# The Iterative Method for Generalized Equilibrium Problems and a Finite Family of Lipschitzian Mappings in Hilbert Spaces

Authors: Atid Kangtunyakarn; Sarawut Suwannaut
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this research, we introduced theS-mapping generated by a finite family of contractive mappings, Lipschitzian mappings and finite real numbers using the results of Kangtunyakarn (2013). Then, we prove the strong convergence theorem for fixed point sets of finite family of contraction and Lipschitzian mapping and solution sets of the modified generalized equilibrium problem introduced by Suwannaut and Kangtunyakarn (2014). Finally, numerical examples are provided to illustrate our main theorem.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H. Let F:C×C⟶ℝ be bifunction. The equilibrium problem for F is to determine its equilibrium point, i.e., the set(1)EPF=x∈C:Fx,y≥0,∀y∈C.Equilibrium problems were first introduced by Muu and Oettli [1] in 1992. It contains various problems such as variational inequality problem, fixed point problem, optimization problem and Nash equilibrium problem. Iterative methods for the equilibrium problems are widely studied, see, for example, [2–9].If we takeFx,y=y−x,Bx, where B:C⟶H is a nonlinear mapping, then the equilibrium problem (1) is equivalent to finding an element x∈C such that(2)y−x,Bx≥0,∀y∈C,which is well-known as the variational inequality problem. The solution set of the problem (2) is denoted by VIC,A.Variational inequality problem were first defined and studied by Stampacchia [10] in 1964. The variational inequality theory is an important tool based on studying a wide class of problems such as economics, optimization, operations research and engineering sciences. Several iterative algorithms have been used for solving variational inequality problem and related optimization problems (see [11–15] and the references therein).LetCBH be the family of all nonempty closed bounded subsets of H and ℋ⋅,⋅ be the Hausdorff metric on CBH defined as(3)ℋM,N=maxsupm∈Mdm,N,supn∈NdM,n,∀M,N∈CBH,where dm,N=infn∈Ndm,n,dM,n=infn∈Ndm,n and dm,n=m−n.A multivalued mappingV:H⟶CBH is said to be ℋ-Lipschitz continuous if there exists a constant ω>0 such that(4)ℋVp,Vq≤ωp−q,∀p,q∈C.LetV:H⟶CBH a multi-valued mapping, ϕ:C⟶ℝ be a real-valued function and Ψ:H×C×C⟶ℝ an equilibrium-like function, that is, Ψz,x,y+Ψz,y,x=0 for every z,x,y∈H×C×C satisfying the following properties:H1z,x↦Ψz,x,y is an upper semicontinuous function from H×C⟶ℝ, for all fixed y∈C, that is, for z,x∈H×C, whenever zn⟶z and xn⟶x as n⟶∞,(5)limsupn⟶∞Ψzn,xn,y≤Ψz,x,y;.H2x↦Ψz,x,y is a concave function, for all fixed z,y∈H×C ;H3y↦Ψz,x,y is a convex function, for all fixed z,x∈H×C.In 2009, Ceng et al. [16] introduced the generalized equilibrium problemGEP as follows:(6)GEPFindx∈Candz∈Vxsuch that,Ψz,x,y+ϕy−ϕx≥0,∀y∈C.Furthermore,GEPsΨ,ϕ denotes the solution set of the generalized equilibrium problem.In 2012, Kangtunyakarn [7] investigated the strong convergence theorem using CQ method for two solution sets of the generalized equilibrium problem GEP and fixed point problem of nonlinear mappings.In 2014, by modifying the generalized equilibrium problem (6), Suwannaut and Kangtunyakarn [17] introduced the modified generalized equilibrium problemMGEP as follows:(7)MGEPFindx∈Candz∈VI−ρAx,∀ρ>0,Ψz,x,y+ϕy−ϕx+y−x,Ax≥0,∀y∈C.where A is a self-mapping on C. Also, MGEPsΨ,ϕ,A represents the solution set of MGEP. If A=0, (7) reduces to (6). They also obtain the strong convergence theorem under some mild conditions.Definition 1.
LetW be a self-mapping on C. Then W is called(i)
nonexpansive if(8)Wu−Wv≤u−v,∀u,v∈C;.(ii)
contractive if there existsτ∈0,1 such that(9)Wu−Wv≤τu−v,∀u,v∈C;.(iii)
inverse-strongly monotone if there exists a real numberω>0 such that(10)u−v,Wu−Wv≥ωWu−Wv2,∀u,v∈C.
It is well-known thatI−W is demiclosed if W is a nonexpansive mapping, see [18]. Moreover, FixW is used to represent the set of fixed points of W.Definition 2.
(see [19]). A mapping W:C⟶C is called ν-strictly pseudo-contractive if there exists a constant ν∈0,1 such that(11)Wu−Wv2≤u−v2+νI−Wu−I−Wv2,∀u,v∈C.
Browder and Petryshyn [19] introduced and studied the class of strictly pseudo-contractive mapping as an important generalization of the class of nonexpansive mappings. It is trivial to prove that every nonexpansive mapping is strictly pseudo-contractive.Definition 3.
A mappingW:C⟶C is called L-Lipschitzian if there exists L>0 satisfying the following inequality:(12)Wu−Wv≤Lu−v,∀u,v∈C.
Note that if0<L<1, W becomes a contractive mapping. If L=1, W is said to be a nonexpansive mapping. In fact, all four classes of mappings mentioned in Definitions 1 and 2 are subclasses of Lipchitzian mapping.
Over the past decades, many mathematicians are interested in studying the fixed point of finite family of nonlinear mappings and their properties, (see [6–8, 17, 20–23]).
In 2009, Kangtunyakarn and Suantai [24] defined K-mapping for a finite family of nonexpansive mappings. Let K:C⟶C be defined by(13)U1=λ1T1+1−λ1I,U2=λ2T2U1+1−λ2U1,U3=λ3T3U2+1−λ3U2,⋮UN−1=λN−1TN−1UN−2+1−λN−1UN−2,K=UN=λNTNUN−1+1−λNUN−1,where Tii=1N is a finite family of nonexpansive mappings and λi∈0,1,i=1,2,…,N. Moreover, under some control conditions, FixK=∩i=1NFixTi and K is a nonexpansive mapping.
Later, Kangtunyakarn and Suantai [6] introduced the S-mapping for a finite family of nonexpansive mappings. Let S:C⟶C be defined by(14)U0=I,U1=α11T1U0+α21U0+α31I,U2=α12T2U1+α22U1+α32I,U3=α13T3U2+α23U2+α33I,⋮UN−1=α1N−1TN−1UN−2+α2N−1UN−2+α3N−1I,S=UN=α1NTNUN−1+α2NUN−1+α3NI,where Tii=1N is a finite family of nonexpansive mappings and αi=α1i,α2i,α3i∈I×I×I, where I=0,1 and α1i+α2i+α3i=1 for every j=1,2,…,N. Moreover, under some control conditions, FixS=∩i=1NFixTi and S is a nonexpansive mapping.
If we takeα3j=0,∀j=1,2,…,N, then the S-mapping reduces to the K-mapping.
In 2013, using the concept ofS-mapping, Kangtunyakarn [25] introduced SA-mapping for a finite family of nonexpansive mappings and strictly pseudo-contractive mappings as follows. Let SA:C⟶C be defined by(15)U0=I,U1=T1α11W1U0+α21U0+α31I,U2=T2α12W2U1+α22U1+α32I,U3=T3α13W3U2+α23U2+α33I,⋮UN−1=TN−1α1N−1WN−1UN−2+α2N−1UN−2+α3N−1I,SA=UN=TNα1NWNUN−1+α2NUN−1+α3NI,where Tii=1N:C⟶C be a finite family of nonexpansive mappings and Wii=1N:C⟶C be a finite family of strictly pseudo-contractive mappings, I is the identity mapping and αi=α1i,α2i,α3i∈I×I×I, where I=0,1 and α1i+α2i+α3i=1 for every j=1,2,…,N. Also, under some control conditions, FixSA=∩i=1NFixTi∩∩i=1NFixWi and SA is a nonexpansive mapping.
IfTi≡I, for every i=1,2,…,N, then the SA-mapping becomes the S-mapping.
Based on the previous research work, we give our theorem for MGEP and S-mapping for Lipschitzian mappings and some important results as follows:(i)
We first establish Lemmas2 and 3 showing fixed point results and some properties of S-mapping for Lipschitzian mappings under some control conditions.(ii)
We prove a strong convergence theorem of the sequences generated by iterative scheme for finding a common solution of generalized equilibrium problems and fixed point problem for a finite family of contractive mappings and Lipschitzian mappings.(iii)
We give some illustrative numerical examples supporting our main theorem and our examples show that our main result is not true is some conditions fail. Moreover, the main theorem can be used to approximate the value of pi.


## 2. Preliminaries

Throughout this work, the notations “⇀” and “⟶” denote weak convergence and strong convergence, respectively.Lemma 1 (see [26]).
Letun be a sequence of nonnegative real numbers satisfying(16)un+1≤1−βnun+ηn,∀n≥0,where βn is a sequence in 0,1 and ηn is a sequence such that(1)
∑n=1∞βn=∞,(2)
limsupn⟶∞ηn/βn≤0 or ∑n=1∞ηn<∞.
Then,limn⟶∞un=0.Theorem 1 (see [16]).
Letϕ:C⟶ℝ be a lower semicontinuous and convex functional. Let V:H⟶CBH be ℋ-Lipschitz continuous with constant ω, and Ψ:H×C×C⟶ℝ be an equilibrium-like function satisfying H1−H3. Let t>0 be a constant. For each z∈C, take pz∈Tz arbitrarily and define a mapping St:C⟶C as follows:(17)Stz=u∈C:Ψpz,u,v+ϕv−ϕu+1tu−z,v−u≥0,∀v∈C.
Then, the following hold:(a)
St is single-valued;(b)
St is firmly nonexpansive (that is, for any u,v∈C, Stu−Stv2≤Stu−Stv,u−v) if(18)Ψp1,Stz1,Stz2+Ψp2,Stz2,Stz1≤0.for allz1,z2∈C×C and all pi∈Vzi,i=1,2;(c)
FixSt=GEPsΨ,ϕ;(d)
GEPsΨ,ϕ is closed and convex.Definition 4 (see [6]).
LetC be a nonempty closed convex subset of a real Banach space. For every i=1,2,…,N, let Tii=1N,Wii=1N be a finite family of a ηi-contractive mapping and Li-Lipschitzian mapping of C into itself, respectively, with Li≥1 and ηiLi≤1. For every i=1,2,…,N, let αi=α1i,α2i,α3i∈I×I×I, where I=0,1 and α1i+α2i+α3i=1. Define a mapping S:C⟶C as follows:(19)U0=I,U1=T1α11W1U0+α21U0+α31I,U2=T2α12W2U1+α22U1+α32I,U3=T3α13W3U2+α23U2+α33I,⋮UN−1=TN−1α1N−1WN−1UN−2+α2N−1UN−2+α3N−1I,S=UN=TNα1NWNUN−1+α2NUN−1+α3NI.
This mappingS is called the S-mapping generated by T1,T2,…,TN, W1,W2,…,WN and α1,α2,…,αN.Lemma 2.
For everyi=1,2,…,N, Tii=1N be a finite family of a ηi-contractive mapping and Wii=1N be Li-Lipschitzian mapping of C into itself, respectively, with Li≥1, ηiLi≤1 and ∩i=1NFixTi∩∩i=1NFixWi≠∅. For every i=1,2,…,N, let αi=α1i,α2i,α3i∈I×I×I, where I=0,1 and α1i+α2i+α3i=1. Let S be the S-mapping generated by W1,W2,…,WN, T1,T2,…,TN and α1,α2,…,αN. Then there hold the following statement:(i)
FixS=∩i=1NFixTi∩∩i=1NFixWi;(ii)
S is a nonexpansive mapping.Proof.
First, it is clear that∩i=1NFixTi∩∩i=1NFixWi⊆FixS.
Next, claim thatFixS⊆∩i=1NFixTi∩∩i=1NFixWi.
Letx∈FixS and y∈∩i=1NFixTi∩∩i=1NFixWi. By Definition 4, we have(20)x−y2=Sx−y2=TNα1NWNUN−1+α2NUN−1+α3NIx−y2≤ηNα1NWNUN−1x−y+α2NUN−1x−y+α3Nx−y2≤ηNα1NWNUN−1x−y2+α2NUN−1x−y2+α3Nx−y2−α1Nα3NWNUN−1x−x2−α2Nα3NUN−1x−x2≤ηNα1NLNUN−1x−y2+α2NUN−1x−y2+α3Nx−y2−α1Nα3NWNUN−1x−x2−α2Nα3NUN−1x−x2=ηNα1NLN+α2NUN−1x−y2+ηNα3Nx−y2−ηNα1Nα3NWNUN−1x−x2−ηNα2Nα3NUN−1x−x2≤ηNLNα1N+α2NUN−1x−y2+ηNα3Nx−y2−ηNα1Nα3NWNUN−1x−x2−ηNα2Nα3NUN−1x−x2≤1−α3NUN−1x−y2+α3Nx−y2−ηNα1Nα3NWNUN−1x−x2−ηNα2Nα3NUN−1x−x2≤1−α3NTN−1α1N−1WN−1UN−2+α2N−1UN−2+α3N−1Ix−y2+α3Nx−y2≤1−α3NηN−1α1N−1WN−1UN−2x−y+α2N−1UN−2x−y+α3N−1x−y2+α3Nx−y2≤1−α3NηN−1α1N−1WN−1UN−2x−y2+α2N−1UN−2x−y2+α3N−1x−y2−α1N−1α3N−1WN−1UN−2x−x2−α2N−1α3N−1UN−2x−x2+α3Nx−y2≤1−α3NηN−1α1N−1LN−1UN−2x−y2+α2N−1UN−2x−y2+α3N−1x−y2−α1N−1α3N−1WN−1UN−2x−x2−α2N−1α3N−1UN−2x−x2+α3Nx−y2≤1−α3NηN−1LN−11−α3N−1UN−2x−y2+ηN−1α3N−1x−y2−ηN−1α1N−1α3N−1WN−1UN−2x−x2−ηN−1α2N−1α3N−1UN−2x−x2+α3Nx−y2≤1−α3N1−α3N−1UN−2x−y2+1−α3N1−1−α3N−1x−y2−1−α3NηN−1α1N−1α3N−1WN−1UN−2x−x2−1−α3NηN−1α2N−1α3N−1UN−2x−x2+α3Nx−y2≤∏i=N−1N1−α3iUN−2x−y2+1−∏i=N−1N1−α3ix−y2−1−α3NηN−1α1N−1α3N−1WN−1UN−2x−x2−1−α3NηN−1α2N−1α3N−1UN−2x−x2≤∏i=N−1N1−α3iUN−2x−y2+1−∏i=N−1N1−α3ix−y2⋮≤∏i=3N1−α3iU2x−y2+1−∏i=3N1−α3ix−y2−∏i=4N1−α3iη3α13α33W3U2x−x2−∏i=4N1−α3iη3α23α33U2x−x2≤∏i=3N1−α3iU2x−y2+1−∏i=3N1−α3ix−y2≤∏i=2N1−α3iU1x−y2+1−∏i=2N1−α3ix−y2−∏i=3N1−α3iη2α12α32W2U1x−x2−∏i=3N1−α3iη2α22α32U1x−x2≤∏i=1N1−α3ix−y2+1−∏i=1N1−α3ix−y2−∏i=2N1−α3iη1α11α31W1x−x2=x−y2−∏i=2N1−α3iη1α11α31W1x−x2.
From (20), it yields that(21)∏i=2N1−α3iη1α11α31W1x−x2≤0.
This implies thatx=W1x, that is,(22)x∈FixW1,Then, by Definition 4, we obtain(23)U1x=T1α11W1U0x+α21U0x+α31x=T1α11x+α21x+α31x=T1x.
Again, from (20), we get(24)x−y2≤∏i=2N1−α3iU1x−y2+1−∏i=2N1−α3ix−y2−∏i=3N1−α3iη2α22α32U1x−x2≤x−y2−∏i=3N1−α3iη2α22α32U1x−x2.
which follows that(25)∏i=3N1−α3iη2α22α32U1x−x2≤0.
We deduce that(26)U1x=x.
that is,x∈FixU1.
From (23) and (26), we have(27)x∈FixT1.
By (22) and (27), it yields that(28)x∈FixT1∩FixW1.
From (20) and (26), we derive that(29)x−y2≤∏i=2N1−α3iU1x−y2+1−∏i=2N1−α3ix−y2−∏i=3N1−α3iη2α12α32W2U1x−x2=x−y2−∏i=3N1−α3iη2α12α32W2x−x2.
which implies that(30)∏i=3N1−α3iη2α12α32W2x−x2≤0,Then we obtain W2x=x, that is,(31)x∈FixW2.
By the definition ofU2, (26) and (31), we get(32)U2x=T2α12W2U1x+α22U1x+α32x=T2α12W2x+α22x+α32x=T2α12x+α22x+α32x=T2x.
By (20), we have that(33)x−y2≤∏i=3N1−α3iU2x−y2+1−∏i=3N1−α3ix−y2−∏i=4N1−α3iη3α23α33U2x−x2≤x−y2−∏i=4N1−α3iη3α23α33U2x−x2.
which follows that(34)∏i=4N1−α3iη3α23α33U2x−x2≤0.
Thus, we get(35)U2x=x.
By (32) and (35), we obtain(36)x∈FixT2.
By (31) and (35), it follows that(37)x∈FixT2∩FixW2.
By using the same method described above, we easily obtain thatx∈FixTi∩FixWi and Uix=x, for each i=1,2,…,N−1.
From (20), we obtain(38)x−y2≤1−α3NUN−1x−y2+α3Nx−y2−ηNα1Nα3NWNUN−1x−x2≤x−y2−ηNα1Nα3NWNx−x2.
which implies that(39)ηNα1Nα3NWNx−x2≤0.
Hence, we haveWNx=x, that is,(40)x∈FixWN.
By the definition ofUN and (40), it yields that(41)x=Sx=UNx=TNα1NWNUN−1x+α2NUN−1x+α3Nx=TNα1NWNx+α2Nx+α3Nx=TNα1Nx+α2Nx+α3Nx=TNx.
which follows that(42)x∈FixTN.Then, we obtain x∈FixTN∩FixWN, that is,(43)x∈∩i=1NFixTi∩∩i=1NFixWi.
Therefore, we can conclude that(44)FixS⊆∩i=1NFixTi∩∩i=1NFixWi.
Finally, by applying the similar method of (20), S is a nonexpansive mapping.Lemma 3.
For eachi=1,2,…,N, let Tii=1N be a finite family of ηi-contractive mappings and Wii=1N be a finite family of Li-Lipschitzian mappings of C into itself, respectively, with ηiLi≤1, η=maxk=2,3,…,Nηk and L=maxk=2,3,…,NLk. For each j=1,2,…,N, let αjn=α1n,j,α2n,j,α3n,j, αj=α1j,α2j,α3j∈I×I×I, where I=0,1, α1n,j+α2n,j+α3n,j=1 and α1j+α2j+α3j=1 satisfying the following conditions: αin,j⟶αij as n⟶∞, for i=1,3 and ∑n=1∞αin+1,j−αin,j<∞, for i=1,3. For every n∈ℕ, let S and Sn be the S-mapping generated by W1,W2,…,WN, T1,T2,…,TN and α1,α2,…,αN and generated by W1,W2,…,WN, T1,T2,…,TN and α1n, α2n,…,αNn, respectively. Then, for any bounded sequences xn in C, there hold the following statement:(i)
limn⟶∞Snxn−Sxn=0;(ii)
∑n=1∞Snxn−1−Sn−1xn−1<∞.Proof.
Letxn be a bounded sequence in C. For fixed k∈ℕ and for all n∈ℕ, let Uk and Un,k be generated by W1,W2,…,WN, T1,T2,…,TN and W1,W2,…,WN, T1,T2,…,TN and α1n, α2n,…,αNn, respectively.
First, we will show that (i) holds. For everyn∈ℕ, we get(45)Un,1xn−U1xn=T1α1n,1W1xn+1−α1n,1xn−T1α11W1xn+1−α11xn≤η1α1n,1W1xn+1−α1n,1xn−α11W1xn+1−α11xn=η1α1n,1−α11W1xn−xn.
Fork∈2,3,…,N, we have(46)Un,1xn−U1xn=T1α1n,1W1xn+1−α1n,1xn−T1α11W1xn+1−α11xn≤η1α1n,1W1xn+1−α1n,1xn−α11W1xn+1−α11xn=η1α1n,1−α11W1xn−xn≤ηkα1n,kWkUn,k−1xn−WkUk−1xn+α1n,k−α1kWkUk−1xnα2n,kUn,k−1xn−Uk−1xn+1−α1n,k−α3n,k−1+α1k+α3kUk−1xn+α3n,k−α3kxn≤ηkα1n,kLkUn,k−1xn−Uk−1xn+α1n,k−α1kWkUk−1xn+α2n,kUn,k−1xn−Uk−1xn+α1n,k−α1k+α3n,k−α3kUk−1xn+α3n,k−α3kxn≤ηkLk+1Un,k−1xn−Uk−1xn+ηkα1n,k−α1kWkUk−1xn+Uk−1xn+ηkα3n,k−α3kUk−1xn+xn≤ηL+1Un,k−1xn−Uk−1xn+ηα1n,k−α1kWkUk−1xn+Uk−1xn+ηα3n,k−α3kUk−1xn+xn.
By (45) and (46), we get(47)Snxn−Sxn=Un,Nxn−UNxn≤ηL+1Un,N−1xn−UN−1xn+ηα1n,N−α1NWNUN−1xn+UN−1xn+ηα3n,N−α3NUN−1xn+xn≤ηL+1ηL+1Un,N−2xn−UN−2xn+ηα1n,N−1−α1N−1WN−1UN−2xn+UN−2xn+ηα3n,N−1−α3N−1UN−2xn+xn+ηα1n,N−α1NWNUN−1xn+UN−1xn+ηα3n,N−α3NUN−1xn+xn=ηL+12Un,N−2xn−UN−2xn+η2L+1α1n,N−1−α1N−1WN−1UN−2xn+UN−2xn+η2L+1α3n,N−1−α3N−1UN−2xn+xn+ηα1n,N−α1NWNUN−1xn+UN−1xn+ηα3n,N−α3NUN−1xn+xn=ηL+12Un,N−2xn−UN−2xn+∑j=N−1NηN−j+1L+1N−jα1n,j−α1jWjUj−1xn+Uj−1xn+∑j=N−1NηN−j+1L+1N−jα3n,j−α3jUj−1xn+xn⋮≤ηL+1N−1Un,1xn−U1xn+∑j=2NηN−j+1L+1N−jα1n,j−α1jWjUj−1xn+Uj−1xn+∑j=2NηN−j+1L+1N−jα3n,j−α3jUj−1xn+xn≤ηL+1N−1η1α1n,1−α11W1xn−xn+∑j=2NηN−j+1L+1N−jα1n,j−α1jWjUj−1xn+Uj−1xn+∑j=2NηN−j+1L+1N−jα3n,j−α3jUj−1xn+xn.
By (47) and the fact that αin,j⟶αij as n⟶∞, for every i=1,3 and j=1,2,…,N, we can deduce that limn⟶∞Snxn−Sxn=0.
Finally, we shall prove that (ii) holds. For anyn∈ℕ, we get(48)Un,1xn−1−Un−1,1xn−1=T1α1n,1W1xn−1+1−α1n,1xn−1−T1α1n−1,1W1xn−1+1−α1n−1,1xn−1≤η1α1n,1W1xn−1+1−α1n,1xn−1−α1n−1,1W1xn−1+1−α1n−1,1xn−1=η1α1n,1−αn−1,11W1xn−1−α1n,1−α1n−1,1xn−1=η1α1n,1−α1n−1,1W1xn−1−xn−1.
Fork∈2,3,…,N and the similar argument as (46), we have(49)Un,kxn−1−Un−1,kxn−1≤ηL+1Un,k−1xn−1−Un−1k−1xn−1+ηα1n,k−α1n−1,kWkUn−1,k−1xn−1+Un−1,k−1xn−1+ηα3n,k−α3n−1,kUn−1,k−1xn−1+xn−1.
From (48), (49) and the same method as (47), we obtain(50)Snxn−1−Sn−1xn−1≤ηL+1N−1η1α1n,1−α1n−1,1W1xn−1−xn−1+∑j=2NηN−j+1L+1N−jα1n,j−α1n−1,jWjUn−1,j−1xn−1+Un−1,j−1xn−1+∑j=2NηN−j+1L+1N−jα3n,j−α3n−1,jUn−1,j−1xn−1+xn−1.
Hence,∑n=1∞αin+1,j−αin,j<∞, for every i=1,3 and j=1,2,…,N, we have ∑n=1∞Snxn−1−Sn−1xn−1<∞.


## 3. Strong Convergence Theorem

Theorem 2.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Tii=1N be a finite family of ηi-contractive mappings and Wii=1N be a finite family of Li-Lipschitzian mappings of C into itself, respectively, with ηiLi≤1, for all i=1,2,…,N. Assume that Ω:=∩i=1NFixTi∩∩i=1NFixWi∩∩i=1NMGEPsΨi,ϕ,A≠∅. For each i=1,2,…,N¯, Vi:H⟶CBH be ℋ-Lipschitz continuous with coefficients μi, Ψi:H×C×C⟶ℝ be equilibrium-like function satisfying (H1)–(H3). Let ϕ:C⟶ℝ be a lower semicontinuous and convex function and A:C⟶C be an λ-inverse strongly monotone mapping. For every n∈ℕ, let Sn be the S-mapping generated by W1,W2,…,Wn, T1,T2,…,TN and α1n, α2n,…,αNn, where αjn=α1n,j,α2n,j,α3n,j∈I×I×I, where I=0,1, α1n,j+α2n,j+α3n,j=1 and α1n,j,α2n,j,α3n,j∈b1,b2⊂0,1, for all j=1,2,…,N. For every i=1,2,…,N¯, let zn be the sequence generated by x1∈C and w1i∈ViI−r1iAx1, there exists sequences pni∈H and zn, cni⊆C such that(51)pni∈ViI−rniAzn,pni−pn+1i≤1+1nℋViI−rniAzn,ViI−rn+1iAzn+1,Ψipni,cni,y+ϕy−ϕcni+1rnicni−zn,y−cni+Azn,y−cni≥0,∀y∈C,vn=∑i=1N¯anicni,zn+1=γnfvn+βnzn+δnSnvn,∀n≥1,where f:C⟶C is a contraction mapping with a constant ξ and γn, βn, δn⊆0,1 with γn+βn+δn=1, ∀n≥1. Suppose the following statement are true:(i)
limn⟶∞γn=0 and ∑n=1∞γn=∞;(ii)
0<τ≤βn,δn≤υ<1;(iii)
0≤η≤ani≤σ<1, for each i=1,2,…,N¯−1 and 0<η≤anN¯≤1 with ∑n=1N¯ani=1;(iv)
0<ε≤rni≤ω<2λ, for every n∈ℕ and i=1,2,…,N¯;(v)
∑n=1∞γn+1−γn<∞, ∑n=1∞βn+1−βn<∞,∑n=1∞δn+1−δn<∞, ∑n=1∞rn+1i−rni<∞,∑n=1∞an+1i−ani<∞, ∑n=1∞α1n+1,j−α1n,j<∞,∑n=1∞α3n+1,j−α3n,j<∞, for each i=1,2,…,N¯ and j=1,2,…,N;(vi)
For eachi=1,2,…,N¯, there exists ρi>0 such that(52)Ψiw1i,Tr1ix1,Tr2ix2+Ψiw2i,Tr2ix2,Tr1ix1≤−ρiTr1ix1−Tr2ix22,
for everyr1i,r2i∈Θi×Θi, x1,x2∈C×C and wji∈Vixj, for j=1,2, where Θi=rni:n≥1.
Thenzn, vn and cni converge strongly to z∗∈Ω, for all i=1,2,…,N¯.Proof.
The proof will be splited into six steps.Step 1.
Claim thatI−rniA is nonexpansive, for each i=1,2,…,N¯.
From (51), we get(53)Ψipni,cni,y+ϕy−ϕcni+1rnicni−I−rniAzn,y−cni≥0,
for everyy∈C and i=1,2,…,N¯. From (53) and Theorem 1, it yields that(54)cni=TrniI−rniAzn,∀i=1,2,…,N¯.
Putri∈Θi for all i=1,2,…,N¯. From (52), we have(55)Ψiw1i,Trix1,Trix2+Ψiw2i,Trix2,Trix1≤−ρiTrix1−Trix22≤0,
for allx1,x2∈C×C and wji∈Vixj,j=1,2.
From (55), it implies that Theorem 1 holds.
Letu,v∈C. Since A is λ-inverse strongly monotone with rni∈0,2λ, it deduces that(56)I−rniAu−I−rniAv2=u−v−rniAu−Av2=u−v2−2rniu−v,Au−Av+rni2Au−Av2≤u−v2−2λrniAu−Av2+rni2Au−Av2=u−v2+rnirni−2λAu−Av2≤u−v2.
ThusI−rniA is a nonexpansive mapping, for all i=1,2,…,N¯.Step 2.
Prove thatzn, vn and cni, ∀i=1,2,…,N¯ are bounded.
Letz∈Ω. From Theorem 1, observe that(57)vn−z≤∑i=1N¯anicni−z=∑i=1N¯aniTrniI−rniAzn−z≤zn−z.
By nonexpansiveness ofSn, we derive that(58)zn+1−z≤γnfvn−z+βnzn−z+δnSnvn−z≤γnfvn−fz+fz−z+βnzn−z+δnvn−z≤γnξzn−z+fz−z+βnzn−z+δnzn−z=1−γn1−ξzn−z+γnfz−z≤maxx1−z,fz−z1−ξ.
By induction, we obtainzn−z≤maxx1−z,fz−z/1−ξ,∀n∈ℕ. It follows that zn is bounded so are vn and cni, ∀i=1,2,…,N¯.Step 3.
Show thatlimn⟶∞zn+1−zn=0.
By the definition ofzn, we obtain(59)zn+1−zn≤γnfvn−fvn−1+γn−γn−1fvn−1+βnzn−zn−1+βn−βn−1zn−1+δnSnvn−Snvn−1+δnSnvn−1−Sn−1vn−1+δn−δn−1Sn−1vn−1≤γnξvn−vn−1+γn−γn−1fvn−1+βnzn−zn−1+βn−βn−1zn−1+δnvn−vn−1+δnSnvn−1−Sn−1vn−1+δn−δn−1Sn−1vn−1≤γnξ∑i=1N¯anicni−cn−1i+∑i=1N¯ani−an−1icn−1i+γn−γn−1fvn−1+βnzn−zn−1+βn−βn−1zn−1+δn∑i=1N¯anicni−cn−1i+∑i=1N¯ani−an−1icn−1i+δnSnvn−1−Sn−1vn−1+δn−δn−1Sn−1vn−1.
By using the same method of proof in Step3 in [17], we obtain(60)cni−cn−1i≤zn−zn−1+rni−rn−1iAzn−1+1εrni−rn−1icni−I−rniAzn.
Substitute (60) into (59), we get(61)zn+1−zn≤γnξ∑i=1N¯anizn−zn−1+rni−rn−1iAzn−1+1εrni−rn−1icni−I−rniAzn+∑i=1N¯ani−an−1icn−1i+γn−γn−1fvn−1+βnzn−zn−1+βn−βn−1zn−1+δn∑i=1N¯anizn−zn−1+rni−rn−1iAzn−1+1εrni−rn−1icni−I−rniAzn+∑i=1N¯ani−an−1icn−1i+δnSnvn−1−Sn−1vn−1+δn−δn−1Sn−1vn−1≤1−γn1−ξzn−zn−1+2∑i=1N¯anirni−rn−1iAzn−1+2ε∑i=1N¯anirni−rn−1icni−I−rniAzn+2∑i=1N¯ani−an−1icn−1i+γn−γn−1fvn−1+βn−βn−1zn−1+Snvn−1−Sn−1vn−1+δn−δn−1Sn−1vn−1.
Using the conditions (i), (v), Lemmas1 and 3 (ii), we obtain(62)limn⟶∞zn+1−zn=0.Step 4.
Claim thatlimn⟶∞cni−zn=limn⟶∞Snvn−vn=0,∀i=1,2,…,N¯.
By following the same method as in Step4 of [17], we deduce that(63)cni−z2≤zn−z2−zn−cni2+2rnizn−cniAzn−Az,and also(64)limn⟶∞Azn−Az=0.
From the definition ofzn and (63), we derive that(65)zn+1−z2≤γnfvn−z2+βnzn−z2+δnSnvn−z2≤γnfvn−z2+βnzn−z2+δn∑i=1N¯anicni−z2≤γnfvn−z2+βnzn−z2+δn∑i=1N¯anizn−z2−zn−cni2+2rnizn−cniAzn−Az≤γnfvn−z2+zn−z2−δn∑i=1N¯anizn−cni2+2δn∑i=1N¯anirnizn−cniAzn−Az.
which implies that(66)δn∑i=1N¯anizn−cni2≤zn−z2−zn+1−z2+γnfvn−z2+2δn∑i=1N¯anirnizn−cniAzn−Az≤zn−z+zn+1−zzn+1−zn+γnfvn−z2+2δn∑i=1N¯anirnizn−cniAzn−Az.
From (62), (64) and the conditions (i), (ii), (iii), we get(67)limn⟶∞zn−cni=0,for alli=1,2,…,N¯.
Consider(68)vn−zn=∑i=1N¯anicni−zn≤∑i=1N¯anicni−zn.Then, by (67), this follows that(69)limn⟶∞vn−zn=0.
Since(70)zn+1−vn≤zn+1−zn+zn−vn,then, from (62) and (69), we obtain(71)limn⟶∞zn+1−vn=0.
By the definition ofzn, we obtain(72)zn+1−vn=γnfvn−vn+βn∑i=1N¯anicni−vn+δnSnvn−vn=γnfvn−vn+βn∑i=1N¯anicni−zn+zn−vn+δnSnvn−vn.
From (67), (69) and (71) and the conditions (i) and (ii), we can conclude that(73)limn⟶∞Snvn−vn=0.Step 5.
Prove thatzn,pni and rni are Cauchy sequences, for each i=1,2,…,N¯.
Leta∈0,1, by (62), there exists N∈ℕ such that(74)zn+1−zn<an,∀n≥N.
Therefore, for anyn≥N∈ℕ and p∈ℕ, we derive that(75)zn+p−zn≤∑k=nn+p−1zk+1−zk≤∑k=nn+p−1ak<∑k=n∞ak=an1−a.
Sincea∈0,1, we get limn⟶∞an=0. From (79), taking n⟶∞, we obtain zn is a Cauchy sequence in a Hilbert space H. Let limn⟶∞zn=z∗. Since Vi:C⟶CBH be ℋ-Lipschitz continuous on H with coefficients μi, for every i=1,2,…,N¯, and (51), we have(76)pni−pn+1i≤1+1nℋViI−rniAzn,ViI−rn+1iAzn+1≤1+1nμiI−rniAzn−I−rn+1iAzn+1≤1+1nμiI−rniAzn−I−rniAzn+1+I−rniAzn+1−I−rn+1iAzn+1≤1+1nμizn−zn+1+rn+1i−rniAzn+1≤1+1nμizn−zn+1+rn+1i−rniM,where M=maxn∈ℕAzn. From (62), (76) and the condition (vi), we obtain(77)limn⟶∞pni−pn+1i=0,for everyi=1,2,…,N¯.
It is obvious thatpni and rni are Cauchy sequences in a Hilbert space H, for all i=1,2,…,N¯. So we let limn⟶∞pni=pi∗, and limn⟶∞rni=ri∗ for every i=1,2,…,N¯.
Next, claim thatpi∗∈ViI−ri∗Az∗, for each i=1,2,…,N¯.
Becausepni∈ViI−rniAzn, we obtain(78)dpni,ViI−ri∗Az∗≤maxdpni,ViI−ri∗Az∗,suppi˜∈ViI−ri∗Az∗dViI−rniAzn,pi˜≤maxsuppi^∈ViI−rniAzndpi^,ViI−ri∗Az∗,suppi˜∈ViI−ri∗Az∗dViI−rniAzn,pi˜=ℋViI−rniAzn,ViI−ri∗Az∗,for eachi=1,2,…,N¯.
Since(79)dpi∗,ViI−ri∗Az∗≤pi∗−pni+dpni,ViI−ri∗Az∗≤pi∗−pni+ℋViI−rniAzn,SiI−ri∗Az∗≤pi∗−pni+μiI−rniAzn−I−ri∗Az∗=pi∗−pni+μizn−z∗−rniAzn−ri∗Az∗.
takingn⟶∞, we have(80)dpi∗,ViI−ri∗Az∗=0,
which implies that(81)pi∗∈ViI−ri∗Az∗,for alli=1,2,…,N¯.Step 6.
Finally, show thatzn, vn and cni converge strongly to z∗∈Ω, for every i=1,2,…,N¯.
Without loss of generality, we can assume thatznk⇀z∗ as k⟶∞. By (69), we easily obtain that vnk⇀z∗ as k⟶∞.
For eachj=1,2,…,N, α1n,j,α2n,j,α3n,j∈b1,b2⊂0,1, without loss of generality, we may assume that(82)αink,j⟶αij∈0,1ask⟶∞,for everyi=1,2,3andj=1,2,…,N.
LetS be the S-mapping generated by T1,T2,…,TN, W1,W2,…,Wn and α1,α2,…,αN. By Lemma 2, we have S is nonexpansive and FixS=∩i=1NFixTi∩∩i=1NFixWi.
From Lemma3 (i), we obtain(83)limk⟶∞Snkvnk−Svnk=0.
Since(84)vnk−Svnk≤vnk−Snkvnk+Snkvnk−Svnk,
by (73) and (83), we have(85)limk⟶∞vnk−Svnk=0.
Sincevnk⇀z∗ as n⟶∞, (85) and I−S is demiclosed, we deduce that(86)z∗∈FixS=∩i=1NFixTi∩∩i=1NFixWi.
Finally, we show thatz∗∈∩i=1N¯GEPsΨi,ϕ,A.
Sinceznk⟶z∗ as k⟶∞ and (67), we get(87)cnki⟶z∗ask⟶∞,for alli=1,2,…,N¯.
From (51), we have(88)Ψipnki,cnki,y+ϕy−ϕcnki+1rnkicnki−znk,y−cnki+Aznk,y−cnki≥0.
for everyy∈C and i=1,2,…,N¯. From (67), (87), the condition H1 and the lower semicontinuity of ϕ, we deduce that(89)Ψipi∗,z∗,y+ϕy−ϕz∗+Az∗,y−z∗≥0.
for everyy∈C and i=1,2,…,N¯, which follows by (81) that(90)z∗∈GEPsΨi,ϕ,A,for everyi=1,2,…,N¯.
It follows that(91)z∗∈∩i=1N¯GEPsΨi,ϕ,A.
From (86) and (91), we have(92)z∗∈Ω.
Therefore, we obtain the sequencezn converges strongly to z∗∈Ω. Moreover, from (67) and (69), we have vn and cni converge strongly to z∗∈Ω, for every i=1,2,…,N¯. This completes the proof.
The following corollary is a direct consequence of Theorem2. Therefore, the proof is omitted.Corollary 1.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let Tii=1N be a finite family of ηi-contractive mappings and Wii=1N be a finite family of Li-Lipschitzian mappings of C into itself, respectively, with ηiLi≤1, for all i=1,2,…,N. Assume that Ω:=∩i=1NFixTi∩∩i=1NFixWi∩∩i=1NGEPsΨi,ϕ≠∅. For each i=1,2,…,N¯, Vi:H⟶CBH be ℋ-Lipschitz continuous with coefficients μi, Ψi:H×C×C⟶ℝ be equilibrium-like function satisfying (H1)–(H3). Let ϕ:C⟶ℝ be a lower semicontinuous and convex function. For every n∈ℕ, let Sn be the S-mapping generated by W1,W2,…,Wn, T1,T2,…,TN and α1n, α2n,…,αNn, where αjn=α1n,j,α2n,j,α3n,j∈I×I×I, where I=0,1, α1n,j+α2n,j+α3n,j=1 and α1n,j,α2n,j,α3n,j∈b1,b2⊂0,1, for all j=1,2,…,N. For every i=1,2,…,N¯, let zn be the sequence generated by x1∈C and w1i∈Vix1, there exists sequences pni∈H and zn, cni⊆C such that(93)pni∈Vizn,pni−pn+1i≤1+1nℋVizn,Vizn+1,Ψipni,cni,y+ϕy−ϕcni+1rnicni−zn,y−cni≥0,∀y∈C,vn=∑i=1N¯anicni,zn+1=γnfvn+βnzn+δnSnvn,∀n≥1,where f:C⟶C is a contraction mapping with a constant ξ and γn, βn, δn⊆0,1 with γn+βn+δn=1, ∀n≥1. Suppose the following statement are true:(i)
limn⟶∞γn=0 and ∑n=1∞γn=∞;(ii)
0<τ≤βn,δn≤υ<1;(iii)
0≤η≤ani≤σ<1, for each i=1,2,…,N¯−1 and 0<η≤anN¯≤1 with ∑n=1N¯ani=1;(iv)
0<ε≤rni≤ω<2λ, for every n∈ℕ and i=1,2,…,N¯;(v)
∑n=1∞γn+1−γn<∞, ∑n=1∞βn+1−βn<∞,∑n=1∞δn+1−δn<∞, ∑n=1∞rn+1i−rni<∞,∑n=1∞an+1i−ani<∞, ∑n=1∞α1n+1,j−α1n,j<∞,∑n=1∞α3n+1,j−α3n,j<∞, for each i=1,2,…,N¯ and j=1,2,…,N;(vi)
For eachi=1,2,…,N¯, there exists ρi>0 such that(94)Ψiw1i,Tr1ix1,Tr2ix2+Ψiw2i,Tr2ix2,Tr1ix1≤−ρiTr1ix1−Tr2ix22.for everyr1i,r2i∈Θi×Θi, x1,x2∈C×C and wji∈Vixj, for j=1,2, where Θi=rni:n≥1. Then zn, vn and cni converge strongly to z∗∈Ω, for all i=1,2,…,N¯.In 2014, Suwannaut and Kangtunyakarn [17] introduced the viscosity approximation method for the modified generalized equilibrium problem and a finite family of strictly pseudo-contractive mappings in Hilbert spaces. Let Tii=1N be a finite family of κi-strictly pseudo-contractive mappings with ℱ:=∩i=1NFTi∩∩i=1NMGEPsΨi,ϕ,A≠∅. For i=1,2,…,N¯, let xn be the sequence generated by x1∈C and w1i∈ViI−r1iAx1, there exists sequences pni∈H and zn, cni⊆C such that(95)pni∈ViI−rniAzn,pni−pn+1i≤1+1nℋViI−rniAzn,ViI−rn+1iAzn+1,Ψipni,cni,y+ϕy−ϕcni+1rnicni−zn,y−cni+Azn,y−cni≥0,∀y∈C,vn=∑i=1N¯anicni,zn+1=γnfvn+βnzn+δnKnvn,∀n≥1,where Kn is a K-mapping generated by a finite family of strictly pseudo-contractive mappings and real numbers. Then, under some control conditions, the sequences zn and cni converge strongly to q=Pℱfq, for all i=1,2,…,N¯.Remark 1.
The iterative method (51) is a modification and extension of the iteration (95) as follows:(1)
S-mapping can be reduced to K-mapping. Therefore, K-mapping is a special case of S-mapping.(2)
In this research, a finite family of Lipschitzian mappings is considered instead of using a finite family of strictly pseudo-contractive mappings.


## 4. Numerical Examples

In this section, we give numerical examples to support our main theorem.Example 1.
Let the mappingsφ:ℝ⟶ℝ, A:ℝ⟶ℝ, f:ℝ⟶ℝ be defined by(96)φu=u2,Au=u2,fu=u4,for allu∈ℝ.
Fori=1,2,…,N, let Ti:ℝ⟶ℝ and Wi:ℝ⟶ℝ be defined by(97)Tiu=u500i,Wiu=2iu,for everyu∈ℝ.
Fori=1,2,…,N¯, let Vi:ℝ⟶ℝ, Φi:ℝ×ℝ⟶ℝ be defined by(98)Viu=u8,Φiw,u,v=iwv−u,for eachw,u,v∈ℝ.
Letγn=1/6n, βn=3n−2/6n, δn=3n+1/6n, rn=5n+2/7n+9 and an1=2n+1/5n+3, an2=3n+2/5n+3 for each n∈ℕ. For every j=1,2,…,N and let α1j=3/5j2+3, α2j=3j2/5j2+3, α3j=2j2/5j2+3. Then, the sequences zn, vn and cni converge strongly to 0, for each i=1,2,…,N¯.
Solution. It is clear that the sequencesγn,βn,δn, and rn satisfy all the conditions of Theorem 2. It is easy to show that Ti is a contractive mapping with coefficient ηi=1/5i and Wi is 3i-Lipschitzian mapping. Since α1j=3/5j2+3, α2j=3j2/5j2+3, α3j=2j2/5j2+3, then σj=3/5j2+3,3j2/5j2+3,2j2/5j2+3 for j=1,2,…,N. Since Sn is S-mapping generated by T1,T2,…,Tn, W1,W2,…,Wn and σ1,σ2,…,σn, we obtain(99)Un,0zn=znUn,1zn=1598Un,0+38Un,0+28znUn,2zn=1251823Un,1+1223Un,1+823zn⋮Un,kzn=15k9k5k2+3Un,k−1+3k25k2+3Un,k−1+2k25k2+3zn⋮Un,N−1zn=15N−19N−15N−12+3Un,N−2+3N−125N−12+3Un,N−2+2N−125N−12+3znSnzn=Un,Nzn=15N9N5N2+3Un,N−1+3N25N2+3Un,N−1+2N25N2+3zn.
From the definition ofTn and Wn, we deduce that(100)0=∩i=1NFixTi∩∩i=1NFixWi.
Fori=1,2,…,N¯, we have that Φi,Vi,φ,A satisfy all conditions in Theorem 2 and(101)∩i=1NFixTi∩∩i=1NFixWi∩∩i=1N¯MGEPsΦi,φ,A=0.
Therefore(102)0≤Φipni,cni,y+φy−φcni+1rniy−cni,cni−zn+Azn,y−cni=ipniy−cni+y2−cni2+1rniy−cnicni−zn+zn2y−cni⇔0≤rniipniy−cni+y2−cni2+1rniy−cnicni−zn+zn2cni−zn=−cni2−rnicni2−irnicnipni+cnizn−12rnicnizn+cniy+irnipniy−zny+12rnizny+rniy2.
LetGy=−cni2−rnicni2−irnicnipni+cnizn−1/2rnicnizn+cniy+irnipniy−zny+1/2rnizny+rniy2, where Gy is a quadratic function of y with coefficients a=rni, b=cni+irnipni−zn+1/2rnizn, and c=−cni2−rnicni2−irnicnipni+cnizn−1/2rnicnizn. Determine the discriminant Δ of G as follows:(103)Δ=b2−4ac=cni+irnipni−zn+12rnizn2−4rni−cni2−rnicni2−irnicnipni+cnizn−12rnicnizn=cni2+4rnicni2+4rni2cni2+2irnicnipni+4irni2cnipni+i2rni2pni2−2cnizn−3rnicnizn+2rni2cnizn−2irnipnizn+irni2pnizn+zn2−rnizn2+14rni2zn2=142cni+4rnicni+2irnipni−2zn+rnizn2.
From (102), we have Gy≥0, for every y∈ℝ. If Gy has most one solution in ℝ, thus we have Δ≤0. This deduces that(104)cni=−2irnipni+2zn−rnizn21+2rni.
For eachn∈ℕ, we rewrite (51) as follows:(105)pni=ViI−rniAzncni=−2irnipni+2zn−rnizn21+2rni,vn=∑i=1N¯anicni,zn+1=γnfvn+βnzn+δnSnvn,∀n∈ℕandi=1,2,…,N¯.
From Theorem2, the sequences zn, vn and cni generated by (105) converge strongly to 0, for every i=1,2,…,N¯.
The numerical values of all sequenceszn, vn and cn1, cn2 are shown in Table 1 and Figure 1, where N¯=2 and n=N=200.
Next, we will give an numerical example for the iterative method (95) in the work of Suwannaut and Kangtunyakarn [17].Table 1
The values of all sequences for the iterative method (51).
ncn1cn2vnzn10.2265620.1926330.2053561.00000020.021764−0.0020270.0071230.1752243−0.007967−0.030669−0.0218410.0585564−0.017253−0.039651−0.0308870.0224695−0.020720−0.042995−0.0342440.009040⋮⋮⋮⋮⋮100−0.022991−0.044964−0.036184−0.000031⋮⋮⋮⋮⋮196−0.022985−0.044950−0.036168−0.000016197−0.022985−0.044949−0.036168−0.000016198−0.022985−0.044949−0.036168−0.000015199−0.022985−0.044949−0.036168−0.000015200−0.022985−0.044949−0.036168−0.000015Figure 1
The convergence of all sequences for the iterative method (51) and (95). (a) The iterative method (51) (b) The iterative method (95).
(a)(b)Example 2.
Fori=1,2,…,N, let the mapping Ti:ℝ⟶ℝ be defined by(106)Tiu=−53u,∀u∈ℝ,and(107)λin=2n250n+i,∀n∈ℕ.
Let all parameters and mappings be defined the same as mentioned in Example1. It is obvious that Ti is 1/4-strictly pseudo-contractive mapping, for each i=1,2,…,N and all parameters and mappings satisfy all conditions of Theorem 2 in [17]. Thus, we get(108)0=∩i=1NFixTi∩∩i=1N¯MGEPsΦi,φ,A.
The numerical values of all sequenceszn, vn and cn1, cn2 are shown in Table 2 and Figure 1, where N¯=2 and n=N=200.Table 2
The values of all sequences for the iterative method (95).
ncn1cn2vnzn10.2265620.1926330.2053561.00000020.023261−0.0005690.0085960.1810573−0.007420−0.030133−0.0213000.0606664−0.017105−0.039506−0.0307400.0230365−0.020743−0.043018−0.0342670.008950⋮⋮⋮⋮⋮100−0.023122−0.045094−0.036314−0.000524⋮⋮⋮⋮⋮196−0.023114−0.045079−0.036298−0.000505197−0.023114−0.045079−0.036298−0.000505198−0.023114−0.045079−0.036298−0.000505199−0.023114−0.045079−0.036297−0.000505200−0.023114−0.045079−0.036297−0.000505Remark 2.
From the above numerical results, we can conclude that(i)
Table1 shows that the sequences cn1, cn2, vn and zn converge to 0, where 0=∩i=1NFixTi∩∩i=1NFixWi∩∩i=1NMGEPsΦ,φ,A and the convergence of all sequences can be guaranteed by Theorem 2.(ii)
Table2 shows that the sequences cn1, cn2, vn and zn converge to 0, where 0=∩i=1NFixTi∩∩i=1NMGEPsΦ,φ,A and the convergence of all sequences can be guaranteed by Theorem 2 in [17].(iii)
From Tables1 and 2, we have that the iterative method (51) converges faster than the iterative method (95).
Similar to Example1, we give another example for Theorem 2. Moreover, we use this numerical example to approximate the value of π.Example 3.
Let the mappingsϕ:ℝ⟶ℝ, A:ℝ⟶ℝ, f:ℝ⟶ℝ be defined by(109)ϕu=u−π2,Au=u−π2,fu=u4,for allu∈ℝ.
For everyi=1,2,…,N, let Ti:ℝ⟶ℝ and Wi:ℝ⟶ℝ be defined by(110)Tiu=u3i+1+3iπ3i+1,Wiu=i+1u−iπ,for allu∈ℝ.
Fori=1,2,…,N¯, let Vi:ℝ⟶ℝ, Ψi:ℝ×ℝ⟶ℝ be defined by(111)Viu=x−π8,Ψiw,u,v=iwv−u,for eachw,u,v∈ℝ.
Let all parameter sequences be defined as in Example1. Then, the sequences zn, vn and cni converge strongly to π, for each i=1,2,…,N¯.
Solution. It is clear that the sequences γn,βn,δn, and rn satisfy all the conditions of Theorem 2. It is obvious that Ti is a contractive mapping with coefficient ηi=1/3i+1 and Wi is i+1-Lipschitzian mapping. From the definition of Tn and Wn, we obtain(112)π=∩i=1NFixTi∩∩i=1NFixWi.
For eachi=1,2,…,N¯, it is obvious that Ψi,Vi,ϕ,A satisfy all conditions in Theorem 2 and π∈∩i=1N¯MGEPsΨi,ϕ,A. Then we have(113)∩i=1NFixTi∩∩i=1NFixWi∩∩i=1N¯MGEPsΨi,ϕ,A=π.Then we deduce that(114)0≤Ψipni,uni,y+ϕy−ϕuni+1rniy−uni,uni−zn+Azn,y−uni=ipniy−uni+y−π2−uni−π2+1rniy−uniuni−zn+zn−π2y−uni⇔=5πrniuni2−uni2−rniuni2−irniunipni+unizn−12rniunizn−5πrniy2+uniy+irnipniy−zny+12rnizny+rniy2.
LetGy=5πrniuni/2−uni2−rniuni2−irniunipni+unizn−1/2rniunizn−5πrniy/2+uniy+irnipniy−zny+1/2rnizny+rniy2, where Gy is a quadratic function of y with coefficients a=rni, b=−5πrni/2+uniy+irnipni−zn+1/2rnizn, and c=5πrniuni/2−uni2−rnuni2−irniunipni+unizn−1/2rniunizn. Determine the discriminant Δ of G as follows:(115)Δ=b2−4ac=−5πrni2+uniy+irnipni−zn+12rnizn2−4rni5πrniuni2−uni2−rniuni2−irniunipni+unizn−12rniunizn=25π2rni24−5πrniuni−10πrni2uni+uni2+4rniuni2+4rni2uni2+5iπrni2pni+2irniunipni+4irni2unipni+i2rni2pni2+5πrnizn−52πrni2zn−2unizn−3rniunizn+2rni2unizn−2irnipnizn+irni2pnizn+zn2−rnizn2+14rni2zn2.=155πrni−2uni−4rniuni−2irnipni+2zn−rnizn2.
From (114), we have Gy≥0, for every y∈ℝ. If Gy has most one solution in ℝ, thus we get Δ≤0. This yields that(116)cni=5πrn−2irnipni+2zn−rnizn21+2rni.
For eachn∈ℕ, (51) becomes(117)pni=ViI−rniAzncni=5πrn−2irnipni+2zn−rnizn21+2rni,vn=∑i=1N¯anicni,zn+1=γnfvn+βnzn+δnSnvn,∀n∈ℕandi=1,2,…,N¯.
From Theorem2, zn, vn and cni generated by (117) converge strongly to π, for every i=1,2,…,N¯.
The numerical values of all sequenceszn, vn and cn1, cn2 are shown in Table 3 and Figure 2, where N¯=2 and n=N=20000.Table 3
The values ofcn1, cn2, vn and zn with x1=3, N¯=2 and n=N=20000.
ncn1cn2vnzn13.1095133.1143173.1125163.00000023.0378973.0439143.0416002.72491833.0576013.0624153.0605432.80498243.0765333.0806913.0790642.87934753.0906483.0944463.0929542.934169⋮⋮⋮⋮⋮2503.1443463.1474553.1462123.139700⋮⋮⋮⋮⋮10003.1446383.1477473.1465043.140807⋮⋮⋮⋮⋮100003.1448253.1479343.1466913.141514⋮⋮⋮⋮⋮199983.1448363.1479453.1467013.141553199993.1448363.1479453.1467013.141553200003.1448363.1479453.1467013.141553Figure 2
The comparison of sequencescn1, cn2, vn and zn with x1=3, N¯=2 and n=N=1000.Remark 3.
(i)
From Table3 and Figure 2, the sequences cn1, cn2, vn and zn converge to π, where π=∩i=1NFixTi∩∩i=1NFixWi∩∩i=1NMGEPsΨ,ϕ,A.(ii)
The convergence ofcn1, cn2, vn and zn can be guaranteed by Theorem 2.(iii)
Using this as an example, Theorem2 can be used to approximate the value of π.


## 5. Conclusion

In this research, we study and analyze the viscosity iterative method for approximating a common solution of the modified generalized equilibrium problems and a common fixed point of a finite family of Lipchitzian mappings. It can be seen as an improvement and modification of some existing algorithms for solving an equilibrium problem and a fixed point problem of Lipchitzian mappings and some related mappings. Some previous research works, for example, [6, 7, 16, 17, 25] can be considered as special cases of Theorem 2. Moreover, some numerical examples for our main theorem are provided.



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