Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

Volume 2011, Article ID 347204,30pages doi:10.1155/2011/347204

Research Article

Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

Poom Kumam

and Chaichana Jaiboon

1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

3Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin (RMUTR), Bangkok 10100, Thailand

Correspondence should be addressed to Chaichana Jaiboon,[email protected] Received 3 October 2010; Accepted 5 March 2011

Academic Editor: Jong Kim

Copyrightq2011 P. Kumam and C. Jaiboon. 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 introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

1. Introduction

Throughout this paper, we denote byNandRthe sets of positive integers and real numbers, respectively. LetHbe a real Hilbert space with inner product·,·and norm·, and letEbe a nonempty closed convex subset ofH. We denote weak convergence and strong convergence by notationsand →, respectively. Recall that a mappingf :E → Eis anα-contraction on Eif there exists a constantα∈0,1such thatfx−fy ≤ αx−yfor allx, y∈E. Let S:E → Ebe a mapping. In the sequel, we will useFSto denote the set of fixed points ofS;

that is,FS {x∈E:Sxx}. In addition, let a mappingS:E → Ebe called nonexpansive, ifSx−Sy ≤ x−y, for allx, y∈E. It is well known that ifE⊂His nonempty, bounded, closed, and convex andSis a nonexpansive self-mapping onE, thenFSis nonempty; see,

for example,1. Recall that a mappingS:E → Eis called strictly pseudo-contraction if there exists a constantk∈0,1such that

Sx−Sy²≤x−y²kI−Sx−I−Sy², ∀x, y∈E, 1.1 where I denotes the identity operator on E. Note that if k 0, then Sis a nonexpansive mapping. The class of strict pseudo-contractions is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strict pseudo- contractions. In 1967, Browder and Petryshyn2introduced a convex combination method to study strict pseudo-contractions in Hilbert spaces. On the other hand, Marino and Xu3 and Zhou4developed some iterative scheme for finding a fixed point of a strict pseudo- contraction mapping. More precisely, takek∈0,1and define a mappingS_kby

Skxkx 1−kSx, ∀x∈E, 1.2

where S is a strict pseudo-contraction. Under appropriate restrictions on k, it is proved that the mappingSk is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudo-contractions.

Let ϕ : E → R ∪ {∞} be a proper extended real-valued function and let φ be a bifunction ofE×EintoRsuch thatE∩domϕ /∅, whereRis the set of real numbers and domϕ{x∈E:ϕx<∞}. Ceng and Yao5considered the following mixed equilibrium problems for findingx∈Esuch that

φ x, y

ϕ y

−ϕx≥0, ∀y∈E. 1.3 The set of solutions of1.3is denoted by MEPφ, ϕ, that is,

MEP φ, ϕ

x∈E:φ x, y

ϕ y

−ϕx≥0,∀y∈E

. 1.4

We see thatxis a solution of a problem1.3that implies thatx ∈domϕ {x∈E : ϕx<∞}.

Special Examples

1Ifϕ0, then the mixed equilibrium problem1.3becomes to be the equilibrium problem which is to findx∈Esuch that

φ x, y

≥0, ∀y∈E. 1.5

The set of solutions of1.5is denoted by EPφ.

2Ifϕ0 andφx, y Bx, y−xfor allx, y∈E, whereB:E → His a nonlinear mapping, then problem1.5becomes to be the variational inequality problems which is to findx∈Esuch that

Bx, y−x

≥0, ∀y∈E. 1.6

The set of solutions of 1.6 is denoted by VIE, B. The variational inequality has been extensively studied in the literature. See, for example,6–8and the references therein.

The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of1.3. Some authors have proposed some useful methods for solving the MEPφ, ϕ and EPφ; see, for instance 5, 9–27. In 1997, Combettes and Hirstoaga 10 introduced an iterative scheme of finding the best approximation to initial data when EPφ is nonempty and proved a strong convergence theorem. Next, we recall some definitions.

Definition 1.1. LetB:E → Hbe nonlinear mappings. ThenBis called 1monotone if

Bx−By, x−y

≥0, ∀x, y∈E, 1.7

2ρ-strongly monotone if there exists a constantρ >0 such that Bx−By, x−y

≥ρx−y², ∀x, y∈E, 1.8

3η-Lipschitz continuous if there exists a constantη >0 such that

Bx−By≤ηx−y, ∀x, y∈E, 1.9

4β-inverse strongly monotone if there exists a constantβ >0 such that Bx−By, x−y

≥βBx−By², ∀x, y∈E. 1.10 Remark 1.2. It is obvious that anyβ-inverse strongly monotone mappingsBis monotone and 1/β-Lipschitz continuous.

5A set-valued mappingT :H → 2^His called a monotone if, for allx, y ∈H,f ∈Tx andg ∈Tyimplyx−y, f−g ≥0.

6A monotone mappingT : H → 2^H is a maximal if the graph ofGTofT is not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for everyy, g∈GTimpliesf∈Tx.

LetBbe a monotone map ofEintoH,η-Lipschitz continuous mapping and letN_Eϑ be the normal cone toEwhenϑ∈E, that is,

NEϑ{w∈H:u−ϑ, w ≥0,∀u∈E} 1.11

and define a mappingTonEby

Tϑ

⎧⎨

⎩

BϑN_Eϑ, ϑ∈E,

∅, ϑ /∈E. 1.12

ThenT is the maximal monotone and 0∈Tϑif and only ifϑ∈VIE, B; see28.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities forβ-inverse strongly monotone, Takahashi and Toyoda29first introduced the following iterative scheme:

x0 ∈E chosen arbitrary,

x_n1α_nx_n 1−α_nSPExn−λ_nBx_n, ∀n≥0, 1.13 whereBis anβ-inverse strongly monotone,{αn}is a sequence in0, 1, and{λn}is a sequence in0,2β. They showed that ifFS∩VIE, Bis nonempty, then the sequence{xn}generated by1.13converges weakly to someq∈FS∩VIE, B.

Further, Y. Yao and J.-C. Yao30introduced the following iterative scheme:

x1x∈E chosen arbitrary, y_nP_Exn−λ_nBx_n, x_n1α_nxβ_nx_nγ_nSP_E

y_n−λ_nBy_n

, ∀n≥1,

1.14

where Bis an β-inverse strongly monotone, {αn},{βn},{γn}are three sequences in 0, 1, and{λn}is a sequence in0,2β. They showed that ifFS∩VIE, Bis nonempty, then the sequence{xn}generated by1.14converges strongly to someq∈FS∩VIE, B.

A mapA:H → His said to be strongly positive if there exists a constantγ >0 such that

Ax, x ≥γx², ∀x∈H. 1.15

A typical problem is to minimize a quadratic function over the set of the fixed points of some nonexpansive mapping on a real Hilbert spaceH:

minx∈E

1

2Ax, x − x, b, 1.16

whereAis some linear,Eis the fixed point set of a nonexpansive mappingSonHandbis a point inH. LetAbe a strongly positive linear bounded map onHwith coefficientγ. In 2006, Marino and Xu31studied the following general iterative method:

x_n1nγfxn 1−nASxn. 1.17

They proved that if the sequencenof parameters appropriate conditions, then the sequence x_ngenerated by1.17converges strongly toqP_FSI−Aγfq. Recently, Plubtieng and Punpaeng32proposed the following iterative algorithm:

φ u_n, y

1 rn

y−u_n, u_n−x_n

≥0, ∀y∈H, x_n1nγfxn I−_nASun.

1.18

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

A−γf

q, x−q

≥0, ∀x∈FS∩EP φ

, 1.19

which is the optimality condition for the minimization problem

x∈FS∩EPφmin 1

2Ax, x −hx, 1.20

wherehis a potential function forγfi.e.,hx γfxforx∈H.

On the other hand, for finding a common element of the set of fixed points of ak- strict pseudo-contraction mapping and the set of solutions of an equilibrium problem in a real Hilbert space, Liu33introduced the following iterative scheme:

φ un, y

1 r_n

y−un, un−xn

≥0, ∀y∈E,

ynβnun 1−βn

Sun,

x_n1nγfxn I−nAun, ∀n≥1,

1.21

whereSis ak-strict pseudo-contraction mapping and{n},{βn}are sequences in0, 1. They proved that under certain appropriate conditions over{n},{βn}, and {rn}, the sequences {xn}and{un}converge strongly to someq ∈ FS∩EPφ, which solves some variational inequality problems.

In 2008, Ceng and Yao5introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem1.3 in Hilbert spaces and obtained the strong convergence theorem which used the following condition:

H K : E → R isη-strongly convex with constantσ > 0 and its derivativeK is sequentially continuous from weak topology to strong topology. We note that the condition Hfor the functionK : E → Ris a very strong condition. We also note that the condition H does not cover the caseKx x²/2 and ηx, y x−y for eachx, y ∈ E×E.

Very recently, R. Wangkeeree and R. Wangkeeree34introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of ak-strict pseudo-contraction mapping, and the set of solutions of

the variational inequality for an inverse strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the conditionHfor the sequences generated by these processes.

In 2009, Qin et al.35introduced a general iterative scheme for finding a common element of the set of common solution of generalized equilibrium problems, the set of a common fixed point of a family of infinite nonexpansive mappings in Hilbert spaces. Let {xn}be the sequence generated iterative by the following algorithm:

x1∈E, un∈E, vn∈E, φ₁un, u Cxn, u−u_n1

ru−u_n, u_n−x_n ≥0, ∀u∈E, φ₂vn, v Bxn, v−v_n1

sv−v_n, v_n−x_n ≥0, ∀v∈E, y_nδ_nu_n 1−δ_nvn,

x_n1nfxn βnxnγnWnyn, ∀n≥1.

1.22

They proved that under certain appropriate conditions imposed on{n},{βn},{γn}and{δn}, the sequence {xn}generated by 1.22converges strongly to q ∈ ∩^∞_n1FTn∩EPφ1, C∩ EPφ2, B, whereqP_∩^∞_{n1FTn∩EPφ1,C∩EPφ2,B}fq.

In the present paper, motivated and inspired by Qin et al. 35, Plubtieng and Punpaeng32, Peng and Yao17, R. Wangkeeree and R. Wangkeeree34, and Y. Yao and J.-C. Yao30, we introduce a new approximation iterative scheme for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem, and the set of common solutions of the variational inequalities with inverse strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. Moreover, we apply our results for solving convex feasibility problems in Hilbert spaces. The results in this paper extend and improve some well-known results in17,30,32,34,35.

2. Preliminaries

LetHbe a real Hilbert space andEbe a closed convex subset ofH. In a real Hilbert spaceH, it is well known that

λx 1−λy²λx² 1−λy²−λ1−λx−y² 2.1 for allx, y∈Handλ∈0,1.

For anyx∈H, there exists a unique nearest point inE, denoted byP_Ex, such that x−P_Ex ≤x−y, ∀y∈E. 2.2 The mappingP_Eis called the metric projection ofHontoE.

It is well known thatPEis a firmly nonexpansive mapping ofHontoE, that is, x−y, P_Ex−P_Ey

≥P_Ex−P_Ey², ∀x, y∈H. 2.3 Further, for anyx∈Handz∈E,zP_Exif and only ifx−z, z−y ≥0, for ally∈E.

Moreover,PExis characterized by the following properties:PEx∈Eand x−P_Ex, y−P_Ex

≤0, 2.4

x−y²≥ x−P_Ex²y−P_Ex² 2.5 for allx∈H, y∈E.

It is easy to see that the following is true:

u∈VIE, B⇐⇒uP_Eu−λBu, λ >0. 2.6

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1see36. LetE,·,·be an inner product space. Then, for allx, y, z∈Eandα, β, γ ∈ 0,1withαβγ1, one has

αxβyγz²αx²βy²γz²−αβx−y²−αγx−z²−βγy−z². 2.7 Lemma 2.2see31. Assume thatA is a strongly positive linear bounded operator onH with coefficientγ >0 and 0< ρ≤ A⁻¹. ThenI−ρA ≤1−ργ.

Lemma 2.3see4. LetEbe a nonempty closed convex subset of a real Hilbert spaceH and let S : E → Ebe ak-strict pseudo-contraction with a fixed point. ThenFS is closed and convex.

DefineS_k : E → EbyS_k kx 1−kSx for eachx ∈ E. ThenS_kis nonexpansive such that FSk FS.

Lemma 2.4see37. LetXbe a uniformly convex Banach spaces,Ebe a nonempty closed convex subset ofXandS:E → Ebe a nonexpansive mapping. ThenI−Sis demi-closed at zero.

Lemma 2.5see38. LetEbe a nonempty closed convex subset of strictly convex Banach spaceX.

Let{Tn:n∈N}be a sequence of nonexpansive mappings onE. Suppose∩^∞_n1FTnis nonempty. Let δ_nbe a sequence of positive numbers with_∞

n1δ_n1. Then a mappingSonEcan be defined by

Sx^∞

n1

δ_nT_nx 2.8

forx∈Eis well defined, nonexpansive andFS ∩^∞_n1FTnholds.

In order to solve the mixed equilibrium problem, the following assumptions are given for the bifunctionφ,ϕand the setE:

A1φx, x 0 for allx∈E;

A2φis monotone, that is,φx, y φy, x≤0 for allx, y∈E;

A3for eachx, y, z∈E, lim_t→0φtz 1−tx, y≤φx, y;

A4for eachx∈E, y→φx, yis convex and lower semicontinuous;

A5for eachy∈E, x→φx, yis weakly upper semicontinuous;

B1for eachx∈Handr >0, there exist abounded subsetDx⊆Eandyx∈Esuch that for anyz∈E\D_x,

φ z, yx

ϕ yx

−ϕz 1 r

yx−z, z−x

<0; 2.9

B2Eis a bounded set.

Lemma 2.6see39. LetEbe a nonempty closed convex subset ofH. Letφ :E×E → Rbe a bifunction satisfies (A1)–(A5) and letϕ : E → R ∪ {∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 andx ∈ H, define a mapping T_r^φ,ϕ:H → Eas follows:

T_r^φ,ϕx

z∈E:φ z, y

ϕ y

−ϕz 1 r

y−z, z−x

≥0,∀y∈E

2.10

for allz∈H. Then, the following holds:

ifor eachx∈H,T_r^φ,ϕx/∅;

iiT_r^φ,ϕis single-valued;

iiiT_r^φ,ϕis firmly nonexpansive, that is, for anyx, y∈H, Tr^φ,ϕx−T_r^φ,ϕy²≤

T_r^φ,ϕx−T_r^φ,ϕy, x−y

; 2.11

ivFTr^φ,ϕ MEPφ, ϕ;

vMEPφ, ϕis closed and convex.

Remark 2.7. Ifϕ0, thenTr^φ,ϕis rewritten asTr^φ.

Remark 2.8. We remark thatLemma 2.6is not a consequence of Lemma 3.1 in5, because the condition of the sequential continuity from the weak topology to the strong topology for the derivativeKof the functionK:E → Rdoes not cover the caseKx x²/2.

Lemma 2.9see40. Let{xn}and{ln}be bounded sequences in a Banach spaceXand let{βn}be a sequence in0,1with 0<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1. Supposex_n1 1−βnlnβnxn

for all integersn≥1 and lim sup_{n→ ∞}l_n1−l_n − x_n1−x_n≤0. Then, lim_{n→ ∞}ln−x_n0.

Lemma 2.10see41. Assume that{an}is a sequence of nonnegative real numbers such that a_n1≤

1−n

anσn, n≥1, 2.12

where{n}is a sequence in0,1and{σn}is a sequence inRsuch that 1_∞

n1n∞,

2lim sup_{n→ ∞}σn/n≤0 or_∞

n1|σn|<∞.

Then lim_{n→ ∞}a_n0.

Lemma 2.11. LetHbe a real Hilbert space. Then for allx, y∈H, xy²≤ x²2

y, xy

. 2.13

3. Main Results

In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudo- contractions, the set of common solutions of the system of a mixed equilibrium problem and the set of a common solutions of the variational inequalities with inverse strongly monotone mappings in a real Hilbert space.

Theorem 3.1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Letφ₁andφ₂be two bifunctions fromE×EtoRsatisfying (A1)–(A5) and letϕ:E → R ∪ {∞}be a proper lower semicontinuous and convex function. LetC :E → Hbe anξ-inverse strongly monotone mapping and B : E → H be anβ-inverse strongly monotone mapping. Letf : E → Ebe a contraction mapping with coefficientα0< α <1and letAbe a strongly positive linear bounded operator onH with coefficientγ >0 and 0< γ < γ/α. LetS:E → Ebe ak-strict pseudo-contraction with a fixed point. Define a mappingS_k:E → EbyS_kxkx 1−kSx, for allx∈E. Assume that

Θ:FS∩VIE, C∩VIE, B∩MEP φ₁, ϕ

∩MEP φ₂, ϕ

/∅. 3.1

Assume that either (B1) or (B2). Let{xn}be a sequence generated by the following iterative algorithm:

x1∈E, un∈E, vn∈E, u_nT_r^φ1,ϕx_n, v_nT_s^φ2,ϕx_n, z_nP_E

u_n−μ_nCu_n , ynPEvn−λnBvn, k_na_nS_kx_nb_ny_nc_nz_n, x_n1nγfxn βnxn

1−βn

I−nA

kn, ∀n≥1,

3.2

where{n},{βn},{an},{bn}, and{cn}are sequences in0,1and{λn},{μn}are positive sequences.

Assume that the control sequences satisfy the following restrictions:

C1anbncn 1, C2limn→ ∞n0 and_∞

n1n∞, C30<lim inf_{n→ ∞}β_n≤lim sup_{n→ ∞}β_n<1, C4lim_{n→ ∞}|λ_n1−λ_n|lim_{n→ ∞}|μ_n1−μ_n|0,

C5d≤λn≤2β,e≤μn≤2ξ, whered, eare two positive constants,

C6limn→ ∞ana, limn→ ∞bnband limn→ ∞cnc, for somea, b, c∈0,1.

Then, {xn} converges strongly to a point q ∈ Θwhich is the unique solution of the variational inequality

A−γf

q, x−q

≥0, ∀x∈Θ 3.3

or equivalentqP_ΘI−Aγfq, wherePis a metric projection mapping formHontoΘ.

Proof. Since_n → 0, asn → ∞, we may assume, without loss of generality, that_n ≤ 1− βnA⁻¹for alln∈N. ByLemma 2.2, we know that if 0≤ρ≤ A⁻¹, thenI−ρA ≤1−ργ.

We will assume thatI−A ≤ 1−γ. SinceAis a strongly positive bounded linear operator onH, we have

Asup{|Ax, x|:x∈H,x1}. 3.4

Observe that

1−βn

I−nA x, x

1−βn−nAx, x

≥1−βn−nA

≥0,

3.5

so this shows that1−βnI−nAis positive. It follows that 1−β_n

I−_nAsup1−β_n

I−_nA

x, x:x∈H,x1 sup

1−β_n−_nAx, x:x∈H,x1

≤1−βn−nγ.

3.6

We divide the proof into seven steps.

Step 1. We claim that the mappingP_ΘI−AγfwhereΘ : FS∩VIE, C∩VIE, B∩ MEPφ1, ϕ∩MEPφ2, ϕhas a unique fixed point.

Sincefbe a contraction ofHinto itself withα∈0,1. Then, we have P_Θ

I−Aγf

x−P_Θ

I−Aγf

y≤I−Aγf x−

I−Aγf y

≤I−Ax−yγfx−f y

≤

1−γx−yγαx−y

1−

γ−γαx−y, ∀x, y∈H.

3.7

Since 0<1−γ−γα<1, it follows thatP_ΘI−Aγfis a contraction ofHinto itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique elementq∈H such thatqP_ΘI−Aγfq.

Step 2. We claim thatI−λ_nBis nonexpansive.

Indeed, from theβ-inverse strongly monotone mapping definition onBand condition C5, we have

I−λnBx−I−λnBy²x−y−λnBx−By² x−y²−2λ_n

x−y, Bx−By

λ²_nBx−By²

≤x−y²−2λnβBx−Byλ²_nBx−By² x−y²λ_n

λ_n−2βBx−By²

≤x−y²,

3.8

whereλ_n ≤ 2β, for alln ∈ N implies that the mappingI−λ_nBis nonexpansive and so is, I−μ_nC.

Step 3. We claim that{xn}is bounded.

Indeed, letp∈ΘandLemma 2.6, we obtain

pPE

p−λnBp PE

p−μnCp

T_r^φ1,ϕpT_s^φ2,ϕp. 3.9

Note thatunT_r^φ1,ϕxn∈domϕandvnT_s^φ2,ϕxn∈domϕ, we have u_n−pT_r^φ1,ϕx_n−T_r^φ1,ϕp≤x_n−p, v_n−pT_s^φ2,ϕx_n−T_s^φ2,ϕp≤x_n−p.

3.10

SinceI−λnBandI−μnCare nonexpansive and from2.6, we have zn−pPE

un−μnCun

−PE

p−μnCp

≤un−μnCun

−

p−μnCp I−μ_nC

u_n−

I−μ_nC p

≤un−p≤xn−p, y_n−pP_Evn−λ_nBv_n−P_E

p−λ_nBp≤v_n−p≤x_n−p.

3.11

FromLemma 2.3, we have thatS_kis nonexpansive withFSk FS. It follows that k_n−pa_nS_kx_nb_ny_nc_nz_n−p

≤anSkxn−pbnyn−pcnzn−p

≤anxn−pbnxn−pcnxn−pxn−p.

3.12

It follows that

x_n1−p_n

γfxn−Ap β_n

x_n−p

1−β_n

I−_nA

k_n−p

≤

1−β_n−_nγk_n−pβ_nx_n−p_nγfxn−Ap

≤

1−βn−nγxn−pβnxn−pnγfxn−Ap

≤

1−_nγx_n−p_nγfxn−f

p_nγf p

−Ap

≤

1−_nγx_n−p_nγαx_n−p_nγf p

−Ap

1− γ−αγ

nxn−p γ−αγ

n

γf p

−Ap γ−αγ

≤max

xn−p,γf p

−Ap γ−αγ

.

3.13

By simple induction, we have

xn−p≤max

x1−p,γf p

−Ap γ−αγ

, ∀n∈N. 3.14

Hence,{xn}is bounded, so are{un},{vn},{zn},{yn},{kn},{fxn},{Cun}, and{Bvn}.

Step 4. We claim that limn→ ∞xn1−xn0.

Observing that un T_r^φ1,ϕxn ∈ domϕ and u_n1 T_r^φ1,ϕx_n1 ∈ domϕ, by the nonexpansiveness ofT_r^φ1,ϕ, we get

u_n1−u_nT_r^φ1,ϕx_n1−T_r^φ1,ϕx_n≤ x_n1−x_n. 3.15

Similarly, letvnT_s^φ2,ϕxn ∈domϕandv_n1T_s^φ2,ϕx_n1∈domϕ, we have

v_n1−v_nT_s^φ2,ϕx_n1−T_s^φ2,ϕx_n≤ x_n1−x_n. 3.16

FromznPEun−μnCunandyn PEvn−λnBvn, we compute zn1−znPE

u_n1−μ_n1Cu_n1

−PE

un−μnCun

≤u_n1−μ_n1Cu_n1

−

un−μnCun u_n1−μ_n1Cu_n1

−

u_n−μ_n1Cu_n

μ_n−μ_n1 Cu_n

≤u_n1−μ_n1Cu_n1

−

un−μ_n1Cunμn−μ_n1Cun I−μ_n1C

u_n1−

I−μ_n1C

u_nμ_n−μ_n1Cun

≤ u_n1−u_nμ_n−μ_n1Cun

≤ x_n1−xnμn−μ_n1Cun.

3.17

Similarly, we have

y_n1−ynPEvn1−λ_n1Bv_n1−PEvn−λnBvn

≤ v_n1−v_n|λn−λ_n1|Bvn

≤ x_n1−x_n|λn−λ_n1|Bvn.

3.18

Observing that

k_n a_nS_kx_nb_ny_nc_nz_n,

k_n1a_n1Skx_n1b_n1y_n1c_n1z_n1, 3.19 we obtain

kn1−k_n ≤a_n1Skx_n1−S_kx_n|an1−a_n|Skx_nb_n1y_n1−y_n |bn1−bn|ync_n1zn1−zn|cn1−cn|zn

≤a_n1x_n1−x_n|a_n1−a_n|Skx_nb_n1y_n1−y_n |bn1−bn|ync_n1zn1−zn|cn1−cn|zn.

3.20

Substituting3.17and3.18into3.20, we have

kn1−kn ≤a_n1xn1−xn|an1−an|Skxnb_n1{xn1−xn|λn−λ_n1|Bvn} c_n1

x_n1−x_nμ_n−μ_n1Cun

|b_n1−b_n|y_n|c_n1−c_n|zn

≤ x_n1−xnM1

|an1−an||bn1−bn||cn1−cn||λn−λ_n1|μn−μ_n1, 3.21 whereM1 is an appropriate constant such that M1 max{sup_n≥1Skxn,yn,zn,Bvn, Cun}.

Puttingx_n1 1−β_nlnβ_nx_n, for alln≥1, we have

ln x_n1−βnxn

1−βn _nγfxn

1−β_n

I−_nA k_n

1−βn . 3.22

Then, we compute

l_n1−l_{n n1}γfxn1

1−β_n1

I−_n1A k_n1 1−β_n1

−nγfxn

1−βn

I−nA kn

1−β_{n n1}

1−β_n1γfxn1− _n

1−β_nγfxn k_n1−k_{n n}

1−β_nAk_n− _n1

1−β_n1Ak_{n1 n1}

1−β_n1

γfx_n1−Ak_{n1 n}

1−β_n

Ak_n−γfxn

k_n1−k_n.

3.23

It follows from3.21and3.23, that ln1−ln − xn1−xn

≤ _n1

1−β_n1γfxn1Akn1 n

1−β_n

Aknγfxn

kn1−kn − xn1−xn

≤ _n1

1−β_n1γfxn1Akn1 n

1−β_n

Aknγfxn M1

|an1−an||bn1−bn||cn1−cn||λn−λ_n1|μn−μ_n1.

3.24

This together withC2,C3,C4, andC6imply that lim sup

n→ ∞ ln1−ln − xn1−xn≤0. 3.25

Hence, byLemma 2.9, we obtainln−x_n → 0 asn → ∞. It follows that

nlim→ ∞x_n1−x_n lim

n→ ∞

1−β_n

ln−x_n0. 3.26

So, we also get

nlim→ ∞u_n1−u_n lim

n→ ∞v_n1−v_n lim

n→ ∞z_n1−z_n lim

n→ ∞y_n1−yn lim

n→ ∞k_n1−kn0.

3.27

Observe that

x_n1−x_nn

γfxn−Ax_n

1−β_n−_nγ

kn−x_n. 3.28

By conditionC2and3.26, we have

lim_{n→ ∞}kn−x_n0. 3.29

Step 5. We claim that the following statements hold:

s1limn→ ∞xn−vn0;

s2limn→ ∞xn−un0;

s3lim_{n→ ∞}xn−y_n0;

s4limn→ ∞xn−zn0.

Indeed, pick anyp∈Θ, to obtain

un−p²Tr^φ1,ϕxn−T_r^φ1,ϕp²

≤

T_r^φ1,ϕx_n−T_r^φ1,ϕp, x_n−p

un−p, xn−p 1

2

u_n−p²x_n−p²− x_n−u_n² .

3.30

Therefore,

un−p²≤xn−p²− xn−un². 3.31

Similarly, we have

vn−p²≤xn−p²− xn−vn². 3.32

Note that

kn−p²anSkxnbnyncnzn−p²

≤a_nS_kx_n−p²b_ny_n−p²c_nz_n−p²

≤a_nx_n−p²b_nv_n−pc_nu_n−p.

3.33

Substituting3.31and3.32into3.33, we obtain k_n−p²≤a_nx_n−p²b_nv_n−pc_nu_n−p

≤anxn−p²bn

xn−p²− xn−vn² cn

xn−p²− xn−un² x_n−p²−b_nxn−v_n²−c_nxn−u_n².

3.34

FromLemma 2.1,3.2and3.34, we obtain

x_n1−p²_nγfxn−Ap β_nxn−p 1−β_nI−_nAkn−p²

≤_nγfxn−Ap²β_nx_n−p²

1−β_n−_nγk_n−p²

≤nγfxn−Ap²βnxn−p²

1−β_n−_nγx_n−p²−b_nxn−v_n²−c_nxn−u_n² _nγfxn−Ap²

1−_nγx_n−p²−

1−β_n−_nγ

b_nxn−v_n²

−

1−β_n−_nγ

c_nxn−u_n²

≤nγfxn−Ap²xn−p²−

1−βn−nγ

bnxn−vn²

−

1−βn−nγ

cnxn−un².

3.35

It follows that 1−β_n−_nγ

c_nxn−u_n²≤_nγfxn−Ap²x_n−p²−x_n1−p²

≤nγfxn−Ap²xn1−xnxn−px_n1−p. 3.36

FromC2,C6, and3.26, we also have

nlim→ ∞xn−un0. 3.37

Similarly, using3.35again, we have 1−β_n−_nγ

b_nxn−v_n²≤_nγfxn−Ap²x_n−p²−x_n1−p²

≤nγfxn−Ap²xn1−xnxn−px_n1−p. 3.38

FromC2,C6, and3.26, we also have

nlim→ ∞xn−v_n0. 3.39

From3.37and3.39, we have

nlim→ ∞un−v_n0. 3.40

Forp∈Θ, we compute

zn−p²PEun−μnCun−PEp−μnCp²

≤un−μ_nCu_n−p−μ_nCp² un−p−μ_nCun−Cp²

≤un−p²−2μn

un−p, Cun−Cp

μ²_nCun−Cp²

≤x_n−p²μ_n

μ_n−2ξCu_n−Cp²

≤xn−p²−μn

2ξ−μnCun−Cp².

3.41

Similarly, we have

yn−p²≤xn−p²−λn

2β−λnBvn−Bp². 3.42