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Volume 2009, Article ID 567147,20pages doi:10.1155/2009/567147

Research Article

A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions,

Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces

Somyot Plubtieng

1, 2

and Wanna Sriprad

1, 2

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2PERDO National Centre of Excellence in Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Correspondence should be addressed to Somyot Plubtieng,somyotp@nu.ac.th Received 12 February 2009; Accepted 18 May 2009

Recommended by William A. Kirk

We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng et al.2008and many others.

Copyrightq2009 S. Plubtieng and W. Sriprad. 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

LetHbe a real Hilbert space whose inner product and norm are denoted by·,·and · , respectively. LetCbe a nonempty closed convex subset ofH, and letF be a bifunction of C×CintoR, whereRis the set of real numbers. The equilibrium problem forF :C×C → R is to findxCsuch that

F x, y

≥0, ∀y∈C. 1.1

The set of solutions of 1.1 is denoted by EPF. Recently, Combettes and Hirstoaga1 introduced an iterative scheme of finding the best approximation to the initial data when EPFis nonempty and proved a strong convergence theorem. LetA:CHbe a nonlinear

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map. The classical variational inequality which is denoted byV IA, Cis to finduCsuch that

Au, v−u ≥0, ∀v∈C. 1.2

The variational inequality has been extensively studied in literature. See, for example,2,3 and the references therein. Recall that a mappingTofCinto itself is called nonexpansive if

Su−Sv ≤ uv, ∀u, v∈C. 1.3

A mappingf:CCis called contractive if there exists a constantβ∈0,1such that fu−fv ≤βuv, ∀u, v∈C. 1.4

We denote byFSthe set of fixed points ofS.

Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping; see, for instance,3–6and the references therein.

Recently, Plubtieng and Punpaeng6introduced the following iterative scheme. Letx1H and let{xn}, and{un}be sequences generated by

F un, y

1 rn

yun, unxn

≥0, ∀y∈H, xn1αnγfxn I−αnASun, ∀n∈N.

1.5

They proved that if the sequences{αn}and{rn}of parameters satisfy appropriate conditions, then the sequences {xn} and {un} both converge strongly to the unique solution of the variational inequality

Aγf

z, zx

≥0, ∀x∈FS∩EPF, 1.6

which is the optimality condition for the minimization problem

x∈FS∩EPFmin 1

2Ax, x −hx, 1.7 wherehis a potential function forγf.

LetA : HHbe a single-valued nonlinear mapping, and letM : H → 2H be a set-valued mapping. We consider the following variational inclusion, which is to find a point uHsuch that

θAu Mu, 1.8

whereθis the zero vector inH. The set of solutions of problem1.8is denoted byIA, M.

IfA 0, then problem1.8becomes the inclusion problem introduced by Rockafellar7.

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IfM ∂δC, whereCis a nonempty closed convex subset ofHandδC :H → 0,∞is the indicator function of C, that is,

δCx

⎧⎨

0, xC,

∞, x /∈C, 1.9

then the variational inclusion problem1.8is equivalent to variational inequality problem 1.2. It is known that1.8provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, com- plementarity, variational inequalities, optimal control, mathematical economics, equilibria, game theory. Also various types of variational inclusions problems have been extended and generalizedsee8and the references therein.

Very recently, Peng et al. 9 introduced the following iterative scheme for finding a common element of the set of solutions to the problem 1.8, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert space.

Starting withx1H, define sequence,{xn},{yn}, and{un}by

F un, y

1 rn

yun, unxn

≥0, ∀y∈H, xn1αnγfxn 1−αnSyn, ynJM,λunλAun, ∀n≥0,

1.10

for all n ∈ N, where λ ∈ 0,2α, {αn} ⊂ 0,1 and {rn} ⊂ 0,∞. They proved that under certain appropriate conditions imposed on{αn}and {rn}, the sequences{xn},{yn}, and {un} generated by 1.10 converge strongly to zFTIA, MEPF, where zPFS∩IA,M∩EPFfz.

Motivated and inspired by Plubtieng and Punpaeng6, Peng et al.9and Aoyama et al.10, we introduce an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem 1.8 with multi-valued maximal monotone mapping and inverse-strongly monotone mappings, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Starting with an arbitrary x1H,define sequences{xn},{yn}and{un}by

F un, y

1 rn

yun, unxn

≥0, ∀y∈H, xn1 αnγfxn I−αnBSnyn,

ynJM,λunλAun, ∀n≥0,

1.11

for alln∈N, whereλ∈0,2α,{αn} ⊂0,1, and let{rn} ⊂0,∞;Bbe a strongly bounded linear operator onH, and{Sn}is a sequence of nonexpansive mappings onH. Under suitable conditions, some strong convergence theorems for approximating to this common elements are proved. Our results extend and improve some corresponding results in3,9and the references therein.

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2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section.

LetHbe a real Hilbert space with inner product·,·and norm · , respectively.

It is wellknown that for allx, yHandλ∈0,1, there holds

λx 1−λy 2λx2 1−λ y 2λ1−λ xy 2. 2.1 LetCbe a nonempty closed convex subset ofH. Then, for anyxH, there exists a unique nearest point ofC, denoted byPCx, such thatx−PCx ≤ xyfor allyC. Such aPCis called the metric projection fromHintoC. We know thatPCis nonexpansive. It is also known that,PCxCand

x−PCx, PCxz ≥0, ∀x∈H andzC. 2.2

It is easy to see that2.2is equivalent to

x−z2xPCx2z−PCx2, ∀x∈H, zC. 2.3

For solving the equilibrium problem for a bifunctionF :C×C → R, let us assume thatFsatisfies the following conditions:

A1Fx, x 0 for allxC;

A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, yC;

A3for eachx, y, zC,

limt→0F

tz 1−tx, y

F x, y

; 2.4

A4for eachxC, yFx, yis convex and lower semicontinuous.

The following lemma appears implicitly in11and1.

Lemma 2.1See1,11. LetCbe a nonempty closed convex subset ofHand letFbe a bifunction ofC×Cin toRsatisfying (A1)–(A4). Letr >0 andxH. Then, there existszCsuch that

F z, y

1

ry−z, zx ≥0, ∀y∈C. 2.5 Define a mappingTr :HCas follows:

Trx

zC:F z, y

1 r

yz, zx

≥0, ∀y∈C

, 2.6

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for allzH. Then, the following hold:

1Tr is single-valued;

2Tr is firmly nonexpansive, that is, for anyx, yH, TrxTry2

TrxTry, xy

; 2.7

3FTr EPF;

4EPFis closed and convex.

We also need the following lemmas for proving our main result.

Lemma 2.2See12. LetHbe a Hilbert space,Ca nonempty closed convex subset ofH,f :HHa contraction with coefficient 0 < α < 1, andBa strongly positive linear bounded operator with coefficientγ >0. Then :

1if 0< γ < γ/α, thenx−y,B−γfx−B−γfy ≥γ−γαxy2, x, yH.

2if 0< ρ <B−1, thenI−ρB ≤1−ργ.

Lemma 2.3See13. Assume{an}is a sequence of nonnegative real numbers such that an1

1−γn

anδn, n≥0, 2.8

wheren}is a sequence in0,1andn}is a sequence inRsuch that 1

n1γn∞;

2lim supn→ ∞δnn0 or

n1n|<∞.

Then limn→ ∞an0.

Recall that a mapping A : HH is calledα-inverse-strongly monotone, if there exists a positive numberαsuch that

Au−Av, uv ≥αAuAv2, ∀u, v∈H. 2.9

LetIbe the identity mapping onH. It is well known that ifA:HHisα-inverse- strongly monotone, thenAis 1/α-Lipschitz continuous and monotone mapping. In addition, if 0< λ≤2α, thenIλAis a nonexpansive mapping.

A set-valuedM : H → 2H is called monotone if for allx, yH, fMxand gMyimplyx−y, fg ≥ 0. A monotone mappingM : H → 2H is maximal if its graph GM :{x, f ∈H×H |fMx}ofMis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingMis maximal if and only if forx, f∈H×H,x−y, fg ≥0 for everyy, g∈GMimpliesfMx.

Let the set-valued mapping M : H → 2H be maximal monotone. We define the resolvent operatorJM,λassociated withMandλas follows:

JM,λu IλM−1u, ∀u∈H, 2.10

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where λ is a positive number. It is worth mentioning that the resolvent operator JM,λ is single-valued, nonexpansive and 1-inverse-strongly monotone, see for example14and that a solution of problem1.8is a fixed point of the operatorJM,λI−λAfor allλ >0, see for instance15.

Lemma 2.4See14. LetM:H → 2Hbe a maximal monotone mapping andA:HHbe a Lipschitz-continuous mapping. Then the mappingSMA:H → 2His a maximal monotone mapping.

Remark 2.5See9. Lemma 2.4implies thatIA, Mis closed and convex ifM:H → 2His a maximal monotone mapping andA:HHbe an inverse strongly monotone mapping.

Lemma 2.6See10. LetCbe a nonempty closed subset of a Banach space and let{Sn}a sequence of mappings ofCinto itself. Suppose that

n1sup{Sn1zSnz :zC} < ∞. Then, for each xC,{Snx}converges strongly to some point ofC. Moreover, letSbe a mapping fromCinto itself defined by

Sx lim

n→ ∞Snx, ∀x∈C. 2.11

Then limn→ ∞sup{Sz−Snz:zC}0.

3. Main Results

We begin this section by proving a strong convergence theorem of an implicit iterative sequence {xn} obtained by the viscosity approximation method for finding a common element of the set of solutions of the variational inclusion, the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.

Throughout the rest of this paper, we always assume thatfis a contraction ofHinto itself with coefficientβ ∈ 0,1, andB is a strongly positive bounded linear operator with coefficientγand 0< γ < γ/β. LetSbe a nonexpansive mapping ofHintoH. LetA:HH be anα-inverse-strongly monotone mapping,M:H → 2Hbe a maximal monotone mapping and letJM,λbe defined as in2.10. Let{Trn}be a sequence of mappings defined asLemma 2.1.

Consider a sequence of mappings{Sn}onHdefined by

Snnγfx I−αnBSJM,λI−λATrnx, xH, n≥1, 3.1

where{αn} ⊂ 0,B−1.By Lemma 2.2, we note thatSn is a contraction. Therefore, by the Banach contraction principle,Snhas a unique fixed pointxnHsuch that

xnαnγfxn I−αnBSJM,λI−λATrnxn. 3.2

Theorem 3.1. LetH be a real Hilbert space, let F be a bifunction from H×H → Rsatisfying (A1)–(A4) and letSbe a nonexpansive mapping onH. LetA :HH be anα-inverse-strongly monotone mapping,M:H → 2Hbe a maximal monotone mapping such thatΩ:FSEPF∩ IA, M/∅.Letfbe a contraction ofHinto itself with a constantβ∈0,1and letBbe a strongly

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bounded linear operator onHwith coefficientγ >0 and 0 < γ < γ/β. Let{xn},{yn}and{un}be sequences generated byx1Hand

F un, y

1

rny−un, unxn ≥0, ∀y∈H xnαnγfxn I−αnBSyn, ynJM,λunλAun ∀n≥0,

3.3

where λ ∈ 0,2α,{rn} ⊂ 0,∞andn} ⊂ 0,1satisfy limn→ ∞αn 0 and lim infn→ ∞rn >

0. Then,{xn}, {yn} and {un} converges strongly to a point z in Ω which solves the variational inequality:

Bγf

z, zx ≤0, x∈Ω. 3.4

Equivalently, we havezPΩI−Bγfz.

Proof. First, we assume thatαn∈0,B−1. ByLemma 2.2, we obtainI−αnB ≤1−αnγ. Let v∈Ω.SinceunTrnxn,we have

unvTrnxnTrnv ≤ xnv ∀n∈N. 3.5

We note fromv∈ΩthatvJM,λv−λAv. AsIλAis nonexpansive, we have ynvJM,λunλAunJM,λv−λAv

≤ unλAun−v−λAv ≤ unv ≤ xnv 3.6

for alln∈N.Thus, we have

xnvαnγfxn I−αnBSynv

αnγfxnBvI−αnBynv

αnγfxnBv

1−αnγ

xnv

αnγ

fxnfv

γfv−Bv

1−αnγ

xnv

αnγβxnnγfv−Bv

1−αnγ

xnv

1−αn

γγβ

xnnγfv−Bv.

3.7

It follows thatxnv ≤ γfvBv/γγβ,∀n ≥ 1.Hence{xn} is bounded and we also obtain that{un},{yn},{fxn},{Syn}and{Aun}are bounded. Next, we show thatynSyn → 0. Sinceαn → 0, we note that

xnSynαnγfxnBSyn −→0 asn−→ ∞. 3.8

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Moreover, it follows fromLemma 2.1that

unv2TrnxnTrnv2≤ TrnxnTrnv, xnvunv, xnv 1

2

unv2xnv2xnun2

, 3.9

and henceunv2≤ xnv2− xnun2.Therefore, we have xnv2 αnγfxn I−αnBSynv 2

I−αnBSynv αnγfxnBv 2

1−αnγ2 Synv 2nγfxnBv, xnv

1−αnγ2 ynv 2nγfxnfv, xnvnγfv−Bv, xnv

1−αnγ2

unv2nγfxnfv, xnvnγfv−Bv, xnv

1−αnγ2

xnv2xnun2

nγβxnv2nγfv−Bvxnv

1−2αn γγβ

αnγ2

xnv2

1−αnγ2

xnun2nγfv−Bvxnv

xnv2αnγ2xnv2

1−αnγ2

xnun2nγfv−Bvxnv, 3.10

and hence

1−αnγ2xnun2αnγ2xnv2nγfv−Bvxnv. 3.11

Since{xn}is bounded andαn → 0,it follows thatxnun → 0 asn → ∞.

PutMsupn≥1{γfv−Bvxnv}. From3.10, it follows by the nonexpansive of JM,λand the inverse strongly monotonicity ofAthat

xnv2

1−αnγ2 ynv 2nγβxnv2nM

1−αnγ2

unλAun−v−λAv2nγβxnv2nM

1−αnγ2

unv2λλ−2αAunAv2

nγβxnv2nM

1−αnγ2xn−v2

1−αnγ2λλ−2αAun−Av2nγβxn−v2nM

1−2αn γ−γβ

αnγ2

xn−v2

1−αnγ2

λλ−2αAun−Av2nM

xnv2αnγ2xnv2

1−αnγ2

λλ−2αAunAv2nM,

3.12

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which implies that

1−αnγ2

λ2α−λAunAv2αnγ2xnv2nM. 3.13

Sinceαn → 0, we haveAun−Av → 0 asn → ∞. SinceJM,λis 1–inverse-strongly monotone andIλAis nonexpansive, we have

ynv 2JM,λunλAunJM,λv−λAv2

unλAun−v−λAv, ynv 1

2

un−λAun−v−λAv2 yn−v 2un−λAun−v−λAv−yn−v 2

≤ 1 2

unv2 ynv 2unynλAunAv 2 1

2

un−v2 yn−v 2un−yn 2

un−yn, Aun−Av

−λ2AunAv2 . 3.14

Thus, we have

ynv 2unv2unyn 2

unyn, AunAv

λ2AunAv2. 3.15

From3.5,3.10, and3.15, we have xnv2

1−αnγ2 ynv 2nγβxnv2nM

1−αnγ2

unv2unyn 2

unyn, AunAv

λ2AunAv2nγβxnv2nM

1−αnγ2

xnv2

1−αnγ2 unyn 22

1−αnγ2 λ

unyn, AunAv

1−αnγ2

λ2AunAv2nγβxnv2nM

1−2αn γγβ

αnγ2

xnv2

1−αnγ2 unyn 2 2

1−αnγ2 λ

unyn, AunAv

1−αnγ2

λ2AunAv2nM

xnv2αnγ2xnv2

1−αnγ2 unyn 2 2

1−αnγ2 λ

unyn, AunAv

1−αnγ2

λ2AunAv2nM.

3.16

Thus, we get

1−αnγ2 unyn 2αnγ2xnv22

1−αnγ2 λ

unyn, AunAv

1−αnγ2

λ2AunAv2nM. 3.17

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Sinceαn → 0,AunAv → 0 asn → ∞, we haveunyn → 0 asn → ∞. It follows from the inequalityynSyn ≤ ynununxnxnSynthatynSyn → 0 as n → ∞. Moreover, we havexnyn ≤ xnununyn → 0 asn → ∞.

PutUSJM,λI−λA. Since bothSandJM,λI−λAare nonexpansive, we haveU is a nonexpansive mapping onHand then we havexnαnγfxn I−αnBUTrnxnfor all n∈N. It follows byTheorem 3.1of Plubtieng and Punpaeng6that{xn}converges strongly tozFUEPF, wherezPFU∩EPFγf I−BzandB−γfz, uz ≥0, for all uFUEPF. We will show thatzFSIA, M. Since{xn}converges strongly to z, we also havexn z. Let us showzFS.Assumez /FS.Sincexnyn → 0 and xn z, we haveyn zSincez /Sz,it follows by the Opial’s condition that

lim inf

n→ ∞ ynz<lim inf

n→ ∞ ynSz ≤lim inf

n→ ∞

ynSynSynSz

≤lim inf

n→ ∞ ynz. 3.18

This is a contradiction. HencezFS.We now show thatzIA, M. In fact, since A isα–inverse-strongly monotone,Ais an 1/α-Lipschitz continuous monotone mapping and DA H. It follows fromLemma 2.4thatMAis maximal monotone. Letp, g∈GMA, that is,gApMp. Again sinceynJM,λunλAun, we haveunλAun∈IλMyn, that is,

1 λ

unynλAun

M yn

. 3.19

By the maximal monotonicity ofMA, we have

pyn, gAp−1 λ

unynλAun

≥0, 3.20

and so

pyn, g

pyn, Ap 1 λ

unynλAun

pyn, ApAynAynAun1 λ

unyn

≥0

pyn, AynAun

pyn,1 λ

unyn .

3.21

It follows fromunyn → 0,AunAyn → 0 andyn zthat

nlim→ ∞

pyn, g

pz, g

≥0. 3.22

SinceAMis maximal monotone, this implies thatθ ∈ MAz,that is,zIA, M.

Hence,z∈Ω:FS∩EPF∩IA, M. SinceFS∩IA, M FS∩FJM,λI−λA⊂FU, we haveΩ⊂FU∩EPF. It implies thatzis the unique solution of the variational inequality 3.4.

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Now we prove the following theorem which is the main result of this paper.

Theorem 3.2. LetH be a real Hilbert space, let F be a bifunction from H×H → Rsatisfying (A1)–(A4) and let{Sn}be a sequence of nonexpansive mappings on H. LetA : HH be an α-inverse-strongly monotone mapping,M : H → 2H be a maximal monotone mapping such that Ω :

n1FSnEPFIA, M/∅.Let f be a contraction ofH into itself with a constant β∈0,1and letBbe a strongly bounded linear operator onHwith coefficientγ >0 and 0< γ < γ/β.

Let{xn},{yn}and{un}be sequences generated byx1Hand F

un, y 1

rny−un, unxn ≥0, ∀y∈H xn1 αnγfxn I−αnBSnyn,

ynJM,λunλAun ∀n≥0,

3.23

for alln∈N, whereλ∈0,2α,{αn} ⊂0,1and{rn} ⊂0,∞satisfy

nlim→ ∞αn0,

n1

αn∞,

n1

n1αn|<∞,

lim inf

n→ ∞ rn>0,

n1

|rn1rn|<∞.

3.24

Suppose that

n1sup{Sn1zSnz :zK} <for any bounded subsetKofH. LetSbe a mapping ofHinto itself defined by Sx limn→ ∞Snx, for all xH and suppose thatFS

n1FSn. Then,{xn},{yn}and{un}converges strongly toz, wherez PΩI−Bγfzis a unique solution of the variational inequalities3.4.

Proof. Sinceαn → 0, we may assume thatαn≤ B−1for alln. First we will prove that{xn}is bonded. Letv∈Ω.Then, we have

xn1vαnγfxn I−αnBSnynv

αnγfxnBvI−αnBynv

αnγfxnBv

1−αnγ

xnv

αnγ

fxnfv

γfv−Bv

1−αnγ

xnv

αnγβxnnγfv−Bv

1−αnγ

xnv

1−αn

γγβ

xnnγfv−Bv

1−αn

γγα

xnn

γγαγfv−Bv γγα .

3.25

It follows from3.25and induction that xnv ≤max

x1v, 1

γγαγf p

B p

, n≥0. 3.26

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Hence {xn} is bounded and therefore {un}, {yn},{fxn},{Snyn} and {Aun} are also bounded. Next, we show that xn1xn → 0. Since IλA is nonexpansive, it follows that

yn1ynJM,λun1λAun1JM,λunλAun

≤ un1λAun1−unλAun ≤ un1un. 3.27

Then, we have

xn2xn1αn1γfxn1 I−αn1BSn1yn1αnγfxn−I−αnBSnyn αn1γfxn1 I−αn1BSn1yn1αnγfxn−I−αnBSnyn

− I−αn1BSn1yn Iαn1BSn1yn−I−αnBSn1yn IαnBSn1yn

I−αn1B

Sn1yn1Sn1yn

αnαn1BSn1yn

IαnB

Sn1yn−Snyn

αn1−αnγfxnαn1γf

xn1−fxn

1−αn1γ

yn1−ynn−αn1| BSn1yn 1−αnγ

Sn1yn−Snynnαn1| γfxnαn1γβxn1xn

1−αn1γ yn1yn αn1γβxn1xnnαn1|

BSn1ynγfxn

Sn1ynSnyn

1−αn1γ

un1unαn1γβxn1xnnαn1|M sup

Sn1zSnz:zyn

,

3.28

whereM:sup{max{BSn1yn,γfxn}:n≥0}<∞. On the other hand, we note that

F un, y

1 rn

yun, unxn

≥0, ∀y∈H, 3.29 F

un1, y 1

rn1

yun1, un1xn1

≥0, ∀y∈H. 3.30

Puttingy un1 in3.29andy un in3.30,Fun, un1 1/rnun1un, unxn ≥0 andFun1, un 1/rn1unun1, un1xn1 ≥0.ByA2, we have

un1un,unxn

rnun1xn1 rn1

≥0 3.31

and hence

un1un, unun1un1xnrn

rn1 un1xn1 ≥0. 3.32

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Since lim infn→ ∞rn >0,we assume that there exists a real numberbsuch thatrn > b >0 for alln∈N.Thus, we have

un1un2

un1un, xn1xn

1− rn

rn1

un1xn1

≤ un1un

xn1xn 1− rn

rn1

un1xn1

,

3.33

and hence

un1un ≤ xn1xn 1

rn1 |rn1rn| un1xn1

xn1xn1

b|rn1rn|L,

3.34

whereLsup{unxn:n∈N}. From3.28, we have

xn2xn1

1−αn1γ xn1xn1

b|rn1rn|L

αn1γβxn1xnnαn1|Msup

Sn1zSnz:zyn

1−αn1γαn1γβ

xn1xn

1−αn1γ

b |rn1rn|Lnαn1|Msup

Sn1zSnz:zyn

1−αn1

γγβ

xn1xnL

b|rn1rn||αnαn1|M sup

Sn1zSnz:zyn

.

3.35

Since{yn}is bounded, it follows that

n1sup{Sn1zSnz : z ∈ {yn}} < ∞. Hence, by Lemma 2.3, we havexn1xn → ∞asn → ∞. From3.34and|rn1rn| → 0, we have limn→ ∞un1un0. Moreover, we have from3.27that limn→ ∞yn1yn0.

We note from3.23thatxnαn−1γfxn−1 1−αn−1BSn−1yn−1. Then, we have xnSnyn ≤ xnSn−1yn−1Sn−1yn−1Sn−1ynSn−1ynSnyn

αn−1γfxn−1BSn−1yn−1yn−1yn sup

Sn−1zSnz:zyn

.

3.36

Sinceαn → 0,yn−1yn → 0 and sup{Sn−1zSnz:z∈ {yn}} → 0, we getxnSnyn → 0. From the proof ofTheorem 3.1, we have

unv2xnv2xnun2, 3.37

(14)

for allv∈Ω. Therefore, we have

xn1v2 αnγfxn I−αnBSnynv 2

I−αnBSnynv αnγfxnBv 2

1−αnγ2 Snynv 2nγfxnBv, xnv

1−αnγ2 ynv 2nγfxnfv, xnvnγfv−Bv, xnv

1−αnγ2

unv2nγfxnfv, xnvnγfvBv, xnv

1−αnγ2

xnv2xnun2

nγβxnv2n γfv−Bv xnv

1−2αn γγβ

αnγ2

xnv2

1−αnγ2

xnun2n γfv−Bv xnv

xnv2αnγ2xnv2

1−αnγ2

xnun2nγfv−Bvxnv 3.38

and hence

1−αnγ2xnun2αnγ2xnv2nγfv−Bvxnvxnv2xn1v2

αnγ2xnv2nγfv−Bvxnv xnxn1xnvxn1v.

3.39 Since{xn}is bounded,αn → 0 andxnxn1 → 0, it follows thatxnun → 0 asn → ∞.

PutM supn≥1{γfv−Bvxnv}. It follows from3.38, the nonexpansive of JM,λand the inverse strongly monotonicity ofAthat

xn1v2

1−αnγ2 ynv 2nγβxnv2nM

1−αnγ2

unλAun−v−λAv2nγβxnv2nM

1−αnγ2

unv2λλ−2αAunAv2

nγβxnv2nM

1−αnγ2xnv2

1−αnγ2λλ−2αAunAv2nγβxnv2nM

1−2αn

γ−γβ

αnγ2

xn−v2

1−αnγ2

λλ−2αAun−Av2nM

xnv2αnγ2xnv2

1−αnγ2

λλ−2αAunAv2nM.

3.40

参照

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