Nonexpansive Mappings

Volume 2012, Article ID 816529,26pages doi:10.1155/2012/816529

Research Article

Viscosity Approximation Method for System of Variational Inclusions Problems and Fixed-Point Problems of a Countable Family of

Nonexpansive Mappings

Chaichana Jaiboon

and Poom Kumam

1Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin (RMUTR), Bangkok 10100, Thailand

2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

Correspondence should be addressed to Chaichana Jaiboon,chaichana.j@rmutr.ac.thand Poom Kumam,poom.kum@kmutt.ac.th

Received 22 January 2012; Accepted 2 March 2012 Academic Editor: Yonghong Yao

Copyrightq2012 C. Jaiboon and P. Kumam. 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.

We propose new iterative schemes for finding the common element of the set of common fixed points of countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous, the set of solutions of system of variational inclusions problem, and the set of solutions of equilibrium problems in a real Hilbert space by using the viscosity approximation method. We prove strong convergence theorem under some parameters. The results in this paper unify and generalize some well-known results in the literature.

1. Introduction

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. A mappingSofCinto itself is called nonexpansive ifSx−Sy ≤ x−yfor allx, y ∈ C. We denote byFSthe set of fixed points ofS; that is, FS {x ∈ C : Sx x}. If C ⊂ H is nonempty, closed and convex and letS :C → Cbe a nonexpansive mapping, thenFSis closed and convex andFS/∅, whenCis bounded; see, for example,1,2. The metric projection,P_C, onto a given nonempty, closed and convex subsetC, satisfies the nonexpansive withFPC C. A mappingB:C → Cis called monotone ifBx−By, x−y ≥0 for allx, y∈C. A mapping B : C → Cis calledβ-inverse-strongly monotone if there exists a constantβ > 0 such that

Bx−By, x−y ≥ βx−y², for all x, y ∈ C. A mapping B : C → Cis called relaxed φ, ω-cocoercive if there existsφ, ω >0 such that

Bx−By, x−y

≥

−φBx−By²ωx−y², ∀x, y∈C. 1.1 A mappingB :C → Cis said to beξ-Lipschitz continuous if there existsξ ≥ 0 such that

Bx−By≤ξx−y, ∀x, y∈C. 1.2 Let B : H → H be a single-valued nonlinear mapping and M : H → 2^H a multivalued mapping. The variational inclusion problem is to findx∈Hsuch that

θ∈Bx Mx, 1.3

whereθis the zero vector inH. The set of solutions of problem1.3is denoted byIB, M.

IfM∂ψ_C, whereCis a nonempty closed convex subset ofHand∂ψ_C :H → 0,∞is the indicator function ofC; that is,

ψ_Cx

0, x∈C,

∞, x /∈C, 1.4 then, the variational inclusion problem 1.3 is equivalent to the variational inequality problems denoted by VIC, Bwhich is to findx∈Csuch that

Bx, y−x

≥0, ∀y∈C. 1.5

In 2003, Takahashi and Toyoda3to findx^∗∈FS∩VIC, Bintroduced the following iterative scheme:

x0∈C chosen arbitrary,

x_n1αnxn 1−αn SPCxn−ξnBxn, ∀n≥0, 1.6 whereBis aβ-inverse-strongly monotone mapping,{αn}is a sequence in0, 1, and{ξn}is a sequence in0,2β. They showed that ifFS∩VIC, Bis nonempty, then the sequence{xn} generated by1.6converges weakly to somex^∗∈FS∩VIC, B.

In 2008, Zhang et al.4to findx^∗ ∈ FS∩IM, B. They introduced the following new iterative scheme:

x₀∈C chosen arbitrary, ynJM,λxn−λBxn, x_n1αnx 1−αnSyn, ∀n≥0,

1.7

whereJM,λ IλM⁻¹is the resolvent operator associated withMand a positive number λ,{αn}is a sequence in the interval0,1.

LetFbe a bifunction ofC×CintoR, whereRis the set of real numbers. The equilibrium problem forF:C×C → Ris to findx∈Csuch that

F x,y

≥0, ∀y∈C. 1.8

The set of solutions of1.8is denoted by EPF. Many problems in applied sciences, such as monotone inclusion problems, variational inequality problems, saddle point problems, Nash equilibria in noncooperative games, as well as certain fixed-point problems reduce to finding some element to EPFin Hilbert and Banach spacessee5–14.

Given anyr >0. The operatorT_r:H → Cdefined by

T_rx z∈C:F z, y

1 r

y−z, z−x

≥0, ∀y∈C

, 1.9

is called the resolvent ofFsee5,6.

It is shown in 6 that, under suitable hypotheses on F to be stated precisely in Section 2,T_r:H → Cis single valued and firmly nonexpansive and satisfies

FTr EPF, ∀r >0. 1.10 Using this result, for finding an element of FS∩VIC, B∩EPF, Su et al. 15 introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces:

x0∈Cchosen arbitrary,

x_n1 α_nfxn 1−α_nSPCI−ξ_nBTrnx_n, ∀n≥0, 1.11 wheref : C → Cis a contractioni.e.,fx−fy ≤ ψx−y,for allx, y ∈ Cand 0 ≤ ψ < 1and{αn} ⊂ 0,1,ξn ⊂ 0,2β, andrn ⊂ 0,∞satisfy some appropriate conditions.

Furthermore, they prove{xn}converges strongly to the same point x^∗ ∈ FS∩VIC, B∩ EPF, wherex^∗PFS∩VIC,B∩EPFfx^∗.

In this paper, motivated and inspired by the above facts, we introduce a new iterative scheme for finding a common element of the set of solutions of the variational inequalities for μ-Lipschitz continuous and relaxedφ, ω-cocoercive mapping, the set of solutions to the variational inclusion for family of α-inverse strongly monotone mappings, the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of an equilibrium problem in a real Hilbert space by using the viscosity approximation method.

Strong convergence results are derived under suitable conditions in a real Hilbert space.

2. Preliminaries

In this section, we will recall some basic notations and collect some conclusions that will be used in the next section.

Let H be a real Hilbert space whose inner product and norm are denoted by ·,· and · , respectively. We denote strong convergence of{xn} to x ∈ H byx_n → xand

weak convergence by xn x. Let C be nonempty closed convex subset of H. Recall that for all x ∈ H there exists a unique nearest point in C to x denoted P_Cx; that is, x−P_Cx ≤ x−y, for ally∈C. The mappingP_C is nonexpansive; that is,PCx−P_Cy ≤ x−y, for allx, y ∈ H. The mappingPC is firmly nonexpansive; that is,PCx−PCy² ≤ PCx−PCy, x−y , for allx, y∈H. It is well known that

x∈VIC, B⇔xPCx−λBx, ∀λ >0. 2.1 A set-valued mappingM : H → 2^H is called monotone if, for allx, y ∈ H,f ∈ Mxand g ∈Myimplyx−y, f−g ≥ 0. A monotone mappingM:H → 2^His called maximal, if its graph of any GraphM:{x, f∈H×H |f ∈Mx}ofMis not properly contained in the graph of any other monotone mapping. It is well known that a monotone mappingM is maximal if and only if for allx, f∈H×H,x−y, f−g ≥0, for ally, g∈GraphM the graph of mappingMimplies thatf∈Mx.

Definition 2.1. Let M : H → 2^H be a multivalued maximal monotone mapping; then the set-valued mappingJ_M,λ:H → Hdefined by

J_M,λx I λM⁻¹x, ∀x∈H, 2.2 is called the resolvent operator associated withM, whereλis any positive number andIis the identity mapping.

Lemma 2.2see16. LetM:H → 2^Hbe a maximal monotone mapping and letB :H → H be a Lipschitz continuous mapping. Then the mappingMB : H → 2^H is a maximal monotone mapping.

Lemma 2.3see16,17.

1The resolvent operator J_M,λis single valued and nonexpansive for allλ >0; that is, JM,λx−JM,λ

y ≤x−y, ∀x, y∈H, ∀λ >0. 2.3

2The resolvent operatorJM,λis 1-inverse-strongly monotone; that is, J_M,λx−J_M,λ

y²≤

x−y, J_M,λx−J_M,λ y

, ∀x, y∈H. 2.4 Lemma 2.4see17.

1Letx∈His a solution of problem1.3if and only ifxJ_M,λI−λBfor allλ >0; that is,

IB, M FJM,λI−λB, ∀λ >0. 2.5

2Ifλ∈0,2β, thenIB, Mis a closed convex subset inH.

Lemma 2.5see18. Each Hilbert spaceH satisfies Opial’s condition; that is, for any sequence {xn} ⊂Hwithx_n x, the inequality

lim inf

n→ ∞ xn−x<lim inf

n→ ∞ x_n−y 2.6

holds for eachy∈Hwithy /x.

Lemma 2.6see19. Let{xn}and{zn}be bounded sequences in a Banach spaceE, and let{βn}be a sequence in0,1with 0<lim inf_{n→ ∞}β_n≤lim sup_{n→ ∞}β_n<1. Supposex_n1 1−β_nznβ_nx_n for all integersn≥1 and lim sup_{n→ ∞}z_n1−zn − x_n1−xn≤0. Then, limn→ ∞zn−xn0.

Lemma 2.7see20. Assume{an}is a sequence of nonnegative real numbers such that

a_n1 ≤1−bnanδn, n≥0, 2.7 where{bn}is a sequence in0,1and{δn}is a sequence inRsuch that

1_∞

n1bn∞,

2lim sup_{n→ ∞}δ_n/b_n≤0 or_∞

n1|δn|<∞.

Then limn→ ∞an0.

Lemma 2.8. LetHbe a real Hilbert space. Then hold the following identities:

itx 1−ty²tx² 1−ty²−t1−tx−y², ∀t∈0,1, ∀x, y∈H, iixy²≤ x²2y, xy , ∀x, y∈H.

Lemma 2.9 see21. LetC be a nonempty closed subset of a Banach space, and let{Sn} be a sequence of mappings ofCinto itself. Suppose that_∞

n1sup{Sn1z−Snz:z ∈C} <∞. Then, for eachy∈C,{Sny}converges strongly to some point ofC. Moreover, letSbe a mapping ofCinto itself defined by

Sy lim

n→ ∞Sny, ∀y∈C. 2.8

Then lim_{n→ ∞}sup{Sz−S_nz:z∈C}0.

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, ∀ x, y∈C;

A3for eachx, y, z∈C, lim_t↓0Ftz 1−tx, y≤Fx, y;

A4for eachx∈C, y→Fx, yis convex and lower semicontinuous.

Lemma 2.10see5. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction of C×CintoRsatisfying (A1)–(A4). Letr >0 andx∈H. Then, there existsz∈Csuch that

F z, y

1 r

y−z, z−x

≥0, ∀y∈C. 2.9

Lemma 2.11see6. Assume thatF :C×C → Rsatisfies (A1)–(A4). Forr > 0 andx ∈H, define a mappingT_r :H → Cas follows:

Trx z∈C:F z, y

1 r

y−z, z−x

≥0, ∀y∈C

, 2.10

for allx∈H. Then, the following hold:

iT_r is single valued;

iiT_r is firmly nonexpansive; that is, for anyx, y∈HTrx−T_ry²≤ Trx−T_ry, x−y ; iiiFTr EPF;

ivEPFis closed and convex.

Lemma 2.12 see22. LetH be a Hilbert space andM a maximal monotone onH. Then, the following holds:

JM,rx−JM,sx²≤ r−s

r JM,rx−JM,sx, JM,rx−x , ∀s,r >0, x∈H, 2.11 whereJ_M,r IrM⁻¹andJ_M,s IsM⁻¹.

3. Main Results

In this section, we will use the viscosity approximation method to prove a strong convergence theorem for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of solutions of the variational inequality problem for relaxed cocoercive and Lipschitz continuous mappings, the set of solutions of system of variational inclusions, and the set of solutions of equilibrium problem in a real Hilbert space.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and letB:C → Hbe relaxedφ, ω-cocoercive andμ-Lipschitz continuous withω > φμ², for someφ, ω, μ >0. Let G{Gk:k1,2,3, . . . , N}be a finite family ofβ-inverse strongly monotone mappings fromCinto H, and letFbe a bifunction fromC×C → Rsatisfying (A1)–(A4). Letf :C → Cbe a contraction with coefficientψ 0 ≤ψ <1, and let{Sn}be a sequence of nonexpansive mappings ofCinto itself such that

Ω: ∞ n1

FSn∩ _N

k1

IGk, M_k

∩VIC, B∩EPF/∅. 3.1

Let the sequences{xn}and{yn}be generated by x1x∈Cchosen arbitrarily,

ynJMN,λN,nI−λN,nGn. . . JM2,λ2,nI−λ2,nG2JM1,λ1,nI−λ1,nG1Trnxn, x_n1αnfxn βnxnγnSnPC

yn−ξnByn

, ∀n≥1,

3.2

where{αn},{βn},{γn} ⊂0,1and{ξn},{rn} ⊂0,∞satisfy the following conditions:

C1α_nβ_nγ_n1, C2limn→ ∞αn0, _∞

n1αn∞,

C30<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1,

C4{ξn} ⊂a, bfor somea,bwith 0≤a≤b≤2ω−φμ²/μ²and lim_{n→ ∞}|ξn1−ξ_n|0, C5{λk,n}^N_k1⊂c, d⊂0,2βand lim_{n→ ∞}|λk,n1−λ_k,n|0, for eachk∈ {1,2, . . . , N}, C6lim inf_{n→ ∞}r_n>0 and lim_{n→ ∞}|r_n1−r_n|0.

Suppose that_∞

n1sup{Sn1z−Snz : z ∈ K} < ∞for any bounded subset K ofC. LetS be a mapping ofCinto itself defined by Sy limn→ ∞Snyfor ally ∈ C and suppose thatFS _∞

n1FSn. Then, the sequences{xn}and{yn}converge strongly to the same pointx^∗ ∈Ω, where x^∗P_Ωfx^∗.

Proof. First, we prove that the mappingP_Ωf :H → Chas a unique fixed point.

In fact, sincef :C → Cis a contraction withψ ∈0,1andP_Ωf : H → Ωis also a contraction, we obtain

P_Ωfx−P_Ωf

y≤fx−f

y≤ψx−y, ∀x, y∈C. 3.3 Therefore, there exists a unique elementx^∗∈Csuch thatx^∗P_Ωfx^∗, where

Ω: ∞ n1

FSn∩ _N

k1

IGk, Mk

∩VIC, B∩EPF. 3.4

Now, we prove thatI−ξ_nBis nonexpansive.

Indeed, for anyx, y ∈ C, sinceB : C → H is aμ-Lipschitz continuous and relaxed φ, ω-cocoercive mappings withω > φμ²andξn≤2ω−φμ²/μ², we obtain

I−ξnBx−I−ξnBy²x−y

−ξn

Bx−By² x−y²−2ξn

x−y, Bx−By

ξ_n²Bx−By²

≤x−y²−2ξn

−φBx−By²ωx−y²

ξ²_nBx−By²

≤x−y²2ξ_nφμ²x−y²−2ξ_nωx−y²ξ²_nμ²x−y²

12ξ_nφμ²−2ξ_nωξ²_nμ²x−y²

1−ξ_nμ²

2

ω−φμ² μ² −ξ_n

x−y²

≤

1−ξ_nμ² 2

ω−φμ² μ² −b

x−y².

3.5 Setting

ζ μ² 2

2

ω−φμ² μ² −b

>0, 3.6

thus,

I−ξ_nBx−I−ξ_nBy²≤1−2ξ_nζx−y²≤1−ξ_nζ²x−y², 3.7

which implies that

I−ξ_nBx−I−ξ_nBy≤1−ξ_nζx−y≤x−y. 3.8

HenceI−ξ_nBis nonexpansive.

We divide the proof ofTheorem 3.1into five steps.

Step 1. We show that the sequence{xn}is bounded.

Now, letx∈Ωand if{Trn}is a sequence of mappings defined as inLemma 2.11, then

xPC x−λn Bx Trnx, and let un Trnxn. So, we have

un−x Trnx_n−T_rnx ≤ x n−x. 3.9

Fork∈ {1,2, . . . , N}and for any positive integer numbern, we define the operatorΥ^k_n:C → Has follows:

Υ⁰_nxx,

Υ^k_nxJ_Mk,λk,nI−λ_k,nG_k. . . J_M2,λ2,nI−λ_2,nG₂JM1,λ1,nI−λ_1,nG₁x , 3.10

for all n, we gety_n Υ^N_nu_n. On the other hand, since G_k : C → H isβ-inverse strongly monotone andλk,n ⊂ c, d ⊂ 0,2β, then JMk,λk,nI −λk,nGkis nonexpansive. Thus Υ^k_n is nonexpansive. FromLemma 2.41, we havex Υ^N_nx. It follows that

y_n−xΥ_n^Nu_n−Υ^N_nx≤ u_n−x ≤ x n−x. 3.11

SettingvnPCyn−ξnBynandI−ξnBis a nonexpansive mapping, we obtain

vn−x P_C

y_n−ξ_nBy_n

−P_Cx−ξ_nBx

≤yn−ξnByn

−x−ξnBx I−ξnByn−I−ξnBx

≤yn−x≤ xn−x.

3.12

From3.2and3.12, we deduce that

xn1−x α_nfxn β_nx_nγ_nS_nv_n−x

≤α_nfxn−xβ_nxn−x γ_nvn−x

≤α_nfxn−fx α_nfx−xβ_nxn−x γ_nxn−x

≤αnψxn−x αnfx −x 1−αnxn−x

≤ 1−αn

1−ψ

xn−x αnfx −x

1−α_n 1−ψ

xn−x α_n

1−ψfx −x 1−ψ

≤max

xn−x, fx−x 1−ψ

.

3.13

It follows from induction that

xn−x ≤ max

x1−x, fx −x 1−ψ

, ∀n≥1. 3.14

Therefore,{xn}is bounded and hence so are{vn},{yn},{un},{Byn}, and{Snvn}.

Step 2. We claim that lim_{n→ ∞}x_n1−x_n0.

By the definition ofTr,unTrnxnandu_n1Tr_n1x_n1, we get

F un, y

1 rn

y−un, un−xn

≥0, ∀y∈H, 3.15

F u_n1, y

1 r_n1

y−u_n1, u_n1−x_n1

≥0, ∀y∈H. 3.16

Takingyu_n1in3.15andyu_nin3.16, we have

Fun, u_n1 1

r_nun1−u_n, u_n−x_n ≥0, 3.17 and hence

Fun1, u_n 1

r_n1un−u_n1, u_n1−x_n1 ≥0. 3.18 So, fromA2we have

u_n1−u_n,u_n−x_n

r_n −u_n1−x_n1 r_n1

≥0, 3.19

and hence

u_n1−u_n, u_n−u_n1u_n1−x_n− rn

r_n1un1−x_n1

≥0. 3.20

Without loss of generality, let us assume that there exists a real numbercsuch thatrn> c >0 for alln∈N. Then, we have

un1−un²≤

u_n1−un, x_n1−xn

1− rn

r_n1

un1−x_n1

≤ u_n1−un xn1−xn 1− rn

r_n1

un1−x_n1

,

3.21

and hence

un1−un ≤ xn1−xn 1

r_n1|rn1−rn|un1−x_n1

≤ x_n1−xnM1

c |rn1−rn|,

3.22

whereM1sup{un−xn:n∈N}.

Notice fromLemma 2.12that y_n1−y_nΥ^N_n1u_n1−Υ^N_nu_n

≤

u_n1−λ_k,n1G_kΥ^k_n1u_n1

−

u_n−λ_k,nG_kΥ^k_nu_n JMk, λ_k,n1

un−λk,nGkΥ^k_nun

−JMk, λk,n

un−λk,nGkΥ^k_nun

≤ u_n1−u_n|λ_k,n1−λ_k,n|G_kΥ^k_nu_n |λk,n1−λ_k,n|

λ_k,n1

JMk, λ_k,n1

un−λk,nGkΥ_n^kun

−

un−λk,nGkΥ^k_nun

≤ u_n1−un2M2|λk,n1−λk,n|

≤ x_n1−x_nM1

c |rn1−r_n|2M₂|λk,n1−λ_k,n|,

3.23

whereM₂is an appropriate constant such that

M₂max

sup

n≥1

G_kΥ_n^ku_n , sup

n≥1

JMk, λ_k,n1

un−λk,nGkΥ^k_nun

−

un−λk,nGkΥ_n^kun JMk, λ_k,n1

. 3.24

SinceI−ξnBis nonexpansive mappings, we have the following estimates:

vn1−vn ≤PC

y_n1−ξ_n1By_n1

−PC

yn−ξnByn

≤y_n1−ξ_n1By_n1

−

yn−ξnByn y_n1−ξ_n1By_n1

−

yn−ξ_n1Byn

ξn−ξ_n1Byn

≤y_n1−ξ_n1By_n1

−

y_n−ξ_n1By_n|ξn−ξ_n1|By_n I−ξ_n1By_n1−I−ξ_n1Byn|ξn−ξ_n1|By_n

≤y_n1−y_n|ξn−ξ_n1|By_n.

3.25

Substituting3.23into3.25, we obtain v_n1−v_n ≤ x_n1−x_nM₁

c |r_n1−r_n|2M₂|λ_k,n1−λ_k,n| |ξn−ξ_n1|Byn.

3.26

Indeed, definex_n1 1−β_nznβ_nx_nfor alln∈N. It follows that zn x_n1−β_nx_n

1−β_n α_nfxn γ_nS_nv_n

1−β_n . 3.27

Thus, we have

zn1−zn

α_n1fxn1 γ_n1S_n1v_n1

1−β_n1 −αnfxn γnSnvn

1−β_n

α_n1 1−β_n1

fxn1−fxn

γ_n1

1−β_n1Sn1v_n1−S_nv_n

α_n1

1−β_n1 − α_n 1−β_n

fxn

γ_n1

1−β_n1 − γ_n 1−β_n

S_nv_n

≤ α_n1

1−β_n1fx_n1−fxn γ_n1

1−β_n1Sn1v_n1−Snvn

α_n1

1−β_n1 − αn

1−β_n

fxn−Snvn

≤ ψα_n1

1−β_n1xn1−x_n γ_n1

1−β_n1Sn1v_n1−S_nv_n

α_n1

1−β_n1 − α_n 1−βn

fxn−S_nv_n.

3.28

Now, compute

Sn1v_n1−S_nv_n ≤ Sn1v_n1−S_n1v_nSn1v_n−S_nv_n

≤ v_n1−v_nSn1v_n−S_nv_n

≤ x_n1−x_n M₁

c |r_n1−r_n||ξn−ξ_n1|By_n 2M2|λk,n1−λk,n|Sn1vn−Snvn.

3.29

Combining3.28and3.29, we have

zn1−zn ≤ ψα_n1

1−β_n1xn1−xn γ_n1

1−β_n1 xn1−xnM1

c |rn1−rn||ξn−ξ_n1|Byn 2M2|λk,n1−λ_k,n|Sn1v_n−S_nv_n

α_n1

1−β_n1 − α_n 1−β_n

fxn−S_nv_n

≤ x_n1−x_n γ_n1 1−β_n1

M₁

c |r_n1−r_n||ξn−ξ_n1|By_n 2M2|λk,n1−λk,n|

γ_n1

1−β_n1Sn1vn−Snvn

α_n1

1−β_n1 − αn

1−β_n

fxn−Snvn.

3.30

It follows that

z_n1−z_n − x_n1−x_n

≤ γ_n1 1−β_n1

M1

c |rn1−rn||ξn−ξ_n1|Byn2M2|λk,n1−λk,n| γ_n1

1−β_n1Sn1v_n−S_nv_n α_n1

1−β_n1 − αn

1−β_n

fxn−S_nv_n

≤ γ_n1 1−β_n1

M₁

c |r_n1−r_n||ξn−ξ_n1|By_n2M₂|λ_k,n1−λ_k,n| γ_n1

1−β_n1sup{S_n1z−Snz:z∈ {vn}}

α_n1

1−β_n1 − α_n 1−βn

fxn−Snvn. 3.31

This together with conditions C1–C6and limn→ ∞sup{S_n1z−Snz : z ∈ {vn}} 0 implies that

lim sup

n→ ∞ z_n1−zn − xn1−xn≤0. 3.32

Hence, byLemma 2.6, we obtainzn−xn → 0 asn → ∞. It then follows that

nlim→ ∞x_n1−x_n lim

n→ ∞

1−β_n

zn−x_n0. 3.33

By3.26, we also have

nlim→ ∞vn1−vn0. 3.34

Step 3. We claim that limn→ ∞Svn−vn0.

Since{Gk:k1,2,3, . . . , N}isβ-inverse strongly monotone mappings, by the choice of{λk,n}for givenx∈Ωandk∈ {0,1,2, . . . , N−1}, we also have

Υ^k1_n u_n−x²

JMk1,λk1,nI−λ_k1,nG_k1Υ^k_nun−JMk1,λk1,nI−λ_k1,nG_k1x²

≤I−λ_k1,nG_k1Υ^k_nu_n−I−λ_k1,nG_k1x²

Υ^k_nu_n−λ_k1,nG_k1Υ^k_nu_n

−x−λ_k1,nG_k1x ²

Υ^k_nun−x

−λ_k1,n

G_k1Υ_n^kun−G_k1x² Υ^k_nu_n−x²−2λ_k1,n

Υ^k_nu_n−x, G _k1Υ^k_nu_n−G_k1x

λ²_k1,nG_k1Υ^k_nu_n−G_k1x²

≤Υ^k_nu_n−x²−2λ_k1,nβG_k1Υ^k_nu_n−G_k1xλ²_k1,nG_k1Υ_n^ku_n−G_k1x²

≤ un−x ²−2λ_k1,nβGk1Υ^k_nun−G_k1xλ²_k1,nGk1Υ^k_nun−G_k1x²

≤ x_n−x ²λ_k1,n

λ_k1,n−2βG_k1Υ^k_nu_n−G_k1x².

3.35

Form3.13, we have

xn1−x ² ≤αnfxn−x²βnxn−x ²γnvn−x ²

≤αnfxn−x²βnxn−x ²γnyn−x² αnfxn−x²βnxn−x ²γnΥ^N_nun−x²

≤αnfxn−x²βnxn−x ²γnΥ^k1_n un−x²

≤αnfxn−x²βnxn−x ² γ_n xn−x ²λ_k1,n

λ_k1,n−2βG_k1Υ^k_nu_n−G_k1x²

≤α_nfxn−x²xn−x ²γ_nλ_k1,n

λ_k1,n−2βG_k1Υ^k_nu_n−G_k1x². 3.36

It follows that

γnλ_k1,n

2β−λ_k1,nGk1Υ^k_nun−G_k1x²

≤γnc

2β−dGk1Υ^k_nun−G_k1x²

≤ xn−x_n1xn−x xn1−x αnfxn−x².

3.37

By conditionC2,3.33, and lim infn→ ∞γn>0, we obtain

nlim→ ∞

Gk1Υ^k_nun−G_k1x0. 3.38

FromLemma 2.32and asI−λ_k1,nG_k1is nonexpansive, we have Υ^k1_n u_n−x²

JMk1,λk1,nI−λ_k1,nG_k1Υ^k_nu_n−J_Mk1,λk1,nI−λ_k1,nG_k1x²

≤

I−λ_k1,nG_k1Υ^k_nun−I−λ_k1,nG_k1x, Υ^k1_n un−x 1

2 I−λ_k1,nG_k1Υ^k_nun−I−λ_k1,nG_k1x²Υ^k1_n un−x²

−I−λ_k1,nG_k1Υ_n^kun−I−λ_k1,nG_k1x−

Υ^k1_n un−x²

≤ 1

2 Υ^knun−x²Υ^k1_n un−x²−

Υ^k_nun−Υ^k1_n un

−λ_k1,n

G_k1Υ^k_nun−G_k1x²

≤ 1

2 Υ^knun−x²Υ^k1_n un−x²−Υ^k_nun−Υ^k1_n un²

−λ²_k1,nGk1Υ^k_nu_n−G_k1x²2λ_k1,n

Υ^k_nu_n−Υ^k1_n u_n, G_k1Υ^k_nu_n−G_k1x ,

3.39 which yields that

Υ^k1_n u_n−x²

≤Υ_n^ku_n−x²−Υ^k_nu_n−Υ^k1_n u_n²2λ_k1,nΥ^k_nu_n−Υ^k1_n u_nGk1Υ_n^ku_n−G_k1x

≤ un−x ²−Υ^k_nun−Υ^k1_n un²2λ_k1,nΥ^k_nun−Υ^k1_n unGk1Υ^k_nun−G_k1x

≤ x_n−x ²−Υ^k_nu_n−Υ^k1_n u_n²2λ_k1,nΥ^k_nu_n−Υ^k1_n u_nG_k1Υ^k_nu_n−G_k1x.

3.40

Substituting3.40into3.36, we obtain

x_n1−x ²≤α_nfxn−x²β_nxn−x ²γ_nΥ^k1_n u_n−x²

≤α_nfxn−x²β_nxn−x ²γ_n xn−x ²−Υ^k_nu_n−Υ^k1_n u_n²

2λk1,nΥ^k_nu_n−Υ^k1_n u_nGk1Υ^k_nu_n−Gk1x

≤αnfxn−x²xn−x ²−γnΥ^k_nun−Υ^k1_n un² 2λ_k1,nγ_nΥ^k_nu_n−Υ^k1_n u_nG_k1Υ^k_nu_n−G_k1x.

3.41