A Hybrid Extragradient Method for Pseudomonotone Equilibrium Problems and Fixed Point Problems

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MALAYSIANMATHEMATICAL

SCIENCESSOCIETY http://math.usm.my/bulletin

A Hybrid Extragradient Method for Pseudomonotone Equilibrium Problems and Fixed Point Problems

PHAMNGOCANH

Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology, Hanoi, Vietnam [email protected]

Abstract. In this paper, we introduce a new hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for pseudomonotone and Lipschitz-type continuous bifunctions. The iterative process is based on two well-known methods: Hybrid and extragradient. We show that the iterative sequences generated by this algorithm converge strongly to the common element in a real Hilbert space.

2010 Mathematics Subject Classification: 65K10, 65K15, 90C25, 90C33

Keywords and phrases: Equilibrium problems, pseudomonotone, Lipschitz-type continuous, strong convergence, nonexpansive mapping, extragradient method.

1. Introduction

LetCbe a nonempty closed convex subset of a real Hilbert spaceHand f be a bifunction fromC×CtoR. We consider the equilibrium problems given as:

Findx^∗∈Csuch that f(x^∗,y)≥0 ∀y∈C. EP(f,C) The set of solutions ofEP(f,C)is denoted by Sol(f,C).

If f(x,y) =hF(x),y−xifor everyx,y∈C, whereF is a mapping fromC toH, then ProblemEP(f,C)becomes the following variational inequalities:

Findx^∗∈Csuch thathF(x^∗),y−x^∗i ≥0 ∀y∈C. V I(F,C) We denote Sol(F,C)which is the set of solutions ofV I(F,C).

In recent years, equilibrium problems become an attractive field for many researchers both theory and applications [1, 2, 4, 11, 17]. There are myriad of literature related to equilibrium problems and their applications in electricity market, transportation, economics and network [3, 5].

For solvingV I(F,C)in the Euclidean spaceRⁿunder the assumption that a subsetC⊆ Rⁿis nonempty closed convex,Fis monotone,L-Lipschitz continuous and Sol(F,C)6=/0,

Communicated byAhmad Izani Md. Ismail.

Received:November 18, 2010;Revised:March 1, 2011.

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Korpelevich in [7] introduced the following extragradient method:





 x⁰∈C,

yⁿ=Pr_C xⁿ−λF(xⁿ) , xⁿ⁺¹=Pr_C xⁿ−λF(yⁿ)

,

for all n≥0, where λ ∈(0,¹_L). The author showed that the sequences {xⁿ} and{yⁿ} converge to the same pointz∈Sol(F,C).

For eachx,y∈C, fϕ(x,y):=f(x,y) +ϕ(y)−ϕ(x), motivated by the results of Peng in [11] introduced a new iterative scheme for finding a common element of the sets Sol(f_ϕ,C), Sol(F,C)and Fix(T)in a real Hilbert space. Let sequences{xⁿ},{yⁿ},{tⁿ}and{zⁿ}be defined by









 x⁰∈H,

fϕ(uⁿ,y) +_r¹

nhy−uⁿ,uⁿ−xⁿi ≥0 ∀y∈C,

yⁿ:=Pr_C uⁿ−λnF(uⁿ) , tⁿ:=Pr_C uⁿ−λ_nF(yⁿ)

, zⁿ:=αntⁿ+ (1−αn)T(tⁿ),

C_n:={z∈C: kzⁿ−zk²≤ kxⁿ−zk²−(1−αn)(α_n−ε)ktⁿ−T(tⁿ)k}, Qn:={z∈H: hxⁿ−z,x−xⁿi ≥0},

xⁿ⁺¹:=Pr_C_n∩Q_n(x⁰).

Then, the author showed that under certain appropriate conditions imposed on{α_n},{λ_n} andε, the sequences{xⁿ},{uⁿ},{tⁿ},{yⁿ}and{zⁿ}converge strongly to Pr_Ω(x⁰), where Ω:=Sol(f_ϕ,C)∩Sol(F,C)∩Fix(T).

Recently, iterative algorithms for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonexpansive mapping in a real Hilbert space have further developed by some authors (see [11, 12, 14, 17, 18]). At each iterationn in all of these algorithms, it requires solving approximation auxiliary equilibrium problems.

In this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone, Lipschitz-type continuous bifunction. This method can be considered as an improvement of the iterative method in [11] via an improvement set of extragradient methods in [1, 2]. At each iterationn, we only solve strongly convex problems onC. We obtain a strong convergence theorem for four sequences generated by this process.

2. Preliminaries

LetH be a real Hilbert space with inner producth·,·iand normk · k, respectively. We list some well known definitions and the projection which will be required in our following analysis.

Definition 2.1. Let C be a closed convex subset in H, we denote the projection on C by Pr_C(·), i.e.,

Pr_C(x) =argmin{ky−xk: y∈C} ∀x∈H.

The bifunction f:C×C→Ris said to be

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(1) γ-strongly monotone on C if for each x,y∈C, we have f(x,y) +f(y,x)≤ −γkx−yk²; (2) monotone on C if for each x,y∈C, we have

f(x,y) +f(y,x)≤0;

(3) pseudomonotone on C if for each x,y∈C, we have f(x,y)≥0⇒f(y,x)≤0;

(4) Lipschitz-type continuous on C with constants c1>0and c2>0, if for each x,y∈C, we have

f(x,y) +f(y,z)≥f(x,z)−c₁kx−yk²−c₂ky−zk². The mapping F:C→H is said to be

(5) monotone on C if for each x,y∈C, we have hF(x)−F(y),x−yi ≥0;

(6) pseudomonotone on C if for each x,y∈C, we have hF(y),x−yi ≥0⇒ hF(x),x−yi ≥0;

(7) L-Lipschitz continuous on C if for each x,y∈C, we have kF(x)−F(y)k ≤Lkx−yk.

If L=1, then F is nonexpansive on C.

Note that if F is L-Lipschitz onC, then for each x,y∈C, f(x,y) =hF(x),y−x) is Lipschitz-type continuous with constantsc₁=c₂=^L₂onC. Indeed,

2f(x,y) +f(y,z)−f(x,z)

=hF(x),y−xi+hF(y),z−yi − hF(x),z−xi

=−hF(y)−F(x),y−zi ≥ −kF(x)−F(y)kky−zk ≥ −Lkx−ykky−zk

≥ −L

2kx−yk²−L

2ky−zk²=−c₁kx−yk²−c₂ky−zk². Thus f is Lipschitz-type continuous onC.

In this paper, for finding a point of the set Sol(f,C)∩Fix(T), we assume that the bifunction f satisfies the following conditions:

(i) f is pseudomonotone onC;

(ii) f is Lipschitz-type continuous onC;

(iii) for eachx∈C,y7→f(x,y)is convex and subdifferentiable onC;

(iv) Sol(f,C)∩Fix(T)6=/0.

Now we are in a position to describe the extragradient algorithm for finding a common of two sets Sol(f,C)and Fix(T).

Algorithm 2.1. Initialization. Choosex⁰∈C, positive sequences{λ_n}and{α_n}satisfy the conditions

({λ_n} ⊂[a,b] for some a,b∈ 0,min{_2c¹

1,_2c¹

2} , {α_n} ⊂[0,c] for some c∈(0,1).

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Step 1.Solve the strongly convex problems:







yⁿ=argmin{¹₂ky−xⁿk²+λ_nf(xⁿ,y): y∈C}, tⁿ=argmin{¹₂kt−xⁿk²+λ_nf(yⁿ,t): t∈C}, zⁿ:=α_nxⁿ+ (1−α_n)T(tⁿ).

Step 2.SetP_n={z∈C: kzⁿ−zk ≤ kxⁿ−zk}andQ_n={z∈C: hxⁿ−z,x⁰−xⁿi ≥0}.

Computexⁿ⁺¹=Pr_P_n∩Q_n(x⁰). Increasekby 1 and go to Step 1.

In order to prove the main result in Section 3, we shall use the following lemma in the sequel.

Lemma 2.1. [5]Let C be a convex subset of a real Hilbert space H and g:C→Rbe convex and subdifferentiable on C. Then, x^∗is a solution to the following convex problem

min{g(x): x∈C}

if and only if0∈∂g(x^∗) +N_C(x^∗), where∂g(·)denotes the subdifferential of g and N_C(x^∗) is the (outward) normal cone of C at x^∗.

3. Main results

In this section, we show a strong convergence theorem of sequences{xⁿ},{yⁿ},{zⁿ} and {tⁿ}defined by Algorithm 2.1 based on the extragradient method which solves the problem of finding a common element of two sets Sol(f,C)and Fix(T)for a monotone, Lipschitz- type continuous bifunction f in a real Hilbert spaceH.

Lemma 3.1. Suppose that x^∗∈Sol(f,C), f(x,·)is convex and subdifferentiable on C for all x∈C, and f is pseudomonotone on C. Then, we have

ktⁿ−x^∗k²≤ kxⁿ−x^∗k²−(1−2λ_nc₁)kxⁿ−yⁿk²−(1−2λ_nc₂)ktⁿ−yⁿk² ∀n≥0.

Proof. Since f(x,·)is convex onCfor eachx∈Cand Lemma 2.1, we obtain tⁿ=argmin{1

2kt−xⁿk²+λ_nf(yⁿ,t): t∈C}

if and only if

(3.1) 0∈∂2{λ_nf(yⁿ,t) +1

2kt−xⁿk²}(tⁿ) +N_C(tⁿ).

Since f(yⁿ,·)is subdifferentiable onC, by the well known Moreau-Rockafellar theorem [5], there existsw∈∂2f(yⁿ,tⁿ)such that

(3.2) f(yⁿ,t)−f(yⁿ,tⁿ)≥ hw,t−tⁿi ∀t∈C.

Witht=x^∗∈C, this inequality becomes

(3.3) f(yⁿ,x^∗)−f(yⁿ,tⁿ)≥ hw,x^∗−tⁿi.

It follows from (3.1) that

0=λ_nw+tⁿ−xⁿ+w,¯ wherew∈∂₂f(yⁿ,tⁿ)and ¯w∈N_C(tⁿ).

By the definition of the normal coneN_C we have, from the latter equality, that (3.4) htⁿ−xⁿ,t−tⁿi ≥λnhw,tⁿ−ti ∀t∈C.

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Witht=x^∗∈Cwe obtain

(3.5) htⁿ−xⁿ,x^∗−tⁿi ≥λnhw,tⁿ−x^∗i.

It follows from (3.3) and (3.5) that

(3.6) htⁿ−xⁿ,x^∗−tⁿi ≥λ_n{f(yⁿ,tⁿ)−f(yⁿ,x^∗)}.

Sincex^∗∈Sol(f,C),f(x^∗,y)≥0 for ally∈C, and f is pseudomonotone onC, we have f(yⁿ,x^∗)≤0. Then, (3.6) implies that

(3.7) htⁿ−xⁿ,x^∗−tⁿi ≥λnf(yⁿ,tⁿ).

Now applying Lipschitzian off withx=xⁿ,y=yⁿandz=tⁿ, we get (3.8) f(yⁿ,tⁿ)≥f(xⁿ,tⁿ)−f(xⁿ,yⁿ)−c1kyⁿ−xⁿk²−c2ktⁿ−yⁿk². Combinating (3.7) and (3.8), we have

(3.9) htⁿ−xⁿ,x^∗−tⁿi ≥λ_n{f(xⁿ,tⁿ)−f(xⁿ,yⁿ)−c₁kyⁿ−xⁿk²−c2ktⁿ−yⁿk²}.

Similarly, sinceyⁿis the unique solution to the strongly convex problem min

1

2ky−xⁿk²+λ_nf(xⁿ,y): y∈C

, we have

(3.10) λn{f(xⁿ,y)−f(xⁿ,yⁿ)} ≥ hyⁿ−xⁿ,yⁿ−yi ∀y∈C.

Asy=tⁿ∈C, we have

(3.11) λn{f(xⁿ,tⁿ)−f(xⁿ,yⁿ)} ≥ hyⁿ−xⁿ,yⁿ−tⁿi.

From (3.9), (3.11) and

2htⁿ−xⁿ,x^∗−tⁿi=kxⁿ−x^∗k²− ktⁿ−xⁿk²− ktⁿ−x^∗k², it implies that

kxⁿ−x^∗k²−ktⁿ−xⁿk²− ktⁿ−x^∗k²

≥2hyⁿ−xⁿ,yⁿ−tⁿi −2λ_nc1kxⁿ−yⁿk²−2λ_nc2ktⁿ−yⁿk². Hence, we have

ktⁿ−x^∗k²

≤kxⁿ−x^∗k²− ktⁿ−xⁿk²−2hyⁿ−xⁿ,yⁿ−tⁿi+2λ_nc₁kxⁿ−yⁿk²+2λ_nc₂ktⁿ−yⁿk²

=kxⁿ−x^∗k²− k(tⁿ−yⁿ) + (yⁿ−xⁿ)k²−2hyⁿ−xⁿ,yⁿ−tⁿi +2λ_nc₁kxⁿ−yⁿk²+2λ_nc₂ktⁿ−yⁿk²

≤kxⁿ−x^∗k²− ktⁿ−yⁿk²− kxⁿ−yⁿk²+2λ_nc₁kxⁿ−yⁿk²+2λ_nc₂ktⁿ−yⁿk²

=kxⁿ−x^∗k²−(1−2λ_nc1)kxⁿ−yⁿk²−(1−2λ_nc2)kyⁿ−tⁿk². The lemma thus is proved.

Lemma 3.2. Suppose that Assumptions(i)-(iv)hold and T is nonexpansive on C. Then, we have

(a) Sol(f,C)∩Fix(T)⊆P_n∩Q_nfor all n≥0.

(b) lim

n→∞kxⁿ⁺¹−xⁿk= lim

n→∞kxⁿ−zⁿk=lim

n→∞kxⁿ−yⁿk=lim

n→∞kxⁿ−tⁿk=0.

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(c) lim

n→∞kT(tⁿ)−tⁿk=0.

Proof. Since Lemma 3.1 andzⁿ=αnxⁿ+ (1−αn)T(tⁿ), for eachx^∗∈Sol(f,C)∩Fix(T) we have

kzⁿ−x^∗k²=kα_nxⁿ+ (1−αn)T(tⁿ)−x^∗k²

=kα_n(xⁿ−x^∗) + (1−α_n){T(tⁿ)−x^∗}k²

≤α_nkxⁿ−x^∗k²+ (1−α_n)kT(tⁿ)−T(x^∗)k²

≤α_nkxⁿ−x^∗k²+ (1−α_n)ktⁿ−x^∗k²≤ kxⁿ−x^∗k². (3.12)

Hencekzⁿ−x^∗k ≤ kxⁿ−x^∗kfor everyn≥0 andx^∗∈P_n. So, we have Sol(f,C)∩Fix(T)⊆P_n ∀n≥0.

Next, we show by mathematical induction that

Sol(f,C)∩Fix(T)⊆Qn ∀n≥0.

Forn=0 we haveQ₀=C, hence we have Sol(f,C)∩Fix(T)⊆Q₀. Now we suppose that Sol(f,C)∩Fix(T)⊆Q_kfor somek≥0. Fromx^k+1=Pr_P_k∩Q_k(x⁰), it follows that

hx^k+1−x,x⁰−x^k+1i ≥0 ∀x∈P_k∩Q_k. Using this and Sol(f,C)∩Fix(T)⊆Q_k, we have

hx^k+1−x,x⁰−x^k+1i ≥0 ∀x∈Sol(f,C)∩Fix(T) and hence Sol(f,C)∩Fix(T)⊆Q_k+1. This proves(a).

It follows from(a)andxⁿ⁺¹=Pr_P_n∩Q_n(x⁰)that

(3.13) kxⁿ⁺¹−x⁰k ≤ kPr_Sol(_f_,C)∩Fix(T)(x⁰)−x⁰k ∀n≥0.

Hence, we get that{xⁿ}is bounded. Otherwise, for eachx∈Q_n, we have hxⁿ−x,x⁰−xⁿi ≥0,

and hencexⁿ=Pr_Q_n(x⁰). Using this andxⁿ⁺¹∈P_n∩Q_n⊆Q_n, we have kxⁿ−x⁰k ≤ kxⁿ⁺¹−x⁰k ∀n≥0.

Therefore, there exists

(3.14) A= lim

n→∞kxⁿ−x⁰k.

Sincexⁿ=Pr_Q_n(x⁰)andxⁿ⁺¹∈Q_n, using

kPr_Q_n(x)−xk²≤ kx−yk²− kPr_Q_n(x)−yk² ∀x∈H,y∈Q_n, we have

kxⁿ⁺¹−xⁿk²≤ kxⁿ⁺¹−x⁰k²− kxⁿ−x⁰k² ∀n≥0.

Combinating this and (3.14), we get

n→∞limkxⁿ⁺¹−xⁿk=0.

It proves the first apart of(b).

Sincexⁿ⁺¹=Pr_P_n∩Q_n(x⁰), we havexⁿ⁺¹∈P_n,kzⁿ−xⁿ⁺¹k ≤ kxⁿ−xⁿ⁺¹kand hence kxⁿ−zⁿk ≤ kxⁿ−xⁿ⁺¹k+kxⁿ⁺¹−zⁿk ≤2kxⁿ−xⁿ⁺¹k ∀n≥0.

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From lim

n→∞kxⁿ⁺¹−xⁿk=0, we have

n→∞limkxⁿ−zⁿk=0.

This proved the second apart of(b).

From (3.12) and Lemma 3.1, it implies that

kzⁿ−x^∗k²≤α_nkxⁿ−x^∗k²+ (1−α_n)ktⁿ−x^∗k²

≤α_nkxⁿ−x^∗k²+ (1−α_n){kxⁿ−x^∗k²−(1−2λ_nc1)kxⁿ−yⁿk²

−(1−2λ_nc₂)ktⁿ−yⁿk²}

≤kxⁿ−x^∗k²−(1−αn)(1−2λ_nc₁)kxⁿ−yⁿk². Therefore, we have

kxⁿ−yⁿk²≤ 1

(1−α_n)(1−2λ_nc1){kxⁿ−x^∗k²− kzⁿ−x^∗k²}

= 1

(1−α_n)(1−2λ_nc1)(kxⁿ−x^∗k − kzⁿ−x^∗k)(kxⁿ−x^∗k+kzⁿ−x^∗k)

≤ 1

(1−α_n)(1−2λ_nc₁)kxⁿ−zⁿk(kxⁿ−x^∗k+kzⁿ−x^∗k) Since lim

n→∞kxⁿ−zⁿk=0 and the sequences{xⁿ},{zⁿ}are bounded, we get

n→∞limkxⁿ−yⁿk=0.

This proves the third apart of(b).

By similar way, we also obtain that lim

n→∞ktⁿ−yⁿk=0. Then we have

n→∞limkxⁿ−tⁿk ≤ lim

n→∞ kxⁿ−yⁿk+kyⁿ−tⁿk

=0, and hence lim

n→∞kxⁿ−tⁿk=0. This proves the last part of(b).

Using(b)andzⁿ=αnxⁿ+ (1−α_n)T(tⁿ), we have

(1−c)kT(tⁿ)−tⁿk ≤(1−α_n)kT(tⁿ)−tⁿk

=kα_n(tⁿ−xⁿ) + (zⁿ−tⁿ)k

≤α_nktⁿ−xⁿk+kzⁿ−tⁿk,

≤(1+αn)ktⁿ−xⁿk+kzⁿ−tⁿk, and hence lim

n→∞ktⁿ−T(tⁿ)k=0.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Sup- pose that Assumptions (i)-(iv) hold and T is nonexpansive on C. Then, the sequences {xⁿ},{yⁿ},{zⁿ}and{tⁿ}generated by Algorithm 2.1 converge strongly to the same point x^∗, where

x^∗=Pr_Sol(_f_,C)∩Fix(T)(x⁰).

Proof. Since {xⁿ}is bounded, there exists a subsequence {xⁿ^j} of {xⁿ} such that {xⁿ^j} converges weakly to some ¯xas j→∞. Then, it follows from(b)of Lemma 3.2 that{tⁿ^j} also converges weakly to some ¯xas j→∞. We can obtain that ¯x∈Sol(f,C)∩Fix(T). First,

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we show ¯x∈Fix(T). Assume that ¯x∈/Fix(T). Since Opial’s condition in [6], i.e., for any sequence{x_n}withx_n*x¯the inequality

lim inf

n→∞ kx_n−xk¯ <lim inf

n→∞ kx_n−yk holds for everyy∈Hwithy6=x, we have¯

lim inf

j→∞ ktⁿ^j−xk¯ <lim inf

j→∞ ktⁿ^j−T(x)k¯

≤lim inf

j→∞ (ktⁿ^j−T(tⁿ^j)k+kT(tⁿ^j)−T(x)k)¯

=lim inf

j→∞ kT(tⁿ^j)−T(¯x)k

≤lim inf

j→∞ ktⁿ^j−xk.¯ This is contradiction. Thus, ¯x=T(x).¯

From(b)of Lemma 3.2 andxⁿ^j*x¯as j→∞, it follows yⁿ^j *x,t¯ ⁿ^j *x¯ as j→∞.

Then, using (3.10),{λ_n} ⊂[a,b]⊂(0,1)and assumptions of f, we have λ_n_j{f(xⁿ^j,y)−f(xⁿ^j,yⁿ^j)} ≥ hyⁿ^j−xⁿ^j,yⁿ^j−yi ∀y∈C.

As j→∞, we get f(¯x,y)≥0 for ally∈C. It means that ¯x∈Sol(f,C). So, we have

¯

x∈Sol(f,C)∩Fix(T).

Sincex^∗=Pr_Sol(_f_,C)∩Fix(T₎(x⁰), ¯x∈Sol(f,C)∩Fix(T)and (3.13), we have kx^∗−x⁰k ≤ kx¯−x⁰k ≤lim inf

j→∞ kxⁿ^j−x⁰k ≤lim sup

j→∞

kxⁿ^j−x⁰k ≤ kx^∗−x⁰k.

(3.15)

So, we get

limj→∞kxⁿ^j−x⁰k=kx¯−x⁰k.

Sincexⁿ^j−x⁰converges weakly to ¯x−x⁰as j→∞, we havexⁿ^j−x⁰converges strongly to

¯

x−x⁰as j→∞. Byxⁿ=Pr_Q_n(x⁰)andx^∗∈Sol(f,C)∩Fix(T)⊂P_n∩Q_n⊂Q_n, we have hx^∗−xⁿ^j,x⁰−x^∗i ≤ hx^∗−xⁿ^j,x⁰−x^∗i+hx^∗−xⁿ^j,xⁿ^j−x⁰i=−kx^∗−xⁿ^jk². As j→∞, we have

hx^∗−x,¯ x⁰−x^∗i ≤ −kx^∗−xk¯ ².

Combinating this, ¯x∈Sol(f,C)∩Fix(T),hx^∗−x,x¯ ⁰−x^∗i ≥0 andx^∗=Pr_Sol(_f_,C)∩Fix(T₎(x⁰), we obtain ¯x=x^∗. This implies that lim

n→∞kxⁿ−x^∗k=0. From(b)of Lemma 3.2, it follows

n→∞limkyⁿ−x^∗k=0 and lim

n→∞ktⁿ−x^∗k=0.

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4. Applications

In this section, we discuss about two applications of Theorem 3.1 to find a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problems for a monotone, Lipschitz continuous mapping.

LetCis a nonempty closed convex subset of a real Hilbert spaceH, for each pairx,y∈C, f(x,y):=hF(x),y−xi,

whereF:C→H.

In Algorithm 2.1, the subproblems needed to solve at Step 1 are of the form (yⁿ=argmin{¹₂ky−xⁿk²+λ_nhF(xⁿ),y−xⁿi: y∈C},

tⁿ=argmin{¹₂kt−xⁿk²+λnhF(yⁿ),t−yⁿi: t∈C}.

Hence, we have

(yⁿ=argmin{¹₂ky− xⁿ−λnF(xⁿ)

k²: y∈C}=Pr_C xⁿ−λnF(xⁿ) , tⁿ=argmin{¹₂kt− xⁿ−λ_nF(yⁿ)

k²: t∈C}=Pr_C xⁿ−λ_nF(yⁿ) . Thus, in this case Algorithm 2.1 and its convergence become the following results:

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F:C→H be a monotone, L-Lipschitz continuous mapping and T:C→C be a nonexpansive mapping such thatFix(T)∩Sol(F,C)6=/0. Let{xⁿ},{yⁿ}and{zⁿ}be the sequences generated by









 x⁰∈C,

yⁿ=Pr_C xⁿ−λnF(xⁿ) , tⁿ=Pr_C xⁿ−λnF(yⁿ)

, zⁿ=αnxⁿ+ (1−αn)T(tⁿ),

P_n={z∈C: kzⁿ−zk ≤ kxⁿ−zk}, Qn={z∈C: hxⁿ−z,x⁰−xⁿi ≥0}, xⁿ⁺¹=Pr_P_n∩Q_n(x⁰),

for every n≥0, where{λ_n} ⊂[a,b]for some a,b∈(0,_L¹)and{α_n} ⊂[0,c]for some c∈ [0,1). Then the sequences{xⁿ},{yⁿ}and{zⁿ}converge strongly toPrSols(F,C)∩Fix(T)(x⁰).

Using Theorem 4.1, we prove the the following theorem proposed by Nakajo and Taka- hashi.

Theorem 4.2. [9]Let C be a nonempty closed convex subset of a real Hilbert space H. Let T :C→C be a nonexpansive mapping such thatFix(T)6=/0. Let{xⁿ} and {yⁿ} be the sequences generated by









 x⁰∈C,

yⁿ=α_nxⁿ+ (1−α_n)T{Pr_C(xⁿ)}, Pn={z∈C: kzⁿ−zk ≤ kxⁿ−zk}, Q_n={z∈C: hxⁿ−z,x−xⁿi ≥0}, xⁿ⁺¹=Pr_P_n_∩Q_n(x⁰),

(10)

for every n≥0, where{α_n} ⊂[0,c]for some c∈[0,1). Then, the sequences{xⁿ}and{yⁿ} converge strongly toPr_Fix(T)(x⁰).

Proof. For f =0, by Theorem 4.1, we have the desired results.

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