# Convergence Theorems for a Maximal Monotone Operator and aV-Strongly Nonexpansive Mapping in a Banach Space

Authors: Hiroko Manaka
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

LetE be a smooth Banach space with a norm ∥·∥. Let V(x,y)=∥x∥2+∥y∥2−2〈x,Jy〉 for any x,y∈E, where 〈·,·〉 stands for the duality
pair and J is the normalized duality mapping. With respect to this bifunction
V(·,·), a generalized nonexpansive mapping and a V-strongly nonexpansive
mapping are defined in E. In this paper, using the properties of generalized
nonexpansive mappings, we prove convergence theorems for common zero
points of a maximal monotone operator and a V-strongly nonexpansive mapping.


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## Body

## 1. Introduction

LetE be a smooth Banach space with a norm ∥·∥ and let C be a nonempty, closed and convex subset of E. We use the following bifunction V(·,·) studied by Alber [1], as well as Kamimura and Takahashi [2]. Let V(·,·):E×E→[0,∞) be defined by V(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2 for any x,y∈E, where 〈·,·〉 stands for the duality pair and J is the normalized duality mapping. Note that the duality mapping is single valued in a smooth Banach space (see [3]). From the definition of V(·,·) the following properties are trivial.Lemma 1.1.
(a) For alla,b,c∈E,
(1.1)V(a,b)≤V(a,b)+V(b,c)=V(a,c)-2〈a-b,Jb-Jc〉.
(b) If a sequence{xn}⊂E satisfies limn→∞V(xn,p)<∞ for some p∈E, then {xn} is bounded.LetF(T) be the fixed points set of T. Ibaraki and Takahashi defined a generalized nonexpansive mapping in a Banach space (see [4]).Definition 1.2.
A mappingT:C→C is said to be generalized nonexpansive if F(T)≠∅ and V(Tx,p)≤V(x,p) for all x∈C and p∈F(T).In this paper, we prove strong convergence theorem for finding common fixed points of a family of generalized nonexpansive mappings. In addition, we prove strong convergence theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone operator. Now, we define aV-strongly nonexpansive mapping as follows.Definition 1.3.
A mappingT:C→E  is called V-strongly nonexpansive if there exists a constant λ>0 such that
(1.2)V(Tx,Ty)≤V(x,y)-λV((I-T)x,(I-T)y),
for all x,y∈C, where I is the identity mapping on E. More explicitly, if (1.2) holds, then T is said to be V-strongly nonexpansive with λ.IfT is V-strongly nonexpansive with λ, then T is V-strongly nonexpansive with any γ∈(0,λ]. It is trivial that a V-strongly nonexpansive mapping is generalized nonexpansive if F(T)≠∅. In the following section, we show that in a Hilbert space H a firmly nonexpansive mapping is V-strongly nonexpansive with λ=1 and a V-strongly nonexpansive mapping is strongly nonexpansive if F(T)≠∅. Motivated by the results of Manaka and Takahashi [5], we prove weak convergence theorem for common zero points of a maximal monotone operator and a V-strongly nonexpansive mapping in a Banach space.


## 2. Preliminaries

LetD be a nonempty subset of a Banach space E. A mapping R:E→D is said to be sunny, if for all x∈E and t≥0,
(2.1)R(Rx+t(x-Rx))=Rx.
A mapping R:E→D is called a retraction if Rx=x for all x∈D (see [6]). It is known that a generalized nonexpansive and sunny retraction of E onto D is uniquely determined if E is a smooth and strictly convex Banach space (cf., [7]). Ibaraki and Takahashi proved the following results in [4].Lemma 2.1 (cf., [4]).
LetE be a reflexive, strictly convex, and smooth Banach space, and let T be a generalized nonexpansive mapping from E into itself. Then there exists a sunny and generalized nonexpansive retraction on F(T).Lemma 2.2 (cf., [4]).
LetD be a nonempty subset of a reflexive, strictly convex, and smooth Banach space E. Let R be a retraction from E onto D. Then R is sunny and generalized nonexpansive if and only if
(2.2)〈x-Rx,JRx-Jy〉≥0,
for all x∈E and y∈D.A generalized resolventJr of a maximal monotone operator B⊂E*×E is defined by Jr=(I+rBJ)-1 for any real number r>0. It is well known that Jr:E→E is single valued if E is reflexive, smooth, and strictly convex (see [8]). It is also known that Jr satisfies
(2.3)〈x-Jrx-(y-Jry),JJrx-JJry〉≥0,∀x,y∈E.
This implies that
(2.4)〈x-Jrx,JJrx-Jp〉≥0,∀x∈E,p∈F(Jr).
Therefore, from Lemma 1.1(a), we obtain the following proposition.Proposition 2.3.
(a)
If a sunny retraction R is generalized nonexpansive, then R satisfies
(2.5)V(x,Rx)+V(Rx,y)=V(x,y)-2〈x-Rx,JRx-Jy〉≤V(x,y),∀x,y∈D.
(b)
For each r>0, a generalized resolvent Jr satisfies
(2.6)V(x,Jrx)+V(Jrx,p)≤V(x,p),∀x∈E,p∈F(Jr).Remark 2.4.
The property in Proposition2.3(b) means that Jr is generalized nonexpansive for any r>0.
We recall some nonlinear mappings in Banach spaces (see, e.g., [9–12]).Definition 2.5.
LetD be a nonempty, closed, and convex subset of E.A mapping T:D→E is said to be firmly nonexpansive if
(2.7)∥Tx-Ty∥2≤〈x-y,j(Tx-Ty)〉,
for all x,y∈D and some j(Tx-Ty)∈J(Tx-Ty).In [12], Reich introduced a class of strongly nonexpansive mappings which is defined with respect to the Bregman distance D(·,·) corresponding to a convex continuous function f in a reflexive Banach space E. Let S be a convex subset of E, and let T:S→S be a self-mapping of S. A point p in the closure of S is said to be an asymptotically fixed point of T if S contains a sequence {xn} which converges weakly to p and the sequence {xn-Txn} converges strongly to 0. F̂(T) denotes the asymptotically fixed points set of T.Definition 2.6.
The Bregman distance corresponding to a functionf:E→R is defined by
(2.8)D(x,y)=f(x)-f(y)-f′(y)(x-y),
where f is the Gâteaux differentiable and f′(x) stands for the derivative of f at the point x. We say that the mapping T is strongly nonexpansive if F̂(T)≠∅ and
(2.9)D(p,Tx)≤D(p,x),∀p∈F̂(T),x∈S,
and if it holds that limn→∞D(Txn,xn)=0 for a bounded sequence {xn} such that limn→∞(D(p,xn)-D(p,Txn))=0 for any p∈F̂(T).We remark that the symbolsxn→u and xn⇀u mean that {xn} converges strongly and weakly to u, respectively. Taking the function ∥·∥2 as the convex, continuous, and Gâteaux differentiable function f, we obtain the fact that the Bregman distance D(·,·) coincides with V(·,·). Especially in a Hilbert space, D(x,y)=V(x,y)=∥x-y∥2. Bruck and Reich defined strongly nonexpansive mappings in a Hilbert space H as follows (cf., [10]).Definition 2.7.
A mappingT:D→H is said to be strongly nonexpansive if T is nonexpansive with F(T)≠∅ and if it holds that
(2.10)(xn-yn)-(Txn-Tyn)→0
when {xn} and {yn} are sequences in D such that {xn-yn} is bounded and limn→∞(∥xn-yn∥-∥Txn-Tyn∥)=0.The relation among firmly nonexpansive mappings, strongly nonexpansive mappings andV-strongly nonexpansive mappings is shown in the following proposition.Proposition 2.8.
In a Hilbert spaceH, the following hold.(a)
A firmly nonexpansive mapping isV-strongly nonexpansive with λ=1.(b)
AV-strongly nonexpansive mapping T with F̂(T)≠∅ is strongly nonexpansive.Proof.
(a) Suppose thatT is firmly nonexpansive. Since J=I in a Hilbert space, it holds that
(2.11)2〈x-y,Tx-Ty〉-∥Tx-Ty∥2=∥x-y∥2-∥(I-T)x-(I-T)y∥2,
for all x,y∈D. Therefore, it is obvious that T is firmly nonexpansive if and only if T satisfies
(2.12)∥x-y∥2-∥(T-I)x-(T-I)y∥2≥∥Tx-Ty∥2,
for all x,y∈D. Hence we obtain (a).
(b) Suppose thatT is V-strongly nonexpansive with λ. Then, it is trivial that T is nonexpansive and (2.9) holds. Suppose that the sequences {xn} and {yn} satisfy the conditions in Definition 2.7. Then {Txn-Tyn} is also bounded. Since T is V-strongly nonexpansive with λ, we have that
(2.13)0≤λ∥xn-yn-(Txn-Tyn)∥2=λV((I-T)xn,(I-T)yn)≤V(xn,yn)-V(Txn,Tyn)=∥xn-yn∥2-∥Txn-Tyn∥2=(∥xn-yn∥+∥Txn-Tyn∥)(∥xn-yn∥-∥Txn-Tyn∥)→0.
Hence, (xn-yn)-(Txn-Tyn)→0 for λ>0. This means that T is strongly nonexpansive.In a Banach space,V-strongly nonexpansive mappings have the following properties.Proposition 2.9.
In a smooth Banach spaceE, the following hold.(a)
Forc∈(-1,1], T=cI is V-strongly nonexpansive. For c=1, T=I is V-strongly nonexpansive for any λ>0. For c∈(-1,1), T=cI is V-strongly nonexpansive for any λ∈(0,(1+c)/(1-c)].(b)
IfT is V-strongly nonexpansive with λ, then, for any α∈[-1,1] with α≠0, αT is also V-strongly nonexpansive with α2λ.(c)
IfT is V-strongly nonexpansive with λ≥1, then A=I-T is V-strongly nonexpansive with λ-1.(d)
Suppose thatT is V-strongly nonexpansive with λ and that α∈[-1,1] satisfies α2λ≥1. Then (I-αT) is V-strongly nonexpansive with (α2λ)-1. Moreover, if Tα=I-αT, then
(2.14)V(Tαx,Tαy)≤V(x,y)-λ-1V(Tx,Ty).Proof.
(a) LetT=cI for any c∈(-1,1], and denote Il=V(Tx,Ty) and Ir=V(x,y)-λV((I-T)x,(I-T)y). Since J(cx)=cJx, we have
(2.15)Il=V(Tx,Ty)=c2{∥x∥2+∥y∥2-2〈x,Jy〉}=c2V(x,y),Ir=V(x,y)-λV((I-T)x,(I-T)y)=∥x∥2-λ∥(1-c)x∥2+∥y∥2-λ∥(1-c)y∥2-2〈x,Jy〉+2λ〈(1-c)x,J((1-c)y)〉={1-λ(1-c)2}(∥x∥2+∥y∥2)-2{1-λ(1-c)2}〈x,Jy〉={1-λ(1-c)2}(∥x∥2+∥y∥2-2〈x,Jy〉)={1-λ(1-c)2}V(x,y).
For c=1, it holds that Il≤Ir for all λ>0. For c∈(-1,1), we obtain
(2.16)Il≤Ir⇔c2≤1-λ(1-c)2⇔0<λ(1-c)2≤1-c2⇔0<λ≤(1-c)(1+c)(1-c)2=(1+c)(1-c).
Therefore, T=cI is V-strongly nonexpansive for any λ∈(0,(1+c)/(1-c)].
(b) IfT is V-strongly nonexpansive with λ>0, then, for α∈[-1,1] with α≠0,
(2.17)V(αTx,αTy)=∥αTx∥2+∥αTy∥2-2〈αTx,J(αTy)〉=α2{∥Tx∥2+∥Ty∥2-2〈Tx,J(Ty)〉}=α2V(Tx,Ty)≤α2{V(x,y)-λV((I-T)x,(I-T)y)}=V(x,y)-(1-α2)V(x,y)-α2λV((I-T)x,(I-T)y)≤V(x,y)-α2λV((I-T)x,(I-T)y).
This means that αT is V-strongly nonexpansive with α2λ.
(c) Suppose thatT is V-strongly nonexpansive with λ≥1 and let A=I-T. Then we have that
(2.18)V((I-A)x,(I-A)y)=V(Tx,Ty)≤V(x,y)-λV((I-T)x,(I-T)y)=V(x,y)-λV(Ax,Ay).
This inequality implies that
(2.19)V(Ax,Ay)≤λ-1{V(x,y)-V((I-A)x,(I-A)y)}=λ-1V(x,y)-λ-1V((I-A)x,(I-A)y)≤V(x,y)-λ-1V((I-A)x,(I-A)y).
Thus A is V-strongly nonexpansive with λ-1.
(d) From (b) and the assumption,αT is V-strongly nonexpansive with α2λ≥1, and from (c) we have that (I-αT) is V-strongly nonexpansive with (α2λ)-1. Furthermore we obtain that
(2.20)V(Tαx,Tαy)≤V(x,y)-(α2λ)-1V((I-Tα)x,(I-Tα)y)=V(x,y)-(α2λ)-1V(αTx,αTy)=V(x,y)-(α2λ)-1α2V(Tx,Ty)=V(x,y)-λ-1V(Tx,Ty).
This completes the proof.In Banach spaces, we have the following example ofV-strongly nonexpansive mappings.Example 2.10.
LetE=R×R  be a Banach space with a norm ∥·∥ defined by
(2.21)∥x∥=|x1|+|x2|,∀x=(x1,x2)∈E.
The normalized duality mapping J is given by
(2.22)Jx=∥x∥(1|x1|x1,1|x2|x2),∀x∈E.
Hence, we have for x,y∈E that
(2.23)V(x,y)=∥x∥2+∥y∥2-2〈x,Jy〉=∥x∥2+∥y∥2-2∥y∥{x1y1|y1|+x2y2|y2|}.
We define a mapping T:E→E as follows:
(2.24)Tx={xif∥x∥≤1,1∥x∥xif∥x∥≥1.
We will show that this mapping is V-strongly nonexpansive for any λ≤1.
(a) Suppose thatx,y∈E with ∥x∥≤1 and ∥y∥≥1. Then, we have
(2.25)V(Tx,Ty)=V(x,Ty)=∥x∥2+∥Ty∥2-2∥Ty∥{x1(Ty)1|(Ty)1|+x2(Ty)2|(Ty)2|}=∥x∥2+1-2{x1y1|y1|+x2y2|y2|}.
Since
(2.26)y-Ty=(∥y∥-1∥y∥y1,∥y∥-1∥y∥y2),
we have that
(2.27)V(x-Tx,y-Ty)=V(0,y-Ty)=∥y-Ty∥2={(∥y∥-1)∥y∥(|y1|+|y2|)}2=(∥y∥-1)2.
Hence, we obtain that
(2.28)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)=∥x∥2+∥y∥2-2∥y∥{x1y1|y1|+x2y2|y2|}-∥x∥2-1+2{x1y1|y1|+x2y2|y2|}-λ(∥y∥-1)2=∥y∥2-1-2(∥y∥-1){x1y1|y1|+x2y2|y2|}-λ(∥y∥-1)2≥(∥y∥-1)(∥y∥+1)-2(∥y∥-1){|x1y1||y1|+|x2y2||y2|}-λ(∥y∥-1)2=(∥y∥-1){∥y∥+1-2∥x∥-λ∥y∥+λ}≥(∥y∥-1){(1-λ)∥y∥+1-2+λ}=(∥y∥-1){(1-λ)(∥y∥-1)}=(1-λ)(∥y∥-1)2≥0,
for any λ∈[0,1]. This means that T is V-strongly nonexpansive for any λ∈[0,1].
(b) Suppose thatx,y∈E with ∥x∥≥1 and ∥y∥≤1. Then, we have
(2.29)V(Tx,Ty)=V(Tx,y)=1+∥y∥2-2∥y∥∥x∥{x1y1|y1|+x2y2|y2|},V(x-Tx,y-Ty)=V((∥x∥-1)∥x∥x,0)=(∥x∥-1)2.
Hence, we have that
(2.30)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)=∥x∥2+∥y∥2-2∥y∥{x1y1|y1|+x2y2|y2|}-1-∥y∥2+2∥y∥∥x∥{x1y1|y1|+x2y2|y2|}-λ(∥x∥-1)2=∥x∥2-1-2∥y∥(1-1∥x∥){x1y1|y1|+x2y2|y2|}≥(∥x∥-1)(∥x∥+1-λ∥x∥+λ)-2∥y∥(∥x∥-1∥x∥)∥x∥=(∥x∥-1){(1-λ)∥x∥+1+λ-2∥y∥}≥(∥x∥-1){(1-λ)∥x∥+1+λ-2}=(1-λ)(∥x∥-1)2≥0,
for any λ∈[0,1]. This means that T is V-strongly nonexpansive for any λ∈[0,1].
(c) Suppose thatx,y∈E with ∥x∥,∥y∥≥1. Then, we have
(2.31)V(Tx,Ty)=1+1-2∥x∥{x1y1|y1|+x2y2|y2|}=2-2∥x∥{x1y1|y1|+x2y2|y2|},V(x-Tx,y-Ty)=(∥x∥-1)2+(∥y∥-1)2-2(∥y∥-1)(∥x∥-1)∥x∥{x1y1|y1|+x2y2|y2|}.
Hence, we have that
(2.32)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)=∥x∥2+∥y∥2-2∥y∥{x1y1|y1|+x2y2|y2|}-2+2∥x∥{x1y1|y1|+x2y2|y2|}-λ{(∥x∥-1)2+(∥y∥-1)2}+2λ(∥y∥-1)(∥x∥-1)∥x∥{x1y1|y1|+x2y2|y2|}=(∥x∥-1)(∥x∥+1)+(∥y∥-1)(∥y∥+1)-λ(∥x∥-1)2-λ(∥y∥-1)2-2∥x∥{x1y1|y1|+x2y2|y2|}{∥x∥∥y∥-1-λ(∥y∥-1)(∥x∥-1)}=(∥x∥-1){∥x∥+1-λ(∥x∥-1)}+(∥y∥-1){∥y∥+1-λ(∥y∥-1)}-2∥x∥{x1y1|y1|+x2y2|y2|}{∥x∥∥y∥-1-λ(∥x∥-1)(∥y∥-1)}.
Now, we note that
(2.33)∥x∥∥y∥-1-λ(∥x∥-1)(∥y∥-1)≥0,
for any ∥x∥,∥y∥≥1 and for any λ∈[0,1]. Therefore, we obtain that
(2.34)V(x,y)-V(Tx,Ty)-λV(x-Tx,y-Ty)≥(∥x∥-1){∥x∥+1-λ(∥x∥-1)}+(∥y∥-1){∥y∥+1-λ(∥y∥-1)}-2∥x∥∥x∥{∥x∥∥y∥-1-λ(∥x∥-1)(∥y∥-1)}=(1-λ)(∥x∥-∥y∥)2≥0,
for any λ∈[0,1]. This means that T is V-strongly nonexpansive for any λ∈[0,1].
It is clear that if∥x∥,∥y∥≤1 then T is V-strongly nonexpansive; therefore, from (a), (b), and (c), we obtain the conclusion that T is V-strongly nonexpansive with λ≤1.Next, we present some lemmas which are used in the proofs of our theorems. Letℕ be the set of natural numbers.Lemma 2.11.
Let{an} and {tn} be sequences of nonnegative real numbers and satisfy the inequality an+1≤(1-tn)an+tnM for any n∈ℕ and a constant M>0. If ∑n∈ℕtn<∞, then limn→∞an exists.Kamimura and Takahashi showed the following useful lemmas (see [2]).Lemma 2.12.
LetE be a smooth and uniformly convex Banach space. Then, there exists a continuous, strictly increasing, and convex function g:[0,∞)→[0,∞) such that g(0)=0, and for each real number r>0,
(2.35)0≤g(∥x-y∥)≤V(x,y),
for all x,y∈Br={z∈E:∥z∥≤r}.From this lemma, it is obvious that the following lemma holds.Lemma 2.13.
LetE be a smooth and uniformly convex Banach space and let {xn} and {yn} be sequences in E such that either {xn} or {yn} is bounded. If limn→∞V(xn,yn)=0, then limn→∞∥xn-yn∥=0.We present the following lemma which plays an important role in our theorems (cf. Butnariu and Resmerita [13] ).Lemma 2.14.
LetE be a smooth and uniformly convex Banach space and C a nonempty, convex, and closed subset of E. Suppose that T:C→E satisfies
(2.36)V(Tx,Ty)≤V(x,y),∀x,y∈C.
If a weakly convergent sequence {zn}n∈ℕ⊂C satisfies that limn→∞V(Tzn,zn)=0, then zn⇀z∈F(T).


## 3. Main Results

In this section, we prove three strong convergence theorems. In the first result, we prove strong convergence theorem for finding common fixed points of a family of generalized nonexpansive mappings. In the next result, we prove strong convergence theorem for finding zeroes of a generalized nonexpansive mapping and a maximal monotone operator. In the last result, we prove weak convergence theorem for finding zeroes of a maximal monotone operator and aV-strongly nonexpansive mapping. As consequence, we prove convergence theorem for common zeroes of a maximal monotone operator and a firmly nonexpansive mapping in a Hilbert space.Theorem 3.1.
LetE be a reflexive, smooth, and strictly convex Banach space, and let {Tn}n∈ℕ be a family of generalized nonexpansive mappings. Suppose that ⋂n∈ℕF(Tn)=F≠∅ and that R is a sunny and generalized nonexpansive retraction from E to F. Let a sequence {xn} be defined as follows. For any x1=x∈E,
(3.1)xn+1=RTnxn,foranyn∈ℕ.
Then, {xn} converges strongly to a point x* in F.Proof.
SinceRxn is a point in F for all n∈ℕ, from Proposition 2.3(a), we have for all n∈ℕ that
(3.2)0≤V(xn+1,Rxn+1)≤V(xn+1,Rxn+1)+V(Rxn+1,Rxn)≤V(xn+1,Rxn)=V(RTnxn,Rxn).
Since R and Tn are generalized nonexpansive, we get that
(3.3)V(RTnxn,Rxn)≤V(Tnxn,Rxn)≤V(xn,Rxn).
Hence, we have that
(3.4)0≤V(xn+1,Rxn+1)≤V(xn,Rxn),∀n∈ℕ,
and therefore, limn→∞V(xn,Rxn)<∞. Furthermore, Proposition 2.3(a) implies that
(3.5)V(xn+k,Rxn+k)+V(Rxn+k,Rxn)≤V(xn+k,Rxn).
This is equivalent to
(3.6)V(Rxn+k,Rxn)≤V(xn+k,Rxn)-V(xn+k,Rxn+k).
Setting m=n+k for all n,k∈ℕ, then we have that
(3.7)V(Rxm,Rxn)≤V(xm,Rxn)-V(xm,Rxm)≤V(RTm-1xm-1,Rxn)-V(xm,Rxm)≤V(Tm-1xm-1,Rxn)-V(xm,Rxm)≤V(xm-1,Rxn)-V(xm,Rxm)≤⋯≤V(xn,Rxn)-V(xm,Rxm)→0,asn,m→∞.
Since V(xn+1,p)=V(RTnxn,p)≤V(x,p) for any p∈F, Lemma 1.1(b) implies that {xn} is bounded. Thus, from Lemma 2.12, we can take the continuous and strictly increasing function g:[0,∞)→[0,∞) with g(0)=0 such that
(3.8)g(∥Rxm-Rxn∥)≤V(Rxm,Rxn)≤V(xn,Rxn)-V(xm,Rxm)→0,asn,m→∞.
Since Rxn=xn for all n≥1, we have g(∥xm-xn∥)=g(∥Rxm-Rxn∥)→0. Therefore, {xn} is a Cauchy sequence. Since E is complete and F is closed, this sequence {xn} converges strongly to point x*∈F.Noting that the generalized resolventJr=(I+rBJ)-1 of a maximal monotone operator B for r>0 is a generalized nonexpansive mapping (see Remark 2.4), we obtain the following result.Theorem 3.2.
LetE be a reflexive, smooth, and strictly convex Banach space. Let T:E→E be  generalized nonexpansive and let B⊂E*×E be a maximal monotone operator. Suppose that F(T)∩(BJ)-1(0)≠∅ and that R is a sunny and generalized nonexpansive retraction from E to F=F(T)∩(BJ)-1(0). Let an iterative sequence {xn} be defined as follows: for any x=x1∈E,
(3.9)xn+1=RTJrnxn,∀n∈ℕ,
where {rn} is a sequence of nonnegative real numbers. Then, the sequence {xn} converges strongly to a point x* in F(T)∩(BJ)-1(0).Proof.
From Propositions2.3(a) and 2.3(b), we have for all n∈N that
(3.10)V(xn+1,Rxn+1)≤V(xn+1,Rxn+1)+V(Rxn+1,Rxn)≤V(xn+1,Rxn)=V(RTJrnxn,Rxn)≤V(TJrnxn,Rxn)≤V(Jrnxn,Rxn)≤V(xn,Rxn).
Thus limn→∞V(xn,Rxn)<∞. Similarly, as in the proof of the previous theorem, we show that {xn} is a Cauchy sequence, and we obtain that {xn} converges strongly to point x* in F=F(T)∩(BJ)-1(0).The duality mappingJ of a Banach space E with the Gâteaux differentiable norm is said to be weakly sequentially continuous if xn⇀x in E implies that {Jxn} converges weak star to Jx in E* (cf., [14]). This happens, for example, if E is a Hilbert space, finite dimensional and smooth, or lp if 1<p<∞ (cf., [15]). Next, we prove the main theorem.Theorem 3.3.
LetE be a reflexive, smooth and strictly convex Banach space. Suppose that the duality mapping J of E is weakly sequentially continuous. Let C be a nonempty, closed, and convex subset of E. Let B:E*→2E be a maximal monotone operator and let Jrn=(I+rnBJ)-1 be a generalized resolvent of B for a sequence {rn}⊂(0,∞). Suppose that A:C→E is a V-strongly nonexpansive mapping with λ≥1 such that C0=A-1(0)∩(BJ)-1(0)≠∅ and that RC:E→C is a sunny and generalized nonexpansive retraction. For an α∈[-1,1] such that α2λ≥1, let an iterative sequence {xn}⊂C be defined as follows: for any x=x1∈C and n∈ℕ,
(3.11)yn=RC(I-αA)xn,xn+1=RC(βnx+(1-βn)Jrnyn),
where {βn}⊂[0,1] and {rn}⊂(0,∞) satisfy that
(3.12)∑n≥1βn<∞,liminfn→∞rn>0.
Then, there exists an element u∈C0 such that
(3.13)xn⇀u,RC0(xn)→u.Proof.
For simplicity, we denoteRC and RC0 by R and R0, respectively. Let zn=βnx+(1-βn)Jrnyn for all n∈ℕ. Since R is generalized nonexpansive, we have for any p∈C0 and all n∈ℕ that
(3.14)V(xn+1,p)=V(Rzn,p)≤V(zn,p).
The convexity of ∥·∥2 implies that
(3.15)V(zn,p)=V(βnx+(1-βn)Jrnyn,p)=∥βnx+(1-βn)Jrnyn∥2+∥p∥2-2〈βnx+(1-βn)Jrnyn,Jp〉≤βn∥x∥2+(1-βn)∥Jrnyn∥2+∥p∥2-2βn〈x,Jp〉-2(1-βn)〈Jrnyn,Jp〉=βn{∥x∥2-2〈x,Jp〉+∥p∥2}+(1-βn){∥Jrnyn∥2-2〈Jrnyn,Jp〉+∥p∥2}=βnV(x,p)+(1-βn)V(Jrnyn,p).
Thus, we obtain that
(3.16)V(zn,p)≤βnV(x,p)+(1-βn)V(Jrnyn,p),
and furthermore, since Jrn is generalized nonexpansive, we have that
(3.17)V(zn,p)≤βnV(x,p)+(1-βn)V(yn,p).
LetAα=(I-αA). Then, from Proposition 2.9(d), Aα is V-strongly nonexpansive with α2λ and Aα is also generalized nonexpansive. Hence, we have that
(3.18)V(yn,p)=V(RAαxn,p)≤V(Aαxn,p)≤V(xn,p).
Thus, we have from (3.14), (3.16), (3.17), and (3.18) that
(3.19)V(xn+1,p)≤V(zn,p)≤βnV(x,p)+(1-βn)V(Jrnyn,p)≤βnV(x,p)+(1-βn)V(yn,p)≤βnV(x,p)+(1-βn)V(Aαxn,p)≤βnV(x,p)+(1-βn)V(xn,p).
From Lemma 2.11, there exists α=limn→∞V(xn,p)<∞. Since limn→∞βn=0, we have that
(3.20)α=limn→∞V(xn,p)=limn→∞V(zn,p),(3.21)=limn→∞V(Jrnyn,p)=limn→∞V(yn,p)=limn→∞V(Aαxn,p).
Hence, {xn}, {zn}, {Jrnyn}, {yn}, and {Aαxn} are bounded from Lemma 1.1(b). Since E is uniformly convex, the boundedness of {xn} implies that there exists a subsequence {xnj}⊂{xn} such that xnj⇀u∈C. Moreover, we can take the index sequence {nj}j≥1 satisfies limj→∞rnj-1>0. We will show that u∈(BJ)-1(0). From Proposition 2.3(a),
(3.22)V(xn+1,p)=V(Rzn,p)≤V(zn,Rzn)+V(Rzn,p)≤V(zn,p),
and furthermore, from (3.16) and Proposition 2.3(b), we obtain that
(3.23)V(zn,p)≤βnV(x,p)+(1-βn)V(Jrnyn,p)≤βnV(x,p)+(1-βn){V(yn,Jrnyn)+V(Jrnyn,p)}≤βnV(x,p)+(1-βn)V(yn,p).
These inequalities and (3.22) imply with limn→∞βn=0 and (3.21) that
(3.24)α≤α+limn→∞V(zn,Rzn)≤α≤limn→∞V(yn,Jrnyn)+α≤α,
that is,
(3.25)limn→∞V(zn,xn+1)=limn→∞V(zn,Rzn)=limn→∞V(yn,Jrnyn)=0.
Lemma 2.13 implies that
(3.26)limn→∞∥xn+1-zn∥=0,limn→∞∥Jrnyn-yn∥=0.
Furthermore, since
(3.27)limn→∞∥zn-Jrnyn∥=limn→∞∥βnx+(1-βn)Jrnyn-Jrnyn∥=limn→∞βn∥x-Jrnyn∥=0,
we have from (3.26) and (3.27) that
(3.28)∥xn+1-Jrnyn∥≤∥xn+1-zn∥+∥zn-Jrnyn∥→0(n→∞).
Hence, for an index sequence {nj}j≥1 of {xnj} such that xnj⇀u∈C and limj→∞rnj-1>0, we obtain that
(3.29)Jrnj-1ynj-1⇀u,ynj-1⇀u,asj⇀∞.
Since (1/r)(Jr-1-I)=(1/r)(I+rBJ-I)=BJ, there exists wnj∈BJ(Jrnj-1ynj-1) such that
(3.30)wnj=1rnj-1(ynj-1-Jrnj-1ynj-1),foranyj≥1.
Since limj→∞rnj-1>0, (3.26) implies that
(3.31)limj→∞∥wnj∥=limj→∞1rnj-1∥Jrnj-1ynj-1-ynj-1∥=0.
For (p,q)∈BJ⊂E×E, the monotonicity of B implies that
(3.32)limj→∞〈q-wnj,Jp-JJrnj-1ynj-1〉≥0,
and we have, since J is weakly sequentially continuous, that
(3.33)〈q,Jp-Ju〉≥0.
The maximality of B implies that u∈(BJ)-1(0).
Now, we will show thatu∈A-1(0). From Proposition 2.9(d) and p∈F(Aα), we get that
(3.34)V(Aαxn,p)=V(Aαxn,Aαp)≤V(xn,p)-λ-1V(Axn,Ap).
Thus, we have from (3.17) that
(3.35)V(xn+1,p)≤V(zn,p)≤βnV(x,p)+(1-βn)V(yn,p)=βnV(x,p)+(1-βn)V(RAαxn,p)≤βnV(x,p)+(1-βn)V(Aαxn,p)≤βnV(x,p)+(1-βn){V(xn,p)-λ-1V(Axn,Ap)}.
This implies that
(3.36)0≤(1-βn)λ-1V(Axn,Ap)≤βnV(x,p)+(1-βn)V(xn,p)-V(xn+1,p)=βn{V(x,p)-V(xn,p)}+V(xn,p)-V(xn+1,p)→0,asn→∞.
Therefore we have that
(3.37)limn→∞V(Axn,Ap)=0.
From Lemma 2.13, we get that limn→∞∥Axn-Ap∥=limn→∞∥Axn∥=0 that is, Axn→0 as n→∞. From Lemma 1.1(a) and the boundedness of {Aαxn}, we have that
(3.38)0≤V(xn,Aαxn)=V(xn,p)-V(Aαxn,p)+2〈xn-Aαxn,JAαxn-Jp〉=V(xn,p)-V(Aαxn,p)+2α〈Axn,JAαxn-Jp〉≤V(xn,p)-V(Aαxn,p)+2α∥Axn∥∥JAαxn-Jp∥≤V(xn,p)-V(Aαxn,p)+2α∥Axn∥M,
for some M>0. From (3.21), we have that limn→∞{V(xn,p)-V(Aαxn,p)}=0, and we obtain that
(3.39)limn→∞V(xn,Aαxn)=0,
and this means that
(3.40)limn→∞∥xn-Aαxn∥=0.
From Lemma 2.14, we obtain xn⇀u0∈F(Aα). Since xnj⇀u, this means that xnj⇀u0=u; hence, we have u∈F(Aα); that is, u∈A-1(0). Therefore, we obtain that u∈A-1(0)∩(BJ)-1(0)=C0.
Letun=R0xn for any n∈ℕ. Since R0 is a sunny generalized nonexpansive retraction,
(3.41)〈xn-un,Jun-Jy〉≥0,∀y∈C0.
Similarly as in the proof of Theorem 3.2, we can show that {un} is a Cauchy sequence, and therefore there exists u*∈C0 such that un→u*. Set y=u in (3.41). Since xn⇀u, we get that
(3.42)〈u-u*,Ju*-Ju〉≥0.
This means that u=u* by the strict convexity of J; that is, R0xn→u. This completes the proof.In a Hilbert space, we obtain the following theorem as a corollary of the main Theorem3.3 by applying Proposition 2.8(a).Corollary 3.4.
LetH be a Hilbert space, and let C be a nonempty, closed, and convex subset of H. Let B:H→2H be a maximal monotone operator, and let Jrn=(I+rnB)-1 be a resolvent of B for a sequence {rn}⊂(0,∞). Suppose that A:C→H is a firmly nonexpansive mapping with C0=A-1(0)∩B-1(0)≠∅. Suppose that RC is a sunny and generalized nonexpansive retraction to C. Let an iterative sequence {xn}⊂C be defined as follows: for any x=x1∈C and n∈ℕ,
(3.43)yn=RC(I-αA)xn,xn+1=RC(βnx+(1-βn)Jrnyn),
where {βn}⊂[0,1] and {rn}⊂(0,∞) satisfy that
(3.44)∑n≥1βn<∞,liminfn→∞rn>0.
Then, there exists an element u∈C0 such that
(3.45)xn⇀u,RC0(xn)→u.



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*Source: 189814-2010-09-01.xml*