An implicit iterative process for solution system of equilibrium problems and fixed point problems of an amenable semigroup and infinite family of non-expansive mappings

J. Semigroup Theory Appl. 2014, 2014:5 ISSN: 2051-2937

AN IMPLICIT ITERATIVE PROCESS FOR SOLUTION SYSTEM OF

EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS OF AN AMENABLE SEMIGROUP AND INFINITE FAMILY OF NON-EXPANSIVE MAPPINGS

HOSSEIN PIRI

Department of Mathematics, Faculty of Basic Science, University of Bonab, Bonab 5551-761167, Iran

Copyright c2014 Hossein Piri. 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.

Abstract. In this paper, usingδ- strongly monotone and λ- strictly pseudo-contractive (in the terminology of Browder-Petryshyn type) mapping F on a real Hilbert space H, we introduce an implicit iterative scheme to find a common element of the set of solutions of a system of equilibrium problems and the set of fixed points of amenable semigroup of non-expansive mappings and infinite family of non-expansive mappings on H, with respect to a sequence of left regular means defined on an appropriate space of bounded real valued functions of semigroup. Then, we prove the convergence of sequence generated by the suggested algorithm to a unique solution of the variational inequality.

Keywords:fixed point; implicit method; non-expansive mapping; amenable semigroup; equilibrium problem.

2010 AMS Subject Classification:47H05, 47H09, 47H20.

1. Introduction

LetH be a real Hilbert space and letCbe a nonempty closed convex subset ofH. Letφ be

a bi-function ofC×Cinto_R, where_Ris the set of real numbers. The equilibrium problem for

Received July 5, 2014

φ:C×C→Ris to determine its equilibrium points, i.e the set

EP(φ) ={x∈C:φ(x,y)≥0,∀y∈C}.

(1)

Let J ={φi}i∈I be a family of bi-functions fromC×C into R. The system of equilibrium

problems forJ ={φi}i∈I is to determine common equilibrium points forJ ={φi}i∈I, i.e the

set

EP(J) ={x∈C:φi(x,y)≥0,∀y∈C, ∀i∈I}.

(2)

Numerous problems in physics, optimization, and economics reduce into finding some element of EP(φ). Some methods have been proposed to solve the equilibrium problem; see, for

in-stance, [3, 9, 10, 21]. The formulation (2), extend this formalism to systems of such problems, covering in particular various forms of feasibility problems [2, 8].

A mappingT ofCinto itself is called non-expansive ifkT x−Tyk≤kx−yk, for allx,y∈C. By Fix(T), we denote the set of fixed point ofT i.e., Fix(T) ={x∈H :T x=x}. It is well known thatFix(T)is closed convex. Recall that a self-mapping f:C→Cis a contraction onC if there is a constantα ∈(0,1)such that

k f x− f yk≤α kx−yk, ∀x,y∈C.

AssumeA: H→H is strongly positive; that is, there is a constantγ >0 with the property hAx,xi ≥γ kxk2, ∀x∈H.

Given anyr>0 the operatorJ_rφ: H→Cdefined by J_rφ(x) ={z∈C:φ(z,y) +1

rhy−z,z−xi ≥0, ∀y∈C},

is called the resolvent ofF, see [9]. It is shown in [9] that, under suitable hypotheses onF (to be stated precisely in Section 2), Jrφ: H →C is single- valued and firmly non-expansive and

satisfies

Fix(Jφ

r) =EP(φ), ∀r>0.

method for finding a common element inEP(φ)∩Fix(T)which solves some certain variational

inequality.

Theorem 1.1. Let H be a real Hilbert space and φ be a bi-functions from H×H into _R

satisfying

(A₁) φ(x,x) =0for all x∈H,

(A2) φ is monotone, i.e;φ(x,y) +φ(y,x)≤0for all x,y∈H,

(A₃) for all x,y,z∈H, lim sup_t→0φ(tz+ (1−t)x,y)≤φ(x,y),

(A4) for all x∈H, y→φ(x,y)is convex and lower semi-continuous.

For r>0, set Jrφ: H→H to be the resolvent ofφ, i.e. Jrφ(x)is the unique z∈H for which

φ(z,y) +1

rhy−z,z−xi ≥0, ∀y∈H.

Let T be a non-expansive mapping on H such that EP(φ)∩Fix(T)6= /0. Let f be a contraction

of H into itself with coefficientα ∈(0,1)and let A be strongly positive bounded linear mapping

on H with coefficientγ >0and0<γ < γ

α. Let{xn}be the sequence generated by

_x

n=αnγf(xn) + (I−αnA)Tyn, ∀n∈N,

φ(yn,y) +_r1_nhy−yn,yn−xni ≥0, ∀y∈H,

where y_n=Jrφn(xn),{rn} ⊂(0,∞)and{αn} ⊂[0,1]satisfyinglimn→∞αn=0andlim infn→∞rn>

0. The sequences{x_n}and{y_n}converge strongly to a unique point x∗∈EP(φ)∩Fix(T)which

solves the variational inequality:

h(γf−A)x∗,x−x∗i ≤0, ∀x∈EP(φ)∩Fix(T).

Let {T_i}∞

i=1 be a sequence of non-expansive mappings ofC into itself and let {λi}∞i=1 be a

itself as follows:

U_n,n+1=I,

U_n,n=λnTnUn,n+1+ (1−λn)I,

U_n,n−1=λn−1Tn−1Un,n+ (1−λn−1)I,

.. . (3)

U_n,k=λkTkUn,k+1+ (1−λk)I,

U_n,k−1=λk−1Tk−1Un,k+ (1−λk−1)I,

.. .

U_n,2=λ2T2Un,3+ (1−λ2)I,

W_n=U_n,1=λ1T1Un,2+ (1−λ1)I.

Such a mappingWn is called theW−mapping generated by T1,T2,· · ·,Tn and λ1,λ2,· · ·,λn.

Then Colao et al. [7] proved the following strong convergence theorem. Theorem 1.2. Let H be a real Hilbert space , {T_i}∞

i=1 an infinite family of non-expansive

mapping of H into H, for k∈ {1,2,· · ·,M} φ_k a bi-function from H×H into_R, A a strongly

positive bounded linear mapping on H with coefficient γ >0and f an α−contraction on H.

Moreover, let{r_k,n}M

k=1,{αn}and{λn}be real sequences such that rk,n>0,0<λn≤b<1,γ

be a real number such that0<γ < _αγ. Assume that,

(B1) for every k∈ {1,2,· · ·,M}, the bifunctionφk satisfies(A1)−(A4),

(B₂) F = (TM

k=1EP(φk))∩(Ti∞=1Fix(Ti))6= /0, (B₃) the sequence{αn}satisfieslimn→∞αn=0,

(B₄) the sequences{r_k,n}M_k=1satisfylimn→∞rk,n=rˆk>0for every k∈ {1,2,· · ·,M}.

Let Wnbe the mapping defined by (3) and the sequence{xn}generated by

x_n=αnγf(xn) + (I−αnA)WnJrφ1,1nJ

φ2

r2,n· · ·J

φM

Then, the sequence{x_n}converges strongly to x∗∈F, where x∗ is the unique solution of the

variational inequality

h(γf−A)x∗,x−x∗i ≤0, ∀x∈F.

A mappingF with domainD(F)and rangeR(F)inH is calledδ-strongly monotone if there

exists a positive real numberδ >0 such that

hFx−Fy,x−yi ≥δ kx−yk2, ∀x,y∈D(F).

(4)

F is called λ-strictly pseudo-contractive in the terminology of Browder and Petryshyn [4] if

there exists a real numberλ ∈[0,1)such that

kFx−Fyk2≤kx−yk2+λ k(I−F)x−(I−F)yk2, ∀x,y∈D(F).

(5)

It is well-known that (5) is equivalent to

hFx−Fy,x−yi ≤kx−yk2−1−λ

2 k(I−F)x−(I−F)yk

2_.

(6)

In this paper, motivated and inspired by Lau et al. [11] Colao et al. [7], Piri [14, 15, 16, 17], Piri and Badali [18] and Marino and Xu [13], we introduce an implicit iterative scheme to fined a common element of the set of solutions of a system of equilibrium problems and the set of fixed points of an amenable semigroup of non-expansive mappings and infinite family of non-expansive mappings on a real Hilbert space. LetF be a mapping on real Hilbert spaceH which is bothδ- strongly monotone and λ- strictly pseudo-contractive of Browder-Petryshyn

type such that δ > 1+₂λ. Assume S be a semigroup and ϕ ={Tt :t ∈S} be a non-expansive

semigroup on H such that Fix(ϕ) = T

t∈S

Fix(T_t)6= /0. Let X be a subspace of B(S) such that 1∈X and the function t → hT_t(x),yi is an element of X for each x,y∈H,. Let {µn} be a

sequence of means onX. We define a sequence{x_n}by xn=αnγf(xn) + (I−αnF)TµnWnJ

φM

rM,n· · ·J

φ2

r2,nJ

φ1

r1,nxn, ∀n∈N,

whereγ ∈ 0,

1−q2−2δ

1−λ

α

!

. We prove that under assumption on parameters like that in Colao et al. [7], the sequence{x_n}strongly converges tox∗∈F =T∞

wherex∗solves the variational inequality

h(γf −F)x∗,x−x∗i ≤0, ∀x∈F.

Our results improve the corresponding results announced by many others and a consequence for commuting pairs of non-expansive mappings is also presented.

2. Preliminaries

LetSbe a semigroup and letB(S)be the space of all bounded real valued functions defined onSwith supremum norm. Fors∈Sand f ∈B(S), we define elementsl_sf andr_sf inB(S)by

(l_sf)(t) = f(st), (r_sf)(t) = f(ts), ∀t ∈S.

LetX be a subspace ofB(S)containing 1 and letX∗be its dual. An element µ inX∗is said to

be a mean onX ifkµ k=µ(1) =1. We often write µt(f(t))instead ofµ(f)forµ ∈X∗and

f ∈X. LetX be left invariant (resp. right invariant), i.e.,l_s(X)⊂X (resp. r_s(X)⊂X) for each s∈S. A mean µ onX is said to be left invariant (resp. right invariant) ifµ(lsf) =µ(f)(resp.

µ(rsf) =µ(f)) for eachs∈Sand f ∈X. X is said to be left (resp. right) amenable ifX has a

left (resp. right) invariant mean. X is amenable ifX is both left and right amenable. As is well known,B(S)is amenable whenSis a commutative semigroup, see [12]. A net{µα}of means onX is said to be strongly left regular if

lim α k

l_s∗µα−µα k=0, for eachs∈S, wherel_s∗is the adjoint operator ofls.

LetC be a nonempty closed and convex subset of a reflexive Banach space E. A family

ϕ ={Tt :t ∈S}of mapping fromCinto itself is said to be a non-expansive semigroup onCif

T_t is non-expansive andT_ts=T_tT_s for each t,s∈S. By Fix(ϕ)we denote the set of common

fixed points ofϕ, i.e.

Fix(ϕ) =

\

t∈S

{x∈C:Tt(x) =x}.

is bounded for some x∈C, let X be a subspace of B(S) such that 1 ∈X and the mapping t→ hT_tx,y∗iis an element of X for each x∈C and y∗∈E∗, andµ is a mean on X . If we write

T_µx instead ofR

Ttxdµ(t), then the followings hold.

(i) T_µ is nonexpansive mapping from C into C. (ii) T_µx=x for each x∈Fix(ϕ).

(iii) T_µx∈co{T_tx:t∈S}for each x∈C.

LetCbe a nonempty subset of a Hilbert spaceH andT:C→H a mapping. ThenT is said to be demiclosed atv∈H if, for any sequence{x_n}inC, the following implication holds:

xn*u∈C, T xn→v imply Tu=v,

where→(resp. *) denotes strong (resp. weak) convergence.

Lemma 2.2. [1]Let C be a nonempty closed convex subset of a Hilbert space H and suppose that T:C→H is non-expansive. Then, the mapping I−T is demiclosed at zero.

LetCbe a nonempty subset of a normed spaceE and letx∈E. An elementy₀∈Cis said to be the best approximation toxif

kx−y₀k=d(x,C),

whered(x,C) =infy∈Ckx−yk. The numberd(x,C)is called the distance fromx toCor the

error in approximatingxbyC. The (possibly empty) set of all best approximation fromxtoC is denoted by

P_C(x) ={y∈C:kx−yk=d(x,C)}.

This defines a mapping P_C fromX into 2C and is called metric (nearest point) projection onto C. It is well-known thatP_Cis a non-expansive mapping ofHontoC.

Lemma 2.3.[23]Let C be a nonempty convex subset of a Hilbert space H and P_Cbe the metric projection mapping from H onto C. Let x∈H and y∈C. Then, the following are equivalent.

(i) y=P_C(x),

Letφ:C×C→_Rbe a bi-function. Given anyr>0, the operatorJrφ: H→Cdefined by

Jφ

rx={z∈C:φ(z,y) +

1

rhy−z,z−xi ≥1,∀y∈C}

is called the resolvent ofφ, see [9]. The equilibrium problem forφ is to determine its

equilib-rium points, i.e., the set

EP(φ) ={x∈C:φ(x,y)≥0,∀y∈C}.

Let J ={φi}i∈I be a family of bi-functions fromC×C into R. The system of equilibrium

problems forJ is to determine common equilibrium points forJ ={φi}i∈I. i.e, the set

EP(J) ={x∈C:φi(x,y)≥0,∀y∈C, ∀i∈I}.

Lemma 2.4. [9]Let C be a nonempty closed convex subset of H andφ:C×C→_Rsatisfy

(A1) φ(x,x) =0for all x∈C,

(A₂) φ is monotone, i.e;φ(x,y) +φ(y,x)≤0for all x,y∈C,

(A3) for all x,y,z∈C, lim supt→0φ(tz+ (1−t)x,y)≤φ(x,y), (A₄) for all x∈C, y→φ(x,y)is convex and lower semi-continuous.

Given r>0, define the operator Jrφ: H→C, the resolvent ofφ, by

Jφ

r (x) ={z∈C:φ(z,y) +

1

rhy−z,z−xi ≥0, ∀y∈C}. Then,

(1) Jrφ is single valued,

(2) J_rφ is firmly non-expansive, i.e,kJ_rφx−J_rφyk2≤ hJ_rφx−J_rφy,x−yifor all x,y∈H,

(3) Fix(J_rφ) =EP(φ),

(4) EP(φ)is closed and convex.

Lemma 2.5. [7]Let C be a nonempty closed convex subset of H and{r_n} ⊂(0,1)be a sequence converging to r>0. For a bifunctionφ: C×C→_R, satisfying conditions(A1)−(A4), define

Jrφn and J

φ

r for n∈Nas in Lemma . Then for every x∈H, we havelimn→∞kJ φ

rnx−J

φ

Let{T_i}∞

i=1be a sequence of non-expansive mappings ofCinto itself, whereCis a nonempty

closed convex subset of a real Hilbert spaceH. Given a sequence{λi}∞_i=1in[0,1], we define a

sequence{Wn}∞_n=1of self mappings onCby (3). Then we have the following results.

Lemma 2.6. [20]Let C be a nonempty closed convex subset of a Hilbert space H,{T_i}∞

i=1be a

sequence of non-expansive mappings of C into itself such thatT∞

i=1Fix(Ti)6= /0,{λi}be a real

sequence such that0<λi≤b<1,∀i≥1. Then

(1) W_nis non-expansive and Fix(W_n) =Tn

i=1Fix(Ti)for each n≥1,

(2) for each x∈C and for each positive integer j, the limitlimn→∞Un,jx exists.

(3) The mapping W:C→C defined by W x:= lim

n→∞

W_nx= lim

n→∞

U_n,1, ∀x∈C,

is a non-expansive mapping satisfying Fix(W) =T∞

i=1Fix(Ti)and it is called the W−mapping

generated by T₁,T₂,· · · andλ1,λ2,· · ·.

Lemma 2.7. [24]Let C be a nonempty closed convex subset of a Hilbert space H,{T_i}∞

i=1be a

sequence of non-expansive mappings of C into itself such thatT∞

i=1Fix(Ti)6= /0,{λi}be a real

sequence such that0<λi≤b<1,∀i≥1. If D is any bounded subset of C, then

lim

n→∞sup_x∈D

kW x−W_nxk=0.

LetK be a nonempty subset of a Banach spaceX and{xn}be a sequence inK. Consider the

functionalr_a(.,{x_n}): X→_Rdefined by r_a(.,{x_n}) =lim sup

n→∞

kx_n−xk, ∀x∈X.

The infimum ofra(.,{xn})overKis to be asymptotic radius of{xn}with respect toKand it is

denoted byr_a(K,{x_n}). A pointx∈Kis said to be asymptotic center of the sequence{x_n}with respect toKif

r_a(x,{x_n}) =inf{r_a(y,{x_n}):y∈K}.

LetK be a nonempty subset of a Banach spaceX and{x_n}be a sequence inK. Consider the functionalr_a(.,{x_n}): X→_Rdefined by

r_a(.,{x_n}) =lim sup

n→∞

kx_n−xk, ∀x∈X.

The infimum ofr_a(.,{x_n})overKis to be asymptotic radius of{x_n}with respect toKand it is denoted byr_a(K,{x_n}). A pointx∈Kis said to be asymptotic center of the sequence{x_n}with respect toKif

ra(x,{xn}) =inf{ra(y,{xn}):y∈K}.

The set of all asymptotic center of {xn}with respect toK is denoted byCa(K,{xn}). This set

may be empty, a singleton, or infinitely many points.

Lemma 2.8. [1]Let X be uniformly convex Banach space satisfying the Opial’s condition and K a nonempty closed convex subset of X . If a sequence{x_n} ⊂K converges weakly to a point

x0, then x0is the asymptotic center of{xn}with respect to K.

The following Lemma will be frequently used throughout the paper. For the sake of com-pleteness, we include its proof.

Lemma 2.9. [6]Let H be a real Hilbert space.

(i) If F: H →H is a mapping which is bothδ-strongly monotone and λ-strictly

pseudo-contractive of Browder-Petryshyn type such thatδ,λ <1andδ > 1+₂λ. Then, I−F is

contractive with constant

q 2−2δ

1−λ .

(ii) If F: H →H is a mapping which is bothδ-strongly monotone and λ-strictly

pseudo-contractive of Browder-Petryshyn type such that δ,λ <1andδ > 1+₂λ. Then, for any

fixed numberτ∈(0,1), I−τF is contractive with constant1−τ

1−

q 2−2δ

1−λ

.

NotationThroughout the rest of this paper, the open ball of radiusrcentered at 0 is denoted by B_r. For subset A of H, by coA, we denote the closed convex hull of A. For ε >0 and

a mapping T: D→H, we let F_ε(T;D) be the set of ε− approximate fixed points of T, i.e.

F_ε(T;D) ={x∈D:kx−T xk≤ε}. Weak convergence is denoted by*and strong convergence

pseudo-contractive of Browder-Petryshyn type such thatδ > 1+₂λ, and f is a contraction with

coefficient 0<α<1. We will also always useγ to mean a number in 0,

1−q2−2δ

1−λ

α

!

.

3. Main results

Consider a mappingΓnonH defined by

Γn(x) =αnγf(x) + (I−αnF)TµnWnJ

φM

rM,n· · ·J

φ2

r2,nJ

φ1

r1,nx, x∈H,n≥1,

Using Lemma , Lemma and Lemma , we have

kΓn(x)−Γn(y)k

=kαnγ(f(x)−f(y)) + (I−αnF)TµnWnJ

φM

rM,n· · ·J

φ2

r2,nJrφ1,1nx

−(I−αnF)TµnWnJ

φM

rM,n· · ·J

φ2

r2,nJ

φ1

r1,nyk

≤αnγ αkx−yk+ 1−αn 1− r

2−2δ

1−λ

!!

kx−yk

= 1−αn 1−

r

2−2δ

1−λ −γ α

!!

kx−yk.

Since 0<1−αn

1−q2−2δ

1−λ −γ α

<1, it follows thatΓnis a contraction. Therefore, by the

Banach contraction principle,Γnhas a unique fixed pointxn∈H such that

x_n=αnγf(xn) + (I−αnF)TµnWnJ

φM

rM,n· · ·J

φ2

r2,nJ

φ1

r1,nxn.

Note thatxnindeed depends on f as well, but we will suppress this dependency ofxnon f for

simplicity of notation throughout the rest of this paper. The following is our main result. Theorem 3.1. Let S be a semigroup andϕ ={Tt :t∈S}a non-expansive semigroup from H

into H such that Fix(ϕ) =T_t∈SFix(Tt)6= /0. Let X be a left invariant subspace of B(S) such

that 1∈X , and the function t → hT_tx,yi is an element of X for each x,y∈H. Let {µn} be

a left regular sequence of means on X . Let J ={φk :k=1,2,· · ·,M} be a finite family of

bi-functions from H×H into_R which satisfy(A₁)−(A₄)and let{T_i}∞

i=1 be an infinite family

F =T∞

i=1Fix(Ti)∩Fix(ϕ)∩EP(J)6= /0. Let{αn}be a sequence in(0,1),{λn}a sequence

in(0,b] for some b∈(0,1), {r_k,n}sequences in (0,∞). Suppose the following conditions are satisfied:

(B₁) limn→∞αn=0,

(B4) limn→∞rk,n=rˆk, for every k∈ {1,2,· · ·,M}.

Let W_nbe the mapping defined by (3) and for every k∈ {1,2,· · ·,M}and n∈_N, let Jφk

rk,nbe the

resolvent generated byφkand rk,nin Lemma . If{xn}is the sequence generated by

(7) x_n=αnγf(xn) + (I−αnF)TµnWnJ

φM

rM,n· · ·J

φ2

r2,nJ

φ1

r1,nxn, ∀n∈N.

Then{x_n}converges strongly to x∗∈F, where x∗ is the unique solution of the variational

inequality:

h(γf −F)x∗,x−x∗i ≤0, ∀x∈F.

Proof. By takingJ_nk=Jφk

rk,n· · ·J

φ2

r2,nJ

φ1

r1,n for k∈ {1,2,· · ·,M} and J

0

n =I for all n∈N, we

shall equivalently write scheme (7) as follows:

x_n=αnγf(xn) + (I−αnF)TµnWnJ

M

n xn, ∀n∈N.

We shall divide the proof into several steps. Step 1. The sequence{x_n}is bounded.

Proof of Step 1. Let p∈F. Using Lemma , Lemma and Lemma , we have

kx_n−pk2

=hαnγf(xn) + (I−αnF)TµnWnJ

M

n xn−p,xn−pi

=αnγhf(xn)−f(p),xn−pi+αnhγf(p)−F(p),xn−pi

+h(I−αnF)TµnWnJ

M

n xn−(I−αnF)p,xn−pi

≤αnγ αkxn−pk2+αnhγf(p)−F(p),xn−pi

+ 1−αn 1−

r

2−2δ

1−λ

! !

Thus,

kx_n−pk2≤ 1

1−q2−2δ

1−λ −γ α

hγf(p)−F(p),xn−pi.

(8)

Hence

kx_n−pk≤ 1

1−q2−2δ

1−λ −γ α

kγf(p)−F(p)k.

(9)

Therefore{x_n}is bounded.

Step 2. For everyk∈ {1,2,· · ·,M}, we have limn→∞kxn−Jrφkk,nxnk=0.

Proof of Step 2. Let p∈F since, for eachk∈ {1,2,· · ·,M}, Jφk

rk,nis firmly non-expansive, we

have

kJφk

rk,nxn−pk

2_=kJφk

rk,nxn−J

φk

rk,npk

2

≤ hJφk

rk,nxn−J

φk

rk,np,xn−pi

=1

2

h

kJφk

rk,nxn−pk

2_+kx

n−pk2− kxn−Jrφkk,nxnk

2i_.

It follows that

kxn−Jrφkk,nxnk

2_≤kx

n−pk2− kJrφkk,nxn−pk

2_.

(10)

Moreover setl_n=2hγf(xn)−FTµnWnJ

M

n xn,xn−piand note that, by using the inequality kx+yk62≤kxk2+2hy,x+yi,

we obtain

kxn−pk2=kαnγf(xn) + (I−αnF)TµnWnJ

M

n xn−pk2

≤kTµnWnJ

M

n xn−pk2+αnln

=kT_µnW_nJφM

rM,n· · ·J

φ2

r2,nJ

φ1

r1,nxn−pk

2₊

αnln

≤kJφ1

r1,nxn−pk

2₊

Applying the last to (10), we obtain

kx_n−Jφ1

r1,nxnk≤αnln

and since{ln}is bounded and limn→∞αn=0, we get

lim

n→∞

kx_n−Jφ1

r1,nxnk=0.

Now we assume thatk∈ {1,2,· · ·,M}and for everyk∈ {1,2,· · ·,k}, limn→∞kxn−Jrφkk,nxnk=

0. We shall prove that limn→∞kxn−J φ_k

r_k,nxnk=0. Indeed

kx_n−pk2≤kT_µnW_nJφM

rM,n· · ·J

φ_k

r_k,n· · ·Jrφ22,nJ

φ1

r1,nxn−pk

2₊

αnln

≤kJφk

r_k,n· · ·Jrφ22,nJ

φ1

r1,nxn−pk

2₊

αnln.

(11)

Observe that

kJφk

r_k,n· · ·Jrφ2,2nJrφ1,1nxn−pk

≤kJφk−1

r_k−1,n· · ·Jrφ2,2nJrφ1,1nxn−xnk+kJ

φ_k

r_k,nxn−pk

≤kJφk−2

r_k−2,n· · ·Jrφ22,nJ

φ1

r1,nxn−xnk+kJ

φ_k−1

r_k−1,nxn−xnk+kJ

φ_k

r_k,nxn−pk

.. .

≤ k−1

∑

kJφk

rk,nxn−xnk+kJ

φ_k

r_k,nxn−pk.

Inequality (11) becomes then

kx_n−pk2

≤

" k−1

∑

kJφk

rk,nxn−xnk+2kJ

φ_k

r_k,nxn−pk #

k−1

∑

kJφk

rk,nxn−xnk

+kJφk

r_k,nxn−pk2+αnln.

It follows from (10) and (12) that

kxn−J

φ_k

r_k,nxnk

≤

" k−1

∑

kJφk

rk,nxn−xnk+2kJ

φ_k

r_k,nxn−pk #

k−1

∑

kJφk

rk,nxn−xnk

+αnln.

By our assumption, we have that

lim

n→∞

k−1

∑

kJφk

rk,nxn−xnk=0.

Then, from the last and condition(B₁), we derive lim

n→∞

kx_n−Jφk

r_k,nxnk=0.

Step 3. limn→∞kxn−TµnWnxnk=0.

Proof of Step 3. putM_n=2hγf(xn)−FTµnWnJ

M

n xn,xn−TµnWnxniand note that

kx_n−T_µnW_nx_nk2=kαnγf(xn) + (I−αnF)TµnWnJ

M

n xn−TµnWnxnk

2

≤kT_µnWnJ_nMx_n−T_µnW_nx_nk2

+2αnhγf(xn)−FTµnWnJ

M

n xn,xn−TµnWnxni

≤kJ_nMxn−xnk2+αnMn.

Moreover,

kJ_nMxn−xnk

=kJφM

rM,n· · ·J

φ2

r2,nJrφ1,1nxn−xnk

≤kJφM−1

rM−1,n· · ·J

φ2

r2,nJrφ1,1nxn−xnk+kJrφMM,nxn−xnk

≤kJφM−2

rM−2,n· · ·J

φ2

r2,nJ

φ1

r1,nxn−xnk+kJ

φM−1

rM−1,nxn−xnk+kJ

φM

rM,nxn−xnk

.. .

≤ M

∑

kJφk

Thus, by Step 2, condition(B₁)and boundedness ofM_n, we have lim

n→∞

kxn−TµnWnxnk=0.

Step 4. limn→∞kxn−Ttxnk=0, for allt ∈S.

Proof of Step 4. Let p∈F and setM₀= 1 1−

q

2−2δ

1−λ−γ α

kγf(p)−F(p)kand D={y∈H :k

y−pk≤M₀}, we remark that D is bounded closed convex set, {x_n} ⊂D and it is invariant under{Jφk

rk,n:k=1,2,· · ·,M, n∈N},ϕ andWnfor alln∈N. We will show that

lim sup

n→∞ sup

y∈D

kT_µny−TtTµnyk=0, ∀t∈S

(13)

Letε >0. By [5, Theorem 1.2], there existsδ >0 such that

coF_δ(T_t;D) +B_δ ⊂F_ε(T_t;D), ∀t∈S.

(14)

Also by [5, Corollary 1.1], there exists a natural numberNsuch that

(15) k 1

N+1

N

∑

T_ti_sy−T_t

1 N+1

N

∑

T_ti_sy

!

k≤δ,

for allt,s∈Sandy∈D. Lett∈S. Since{µn}is strongly left regular, there existsN0∈Nsuch

thatkµn−l_t∗iµnk≤ _(M δ

0+kpk) forn≥N0andi=1,2,· · ·,N. Then, we have

sup

y∈D

kT_µny−

Z ₁ N+1

N

∑

T_ti_sydµ_n(s)k

=sup

y∈D

sup kzk=1

| hT_µny,zi − h

Z ₁ N+1

N

∑

T_ti_sydµ_n(s),zi |

=sup

y∈D

sup kzk=1

| 1

N+1

N

∑

(µn)shTsy,zi −

1 N+1

N

∑

(µn)shTti_sy,zi |

≤ 1

N+1

N

∑

sup

y∈D

sup kzk=1

|(µn)shTsy,zi −(l_t∗iµn)shTsy,zi |

≤ max

i=0,1,2,···,Nkµn−l

∗

tiµnk(M0+kpk)≤δ, ∀n≥N0.

(16)

By Lemma , we have Z ₁

N+1

N

∑

T_ti_sydµ_n(s)∈co

(

1 N+1

N

∑

T_ti(T_sy):s∈S

)

.

It follows from (14), (15), (16) and (17) that

T_µny∈co

(

1 N+1

N

∑

T_ti_sy:s∈S

)

+B_δ

⊂coF_δ(T_t;D) +B_δ ⊂F_ε(T_t;D),

for ally∈Dandn≥N₀. Therefore, lim sup

n→∞ sup

y∈D

kTt(Tµny)−Tµnyk≤ε.

Sinceε >0 is arbitrary, we get (13).

Lett∈Sandε>0. Then, there existsδ >0, which satisfies (14). Take

L₀= "

1+ r

2−2δ

1−λ +γ α

!

M₀+kγf(p)−F(p)k

#

.

From (13) and condition (B₁), there exists N₁ ∈ _N such that T_µny∈ F_δ(T_t;D), ∀y∈D and

αn≤_Lδ₀ for alln≥N1. By Lemma , Lemma and Lemma , we have kγf(xn)−FTµnWnJ

M n xnk

≤γ k f(xn)−f(p)k+kγf(p)−F(p)k+kF(p)−FTµnWnJ

M n xnk

≤γ αkxn−pk+kγf(p)−F(p)k+ 1+ r

2−2δ

1−λ

!

kx_n−pk

= 1+

r

2−2δ

1−λ +γ α

!

kx_n−pk+kγf(p)−F(p)k=L0.

It follows that

αnkγf(xn)−FTµnWnJ

M

n xnk≤αnL0≤δ ∀n≥N1.

Therefore, we have

x_n=T_µnWnJ_nMx_n+αn[γf(xn)−FTµnWnJ

M n xn]

∈F_δ(Tt;D) +Bδ ⊂Fε(Tt;D), for alln≥N₁. This shows that

Sinceε >0 is arbitrary the proof of Step 4 is complete.

Step 5. There exists a uniquex∗∈F such that

lim sup

n→∞

h(γf−F)x∗,xn−x∗i ≤0.

(18)

Proof of Step 5. LetQ=P_F. ThenQ(I−F+γf)is a contraction ofH into itself. In fact, we

see that

kQ(I−F+γf)x−Q(I−F+γf)yk

≤k(I−F+γf)x−(I−F+γf)yk

≤k(I−F)x−(I−F)yk+γk f(x)−f(y)k

≤

r

1−δ

λ +γ α

!

kx−yk,

and henceQ(I−F+γf)is a contraction due to

q 1−δ

λ +γ α

∈(0,1).

Therefore, by Banachs contraction principal,P_F(I−F+γf)has a unique fixed pointx∗. Then

using Lemma ,x∗is the unique solution of the variational inequality:

h(γf −F)x∗,x−x∗i ≤0, ∀x∈F.

(19)

We can choose a subsequence{xnj}of{xn}such that

lim sup

n→∞

h(γf−F)x∗,xn−x∗i= lim j→∞

h(γf−F)x∗,xnj−x

∗_i.

Without loss of generality, we may assume that xnj *z

∗_{. In terms of Lemma and Step 4, we} conclude thatz∗∈Fix(ϕ).

Consider the set of the asymptotic center of{x_nj}with respect toH,

C_a(H,{x_nj}) ={x∈H: lim sup

j→∞

kx_nj−xk= inf

y∈Hlim sup_j→∞ kxnj−yk}.

Forz∈Ca(H,{xnj}), we have

lim sup

j→∞

kxnj−zk≤lim sup

j→∞

By Step 4, we get x_nj → z. It follows from Step 4 and Lemma that z ∈Fix(ϕ). By our

assumption, we haveT_iz∈Fix(ϕ) for all i∈_Nand thenWnz∈Fix(ϕ), henceTµnWnz=Wnz.

Using Lemma and Step 3, we get

lim sup

j→∞

kx_nj−W zk

≤lim sup

j→∞

kxnj−Tµ_{n j}Wnjxnj k+lim sup

j→∞

kT_{µn j}Wnjxnj−Tµ_{n j}Wnjzk

+lim sup

j→∞

kT_µ

n jWnjz−W zk

≤lim sup

j→∞

kx_nj−T_µ

n jWnjxnj k+lim sup

j→∞

kx_nj−zk

+lim sup

j→∞

kW_njz−W zk

≤lim sup

j→∞

kxnj−zk.

This is enough to prove thatW(C_a(H,{x_nj}))⊂C_a(H,{x_nj}). Using Lemma , Lemma and Step 2, we have

lim sup

j→∞

kx_nj−Jφk

rkzk

≤lim sup

j→∞

kx_nj−Jφk

rk,njxnj k+lim sup

j→∞

kJφk

rk,njxnj−J

φk

rk,njzk

+lim sup

j→∞

kJφk

rk,njz−J

φk

ˆ rkzk

≤lim sup

j→∞

kx_nj−zk.

This is enough to prove thatJφk

rk(Ca(H,{xnj}))⊂Ca(H,{xnj}).

Sincexnj*z

∗_{Lemma implies thatC}

a(H,{xnj}) ={z

∗_{}; therefore,z}∗_{∈Fix(W)∩(}TM

k=1Fix(J

φk

rk,n)).

In terms of Lemma and Lemma, we conclude that z∗ ∈ (T∞

i=1Fix(Ti))∩EP(J). Since

z∗∈Fix(ϕ); therefore,z∗∈F. Applying (19), we have

lim sup

n→∞

Step 6. The sequence{x_n}converges strongly tox∗. Proof of Step 6. Sincex∗∈F from (8), we have

lim sup

n→∞

kx_n−x∗k2≤ 1

1−q2−2δ

1−λ −γ α

lim sup

n→∞

h(γf−F)x∗,xn−x∗i.

Then, using Step 5, we havex_n→x∗asn→∞.

Corollary 3.2. Let H be a real Hilbert space H, {T_i}∞

i=1 an infinite family of non-expansive

mapping of H into H, for k ∈ {1,2,· · ·,M} φk a bi-functions from H×H into R, λ a real

number in [0.1), A a strongly positive bounded linear operator on H with coefficient γ such

that γ > 1+₂λ, ζ a real number in(0,

1−q2−2γ

1−λ

α ). Moreover, let{rk,n}

M

k=1, {εn} and λn be real

sequences such that r_k,n>0and0<λn≤b<1. Assume that,

(B₁) for every k∈ {1,2,· · ·,M}, the bifunctionφk satisfies(A1)−(A4).

(B₂) F = (TM

k=1EP(φk))∩(Ti∞=1Fix(Ti))6= /0. (B₃) the sequence{αn}satisfieslimn→∞αn=0,

(B4) the sequences{rk,n}Mk=1satisfylimn→∞rk,n=rˆk>0for every k∈ {1,2,· · ·,M}.

Let W_nbe the mapping defined by (3) and the sequence{x_n}generated by

x_n=αnζf(xn) + (I−αnA)WnJrφMM,n· · ·J

φ2

r2,nJ

φ1

r1,nxn, ∀n∈N,

Then, the sequence{x_n}converges strongly to x∗∈F, where x∗ is the unique solution of the

variational inequality

h(γf−A)x∗,x−x∗i ≤0, ∀x∈F.

Proof. Take ϕ ={I} then, using Lemma we have Tµn =I. Because A is strongly positive

bounded linear operator onH with coefficientγ, we have

Therefore,Aisγ−strongly monotone. On the other hand

k(I−A)x−(I−A)yk2

=h(x−y)−(Ax−Ay),(x−y)−(Ax−Ay)i

=hx−y,x−yi −2hAx−Ay,x−yi+hAx−Ay,Ax−Ayi ≤kx−yk2−2hAx−Ay,x−yi+kAkkx−yk2

SinceA is strongly positive if and only if _kA1_kAis strongly positive. We may assume, with no loss of generality, thatkAk=1. Therefore,

hAx−Ay,x−yi ≤kx−yk2−1

2k(I−A)x−(I−A)yk

2

≤kx−yk2−1−λ

2 k(I−A)x−(I−A)yk

2

This show that A is λ− strictly pseudo-contractive of Browder-Petryshyn type. Now apply

Theorem to conclude the result.

Corollary 3.3. Let S and T be non-expansive mappings on H with ST =T S,J ={φ_k:k=

1,2,· · ·,M}a finite family of bi-functions from H×H into_Rwhich satisfy(A₁)−(A₄),{T_i}∞

i=1

an infinite family of non-expansive mappings of H into H such that T_i(Fix(T)∩Fix(S))⊂

Fix(T)∩Fix(S) for each i∈_N, F =T∞

i=1Fix(Ti)∩Fix(T)∩Fix(S)∩EP(J)6= /0, {αn} a

sequence in (0,1), {λn} a sequence in (0,b] for some b ∈(0,1), {rk,n} sequences in (0,∞).

Suppose the following conditions are satisfied:

(B₁) limn→∞αn=0,

(B4) limn→∞rk,n=rˆk, for every k∈ {1,2,· · ·,M}.

Let W_nbe the mapping defined by (3) and for every k∈ {1,2,· · ·,M}and n∈_N, let Jφk

rk,nbe the

resolvent generated byφkand rk,nin Lemma . If{xn}is the sequence generated by

xn=αnγf(xn) + (I−αnF)

1 n2

n−1

∑

n−1

∑

SiTjWnJrφMM,n· · ·J

φ2

Then{x_n}converges strongly to x∗∈F, where x∗ is the unique solution of the variational

inequality:

h(γf −F)x∗,x−x∗i ≤0, ∀x∈F.

Proof. Let S =_NS

{o} ×_NS

{o} and let T(i,j) =SiTj for all (i,j)∈S. Since Si and Tj are non-expansive for each (i,j)∈S and ST =T S. Therefore, ϕ ={T(i,j):(i,j)∈S} is a

non-expansive semigroup onH. Now for eachn∈_N, define µn(f) = _n12

n−1

∑

i=0 n−1

∑

j=0

f(i,j)for each f ∈B(S). Then, {µn}is strongly left regular sequence of means onB(S); for more details, see

[22]. Next, for eachy∈H andn∈_N, we have

T_µn(y) = 1

n2 n−1

∑

n−1

∑

SiTj(y).

Therefore, it follows from Theorem that the sequence{x_n}converges strongly, asn→∞to a pointz∈F, which solves the variational inequality:

h(γf −F)x∗,x−x∗i ≤0, ∀x∈F.

Conflict of Interests

The author declares that there is no conflict of interests.

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