which completes the proof.Lemma 27.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {βk}k∈ℕ⊂(0,1) and {γk}k∈ℕ⊂(0,1) be sequences of real numbers such that liminfk→∞γk(1-γk)>0 and liminfk→∞βk(1-βk)>0. Let w∈F(T) and gI(T,{γk}k∈ℕ,{βk}k∈ℕ,{nk}k∈ℕ) be a generalized Ishikawa iteration process. Then
(97)limk→∞D(Tnk(yk),xk)=0,limk→∞Dg(xk+1,xk)=0.Proof.
In view of Proposition13, F(T)≠∅. Take any w∈F(T) arbitrarily chosen. Then, by Lemma 26, there exists θ∈ℝ such that limk→∞Dg(xk,w)=θ. By the same arguments, as in the proof of Lemma 26, we conclude that
(98)Dg(xk+1,w)≤M7bnk(xk)Bdiam(C)+Dg(xk,w)-γkρs4(∥Tnk(xk)-xk∥).
This implies that
(99)γkρs4(∥Tnk(xk)-xk∥)≤M7bnk(xk)Bdiam(C)+Dg(xk,w)-Dg(xk+1,w).
This implies that
(100)limk→∞ρs4(∥Tnk(xk)-xk∥)=0.
Therefore, from the property of ρs4 we deduce that
(101)limk→∞∥Tnk(xk)-xk∥=0.
In a similar way, as in the proof of Lemma 20, we can prove that
(102)limk→∞∥T(xk)-xk∥=0.
This completes the proof.Lemma 28.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {γk}k∈ℕ⊂(0,1) such that liminfk→∞γk(1-γk)>0. Let {nk}k∈ℕ⊂ℕ be such that the generalized Ishikawa process gI(T,{γk}k∈ℕ,{βk}k∈ℕ,{nk}k∈ℕ) is well defined. If, in addition, the set of indices 𝒥={j∈ℕ:nj+1=1+nj} is quasi-periodic, then {xk}k∈ℕ is an approximate fixed point sequence; that is,
(103)limk→∞∥T(xk)-xk∥=0.Theorem 29.
LetE be a reflexive Banach space and let g:E→ℝ be a convex, continuous, strongly coercive, and Gâteaux differentiable function which is bounded on bounded subsets and uniformly convex on bounded subsets. Let C be a nonempty, bounded, closed, and convex subset of E. Let T∈ℬ𝒯r(C) and let {nk}k∈ℕ⊂ℕ. Let {γk}k∈ℕ⊂(0,1) such that liminfk→∞γk(1-γk)>0. Let {nk}k∈ℕ⊂ℕ be such that the generalized Ishikawa process gI(T,{γk}k∈ℕ,{βk}k∈ℕ,{nk}k∈ℕ) is well defined. If, in addition, the set of indices 𝒥={j∈ℕ:kj+1=1+kj} is quasi-periodic, then the sequence {nk}k∈ℕ generated by gI(T,{γk}k∈ℕ,{βk}k∈ℕ,{nk}k∈ℕ) converges weakly to a fixed point of T.Remark 30.
Theorem29 improves Theorems 3.1 and 4.1 of [18] in the following aspects.(1)
For the structure of Banach spaces, we extend the duality mapping to more general case, that is, a convex, continuous, strongly coercive Bregman function which is bounded on bounded sets and uniformly convex and uniformly smooth on bounded sets.(2)
For the mappings, we extend the mapping from an asymptotic pointwise nonexpansive mapping to a Bregman asymptotic pointwise nonexpansive mapping.(3)
Since we do not need the weak sequential continuity of the duality mapping in Theorems23 and 29 as was the case in [18], we can apply Theorem 29 in the Lebesgue space Lp where 1<p<∞ and p≠2 while this space is not applicable for Theorems 3.1, 4.1, and 5.1 of [18].



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*Source: 316813-2013-12-25.xml*