# New Iterative Manner Involving Sunny Nonexpansive Retractions for Pseudocontractive Mappings

Authors: Yaqin Zheng; Jinwei Shi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Iterative methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.


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## Body

## 1.  Introduction

LetC be a nonempty closed convex subset of a real Banach space E. A mapping T:C→C is said to be nonexpansive, if
(1.1)∥Tx-Ty∥≤∥x-y∥,
for all x,y∈C.Now we know that the involved operators in the many practical applications can be reduced to the nonexpansive mappings, that is, there are a large number of applied areas which are closely related to the nonexpansive mappings, for example, inverse problem, partial differential equations, image recovery, and signal processing. Based on these facts, recently, iterative methods for finding fixed points of nonexpansive mappings have received vast investigations. For related works, please see [1–26] and the references therein.In the present paper, we focus on a class of strictly pseudocontractive mappings which strictly includes the class of nonexpansive mappings. Recall that a mappingT:C→C is said to be strictly pseudocontractive if there exists a constant λ>0 and j(x-y)∈J(x-y) such that
(1.2)〈Tx-Ty,j(x-y)〉≤∥x-y∥2-λ∥x-y-(Tx-Ty)∥2,
for all x,y∈C. We use Fix(T) to denote the set of fixed points of T.We know that the strict pseudocontractions have more powerful applications than nonexpansive mappings in solving inverse problems. There are some related references in the literature for strictly pseudocontractive mappings; see, for example, [27–30]. Motivated and inspired by the works in the literature, in the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.


## 2. Preliminaries

LetE* be the dual space of a Banach space E. Let Jq (q>1) be the generalized duality mapping from E into 2E* given by
(2.1)Jq(x)={g∈E*:〈x,g〉=∥x∥∥g∥,∥g∥=∥x∥q-1},
where 〈·,·〉 denotes the generalized duality pairing. In particular, J2 is called the normalized duality mapping and it is usually denoted by j. It is well known that E is a uniformly smooth Banach space if and only if Jq is single valued and uniformly continuous on any bounded subset of E.LetC be a nonempty closed convex subset of a Banach space E, and let D be a nonempty subset of C. Recall that a mapping Q:C→D is called a retraction from C onto D provided Q(x)=x for all x∈D. A retraction Q:C→D is sunny provided Q(x+t(x-Q(x)))=Q(x) for all x∈C and t≥0 whenever x+t(x-Q(x))∈C. A sunny nonexpansive retraction is a sunny retraction which is also nonexpansive. Sunny non-expansive retractions are characterized as follows.Lemma 2.1.
IfE is a smooth Banach space, then Q:C→D is a sunny nonexpansive retraction if and only if there holds the inequality
(2.2)〈x-Qx,J(y-Qx)〉≤0,
for all x∈C and y∈D.Lemma 2.2 (see [31]).
LetE be a real q-uniformly smooth Banach space, and let 1<q≤2. Then, one has
(2.3)∥x+y∥q≤∥x∥q+q〈y,Jq(x)〉+2∥Ky∥q,
for all x,y∈E.Lemma 2.3 (see [16]).
Let{xn} and {zn} be two bounded sequences in Banach spaces, and let {βn} be a sequence in [0,1] with 0<liminfn→∞βn≤limsupn→∞βn<1. Suppose that xn+1=(1-βn)xn+βnzn  for all n≥0 and limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then limn→∞∥zn-xn∥=0.Lemma 2.4 (see [14]).
LetC be a nonempty closed convex subset of a real q-uniformly smooth and uniformly convex Banach space E. Let T:C→C be a strictly pseudocontractive mapping. Then I-T is demiclosed.Lemma 2.5 (see [32]).
Assume{an} is a sequence of nonnegative real numbers such that an+1≤(1-γn)an+γnδn  for n≥0 where {γn} is a sequence in (0,1) and {δn} is a sequence in R such that (i)
∑n=0∞‍γn=∞;(ii)
limsupn→∞δn≤0 or ∑n=0∞‍|δnγn|<∞.
Thenlimn→∞an=0.


## 3. Main Results

In this section, we will give our main results. In the sequel, we assume the following:(C1)
E is a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping j from E to E*;(C2)
C is a nonempty closed convex subset of E;(C3)
QC is a sunny nonexpansive retraction from E onto C;(C4)
T:C→C is a λ-strict pseudocontraction;(C5)
ϑ:E→E is a ρ-contraction;(C6)
A:E→E is strongly positive (i.e., 〈Ax,j(x)〉≥γ∥x∥2 for some 0<γ<1) and linear bounded operator with ∥(1-ζ)I-θA∥≤1-ζ-θ for all ζ>0,θ>0 and 0<ζ+θ<1;(C7)
Fix(T)≠∅.First, we consider the followingVI: finding x†∈Fix(T) such that
(3.1)〈(A-δϑ)x†,j(x~-x†)〉≥0,x~∈Fix(T).
The set of solutions of (3.1) is denoted by VI(Fix(T),A). In the sequel, we assume that VI(Fix(T),A)≠∅. Note that (3.1) has the unique solution.Next, we propose our algorithm.Algorithm 3.1.
For the initial pointx0∈C, we generate a sequence {xn} via the following manner:
(3.2)xn+1=αnδϑ(xn)+[(1-η)I-αnA]xn+η[(1-k)QCxn+kTQCxn],n≥0,
where {αn} is a sequence in (0,1) and δ>0,0<η<1,δρ<γ,0<k<λ/K2.Theorem 3.2.
If the sequence{αn} satisfies limn→∞an=0 and ∑n=0∞‍αn=∞, then the sequence {xn} generated by (3.2) converges strongly to the unique solution x† of VI (3.1).Proof.
First, by using Lemma2.2, we know that (1-k)I+kT is nonexpansive. QC is sunny nonexpansive. Thus, (1-k)QC+kTQC is nonexpansive. Let x†∈Fix(T). From (3.2), we have
(3.3)∥xn+1-x†∥=∥αnδϑ(xn)+[(1-η)I-αnA]xn+η[(1-k)QCxn+kTQCxn]-x†∥=∥αn(δϑ(xn)-Ax†)+[(1-η)I-αnA](xn-x†)+η[(1-k)QCxn+kTQCxn-x†]∥≤αn∥δϑ(xn)-Ax†∥+∥(1-η)I-αnA∥∥xn-x†∥+η∥(1-k)QCxn+kTQCxn-x†∥≤αnδ∥ϑ(xn)-ϑ(x†)∥+αn∥δϑ(x†)-Ax†∥+(1-η-αnγ)∥xn-x†∥+η∥xn-x†∥≤αnδρ∥xn-x†∥+αn∥δϑ(x†)-Ax†∥+(1-η-αnγ)∥xn-x†∥+η∥xn-x†∥=[1-(γ-δρ)αn]∥xn-x†∥+αn∥δϑ(x†)-Ax†∥.
Thus,
(3.4)∥xn+1-x†∥≤max{∥xn-x†∥,∥δϑ(x†)-Ax†∥γ-δρ}≤max{∥x0-x†∥,∥δϑ(x†)-Ax†∥γ-δρ}.
This indicates that {xn} is bounded.
We writexn+1=(1-η)xn+ηun for all n≥0. So,
(3.5)un=xn+1-(1-η)xnη=αnδϑ(xn)η-αnAxnη+(1-k)QCxn+kTQCxn=αnδϑ-Aηxn+[(1-k)QC+kTQC]xn.
Hence,
(3.6)un+1-un=αn+1δϑ-Aηxn+1+[(1-k)QC+kTQC]xn+1-αnδϑ-Aηxn-[(1-k)QC+kTQC]xn.
It follows that
(3.7)∥un+1-un∥≤αn+1∥δϑ-Aηxn+1∥+αn∥δϑ-Aηxn∥+∥[(1-k)QC+kTQC]xn+1-[(1-k)QC+kTQC]xn∥≤(αn+αn+1)M+∥xn+1-xn∥,
where M>0 is a constant satisfying sup{∥((δϑ-A)/η)xn∥}≤M. This implies that
(3.8)limsupn→∞(∥un+1-un∥-∥xn+1-xn∥)≤0.
By Lemma 2.3, we deduce
(3.9)limn→∞∥un-xn∥=0.
Therefore,
(3.10)limn→∞∥xn+1-xn∥=limn→∞η∥un-xn∥=0.
Note that Fix(T)=Fix(TQC). As a matter of fact, if p∈Fix(T), that is p=Tp, then p∈Fix(TQC). Since T is a self-mapping, it is clear that p∈C. So, QCp=p. Therefore, TQCp=Tp=p. Conversely, if q∈Fix(TQC), that is q=TQCq, we also have q∈C. Thus, q=Tq. Set S=(1-k)I+kT. We observe that Fix(S)=Fix(T)=Fix(TQC)=Fix(SQC). Next, we estimate ∥xn-SQCxn∥.
Since(3.11)∥xn-SQCxn∥≤∥xn-xn+1∥+∥xn+1-SQCxn∥≤∥xn-xn+1∥+(1-η)∥xn-SQCxn∥+η∥un-SQCxn∥≤∥xn-xn+1∥+(1-η)∥xn-SQCxn∥+αn∥δϑ(xn)-Axn∥,
we have
(3.12)∥xn-SQCxn∥≤1η∥xn+1-xn∥+αnη∥δϑ(xn)-Axn∥→0.
Next, we show(3.13)limsupn→∞〈δϑ(x†)-Ax†,j(xn+1-x†)〉≤0,
where x†∈VI(Fix(T),A).
First, we have(3.14)limsupn→∞〈δϑ(x†)-Ax†,j(xn+1-x†)〉=limi→∞〈δϑ(x†)-Ax†,j(xni+1-x†)〉.
Since the sequence {xn} is bounded, hence {xni} is bounded. Thus, we can take a subsequence {xnij} of {xn} such that xnij→x~ weakly. Without loss of generality, we may assume that xni→x~ weakly. Note that SQC is nonexpansive and ∥xni-SQCxni∥→0. By using the demiclosed principle of nonexpansive mappings (see Lemma 2.4), we get x~∈Fix(SQC)=Fix(T). At the same time, j is weakly sequentially continuous. Therefore,
(3.15)limsupn→∞〈δϑ(x†)-Ax†,j(xn+1-x†)〉=limi→∞〈δϑ(x†)-Ax†,j(xni+1-x†)〉=〈δϑ(x†)-Ax†,j(x~-x†)〉≤0.
Finally we show that xn→x†. From (3.2), we have
(3.16)∥xn+1-x†∥2=∥αn(δϑ(xn)-Ax†)+((1-η)I-αnA)(xn-x†)+η((1-k)QCxn+kTQCxn-x†)∥2≤∥((1-η)I-αnA)(xn-x†)+η((1-k)QCxn+kTQCxn-x†)∥2+2αn〈δϑ(xn)-Ax†,j(xn+1-x†)〉≤(∥((1-η)I-αnA)(xn-x†)∥2+∥η((1-k)QCxn+kTQCxn-x†)∥)2+2αnδ〈ϑ(xn)-ϑ(x†),j(xn+1-x†)〉+2αn〈δϑ(x†)-Ax†,j(xn+1-x†)〉≤[(1-η-αnγ)∥xn-x†∥+η∥xn-x†∥]2+2αnδρ∥xn-x†∥∥xn+1-x†∥+2αn〈δϑ(x†)-Ax†,j(xn+1-x†)〉≤(1-αnγ)2∥xn-x†∥2+αnδρ(∥xn-x†∥2+∥xn+1-x†∥2)+2αn〈δϑ(x†)-Ax†,j(xn+1-x†)〉.
It follows that
(3.17)∥xn+1-x†∥2≤1-2αnγ+αn2γ2+αnδρ1-αnδρ∥xn-x†∥2+2αn1-αnδρ〈δϑ(x†)-Ax†,j(xn+1-x†)〉=[1-2(γ-δρ)αn1-αnδρ]∥xn-x†∥2+αn2γ21-αnδρ∥xn-x†∥2+2αn1-αnδρ〈δϑ(x†)-Ax†,j(xn+1-x†)〉=[1-2(γ-δρ)αn1-αnδρ]∥xn-x†∥2+2(γ-δρ)αn1-αnδρ×{αnγ2∥xn-x†∥22(γ-δρ)+1γ-δρ〈δϑ(x†)-Ax†,j(xn+1-x†)〉}.
It can be checked easily that ∑n‍(2(γ-δρ)αn/(1-αnδρ))=∞ and limsupn→∞(αnγ2∥xn-x†∥2/2(γ-δρ))+(1/(γ-δρ))〈δϑ(x†)-Ax†,j(xn+1-x†)〉≤0. From Lemma 2.5, we deduce xn→x†. This completes the proof.Algorithm 3.3.
For the initial pointx0∈C, we generate a sequence {xn} via the following manner:
(3.18)xn+1=αnδϑ(xn)+(1-η-αn)xn+η[(1-k)QCxn+kTQCxn],n≥0,
where {αn} is a sequence in (0,1) and δ>0,0<η<1,δρ<1,0<k<λ/K2.Corollary 3.4.
If the sequence{αn} satisfies limn→∞an=0 and ∑n=0∞‍αn=∞, then the sequence {xn} generated by (3.18) converges strongly to the unique solution x† of VI: finding x†∈Fix(T) such that
(3.19)〈(I-δϑ)x†,j(x~-x†)〉≥0,x~∈Fix(T).Algorithm 3.5.
For the initial pointx0∈C and u∈E, we generate a sequence {xn} via the following manner:
(3.20)xn+1=αnu+(1-η-αn)xn+η[(1-k)QCxn+kTQCxn],n≥0,
where {αn} is a sequence in (0,1) and 0<η<1,0<k<λ/K2.Corollary 3.6.
If the sequence{αn} satisfies limn→∞αn=0 and ∑n=0∞‍αn=∞, then the sequence {xn} generated by (3.20) converges strongly to the unique solution x† of VI: finding x†∈Fix(T) such that
(3.21)〈(I-u)x†,j(x~-x†)〉≥0,x~∈Fix(T).Algorithm 3.7.
For the initial pointx0∈C, we generate a sequence {xn} via the following manner:
(3.22)xn+1=(1-η-αn)xn+η[(1-k)QCxn+kTQCxn],n≥0,
where {αn} is a sequence in (0,1) and 0<η<1,0<k<λ/K2.Corollary 3.8.
If the sequence{αn} satisfies limn→∞αn=0 and ∑n=0∞‍αn=∞, then the sequence {xn} generated by (3.22) converges strongly to QFix(T)(0).



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*Source: 207896-2012-11-01.xml*