A Viscosity of Ces `aro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems

Fixed Point Theory and Applications Volume 2011, Article ID 945051,24pages doi:10.1155/2011/945051

Research Article

A Viscosity of Ces `aro Mean Approximation Methods for a Mixed Equilibrium, Variational Inequalities, and Fixed Point Problems

Thanyarat Jitpeera, Phayap Katchang, and Poom Kumam

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

Correspondence should be addressed to Poom Kumam,poom.kum@kmutt.ac.th Received 6 September 2010; Accepted 15 October 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 Thanyarat Jitpeera 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.

We introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a β-inverse- strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Ces`aro mean approximation method.

We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang2009, Peng and Yao2009, Shimizu and Takahashi1997, and some authors.

1. Introduction

Throughout this paper, we assume thatHis a real Hilbert space with inner product and norm are denoted by·,·and · , respectively and letCbe a nonempty closed convex subset of H. A mappingT :C → Cis callednonexpansiveifTx−Ty ≤ x−y, for allx, y ∈C.

We useFTto denote the set of fixed points ofT, that is,FT {x ∈ C : Tx x}. It is assumed throughout the paper thatT is a nonexpansive mapping such thatFT/∅. Recall that a self-mappingf :C → Cis acontractiononCif there exists a constantα∈0,1and x, y∈Csuch thatfx−fy ≤αx−y.

Letϕ:C → R∪{∞}be a proper extended real-valued function andφbe a bifunction ofC×CintoR, whereRis the set of real numbers. Ceng and Yao1considered the following mixed equilibrium problemfor findingx∈Csuch that

φ x, y

ϕ y

≥ϕx, ∀y∈C. 1.1

The set of solutions of1.1is denoted by MEPφ, ϕ. We see that x is a solution of problem 1.1implies thatx ∈ domϕ {x ∈ C| ϕx < ∞}. If ϕ 0, then the mixed equilibrium problem1.1becomes the following equilibrium problem is to findx∈Csuch that

φ x, y

≥0, ∀y∈C. 1.2

The set of solutions of 1.2 is denoted by EPφ. 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 of1.2. Some methods have been proposed to solve the equilibrium problemsee2–14.

Let B : C → H be a mapping. The variational inequality problem, denoted by VIC, B, is to findx∈Csuch that

Bx, y−x

≥0, 1.3

for ally∈C.The variational inequality problem has been extensively studied in the literature.

See, for example, 15, 16 and the references therein. A mapping B of C into H is called monotone if

Bx−By, x−y

≥0, 1.4

for allx, y∈C. B is calledβ-inverse-strongly monotoneif there exists a positive real number β >0 such that for allx, y∈C

Bx−By, x−y

≥βBx−By². 1.5 LetAbe a strongly positive linear bounded operator onH: that is, there is a constantγ >0 with property

Ax, x ≥γx², ∀x∈H. 1.6 A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:

x∈FTmin 1

2Ax, x −hx, 1.7

whereAis strongly positive linear bounded operator andhis a potential function forγfi.e., hx γfxforx∈H. Moreover, it is shown in17that the sequence{xn}defined by the scheme

x_n1nγfxn l−_nATxn, 1.8 converges strongly tozP_FTI−Aγfz.

In 1997, Shimizu and Takahashi18originally studied the convergence of an iteration process{xn}for a family of nonexpansive mappings in the framework of a real Hilbert space.

They restate the sequence{xn}as follows:

x_n1α_nx 1−α_n 1 n1

n j0

T^jx_n, forn0,1,2, . . . , 1.9

wherex₀andxare all elements ofCandα_nis an appropriate in0,1.They proved that{xn} converges strongly to an element of fixed point ofTwhich is the nearest tox.

In 2007, Plubtieng and Punpaeng19proposed the following iterative algorithm:

φ u_n, y

1 rn

y−u_n, u_n−x_n

≥0, ∀y∈H, x_n1nγfxn I−nATun.

1.10

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

A−γf

z, x−z

≥0, ∀x∈FT∩EP φ

, 1.11

which is the optimality condition for the minimization problem

x∈FT∩EPφmin 1

2Ax, x −hx, 1.12 wherehis a potential function forγfi.e.,hx γfxforx∈H.

In 2008, Peng and Yao20introduced an iterative algorithm based on extragradient method which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. The sequences generated byv∈C,

x₁x∈C, φ

un, y ϕ

y

−ϕun

1 rn

y−un, un−xn

≥0, ∀y∈C,

ynPC

un−γnBun

, x_n1α_nvβ_nx_n

1−α_n−β_n W_nP_C

u_n−λ_nBy_n ,

1.13

for alln≥1, whereW_nis W-mapping. They proved the strong convergence theorems under some mind conditions.

In this paper, motivated by the above results and the iterative schemes considered in 9,18–20, we introduce a new iterative process below based on viscosity and Ces`aro mean

approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for aβ-inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we prove strong convergence theorems which are connected with5,21–24. We extend and improve the corresponding results of Kumam and Katchang9, Peng and Yao20, Shimizu and Takahashi18and some authors.

2. Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · and letCbe a nonempty closed convex subset ofH. Then

x−y²x²−y²−2

x−y, y

, 2.1

λx 1−λy²λx² 1−λy²−λ1−λx−y², 2.2 for allx, y∈Handλ∈0,1. For every pointx∈H, there exists a uniquenearest pointin C, denoted byP_Cx, such that

x−PCx ≤ x−y, ∀y∈C. 2.3

PC is called the metric projection ofH ontoC. It is well known that PC is a nonexpansive mapping ofHontoCand satisfies

x−y, P_Cx−P_Cy

≥P_Cx−P_Cy², 2.4 for everyx, y∈H.Moreover,PCxis characterized by the following properties:PCx∈Cand

x−P_Cx, y−P_Cx

≤0, 2.5

x−y²≥ x−PCx²y−PCx², 2.6 for allx∈H, y∈C. LetBbe a monotone mapping of C into H. In the context of the variational inequality problem the characterization of projection2.5implies the following:

u∈VIC, B⇐⇒uPCu−λBu, λ >0. 2.7 It is also known that H satisfies the Opial condition25, that is, for any sequence{xn} ⊂H withx_n x, the inequality

lim inf

n→ ∞ xn−x<lim inf

n→ ∞ xn−y, 2.8

holds for everyy∈Hwithx /y.

A set-valued mappingU:H → 2^His calledmonotoneif for allx, y∈H,f∈Uxand g∈Uyimplyx−y, f−g ≥0. A monotone mappingU:H → 2^Hismaximalif the graph of GUofUis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingUis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for everyy, g∈GUimpliesf∈Ux. LetBbe a monotone mapping ofCintoHand letNCy be thenormal conetoCaty∈C, that is,N_Cy{w∈H:u−y, w ≤0,∀u∈C}and define

Uy

⎧⎨

⎩

ByNCy, y∈C,

∅, y /∈C. 2.9

ThenUis the maximal monotone and 0∈Uyif and only ify∈VIC, B; see26.

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunctionφ:C×C → Rand a proper extended real-valued functionϕ:C → R∪ {∞}

satisfies the following conditions:

A1φx, x 0 for allx∈C;

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

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

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

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

B1for eachx∈Handr >0, there exist a bounded subsetDx⊆Candyx∈Csuch that for anyz∈C\Dx,

φ z, y_x

ϕ y_x

1 r

y_x−z, z−x

< ϕz; 2.10

B2C is a bounded set.

We need the following lemmas for proving our main results.

Lemma 2.1Peng and Yao20. LetCbe a nonempty closed convex subset of H. Letφ:C×C → R be a bifunction satisfies (A1)–(A5) and letϕ:C → R∪ {∞}be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0 andx ∈ H, define a mapping Tr :H → Cas follows:

T_rx

z∈C:φ z, y

ϕ y

1 r

y−z, z−x

≥ϕz, ∀y∈C

, 2.11

for allx∈H. Then, the following hold.

1For eachx∈H, Trx/∅;

2T_r is single-valued;

3Tr is firmly nonexpansive, that is, for anyx, y∈H,Trx−Try² ≤ Trx−Try, x−y;

4FTr MEPφ, ϕ;

5MEPφ, ϕis closed and convex.

Lemma 2.2Xu27. Assume{an}is a sequence of nonnegative real numbers such that

a_n1≤1−α_nanδ_n, n≥0, 2.12

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

n1α_n∞,

2lim sup_{n→ ∞}δn/α_n≤0 or_∞

n1|δn|<∞.

Then lim_{n→ ∞}an0.

Lemma 2.3 Osilike and Igbokwe 28. LetC,·,· be an inner product space. Then for all x, y, z∈Candα, β, γ∈0,1withαβγ 1,we have

αxβyγz² αx²βy²γz²−αβx−y²−αγx−z²−βγy−z². 2.13

Lemma 2.4 Suzuki 29. Let {xn} and {yn} be bounded sequences in a Banach space X and let{βn} be a sequence in0,1 with 0 < lim inf_{n→ ∞}β_n ≤ lim sup_{n→ ∞}β_n < 1.Supposex_n1 1−βnynβnxnfor all integersn ≥ 0 and lim sup_{n→ ∞}yn1−yn − xn1−xn≤ 0.Then, lim_{n→ ∞}yn−x_n0.

Lemma 2.5Marino and Xu17. AssumeAis a strongly positive linear bounded operator on a Hilbert spaceHwith coefficientγ >0 and 0< ρ≤ A⁻¹. ThenI−ρA ≤1−ργ.

Lemma 2.6Bruck30. LetCbe a nonempty bounded closed convex subset of a uniformly convex Banach spaceEandT : C → Ca nonexpansive mapping. For eachx ∈ Cand the Ces`aro means Tnx 1/n1_n

i0Tⁱx,then lim sup_{n→ ∞}Tnx−TTnx0.

3. Main Results

In this section, we show a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of mixed equilibrium problem and the set of solutions of a variational inequality problem for a β-inverse-strongly monotone mapping in a real Hilbert space by using the viscosity of Ces`aro mean approximation method.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ be a bifunction of C×Cinto real numbersRsatisfying (A1)–(A5) and let ϕ : C → R∪ {∞} be a proper lower semicontinuos and convex function. LetTⁱ:C → Cbe a nonexpansive mappings for all i1,2,3, . . . , n, such thatΘ:_n

i1FTⁱ∩VIC, B∩MEPφ, ϕ/∅. Letfbe a contraction ofC

into itself with coefficientα∈0,1and letBbe aβ-inverse-strongly monotone mapping ofCintoH.

LetAbe a strongly positive bounded linear self-adjoint onHwith coefficientγ >0 and 0< γ < γ/α.

Assume that eitherB₁orB₂holds. Let{xn},{yn}and{un}be sequences generated byx₀∈C,u_n∈C and

φ u_n, y

ϕ y

−ϕun

1 rn

y−u_n, u_n−x_n

≥0, ∀y∈C, ynδnun 1−δnPCun−λnBun,

x_n1 αnγfxn βnxn 1−βn

I−αnA 1 n1

n i0

Tⁱyn, ∀n≥0,

3.1

where{αn},{βn}andδn⊂0,1,{λn} ⊂0,2βand{rn} ⊂0,∞satisfy the following conditions:

i_∞

n0αn∞and lim_{n→ ∞}αn0, iilim_{n→ ∞}δ_n 0,

iii0<lim inf_{n→ ∞}β_n≤lim sup_{n→ ∞}β_n<1,

iv{λn} ⊂a, b,∃a, b∈0,2βand limn→ ∞|λn1−λn|0, vlim inf_{n→ ∞}r_n>0 and lim_{n→ ∞}|r_n1−r_n|0.

Then,{xn}converges strongly toz∈Θ,wherezP_ΘI−Aγfz, which is the unique solution of the variational inequality

A−γf

z, x−z

≥0, ∀x∈Θ. 3.2

Proof. Now, we have1−βnI−αnA ≤1−βn−αnγ see9, page 479. Sinceλn ∈0,2β andBis aβ-inverse-strongly monotone mapping. For anyx, y∈C, we have

I−λnBx−I−λnBy²x−y−λnBx−By² x−y²−2λ_n

x−y, Bx−By

λ²_nBx−By²

≤x−y²−2λnβBx−By²λ²_nBx−By² x−y²λ_n

λ_n−2βBx−By²

≤x−y².

3.3

It follows thatI−λ_nBx−I−λ_nBy ≤ x−y, henceI−λ_nBis nonexpansive.

Letx^∗∈Θ, Trn be a sequence of mapping defined as inLemma 2.1andunTrnxn, for alln≥0,we have

un−x^∗Trnx_n−T_rnx^∗ ≤ xn−x^∗. 3.4

By the fact thatP_CandI−λ_nBare nonexpansive andx^∗P_Cx^∗−λ_nBx^∗, we get

yn−x^∗δnun 1−δnPCun−λnBun−x^∗

≤δ_nun−x^∗ 1−δ_nPCun−λ_nBu_n−P_Cx^∗−λ_nBx^∗

≤δ_nun−x^∗ 1−δ_nI−λ_nBun−I−λ_nBx^∗

≤δ_nun−x^∗ 1−δ_nun−x^∗ un−x^∗

≤ xn−x^∗.

3.5

LetTn 1/n1_n

i0Tⁱ; it follows that

Tnx−T_ny

1 n1

n i0

Tⁱx− 1 n1

n i0

Tⁱy

≤ 1 n1

n i0

Tⁱx−Tⁱy

≤ 1 n1

n i0

x−y

n1 n1x−y x−y,

3.6

which implies thatT_nis nonexpansive. Sincex^∗∈Θ,we haveT_nx^∗ 1/n1_n

i0Tⁱx^∗ 1/n1_n

i0x^∗x^∗, for allx, y∈C.By3.5and3.6, we have x_n1−x^∗αn

γfxn−Ax^∗

β_nxn−x^∗ 1−β_n

I−α_nA

T_ny_n−x^∗

≤α_nγfxn−Ax^∗β_nxn−x^∗

1−β_n−α_nγ

yn−x^∗

≤αnγfxn−Ax^∗βnxn−x^∗

1−βn−αnγ

xn−x^∗

≤αnγfxn−fx^∗αnγfx^∗−Ax^∗

1−αnγ

xn−x^∗

≤α_nγαxn−x^∗α_nγfx^∗−Ax^∗

1−α_nγ

xn−x^∗

1−α_n

γ−γα

xn−x^∗α_n

γ−γαγfx^∗−Ax^∗ γ−γα

≤max

x0−x^∗,γfx^∗−Ax^∗ γ−γα

.

3.7

Hence{xn}is bounded and also{un},{yn}and{Tny_n}are bounded.

Next, we show that limn→ ∞xn1−xn 0.Observing thatun Trnxn ∈domϕand u_n1Trn1x_n1∈domϕ, we get

φ u_n, y

ϕ y

−ϕun

1 r_n

y−u_n, u_n−x_n

≥0, ∀y∈C, 3.8

φ u_n1, y

ϕ y

−ϕun1 1 r_n1

y−u_n1, u_n1−x_n1

≥0, ∀y∈C. 3.9

Takeyu_n1in3.8andyunin3.9, by using conditionA2; it follows that

u_n1−u_n,u_n−x_n

rn −u_n1−x_n1 r_n1

≥0. 3.10

Thusun1−un, un−u_n1x_n1−xn1−rn/r_n1un1−xn1 ≥0. Without loss of generality, let us assume that there exists a nonnegative real numbercsuch thatrn > c, for alln ≥ 1.

Then, we have

un1−u_n²≤ un1−u_n

xn1−x_n 1− r_n

r_n1

un1−x_n1

3.11

and hence

u_n1−u_n ≤ x_n1−x_n 1

r_n1|r_n1−r_n|u_n1−x_n1

≤ xn1−xn1

c|rn1−rn|M1,

3.12

whereM1 sup{un−xn :n∈N}. On the other hand, letvn PCun−λnBun; it follows from the definition of{yn}that

y_n1−yn{δn1u_n1 1−δ_n1PCun1−λ_n1Bu_n1}

− {δ_nu_n 1−δ_nPCun−λ_nBu_n}

δn1un1−un δn1−δnun 1−δ_n1PCun1−λ_n1Bu_n1

−1−δ_n1PCun−λ_nBu_n 1−δ_n1PCun−λ_nBu_n

−1−δ_nPCun−λ_nBu_n δn1un1−un δn1−δnun

1−δ_n1{PCu_n1−λ_n1Bu_n1−P_Cun−λ_nBu_n} δ_n−δ_n1PCun−λ_nBu_n

≤δ_n1u_n1−un|δn1−δn|unvn 1−δ_n1u_n1−λ_n1Bu_n1−un−λ_nBu_n δ_n1un1−un|δn1−δn|unvn

1−δ_n1un1−λ_n1Bu_n1−un−λ_n1Bun u_n−λ_n1Bu_n−un−λ_nBu_n

≤δ_n1un1−un|δn1−δn|unvn 1−δ_n1{u_n1−u_n|λ_n1−λ_n|Bun}

|δn1−δ_n|unvn un1−u_n 1−δ_n1|λn1−λ_n|Bun

≤ |δ_n1−δn|unvn un1−un|λn1−λn|Bun

≤ |δ_n1−δ_n|unvn x_n1−x_n 1

c|r_n1−r_n|M1

|λ_n1−λ_n|Bun.

3.13

We compute that

T_n1y_n1−T_ny_n ≤ T_n1y_n1−T_n1y_nT_n1y_n−T_ny_n

≤ yn1−yn

1 n2

n1

i0

Tⁱyn− 1 n1

n i0

Tⁱyn

yn1−yn

1 n2

n i0

Tⁱyn 1

n2Tⁿ¹yn− 1 n1

n i0

Tⁱyn

yn1−yn

− 1 n1n2

n i0

Tⁱyn 1

n2Tⁿ¹yn

≤ yn1−y_n 1 n1n2

n i0

Tⁱy_n 1

n2Tⁿ¹y_n

≤ y_n1−y_n 1 n1n2

n i0

Tⁱy_n−Tⁱx^∗x^∗

1 n2

Tⁿ¹yn−Tⁿ¹x^∗x^∗

≤ yn1−yn 1 n1n2

n i0

yn−x^∗x^∗

1 n2

yn−x^∗x^∗

≤ yn1−y_n n1 n1n2

yn−x^∗x^∗

1

n2yn−x^∗ 1 n2x^∗ yn1−yn 2

n2yn−x^∗ 2 n2x^∗

≤ |δ_n1−δ_n|unvn x_n1−x_n1

c|r_n1−r_n|M1

|λn1−λn|Bun 2

n2yn−x^∗ 2 n2x^∗.

3.14

Letx_n1 1−β_nznβ_nx_n; it follows that

z_n x_n1−β_nx_n 1−β_n α_nγfxn

1−β_n

I−α_nA T_ny_n

1−βn ,

3.15

and hence

zn1−zn

α_n1γfx_n1

1−β_n1

I−α_n1A

T_n1y_n1 1−β_n1

− α_nγfxn

1−β_n

I−α_nA T_ny_n 1−β_n

α_n1γfxn1 1−β_n1

1−β_n1

T_n1y_n1 1−β_n1

−α_n1AT_n1y_n1

1−β_n1 −α_nγfxn 1−β_n

−

1−β_n T_ny_n

1−βn αnATnyn

1−βn

α_n1 1−β_n1

γfxn1−AT_n1y_n1 αn

1−β_n

ATnyn−γfxn

T_n1y_n1−Tnyn

≤ α_n1

1−β_n1γfx_n1−AT_n1y_n1 α_n

1−βnATny_n−γfxnT_n1y_n1−T_ny_n

≤ α_n1

1−β_n1γfxn1−AT_n1y_n1 αn

1−β_nATnyn−γfxn |δn1−δn|unvn xn1−xn 1

c|rn1−rn|M1

|λn1−λn|Bun 2

n2yn−x^∗ 2 n2x^∗.

3.16

Therefore,

z_n1−zn − x_n1−xn ≤ α_n1

1−β_n1γfx_n1−AT_n1y_n1 α_n

1−βnATnyn−γfxn |δn1−δ_n|unvn

1

crn1−r_nM1

|λ_n1−λ_n|Bun 2

n2yn−x^∗ 2 n2x^∗.

3.17

It follows fromn → ∞and the conditionsi–v, that

lim sup

n→ ∞ zn1−zn − xn1−xn≤0. 3.18

FromLemma 2.4and3.18, we obtain limn→ ∞zn−x_n0 and also

nlim→ ∞xn1−xn lim

n→ ∞

1−βn

zn−xn0. 3.19

Next, we show thatxn−un → 0 asn → ∞.Forx^∗∈Θ,we obtain un−x^∗2Trnx_n−T_rnx^∗2

≤ T_rnx_n−T_rnx^∗, x_n−x^∗ un−x^∗, xn−x^∗ 1

2

un−x^∗2xn−x^∗2− un−x^∗−x_nx^∗2

1 2

un−x^∗2xn−x^∗2− un−xn² ,

3.20

and hence

un−x^∗2≤ xn−x^∗2− un−x_n². 3.21

Sinceyn−x^∗ ≤ un−x^∗and fromLemma 2.3and3.21, we obtain xn1−x^∗2α_nγfxn β_nx_n

1−β_n

I−α_nA

T_ny_n−x^∗2 α_n

γfxn−Ax^∗

β_nxn−x^∗ 1−β_n

I−α_nA

T_ny_n−x^∗2

≤αnγfxn−Ax^∗2βnxn−x^∗2

1−βn−αnγyn−x^∗2

≤α_nγfxn−Ax^∗2β_nxn−x^∗2

1−β_n−α_nγ

un−x^∗2

≤αnγfxn−Ax^∗2βnxn−x^∗2

1−βn−αnγ

xn−x^∗2− xn−un²

≤αnγfxn−Ax^∗2xn−x^∗2−

1−βn−αnγ

xn−un².

3.22

Then, we have 1−βn−αnγ

xn−un²≤αnγfxn−Ax^∗2xn−x^∗2− x_n1−x^∗2

α_nγfxn−Ax^∗2xn−x_n1xn−x^∗x_n1−x^∗. 3.23

By lim_{n→ ∞}x_n1−x_n0,iandiv, imply that

nlim→ ∞un−xn0. 3.24

Since lim infn→ ∞rn>0,we obtain

nlim→ ∞

x_n−u_n r_n

lim

n→ ∞

1

r_nxn−u_n0. 3.25

Next, we show that limn→ ∞Tnyn−xn0.Indeed, observe that

xn−T_ny_n ≤ xn−x_n1x_n1−T_ny_n

xn−x_n1αnγfxn β_nx_n 1−β_n

I−α_nA

T_ny_n−T_ny_n

xn−x_n1αnγfxn−αnATnynαnATnynβnxn−βnTnynβnTnyn

1−βn

I−αnA

Tnyn−Tnyn

≤ xn−x_n1αnγfxn−ATnynβnxn−Tnyn

3.26

and then

xn−T_ny_n ≤ 1

1−βnxn−x_n1 α_n

1−βnγfxn−AT_ny_n. 3.27

Since limn→ ∞x_n1−xn0,iandiv, we get limn→ ∞xn−Tnyn0.

Next, we show that lim_{n→ ∞}yn−v_n0,wherev_nP_Cun−λ_nBu_n.FromLemma 2.3 and3.3, we obtain

x_n1−x^∗2≤α_nγfxn−Ax^∗2β_nxn−x^∗2 1−β_n

I−α_nAT_ny_n−x^∗2

≤α_nγfxn−Ax^∗2β_nxn−x^∗2

1−β_n−α_nγy_n−x^∗2 α_nγfxn−Ax^∗2β_nxn−x^∗2

1−βn−αnγ

δnun−x^∗ 1−δn{PCun−λnBun−PCx^∗−λnBx^∗}²

≤αnγfxn−Ax^∗2βnxn−x^∗2

1−βn−αnγ

δnun−x^∗2

1−β_n−α_nγ

1−δ_nun−λ_nBu_n−x^∗−λ_nBx^∗2 αnγfxn−Ax^∗2βnxn−x^∗2

1−βn−αnγ

δnun−x^∗2

1−βn−αnγ

1−δnun−x^∗−λnBun−Bx^∗2

≤α_nγfxn−Ax^∗2β_nxn−x^∗2

1−β_n−α_nγ

δ_nun−x^∗2

1−β_n−α_nγ

1−δ_n

xn−x^∗2λ_n

λ_n−2β

Bun−Bx^∗2

≤α_nγfxn−Ax^∗2

1−α_nγ

xn−x^∗2

1−βn−αnγ

1−δnλn

λn−2β

Bun−Bx^∗2

≤α_nγfxn−Ax^∗2xn−x^∗2

1−β_n−α_nγ

1−δ_na b−2β

Bun−Bx^∗2.

3.28

It follows that 0≤

1−βn−αnγ

1−δna 2β−b

Bun−Bx^∗2

≤α_nγfxn−Ax^∗2xn−x^∗2− x_n1−x^∗2

≤αnγfxn−Ax^∗2xn1−xnxn−x^∗xn1−x^∗.

3.29

Sinceα_n → 0 andxn1−x_n → 0,asn → ∞,we obtainBun−Bx^∗ → 0 asn → ∞.Using 2.1, we have

vn−x^∗2PCun−λ_nBu_n−P_Cx^∗−λ_nBx^∗2

≤ un−λ_nBu_n−x^∗−λ_nBx^∗, vn−x^∗ 1

2

un−λnBun−x^∗−λnBx^∗2vn−x^∗2

−1 2

un−λnBun−x^∗−λnBx^∗−vn−x^∗2

1 2

un−x^∗2vn−x^∗2− un−v_n−λ_nBun−Bx^∗2

1 2

un−x^∗2vn−x^∗2

−

un−vn²λ²_nBun−Bx^∗2−2λnun−vn, Bun−Bx^∗

≤ 1 2

un−x^∗2vn−x^∗2− un−vn²−λ²_nBun−Bx^∗2

2λnun−v_n, Bu_n−Bx^∗ ,

3.30

so, we obtain

vn−x^∗2≤ un−x^∗2− un−vn²−λ²_nBun−Bx^∗22λnun−vn, Bun−Bx^∗, 3.31