# Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

Authors: Lu-Chuan Ceng; Cheng-Wen Liao; Chin-Tzong Pang; Ching-Feng Wen
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.


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## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. Let S:C→H be a nonlinear mapping on C. We denote by Fix⁡(S) the set of fixed points of S and by R the set of all real numbers. A mapping S:C→H is called L-Lipschitz continuous if there exists a constant L≥0 such that
(1)∥Sx-Sy∥≤L∥x-y∥,∀x,y∈C.
In particular, if L=1, then S is called a nonexpansive mapping; if L∈[0,1), then S is called a contraction.LetA:C→H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x∈C such that
(2)〈Ax,y-x〉≥0,∀y∈C.
The solution set of VIP (2) is denoted by VI⁡(C,A).The VIP (2) was first discussed by Lions [1]. There are many applications of VIP (2) in various fields; see, for example, [2–5]. It is well known that if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (2) has a unique solution. In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (2) in Euclidean space Rn:
(3)yn=PC(xn-τAxn),xn+1=PC(xn-τAyn),∀n≥0,
with τ>0 a given number, which is known as the extragradient method. The literature on the VIP is vast, among which, Korpelevich's extragradient method has received great attention in various applications and undergone improvements in many ways; see, for example, [7–20] and references therein, to name but a few.Letφ:C→R be a real-valued function, let A:H→H be a nonlinear mapping, and let Θ:C×C→R be a bifunction. In 2008, Peng and Yao [8] introduced the following generalized mixed equilibrium problem (GMEP) of finding x∈C such that
(4)Θ(x,y)+φ(y)-φ(x)+〈Ax,y-x〉≥0,Θ(x,y)+φ(y)-φ(x)+RQDOK∀y∈C.
We denote the set of solutions of GMEP (4) by GMEP⁡(Θ,φ,A). The GMEP⁡ (4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied; see, for example, [10, 16, 18, 19, 21–23]. In particular, if φ=0, then GMEP (4) is reduced to the generalized equilibrium problem (GEP) which is to find x∈C such that
(5)Θ(x,y)+〈Ax,y-x〉≥0,∀y∈C.
It was introduced and studied by S. Takahashi and W. Takahashi [24]. The set of solutions of GEP is denoted by GEP⁡(Θ,A).IfA=0, then GMEP (4) is reduced to the mixed equilibrium problem (MEP), which is to find x∈C such that
(6)Θ(x,y)+φ(y)-φ(x)≥0,∀y∈C.
It was considered and studied in [25]. The set of solutions of MEP is denoted by MEP⁡(Θ,φ).Ifφ=0 and A=0, then GMEP (4) is reduced to the equilibrium problem (EP), which is to find x∈C such that
(7)Θ(x,y)≥0,∀y∈C.
It was considered and studied in [26, 27]. The set of solutions of EP is denoted by EP⁡(Θ). It is worth mentioning that the EP is a unified model of several problems, namely, the variational inequality problems, the optimization problems, the saddle point problems, the complementarity problems, the fixed point problems, the Nash equilibrium problems, and so forth.It was assumed in [8] that Θ:C×C→R is a bifunction satisfying conditions (A1)–(A4) and φ:C→R is a lower semicontinuous and convex function with a restriction (B1) or (B2), where(A1)
Θ(x,x)=0 for all x∈C;(A2)
Θ is monotone; that is, Θ(x,y)+Θ(y,x)≤0 for any x,y∈C;(A3)
Θ is upper-hemicontinuous; that is, for each x,y,z∈C,
(8)lim⁡⁡sup⁡⁡t→0+Θ(tz+(1-t)x,y)≤Θ(x,y);(A4)
Θ(x,·) is convex and lower semicontinuous for each x∈C;(B1)
for eachx∈H and r>0, there exists a bounded subset Dx⊂C and yx∈C such that for any z∈C∖Dx,
(9)Θ(z,yx)+φ(yx)-φ(z)+1r〈yx-z,z-x〉<0;(B2)
C is a bounded set.Given a positive numberr>0. Let Tr(Θ,φ):H→C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x∈H,
(10)Tr(Θ,φ)(x)≔{+1r〈y-x,z-y〉≥0,∀z∈C}y∈C:Θ(y,z)+φ(z)-φ(y)+1r〈y-x,z-y〉≥0,∀z∈C}.LetF1,F2:C→H be two mappings. Consider the following general system of variational inequalities (GSVI) of finding (x*,y*)∈C×C such that
(11)〈ν1F1y*+x*-y*,x-x*〉≥0,∀x∈C,〈ν2F2x*+y*-x*,x-y*〉≥0,∀x∈C,
where ν1>0 and ν2>0 are two constants, which was considered and studied in [9, 11, 28]. In particular, if F1=F2=A, then the GSVI (11) is reduced to the following problem of finding (x*,y*)∈C×C such that
(12)〈ν1Ay*+x*-y*,x-x*〉≥0,∀x∈C,〈ν2Ax*+y*-x*,x-y*〉≥0,∀x∈C,
which is defined by Verma [29] and called as a new system of variational inequalities (NSVI). Furthermore, if x*=y*, then the NSVI reduces to the classical VIP (2). In 2008, Ceng et al. [9] transformed the GSVI (11) into a fixed point problem as follows.Proposition CWY (see [9]).For givenx¯,y¯∈C,(x¯,y¯)is a solution of the GSVI (11) if and only ifx¯is a fixed point of the mappingG:C→Cdefined by(13)Gx=PC(I-ν1F1)PC(I-ν2F2)x,∀x∈C,wherey¯=PC(I-ν2F2)x¯.In particular, if the mappingFj:C→H is ζj-inverse-strongly monotone for j=1,2, then the mapping G is nonexpansive for all νj∈(0,2ζj], j=1,2. We denote by GSVI(G) the fixed point set of the mapping G.Letf:C→R be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing f over the constraint set C(14)minimize{f(x):x∈C},
as considered and studied in [13, 14, 30–32]. We denote by Γ the set of minimizers of CMP (14). The gradient-projection algorithm (GPA) generates a sequence {xn} determined by the gradient ∇f and the metric projection PC:
(15)xn+1≔PC(xn-λ∇f(xn)),∀n≥0
or more generally
(16)xn+1≔PC(xn-λn∇f(xn)),∀n≥0,
where, in both (15) and (16), the initial guess x0 is taken from C arbitrarily and the parameters λ or λn are positive real numbers. The convergence of algorithms (15) and (16) depends on the behavior of the gradient ∇f. As a matter of fact, it is known that if ∇f is α-strongly monotone and L-Lipschitz continuous, then, for 0<λ<2α/L2, the operator PC(I-λ∇f) is a contraction; hence, the sequence {xn} defined by the GPA (15) converges in a norm to the unique solution of CMP (14). More generally, if {λn} is chosen to satisfy the property
(17)0<lim⁡⁡inf⁡⁡n→∞λn≤lim⁡⁡sup⁡⁡n→∞λn<2αL2,
then the sequence {xn} defined by the GPA (16) converges in a norm to the unique minimizer of CMP (14). If the gradient ∇f is only assumed to be Lipschitz continuous, then {xn} can only be weakly convergent if H is infinite-dimensional (a counterexample is given in Section 5 of Xu [31]). Recently, Xu [31] used averaged mappings to study the convergence analysis of the GPA, which is hence an operator-oriented approach.Very recently, Ceng and Al-Homidan [23] introduced and analyzed the following iterative algorithm by hybrid steepest-descent viscosity method and derived its strong convergence under appropriate conditions.Theorem CA (see [23, Theorem 21]).LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letf:C→Rbe a convex functional withL-Lipschitz continuous gradient∇f. LetM,Nbe two integers. LetΘkbe a bifunction fromC×CtoRsatisfying (A1)–(A4) and letφk:C→R∪{+∞}be a proper lower semicontinuous and convex function, wherek∈{1,2,…,M}. LetBk:H→HandAi:C→Hbeμk-inverse-strongly monotone andηi-inverse-strongly monotone, respectively, wherek∈{1,2,…,M}andi∈{1,2,…,N}. LetF:H→Hbe aκ-Lipschitzian andη-strongly monotone operator with positive constantsκ,η>0. LetV:H→Hbe anl-Lipschitzian mapping with a constantl≥0. Let0<μ<2η/κ2and0≤γl<τ, whereτ=1-1-μ(2η-μκ2). Assume thatΩ≔∩k=1MGMEP(Θk,φk,Bk)∩∩i=1NVI(C,Ai)∩Γ≠∅and that either (B1) or (B2) holds. For arbitrarily givenx1∈H, let{xn}be a sequence generated by(18)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM-1,n(ΘM-1,φM-1)×(I-rM-1,nBM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,vn=PC(I-λN,nAN)PC(I-λN-1,nAN-1)⋯×PC(I-λ2,nA2)PC(I-λ1,nA1)un,xn+1=snγVxn+βnxn+((1-βn)I-snμF)Tnvn,=snγVxn+βnxn+ROQQQKOPRD∀n≥1,wherePC(I-λn∇f)=snI+(1-sn)Tn(hereTnis nonexpansive andsn=(2-λnL)/4∈(0,1/2)for eachλn∈(0,2/L)). Assume that the following conditions hold:(i)
sn∈(0,1/2)for eachλn∈(0,2/L),andlim⁡n→∞sn=0(⇔lim⁡n→∞λn=2/L);(ii)
{βn}⊂(0,1)and0<lim⁡⁡inf⁡n→∞βn≤lim⁡⁡sup⁡n→∞βn<1;(iii)
{λi,n}⊂[ai,bi]⊂(0,2ηi)andlim⁡n→∞|λi,n+1-λi,n|=0for alli∈{1,2,…,N};(iv)
{rk,n}⊂[ek,fk]⊂(0,2μk)andlim⁡n→∞|rk,n+1-rk,n|=0for allk∈{1,2,…,M}.Then{xn}converges strongly asλn→(2/L)(⇔sn→0)to a pointx*∈Ω, which is a unique solution inΩto the VIP:(19)〈(μF-γV)x*,p-x*〉≥0,∀p∈Ω.Equivalently,x*=PΩ(I-(μF-γV))x*.In 2009, Yao et al. [33] considered the following hierarchical fixed point problem (HFPP): find hierarchically a fixed point of a nonexpansive mapping T with respect to another nonexpansive mapping S; namely, find x~∈Fix⁡(T) such that
(20)〈x~-Sx~,x~-x〉≤0,∀x∈Fix⁡(T).
The solution set of HFPP (20) is denoted by Λ. It is obvious to see that solving HFPP (20) is equivalent to the fixed point problem of the composite mapping PFix⁡(T)S; that is, find x~∈C such that x~=PFix⁡(T)Sx~. The authors [33] introduced and analyzed the following iterative algorithm for solving HFPP (20):
(21)yn=βnSxn+(1-βn)xn,xn+1=αnVxn+(1-αn)Tyn,∀n≥0.Theorem YLM (see [33, Theorem 3.2]).LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetSandTbe two nonexpansive mappings ofCinto itself. LetV:C→Cbe a fixed contraction withα∈(0,1). Let{αn}and{βn}be two sequences in(0,1). For any givenx0∈C, let{xn}be the sequence generated by (21). Assume that the sequence{xn}is bounded and that(i)
∑n=0∞‍αn=∞;(ii)
lim⁡n→∞(1/αn)|(1/βn)-(1/βn-1)|=0, lim⁡n→∞(1/βn)|1-(αn-1/αn)|=0;(iii)
lim⁡n→∞βn=0, lim⁡n→∞(αn/βn)=0andlim⁡n→∞(βn2/αn)=0;(iv)
Fix⁡(T)∩int⁡C≠∅;(v)
there exists a constantk>0such that∥x-Tx∥≥kDist(x,Fix⁡(T))for eachx∈C,whereDist(x,Fix⁡(T))=inf⁡y∈Fix⁡(T)∥x-y∥. Then{xn}converges strongly tox~=PΛVx~which solves the VIP〈x~-Sx~,x~-x〉≤0,  for allx∈Fix⁡(T).Very recently, Iiduka [34, 35] considered a variational inequality with a variational inequality constraint over the set of fixed points of a nonexpansive mapping. Since this problem has a triple structure in contrast with the bilevel programming problems or the hierarchical constrained optimization problems or the hierarchical fixed point problem, it is referred to as the triple hierarchical constrained optimization problem (THCOP). He presented some examples of THCOP and developed iterative algorithms to find the solution of such problem. The convergence analysis of the proposed algorithms is also studied in [34, 35]. Since the original problem is of a variational inequality, in this paper, we call it triple hierarchical variational inequality (THVI). Subsequently, Ceng et al. [36] introduced and considered the following triple hierarchical variational inequality (THVI).Problem 1.
LetS,T:C→C be two nonexpansive mappings with Fix⁡(T)≠∅, let V:C→H be a ρ-contractive mapping with a constant ρ∈[0,1), and let F:C→H be a κ-Lipschitzian and η-strongly monotone mapping with constants κ,η>0. Let 0<μ<2η/κ2 and 0<γ≤τ, where τ=1-1-μ(2η-μκ2). Consider the following THVI: find x*∈Ξ such that
(22)〈(μF-γV)x*,x-x*〉≥0,∀x∈Ξ,
which Ξ denotes the solution set of the following hierarchical variational inequality (HVI): find z*∈Fix⁡(T) such that
(23)〈(μF-γS)z*,z-z*〉≥0,∀z∈Fix⁡(T),
where the solution set Ξ is assumed to be nonempty.The authors [36] proposed both implicit and explicit iterative methods and studied the convergence analysis of the sequences generated by the proposed methods. In this paper, we introduce and study the following triple hierarchical variational inequality (THVI) with constraints of mixed equilibria, variational inequalities, and convex minimization problem.Problem 2.
LetM,N be two integers. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θk be a bifunction from C×C to R and let φk:C→R∪{+∞} be a proper lower semicontinuous and convex function, where k∈{1,2,…,M}. Let Bk,Ai:H→H and Fj:C→H be inverse-strongly monotone mappings, where k∈{1,2,…,M}, i∈{1,2,…,N}, and j∈{1,2}. Let S:H→H be a nonexpansive mapping and let {Sn}n=1∞ be a sequence of nonexpansive mappings on H. Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with a constant l≥0. Let 0<μ<2η/κ2, 0<γ≤τ, and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Consider the following triple hierarchical variational inequality (THVI): find x*∈Ξ such that
(24)〈(μF-γV)x*,x-x*〉≥0,∀x∈Ξ,
where Ξ denotes the solution set of the following hierarchical variational inequality (HVI): find z*∈Ω≔(∩n=1∞Fix⁡(Sn))∩(∩k=1MGMEP⁡(Θk,φk,Bk))∩(∩i=1NVI⁡(C,Ai))∩GSVI(G)∩Γ such that
(25)〈(μF-γS)z*,z-z*〉≥0,∀z∈Ω,
where the solution set Ξ is assumed to be nonempty.Motivated and inspired by the above facts, we introduce and analyze a hybrid iterative algorithm by the virtue of Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the averaged mapping approach to the GPA. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common elementx*∈Ω≔(∩n=1∞Fix⁡(Sn))∩(∩k=1MGMEP⁡(Θk,φk,Bk))∩(∩i=1NVI⁡(C,Ai))∩GSVI(G)∩Γ of the fixed point set of infinitely many nonexpansive mappings {Sn}n=1∞, the solution set of finitely many GMEPs, the solution set of finitely many VIPs, the solution set of GSVI (11), and the set of minimizers of CMP (14), which is just a unique solution of the THVI (24). In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI (11), and CMP (14). That is, under very mild conditions, it is proven that the proposed algorithm converges strongly to a unique solution x*∈Ω of the VIP: 〈(γV-μF)x*,x-x*〉≤0, for all x∈Ω; equivalently, PΩ(I-(μF-γV))x*=x*. The results obtained in this paper improve and extend the corresponding results announced by many others.


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let C be a nonempty closed convex subset of H. We write xn⇀x to indicate that the sequence {xn} converges weakly to x and xn→x to indicate that the sequence {xn} converges strongly to x. Moreover, we use ωw(xn) to denote the weak ω-limit set of the sequence {xn}; that is,
(26)ωw(xn)≔{forsomesubsequence{xni}of{xn}}x∈H:xni⇀xforsomesubsequence{xni}of{xn}}.Recall that a mappingA:C→H is called(i)
monotone if(27)〈Ax-Ay,x-y〉≥0,∀x,y∈C,(ii)
η-strongly monotone if there exists a constant η>0 such that
(28)〈Ax-Ay,x-y〉≥η∥x-y∥2,∀x,y∈C,(iii)
α-inverse-strongly monotone if there exists a constant α>0 such that
(29)〈Ax-Ay,x-y〉≥α∥Ax-Ay∥2,∀x,y∈C.It is obvious that ifA is α-inverse-strongly monotone, then A is monotone and (1/α)-Lipschitz continuous. Moreover, we also have that, for all u,v∈C and λ>0,
(30)∥(I-λA)u-(I-λA)v∥2=∥(u-v)-λ(Au-Av)∥2=∥u-v∥2-2λ〈Au-Av,u-v〉+λ2∥Au-Av∥2≤∥u-v∥2+λ(λ-2α)∥Au-Av∥2.
So, if λ≤2α, then I-λA is a nonexpansive mapping from C to H.The metric projection fromH onto C is the mapping PC:H→C which assigns to each point x∈H the unique point PCx∈C satisfying the property
(31)∥x-PCx∥=inf⁡y∈C∥x-y∥≕d(x,C).Some important properties of projections are gathered in the following proposition.Proposition 1.
For givenx∈H and z∈C:(i)
z=PCx⇔〈x-z,y-z〉≤0, for ally∈C;(ii)
z=PCx⇔∥x-z∥2≤∥x-y∥2-∥y-z∥2, for ally∈C;(iii)
〈PCx-PCy,x-y〉≥∥PCx-PCy∥2, for ally∈H.Consequently, PC is nonexpansive and monotone.Definition 2.
A mappingT:H→H is said to be(a)
nonexpansive if(32)∥Tx-Ty∥≤∥x-y∥,∀x,y∈H,(b)
firmly nonexpansive if2T-I is nonexpansive or equivalently if T is 1-inverse-strongly monotone (1-ism), then
(33)〈x-y,Tx-Ty〉≥∥Tx-Ty∥2,∀x,y∈H;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(34)T=12(I+S),
where S:H→H is nonexpansive; projections are firmly nonexpansive.It can be easily seen that ifT is nonexpansive, then I-T is monotone and a projection PC is 1-ism. The inverse-strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.Definition 3.
A mappingT:H→H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping; that is,
(35)T≡(1-α)I+αS,
where α∈(0,1) and S:H→H is nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are (1/2)-averaged mappings.Proposition 4 (see [37]).
LetT:H→H be a given mapping.(i)
Tis nonexpansive if and only if the complementI-Tis(1/2)-ism.(ii)
IfTisν-ism, then forγ>0,γTis(ν/γ)-ism.(iii)
Tis averaged if and only if the complementI-Tisν-ism for someν>1/2. Indeed, forα∈(0,1),Tisα-averaged if and only ifI-Tis(1/2α)-ism.Proposition 5 (see [37, 38]).
LetS,T,V:H→H be given operators.(i)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis averaged andVis nonexpansive, thenTis averaged.(ii)
Tis firmly nonexpansive if and only if the complementI-Tis firmly nonexpansive.(iii)
IfT=(1-α)S+αVfor someα∈(0,1)and ifSis firmly nonexpansive andVis nonexpansive, thenTis averaged.(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings{Ti}i=1Nis averaged, then so is the compositeT1⋯TN.In particular, ifT1isα1-averaged andT2isα2-averaged, whereα1,α2∈(0,1), then the compositeT1T2isα-averaged, whereα=α1+α2-α1α2.(v)
If the mappings{Ti}i=1Nare averaged and have a common fixed point, then(36)⋂i=1NFix⁡(Ti)=Fix⁡(T1⋯TN).The notation Fix⁡(T) denotes the set of all fixed points of the mapping T; that is, Fix⁡(T)={x∈H:Tx=x}.Next we list some elementary conclusions for the MEP.Proposition 6 (see [25]).
Assume thatΘ:C×C→R satisfies(A1)–(A4) and let φ:C→R be a proper lower semicontinuous and convex function. Assume that either(B1) or(B2) holds. For r>0 and x∈H, define a mapping Tr(Θ,φ):H→C as follows:
(37)Tr(Θ,φ)(x)={+1r〈y-z,z-x〉≥0,∀y∈C}z∈C:Θ(z,y)+φ(y)-φ(z)+1r〈y-z,z-x〉≥0,∀y∈C}
for all x∈H. Then the following hold:(i)
for eachx∈H,Tr(Θ,φ)(x)is nonempty and single-valued;(ii)
Tr(Θ,φ)is firmly nonexpansive; that is, for anyx,y∈H,(38)∥Tr(Θ,φ)x-Tr(Θ,φ)y∥2≤〈Tr(Θ,φ)x-Tr(Θ,φ)y,x-y〉;(iii)
Fix⁡(Tr(Θ,φ))=MEP⁡(Θ,φ);(iv)
MEP⁡(Θ,φ)is closed and convex;(v)
∥Ts(Θ,φ)x-Tt(Θ,φ)x∥2≤((s-t)/s)〈Ts(Θ,φ)x-Tt(Θ,φ)x,Ts(Θ,φ)x-x〉for alls,t>0andx∈H.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 7.
LetX be a real inner product space. Then there holds the following inequality:
(39)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈X.Lemma 8.
LetA:C→H be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection(see Proposition 1(i)) implies that
(40)u∈VI(C,A)⟺u=PC(u-λAu),λ>0.Lemma 9 (see [39, Demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then I-T is demiclosed. That is, whenever {xn} is a sequence in C weakly converging to some x∈C and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y. Here I is the identity operator of H.Let{Sn}n=1∞ be an infinite family of nonexpansive mappings on H and let {λn}n=1∞ be a sequence of nonnegative numbers in [0,1]. For any n≥1, define a mapping Wn on H as follows:
(41)Un,n+1=I,Un,n=λnSnUn,n+1+(1-λn)I,Un,n-1=λn-1Sn-1Un,n+(1-λn-1)I,⋮Un,k=λkSkUn,k+1+(1-λk)I,Un,k-1=λk-1Sk-1Un,k+(1-λk-1)I,⋮Un,2=λ2S2Un,3+(1-λ2)I,Wn=Un,1=λ1S1Un,2+(1-λ1)I.
Such a mapping Wn is called theW-mapping generated by Sn,Sn-1,…,S1 and λn,λn-1,…,λ1.Lemma 10 (see [40, Lemma 3.2]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Sn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix⁡(Sn)≠∅ and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, for every x∈C and k≥1, the limit lim⁡n→∞Un,kx exists where Un,k is defined as in (41).Remark 11 (see [41, Remark 3.1]).
It can be known from Lemma10 that if D is a nonempty bounded subset of C, then for ϵ>0 there exists n0≥k such that for all n>n0(42)sup⁡x∈D∥Un,kx-Ukx∥≤ϵ.Remark 12 (see [41, Remark 3.2]).
Utilizing Lemma10, we define a mapping W:C→C as follows:
(43)Wx=lim⁡n→∞Wnx=lim⁡n→∞Un,1x,∀x∈C.
Such a W is called theW-mapping generated by S1,S2,…  and λ1,λ2,…. Since Wn is nonexpansive, W:C→C is also nonexpansive. If {xn} is a bounded sequence in C, then it is clear from Remark 11 that
(44)lim⁡n→∞∥Wnxn-Wxn∥=0.Lemma 13 (see [40, Lemma 3.3]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {Sn}n=1∞ be a sequence of nonexpansive self-mappings on C such that ∩n=1∞Fix⁡(Sn)≠∅, and let {λn} be a sequence in (0,b] for some b∈(0,1). Then, Fix⁡(W)=∩n=1∞Fix⁡(Sn).The following lemma can be easily proven, and therefore, we omit the proof.Lemma 14.
LetV:H→H be an l-Lipschitzian mapping with constant l≥0 and let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Then for 0≤γl<μη,
(45)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)∥x-y∥2,∀x,y∈H.
That is, μF-γV is strongly monotone with the constant μη-γl>0.
LetC be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in (0,1] and let μ>0. Associating with a nonexpansive mapping T:C→H, we define the mapping Tλ:C→H by
(46)Tλx≔Tx-λμF(Tx),∀x∈C,
where F:H→H is an operator such that, for some positive constants κ,η>0, F is κ-Lipschitzian and η-strongly monotone on H; that is, F satisfies the following conditions:
(47)∥Fx-Fy∥≤κ∥x-y∥,〈Fx-Fy,x-y〉≥η∥x-y∥2
for all x,y∈H.Lemma 15 (see [42, Lemma 3.1]).
T
λ is a contraction provided that 0<μ<2η/κ2; that is,
(48)∥Tλx-Tλy∥≤(1-λτ)∥x-y∥,∀x,y∈C,
where τ=1-1-μ(2η-μκ2)∈(0,1].Lemma 16 (see [42]).
Let{sn} be a sequence of nonnegative numbers satisfying the conditions
(49)sn+1≤(1-αn)sn+αnβn,∀n≥1,
where {αn} and {βn} are sequences of real numbers such that(i)
{αn}⊂[0,1] and ∑n=1∞‍αn=∞ or equivalently
(50)∏n=1∞(1-αn)≔lim⁡n→∞∏k=1n(1-αk)=0;(ii)
lim⁡⁡sup⁡n→∞βn≤0, or ∑n=1∞‍|αnβn|<∞.Then lim⁡n→∞sn=0.Lemma 17 (see [39]).
LetH be a real Hilbert space. Then the following hold:(a)
∥x-y∥2=∥x∥2-∥y∥2-2〈x-y,y〉 for all x,y∈H;(b)
∥λx+μy∥2=λ∥x∥2+μ∥y∥2-λμ∥x-y∥2 for all x,y∈H and λ,μ∈[0,1] with λ+μ=1;(c)
if{xn} is a sequence in H such that xn⇀x, it follows that
(51)lim⁡⁡sup⁡⁡n→∞∥xn-y∥2=lim⁡⁡sup⁡⁡n→∞∥xn-x∥2+∥x-y∥2,lim⁡⁡sup⁡⁡n→∞∥xn-y∥2=lim⁡⁡sup⁡⁡n→∞∥xn-x∥2∀y∈H.
A set-valued mappingT~:H→2H is called monotone if for all x,y∈H, f∈T~x and g∈T~y imply that 〈x-y,f-g〉≥0. A monotone set-valued mapping T~:H→2H is called maximal if its graph Gph(T~) is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping T~:H→2H is maximal if and only if for (x,f)∈H×H, 〈x-y,f-g〉≥0 for every (y,g)∈Gph(T~) implies that f∈T~x. Let A:C→H be a monotone and Lipschitz continuous mapping and let NCv be the normal cone to C at v∈C, that is,
(52)NCv={w∈H:〈v-u,w〉≥0,∀u∈C}.
Define
(53)T~v={Av+NCv,ifv∈C,∅,ifv∉C.Lemma 18 (see [43]).
LetA:C→H be a monotone mapping. Then there hold the following statements:(i)
T~ is maximal monotone;(ii)
v∈T~-10⇔v∈VI⁡(C,A).


## 3. Strong Convergence Criteria for the THVI and HFPP

In this section, we will introduce and analyze an iterative algorithm for finding a solution of the THVI (24) with constraints of several problems: the finitely many GMEPs, the finitely many VIPs, GSVI (11), and CMP (14) in a real Hilbert space. This algorithm is based on the Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, and the averaged mapping approach to the GPA. We prove the strong convergence of the proposed algorithm to a unique solution of THVI (24) under suitable conditions. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with the same constraints.Theorem 19.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M,N be two integers. Let f:C→R be a convex functional with L-Lipschitz continuous gradient ∇f. Let Θk be a bifunction from C×C to R satisfying(A1)–(A4) and let φk:C→R∪{+∞} be a proper lower semicontinuous and convex function, where k∈{1,2,…,M}. Let Bk,Ai:H→H and Fj:C→H be μk-inverse-strongly monotone, ηi-inverse-strongly monotone, and ζj-inverse-strongly monotone, respectively, where k∈{1,2,…,M}, i∈{1,2,…,N}, and j∈{1,2}. Let S:H→H be a nonexpansive mapping, let {Sn}n=1∞ be a sequence of nonexpansive mappings on H, and let {λn} be a sequence in (0,b] for some b∈(0,1). Let F:H→H be a κ-Lipschitzian and η-strongly monotone operator with positive constants κ,η>0. Let V:H→H be an l-Lipschitzian mapping with the constant l≥0. Let 0<μ<2η/κ2, 0<γ≤τ, and 0≤γl<τ, where τ=1-1-μ(2η-μκ2). Assume that either(B1) or(B2) holds. For arbitrarily given x1∈H, let {xn} be a sequence generated by
(54)un=TrM,n(ΘM,φM)(I-rM,nBM)TrM-1,n(ΘM-1,φM-1)×(I-rM-1,nBM-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn,vn=PC(I-λN,nAN)×PC(I-λN-1,nAN-1)⋯PC(I-λ2,nA2)×PC(I-λ1,nA1)un,yn=αnγSvn+(I-αnμF)WnGvn,xn+1=snγVxn+(I-snμF)Tnyn,∀n≥1,
where PC(I-θn∇f)=snI+(1-sn)Tn(here Tn is nonexpansive, sn=(2-θnL)/4∈(0,1/2) for each θn∈(0,2/L)), {αn}⊂(0,1], νj∈(0,2ζj) for j=1,2, and Wn is theW-mapping defined by (41). Suppose that the following conditions are satisfied:(H1)
sn∈(0,1/2) for each θn∈(0,2/L), and lim⁡n→∞sn=0  (⇔lim⁡n→∞θn=2/L);(H2)
∑n=1∞‍sn=∞ and lim⁡n→∞(1/sn)|1-(αn-1/αn)|=0;(H3)
lim⁡n→∞(bn/snαn)=0, lim⁡n→∞(1/sn)|(1/αn)-(1/αn-1)|=0, and lim⁡n→∞(1/αn)|1-(sn-1/sn)|=0;(H4)
lim⁡n→∞(αn/sn)=0;(H5)
{λi,n}⊂[ai,bi]⊂(0,2ηi) and lim⁡n→∞(|λi,n-λi,n-1|/snαn)=0 for all i∈{1,2,…,N};(H6)
{rk,n}⊂[ek,fk]⊂(0,2μk) and lim⁡n→∞(|rk,n-rk,n-1|/snαn)=0 for all k∈{1,2,…,M}. Then there hold the following:(i)
lim⁡n→∞(∥xn+1-xn∥/αn)=0;(ii)
ωw(xn)⊂Ω provided lim⁡n→∞∥xn-yn∥=0;(iii)
ωw(xn)⊂Ξ provided ∥xn-yn∥=o(αn) additionally.Proof.
Since∇f is L-Lipschitzian, it follows that ∇f is 1/L-ism; see [44] (see also [31]). By Proposition 4(ii) we know that for θ>0, θ∇f is (1/θL)-ism. So by Proposition 4(iii) we deduce that I-θ∇f is (θL/2)-averaged. Now since the projection PC is (1/2)-averaged, it is easy to see from Proposition 5(iv) that the composite PC(I-θ∇f) is ((2+θL)/4)-averaged for θ∈(0,2/L). Hence we obtain that for each n≥1, PC(I-θn∇f) is ((2+θnL)/4)-averaged for each θn∈(0,2/L). Therefore, we can write
(55)PC(I-θn∇f)=2-θnL4I+2+θnL4Tn=snI+(1-sn)Tn,
where Tn is nonexpansive and sn≔sn(θn)=(2-θnL)/4∈(0,1/2) for each θn∈(0,2/L). It is clear that
(56)θn⟶2L⟺sn⟶0.
Put
(57)Δnk=Trk,n(Θk,φk)(I-rk,nBk)Trk-1,n(Θk-1,φk-1)×(I-rk-1,nBk-1)⋯Tr1,n(Θ1,φ1)(I-r1,nB1)xn
for all k∈{1,2,…,M} and n≥1,
(58)Λni=PC(I-λi,nBi)×PC(I-λi-1,nBi-1)⋯PC(I-λ1,nB1)
for all i∈{1,2,…,N}, Δn0=I and Λn0=I, where I is the identity mapping on H. Then we have un=ΔnMxn and vn=ΛnNun.
We divide the rest of the proof into several steps.
Step
1. Let us show that {xn} is bounded. Indeed, taking into account the assumption Ξ≠∅ in Problem 2, we know that Ω≠∅. By (H4), we may assume, without loss of generality, that αn≤sn for all n≥1. Taking p∈Ω arbitrarily. Then from (30) and Proposition 6(ii) we have
(59)∥un-p∥=∥TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1p∥≤∥(I-rM,nBM)ΔnM-1xn-(I-rM,nBM)ΔnM-1p∥≤∥ΔnM-1xn-ΔnM-1p∥⋮≤∥Δn0xn-Δn0p∥=∥xn-p∥.
Similarly, we have
(60)∥vn-p∥=∥PC(I-λN,nAN)ΛnN-1un-PC(I-λN,nAN)ΛnN-1p∥≤∥(I-λN,nAN)ΛnN-1un-(I-λN,nAN)ΛnN-1p∥≤∥ΛnN-1un-ΛnN-1p∥⋮≤∥Λn0un-Λn0p∥=∥un-p∥.
Combining (59) and (60), we have
(61)∥vn-p∥≤∥xn-p∥.
Since p=Gp=PC(I-ν1F1)PC(I-ν2F2)p,Fj is ζj-inverse-strongly monotone for j=1,2, and 0≤νj≤2ζj for j=1,2, we deduce that, for any n≥1,
(62)∥Gvn-p∥2=∥PC(I-ν1F1)PC(I-ν2F2)vn-PC(I-ν1F1)PC(I-ν2F2)p∥2≤∥(I-ν1F1)PC(I-ν2F2)vn-(I-ν1F1)PC(I-ν2F2)p∥2=∥[PC(I-ν2F2)vn-PC(I-ν2F2)p]-ν1[F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p]∥2≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2+ν1(ν1-2ζ1)∥F1PC(I-ν2F2)vn-F1PC(I-ν2F2)p∥2≤∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2≤∥(I-ν2F2)vn-(I-ν2F2)p∥2=∥(vn-p)-ν2(F2vn-F2p)∥2≤∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2≤∥vn-p∥2.
Utilizing Lemma 15 and the relation αn≤sn, from (54), (61), and (62), we obtain that
(63)∥yn-p∥=∥αnγ(Svn-Sp)+(I-αnμF)WnGvn-(I-αnμF)p+αn(γS-μF)p∥≤αnγ∥Svn-Sp∥+∥(I-αnμF)WnGvn-(I-αnμF)p∥+αn∥(γS-μF)p∥≤αnγ∥vn-p∥+(1-αnτ)∥Gvn-p∥+αn∥(γS-μF)p∥≤αnγ∥vn-p∥+(1-αnτ)∥vn-p∥+αn∥(γS-μF)p∥≤αnγ∥xn-p∥+(1-αnτ)∥xn-p∥+αn∥(γS-μF)p∥=(1-αn(τ-γ))∥xn-p∥+αn∥(γS-μF)p∥≤∥xn-p∥+αn∥(γS-μF)p∥≤∥xn-p∥+sn∥(γS-μF)p∥,
and hence
(64)∥xn+1-p∥=∥snγ(Vxn-Vp)+(I-snμF)Tnyn-(I-snμF)p+sn(γS-μF)p∥≤snγ∥Vxn-Vp∥+∥(I-snμF)Tnyn-(I-snμF)p∥+sn∥(γS-μF)p∥≤snγl∥xn-p∥+(1-snτ)∥yn-p∥+sn∥(γS-μF)p∥≤snγl∥xn-p∥+(1-snτ)(∥xn-p∥+sn∥(γS-μF)p∥)+sn∥(γS-μF)p∥≤snγl∥xn-p∥+(1-snτ)∥xn-p∥+sn∥(γS-μF)p∥+sn∥(γS-μF)p∥=(1-sn(τ-γl))∥xn-p∥+sn(∥(γS-μF)p∥+∥(γS-μF)p∥)=(1-sn(τ-γl))∥xn-p∥+sn(τ-γl)∥(γS-μF)p∥+∥(γS-μF)p∥τ-γl≤max⁡{∥xn-p∥,∥(γS-μF)p∥+∥(γS-μF)p∥τ-γl}.
By induction, we get
(65)∥xn-p∥≤max⁡⁡{∥x1-p∥,∥(γS-μF)p∥+∥(γS-μF)p∥τ-γl},DQGQQQOPKRRRRRRRRDARRRRBMMMMM∀n≥1.
Hence {xn} is bounded and so are the sequences {un},{vn},{yn}.
Step
2. Let us show that lim⁡n→∞(∥xn+1-xn∥/αn)=0.
Indeed, taking into account the(1/L)-inverse-strong monotonicity of ∇f, we know that PC(I-θn∇f) is nonexpansive for θn∈(0,2/L). Hence it follows that for any given p∈Ω,
(66)∥PC(I-θn+1∇f)yn∥≤∥PC(I-θn+1∇f)yn-p∥+∥p∥=∥PC(I-θn+1∇f)yn-PC(I-θn+1∇f)p∥+∥p∥≤∥yn-p∥+∥p∥≤∥yn∥+2∥p∥.
This together with the boundedness of {yn} implies that {PC(I-λn+1∇f)yn} is bounded. Also, observe that
(67)∥Tn+1yn-Tnyn∥=∥4PC(I-θn+1∇f)-(2-θn+1L)I2+θn+1Lyn-4PC(I-θn∇f)-(2-θnL)I2+θnLyn∥≤∥4PC(I-θn+1∇f)2+θn+1Lyn-4PC(I-θn∇f)2+θnLyn∥+∥2-θnL2+θnLyn-2-θn+1L2+θn+1Lyn∥=∥×((2+θn+1L)(2+θnL))-1∥(4(2+θnL)PC(I-θn+1∇f)yn-4(2+θn+1L)PC(I-θn∇f)Gyn)×((2+θn+1L)(2+θnL))-1∥+4L|θn+1-θn|(2+θn+1L)(2+θnL)∥yn∥=∥×((2+θn+1L)(2+θnL))-1∥((PC(I-θn+1∇f)yn-PC(I-θn∇f)yn))4L(θn-θn+1)PC(I-θn+1∇f)yn+4(2+θn+1L)×(PC(I-θn+1∇f)yn-PC(I-θn∇f)yn))×((2+θn+1L)(2+θnL))-1∥+4L|θn+1-θn|(2+θn+1L)(2+θnL)∥yn∥≤4L|θn-θn+1|∥PC(I-θn+1∇f)yn∥(2+θn+1L)(2+θnL)+(((2+θn+1L)(2+θnL))-1)(×∥PC(I-θn+1∇f)yn-PC(I-θn∇f)yn∥)4(2+θn+1L)×∥PC(I-θn+1∇f)yn-PC(I-θn∇f)yn∥)×((2+θn+1L)(2+θnL))-1)+4L|θn+1-θn|(2+θn+1L)(2+θnL)∥yn∥≤|θn+1-θn|×[L∥PC(I-θn+1∇f)yn∥+4∥∇f(yn)∥+L∥yn∥]≤M~|θn+1-θn|,
where sup⁡n≥1{L∥PC(I-θn+1∇f)yn∥+4∥∇f(yn)∥+L∥yn∥}≤M~ for some M~>0. So, by (67), we have that
(68)∥Tn+1yn+1-Tnyn∥≤∥Tn+1yn+1-Tn+1yn∥+∥Tn+1yn-Tnyn∥≤∥yn+1-yn∥+M~|θn+1-θn|≤∥yn+1-yn∥+4M~L|sn+1-sn|.
Note that
(69)∥vn+1-vn∥=∥Λn+1Nun+1-ΛnNun∥=∥PC(I-λN,n+1AN)Λn+1N-1un+1-PC(I-λN,nAN)ΛnN-1un∥≤∥PC(I-λN,n+1AN)Λn+1N-1un+1-PC(I-λN,nAN)Λn+1N-1un+1∥+∥PC(I-λN,nAN)Λn+1N-1un+1-PC(I-λN,nAN)ΛnN-1un∥≤∥(I-λN,n+1AN)Λn+1N-1un+1-(I-λN,nAN)Λn+1N-1un+1∥+∥(I-λN,nAN)Λn+1N-1un+1-(I-λN,nAN)ΛnN-1un∥≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+∥Λn+1N-1un+1-ΛnN-1un∥≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+|λN-1,n+1-λN-1,n|∥AN-1Λn+1N-2un+1∥+∥Λn+1N-2un+1-ΛnN-2un∥⋮≤|λN,n+1-λN,n|∥ANΛn+1N-1un+1∥+|λN-1,n+1-λN-1,n|∥AN-1Λn+1N-2un+1∥+⋯+|λ1,n+1-λ1,n|∥A1Λn+10un+1∥+∥Λn+10un+1-Λn0un∥≤M~0∑i=1N|λi,n+1-λi,n|+∥un+1-un∥,
where sup⁡n≥1{∑i=1N∥AiΛn+1i-1un+1∥}≤M~0 for some M~0>0. Also, utilizing Proposition 6(ii), (v) we deduce that
(70)∥un+1-un∥=∥Δn+1Mxn+1-ΔnMxn∥=∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥+∥TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)ΔnM-1xn∥≤∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥+∥TrM,n(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-TrM,n(ΘM,φM)(I-rM,nBM)Δn+1M-1xn+1∥+∥(I-rM,nBM)Δn+1M-1xn+1-(I-rM,nBM)ΔnM-1xn∥≤|rM,n+1-rM,n|rM,n+1×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-(I-rM,n+1BM)Δn+1M-1xn+1∥+|rM,n+1-rM,n|∥BMΔn+1M-1xn+1∥+∥Δn+1M-1xn+1-ΔnM-1xn∥=|rM,n+1-rM,n|×[∥BMΔn+1M-1xn+1∥+1rM,n+1×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-(I-rM,n+1BM)Δn+1M-1xn+1∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥[∥BMΔn+1M-1xn+1∥+1rM,n+1]+∥Δn+1M-1xn+1-ΔnM-1xn∥⋮≤|rM,n+1-rM,n|×[∥BMΔn+1M-1xn+1∥+1rM,n+1×∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1-(I-rM,n+1BM)Δn+1M-1xn+1∥TrM,n+1(ΘM,φM)(I-rM,n+1BM)Δn+1M-1xn+1∥[∥BMΔn+1M-1xn+1∥+1rM,n+1]+⋯+|r1,n+1-r1,n|×[∥B1Δn+10xn+1∥+1r1,n+1×∥Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1-(I-r1,n+1B1)Δn+10xn+1∥Tr1,n+1(Θ1,φ1)(I-r1,n+1B1)Δn+10xn+1∥[∥B1Δn+10xn+1∥+1r1,n+1]+∥Δn+10xn+1-Δn0xn∥≤M~1∑k=1M|rk,n+1-rk,n|+∥xn+1-xn∥,
where M~1>0 is a constant such that for each n≥1(71)∑k=1M[∥BkΔn+1k-1xn+1∥+1rk,n+1×∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1-(I-rk,n+1Bk)Δn+1k-1xn+1∥Trk,n+1(Θk,φk)(I-rk,n+1Bk)Δn+1k-1xn+1∥[∥BkΔn+1k-1xn+1∥+1rk,n+1]≤M~1.
Also, from (54) we have
(72)yn+1=αn+1γSvn+1+(I-αn+1μF)Wn+1Gvn+1,yn=αnγSvn+(I-αnμF)WnGvn,∀n≥1.
Simple calculation shows that
(73)yn+1-yn=αn+1γ(Svn+1-Svn)+(αn+1-αn)(γSvn-μFWnGvn)+(I-αn+1μF)Wn+1Gvn+1-(I-αn+1μF)WnGvn.
In the meantime, from (41), since Wn, Sn, and Un,i are all nonexpansive, we have
(74)∥Wn+1Gvn+1-WnGvn∥≤∥Wn+1Gvn+1-Wn+1Gvn∥+∥Wn+1Gvn-WnGvn∥≤∥vn+1-vn∥+∥Wn+1Gvn-WnGvn∥=∥vn+1-vn∥+∥λ1T1Un+1,2Gvn-λ1T1Un,2Gvn∥≤∥vn+1-vn∥+λ1∥Un+1,2Gvn-Un,2Gvn∥=∥vn+1-vn∥+λ1∥λ2T2Un+1,3Gvn-λ2T2Un,3Gvn∥≤∥vn+1-vn∥+λ1λ2∥Un+1,3Gvn-Un,3Gvn∥⋮≤∥vn+1-vn∥+λ1λ2⋯λn×∥Un+1,n+1Gvn-Un,n+1Gvn∥≤∥vn+1-vn∥+M~2∏i=1nλi,
where M~2 is a constant such that ∥Un+1,n+1Gvn∥+∥Un,n+1Gvn∥≤M~2 for each n≥1. Therefore, by utilizing Lemma 15, from (69)–(74) and {λn}⊂(0,b]⊂(0,1) it follows that
(75)∥yn+1-yn∥≤αn+1γ∥Svn+1-Svn∥+|αn+1-αn|∥γSvn-μFWnGvn∥+∥(I-αn+1μF)Wn+1Gvn+1-(I-αn+1μF)WnGvn∥≤αn+1γ∥vn+1-vn∥+|αn+1-αn|×∥γSvn-μFWnGvn∥+(1-αn+1τ)∥Wn+1Gvn+1-WnGvn∥≤αn+1γ∥vn+1-vn∥+|αn+1-αn|×∥γSvn-μFWnGvn∥+(1-αn+1τ)[∥vn+1-vn∥+M~2∏i=1n‍λi]≤(1-αn+1(τ-γ))∥vn+1-vn∥+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1n‍λi≤(1-αn+1(τ-γ))×[M~0∑i=1N|λi,n+1-λi,n|+∥un+1-un∥]+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi≤(1-αn+1(τ-γ))×[M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n|+∥xn+1-xn∥]+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi≤(1-αn+1(τ-γ))∥xn+1-xn∥+M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n|+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi.
On the other hand, from (54) we have
(76)xn+2=sn+1γVxn+1+(I-sn+1μF)Tn+1yn+1,xn+1=snγVxn+(I-snμF)Tnyn,∀n≥1.
The simple calculations show that
(77)xn+2-xn+1=(I-sn+1μF)Tn+1yn+1-(I-sn+1μF)Tnyn+(sn+1-sn)(γVxn-μFTnyn)+sn+1γ(Vxn+1-Vxn).
Utilizing Lemma 15 we deduce from (68), (75), and (77) that
(78)∥xn+2-xn+1∥≤∥(I-sn+1μF)Tn+1yn+1-(I-sn+1μF)Tnyn∥+|sn+1-sn|∥γVxn-μFTnyn∥+sn+1γ∥Vxn+1-Vxn∥≤(1-sn+1τ)∥Tn+1yn+1-Tnyn∥+|sn+1-sn|∥γVxn-μFTnyn∥+sn+1γl∥xn+1-xn∥≤(1-sn+1τ)[∥yn+1-yn∥+4M~L|sn+1-sn|]+|sn+1-sn|∥γVxn-μFTnyn∥+sn+1γl∥xn+1-xn∥≤(1-sn+1τ)×[M~2∏i=1nλi+4M~L|sn+1-sn|](1-αn+1(τ-γ))∥xn+1-xn∥+M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n|+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2∏i=1nλi+4M~L|sn+1-sn|]+|sn+1-sn|∥γVxn-μFTnyn∥+sn+1γl∥xn+1-xn∥≤(1-sn+1τ)∥xn+1-xn∥+M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n|+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2bn+(4M~L+∥γVxn-μFTnyn∥)|sn+1-sn|+sn+1γl∥xn+1-xn∥=(1-sn+1(τ-γl))∥xn+1-xn∥+M~0∑i=1N|λi,n+1-λi,n|+M~1∑k=1M|rk,n+1-rk,n|+|αn+1-αn|∥γSvn-μFWnGvn∥+M~2bn+(4M~L+∥γVxn-μFTnyn∥)|sn+1-sn|≤(1-sn+1(τ-γl))∥xn+1-xn∥+M~3(∑i=1N|λi,n+1-λi,n|+∑k=1M|rk,n+1-rk,n|+|αn+1-αn|+bn+|sn+1-sn|(∑i=1N|λi,n+1-λi,n|+∑k=1M|rk,n+1-rk,n|),
where M~3>0 is a constant such that for each n≥1(79)M~0+M~1+M~2+4M~L+∥γVxn-μFTnyn∥+∥γSvn-μFWnGvn∥≤M~3.
Therefore,
(80)∥xn+1-xn∥αn≤(1-sn(τ-γl))∥xn-xn-1∥αn+M~3(∑i=1N|λi,n-λi,n-1|αn+∑k=1M|rk,n-rk,n-1|αn+|αn-αn-1|αn+bn-1αn+|sn-sn-1|αn(∑i=1N|λi,n-λi,n-1|αn+∑k=1M|rk,n-rk,n-1|αn)=(1-sn(τ-γl))∥xn-xn-1∥αn-1+(1-sn(τ-γl))∥xn-xn-1∥(1αn-1αn-1)+M~3(∑i=1N‍|λi,n-λi,n-1|αn+∑k=1M‍|rk,n-rk,n-1|αn+|αn-αn-1|αn+bn-1αn+|sn-sn-1|αn(∑i=1N‍|λi,n-λi,n-1|αn+∑k=1M‍|rk,n-rk,n-1|αn)≤(1-(τ-γl)sn)∥xn-xn-1∥αn-1+(τ-γl)sn·1τ-γl×{|αn-αn-1|snαn+bn-1snαn+|sn-sn-1|snαn(∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn)}∥xn-xn-1∥1sn|1αn-1αn-1|+M~3(∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn+|αn-αn-1|snαn+bn-1snαn+|sn-sn-1|snαn(∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn)}≤(1-(τ-γl)sn)∥xn-xn-1∥αn-1+(τ-γl)sn·M~4τ-γl×{1sn|1-αn-1αn|+1αn|1-sn-1sn|+bn-1snαn}1sn|1αn-1αn-1|+∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn+1sn|1-αn-1αn|+1αn|1-sn-1sn|+bn-1snαn},
where M~3+∥xn-xn-1∥≤M~4, for all n>1 for some M~4>0. From (H2), (H3), (H5), and (H6), it follows that ∑n=1∞(τ-γl)sn=∞ and
(81)lim⁡n→∞M~4τ-γl{1sn|1αn-1αn-1|+∑i=1N|λi,n-λi,n-1|snαn+∑k=1M|rk,n-rk,n-1|snαn+1sn|1-αn-1αn|+1αn|1-sn-1sn|+bn-1snαn{1sn|1αn-1αn-1|+∑i=1N‍|λi,n-λi,n-1|snαn}=0.
Thus, applying Lemma 16 to (80), we immediately conclude that
(82)lim⁡n→∞∥xn+1-xn∥αn=0.
So, from lim⁡n→∞αn=0 it follows that
(83)lim⁡n→∞∥xn+1-xn∥=0.
Step
3. We prove that ωw(xn)⊂Ω provided lim⁡n→∞∥xn-yn∥=0.
Indeed, first of all, let us show that∥yn-PC(I-(2/L)∇f)yn∥→0, ∥xn-un∥→0, ∥xn-vn∥→0, ∥vn-Gvn∥→0, and ∥vn-Wvn∥→0 as n→∞. As a matter of fact, utilizing Lemmas 7 and 15 we obtain from (54), (61), and (62) that
(84)∥yn-p∥2=∥αnγ(Svn-Sp)+(I-αnμF)WnGvn-(I-αnμF)p+αn(γS-μF)p∥2≤∥αnγ(Svn-Sp)+(I-αnμF)WnGvn-(I-αnμF)p∥2+2αn〈(γS-μF)p,yn-p〉≤[αnγ∥Svn-Sp∥+∥(I-αnμF)WnGvn-(I-αnμF)p∥]2+2αn〈(γV-μF)p,yn-p〉≤[αnγ∥vn-p∥+(1-αnτ)∥Gvn-p∥]2+2αn〈(γS-μF)p,yn-p〉≤[αnτ∥vn-p∥+(1-αnτ)∥Gvn-p∥]2+2αn〈(γS-μF)p,yn-p〉≤αnτ∥vn-p∥2+(1-αnτ)∥Gvn-p∥2+2αn〈(γS-μF)p,yn-p〉≤αnτ∥vn-p∥2+(1-αnτ)∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉=∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉≤∥xn-p∥2+2αn〈(γS-μF)p,yn-p〉.
Note thatxn+1=snγVxn+(I-snμF)Tnyn. Hence we have
(85)xn+1-yn=sn(γVxn-μFTnyn)+Tnyn-yn,
which yields
(86)∥Tnyn-yn∥≤∥xn+1-yn-sn(γVxn-μFTnyn)∥≤∥xn+1-yn∥+sn∥γVxn-μFTnyn∥≤∥xn+1-xn∥+∥xn-yn∥+sn∥γVxn-μFTnyn∥.
Since lim⁡n→∞sn=0 and lim⁡n→∞∥xn-xn+1∥=0, from the assumption lim⁡n→∞∥xn-yn∥=0 and the boundedness of {xn}, {yn}, we obtain
(87)lim⁡n→∞∥yn-Tnyn∥=0.
It is clear that
(88)∥PC(I-θn∇f)yn-yn∥=∥snyn+(1-sn)Tnyn-yn∥=(1-sn)∥Tnyn-yn∥≤∥Tnyn-yn∥,
where sn=((2-θnL)/4)∈(0,1/2) for each θn∈(0,2/L). Hence we have
(89)∥PC(I-2L∇f)yn-yn∥≤∥PC(I-2L∇f)yn-PC(I-θn∇f)yn∥+∥PC(I-θn∇f)yn-yn∥≤∥(I-2L∇f)yn-(I-θn∇f)yn∥+∥PC(I-θn∇f)yn-yn∥≤(2L-θn)∥∇f(yn)∥+∥Tnyn-yn∥.
From the boundedness of {yn},sn→0  (⇔θn→2/L) and ∥Tnyn-yn∥→0 (due to (87)), it follows that
(90)lim⁡n→∞∥yn-PC(I-2L∇f)yn∥=0.
Also, from (30) it follows that for all i∈{1,2,…,N} and k∈{1,2,…,M}(91)∥vn-p∥2=∥ΛnNun-p∥2≤∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2≤∥Λni-1un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2≤∥xn-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2,∥un-p∥2=∥ΔnMxn-p∥2≤∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2≤∥Δnk-1xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2.
So, from (84) and (91) it follows that
(92)∥yn-p∥2≤∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉≤∥un-p∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+2αn〈(γS-μF)p,yn-p〉≤∥xn-p∥2+rk,n(rk,n-2μk)∥BkΔnk-1xn-Bkp∥2+λi,n(λi,n-2ηi)∥AiΛni-1un-Aip∥2+2αn∥(γS-μF)p∥∥yn-p∥,
which hence leads to
(93)rk,n(2μk-rk,n)∥BkΔnk-1xn-Bkp∥2+λi,n(2ηi-λi,n)∥AiΛni-1un-Aip∥2≤∥xn-p∥2-∥yn-p∥2+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2αn∥(γS-μF)p∥∥yn-p∥.
Since lim⁡n→∞αn=0,{λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk) for all i∈{1,2,…,N} and k∈{1,2,…,M}, by the assumption lim⁡n→∞∥xn-yn∥=0 and the boundedness of {xn}, {yn}, we conclude immediately that
(94)lim⁡n→∞∥AiΛni-1un-Aip∥=0,lim⁡n→∞∥BkΔnk-1xn-Bkp∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}.
Furthermore, by Proposition6(ii) we obtain that for each k∈{1,2,…,M}(95)∥Δnkxn-p∥2=∥Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn-Trk,n(Θk,φk)(I-rk,nBk)p∥2≤〈(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p,Δnkxn-p〉=12(∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p∥2+∥Δnkxn-p∥2-∥(I-rk,nBk)Δnk-1xn-(I-rk,nBk)p-(Δnkxn-p)∥2)≤12(∥Δnk-1xn-p∥2+∥Δnkxn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2),
which implies that
(96)∥Δnkxn-p∥2≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn-rk,n(BkΔnk-1xn-Bkp)∥2=∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2-rk,n2∥BkΔnk-1xn-Bkp∥2+2rk,n〈Δnk-1xn-Δnkxn,BkΔnk-1xn-Bkp〉≤∥Δnk-1xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥.
Also, by Proposition 1(iii), we obtain that for each i∈{1,2,…,N}(97)∥Λniun-p∥2=∥PC(I-λi,nAi)Λni-1un-PC(I-λi,nAi)p∥2≤〈(I-λi,nAi)Λni-1un-(I-λi,nAi)p,Λniun-p〉=12(∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2+∥Λniun-p∥2-∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p-(Λniun-p)∥2(∥(I-λi,nAi)Λni-1un-(I-λi,nAi)p∥2)≤12(∥Λni-1un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)≤12(-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2)∥un-p∥2+∥Λniun-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2),
which implies
(98)∥Λniun-p∥2≤∥un-p∥2-∥Λni-1un-Λniun-λi,n(AiΛni-1un-Aip)∥2=∥un-p∥2-∥Λni-1un-Λniun∥2-λi,n2∥AiΛni-1un-Aip∥2+2λi,n〈Λni-1un-Λniun,AiΛni-1un-Aip〉≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥.
Thus, from (84), (96), and (98), we have
(99)∥yn-p∥2≤∥vn-p∥2+2αn〈(γS-μF)p,yn-p〉≤∥Λniun-p∥2+2αn〈(γS-μF)p,yn-p〉≤∥un-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2αn〈(γS-μF)p,yn-p〉≤∥Δnkxn-p∥2-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2αn〈(γS-μF)p,yn-p〉≤∥xn-p∥2-∥Δnk-1xn-Δnkxn∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥-∥Λni-1un-Λniun∥2+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2αn∥(γS-μF)p∥∥yn-p∥,
which yields
(100)∥Δnk-1xn-Δnkxn∥2+∥Λni-1un-Λniun∥2≤∥xn-p∥2-∥yn-p∥2+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2rk,n∥Δnk-1xn-Δnkxn∥∥BkΔnk-1xn-Bkp∥+2λi,n∥Λni-1un-Λniun∥∥AiΛni-1un-Aip∥+2αn∥(γS-μF)p∥∥yn-p∥.
Since lim⁡n→∞αn=0, {xn}, {yn}, and {un} are bounded. For all i∈{1,2,…,N} and k∈{1,2,…,M}, we have {λi,n}⊂[ai,bi]⊂(0,2ηi) and {rk,n}⊂[ek,fk]⊂(0,2μk), then by (94) and the assumption lim⁡n→∞∥xn-yn∥=0, we conclude immediately that
(101)lim⁡n→∞∥Λni-1un-Λniun∥,lim⁡n→∞∥Δnk-1xn-Δnkxn∥=0,
for all i∈{1,2,…,N} and k∈{1,2,…,M}. Note that
(102)∥xn-un∥=∥Δn0xn-ΔnMxn∥≤∥Δn0xn-Δn1xn∥+∥Δn1xn-Δn2xn∥+⋯+∥ΔnM-1xn-ΔnMxn∥,∥un-vn∥=∥Λn0un-ΛnNun∥≤∥Λn0un-Λn1un∥+∥Λn1un-Λn2un∥+⋯+∥ΛnN-1un-ΛnNun∥.
Thus, from (101) we have
(103)lim⁡n→∞∥xn-un∥=0,lim⁡n→∞∥un-vn∥=0.
It is easy to see that as n→∞(104)∥xn-vn∥≤∥xn-un∥+∥un-vn∥⟶0.
On the other hand, for simplicity, we writep~=PC(I-ν2F2)p, v~n=PC(I-ν2F2)vn, and wn=Gvn=PC(I-ν1F1)v~n for all n≥1. Then
(105)p=Gp=PC(I-ν1F1)p~=PC(I-ν1F1)PC(I-ν2F2)p.
We now show thatlim⁡n→∞∥Gvn-vn∥=0; that is, lim⁡n→∞∥wn-vn∥=0. As a matter of fact, for p∈Ω, it follows from (61), (62), and (84) that
(106)∥yn-p∥2≤αnτ∥vn-p∥2+(1-αnτ)∥Gvn-p∥2+2αn〈(γS-μF)p,yn-p〉≤αnτ∥vn-p∥2+(1-αnτ)∥wn-p∥2+2αn∥(γS-μF)p∥∥yn-p∥≤αnτ∥vn-p∥2+(1-αnτ)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2αn∥(γS-μF)p∥∥yn-p∥≤αnτ∥vn-p∥2+(1-αnτ)×[∥vn-p∥2+ν2(ν2-2ζ2)∥F2vn-F2p∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2αn∥(γS-μF)p∥∥yn-p∥=∥vn-p∥2+(1-αnτ)×[ν2(ν2-2ζ2)∥F2vn-F2p∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-p∥2+(1-αnτ)×[ν2(ν2-2ζ2)∥F2vn-F2p∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2αn∥(γS-μF)p∥∥yn-p∥,
which immediately yields
(107)(1-αnτ)[ν2(2ζ2-ν2)∥F2vn-F2p∥2+ν1(2ζ1-ν1)∥F1v~n-F1p~∥2]≤∥xn-p∥2-∥yn-p∥2+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2αn∥(γS-μF)p∥∥yn-p∥.
Since lim⁡n→∞αn=0, νj∈(0,2ζj) for j=1,2 and {xn} and {yn} are bounded, by the assumption lim⁡n→∞∥xn-yn∥=0, we get
(108)lim⁡n→∞∥F2vn-F2p∥=0,lim⁡n→∞∥F1v~n-F1p~∥=0.
Also, in terms of the firm nonexpansivity of PC and the ζj-inverse-strong monotonicity of Fj for j=1,2, we obtain from νj∈(0,2ζj), j=1,2 and (67) that
(109)∥v~n-p~∥2=∥PC(I-ν2F2)vn-PC(I-ν2F2)p∥2≤〈(I-ν2F2)vn-(I-ν2F2)p,v~n-p~〉=12[∥(I-ν2F2)vn-(I-ν2F2)p∥2+∥v~n-p~∥2-∥(I-ν2F2)vn-(I-ν2F2)p-(v~n-p~)∥2]≤12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-ν2(F2vn-F2p)-(p-p~)∥2]=12[∥vn-p∥2+∥v~n-p~∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉-ν22∥F2vn-F2p∥2],∥wn-p∥2=∥PC(I-ν1F1)v~n-PC(I-ν1F1)p~∥2≤〈(I-ν1F1)v~n-(I-ν1F1)p~,wn-p〉=12[∥(I-ν1F1)v~n-(I-ν1F1)p~∥2+∥wn-p∥2-∥(I-ν1F1)v~n-(I-ν1F1)p~-(wn-p)∥2]≤12[∥v~n-p~∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈F1v~n-F1p~,(v~n-wn)+(p-p~)〉-ν12∥F1v~n-F1p~∥2]≤12[∥vn-p∥2+∥wn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1〈F1v~n-F1p~,(v~n-wn)+(p-p~)〉[∥vn-p∥2+∥wn-p∥2].
Thus, we have
(110)∥v~n-p~∥2≤∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉-ν22∥F2vn-F2p∥2,(111)∥wn-p∥2≤∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥.
Consequently, from (61), (106), and (110) it follows that
(112)∥yn-p∥2≤αnτ∥vn-p∥2+(1-αnτ)×[∥v~n-p~∥2+ν1(ν1-2ζ1)∥F1v~n-F1p~∥2]+2αn∥(γS-μF)p∥∥yn-p∥≤αnτ∥vn-p∥2+(1-αnτ)∥v~n-p~∥2+2αn∥(γS-μF)p∥∥yn-p∥≤αnτ∥vn-p∥2+(1-αnτ)×[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2〈(vn-v~n)-(p-p~),F2vn-F2p〉-ν22∥F2vn-F2p∥2]+2αn∥(γS-μF)p∥∥yn-p∥≤αnτ∥vn-p∥2+(1-αnτ)×[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥[∥vn-p∥2-∥(vn-v~n)-(p-p~)∥2]+2αn∥(γS-μF)p∥∥yn-p∥≤∥vn-p∥2-(1-αnτ)∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-p∥2-(1-αnτ)∥(vn-v~n)-(p-p~)∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2αn∥(γS-μF)p∥∥yn-p∥,
which hence leads to
(113)(1-αnτ)∥(vn-v~n)-(p-p~)∥2≤∥xn-p∥2-∥yn-p∥2+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2ν2∥(vn-v~n)-(p-p~)∥∥F2vn-F2p∥+2αn∥(γS-μF)p∥∥yn-p∥.
Since lim⁡n→∞αn=0, {xn}, {yn}, {vn}, and {v~n} are bounded sequences, by the assumption lim⁡n→∞∥xn-yn∥=0, we conclude from (108) that
(114)lim⁡n→∞∥(vn-v~n)-(p-p~)∥=0.
Furthermore, from (62), (106), and (111) it follows that
(115)∥yn-p∥2≤αnτ∥vn-p∥2+(1-αnτ)∥wn-p∥2+2αn∥(γS-μF)p∥∥yn-p∥≤αnτ∥vn-p∥2+(1-αnτ)×[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥[∥vn-p∥2-∥(v~n-wn)+(p-p~)∥2]+2αn∥(γS-μF)p∥∥yn-p∥≤∥vn-p∥2-(1-αnτ)∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-p∥2-(1-αnτ)∥(v~n-wn)+(p-p~)∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2αn∥(γS-μF)p∥∥yn-p∥,
which hence yields
(116)(1-αnτ)∥(v~n-wn)+(p-p~)∥2≤∥xn-p∥2-∥yn-p∥2+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2αn∥(γS-μF)p∥∥yn-p∥≤∥xn-yn∥(∥xn-p∥+∥yn-p∥)+2ν1∥F1v~n-F1p~∥∥(v~n-wn)+(p-p~)∥+2αn∥(γS-μF)p∥∥yn-p∥.
Since lim⁡n→∞αn=0 and {xn}, {yn}, {wn}, and {v~n} are bounded sequences, by the assumption lim⁡n→∞∥xn-yn∥=0, we conclude from (108) that
(117)lim⁡n→∞∥(v~n-wn)+(p-p~)∥=0.
Note that
(118)∥vn-wn∥≤∥(vn-v~n)-(p-p~)∥+∥(v~n-wn)+(p-p~)∥.
Hence from (114) and (117) we get
(119)lim⁡n→∞∥vn-Gvn∥=lim⁡n→∞∥vn-wn∥=0.
Also, observe that
(120)yn=αnγSvn+(I-αnμF)WnGvn.
Hence we get
(121)yn-WnGvn=αn(γSvn-μFWnGvn).
So, from lim⁡n→∞αn=0 and the boundedness of {vn} we deduce that
(122)lim⁡n→∞∥yn-WnGvn∥=0.
In addition, it is readily found that
(123)∥Wnvn-vn∥≤∥Wnvn-WnGvn∥+∥WnGvn-vn∥≤∥vn-Gvn∥+∥WnGvn-vn∥≤∥vn-Gvn∥+∥WnGvn-yn∥+∥yn-vn∥≤∥vn-Gvn∥+∥WnGvn-yn∥+∥yn-xn∥+∥xn-vn∥.
Thus, by the assumption lim⁡n→∞∥xn-yn∥=0, from (103) and (119)–(123) we have
(124)lim⁡n→∞∥Wnvn-vn∥=0.
Taking into account that ∥vn-Wvn∥≤∥vn-Wnvn∥+∥Wnvn-Wvn∥, we obtain from ∥vn-Wnvn∥→0 and Remark 12 that
(125)lim⁡n→∞∥vn-Wvn∥=0.
Next, let us show that ωw(xn)⊂Ω. In fact, since H is reflexive and {xn} is bounded, there exists at least a weak convergence subsequence of {xn}. Hence it is known that ωw(xn)≠∅. Now, take an arbitrary w∈ωw(xn). Then there exists a subsequence {xni} of {xn} such that xni⇀w. From (101) and (103) and the assumption lim⁡n→∞∥xn-yn∥=0, we have that yni⇀w, uni⇀w, vni⇀w, Δnikxni⇀w, and Λnimuni⇀w, where k∈{1,2,…,M} and m∈{1,2,…,N}. Utilizing Lemma 9, we deduce from xni⇀w, vni⇀w, (90), (119), and (125) that w∈Fix⁡(PC(I-(2/L)∇f))=VI⁡(C,∇f)=Γ, w∈GSVI(G), and w∈Fix⁡(W)=∩n=1∞Fix⁡(Sn) (due to Lemma 13). Thus, we get w∈∩n=1∞Fix⁡(Sn)∩GSVI(G)∩Γ. Next we prove that w∈∩m=1NVI⁡(C,Am). Let
(126)T~mv={Amv+NCv,v∈C,∅,v∉C,
where m∈{1,2,…,N}. Let (v,u)∈G(T~m). Since u-Amv∈NCv and Λnmun∈C, we have
(127)〈v-Λnmun,u-Amv〉≥0.
On the other hand, from Λnmun=PC(I-λm,nAm)Λnm-1un and v∈C, we have
(128)〈v-Λnmun,Λnmun-(Λnm-1un-λm,nAmΛnm-1un)〉≥0,
and hence
(129)〈v-Λnmun,Λnmun-Λnm-1unλm,n+AmΛnm-1un〉≥0.
Therefore we have
(130)〈v-Λnimuni,u〉≥〈v-Λnimuni,Amv〉≥〈v-Λnimuni,Amv〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni+AmΛnim-1uni〉=〈v-Λnimuni,Amv-AmΛnimuni〉+〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉≥〈v-Λnimuni,AmΛnimuni-AmΛnim-1uni〉-〈v-Λnimuni,Λnimuni-Λnim-1uniλm,ni〉.
From (101) and since Am is Lipschitz continuous, we obtain that lim⁡n→∞∥AmΛnmun-AmΛnm-1un∥=0. From Λnimuni⇀w, {λi,n}⊂[ai,bi]⊂(0,2ηi), for all i∈{1,2,…,N} and (101), we have
(131)〈v-w,u〉≥0.
Since T~m is maximal monotone, we have w∈T~m-10 and hence w∈VI⁡(C,Am), m=1,2,…,N, which implies w∈∩m=1NVI⁡(C,Am). Next we prove that w∈∩k=1MGMEP⁡(Θk,φk,Bk). Since Δnkxn=Trk,n(Θk,φk)(I-rk,nBk)Δnk-1xn, n≥1, k∈{1,2,…,M}, we have
(132)Θk(Δnkxn,y)+φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥0.
By (A2), we have
(133)φk(y)-φk(Δnkxn)+〈BkΔnk-1xn,y-Δnkxn〉+1rk,n〈y-Δnkxn,Δnkxn-Δnk-1xn〉≥Θk(y,Δnkxn).
Let zt=ty+(1-t)w for all t∈(0,1] and y∈C. This implies that zt∈C. Then, we have
(134)〈zt-Δnkxn,Bkzt〉≥φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt〉-〈zt-Δnkxn,BkΔnk-1xn〉-〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn)=φk(Δnkxn)-φk(zt)+〈zt-Δnkxn,Bkzt-BkΔnkxn〉+〈zt-Δnkxn,BkΔnkxn-BkΔnk-1xn〉-〈zt-Δnkxn,Δnkxn-Δnk-1xnrk,n〉+Θk(zt,Δnkxn).
By (101), we have ∥BkΔnkxn-BkΔnk-1xn∥→0 as n→∞. Furthermore, by the monotonicity of Bk, we obtain 〈zt-Δnkxn,Bkzt-BkΔnkxn〉≥0. Then, by (A4) we obtain
(135)〈zt-w,Bkzt〉≥φk(w)-φk(zt)+Θk(zt,w).
Utilizing (A1), (A4), and (135), we obtain
(136)0=Θk(zt,zt)+φk(zt)-φk(zt)≤tΘk(zt,y)+(1-t)Θk(zt,w)+tφk(y)+(1-t)φk(w)-φk(zt)≤t[Θk(zt,y)+φk(y)-φk(zt)]+(1-t)〈zt-w,Bkzt〉=t[Θk(zt,y)+φk(y)-φk(zt)]+(1-t)t〈y-w,Bkzt〉,
and hence
(137)0≤Θk(zt,y)+φk(y)-φk(zt)+(1-t)〈y-w,Bkzt〉.
Letting t→0, we have, for each y∈C,
(138)0≤Θk(w,y)+φk(y)-φk(w)+〈y-w,Bkw〉.
This implies that w∈GMEP⁡(Θk,φk,Bk) and hence w∈∩k=1MGMEP⁡(Θk,φk,Bk). Consequently, w∈(∩n=1∞Fix⁡(Sn))∩(∩k=1MGMEP⁡(Θk,φk,Bk))∩(∩i=1NVI⁡(C,Ai))∩GSVI(G)∩Γ≕Ω. (This shows that ωw(xn)⊂Ω.)
Step
4. We prove that ωw(xn)⊂Ξ provided that ∥xn-yn∥=o(αn) additionally.
Indeed, letw∈ωw(xn) be the same as mentioned in Step 3. Then we get xni⇀w. In addition, from (84) we have that for every p∈Ω(139)∥yn-p∥2≤∥xn-p∥2+2αn〈(γS-μF)p,yn-p〉,
which immediately implies that
(140)2〈(γS-μF)p,p-yn〉≤1αn(∥xn-p∥2-∥yn-p∥2)≤∥xn-yn∥αn(∥xn-p∥+∥yn-p∥).
This together with ∥xn-yn∥=o(αn) leads to
(141)lim⁡⁡sup⁡⁡n→∞〈(γS-μF)p,p-yn〉≤0.
Observe that
(142)lim⁡⁡sup⁡⁡n→∞〈(γS-μF)p,p-xn〉=lim⁡⁡sup⁡⁡n→∞(〈(γS-μF)p,yn-xn〉+〈(γS-μF)p,p-yn〉)=lim⁡⁡sup⁡⁡n→∞〈(γS-μF)p,p-yn〉≤0.
So, it follows from xni⇀w that
(143)〈(γS-μF)p,p-w〉≤0,∀p∈Ω.
Also, note that 0<γ≤τ and
(144)μη≥τ⟺μη≥1-1-μ(2η-μκ2)⟺1-μ(2η-μκ2)≥1-μη⟺1-2μη+μ2κ2≥1-2μη+μ2η2⟺κ2≥η2⟺κ≥η.
It is clear that
(145)〈(μF-γS)x-(μF-γS)y,x-y〉≥(μη-γ)∥x-y∥2,∀x,y∈H.
Hence, it follows from 0<γ≤τ≤μη that μF-γS is monotone. Since w∈ωw(xn)⊂Ω, by Minty’s lemma [39] we have
(146)〈(γS-μF)w,p-w〉≤0,∀p∈Ω;
that is, w∈Ξ. Therefore, ωw(xn)⊂Ξ. This completes the proof.Theorem 20.
Assume that there hold all the conditions in Theorem19. Then we have the following.(i)
{xn} converges strongly to a point x*∈Ω provided that lim⁡n→∞∥xn-yn∥=0, which is a unique solution of the VIP: 〈(γV-μF)x*,p-x*〉≤0, for all p∈Ω; equivalently,
(147)PΩ(I-(μF-γV))x*=x*;(ii)
{xn} converges strongly to a unique solution of THVI (24) provided that ∥xn-yn∥=o(αn) additionally.Proof .
Observe that(148)〈(μF-γV)x-(μF-γV)y,x-y〉≥(μη-γl)∥x-y∥2,∀x,y∈H.
Hence we know that μF-γV is (μη-γl)-strongly monotone with constant (μη-γl)>0. In the meantime, it is easy to see that μF-γV is μκ+γl-Lipschitzian with constant μκ+γl>0. Thus, there exists a unique solution x*∈Ω of the VIP
(149)〈(γV-μF)x*,p-x*〉≤0,∀p∈Ω.
Equivalently, x*=PΩ(I-(μF-γV))x*. Now, let us show that
(150)lim⁡⁡sup⁡⁡n→∞〈(γV-μF)x*,xn-x*〉≤0.
Since {xn} is bounded, we may assume, without loss of generality, that there exists a subsequence {xni} of {xn} such that xni⇀w and
(151)lim⁡⁡sup⁡⁡n→∞〈(γV-μF)x*,xn-x*〉=lim⁡i→∞〈(γV-μF)x*,xni-x*〉=〈(γV-μF)x*,w-x*〉.
In terms of Theorem 19(ii), we know that w∈ωw(xn)⊂Ω. So, from (149) it follows that
(152)lim⁡⁡sup⁡⁡n→∞〈(γV-μF)x*,xn-x*〉=〈(γV-μF)x*,w-x*〉≤0.
Next, let us show thatlim⁡n→∞∥xn-x*∥=0. In fact, put p=x* in (84). Then from (54) we get
(153)∥xn+1-x*∥2=∥snγ(Vxn-Vx*)+(I-snμF)Tnyn-(I-snμF)x*+sn(γV-μF)x*∥snγ(Vxn-Vx*)+(I-snμF)Tnyn∥2≤∥snγ(Vxn-Vx*)+(I-snμF)Tnyn-(I-snμF)x*)∥2+2sn〈(γV-μF)x*,xn+1-x*〉≤[+∥(I-snμF)Tnyn-(I-snμF)x*∥]snγ∥Vxn-Vx*∥+∥(I-snμF)Tnyn-(I-snμF)x*∥]2+2sn〈(γV-μF)x*,xn+1-x*〉≤[snγl∥xn-x*∥+(1-snτ)∥yn-x*∥]2+2sn〈(γV-μF)x*,xn+1-x*〉=[snτγlτ∥xn-x*∥+(1-snτ)∥yn-x*∥]2+2sn〈(γV-μF)x*,xn+1-x*〉≤snτ(γl)2τ2∥xn-x*∥2+(1-snτ)∥yn-x*∥2+2sn〈(γV-μF)x*,xn+1-x*〉≤sn(γl)2τ∥xn-x*∥2+(1-snτ)×[∥xn-x*∥2+2αn〈(γS-μF)x*,yn-x*〉]+2sn〈(γV-μF)x*,xn+1-x*〉=(1-snτ2-(γl)2τ)∥xn-x*∥2+2(1-snτ)αn〈(γS-μF)x*,yn-x*〉+2sn〈(γV-μF)x*,xn+1-x*〉=(1-snτ2-(γl)2τ)∥xn-x*∥2+snτ2-(γl)2τ·ττ2-(γl)2×[2(1-snτ)αnsn〈(γS-μF)x*,yn-x*〉+2〈(γV-μF)x*,xn+1-x*〉[2(1-snτ)αnsn〈(γS-μF)x*,yn-x*〉].
Since ∑n=1∞‍sn=∞, lim⁡n→∞(αn/sn)=0, and limsup⁡n→∞〈(γV-μF)x*,xn+1-x*〉≤0 (due to (152)), we deduce that ∑n=1∞‍sn((τ2-(γl)2)/τ)=∞ and
(154)lim⁡⁡sup⁡⁡n→∞ττ2-(γl)2×[2(1-snτ)αnsn〈(γS-μF)x*,yn-x*〉+2〈(γV-μF)x*,xn+1-x*〉[2(1-snτ)αnsn〈(γS-μF)x*,yn-x*〉]≤0.
Therefore, applying Lemma 16 to (153) we infer that lim⁡n→∞∥xn-x*∥=0.
On the other hand, let us suppose that∥xn-yn∥=o(αn). Then by Theorem 19(iii) we know that ωw(xn)⊂Ξ. Since μF-γV:H→H is (μκ+γl)-Lipschitzian and (μη-γl)-strongly monotone, there exists a unique solution x*∈Ξ of the VIP
(155)〈γVx*-μFx*,x-x*〉≤0,∀x∈Ξ.
Since the sequence {xn} is bounded, there exists a subsequence {xni} of {xn} such that
(156)lim⁡⁡sup⁡⁡n→∞〈(γV-μF)x*,xn-x*〉=lim⁡i→∞〈(γV-μF)x*,xni-x*〉.
Also, since H is reflexive and {xn} is bounded, without loss of generality we may assume that xni⇀x¯∈Ξ (due to Theorem 19(iii)). Taking into account that x* is the unique solution of the VIP (155), we deduce from (156) that
(157)lim⁡⁡sup⁡⁡n→∞〈(γV-μF)x*,xn+1-x*〉≤〈(γV-μF)x*,x¯-x*〉≤0.
Repeating the same argument as in (153) we immediately conclude that
(158)∥xn+1-x*∥2≤(1-snτ2-(γl)2τ)∥xn-x*∥2+snτ2-(γl)2τ·ττ2-(γl)2×[2(1-snτ)αnsn〈(γS-μF)x*,yn-x*〉+2〈(γV-μF)x*,xn+1-x*〉[2(1-snτ)αnsn〈(γS-μF)x*,yn-x*〉].
Repeating the same arguments as above, we can readily see that lim⁡n→∞∥xn-x*∥=0. This completes the proof.Remark 21.
It is obvious that our iterative algorithm (54) is very different from Ceng and Al-Homidan’s iterative one in [23, Theorem 21] and Yao et al.’s iterative one (21). Here, the two-step iterative scheme in [33, Theorem 3.2] and the three-step iterative scheme in [23, Theorem 21] are combined to develop our four-step iterative scheme (54) for the THVI (24). It is worth pointing out that under the lack of the assumptions similar to those in [33, Theorem 3.2], for example, {xn} is bounded, Fix⁡(T)∩int⁡C≠∅ and ∥x-Tx∥≥kDist(x,Fix⁡(T)), for all x∈C for some k>0, the sequence {xn} generated by (54) converges strongly to a point x*∈(∩n=1∞Fix⁡(Sn))∩(∩k=1MGMEP⁡(Θk,φk,Bk))∩(∩i=1NVI⁡(C,Ai))∩GSVI(G)∩Γ≕Ω, which is a unique solution of the VIP: 〈γVx*-μFx*,x-x*〉≤0, for all x∈Ω; equivalently, PΩ(I-(μF-γV))x*=x* (see Theorem 20(i)).Remark 22.
Our Theorems19 and 20 improve, extend, supplement, and develop Yao et al. [33, Theorems 3.1 and 3.2] and Ceng and Al-Homidan [23, Theorem 21] in the following aspects.(a)
Our THVI (24) with the unique solution x*∈Ξ satisfying
(159)x*=P(∩n=1∞Fix⁡(Sn))∩(∩k=1MGMEP⁡(Θk,φk,Bk))∩(∩i=1NVI⁡(C,Ai))∩GSVI(G)∩Γ×(I-(μF-γS))x*is more general than the problem of finding a pointx~∈C satisfying x~=PFix⁡(T)Sx~ in [33] and than the problem of finding a point x*∈(∩k=1MGMEP⁡(Θk,φk,Bk))∩(∩i=1NVI⁡(C,Ai))∩Γ in [23, Theorem 21].(b)
Our four-step iterative scheme (54) for THVI (24) is more flexible, more advantageous, and more subtle than Ceng and Al-Homidan’s three-step iterative one in [23, Theorem 21] and than Yao et al.’s two-step iterative one (21) because it can be used to solve several kinds of problems, for example, the THVI, the HFPP, and the problem of finding a common point of five sets: ∩n=1∞Fix⁡(Sn), ∩k=1MGMEP⁡(Θk,φk,Bk), ∩i=1NVI⁡(C,Ai), GSVI(G), and Γ. In addition, it also drops the crucial requirements that Fix⁡(T)∩int⁡C≠∅ and ∥x-Tx∥≥kDist(x,Fix⁡(T)), for all x∈C for some k>0 in [33, Theorem 3.2(v)].(c)
The argument techniques in our Theorems19 and 20 are very different from the argument ones in [33, Theorems 3.1 and 3.2] and from the argument ones in [23, Theorem 21] because we use the W-mapping approach to find the fixed points of infinitely many nonexpansive mappings {Sn}n=1∞ (see Lemmas 10 and 13), the properties of resolvent operators and maximal monotone mappings (see Proposition 6 and Lemma 18), the fixed point equation equivalent to the GSVI (11) (see Proposition CWY), and the contractive coefficient estimates for the contractions associating with nonexpansive mappings (see Lemma 15);(d)
Compared with the proof in [23, Theorem 21], our proof (see the arguments in Theorem 19) makes use of Minty’s Lemma [39] to derive ωw(xn)⊂Ξ because our Theorem 19 involves a quite complex problem, that is, the THVI (24). The THVI (24) involves the HFPP for the nonexpansive mapping S and infinitely many nonexpansive mappings {Sn}n=1∞, but the problem in [23, Theorem 21] involves no HFPP for nonexpansive mappings.



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*Source: 129379-2014-05-26.xml*