Approximation of Fixed Points of Nonexpansive Mappings and Solutions of Variational Inequalities

Journal of Inequalities and Applications Volume 2008, Article ID 284345,12pages doi:10.1155/2008/284345

Research Article

Approximation of Fixed Points of Nonexpansive

Mappings and Solutions of Variational Inequalities

C. E. Chidume,1 _{C. O. Chidume,}2_{and Bashir Ali}3

1_{The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy}

2_{Department of Mathematics and Statistics, College of Sciences and Mathematics, Auburn University,}

Auburn, AL 36849, USA

3_{Department of Mathematical Sciences, Bayero University, 3011 Kano, Nigeria}

Correspondence should be addressed to C. E. Chidume,[email protected]

Received 3 July 2007; Accepted 17 October 2007

Recommended by Siegfried Carl

Let Ebe a real q-uniformly smooth Banach space with constantdq, q ≥ 2. Let T : E → Eand

G :E →Ebe a nonexpansive map and anη-strongly accretive map which is alsoκ-Lipschitzian, respectively. Let{λn}be a real sequence in0,1that satisfies the following condition:C1: limλn0 andλn ∞. Forδ ∈0,qη/dqkq1/q−1andσ ∈0,1, define a sequence{xn}iteratively inE byx0∈E, xn1Tλn1xn 1−σxnσTxn−δλn1GTxn,n≥0. Then,{xn}converges strongly to the unique solutionx∗of the variational inequality problem VIG, K search forx∗ ∈ Ksuch thatGx∗, jqy−x∗ ≥0 for ally∈K, whereK:FixT {x∈E:Txx}/∅. A convergence theorem related to finite family of nonexpansive maps is also proved.

Copyrightq2008 C. E. Chidume et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetEbe a real-normed space and letE∗be its dual space. For some real numberq1< q <∞, the generalized duality mappingJq:E→2E

∗

is defined by

Jqx f∗∈E∗:x, f∗ x q,f∗ x q−1, 1.1

where·,·denotes the pairing between elements ofEand elements ofE∗.

Let K be a nonempty closed convex subset of E, and let S : E → E be a nonlinear operator. The variational inequality problem is formulated as follows. Find a point x∗ ∈ K such that

VIS, K:Sx∗, jq

IfEH,a real Hilbert space, the variational inequality problem reduces to the following. Find a pointx∗∈Ksuch that

VIS, K:Sx∗, y−x∗ ≥0 ∀y∈K. 1.3

A mappingG:DG⊂E→Eis said to beaccretiveif for allx, y ∈DG,there existsjqx−y∈ Jqx−ysuch that

Gx−Gy, jqx−y ≥0, 1.4

whereDGdenotes the domain ofG.For some real numberη >0, Gis calledη-strongly accre-tiveif for allx, y ∈DG,there existsjqx−y∈Jqx−ysuch that

Gx−Gy, jqx−y ≥η x−y q. 1.5

Gisκ-Lipschitzianif for someκ >0, Gx−Gy ≤κ x−y for allx, y∈DGandGis called nonexpansiveifk1.

In Hilbert spaces, accretive operators are called monotonewhere inequalities 1.4 and

1.5hold withjqreplaced by the identity map ofH.

It is known that ifSis Lipschitz andstrongly accretive, then VIS, Khas a unique solu-tion. An important problem is how to find a solution of VIS, Kwhenever it exists. Consid-erable efforts have been devoted to this problemsee, e.g.,1,2and the references contained therein.

It is known that in a real Hilbert space, the VIS, Kis equivalent to the following fixed-point equation:

x∗PK

x∗−δSx∗, 1.6

where δ > 0 is an arbitrary fixed constant and PK is the nearest point projection map from H

onto K, that is, PKx y, where x − y infu∈K x− u for x ∈ H. Consequently,

un-der appropriate conditions on S and δ,fixed-point methods can be used to find or approx-imate a solution of VIS, K. For instance, if S is strongly monotone and Lipschitz, then a mappingG:H →H, defined by Gx PKx− δSx, x ∈ H with δ > 0 sufficiently small,

is a strict contraction. Hence, the Picard iteration,x0 ∈ H, xn1 Gxn, n ≥ 0 of the classical

Banach contraction mapping principle, converges to the unique solution of the VIK, S. It has been observed that the projection operatorPKin the fixed-point formulation1.6

may make the computation of the iterates difficult due to possible complexity of the convex set K.In order to reduce the possible difficulty with the use ofPK,Yamada2recently introduced

a hybrid descent method for solving the VIK, S. LetT : H → H be a map and letK:{x∈ H : Tx x} /∅.LetS beη-strongly monotone andκ-Lipschitz onH.Letδ ∈ 0,2η/κ2_be arbitrary but fixed real number and let a sequence{λn}in0,1satisfy the following conditions:

C1: limλn0; C2:

λn∞; C3: limλn−λn1

λ2

n

0. 1.7

Starting with an arbitrary initial guessx0∈H,let a sequence{xn}be generated by the

follow-ing algorithm:

xn1 Txn−λn1δSTxn

, n≥0. 1.8

In the case that K r_i1FTi/∅, where {Ti}ri1 is a finite family of nonexpansive mappings, Yamada2studied the following algorithm:

xn1Tn1xn1−λn1δS

Tn1xn

, n≥0, 1.9

whereTkTkmodrfork≥1,with the modfunctiontaking values in the set{1,2, . . . , r},where

the sequence {λn} satisfies the conditions C1, C2, andC4:

|λn −λnN| < ∞. Under these

conditions, he proved the strong convergence of{xn}to the unique solution of the VIK, S.

Recently, Xu and Kim1studied the convergence of the algorithms1.8and1.9, still in the framework of Hilbert spaces, and proved strong convergence with conditionC3 replaced byC5: limλn−λn1/λn1 0 and with conditionC4 replaced byC6: limλn−λnr/λnr 0.These are improvements on the results of Yamada. In particular, the canonical choice λn :

1/n1is applicable in the results of Xu and Kim but is not in the result of Yamada2. For further recent results on the schemes1.8and1.9, still in the framework of Hilbert spaces, the reader my consult Wang3, Zeng and Yao4, and the references contained in them.

Recently, the present authors5extended the results of Xu and Kim1toq-uniformly smooth Banach spaces,q ≥ 2.In particular, they proved theorems which are applicable inLp

spaces, 2≤p <∞under conditionsC1,C2, andC5 orC6 as in the result of Xu and Kim. It is our purpose in this paper to modify the schemes1.8and1.9and prove strong convergence theorems for the unique solution of the variational inequality VIK, S. Further-more, in the case Ti : E → E, i 1,2, . . . , r, is a family of nonexpansive mappings with

K r_i1FTi/ ∅, we prove a convergence theorem where condition C6 is replaced by limn→∞ Tn1xn−Tnxn 0.An example satisfying this condition is given see, for example, 6. All our theorems are proved inq-uniformly smooth spaces,q ≥ 2.In particular, our theo-rems are applicable inLpspaces, 2≤p <∞.

2. Preliminaries

LetEbe a real Banach space and letK be a nonempty, closed, and convex subset ofE.LetP be a mapping ofEontoK.Then,P is said to besunnyifPP xtx−P x P xfor allx∈ E andt≥ 0.A mappingP ofEintoEis said to be aretractionifP2 _{P.A subsetKis said to be} sunny nonexpansive retractof Eif there exists a sunny nonexpansive retraction ofEontoK. A retractionP is said to beorthogonalif for eachx,x−Pxis normal toKin the sense of James

7.

It is well knownsee8that ifEis uniformly smooth and there exists a nonexpansive retraction ofEontoK, then there exists a nonexpansive projection of EontoK. IfEis a real smooth Banach space, thenPis an orthogonal retraction ofEontoKif and only ifPx∈Kand Px−x, jqPx−y ≤ 0 for ally ∈K.It is also knownsee, e.g.,9that ifKis a convex

subset of a uniformly convex Banach space whose norm is uniformly Gˆateaux differentiable andT :K→Kis nonexpansive withFT/∅,thenFTis a nonexpansive retract ofK.

uniformly convex and uniformly smooth Banach spaces have uniform normal structuresee, e.g.,11,12.

We will denote a Banach limit byμ. Recall thatμis an element ofl∞∗such that μ 1, lim infn→∞an ≤ μnan ≤ lim sup_n→∞an and μnan μn1an for all {an}n≥0 ∈ l∞ see, e.g.,

11,13.

LetEbe a normed space with dimE≥ 2. Themodulus of smoothnessofEis the function ρE:0,∞→0,∞defined by

ρEτ:sup

xy x−y

2 −1 : x 1; y τ

. 2.1

The space E is called uniformly smoothif and only if limt→0ρEt/t 0. For some positive

constantq,Eis calledq-uniformly smoothif there exists a constantc > 0 such thatρEt≤ ctq,

t >0.It is known that

Lporlpspaces are

2-uniformly smooth if 2≤p <∞,

p-uniformly smooth if 1< p≤2 2.2

see, e.g.,13. It is well known that ifEis smooth, then the duality mapping is singled-valued, and ifEis uniformly smooth, then the duality mapping is norm-to-norm uniformly continuous on bounded subset ofE.

We will make use of the following well-known results.

Lemma 2.1. LetEbe a real-normed linear space. Then, the following inequality holds:

xy 2 ≤ x 22y, jxy ∀x, y∈E, ∀jxy∈Jxy. 2.3

In the sequel, we will also make use of the following lemmas.

Lemma 2.2 see 14. Let a0, a1, . . . ∈ l∞ such that μnan ≤ 0 for all Banach limit μ and

lim sup_n→∞an1−an≤0.Then,lim sup_n→∞an≤0.

Lemma 2.3see15. Let{xn}and{yn}be bounded sequences in a Banach spaceEand let{βn}be

a sequence in0,1with0 < lim infβn ≤ lim supβn < 1. Supposexn1 βnyn 1−βnxnfor all

integersn≥0andlim sup yn1−yn − xn1−xn ≤0. Then,lim yn−xn 0.

Lemma 2.4 see16. Let {an} be a sequence of nonnegative real numbers satisfying the following

relation:

an1≤

1−αn

anαnσnγn, n≥0, 2.4

wherei{αn} ⊂0,1,αn∞;iilim supσn≤0;iiiγn≥0;n≥0,γn<∞.Then,an→0

asn→ ∞.

Lemma 2.5 see17. LetE be a real q-uniformly smooth Banach space for someq > 1, then there exists some positive constantdqsuch that

Lemma 2.6see12, Theorem 1. SupposeEis a Banach space with uniformly normal structure, K is a nonempty bounded subset of E, andT : K → K is uniformly k-Lipschitzian mapping with k < NE1/2.Suppose also that there exists a nonempty bounded closed convex subset ofCofKwith the following propertyP:

x∈C impliesωwx⊂C, P

whereωwxis theω-limi set ofT atx,that is, the set

y∈E:yweak-lim

j T

nj_{x for somen}

j −→ ∞

. 2.6

Then,T has a fixed point inC.

3. Main results

We first prove the following lemma which will be central in the sequel.

Lemma 3.1. LetEbe a realq-uniformly smooth Banach space with constantdq,q≥2.LetT :E→ E

andG :E → Ebe a nonexpansive map and anη-strongly accretive map which is alsoκ-Lipschitzian, respectively. Forδ∈0,qη/dqκq1/q−1,σ ∈0,1, andλ∈0,2/pp−1,define a mapTλ :E→

EbyTλ_{x 1−σxσTx−λδGTx,x∈E.Then,T}λ_{is a strict contraction. Furthermore, T}λ_x−Tλ_{y≤1−λα x−y , x, y ∈E, 3.1}

whereαq/2−

q2_{/4−σδqη−δ}q−1_d

qκq∈0,1.

Proof. Forx, y ∈E,

_Tλ_x−Tλ_yq_{1−σx−y σTx−Ty−λδGTx−GTy}q

≤1−σ x−y qσ Tx−Ty q−qλδGTx−GTy, jqTx−Ty

dqλqδqGTx−GTyq

≤1−σ x−y qσ Tx−Ty q−qλδη Tx−Ty qdqλqδqκq Tx−Ty q

≤1−σλδqη−dqλq−1δq−1κq

x−y q ≤1−σλδqη−dqδq−1κq

x−y q.

3.2

Define

fλ:1−σλδqη−dqδq−1κq

1−λτq for some τ∈0,1say. 3.3

Then, there existsξ∈0, λsuch that

1−σλδqη−dqδq−1κq

1−qτλ1

2qq−11−ξτ

This implies that

1−σλδqη−dqδq−1κq

≤1−qτλ1

2qq−1λ

2_τ2_{. 3.5}

Then, we haveτ ≤q/2−

q2_{/4−σδqη−d}

qδq−1κq.

Set

α: q 2 −

q2 4 −σδ

qη−dqδq−1κq

, 3.6

and the proof is complete.

We note that inLpspaces, 2≤p <∞,the following inequality holdssee, e.g.,13. For

eachx, y ∈Lp, 2≤p <∞,

xy 2≤ x 22y, jx p−1 y 2. 3.7

Using this inequality and following the method of proof ofLemma 3.1, the following corollary is easily proved.

Corollary 3.2. LetE Lp,2 ≤ p <∞.LetT : E → E, G : E → Ebe a nonexpansive map, an η

-strongly monotone, andκ-Lipschitzian map, respectively. Forλ, σ ∈0,1andδ ∈0,2η/p−1κ2_, define a mapTλ :E→EbyTλx 1−σxσTx−λδGTx,x∈E.Then,Tλis a contraction. In particular,

_Tλ_x−Tλ_{y≤1−λα x−y , x, y ∈H, 3.8}

whereα1−1−σδ2η−p−1δκ2_∈0,1.

Corollary 3.3. LetHbe a real Hilbert space,T :H →H,G:H→Ha nonexpansive map and anη -strongly monotone map which is alsoκ-Lipschitzian, respectively. Forλ, σ∈0,1andδ∈0,2η/κ2, define a mapTλ :H →HbyTλx 1−σxσTx−λδGTx,x∈H.Then,Tλis a contraction. In particular,

_Tλ_x−Tλ_{y≤1−λα x−y , x, y ∈H, 3.9}

whereα1−1−σδ2η−δκ2_∈0,1.

Proof. Setp2 inCorollary 3.2and the result follows.

Corollary 3.3is a result of Yamada2and is the main tool used in1–4.

We now prove our main theorems.

Theorem 3.4. LetEbe a realq-uniformly smooth Banach space with constantdq,q≥2. LetT :E→E

andG :E → Ebe a nonexpansive map and anη-strongly accretive map which is alsoκ-Lipschitzian, respectively. Let{λn}be a real sequence in0,1satisfying

C1:limλn0; C2:

λn∞. 3.10

Forδ∈0,qη/dqκq1/q−1andσ ∈0,1, define a sequence{xn}iteratively inEbyx0∈E,

xn1Tλn1xn 1−σxnσ

Txn−δλn1GTxn

, n≥0. 3.11

Proof. Letx∗∈K:FixT,then the sequence{xn}satisfies

_xn−x∗ _≤maxx0−x∗_,δ

αG

x∗

, n≥0. 3.12

It is obvious that this is true forn0.Assume that it is true fornkfor somek∈N. From the recursion formula3.11, we have

_xk1−x∗_Tλk1_x

k−x∗

≤Tλk1_x

k−Tλk1x∗Tλk1x∗−x∗

≤1−λk1αxk−x∗λk1δG

x∗

≤maxx0−x∗,δ αG

x∗

,

3.13

and the claim follows by induction. Thus, the sequence{xn}is bounded and so are{Txn}and

{GTxn}.

Define two sequences{βn}and{yn}byβn: 1−σλn1σandyn: xn1−xnβnxn/βn.

Then,

yn

1−σλn1xnσ

Txn−λn1δG

Txn

βn .

3.14

Observe that{yn}is bounded and that _{yn1−yn−xn1−xn}

≤ σ βn1 −1

xn1−xn

_βσ_n1− σ βn

Txn λn21−σ

βn1

_xn1−xn

1−σλn2 βn1 −

λn1 βn

xn λn1σδ

βn _G

Txn

−GTxn1σδ

λn1

βn −

λn2 βn1

GTxn1.

3.15

This implies that lim sup_n→∞||yn1−yn|| − ||xn1−xn||≤0,and byLemma 2.3,

lim

n→∞yn−xn0. 3.16

Hence,

_{xn1−xnβnyn−xn−→0 asn−→ ∞. 3.17}

From the recursion formula3.11, we have that

σxn1−Txn≤1−σxn1−xnλn1σδG

Txn−→0 asn−→ ∞, 3.18

which implies that

From3.17and3.19, we have

_{xn−Txn≤xn−xn1xn1−Txn−→0 asn−→ ∞. 3.20}

We now prove that lim sup_n→∞−Gx∗, jxn1−x∗ ≤0. Define a mapφ:E→Rby

φx μnxn−x2 ∀x∈E. 3.21

Then, φx → ∞ as x → ∞, φ is continuous and convex, so as E is reflexive, there exists y∗∈Esuch thatφy∗ minu∈Eφu.Hence, the set

K∗:x∈E:φx min

u∈E φu

/

∅. 3.22

By Lemma 2.6, K∗∩ K /∅. Without loss of generality, assume that y∗ x∗ ∈ K∗∩ K. Let

t∈0,1.Then, it follows thatφx∗≤φx∗−tGx∗and usingLemma 2.1, we obtain that _xn−x∗_tGx∗2_≤

xn−x∗22t

Gx∗, jxn−x∗tG

x∗ 3.23

which implies that

μn

−Gx∗, jxn−x∗tG

x∗ ≤0. 3.24

Moreover,

μn

−Gx∗, jxn−x∗ μn

−Gx∗, jxn−x∗

−jxn−x∗tG

x∗

μn

−Gx∗, jxn−x∗tG

x∗ ≤μn

−Gx∗, jxn−x∗

−jxn−x∗tG

x∗ .

3.25

Sincejis norm-to-norm uniformly continuous on bounded subsets ofE,we have that

μn

−Gx∗, jxn−x∗ ≤0. 3.26

Furthermore, since xn1−xn →0,asn→ ∞,we also have

lim sup

n→∞

−Gx∗, jxn−x∗ −

−Gx∗, jxn1−x∗ ≤0, 3.27

and so we obtain byLemma 2.2that lim sup_n→∞−Gx∗, jxn−x∗ ≤0.

From the recursion formula3.11andLemma 2.1, we have _xn1−x∗2

Tλn1_x

n−Tλn1x∗Tλn1x∗−x∗2

≤Tλn1_x

n−Tλn1x∗22λn1δ−Gx∗, jxn1−x∗ ≤1−λn1αxn−x∗22λn1δ−Gx∗, jxn1−x∗ ,

3.28

The following corollaries follow fromTheorem 3.4.

Corollary 3.5. LetELp,2≤p <∞. LetT :E→EandG:E→ Ebe a nonexpansive map and an

η-strongly accretive map which is alsoκ-Lipschitzian, respectively. Let{λn}be a real sequence in0,1

that satisfies conditionsC1andC2as inTheorem 3.4. Forδ∈0,2η/p−1κ2_{andσ ∈0,1,define} a sequence{xn}iteratively inEby3.11. Then,{xn}converges strongly to the unique solutionx∗of

the variational inequalityVIG, K.

Corollary 3.6. LetEHbe a real Hilbert space. LetT :H →HandG:H→H be a nonexpansive map and an η-strongly monotone map which is also κ-Lipschitzian, respectively. Let {λn} be a real

sequence in 0,1 that satisfies conditions C1 and C2 as in Theorem 3.4. For δ ∈ 0,2η/κ2 and σ ∈ 0,1,define a sequence {xn}iteratively in H by3.11. Then, {xn}converges strongly to the

unique solutionx∗of the variational inequalityVIG, K.

Finally, we prove the following more general theorem.

Theorem 3.7. Let E be a real q-uniformly smooth Banach space with constant dq, q ≥ 2. Let Ti :

E → E, i 1,2, . . . , r,be a finite family of nonexpansive mappings withK : r_i1FixTi/ ∅.Let G :E→ Ebe anη-strongly accretive map which is alsoκ-Lipschitzian. Let{λn}be a real sequence in 0,1satisfying

C1:limλn0; C2:

λn∞. 3.29

For a fixed real numberδ∈0,qη/dqκq1/q−1,define a sequence{xn}iteratively inEbyx0∈E:

xn1Tλ_nn1₁xn 1−σxnσ

Tn1xn−δλnG

Tn1xn

, n≥0, 3.30

whereTnTnmodr.Assume also that

KFixTrTr−1· · ·T1

FixT1Tr· · ·T2

· · ·FixTr−1Tr−2· · ·Tr

3.31

and limn→∞ Tn1xn−Tnxn 0. Then, {xn} converges strongly to the unique solution x∗ of the

variational inequalityVIG, K.

Proof. Letx∗∈K,then the sequence{xn}satisfies that

_xn−x∗_≤maxx0−x∗_,δ

αG

x∗

, n≥0. 3.32

It is obvious that this is true forn0.Assume it is true fornkfor somek∈N. From the recursion formula3.30, we have

_xk1−x∗_Tλk1

k1xk−x

∗

≤Tλk1

k1xk−Tλ_kk11x∗Tλkk11x∗−x∗ ≤1−λk1αxk−x∗λk1δG

x∗

≤maxx0−x∗,δ αG

x∗

,

and the claim follows by induction. Thus, the sequence {xn}is bounded and so are{Tnxn}

and{GTnxn}.

Define two sequences{βn}and{yn}byβn: 1−σλn1σandyn: xn1−xnβnxn/βn.

Then,

yn

1−σλn1xnσ

Tn1xn−λn1δG

Tn1xn

βn .

3.34

Observe that{yn}is bounded and that

_{yn1−yn−xn1−xn≤} σ βn1 −1

xn1−xn

σ

βn1

_{Tn2xn−Tn1xn} σ βn1 −

σ βn

Tn1xn λn21−σ

βn1

_{xn1−xn 1−σ}λn2 βn1 −

λn1 βn

xn λn1σδ

βn _G

Tn1xn

−GTn2xn1

σδλn1 βn −

λn2 βn1

GTn2xn1.

3.35

This implies that lim sup_n→∞ yn1−yn − xn1−xn ≤0,and byLemma 2.3,

lim

n→∞yn−xn0. 3.36

Hence,

_{xn1−xnβnyn−xn−→0 asn−→ ∞. 3.37}

From the recursion formula3.30, we have that

σxn1−Tn1xn≤1−σxn1−xnλn1σδG

Tn1xn−→0 as n−→ ∞ 3.38

which implies that

_{xn1−Tn1xn−→0 asn−→ ∞. 3.39}

From3.37and3.39, we have

_{xn−Tn1xn≤xn−xn1xn1−Tn1xn−→0 asn−→ ∞. 3.40}

Also,

_{xnr−xn≤xnr−xnr−1xnr−1−xnr−2· · ·xn1−xn, 3.41}

and so

Using the fact thatTiis nonexpansive for eachi,we obtain the following finite table:

xnr−Tnrxnr−1−→0 asn−→ ∞;

Tnrxnr−1−TnrTnr−1xnr−2−→0 as n−→ ∞; ..

.

TnrTnr−1· · ·Tn2xn1−TnrTnr−1· · ·Tn2Tn1xn−→0 asn−→ ∞;

3.43

and adding up the table yields

xnr−TnrTnr−1· · ·Tn1xn−→0 as n−→ ∞. 3.44

Using this and3.42, we get that limn→∞ xn−TnrTnr−1· · ·Tn1xn 0.

Carrying out similar arguments as in the proof ofTheorem 3.4, we easily get that

lim sup

n→∞

−Gx∗, jxn1−x∗ ≤0. 3.45

From the recursion formula3.30, andLemma 2.1, we have _xn1−x∗2

Tλn1

n1xn−T

λn1

n x

∗_Tλn1

n1x

∗_−x∗2

≤Tλn1

n1xn−T

λn1

n1x∗ 2

2λn1σδ−Gx∗, jxn1−x∗ ≤1−λn1αxn−x∗22λn1σδ

−Gx∗, jxn1−x∗ ,

3.46

and byLemma 2.4, we have thatxn→ x∗ as n→ ∞.This completes the proof.

The following corollaries follow fromTheorem 3.7.

Corollary 3.8. LetE Lp,2 ≤ p < ∞.LetTi : E → E, i 1,2, . . . , r,be a finite family of

nonex-pansive mappings withK r_i1FixTi/∅.LetG : E → Ebe an η-strongly accretive map which is alsoκ-Lipschitzian. Let{λn}be a real sequence in 0,1that satisfies conditionsC1 andC2 as in Theorem 3.7and alsolimn→∞ Tn1xn−Tnxn 0.Forδ∈0,2η/p−1κ2,define a sequence{xn}

iteratively inEby3.30. Then,{xn} converges strongly to the unique solutionx∗of the variational

inequalityVIG, K.

Corollary 3.9. LetEHbe a real Hilbert space. LetTi:H →H, i1,2, . . . , r,be a finite family of

nonexpansive mappings withKr_i1FixTi/∅.LetG:H → Hbe anη-strongly monotone map

which is alsoκ-Lipschitzian. Let{λn}be a real sequence in0,1that satisfies conditionsC1 andC2

as inTheorem 3.7and alsolimn→∞ Tn1xn−Tnxn 0.Forδ ∈ 0,2η/κ2,define a sequence{xn}

iteratively inH by3.30. Then,{xn}converges strongly to the unique solution x∗of the variational

inequalityVIG, K.

Remark 3.10. Observe that conditionC6 in Theorem 3.2 of1is dropped inCorollary 3.9, being replaced by condition limn→∞ Tn1xn−Tnxn 0 on the mappings{Ti}ri1.

Acknowledgment

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