A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

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Volume 2011, Article ID 619813,25pages doi:10.1155/2011/619813

Research Article

A General Iterative Algorithm for Generalized

Mixed Equilibrium Problems and Variational

Inclusions Approach to Variational Inequalities

Thanyarat Jitpeera and Poom Kumam

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

Correspondence should be addressed to Poom Kumam,[email protected]

Received 1 December 2010; Revised 28 January 2011; Accepted 17 February 2011

Academic Editor: Vittorio Colao

Copyrightq2011 T. Jitpeera 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 introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu2006, Su et al.2008, Klin-eam and Suantai2009, Tan and Chang2011, and some other authors.

1. Introduction

LetCbe a closed convex subset of a real Hilbert spaceHwith the inner product·,·and the norm·. LetFbe a bifunction ofC×CintoR, whereRis the set of real numbers,Ψ:C → H a mapping, andϕ:C → Ra real-valued function. The generalized mixed equilibrium problem is for findingx∈Csuch that

Fx, yΨx, y−xϕy−ϕx≥0, ∀y∈C. 1.1

The set of solutions of1.1is denoted by GMEPF, ϕ,Ψ, that is,

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IfF ≡ 0, the problem1.1is reduced into the mixed variational inequality of Browder type1

for findingx∈Csuch that

Ψx, y−xϕy−ϕx≥0, ∀y∈C. 1.3

The set of solutions of1.3is denoted by MVIC, ϕ,Ψ.

IfΨ ≡ 0 andϕ ≡ 0, the problem1.1is reduced into the equilibrium problem 2for findingx∈Csuch that

Fx, y≥0, ∀y∈C. 1.4

The set of solutions of1.4is denoted by EPF. This problem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; see3–5.

If F ≡ 0 and ϕ ≡ 0, the problem 1.1 is reduced into the Hartmann-Stampacchia

variational inequality6for findingx∈Csuch that

Ψx, y−x≥0, ∀y∈C. 1.5

The set of solutions of 1.5 is denoted by VIC,Ψ. The variational inequality has been extensively studied in the literature7.

IfF ≡ 0 andΨ≡0, the problem1.1is reduced into the minimize problem for finding x∈Csuch that

ϕy−ϕx≥0, ∀y∈C. 1.6

The set of solutions of1.6is denoted by Arg minϕ.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. 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:

θx 1

2Ax, x −

x, y, ∀x∈FS, 1.7

whereAis a linear bounded operator,FSis the fixed point set of a nonexpansive mapping S, andyis a given point inH8.

Recall that a mappingS:C → Cis said to be nonexpansive if

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for allx, y ∈ C. IfCis bounded closed convex andSis a nonexpansive mapping ofCinto itself, thenFSis nonempty9. We denote weak convergence and strong convergence by notationsand →, respectively. A mappingAofCintoHis called monotone if

Ax−Ay, x−y≥0, 1.9

for allx, y∈C. A mappingAofCintoHis calledα-inverse-strongly monotone if there exists a positive real numberαsuch that

Ax−Ay, x−y≥αAx−Ay2, 1.10

for allx, y∈C. It is obvious that anyα-inverse-strongly monotone mappingAis monotone and Lipschitz continuous mapping. A linear bounded operatorAis strongly positive if there exists a constantγ >0 with the property

Ax, x ≥γx2, 1.11

for all x ∈ H. A self mapping f : C → Cis a contractions onC if there exists a constant α∈0,1such that

fx−fy≤αx−y, 1.12

for allx, y ∈ C. We use _C to denote the collection of all contraction on C. Note that each f∈ _Chas a unique fixed point inC.

LetB:H → Hbe a single-valued nonlinear mapping andM:H → 2H_{a set-valued}

mapping. The variational inclusion problem is to findx∈Hsuch that

θ∈Bx Mx, 1.13

whereθis the zero vector inH. The set of solutions of problem1.13is denoted byIB, M. The variational inclusion has been extensively studied in the literature, see, for example,10–

13and the reference therein.

A set-valued mappingM:H → 2H_{is called monotone if for all}_{x, y} _∈_H,_f _∈_M_x_,

and g ∈ Myimply x−y, f −g ≥ 0. A monotone mapping Mis maximal if its graph GM : {f, x ∈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 ally, g∈GMimplyf∈Mx.

LetBbe an inverse-strongly monotone mapping ofCintoH, and letNCvbe normal

cone toCatv∈C, that is,NCv{w∈H:v−u, w ≥0,∀u∈C}, and define

Tv

⎧ ⎨ ⎩

BvNCv, ifv∈C,

∅, ifv /∈C.

1.14

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LetM:H → 2H_{be a set-valued maximal monotone mapping; then the single-valued}

mappingJM,λ:H → Hdefined by

JM,λx IλM−1x, x∈H 1.15

is called the resolvent operator associated with M, whereλis any positive number and I is the identity mapping. It is worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone and that a solution of problem 1.13 is a fixed point of the operatorJM,λI−λBfor allλ >015.

In 2000, Moudafi16introduced the viscosity approximation method for nonexpan-sive mapping and proved that, ifHis a real Hilbert space, the sequence{xn}defined by the iterative method below, with the initial guessx0∈C, is chosen arbitrarily,

xn1αnfxn 1−αnSxn, n≥0, 1.16

where{αn} ⊂ 0,1satisfies certain conditions and converges strongly to a fixed point ofS

sayx∈Cwhich is the unique solution of the following variational inequality:

I−fx, x−x≥0, ∀x∈FS. 1.17

In 2006, Marino and Xu8introduced a general iterative method for nonexpansive mapping. They defined the sequence{xn}generated by the algorithmx0∈C:

xn1αnγfxn I−αnASxn, n≥0, 1.18

where{αn} ⊂ 0,1andAis a strongly positive linear bounded operator. They proved that, if C H and the sequence {αn} satisfies appropriate conditions, then the sequence {xn} generated by1.18converges strongly to a fixed point ofSsayx∈Hwhich is the unique solution of the following variational inequality:

A−γfx, x−x ≥0, ∀x∈FS, 1.19

which is the optimality condition for the minimization problem

min

x∈FS∩EPF

1

2Ax, x −hx, 1.20

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For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities, letPCbe the projection ofHontoC. In

2005, Iiduka and Takahashi17introduced following iterative process forx0∈C:

xn1αnu 1−αnSPCxn−λnAxn, ∀n≥0, 1.21

where u ∈ C, {αn} ⊂ 0,1, and{λn} ⊂ a, b for some a, b with 0 < a < b < 2β. They proved that under certain appropriate conditions imposed on{αn}and{λn}, the sequence

{xn}generated by1.21converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mappingsayx∈Cwhich solve some variational inequality

x−u, x−x ≥0, ∀x∈FS∩VIC, A. 1.22

In 2008, Su et al. 18 introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space:x1, un∈H

Fun, y

1

rn

y−un, un−xn ≥0, ∀y∈C,

xn1αnfxn 1−αnSPCun−λnAun,

1.23

for alln ∈ N, where {αn} ⊂ 0,1and {rn} ⊂ 0,∞ satisfy some appropriate conditions. Furthermore, they proved that{xn}and{un}converge strongly to the same pointz∈FS∩ VIC, A∩EPF, wherezPFS∩VIC,A∩EPFfz.

In 2011, Tan and Chang12introduced following iterative process for{Tn :C → C}

which is a sequence of nonexpansive mappings. Let{xn}be the sequence defined by

x_n1αnxn 1−αn

SPC

1−tnJM,λI−λATμ

I−μBxn

, ∀n≥0, 1.24

where {αn} ⊂ 0,1, λ ∈ 0,2α, and μ ∈ 0,2β. The sequence {xn} defined by 1.24

converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solutions of the variational inequality, and the generalized equilibrium problem.

In this paper, we modify the iterative methods1.18,1.23, and1.24by proposing the following new general viscosity iterative method:x0, un∈C,

Fun, y

ϕy−ϕun

Ψxn, y−un

1

rn

y−un, un−xn

≥0, ∀y∈C,

x_n1PC

αnγfxn I−αnASJM,λun−λBun

,

1.25

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2. Preliminaries

LetHbe a real Hilbert space andCa nonempty closed convex subset ofH. Recall that the

nearest point projection PC fromH onto C assigns to each x ∈ H the unique point in

PCx∈Csatisfying the property

x−PCxmin

y∈Cx−y. 2.1

The following characterizes the projectionPC. We recall some lemmas which will be needed

in the rest of this paper.

Lemma 2.1. The functionu∈Cis a solution of the variational inequality1.5if and only ifu∈C

satisfies the relationuPCu−λΨufor allλ >0.

Lemma 2.2. For a givenz∈H,u∈C,uPCz⇔ u−z, v−u ≥0, for allv∈C.

It is well known thatPCis a firmly nonexpansive mapping ofHontoCand satisfies

PCx−PCy2≤

PCx−PCy, x−y

, ∀x, y∈H. 2.2

Moreover,PCxis characterized by the following properties:PCx∈Cand, for allx∈H,y∈C,

x−PCx, y−PCx

≤0. 2.3

Lemma 2.3see19. LetM:H → 2Hbe a maximal monotone mapping, and letB:H → Hbe

a monotone and Lipshitz continuous mapping. Then the mappingLMB:H → 2H_{is a maximal}

monotone mapping.

Lemma 2.4see20. Each Hilbert spaceH satisfies Opial’s condition, that is, for any sequence

{xn} ⊂Hwithxn x, the inequality lim infn→ ∞xn−x<lim infn→ ∞xn−yholds for each

y∈Hwithy /x.

Lemma 2.5see21. Assume that{an}is a sequence of nonnegative real numbers such that

a_n1≤

1−γn

anδn, ∀n≥0, 2.4

where{γn} ⊂0,1and{δn}is a sequence inRsuch that

i∞_n₁γn∞,

iilim sup_n_{→ ∞}δn/γn≤0 or

_∞

n1|δn|<∞.

Then limn→ ∞an0.

Lemma 2.6see22. LetCbe a closed convex subset of a real Hilbert spaceH, and letT :C → C

be a nonexpansive mapping. ThenI−Tis demiclosed at zero, that is,

xn x, xn−Txn−→0 2.5

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For solving the generalized mixed equilibrium problem, let us assume that the bifunctionF : C×C → R, the nonlinear mapping Ψ : C → H is continuous monotone, andϕ:C → Rsatisfies the following conditions:

A1Fx, x 0 for allx∈C;

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

A3for each fixedy∈C,x→Fx, yis weakly upper semicontinuous;

A4for each fixedx∈C,y→Fx, yis convex and lower semicontinuous;

B1for eachx∈Candr >0, there exist a bounded subsetDx⊆Candyx ∈Csuch that,

for anyz∈C\Dx,

Fz, yx

ϕyx

−ϕz 1

r

yx−z, z−x

<0; 2.6

B2Cis a bounded set.

Lemma 2.7see23. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let

F :C×C → Rbe a bifunction mapping satisfying (A1)–(A4), and letϕ :C → Rbe convex and lower semicontinuous such thatC∩domϕ /∅. Assume that either (B1) or (B2) holds. Forr >0 and x∈H, there existsu∈Csuch that

Fu, yϕy−ϕu 1 r

y−u, u−x. 2.7

Define a mappingKr :H → Cas follows:

Krx

u∈C:Fu, yϕy−ϕu 1 r

y−u, u−x≥0, ∀y∈C

2.8

for allx∈H. Then, the following hold:

iKris single valued;

iiKris firmly nonexpansive, that is, for anyx, y∈H, Krx−Kry2 ≤ Krx−Kry, x−y;

iiiFKr MEPF, ϕ;

ivMEPF, ϕis closed and convex.

Lemma 2.8see8. Assume that Ais a strongly positive linear bounded operator on a Hilbert

spaceHwith coeﬃcientγ >0 and 0< ρ≤ A−1; thenI−ρA ≤1−ργ.

3. Strong Convergence Theorems

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Theorem 3.1. Let H be a real Hilbert space, C a closed convex subset ofH, B,Ψ : C → H

beβ,σ-inverse-strongly monotone mappings, respectively. Letϕ : C → Rbe a convex and lower semicontinuous function,f :C → Ca contraction with coeﬃcientα0< α <1,M:H → 2H_a

maximal monotone mapping, andAa strongly positive linear bounded operator ofHinto itself with coeﬃcientγ >0. Assume that 0< γ < γ/α. LetSbe a nonexpansive mapping ofCinto itself such that

Ω:FS∩GMEPF, ϕ,Ψ∩IB, M/∅. 3.1

Suppose that{xn}is a sequences generated by the following algorithm forx0 ∈Carbitrarily:

Fun, y

ϕy−ϕun

Ψxn, y−un

1

rn

y−un, un−xn

≥0, ∀y∈C,

xn1PC

αnγfxn I−αnASJM,λun−λBun

3.2

for alln0,1,2, . . ., where

C1{αn} ⊂0,1, limn→0αn0,

_∞

n1αn∞, and

_∞

n1|αn1−αn|<∞,

C2{rn} ⊂c, dwithc, d∈0,2σand∞_n₁|rn1−rn|<∞,

C3λ∈0,2β.

Then{xn} converges strongly toq ∈ Ω, whereq PΩγf I −Aq which solves the

following variational inequality:

γf−Aq, p−q≤0, ∀p∈Ω 3.3

which is the optimality condition for the minimization problem

min

q∈Ω

1

2Aq, q −h

q, 3.4

wherehis a potential function forγf(i.e.,hq γfqforq∈H).

Proof. Due to condition C1, we may assume without loss of generality, then, that αn ∈

0,A−1_{for all}_{n. By}_{Lemma 2.8}_{, we have that}_I₋_α

nA ≤ 1−αnγ. Next, we will assume

thatI−A ≤ 1−γ.

Next, we will divide the proof into six steps.

Step 1. We will show that{xn},{un}are bounded.

SinceB,Ψareβ,σ-inverse-strongly monotone mappings, we have that

_I₋_λB_x₋_I₋_λB_y2

x−y−λBx−By2

x−y2−2λx−y, Bx−Byλ2Bx−By2

≤x−y2λλ−2βBx−By2

≤x−y2.

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In a similar way, we can obtain

I−rnΨx−I−rnΨy2≤x−y2. 3.6

It is clear that if 0< λ <2β, 0< rn<2σ, thenI−λB,I−rnΨare all nonexpansive.

PutynJM,λun−λBun,n≥0. It follows that

yn−qJM,λun−λBun−JM,λ

q−λBq

≤un−q.

3.7

ByLemma 2.7, we have thatunKrnxn−rnΨxnfor alln≥0. Then, we have that

_u_n₋_q2

Krnxn−rnΨxn−Krnq−rnΨq2

≤xn−rnΨxn−q−rnΨq2

≤xn−q2rnrn−2σΨxn−Ψq2

xn−q2.

3.8

Hence, we have that

_y_n₋_q_≤_x_n₋_q_. _3.9

From3.2, we deduce that

_x_n₁₋_q_P_C

αnγfxn I−αnASyn

−PC

q

≤αn

γfxn−Aq

I−αnA

Syn−q

≤αnγfxn−Aq

1−αnγyn−q

≤αnγfxn−γf

qαnγf

q−Aq1−αnγyn−q

≤αγαnxn−qαnγf

q−Aq1−αnγxn−q

1−γ−γααnxn−qαnγf

q−Aq

1−γ−γααnxn−q

γ−γααn

γfq−Aq γ−γα

≤max

xn−q,

γfq−Aq γ−γα

.

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It follows by induction that

xn−q≤max

x0−q,

_γf

q−Aq γ−γα

, n≥0. 3.11

Therefore{xn}is bounded, so are{yn},{Syn},{Bun},{fxn}, and{ASyn}.

Step 2. We claim that limn→ ∞xn1−xn0. From3.2, we have that

xn1−xnPC

−PC

αn−1γfxn−1 I−αn−1ASyn−1

≤I−αnA

Syn−Syn−1

−αn−αn−1ASyn−1

γαn

fxn−fxn−1

γαn−αn−1fxn−1

≤1−αnγyn−yn−1|αn−αn−1|ASyn−1

γααnxn−xn−1γ|αn−αn−1|fxn−1.

3.12

SinceI−λBare nonexpansive, we also have that

yn−yn−1JM,λun−λBun−JM,λun−1−λBun−1

≤ un−λBun−un−1−λBun−1

≤ un−un−1.

3.13

On the other hand, fromu_n−1 Krn−1xn−1−rn−1Ψxn−1andun Krnxn−rnΨxn, it follows

that

Fun−1, yΨxn−1, y−un−1

ϕy−ϕun−1 1 r_n−1

y−un−1, un−1−xn−1

≥0, ∀y∈C,

3.14

Fun, y

Ψxn, y−unϕ

y−ϕun 1

rn

y−un, un−xn ≥0, ∀y∈C. 3.15

Substitutingyuninto3.14and yun−1into3.15, we get

Fun−1, un Ψxn−1, un−un−1ϕun−ϕun−1 1 r_n−1

un−un−1, un−1−xn−1 ≥0,

Fun, un−1 Ψxn, un−1−unϕun−1−ϕun 1

rn

un−1−un, un−xn ≥0.

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FromA2, we obtain

un−un−1,Ψxn−1−Ψxn

un−1−xn−1 r_n−1 −

un−xn

rn

≥0, 3.17

and then

un−un−1, rn−1Ψxn−1−Ψxn un−1−xn−1−rn−1

rn un−xn

≥0. 3.18

So

un−un−1, rn−1Ψxn−1−rn−1Ψxnun−1−unun−xn−1−

rn−1 rn

un−xn ≥0. 3.19

It follows that

un−un−1,I−rn−1Ψxn−I−rn−1Ψxn−1un−1−unun−xn−

rn−1 rn

un−xn

≥0,

un−un−1, un−1−unun−un−1, xn−xn−1

1−rn−1 rn

un−xn ≥0.

3.20

Without loss of generality, let us assume that there exists a real numbercsuch thatrn−1> c >0, for alln∈N. Then, we have that

un−un−12≤

un−un−1, xn−xn−1

1− rn−1 rn

un−xn

≤ un−un−1

xn−xn−1

1− rn−1

rn

un−xn

,

3.21

and hence

un−un−1 ≤ xn−xn−1 1

rn|

rn−rn−1|un−xn

≤ xn−xn−1 M1

c |rn−rn−1|,

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whereM1sup{un−xn:n∈N}. Substituting3.22into3.13, we have that

yn−yn−1≤ xn−xn−1 M1

c |rn−rn−1|. 3.23

Substituting3.23into3.12, we get

xn1−xn ≤

1−αnγ

xn−xn−1

M1

c |rn−rn−1|

|αn−αn−1|ASyn−1

γααnxn−xn−1γ|αn−αn−1|fxn−1

1−αnγ

xn−xn−1

1−αnγ

M1

c |rn−rn−1||αn−αn−1|ASyn−1

γααnxn−xn−1γ|αn−αn−1|fxn−1

≤1−γ−γααn

xn−xn−1

M1

c |rn−rn−1||αn−αn−1|ASyn−1

γ|αn−αn−1|fxn−1

≤1−γ−γααn

xn−xn−1

M1

c |rn−rn−1|M2|αn−αn−1|,

3.24

whereM2 sup{max{ASyn−1,fxn−1 :n ∈N}}. By conditionsC1-C2andLemma

2.5, we have thatxn1−xn → 0 asn → ∞. From3.23, we also have thatyn1−yn → 0 asn → ∞.

Step 3. We show the following:

ilimn→ ∞Bun−Bq0;

iilimn→ ∞Ψxn−Ψq0.

Forq∈ΩandqJM,λq−λBq, by3.5and3.8, we get

yn−q2JM,λun−λBun−JM,λq−λBq2

≤un−λBun−q−λBq2

≤un−q2λ

λ−2βBun−Bq2

≤xn−q2λ

λ−2βBun−Bq2.

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It follows that

xn1−q2PCαnγfxn I−αnASyn−PCq2

≤αnγfxn−Aq I−αnASyn−q2

≤αnγfxn−Aq

1−αnγyn−q2

≤αnγfxn−Aq2

1−αnγyn−q2

2αn

1−αnγγfxn−Aqyn−q

≤αnγfxn−Aq22αn

1−αnγ

xn−q2λ

λ−2βBun−Bq2

xn−q2

1−αnγ

λλ−2βBun−Bq2.

3.26

So, we obtain

1−αnγ

λ2β−λBun−Bq2

≤αnγfxn−Aq2xn−xn1xn−qxn1−qn,

3.27

wheren2αn1−αnγγfxn−Aqyn−q. By conditionsC1andC3and limn→ ∞xn1− xn0, we obtain thatBun−Bq → 0 asn → ∞.

_y_n₋_q2_≤

un−q2λ

λ−2βBun−Bq2

≤xn−q2rnrn−2σΨxn−Ψq2

3.28

From3.26, we have that

_x_n₁₋_q2_≤

αnγfxn−Aq22αn

1−αnγxn−q2rnrn−2σΨxn−Ψq2λ

λ−2βBun−Bq2

1−αnγγfxn−Aqyn−qxn−q2

1−αnγ

rnrn−2σΨxn−Ψq2

1−αnγ

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So, we also have that

1−αnγ

rn2σ−rnΨxn−Ψq2

≤αnγfxn−Aq2xn−xn1xn−qxn1−q

n

1−αnγ

λλ−2βBun−Bq2,

3.30

wheren2αn1−αnγγfxn−Aqyn−q. By conditionsC1–C3, limn→ ∞xn1−xn0 and limn→ ∞Bun−Bq0, we obtain thatΨxn−Ψq → 0 asn → ∞.

Step 4. We show the following:

ilimn→ ∞xn−un0;

iilimn→ ∞un−yn0;

iiilimn→ ∞yn−Syn0.

SinceKrn is firmly nonexpansive and by2.2, we observe that

un−q2Krnxn−rnΨxn−Krnq−rnΨq2

≤xn−rnΨxn−

q−rnΨq

, un−q

1

2

xn−rnΨxn−q−rnΨq2un−q2

−xn−rnΨxn−q−rnΨq−un−q2

≤ 1

2

xn−q2un−q2−xn−un−rnΨxn−Ψq2

1

2

xn−q2un−q2− xn−un2

2rn

Ψxn−Ψq, xn−un

−r_n2Ψxn−Ψq2

.

3.31

Hence, we have that

_u_n₋_q2_≤

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SinceJM,λis 1-inverse-strongly monotone and by2.2, we compute

_y_n₋_q2

JM,λun−λBun−JM,λq−λBq2

≤un−λBun−

q−λBq, yn−q

1

2

un−λBun−

q−λBq2yn−q2

−un−λBun−

q−λBq−yn−q2

≤ 1

2

un−q2yn−q2−un−yn

−λBun−Bq2

1

2

un−q2yn−q2−un−yn2

2λun−yn, Bun−Bq

−λ2Bun−Bq2

,

3.33

which implies that

_y_n₋_q2_≤

un−q2−un−yn22λun−ynBun−Bq. 3.34

Substituting3.32into3.34, we have that

yn−q2≤

xn−q2− xn−un22rnΨxn−Ψqxn−un

−un−yn22λun−ynBun−Bq.

3.35

x_n1−q2≤αnγfxn−Aq2yn−q22αn

≤αnγfxn−Aq2

xn−q2− xn−un22rnΨxn−Ψqxn−un

−un−yn22λnun−ynBun−Bq

2αn

1−αnγγfxn−Aqyn−q.

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Then, we derive

xn−un2un−yn2≤αnγfxn−Aq2xn−q2−xn1−q2

2rnΨxn−Ψqxn−un2λun−ynBun−Bq

2αn

αnγfxn−Aq2xn−xn1xn−qxn1−q

2rnΨxn−Ψqxn−un2λun−ynBun−Bq

2αn

1−αnγ

γfxn−Aqyn−q.

3.37

By conditionC1, limn→ ∞xn−xn10, limn→ ∞Ψxn−Ψq0, and limn→ ∞Bun−Bq0.

So, we have thatxn−un → 0,un−yn → 0 asn → ∞. It follows that

xn−yn≤ xn−unun−yn−→0, asn−→ ∞. 3.38

From3.2, we have that

xn−Syn≤xn−Syn−1Syn−1−Syn

≤PC

αn−1γfxn−1 I−αn−1ASyn−1

−PC

Syn−1yn−1−yn

≤α_n−1γfxn−1−ASyn−1yn−1−yn.

3.39

By conditionC1and limn→ ∞yn−1−yn 0, we obtain thatxn−Syn → 0 asn → ∞.

Next, we observe that

x_n1−SynPC

−PC

Syn

≤αnγfxn I−αnASyn−Syn

αnγfxn−ASyn.

3.40

Since{ASyn}is bounded and by conditionC1, we have thatxn1−Syn → 0 asn → ∞, and

_x_n₋_Sy_n_≤_x_n₋_x_n₁_x_n₁₋_Sy_n_. _3.41

Since limn→ ∞xn−xn10 and limn→ ∞xn1−Syn0, it implies thatxn−Syn → 0 as

n → ∞. Hence, we have that

xn−Sxn ≤xn−SynSyn−Sxn

≤xn−Synyn−xn.

(17)

By3.38and limn→ ∞xn−Syn 0, we obtainxn−Sxn → 0 asn → ∞. Moreover, we

also have that

yn−Syn≤yn−xnxn−Syn. 3.43

By3.38and limn→ ∞xn−Syn0, we obtainyn−Syn → 0 asn → ∞.

Step 5. We show that q ∈ Ω : FS∩GMEPF, ϕ,Ψ∩IB, M and lim sup_n_{→ ∞}γf − Aq, Syn −q ≤ 0. It is easy to see that PΩγf I −Ais a contraction of H into itself.

Indeed, since 0< γ < γ/α, we have that

P_Ωγf I−Ax−P_Ωγf I−Ay≤γf I−Ax−γf−I−Ay

≤γfx−fyI−Ax−y

≤γαx−y1−γx−y

≤1−γγαx−y.

3.44

HenceHis complete, and there exists a unique fixed pointq∈Hsuch thatqPΩγf I−

Aq. ByLemma 2.2, we obtain thatγf−Aq, w−q ≤0 for allw∈Ω.

Next, we show that lim sup_n_{→ ∞}γf−Aq, Syn−q ≤0, whereqPΩγfI−Aq

is the unique solution of the variational inequalityγf−Aq, p−q ≥0, for allp∈Ω. We can choose a subsequence{yni}of{yn}such that

lim sup

n→ ∞

γf−Aq, Syn−q lim

i→ ∞

γf−Aq, Syni −q

. _3.45

As{yni}is bounded, there exists a subsequence{yn_ij}of{yni}which converges weakly tow.

We may assume without loss of generality thatyni w.

We claim thatw∈Ω. Sinceyn−Syn → 0,xn−Sxn → 0, andxn−yn → 0 and

byLemma 2.6, we have thatw∈FS.

Next, we show thatw∈GMEPF, ϕ,Ψ. SinceunKrnxn−rnΨxn, we know that

Fun, y

ϕy−ϕun Ψxn, y−un

1

rny−un, un−xn ≥0, ∀y∈C.

3.46

It follows byA2that

ϕy−ϕun Ψxn, y−un 1

rny−un, un−xn ≥F

y, un

, ∀y∈C. 3.47

Hence,

ϕy−ϕuni

Ψxni, y−uni

1

rni

y−uni, uni −xni

≥Fy, uni

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Fort∈0,1andy∈H, letytty 1−tw. From3.48, we have that

yt−uni,Ψyt ≥ yt−uni,Ψyt −ϕ

yt

ϕuni− Ψxni, yt−uni

− 1

rni

yt−uni, uni−xniF

yt, uni

yt−uni,Ψyt−Ψuni

yt−uni,Ψuni −Ψxni

−ϕyt

ϕuni

− 1

rni

yt−uni, uni−xni

Fyt, uni

.

3.49

Fromuni−xni → 0, we have thatΨuni −Ψxni → 0. Further, fromA4and the weakly

lower semicontinuity ofϕ,uni−xni/rni → 0 anduni w, we have that

yt−w,Ψyt

≥ −ϕyt

ϕw Fyt, w

. 3.50

FromA1,A4, and3.50, we have that

0Fyt, yt

−ϕyt

ϕyt

≤tFyt, y

1−tFyt, w

tϕy 1−tϕw−ϕyt

tFyt, y

ϕy−ϕyt

1−tFyt, w

ϕw−ϕyt

≤tFyt, y

ϕy−ϕyt

1−tyt−w,Ψyt

tFyt, y

ϕy−ϕyt

1−tty−w,Ψyt,

3.51

and hence

0≤Fyt, y

ϕy−ϕyt

1−ty−w,Ψyt

. 3.52

Lettingt → 0, we have, for eachy∈C, that

Fw, yϕy−ϕw y−w,Ψw ≥0. 3.53

This implies thatw∈GMEPF, ϕ,Ψ.

Lastly, we show thatw∈IB, M. In fact, sinceBis aβ-inverse-strongly monotone,B is monotone and Lipschitz continuous mapping. It follows fromLemma 2.3thatMBis a maximal monotone. Letv, g∈GMB, sinceg−Bv∈Mv. Again sinceyni JM,λuni−

λBuni, we have thatuni−λBuni ∈IλMyni, that is,1/λuni−yni−λBuni∈Myni. By

virtue of the maximal monotonicity ofMB, we have that

v−yni, g−Bv−

1 λ

uni−yni−λBuni

(19)

and hence

v−yni, g

≥

v−yni, Bv

1 λ

uni−yni−λBuni

v−yni, Bv−Byni

v−yni, Byni −Buni

v−yni,

1 λ

uni−yni

.

3.55

It follows from limn→ ∞un−yn0, limn→ ∞Bun−Byn0, andyni wthat

lim sup

n→ ∞

v−yn, g

v−w, g≥0. _3.56

It follows from the maximal monotonicity ofBMthatθ∈MBw, that is,w∈IB, M. Therefore,w∈Ω. It follows that

lim sup

n→ ∞

γf−Aq, Syn−q

lim

i→ ∞

γf−Aq, Syni −q

γf−Aq, w−q≤0. _3.57

Step 6. We prove thatxn → q. By using3.2and together with Schwarz inequality, we have

that

xn1−q2PC

−PCq2

≤αnγfxn−Aq I−αnA

Syn−q2

≤I−αnA2Syn−q2α2nγfxn−Aq2

2αn

I−αnA

Syn−q

, γfxn−Aq

≤1−αnγ

2

yn−q2α2nγfxn−Aq2

2αn

Syn−q, γfxn−Aq

−2α2_nASyn−q

, γfxn−Aq

≤1−αnγ

2

xn−q2α2nγfxn−Aq22αnSyn−q, γfxn−γf

q

2αnSyn−q, γf

q−Aq −2α2_nASyn−q

, γfxn−Aq

≤1−αnγ

2

xn−q2αn2γfxn−Aq22αnSyn−qγfxn−γf

q

2αn

Syn−q, γf

q−Aq−2α2_nASyn−q

, γfxn−Aq

≤1−αnγ

2

xn−q2α2nγfxn−Aq22γααnyn−qxn−q

2αn

Syn−q, γf

q−Aq−2α2_nASyn−q

, γfxn−Aq

≤1−αnγ

2

xn−q2α2nγfxn−Aq22γααnxn−q2

2αnSyn−q, γf

q−Aq −2α2

n

ASyn−q

, γfxn−Aq

(20)

≤1−αnγ

2_2γαα

n

xn−q2

αn

αnγfxn−Aq22

Syn−q, γf

q−Aq

−2αnA

Syn−qγfxn−Aq

1−2γ−γααnxn−q2

αn

αnγfxn−Aq22

Syn−q, γf

q−Aq

−2αnA

Syn−qγfxn−Aqαnγ2xn−q2

.

3.58

Since{xn}is bounded, whereη ≥ γfxn−Aq2 −2ASyn−qγfxn−Aq

γ2xn−q2for alln≥0, it follows that

x_n1−q2≤

1−2γ−γααnxn−q2αnςn, 3.59

whereςn 2Syn −q, γfq−Aqηαn. By lim sup_n_{→ ∞}γf −Aq, Syn −q ≤ 0, we get

lim sup_n_{→ ∞}ςn ≤ 0. ApplyingLemma 2.5, we can conclude thatxn → q. This completes the

proof.

Corollary 3.2. Let H be a real Hilbert space andCa closed convex subset ofH. LetB,Ψ:C → H

beβ, σ-inverse-strongly monotone mappings andϕ : C → Ra convex and lower semicontinuous function. Letf :C → Cbe a contraction with coeﬃcientα0< α <1,M:H → 2H_{a maximal}

monotone mapping, andSa nonexpansive mapping ofCinto itself such that

Ω:FS∩GMEPF, ϕ,Ψ∩IB, M/∅. 3.60

Suppose that{xn}is a sequence generated by the following algorithm forx0, un∈Carbitrarily:

Fun, y

1 rn

y−un, un−xn ≥0, ∀y∈C,

x_n1 PC

αnfxn 1−αnSJM,λun−λBun

3.61

for alln0,1,2, . . ., by (C1)–(C3) inTheorem 3.1.

Then{xn}converges strongly toq∈ Ω, whereqP_ΩfIqwhich solves the following variational inequality:

f−Iq, p−q≤0, ∀p∈Ω. 3.62

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Corollary 3.3. LetHbe a real Hilbert space andCa closed convex subset ofH. LetB,Ψ:C → Hbe

β,σ-inverse-strongly monotone mappings,ϕ:C → Ra convex and lower semicontinuous function, andM:H → 2H_{a maximal monotone mapping. Let}_S_{be a nonexpansive mapping of}_C_{into itself}

such that

Ω:FS∩GMEPF, ϕ,Ψ∩IB, M/∅. 3.63

Suppose that{xn}is a sequence generated by the following algorithm forx0, u∈Candun∈C:

Fun, y

rny−un, un−xn ≥0, ∀y∈C,

xn1PCαnu 1−αnSJM,λun−λBun

3.64

Then {xn} converges strongly to q ∈ Ω, where q PΩq which solves the following

variational inequality:

u−q, p−q ≤0, ∀p∈Ω. 3.65

Proof. Puttingfx ≡u, for allx ∈C, inCorollary 3.2, we can obtain the desired conclusion immediately.

Corollary 3.4. LetH be a real Hilbert space, Ca closed convex subset ofH,B : C → H beβ

-inverse-strongly monotone mappings, andAa strongly positive linear bounded operator ofH into itself with coeﬃcient γ > 0. Assume that 0 < γ < γ/α. Let f : C → C be a contraction with coeﬃcientα0< α <1andSa nonexpansive mapping ofCinto itself such that

Ω:FS∩VIC, B/∅. 3.66

Suppose that{xn}is a sequence generated by the following algorithm forx0∈Carbitrarily:

xn1PC

αnγfxn I−αnASPCxn−λBxn

3.67

Then{xn} converges strongly toq ∈ Ω, whereq PΩγf I −Aq which solves the

following variational inequality:

γf−Aq, p−q≤0, ∀p∈Ω. 3.68

Proof. TakingF≡0, Ψ≡0, ϕ≡0, unxn, andJM,λ PCinTheorem 3.1, we can obtain the

desired conclusion immediately.

Remark 3.5. InCorollary 3.4we generalize and improve the result of Klin-eam and Suantai

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4. Applications

In this section, we apply the iterative scheme 1.25 for finding a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping and also applyTheorem 3.1

for finding a common fixed point of nonexpansive mappings and inverse-strongly monotone mappings.

Definition 4.1. A mapping T : C → C is called strictly pseudocontraction if there exists a constant 0≤κ <1 such that

_Tx₋_Ty2_≤

x−y2κI−Tx−I−Ty2, ∀x, y∈C. 4.1

If κ 0, then S is nonexpansive. In this case, we say that T : C → C is a κ-strictly pseudocontraction. PuttingBI−T. Then, we have that

_I₋_B_x₋_I₋_B_y2 _≤

x−y2κBx−By2, ∀x, y∈C. 4.2

Observe that

I−Bx−I−By2x−y2Bx−By2−2x−y, Bx−By, ∀x, y∈C. 4.3

Hence, we obtain

x−y, Bx−By≥ 1−κ

2 Bx−By 2

, ∀x, y∈C. 4.4

Then,Bis1−κ/2-inverse-strongly monotone mapping.

UsingTheorem 3.1, we first prove a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strict pseudocontraction.

Theorem 4.2. LetHbe a real Hilbert space,Ca closed convex subset ofH,B,Ψ :C → Hbeβ,

σ-inverse-strongly monotone mappings,ϕ : C → Ra convex and lower semicontinuous function,

f :C → Ca contraction with coeﬃcientα0 < α <1, andAa strongly positive linear bounded operator ofHinto itself with coeﬃcientγ >0. Assume that 0 < γ < γ/α. LetSbe a nonexpansive mapping ofCinto itself, and letT be aκ-strictly pseudocontraction ofCinto itself such that

Ω:FS∩FT∩GMEPF, ϕ,Ψ/∅. 4.5

Suppose that{xn}is a sequence generated by the following algorithm forx0, un∈Carbitrarily:

Fun, y

1 rn

y−un, un−xn

≥0, ∀y∈C,

xn1PC

αnγfxn I−αnAS1−λunλTun

4.6

(23)

Then{xn}converges strongly toq∈Ω, whereqPΩγfI−Aqwhich solves the following

γf−Aq, p−q≤0, ∀p∈Ω 4.7

min

q∈Ω

1 2

Aq, q−hq, 4.8

Proof. PutB ≡ I−T, thenBis1−κ/2-inverse-strongly monotone,FT IB, M, and JM,λxn−λBxn 1−λxnλTxn. So byTheorem 3.1, we obtain the desired result.

Corollary 4.3. LetHbe a real Hilbert space,Ca closed convex subset ofH,B,Ψ:C → Hbeβ,σ

-inverse-strongly monotone mappings, andϕ:C → Ra convex and lower semicontinuous function. Letf :C → Cbe a contraction with coeﬃcientα0 < α <1andSa nonexpansive mapping ofC

into itself, and letT be aκ-strictly pseudocontraction ofCinto itself such that

Ω:FS∩FT∩GMEPF, ϕ,Ψ/∅. 4.9

Suppose that{xn}is a sequence generated by the following algorithm forx0∈Carbitrarily:

Fun, y

rn

y−un, un−xn ≥0, ∀y∈C,

xn1PC

αnfxn I−αnS1−λunλTun

4.10

Then{xn}converges strongly toq∈ Ω, whereqPΩfIqwhich solves the following

f−Iq, p−q ≤0, ∀p∈Ω 4.11

min

q∈Ω

1

2Aq, q −h

q, 4.12

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Acknowledgments

The authors would like to thank the National Research University Project of Thailand’s Oﬃce of the Higher Education Commission for financial support under the project NRU-CSEC no. 54000267. Furthermore, they also would like to thank the Faculty of ScienceKMUTT and the National Research Council of Thailand. Finally, the authors would like to thank Professor Vittorio Colao and the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

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