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

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

and Wanna Sriprad

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,[email protected] 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 findx∈Csuch 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:C → Hbe a nonlinear

map. The classical variational inequality which is denoted byV IA, Cis to findu∈Csuch 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 ≤ u−v, ∀u, v∈C. 1.3

A mappingf:C → Cis called contractive if there exists a constantβ∈0,1such that fu−fv ≤βu−v, ∀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. Letx1∈H and let{xn}, and{un}be sequences generated by

F un, y

1 rn

y−un, un−xn

≥0, ∀y∈H, x_n1αnγfxn 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

A−γf

z, z−x

≥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 : H → Hbe a single-valued nonlinear mapping, and letM : H → 2^H be a set-valued mapping. We consider the following variational inclusion, which is to find a point u∈Hsuch that

θ∈Au 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.

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

δ_Cx

⎧⎨

⎩

0, x∈C,

∞, 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 withx₁∈H, define sequence,{xn},{yn}, and{un}by

F u_n, y

1 r_n

y−u_n, u_n−x_n

≥0, ∀y∈H, x_n1α_nγfx_n 1−α_nSy_n, y_nJ_M,λun−λAu_n, ∀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 z ∈ FT ∩IA, M ∩EPF, 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 x₁∈H,define sequences{xn},{yn}and{un}by

F u_n, y

1 rn

y−u_n, u_n−x_n

≥0, ∀y∈H, x_n1 αnγfxn I−αnBSnyn,

y_nJ_M,λun−λAu_n, ∀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.

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, y∈Handλ∈0,1, there holds

λx 1−λy ²λx² 1−λ y ²−λ1−λ x−y ². 2.1 LetCbe a nonempty closed convex subset ofH. Then, for anyx∈H, there exists a unique nearest point ofC, denoted byP_Cx, such thatx−P_Cx ≤ x−yfor ally∈C. Such aP_Cis called the metric projection fromHintoC. We know thatPCis nonexpansive. It is also known that,PCx∈Cand

x−P_Cx, P_Cx−z ≥0, ∀x∈H andz∈C. 2.2

It is easy to see that2.2is equivalent to

x−z²≥ x−PCx²z−PCx², ∀x∈H, z∈C. 2.3

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

A1Fx, x 0 for allx∈C;

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

A3for eachx, y, z∈C,

limt→0F

tz 1−tx, y

≤F x, y

; 2.4

A4for eachx∈C, y→Fx, 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 andx∈H. Then, there existsz∈Csuch that

F z, y

1

ry−z, z−x ≥0, ∀y∈C. 2.5 Define a mappingTr :H → Cas follows:

Trx

z∈C:F z, y

1 r

y−z, z−x

≥0, ∀y∈C

, 2.6

for allz∈H. Then, the following hold:

1Tr is single-valued;

2T_r is firmly nonexpansive, that is, for anyx, y∈H, Trx−Try²≤

Trx−Try, x−y

; 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 :H → Ha 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−γαx−y², x, y∈H.

2if 0< ρ <B⁻¹, thenI−ρB ≤1−ργ.

Lemma 2.3See13. Assume{an}is a sequence of nonnegative real numbers such that a_n1 ≤

1−γ_n

a_nδ_n, n≥0, 2.8

where{γn}is a sequence in0,1and{δn}is a sequence inRsuch that 1_∞

n1γ_n∞;

2lim sup_{n→ ∞}δn/γn≤0 or_∞

n1|δn|<∞.

Then lim_{n→ ∞}a_n0.

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

Au−Av, u−v ≥αAu−Av², ∀u, v∈H. 2.9

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

A set-valuedM : H → 2^H is called monotone if for allx, y ∈ H, f ∈ Mxand g ∈ Myimplyx−y, f −g ≥ 0. A monotone mappingM : H → 2^H is maximal if its graph GM :{x, f ∈H×H |f ∈ Mx}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, f−g ≥0 for everyy, g∈GMimpliesf∈Mx.

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

J_M,λu IλM⁻¹u, ∀u∈H, 2.10

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 → 2^Hbe a maximal monotone mapping andA:H → Hbe a Lipschitz-continuous mapping. Then the mappingSMA:H → 2^His a maximal monotone mapping.

Remark 2.5See9. Lemma 2.4implies thatIA, Mis closed and convex ifM:H → 2^His a maximal monotone mapping andA:H → Hbe 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{S_n1z−S_nz :z ∈ C} < ∞. Then, for each x∈C,{Snx}converges strongly to some point ofC. Moreover, letSbe a mapping fromCinto itself defined by

Sx lim

n→ ∞Snx, ∀x∈C. 2.11

Then lim_{n→ ∞}sup{Sz−S_nz:z∈C}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:H → H be anα-inverse-strongly monotone mapping,M:H → 2^Hbe 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

Snxαnγfx I−αnBSJM,λI−λATrnx, x∈H, 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,S_nhas a unique fixed pointx_n∈Hsuch 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 :H → H be anα-inverse-strongly monotone mapping,M:H → 2^Hbe a maximal monotone mapping such thatΩ:FS∩EPF∩ IA, M/∅.Letfbe a contraction ofHinto itself with a constantβ∈0,1and letBbe a strongly

bounded linear operator onHwith coefficientγ >0 and 0 < γ < γ/β. Let{xn},{yn}and{un}be sequences generated byx₁∈Hand

F u_n, y

1

r_ny−u_n, u_n−x_n ≥0, ∀y∈H x_nα_nγfxn I−α_nBSy_n, ynJM,λun−λAun ∀n≥0,

3.3

where λ ∈ 0,2α,{rn} ⊂ 0,∞and {αn} ⊂ 0,1satisfy lim_{n→ ∞}α_n 0 and lim inf_{n→ ∞}r_n >

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

B−γf

z, z−x ≤0, x∈Ω. 3.4

Equivalently, we havezP_ΩI−Bγfz.

Proof. First, we assume thatα_n∈0,B⁻¹. ByLemma 2.2, we obtainI−α_nB ≤1−α_nγ. Let v∈Ω.Sinceu_nT_rnx_n,we have

un−vTrnx_n−T_rnv ≤ xn−v ∀n∈N. 3.5

We note fromv∈ΩthatvJ_M,λv−λAv. AsI−λAis nonexpansive, we have yn−vJM,λun−λAun−JM,λv−λAv

≤ un−λAu_n−v−λAv ≤ un−v ≤ xn−v 3.6

for alln∈N.Thus, we have

xn−vαnγfxn I−αnBSyn−v

≤αnγfxn−BvI−αnByn−v

≤α_nγfxn−Bv

1−α_nγ

xn−v

≤αnγ

fxn−fv

γfv−Bv

1−αnγ

xn−v

≤αnγβxn−vαnγfv−Bv

1−αnγ

xn−v

1−α_n

γ−γβ

xn−vα_nγfv−Bv.

3.7

It follows thatxn −v ≤ γfv−Bv/γ−γβ,∀n ≥ 1.Hence{xn} is bounded and we also obtain that{un},{yn},{fxn},{Syn}and{Aun}are bounded. Next, we show thatyn− Syn → 0. Sinceαn → 0, we note that

xn−Sy_nα_nγfxn−BSy_n −→0 asn−→ ∞. 3.8

Moreover, it follows fromLemma 2.1that

u_n−v²T_rnx_n−T_rnv²≤ Trnx_n−T_rnv, x_n−vun−v, x_n−v 1

2

un−v²xn−v²− x_n−u_n²

, 3.9

and henceun−v²≤ xn−v²− xn−un².Therefore, we have xn−v² α_nγfxn I−α_nBSyn−v ²

I−α_nBSyn−v α_nγfxn−Bv ²

≤

1−αnγ2 Syn−v ²2αnγfxn−Bv, xn−v

≤

1−α_nγ₂ y_n−v ²2α_nγfxn−fv, x_n−v2α_nγfv−Bv, x_n−v

≤

1−αnγ₂

un−v²2αnγfxn−fv, xn−v2αnγfv−Bv, xn−v

≤

1−α_nγ₂

x_n−v²− x_n−u_n²

2α_nγβx_n−v² 2α_nγfv−Bvxn−v

1−2αn γ−γβ

αnγ₂

xn−v²−

1−αnγ₂

xn−un² 2α_nγfv−Bvxn−v

≤ xn−v²αnγ²xn−v²−

1−αnγ₂

xn−un²2αnγfv−Bvxn−v, 3.10

and hence

1−α_nγ2xn−u_n²≤α_nγ²xn−v²2α_nγfv−Bvxn−v. 3.11

Since{xn}is bounded andα_n → 0,it follows thatxn−u_n → 0 asn → ∞.

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

xn−v²≤

1−αnγ2 yn−v ²2αnγβxn−v²2αnM

≤

1−α_nγ₂

un−λAu_n−v−λAv²2α_nγβx_n−v²2α_nM

≤

1−α_nγ2

un−v²λλ−2αAun−Av²

2α_nγβxn−v²2α_nM

≤

1−αnγ2xn−v²

1−αnγ2λλ−2αAun−Av²2αnγβxn−v²2αnM

1−2αn γ−γβ

αnγ₂

xn−v²

1−αnγ₂

λλ−2αAun−Av²2αnM

≤ xn−v²αnγ²xn−v²

1−αnγ₂

λλ−2αAun−Av²2αnM,

3.12

which implies that

1−αnγ₂

λ2α−λAun−Av² ≤αnγ²xn−v²2αnM. 3.13

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

y_n−v ²JM,λun−λAu_n−J_M,λv−λAv²≤

u_n−λAu_n−v−λAv, y_n−v 1

2

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

≤ 1 2

un−v² y_n−v ²− u_n−y_n−λAun−Av ² 1

2

u_n−v² y_n−v ²− u_n−yn ²2λ

u_n−yn, Au_n−Av

−λ²Au_n−Av² . 3.14

Thus, we have

y_n−v ² ≤ u_n−v²− u_n−y_n ²2λ

u_n−y_n, Au_n−Av

−λ²Aun−Av². 3.15

From3.5,3.10, and3.15, we have xn−v²≤

1−α_nγ₂ y_n−v ²2α_nγβxn−v²2α_nM

≤

1−αnγ2

un−v²− un−yn ²2λ

un−yn, Aun−Av

−λ²Aun−Av² 2α_nγβx_n−v²2α_nM

≤

1−αnγ₂

xn−v²−

1−αnγ₂ un−yn ²2

1−αnγ₂ λ

un−yn, Aun−Av

−

1−α_nγ₂

λ²Aun−Av²2α_nγβxn−v²2α_nM

1−2αn γ−γβ

αnγ2

xn−v²−

1−αnγ2 un−yn ² 2

1−αnγ₂ λ

un−yn, Aun−Av

−

1−αnγ₂

λ²Aun−Av²2αnM

≤ x_n−v²α_nγ²xn−v²−

1−α_nγ₂ u_n−y_n ² 2

1−αnγ₂ λ

un−yn, Aun−Av

−

1−αnγ₂

λ²Aun−Av²2αnM.

3.16

Thus, we get

1−α_nγ₂ u_n−y_n ²≤α_nγ²xn−v²2

1−α_nγ₂ λ

u_n−y_n, Au_n−Av

−

1−αnγ₂

λ²Aun−Av²2αnM. 3.17

Sinceαn → 0,Aun−Av → 0 asn → ∞, we haveun−yn → 0 asn → ∞. It follows from the inequalityyn−Sy_n ≤ yn−u_nun−x_nxn−Sy_nthatyn−Sy_n → 0 as n → ∞. Moreover, we havexn−yn ≤ xn−unun−yn → 0 asn → ∞.

PutU ≡SJM,λI−λA. Since bothSandJM,λI−λAare nonexpansive, we haveU is a nonexpansive mapping onHand then we havex_nα_nγfxn I−α_nBUTrnx_nfor all n∈N. It follows byTheorem 3.1of Plubtieng and Punpaeng6that{xn}converges strongly toz∈FU∩EPF, wherezP_FU∩EPFγf I−BzandB−γfz, u−z ≥0, for all u∈FU∩EPF. We will show thatz ∈FS∩IA, M. Since{xn}converges strongly to z, we also havex_n z. Let us showz ∈FS.Assumez /∈FS.Sincexn−y_n → 0 and xn z, we haveyn zSincez /Sz,it follows by the Opial’s condition that

lim inf

n→ ∞ yn−z<lim inf

n→ ∞ yn−Sz ≤lim inf

n→ ∞

yn−Sy_nSyn−Sz

≤lim inf

n→ ∞ yn−z. 3.18

This is a contradiction. Hencez ∈ FS.We now show thatz ∈ IA, 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,g−Ap∈Mp. Again sinceynJM,λun−λAun, we haveun−λAun∈IλMyn, that is,

1 λ

u_n−y_n−λAu_n

∈M y_n

. 3.19

By the maximal monotonicity ofMA, we have

p−y_n, g−Ap−1 λ

u_n−y_n−λAu_n

≥0, 3.20

and so

p−yn, g

≥

p−yn, Ap 1 λ

un−yn−λAun

p−yn, Ap−AynAyn−Aun1 λ

un−yn

≥0

p−y_n, Ay_n−Au_n

p−y_n,1 λ

u_n−y_n .

3.21

It follows fromun−yn → 0,Aun−Ayn → 0 andyn zthat

nlim→ ∞

p−y_n, g

p−z, g

≥0. 3.22

SinceAMis maximal monotone, this implies thatθ ∈ MAz,that is,z ∈ IA, 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.

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 : H → H be an α-inverse-strongly monotone mapping,M : H → 2^H be a maximal monotone mapping such that Ω : _∞

n1FSn∩EPF∩IA, 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 byx1∈Hand F

un, y 1

r_ny−un, un−xn ≥0, ∀y∈H x_n1 α_nγfxn I−α_nBS_ny_n,

ynJM,λun−λAun ∀n≥0,

3.23

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

nlim→ ∞αn0, ^∞

n1

αn∞, ^∞

n1

|αn1−αn|<∞,

lim inf

n→ ∞ r_n>0, ^∞

n1

|r_n1−r_n|<∞.

3.24

Suppose that_∞

n1sup{S_n1z−S_nz :z ∈ K} < ∞for any bounded subsetKofH. LetSbe a mapping ofHinto itself defined by Sx lim_{n→ ∞}S_nx, for all x ∈ H and suppose thatFS _∞

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

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

xn1−vαnγfxn I−αnBSnyn−v

≤α_nγfxn−BvI−α_nByn−v

≤αnγfxn−Bv

1−αnγ

xn−v

≤αnγ

fxn−fv

γfv−Bv

1−αnγ

xn−v

≤α_nγβxn−vα_nγfv−Bv

1−α_nγ

xn−v

1−αn

γ−γβ

xn−vαnγfv−Bv

1−αn

γ−γα

xn−vαn

γ−γαγfv−Bv γ−γα .

3.25

It follows from3.25and induction that xn−v ≤max

x1−v, 1

γ−γαγf p

−B p

, n≥0. 3.26

Hence {xn} is bounded and therefore {un}, {yn},{fxn},{Snyn} and {Aun} are also bounded. Next, we show that x_n1 −x_n → 0. Since I−λA is nonexpansive, it follows that

y_n1−y_nJM,λu_n1−λAu_n1−J_M,λun−λAu_n

≤ un1−λAu_n1−un−λAun ≤ un1−un. 3.27

Then, we have

x_n2−x_n1α_n1γfx_n1 I−α_n1BS_n1y_n1−α_nγfxn−I−α_nBS_ny_n α_n1γfxn1 I−α_n1BS_n1y_n1−αnγfxn−I−αnBSnyn

− I−α_n1BS_n1y_n I−α_n1BS_n1y_n−I−α_nBS_n1y_n I−αnBS_n1yn

I−α_n1B

S_n1y_n1−S_n1yn

αn−α_n1BS_n1yn

I−α_nB

S_n1y_n−Sny_n

α_n1−αnγfxnα_n1γf

x_n1−fxn

≤

1−αn1γ

yn1−yn|αn−αn1| BSn1yn 1−αnγ

Sn1yn−Snyn |αn−α_n1| γfxnα_n1γβxn1−xn

≤

1−α_n1γ y_n1−y_n α_n1γβx_n1−x_n |αn−α_n1|

BSn1ynγfxn

Sn1yn−Snyn

≤

1−α_n1γ

un1−unα_n1γβxn1−xn|αn−α_n1|M sup

S_n1z−S_nz:z∈ y_n

,

3.28

whereM:sup{max{BS_n1y_n,γfxn}:n≥0}<∞. On the other hand, we note that

F un, y

1 r_n

y−un, un−xn

≥0, ∀y∈H, 3.29 F

u_n1, y 1

r_n1

y−u_n1, u_n1−x_n1

≥0, ∀y∈H. 3.30

Puttingy u_n1 in3.29andy u_n in3.30,Fun, u_n1 1/rnu_n1−u_n, u_n−x_n ≥0 andFun1, un 1/rn1un−u_n1, u_n1−x_n1 ≥0.ByA2, we have

u_n1−un,un−xn

r_n −u_n1−x_n1 r_n1

≥0 3.31

and hence

un1−un, un−u_n1u_n1−xn− rn

r_n1 un1−x_n1 ≥0. 3.32

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

un1−un² ≤

u_n1−un, x_n1−xn

1− rn

r_n1

un1−x_n1

≤ u_n1−u_n

x_n1−x_n 1− r_n

r_n1

u_n1−x_n1

,

3.33

and hence

u_n1−u_n ≤ x_n1−x_n 1

r_n1 |r_n1−r_n| u_n1−x_n1

≤ x_n1−xn1

b|rn1−rn|L,

3.34

whereLsup{un−x_n:n∈N}. From3.28, we have

x_n2−x_n1 ≤

1−α_n1γ x_n1−x_n1

b|r_n1−r_n|L

α_n1γβx_n1−x_n |αn−α_n1|Msup

S_n1z−S_nz:z∈ y_n

1−α_n1γα_n1γβ

xn1−xn

1−α_n1γ

b |rn1−rn|L |αn−α_n1|Msup

Sn1z−Snz:z∈ yn

≤

1−α_n1

γ−γβ

xn1−xnL

b|rn1−rn||αn−α_n1|M sup

Sn1z−Snz:z∈ yn

.

3.35

Since{yn}is bounded, it follows that _∞

n1sup{S_n1z−S_nz : z ∈ {yn}} < ∞. Hence, by Lemma 2.3, we havexn1−xn → ∞asn → ∞. From3.34and|rn1−rn| → 0, we have limn→ ∞u_n1−u_n0. Moreover, we have from3.27that limn→ ∞y_n1−y_n0.

We note from3.23thatx_nα_n−1γfx_n−1 1−α_n−1BS_n−1y_n−1. Then, we have xn−Snyn ≤ xn−S_n−1y_n−1Sn−1y_n−1−S_n−1ynSn−1yn−Snyn

≤α_n−1γfx_n−1−BS_n−1y_n−1y_n−1−y_n sup

Sn−1z−Snz:z∈ yn

.

3.36

Sinceα_n → 0,y_n−1−y_n → 0 and sup{S_n−1z−S_nz:z∈ {yn}} → 0, we getxn−S_ny_n → 0. From the proof ofTheorem 3.1, we have

un−v²≤ x_n−v²− x_n−u_n², 3.37

for allv∈Ω. Therefore, we have

xn1−v² αnγfxn I−αnBSnyn−v ²

I−α_nBSny_n−v α_nγfxn−Bv ²

≤

1−αnγ₂ Snyn−v ²2αnγfxn−Bv, xn−v

≤

1−αnγ2 yn−v ²2αnγfxn−fv, xn−v2αnγfv−Bv, xn−v

≤

1−α_nγ₂

u_n−v²2α_nγfx_n−fv, x_n−v2α_nγfv−Bv, x_n−v

≤

1−α_nγ2

xn−v²− x_n−u_n²

2α_nγβxn−v² 2αn γfv−Bv xn−v

1−2α_n γ−γβ

α_nγ₂

xn−v²−

1−α_nγ₂

xn−u_n² 2αn γfv−Bv xn−v

≤ x_n−v²α_nγ²x_n−v²−

1−α_nγ₂

x_n−u_n²2α_nγfv−Bvxn−v 3.38

and hence

1−αnγ2xn−un²≤αnγ²xn−v²2αnγfv−Bvxn−vxn−v²− x_n1−v²

≤α_nγ²x_n−v²2α_nγfv−Bvxn−v xn−x_n1xn−vx_n1−v.

3.39 Since{xn}is bounded,αn → 0 andxn−x_n1 → 0, it follows thatxn−un → 0 asn → ∞.

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

xn1−v²≤

1−αnγ2 yn−v ²2αnγβxn−v²2αnM

≤

1−α_nγ₂

un−λAu_n−v−λAv²2α_nγβx_n−v²2α_nM

≤

1−α_nγ2

un−v²λλ−2αAun−Av²

2α_nγβxn−v²2α_nM

≤

1−αnγ2xn−v²

1−αnγ2λλ−2αAun−Av² 2αnγβxn−v²2αnM

1−2αn

γ−γβ

αnγ₂

xn−v²

1−αnγ₂

λλ−2αAun−Av²2αnM

≤ xn−v²αnγ²xn−v²

1−αnγ₂

λλ−2αAun−Av²2αnM.

3.40