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

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.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑃_𝐶 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 𝑆𝑥−𝑆𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1) In particular, if 𝐿=1, then 𝑆 is called a nonexpansive mapping; if 𝐿∈[0,1), then 𝑆 is called a contraction.

Let 𝐴∶𝐶→𝐻 be a nonlinear mapping on 𝐶. We consider the following variational inequality problem (VIP): find a point 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (2) 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 𝐑^𝑛: 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑦_𝑛,∀𝑛≥0, (3) 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.

Let 𝜑∶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 Θ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)+𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (4) 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 Θ𝑥,𝑦+⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶. (5) It was introduced and studied by S. Takahashi and W. Takahashi [24]. The set of solutions of GEP is denoted by GEP(Θ,𝐴).

If 𝐴=0, then GMEP (4) is reduced to the mixed equilibrium problem (MEP), which is to find 𝑥∈𝐶 such that Θ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (6) It was considered and studied in [25]. The set of solutions of MEP is denoted by MEP(Θ,𝜑).

If 𝜑=0 and 𝐴=0, then GMEP (4) is reduced to the equilibrium problem (EP), which is to find 𝑥∈𝐶 such that Θ𝑥,𝑦≥0,∀𝑦∈𝐶. (7) 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, limsup_𝑡→0+Θ𝑡𝑧+(1−𝑡)𝑥,𝑦≤Θ𝑥,𝑦; (8)(A4)Θ(𝑥,⋅) is convex and lower semicontinuous for each 𝑥∈𝐶;(B1)for each 𝑥∈𝐻 and 𝑟>0, there exists a bounded subset 𝐷_𝑥⊂𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, Θ𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0; (9)(B2)𝐶 is a bounded set.

Given a positive number 𝑟>0. Let 𝑇_𝑟^(Θ,𝜑)∶𝐻→𝐶 be the solution set of the auxiliary mixed equilibrium problem; that is, for each 𝑥∈𝐻, 𝑇_𝑟^(Θ,𝜑)(𝑥)(𝑧)𝑦+1∶=𝑦∈𝐶∶Θ𝑦,𝑧+𝜑−𝜑𝑟⟨𝑦−𝑥,𝑧−𝑦⟩≥0,∀𝑧∈𝐶. (10)

Let 𝐹₁,𝐹₂∶𝐶→𝐻 be two mappings. Consider the following general system of variational inequalities (GSVI) of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜈₁𝐹₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜈≥0,∀𝑥∈𝐶,₂𝐹₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (11) 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 𝜈₁𝐴𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜈≥0,∀𝑥∈𝐶,₂𝐴𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (12) 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 given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of the GSVI (11) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by𝐺𝑥=𝑃_𝐶𝐼−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑥,∀𝑥∈𝐶, (13)where 𝑦=𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑥.

In particular, if the mapping 𝐹_𝑗∶𝐶→𝐻 is 𝜁_𝑗-inverse-strongly monotone for 𝑗=1,2, then the mapping 𝐺 is nonexpansive for all 𝜈_𝑗∈(0,2𝜁_𝑗], 𝑗=1,2. We denote by GSVI(𝐺) the fixed point set of the mapping 𝐺.

Let 𝑓∶𝐶→𝐑 be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing 𝑓 over the constraint set 𝐶minimize𝑓(𝑥)∶𝑥∈𝐶, (14) 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 {𝑥_𝑛} determined by the gradient ∇𝑓 and the metric projection 𝑃_𝐶: 𝑥_𝑛+1∶=𝑃_𝐶𝑥_𝑛−𝜆∇𝑓𝑥_𝑛,∀𝑛≥0 (15) or more generally 𝑥_𝑛+1∶=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛,∀𝑛≥0, (16) where, in both (15) and (16), the initial guess 𝑥₀ is taken from 𝐶 arbitrarily and the parameters 𝜆 or 𝜆_𝑛 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 𝑃_𝐶(𝐼−𝜆∇𝑓) is a contraction; hence, the sequence {𝑥_𝑛} defined by the GPA (15) converges in a norm to the unique solution of CMP (14). More generally, if {𝜆_𝑛} is chosen to satisfy the property 0<liminf_𝑛→∞𝜆_𝑛≤limsup_𝑛→∞𝜆_𝑛<2𝛼𝐿², (17) then the sequence {𝑥_𝑛} 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 {𝑥_𝑛} 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]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝐑 be a convex functional with 𝐿-Lipschitz continuous gradient ∇𝑓. Let 𝑀,𝑁 be two integers. Let Θ_𝑘 be a bifunction from 𝐶×𝐶 to 𝐑 satisfying (A1)–(A4) and let 𝜑_𝑘∶𝐶→𝐑∪{+∞} be a proper lower semicontinuous and convex function, where 𝑘∈{1,2,…,𝑀}. Let 𝐵_𝑘∶𝐻→𝐻 and 𝐴_𝑖∶𝐶→𝐻 be 𝜇_𝑘-inverse-strongly monotone and 𝜂_𝑖-inverse-strongly monotone, respectively, where 𝑘∈{1,2,…,𝑀} and 𝑖∈{1,2,…,𝑁}. 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 and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=∩^𝑀_𝑘=1𝐺𝑀𝐸𝑃(Θ_𝑘,𝜑_𝑘,𝐵_𝑘)∩∩^𝑁_𝑖=1𝑉𝐼(𝐶,𝐴_𝑖)∩Γ≠0/ and that either (B1) or (B2) holds. For arbitrarily given 𝑥₁∈𝐻, let {𝑥_𝑛} be a sequence generated by𝑢_𝑛=𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀𝑇^{(Θ𝑀−1,𝜑𝑀−1)}_{𝑟𝑀−1,𝑛}×𝐼−𝑟_{𝑀−1,𝑛}𝐵_𝑀−1⋯𝑇^(Θ1,𝜑1)_𝑟1,𝑛𝐼−𝑟_1,𝑛𝐵₁𝑥_𝑛,𝑣_𝑛=𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁𝑃_𝐶𝐼−𝜆_{𝑁−1,𝑛}𝐴_𝑁−1⋯×𝑃_𝐶𝐼−𝜆_2,𝑛𝐴₂𝑃_𝐶𝐼−𝜆_1,𝑛𝐴₁𝑢_𝑛,𝑥_𝑛+1=𝑠_𝑛𝛾𝑉𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑣_𝑛,∀𝑛≥1, (18)where 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛 (here 𝑇_𝑛 is nonexpansive and 𝑠_𝑛=(2−𝜆_𝑛𝐿)/4∈(0,1/2) for each 𝜆_𝑛∈(0,2/𝐿)). Assume that the following conditions hold:(i)𝑠_𝑛∈(0,1/2) for each 𝜆_𝑛∈(0,2/𝐿), and lim_𝑛→∞𝑠_𝑛=0 (⇔lim_𝑛→∞𝜆_𝑛=2/𝐿);(ii){𝛽_𝑛}⊂(0,1) and 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;(iii){𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖) and lim_𝑛→∞|𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛|=0 for all 𝑖∈{1,2,…,𝑁};(iv){𝑟_𝑘,𝑛}⊂[𝑒_𝑘,𝑓_𝑘]⊂(0,2𝜇_𝑘) and lim_𝑛→∞|𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛|=0 for all 𝑘∈{1,2,…,𝑀}.Then {𝑥_𝑛} converges strongly as 𝜆_𝑛→(2/𝐿)(⇔𝑠_𝑛→0) to a point 𝑥^∗∈Ω, which is a unique solution in Ω to the VIP:𝜇𝐹−𝛾𝑉𝑥^∗,𝑝−𝑥^∗≥0,∀𝑝∈Ω. (19)Equivalently, 𝑥^∗=𝑃_Ω(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 ⟨𝑥−𝑆𝑥,𝑥−𝑥⟩≤0,∀𝑥∈Fix(𝑇). (20) 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 𝑃_Fix(𝑇)𝑆; that is, find 𝑥∈𝐶 such that 𝑥=𝑃_Fix(𝑇)𝑆𝑥. The authors [33] introduced and analyzed the following iterative algorithm for solving HFPP (20): 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑉𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥0. (21)

Theorem YLM (see [33, Theorem 3.2]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆 and 𝑇 be two nonexpansive mappings of 𝐶 into itself. Let 𝑉∶𝐶→𝐶 be a fixed contraction with 𝛼∈(0,1). Let {𝛼_𝑛} and {𝛽_𝑛} be two sequences in (0,1). For any given 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated by (21). Assume that the sequence {𝑥_𝑛} is bounded and that(i)∑^∞_𝑛=0𝛼_𝑛=∞;(ii)lim_𝑛→∞(1/𝛼_𝑛)|(1/𝛽_𝑛)−(1/𝛽_𝑛−1)|=0, lim_𝑛→∞(1/𝛽_𝑛)|1−(𝛼_𝑛−1/𝛼_𝑛)|=0;(iii)lim_𝑛→∞𝛽_𝑛=0, lim_𝑛→∞(𝛼_𝑛/𝛽_𝑛)=0 and lim_𝑛→∞(𝛽²_𝑛/𝛼_𝑛)=0;(iv)Fix(𝑇)∩int𝐶≠0/;(v) there exists a constant 𝑘>0 such that ‖𝑥−𝑇𝑥‖≥𝑘Dist(𝑥,Fix(𝑇)) for each 𝑥∈𝐶, where Dist(𝑥,Fix(𝑇))=inf_{𝑦∈Fix(𝑇)}‖𝑥−𝑦‖. Then {𝑥_𝑛} converges strongly to 𝑥=𝑃_Λ𝑉𝑥 which solves the VIP ⟨𝑥−𝑆𝑥,𝑥−𝑥⟩≤0,  for all 𝑥∈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. Let 𝑆,𝑇∶𝐶→𝐶 be two nonexpansive mappings with Fix(𝑇)≠0/, 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 𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ, (22) which Ξ denotes the solution set of the following hierarchical variational inequality (HVI): find 𝑧^∗∈Fix(𝑇) such that 𝜇𝐹−𝛾𝑆𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Fix(𝑇), (23) 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. Let 𝑀,𝑁 be two integers. Let 𝑓∶𝐶→𝐑 be a convex functional with 𝐿-Lipschitz continuous gradient ∇𝑓. Let Θ_𝑘 be a bifunction from 𝐶×𝐶 to 𝐑 and let 𝜑_𝑘∶𝐶→𝐑∪{+∞} be a proper lower semicontinuous and convex function, where 𝑘∈{1,2,…,𝑀}. Let 𝐵_𝑘,𝐴_𝑖∶𝐻→𝐻 and 𝐹_𝑗∶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 {𝑆_𝑛}^∞_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 𝜇𝐹−𝛾𝑉𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Ξ, (24) where Ξ denotes the solution set of the following hierarchical variational inequality (HVI): find 𝑧^∗∈Ω∶=(∩^∞_𝑛=1Fix(𝑆_𝑛))∩(∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘))∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))∩GSVI(𝐺)∩Γ such that 𝜇𝐹−𝛾𝑆𝑧^∗,𝑧−𝑧^∗≥0,∀𝑧∈Ω, (25) 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 element 𝑥^∗∈Ω∶=(∩^∞_𝑛=1Fix(𝑆_𝑛))∩(∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘))∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))∩GSVI(𝐺)∩Γ of the fixed point set of infinitely many nonexpansive mappings {𝑆_𝑛}^∞_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, 𝑃_Ω(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 that 𝐻 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 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥. Moreover, we use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛}; that is, 𝜔_𝑤𝑥_𝑛∶=𝑥∈𝐻∶𝑥_𝑛𝑖𝑥⇀𝑥forsomesubsequence_𝑛𝑖of𝑥_𝑛. (26)

Recall that a mapping 𝐴∶𝐶→𝐻 is called(i)monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶, (27)(ii)𝜂-strongly monotone if there exists a constant 𝜂>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (28)(iii)𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (29)

It is obvious that if 𝐴 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, ‖(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. (30) So, if 𝜆≤2𝛼, then 𝐼−𝜆𝐴 is a nonexpansive mapping from 𝐶 to 𝐻.

The metric projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐻 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the property 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦=∶𝑑(𝑥,𝐶). (31)

Some important properties of projections are gathered in the following proposition.

Proposition 1. For given 𝑥∈𝐻 and 𝑧∈𝐶:(i)𝑧=𝑃_𝐶𝑥⇔⟨𝑥−𝑧,𝑦−𝑧⟩≤0, for all 𝑦∈𝐶;(ii)𝑧=𝑃_𝐶𝑥⇔‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖², for all 𝑦∈𝐶;(iii)⟨𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖², for all 𝑦∈𝐻.Consequently, 𝑃_𝐶 is nonexpansive and monotone.

Definition 2. A mapping 𝑇∶𝐻→𝐻 is said to be(a)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻, (32)(b)firmly nonexpansive if 2𝑇−𝐼 is nonexpansive or equivalently if 𝑇 is 1-inverse-strongly monotone (1-ism), then ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐻; (33)alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (34) where 𝑆∶𝐻→𝐻 is nonexpansive; projections are firmly nonexpansive.

It can be easily seen that if 𝑇 is nonexpansive, then 𝐼−𝑇 is monotone and a projection 𝑃_𝐶 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 mapping 𝑇∶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, 𝑇≡(1−𝛼)𝐼+𝛼𝑆, (35) 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]). Let 𝑇∶𝐻→𝐻 be a given mapping.(i)𝑇 is nonexpansive if and only if the complement 𝐼−𝑇 is (1/2)-ism.(ii) If 𝑇 is 𝜈-ism, then for 𝛾>0, 𝛾𝑇 is (𝜈/𝛾)-ism.(iii)𝑇 is averaged if and only if the complement 𝐼−𝑇 is 𝜈-ism for some 𝜈>1/2. Indeed, for 𝛼∈(0,1),𝑇 is 𝛼-averaged if and only if 𝐼−𝑇 is (1/2𝛼)-ism.

Proposition 5 (see [37, 38]). Let 𝑆,𝑇,𝑉∶𝐻→𝐻 be given operators.(i) If 𝑇=(1−𝛼)𝑆+𝛼𝑉 for some 𝛼∈(0,1) and if 𝑆 is averaged and 𝑉 is nonexpansive, then 𝑇 is averaged.(ii)𝑇 is firmly nonexpansive if and only if the complement 𝐼−𝑇 is firmly nonexpansive.(iii) If 𝑇=(1−𝛼)𝑆+𝛼𝑉 for some 𝛼∈(0,1) and if 𝑆 is firmly nonexpansive and 𝑉 is nonexpansive, then 𝑇 is averaged.(iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {𝑇_𝑖}^𝑁_𝑖=1 is averaged, then so is the composite 𝑇₁⋯𝑇_𝑁. In particular, if 𝑇₁ is 𝛼₁-averaged and 𝑇₂ is 𝛼₂-averaged, where 𝛼₁,𝛼₂∈(0,1), then the composite 𝑇₁𝑇₂ is 𝛼-averaged, where 𝛼=𝛼₁+𝛼₂−𝛼₁𝛼₂.(v) If the mappings {𝑇_𝑖}^𝑁_𝑖=1 are averaged and have a common fixed point, thenFix𝑇^𝑁_𝑖=1𝑖=Fix𝑇₁⋯𝑇_𝑁. (36)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 that Θ∶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 𝑇_𝑟^(Θ,𝜑)∶𝐻→𝐶 as follows: 𝑇_𝑟^(Θ,𝜑)(𝑥)=𝑦(𝑧)+1𝑧∈𝐶∶Θ𝑧,𝑦+𝜑−𝜑𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶 (37) for all 𝑥∈𝐻. Then the following hold:(i) for each 𝑥∈𝐻, 𝑇_𝑟^(Θ,𝜑)(𝑥) is nonempty and single-valued;(ii)𝑇_𝑟^(Θ,𝜑) is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐻,𝑇_𝑟^(Θ,𝜑)𝑥−𝑇_𝑟^(Θ,𝜑)𝑦²≤𝑇_𝑟^(Θ,𝜑)𝑥−𝑇_𝑟^(Θ,𝜑)𝑦,𝑥−𝑦; (38)(iii)Fix(𝑇_𝑟^(Θ,𝜑))=MEP(Θ,𝜑);(iv)MEP(Θ,𝜑) is closed and convex;(v)‖𝑇_𝑠^(Θ,𝜑)𝑥−𝑇_𝑡^(Θ,𝜑)𝑥‖²≤((𝑠−𝑡)/𝑠)⟨𝑇_𝑠^(Θ,𝜑)𝑥−𝑇_𝑡^(Θ,𝜑)𝑥, 𝑇_𝑠^(Θ,𝜑)𝑥−𝑥⟩ for all 𝑠,𝑡>0 and 𝑥∈𝐻.

We need some facts and tools in a real Hilbert space 𝐻 which are listed as lemmas below.

Lemma 7. Let 𝑋 be a real inner product space. Then there holds the following inequality: 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝑋. (39)

Lemma 8. Let 𝐴∶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 𝑢∈V𝐼(𝐶,𝐴)⟺𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢),𝜆>0. (40)

Lemma 9 (see [39, Demiclosedness principle]). Let 𝐶 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 {𝑥_𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑇)𝑥_𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑇)𝑥=𝑦. Here 𝐼 is the identity operator of 𝐻.

Let {𝑆_𝑛}^∞_𝑛=1 be an infinite family of nonexpansive mappings on 𝐻 and let {𝜆_𝑛}^∞_𝑛=1 be a sequence of nonnegative numbers in [0,1]. For any 𝑛≥1, define a mapping 𝑊_𝑛 on 𝐻 as follows: 𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜆_𝑛𝑆_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜆_𝑛−1𝑆_𝑛−1𝑈_𝑛,𝑛+1−𝜆_𝑛−1⋮𝑈𝐼,_𝑛,𝑘=𝜆_𝑘𝑆_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜆_𝑘−1𝑆_𝑘−1𝑈_𝑛,𝑘+1−𝜆_𝑘−1⋮𝑈𝐼,_𝑛,2=𝜆₂𝑆₂𝑈_𝑛,3+1−𝜆₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝜆₁𝑆₁𝑈_𝑛,2+1−𝜆₁𝐼. (41) Such a mapping 𝑊_𝑛 is called the W-mapping generated by 𝑆_𝑛,𝑆_𝑛−1,…,𝑆₁ and 𝜆_𝑛,𝜆_𝑛−1,…,𝜆₁.

Lemma 10 (see [40, Lemma 3.2]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑆_𝑛}^∞_𝑛=1 be a sequence of nonexpansive self-mappings on 𝐶 such that ∩^∞_𝑛=1Fix(𝑆_𝑛)≠0/ and let {𝜆_𝑛} be a sequence in (0,𝑏] for some 𝑏∈(0,1). Then, for every 𝑥∈𝐶 and 𝑘≥1, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists where 𝑈_𝑛,𝑘 is defined as in (41).

Remark 11 (see [41, Remark 3.1]). It can be known from Lemma 10 that if 𝐷 is a nonempty bounded subset of 𝐶, then for 𝜖>0 there exists 𝑛₀≥𝑘 such that for all 𝑛>𝑛₀sup_𝑥∈𝐷𝑈_𝑛,𝑘𝑥−𝑈_𝑘𝑥≤𝜖. (42)

Remark 12 (see [41, Remark 3.2]). Utilizing Lemma 10, we define a mapping 𝑊∶𝐶→𝐶 as follows: 𝑊𝑥=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,∀𝑥∈𝐶. (43) Such a 𝑊 is called the W-mapping generated by 𝑆₁,𝑆₂,…  and 𝜆₁,𝜆₂,…. Since 𝑊_𝑛 is nonexpansive, 𝑊∶𝐶→𝐶 is also nonexpansive. If {𝑥_𝑛} is a bounded sequence in 𝐶, then it is clear from Remark 11 that lim_𝑛→∞𝑊_𝑛𝑥_𝑛−𝑊𝑥_𝑛=0. (44)

Lemma 13 (see [40, Lemma 3.3]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑆_𝑛}^∞_𝑛=1 be a sequence of nonexpansive self-mappings on 𝐶 such that ∩^∞_𝑛=1Fix(𝑆_𝑛)≠0/, and let {𝜆_𝑛} be a sequence in (0,𝑏] for some 𝑏∈(0,1). Then, Fix(𝑊)=∩^∞_𝑛=1Fix(𝑆_𝑛).

The following lemma can be easily proven, and therefore, we omit the proof.

Lemma 14. Let 𝑉∶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, ≥𝜇𝐹−𝛾𝑉𝑥−𝜇𝐹−𝛾𝑉𝑦,𝑥−𝑦𝜇𝜂−𝛾𝑙𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (45) That is, 𝜇𝐹−𝛾𝑉 is strongly monotone with the constant 𝜇𝜂−𝛾𝑙>0.
Let 𝐶 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 𝑇^𝜆∶𝐶→𝐻 by 𝑇^𝜆𝑥∶=𝑇𝑥−𝜆𝜇𝐹(𝑇𝑥),∀𝑥∈𝐶, (46) 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: 𝐹𝑥−𝐹𝑦≤𝜅𝑥−𝑦,⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂𝑥−𝑦² (47) 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, 𝑇^𝜆𝑥−𝑇^𝜆𝑦≤(1−𝜆𝜏)𝑥−𝑦,∀𝑥,𝑦∈𝐶, (48) where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²)∈(0,1].

Lemma 16 (see [42]). Let {𝑠_𝑛} be a sequence of nonnegative numbers satisfying the conditions 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛽_𝑛,∀𝑛≥1, (49) where {𝛼_𝑛} and {𝛽_𝑛} are sequences of real numbers such that(i){𝛼_𝑛}⊂[0,1] and ∑^∞_𝑛=1𝛼_𝑛=∞ or equivalently ^∞_𝑛=11−𝛼_𝑛∶=lim^𝑛_{𝑛→∞𝑘=1}1−𝛼_𝑘=0; (50)(ii)limsup_𝑛→∞𝛽_𝑛≤0, or ∑^∞_𝑛=1|𝛼_𝑛𝛽_𝑛|<∞.Then lim_𝑛→∞𝑠_𝑛=0.

Lemma 17 (see [39]). Let 𝐻 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 {𝑥_𝑛} is a sequence in 𝐻 such that 𝑥_𝑛⇀𝑥, it follows that limsup_𝑛→∞𝑥_𝑛−𝑦²=limsup_𝑛→∞𝑥_𝑛−𝑥²+𝑥−𝑦²,∀𝑦∈𝐻. (51)
A set-valued mapping 𝑇∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻, 𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply that ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone set-valued mapping 𝑇∶𝐻→2^𝐻 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→2^𝐻 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∈𝑇𝑥. Let 𝐴∶𝐶→𝐻 be a monotone and Lipschitz continuous mapping and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐶}. (52) Define 𝑇𝑣=𝐴𝑣+𝑁_𝐶𝑣,𝑖𝑓𝑣∈𝐶,0/,𝑖𝑓𝑣∉𝐶. (53)

Lemma 18 (see [43]). Let 𝐴∶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. Let 𝐶 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 Θ_𝑘 be a bifunction from 𝐶×𝐶 to 𝐑 satisfying (A1)–(A4) and let 𝜑_𝑘∶𝐶→𝐑∪{+∞} be a proper lower semicontinuous and convex function, where 𝑘∈{1,2,…,𝑀}. Let 𝐵_𝑘,𝐴_𝑖∶𝐻→𝐻 and 𝐹_𝑗∶𝐶→𝐻 be 𝜇_𝑘-inverse-strongly monotone, 𝜂_𝑖-inverse-strongly monotone, and 𝜁_𝑗-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 {𝑆_𝑛}^∞_𝑛=1 be a sequence of nonexpansive mappings on 𝐻, and let {𝜆_𝑛} 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 {𝑥_𝑛} be a sequence generated by 𝑢_𝑛=𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀𝑇^{(Θ𝑀−1,𝜑𝑀−1)}_{𝑟𝑀−1,𝑛}×𝐼−𝑟_{𝑀−1,𝑛}𝐵_𝑀−1⋯𝑇^(Θ1,𝜑1)_𝑟1,𝑛𝐼−𝑟_1,𝑛𝐵₁𝑥_𝑛,𝑣_𝑛=𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁×𝑃_𝐶𝐼−𝜆_{𝑁−1,𝑛}𝐴_𝑁−1⋯𝑃_𝐶𝐼−𝜆_2,𝑛𝐴₂×𝑃_𝐶𝐼−𝜆_1,𝑛𝐴₁𝑢_𝑛,𝑦_𝑛=𝛼_𝑛𝛾𝑆𝑣_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛,𝑥_𝑛+1=𝑠_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑦_𝑛,∀𝑛≥1, (54) where 𝑃_𝐶(𝐼−𝜃_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛 (here 𝑇_𝑛 is nonexpansive, 𝑠_𝑛=(2−𝜃_𝑛𝐿)/4∈(0,1/2) for each 𝜃_𝑛∈(0,2/𝐿)), {𝛼_𝑛}⊂(0,1], 𝜈_𝑗∈(0,2𝜁_𝑗) for 𝑗=1,2, and 𝑊_𝑛 is the W-mapping defined by (41). Suppose that the following conditions are satisfied:(H1)𝑠_𝑛∈(0,1/2) for each 𝜃_𝑛∈(0,2/𝐿), and lim_𝑛→∞𝑠_𝑛=0  (⇔lim_𝑛→∞𝜃_𝑛=2/𝐿);(H2)∑^∞_𝑛=1𝑠_𝑛=∞ and lim_𝑛→∞(1/𝑠_𝑛)|1−(𝛼_𝑛−1/𝛼_𝑛)|=0;(H3)lim_𝑛→∞(𝑏^𝑛/𝑠_𝑛𝛼_𝑛)=0, lim_𝑛→∞(1/𝑠_𝑛)|(1/𝛼_𝑛)−(1/𝛼_𝑛−1)|=0, and lim_𝑛→∞(1/𝛼_𝑛)|1−(𝑠_𝑛−1/𝑠_𝑛)|=0;(H4)lim_𝑛→∞(𝛼_𝑛/𝑠_𝑛)=0;(H5){𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖) and lim_𝑛→∞(|𝜆_𝑖,𝑛−𝜆_{𝑖,𝑛−1}|/𝑠_𝑛𝛼_𝑛)=0 for all 𝑖∈{1,2,…,𝑁};(H6){𝑟_𝑘,𝑛}⊂[𝑒_𝑘,𝑓_𝑘]⊂(0,2𝜇_𝑘) and lim_𝑛→∞(|𝑟_𝑘,𝑛−𝑟_{𝑘,𝑛−1}|/𝑠_𝑛𝛼_𝑛)=0 for all 𝑘∈{1,2,…,𝑀}. Then there hold the following:(i)lim_𝑛→∞(‖𝑥_𝑛+1−𝑥_𝑛‖/𝛼_𝑛)=0;(ii)𝜔_𝑤(𝑥_𝑛)⊂Ω provided lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0;(iii)𝜔_𝑤(𝑥_𝑛)⊂Ξ provided ‖𝑥_𝑛−𝑦_𝑛‖=𝑜(𝛼_𝑛) 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 𝑃_𝐶 is (1/2)-averaged, it is easy to see from Proposition 5(iv) that the composite 𝑃_𝐶(𝐼−𝜃∇𝑓) is ((2+𝜃𝐿)/4)-averaged for 𝜃∈(0,2/𝐿). Hence we obtain that for each 𝑛≥1, 𝑃_𝐶(𝐼−𝜃_𝑛∇𝑓) is ((2+𝜃_𝑛𝐿)/4)-averaged for each 𝜃_𝑛∈(0,2/𝐿). Therefore, we can write 𝑃_𝐶𝐼−𝜃_𝑛∇𝑓=2−𝜃_𝑛𝐿4𝐼+2+𝜃_𝑛𝐿4𝑇_𝑛=𝑠_𝑛𝐼+1−𝑠_𝑛𝑇_𝑛, (55) where 𝑇_𝑛 is nonexpansive and 𝑠_𝑛∶=𝑠_𝑛(𝜃_𝑛)=(2−𝜃_𝑛𝐿)/4∈(0,1/2) for each 𝜃_𝑛∈(0,2/𝐿). It is clear that 𝜃_𝑛⟶2𝐿⟺𝑠_𝑛⟶0. (56) Put Δ^𝑘_𝑛=𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝐼−𝑟_𝑘,𝑛𝐵_𝑘𝑇^{(Θ𝑘−1,𝜑𝑘−1)}_{𝑟𝑘−1,𝑛}×𝐼−𝑟_{𝑘−1,𝑛}𝐵_𝑘−1⋯𝑇^(Θ1,𝜑1)_𝑟1,𝑛𝐼−𝑟_1,𝑛𝐵₁𝑥_𝑛 (57) for all 𝑘∈{1,2,…,𝑀} and 𝑛≥1, Λ^𝑖_𝑛=𝑃_𝐶𝐼−𝜆_𝑖,𝑛𝐵_𝑖×𝑃_𝐶𝐼−𝜆_{𝑖−1,𝑛}𝐵_𝑖−1⋯𝑃_𝐶𝐼−𝜆_1,𝑛𝐵₁ (58) for all 𝑖∈{1,2,…,𝑁}, Δ⁰_𝑛=𝐼 and Λ⁰_𝑛=𝐼, where 𝐼 is the identity mapping on 𝐻. Then we have 𝑢_𝑛=Δ^𝑀_𝑛𝑥_𝑛 and 𝑣_𝑛=Λ^𝑁_𝑛𝑢_𝑛.
We divide the rest of the proof into several steps.
Step 1. Let us show that {𝑥_𝑛} is bounded. Indeed, taking into account the assumption Ξ≠0/ in Problem 2, we know that Ω≠0/. By (H4), we may assume, without loss of generality, that 𝛼_𝑛≤𝑠_𝑛 for all 𝑛≥1. Taking 𝑝∈Ω arbitrarily. Then from (30) and Proposition 6(ii) we have 𝑢_𝑛=𝑇−𝑝_{(Θ𝑀,𝜑𝑀)𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑥_𝑛−𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑝≤𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑥_𝑛−𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑝≤Δ_𝑛^𝑀−1𝑥_𝑛−Δ_𝑛^𝑀−1𝑝⋮≤Δ⁰_𝑛𝑥_𝑛−Δ⁰_𝑛𝑝=𝑥_𝑛−𝑝. (59) Similarly, we have 𝑣_𝑛=𝑃−𝑝_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑝≤𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑢_𝑛−𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑝≤Λ_𝑛^𝑁−1𝑢_𝑛−Λ_𝑛^𝑁−1𝑝⋮≤Λ⁰_𝑛𝑢_𝑛−Λ⁰_𝑛𝑝=𝑢_𝑛−𝑝. (60) Combining (59) and (60), we have 𝑣_𝑛−𝑝≤𝑥_𝑛−𝑝. (61) Since 𝑝=𝐺𝑝=𝑃_𝐶(𝐼−𝜈₁𝐹₁)𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑝,𝐹_𝑗 is 𝜁_𝑗-inverse-strongly monotone for 𝑗=1,2, and 0≤𝜈_𝑗≤2𝜁_𝑗 for 𝑗=1,2, we deduce that, for any 𝑛≥1, 𝐺𝑣_𝑛−𝑝²=𝑃_𝐶𝐼−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑣_𝑛−𝑃_𝐶𝐼−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑝²≤𝐼−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑣_𝑛−𝐼−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑝²=𝑃_𝐶𝐼−𝜈₂𝐹₂𝑣_𝑛−𝑃_𝐶𝐼−𝜈₂𝐹₂𝑝−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑣_𝑛−𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑝²≤𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑣_𝑛−𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑝²+𝜈₁𝜈₁−2𝜁₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑣_𝑛−𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑝²≤𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑣_𝑛−𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑝²≤(𝐼−𝜈₂𝐹₂)𝑣_𝑛−(𝐼−𝜈₂𝐹₂)𝑝²=(𝑣_𝑛−𝑝)−𝜈₂(𝐹₂𝑣_𝑛−𝐹₂𝑝)²≤𝑣_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐹₂𝑣_𝑛−𝐹₂𝑝²≤𝑣_𝑛−𝑝². (62) Utilizing Lemma 15 and the relation 𝛼_𝑛≤𝑠_𝑛, from (54), (61), and (62), we obtain that 𝑦_𝑛=𝛼−𝑝_𝑛𝛾𝑆𝑣_𝑛−𝑆𝑝+𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑝+𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝛼_𝑛𝛾𝑆𝑣_𝑛+−𝑆𝑝𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑝+𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝛼_𝑛𝛾𝑣_𝑛−𝑝+1−𝛼_𝑛𝜏𝐺𝑣_𝑛−𝑝+𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝛼_𝑛𝛾𝑣_𝑛−𝑝+1−𝛼_𝑛𝜏𝑣_𝑛−𝑝+𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝛼_𝑛𝛾𝑥_𝑛−𝑝+1−𝛼_𝑛𝜏𝑥_𝑛−𝑝+𝛼_𝑛𝑝=𝛾𝑆−𝜇𝐹1−𝛼_𝑛𝜏−𝛾𝑥_𝑛−𝑝+𝛼_𝑛𝑝≤𝑥𝛾𝑆−𝜇𝐹_𝑛−𝑝+𝛼_𝑛𝑝≤𝑥𝛾𝑆−𝜇𝐹_𝑛−𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝, (63) and hence 𝑥_𝑛+1=𝑠−𝑝_𝑛𝛾𝑉𝑥_𝑛−𝑉𝑝+𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑦_𝑛−𝐼−𝑠_𝑛𝜇𝐹𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝑠_𝑛𝛾𝑉𝑥_𝑛+−𝑉𝑝𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑦_𝑛−𝐼−𝑠_𝑛𝜇𝐹𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝑠_𝑛𝛾𝑙𝑥_𝑛−𝑝+1−𝑠_𝑛𝜏𝑦_𝑛−𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝑠_𝑛𝛾𝑙𝑥_𝑛+−𝑝1−𝑠_𝑛𝜏𝑥_𝑛−𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝≤𝑠_𝑛𝛾𝑙𝑥_𝑛−𝑝+1−𝑠_𝑛𝜏𝑥_𝑛−𝑝+𝑠_𝑛𝛾𝑆−𝜇𝐹𝑝+𝑠_𝑛𝑝=𝛾𝑆−𝜇𝐹1−𝑠_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑝+𝑠_𝑛𝑝+𝑝=𝛾𝑆−𝜇𝐹𝛾𝑆−𝜇𝐹1−𝑠_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑝+𝑠_𝑛𝑝+𝑝𝜏−𝛾𝑙𝛾𝑆−𝜇𝐹𝛾𝑆−𝜇𝐹𝑥𝜏−𝛾𝑙≤max_𝑛,𝑝+𝑝−𝑝𝛾𝑆−𝜇𝐹𝛾𝑆−𝜇𝐹.𝜏−𝛾𝑙 (64) By induction, we get 𝑥_𝑛𝑥−𝑝≤max₁,𝑝+𝑝−𝑝𝛾𝑆−𝜇𝐹𝛾𝑆−𝜇𝐹,𝜏−𝛾𝑙∀𝑛≥1. (65) Hence {𝑥_𝑛} is bounded and so are the sequences {𝑢_𝑛},{𝑣_𝑛},{𝑦_𝑛}.
Step 2. Let us show that lim_𝑛→∞(‖𝑥_𝑛+1−𝑥_𝑛‖/𝛼_𝑛)=0.
Indeed, taking into account the (1/𝐿)-inverse-strong monotonicity of ∇𝑓, we know that 𝑃_𝐶(𝐼−𝜃_𝑛∇𝑓) is nonexpansive for 𝜃_𝑛∈(0,2/𝐿). Hence it follows that for any given 𝑝∈Ω, 𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛≤𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛+𝑝=𝑃−𝑝_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛−𝑃_𝐶𝐼−𝜃_𝑛+1𝑝+𝑝≤𝑦∇𝑓_𝑛+𝑝≤𝑦−𝑝_𝑛+2𝑝. (66) This together with the boundedness of {𝑦_𝑛} implies that {𝑃_𝐶(𝐼−𝜆_𝑛+1∇𝑓)𝑦_𝑛} is bounded. Also, observe that 𝑇_𝑛+1𝑦_𝑛−𝑇_𝑛𝑦_𝑛=4𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓−2−𝜃_𝑛+1𝐿𝐼2+𝜃_𝑛+1𝐿𝑦_𝑛−4𝑃_𝐶𝐼−𝜃_𝑛∇𝑓−2−𝜃_𝑛𝐿𝐼2+𝜃_𝑛𝐿𝑦_𝑛≤4𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓2+𝜃_𝑛+1𝐿𝑦_𝑛−4𝑃_𝐶𝐼−𝜃_𝑛∇𝑓2+𝜃_𝑛𝐿𝑦_𝑛+2−𝜃_𝑛𝐿2+𝜃_𝑛𝐿𝑦_𝑛−2−𝜃_𝑛+1𝐿2+𝜃_𝑛+1𝐿𝑦_𝑛=42+𝜃_𝑛𝐿𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛−42+𝜃_𝑛+1𝐿𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝐺𝑦_𝑛×2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿⁻¹+4𝐿𝜃_𝑛+1−𝜃_𝑛2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿𝑦_𝑛=4𝐿𝜃_𝑛−𝜃_𝑛+1𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛+42+𝜃_𝑛+1𝐿×𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛−𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝑦_𝑛×2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿⁻¹+4𝐿𝜃_𝑛+1−𝜃_𝑛2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿𝑦_𝑛≤4𝐿𝜃_𝑛−𝜃_𝑛+1𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿+42+𝜃_𝑛+1𝐿×𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛−𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝑦_𝑛×2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿⁻¹+4𝐿𝜃_𝑛+1−𝜃_𝑛2+𝜃_𝑛+1𝐿2+𝜃_𝑛𝐿𝑦_𝑛≤𝜃_𝑛+1−𝜃_𝑛×𝐿𝑃_𝐶𝐼−𝜃_𝑛+1∇𝑓𝑦_𝑛+4∇𝑓𝑦_𝑛+𝐿𝑦_𝑛≤𝑀𝜃_𝑛+1−𝜃_𝑛, (67) where sup_𝑛≥1{𝐿‖𝑃_𝐶(𝐼−𝜃_𝑛+1∇𝑓)𝑦_𝑛‖+4‖∇𝑓(𝑦_𝑛)‖+𝐿‖𝑦_𝑛‖}≤𝑀 for some 𝑀>0. So, by (67), we have that 𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛≤𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛+1𝑦_𝑛+𝑇_𝑛+1𝑦_𝑛−𝑇_𝑛𝑦_𝑛≤𝑦_𝑛+1−𝑦_𝑛+𝑀𝜃_𝑛+1−𝜃_𝑛≤𝑦_𝑛+1−𝑦_𝑛+4𝑀𝐿𝑠_𝑛+1−𝑠_𝑛. (68) Note that 𝑣_𝑛+1−𝑣_𝑛=Λ^𝑁_𝑛+1𝑢_𝑛+1−Λ^𝑁_𝑛𝑢_𝑛=𝑃_𝐶𝐼−𝜆_𝑁,𝑛+1𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1−𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑢_𝑛≤𝑃_𝐶𝐼−𝜆_𝑁,𝑛+1𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1−𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1+𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1−𝑃_𝐶𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑢_𝑛≤𝐼−𝜆_𝑁,𝑛+1𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1−𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1+𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1−𝐼−𝜆_𝑁,𝑛𝐴_𝑁Λ_𝑛^𝑁−1𝑢_𝑛≤𝜆_𝑁,𝑛+1−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1+Λ^𝑁−1_𝑛+1𝑢_𝑛+1−Λ_𝑛^𝑁−1𝑢_𝑛≤𝜆_𝑁,𝑛+1−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1+𝜆_{𝑁−1,𝑛+1}−𝜆_{𝑁−1,𝑛}𝐴_𝑁−1Λ^𝑁−2_𝑛+1𝑢_𝑛+1+Λ^𝑁−2_𝑛+1𝑢_𝑛+1−Λ_𝑛^𝑁−2𝑢_𝑛⋮≤𝜆_𝑁,𝑛+1−𝜆_𝑁,𝑛𝐴_𝑁Λ^𝑁−1_𝑛+1𝑢_𝑛+1+𝜆_{𝑁−1,𝑛+1}−𝜆_{𝑁−1,𝑛}𝐴_𝑁−1Λ^𝑁−2_𝑛+1𝑢_𝑛+1+⋯+𝜆_1,𝑛+1−𝜆_1,𝑛𝐴₁Λ⁰_𝑛+1𝑢_𝑛+1+Λ⁰_𝑛+1𝑢_𝑛+1−Λ⁰_𝑛𝑢_𝑛≤𝑀₀^𝑁_𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑢_𝑛+1−𝑢_𝑛, (69) where sup_𝑛≥1{∑^𝑁_𝑖=1‖𝐴_𝑖Λ^𝑖−1_𝑛+1𝑢_𝑛+1‖}≤𝑀₀ for some 𝑀₀>0. Also, utilizing Proposition 6(ii), (v) we deduce that 𝑢_𝑛+1−𝑢_𝑛=Δ^𝑀_𝑛+1𝑥_𝑛+1−Δ^𝑀_𝑛𝑥_𝑛=𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛+1}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑥_𝑛≤𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛+1}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑥_𝑛≤𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛+1}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝐼−𝑟_𝑀,𝑛𝐵_𝑀Δ_𝑛^𝑀−1𝑥_𝑛≤𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛𝑟_𝑀,𝑛+1×𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛+1}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+Δ^𝑀−1_𝑛+1𝑥_𝑛+1−Δ_𝑛^𝑀−1𝑥_𝑛=𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛×𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+1𝑟_𝑀,𝑛+1×𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛+1}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+Δ^𝑀−1_𝑛+1𝑥_𝑛+1−Δ_𝑛^𝑀−1𝑥_𝑛⋮≤𝑟_𝑀,𝑛+1−𝑟_𝑀,𝑛×𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+1𝑟_𝑀,𝑛+1×𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛+1}𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1−𝐼−𝑟_𝑀,𝑛+1𝐵_𝑀Δ^𝑀−1_𝑛+1𝑥_𝑛+1+⋯+𝑟_1,𝑛+1−𝑟_1,𝑛×𝐵₁Δ⁰_𝑛+1𝑥_𝑛+1+1𝑟_1,𝑛+1×𝑇^(Θ1,𝜑1)_𝑟1,𝑛+1𝐼−𝑟_1,𝑛+1𝐵₁Δ⁰_𝑛+1𝑥_𝑛+1−𝐼−𝑟_1,𝑛+1𝐵₁Δ⁰_𝑛+1𝑥_𝑛+1+Δ⁰_𝑛+1𝑥_𝑛+1−Δ⁰_𝑛𝑥_𝑛≤𝑀₁^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝑥_𝑛+1−𝑥_𝑛, (70) where 𝑀₁>0 is a constant such that for each 𝑛≥1^𝑀_𝑘=1𝐵_𝑘Δ^𝑘−1_𝑛+1𝑥_𝑛+1+1𝑟_𝑘,𝑛+1×𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛+1}𝐼−𝑟_𝑘,𝑛+1𝐵_𝑘Δ^𝑘−1_𝑛+1𝑥_𝑛+1−𝐼−𝑟_𝑘,𝑛+1𝐵_𝑘Δ^𝑘−1_𝑛+1𝑥_𝑛+1≤𝑀₁. (71) Also, from (54) we have 𝑦_𝑛+1=𝛼_𝑛+1𝛾𝑆𝑣_𝑛+1+𝐼−𝛼_𝑛+1𝜇𝐹𝑊_𝑛+1𝐺𝑣_𝑛+1,𝑦_𝑛=𝛼_𝑛𝛾𝑆𝑣_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛,∀𝑛≥1. (72) Simple calculation shows that 𝑦_𝑛+1−𝑦_𝑛=𝛼_𝑛+1𝛾𝑆𝑣_𝑛+1−𝑆𝑣_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝐼−𝛼_𝑛+1𝜇𝐹𝑊_𝑛+1𝐺𝑣_𝑛+1−𝐼−𝛼_𝑛+1𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛. (73) In the meantime, from (41), since 𝑊_𝑛, 𝑆_𝑛, and 𝑈_𝑛,𝑖 are all nonexpansive, we have 𝑊_𝑛+1𝐺𝑣_𝑛+1−𝑊_𝑛𝐺𝑣_𝑛≤𝑊_𝑛+1𝐺𝑣_𝑛+1−𝑊_𝑛+1𝐺𝑣_𝑛+𝑊_𝑛+1𝐺𝑣_𝑛−𝑊_𝑛𝐺𝑣_𝑛≤𝑣_𝑛+1−𝑣_𝑛+𝑊_𝑛+1𝐺𝑣_𝑛−𝑊_𝑛𝐺𝑣_𝑛=𝑣_𝑛+1−𝑣_𝑛+𝜆₁𝑇₁𝑈_𝑛+1,2𝐺𝑣_𝑛−𝜆₁𝑇₁𝑈_𝑛,2𝐺𝑣_𝑛≤𝑣_𝑛+1−𝑣_𝑛+𝜆₁𝑈_𝑛+1,2𝐺𝑣_𝑛−𝑈_𝑛,2𝐺𝑣_𝑛=𝑣_𝑛+1−𝑣_𝑛+𝜆₁𝜆₂𝑇₂𝑈_𝑛+1,3𝐺𝑣_𝑛−𝜆₂𝑇₂𝑈_𝑛,3𝐺𝑣_𝑛≤𝑣_𝑛+1−𝑣_𝑛+𝜆₁𝜆₂𝑈_𝑛+1,3𝐺𝑣_𝑛−𝑈_𝑛,3𝐺𝑣_𝑛⋮≤𝑣_𝑛+1−𝑣_𝑛+𝜆₁𝜆₂⋯𝜆_𝑛×𝑈_{𝑛+1,𝑛+1}𝐺𝑣_𝑛−𝑈_𝑛,𝑛+1𝐺𝑣_𝑛≤𝑣_𝑛+1−𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖, (74) where 𝑀₂ is a constant such that ‖𝑈_{𝑛+1,𝑛+1}𝐺𝑣_𝑛‖+‖𝑈_𝑛,𝑛+1𝐺𝑣_𝑛‖≤𝑀₂ for each 𝑛≥1. Therefore, by utilizing Lemma 15, from (69)–(74) and {𝜆_𝑛}⊂(0,𝑏]⊂(0,1) it follows that 𝑦_𝑛+1−𝑦_𝑛≤𝛼_𝑛+1𝛾𝑆𝑣_𝑛+1−𝑆𝑣_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝐼−𝛼_𝑛+1𝜇𝐹𝑊_𝑛+1𝐺𝑣_𝑛+1−𝐼−𝛼_𝑛+1𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛≤𝛼_𝑛+1𝛾𝑣_𝑛+1−𝑣_𝑛+𝛼_𝑛+1−𝛼_𝑛×𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+1−𝛼_𝑛+1𝜏𝑊_𝑛+1𝐺𝑣_𝑛+1−𝑊_𝑛𝐺𝑣_𝑛≤𝛼_𝑛+1𝛾𝑣_𝑛+1−𝑣_𝑛+𝛼_𝑛+1−𝛼_𝑛×𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+1−𝛼_𝑛+1𝜏𝑣_𝑛+1−𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖≤1−𝛼_𝑛+1𝜏−𝛾𝑣_𝑛+1−𝑣_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖≤1−𝛼_𝑛+1×𝑀𝜏−𝛾_0𝑁𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑢_𝑛+1−𝑢_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖≤1−𝛼_𝑛+1×𝑀𝜏−𝛾_0𝑁𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑀₁^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖≤1−𝛼_𝑛+1𝜏−𝛾𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑁_𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑀₁^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖. (75) On the other hand, from (54) we have 𝑥_𝑛+2=𝑠_𝑛+1𝛾𝑉𝑥_𝑛+1+𝐼−𝑠_𝑛+1𝜇𝐹𝑇_𝑛+1𝑦_𝑛+1,𝑥_𝑛+1=𝑠_𝑛𝛾𝑉𝑥_𝑛+𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑦_𝑛,∀𝑛≥1. (76) The simple calculations show that 𝑥_𝑛+2−𝑥_𝑛+1=𝐼−𝑠_𝑛+1𝜇𝐹𝑇_𝑛+1𝑦_𝑛+1−𝐼−𝑠_𝑛+1𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1−𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1𝛾𝑉𝑥_𝑛+1−𝑉𝑥_𝑛. (77) Utilizing Lemma 15 we deduce from (68), (75), and (77) that 𝑥_𝑛+2−𝑥_𝑛+1≤𝐼−𝑠_𝑛+1𝜇𝐹𝑇_𝑛+1𝑦_𝑛+1−𝐼−𝑠_𝑛+1𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1−𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1𝛾𝑉𝑥_𝑛+1−𝑉𝑥_𝑛≤1−𝑠_𝑛+1𝜏𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1−𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1𝛾𝑙𝑥_𝑛+1−𝑥_𝑛≤1−𝑠_𝑛+1𝜏𝑦_𝑛+1−𝑦_𝑛+4𝑀𝐿𝑠_𝑛+1−𝑠_𝑛+𝑠_𝑛+1−𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1𝛾𝑙𝑥_𝑛+1−𝑥_𝑛≤1−𝑠_𝑛+1𝜏×1−𝛼_𝑛+1𝜏−𝛾𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑁_𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑀₁^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂^𝑛_𝑖=1𝜆_𝑖+4𝑀𝐿𝑠_𝑛+1−𝑠_𝑛+𝑠_𝑛+1−𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑠_𝑛+1𝛾𝑙𝑥_𝑛+1−𝑥_𝑛≤1−𝑠_𝑛+1𝜏𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑁_𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑀₁^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂𝑏^𝑛+4𝑀𝐿+𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛𝑠_𝑛+1−𝑠_𝑛+𝑠_𝑛+1𝛾𝑙𝑥_𝑛+1−𝑥_𝑛=1−𝑠_𝑛+1𝜏−𝛾𝑙𝑥_𝑛+1−𝑥_𝑛+𝑀₀^𝑁_𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+𝑀₁^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛+𝑀₂𝑏^𝑛+4𝑀𝐿+𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛𝑠_𝑛+1−𝑠_𝑛≤1−𝑠_𝑛+1𝜏−𝛾𝑙𝑥_𝑛+1−𝑥_𝑛+𝑀₃^𝑁_𝑖=1𝜆_𝑖,𝑛+1−𝜆_𝑖,𝑛+^𝑀_𝑘=1𝑟_𝑘,𝑛+1−𝑟_𝑘,𝑛+𝛼_𝑛+1−𝛼_𝑛+𝑏^𝑛+𝑠_𝑛+1−𝑠_𝑛, (78) where 𝑀₃>0 is a constant such that for each 𝑛≥1𝑀₀+𝑀₁+𝑀₂+4𝑀𝐿+𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛≤𝑀₃. (79) Therefore, 𝑥_𝑛+1−𝑥_𝑛𝛼_𝑛≤1−𝑠_𝑛𝑥𝜏−𝛾𝑙^𝑛−𝑥_𝑛−1𝛼_𝑛+𝑀₃^𝑁_𝑖=1𝜆_𝑖,𝑛−𝜆_{𝑖,𝑛−1}𝛼_𝑛+^𝑀_𝑘=1𝑟_𝑘,𝑛−𝑟_{𝑘,𝑛−1}𝛼_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛼_𝑛+𝑏^𝑛−1𝛼_𝑛+𝑠_𝑛−𝑠_𝑛−1𝛼_𝑛=1−𝑠_𝑛𝑥𝜏−𝛾𝑙^𝑛−𝑥_𝑛−1𝛼_𝑛−1+1−𝑠_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑥_𝑛−11𝛼_𝑛−1𝛼_𝑛−1+𝑀₃^𝑁_𝑖=1𝜆_𝑖,𝑛−𝜆_{𝑖,𝑛−1}𝛼_𝑛+^𝑀_𝑘=1𝑟_𝑘,𝑛−𝑟_{𝑘,𝑛−1}𝛼_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛼_𝑛+𝑏^𝑛−1𝛼_𝑛+𝑠_𝑛−𝑠_𝑛−1𝛼_𝑛≤1−𝜏−𝛾𝑙𝑠_𝑛𝑥_𝑛−𝑥_𝑛−1𝛼_𝑛−1+𝜏−𝛾𝑙𝑠_𝑛⋅1×𝑥𝜏−𝛾𝑙_𝑛−𝑥_𝑛−11𝑠_𝑛1𝛼_𝑛−1𝛼_𝑛−1+𝑀₃^𝑁_𝑖=1𝜆_𝑖,𝑛−𝜆_{𝑖,𝑛−1}𝑠_𝑛𝛼_𝑛+^𝑀_𝑘=1𝑟_𝑘,𝑛−𝑟_{𝑘,𝑛−1}𝑠_𝑛𝛼_𝑛+𝛼_𝑛−𝛼_𝑛−1𝑠_𝑛𝛼_𝑛+𝑏^𝑛−1𝑠_𝑛𝛼_𝑛+𝑠_𝑛−𝑠_𝑛−1𝑠_𝑛𝛼_𝑛≤1−𝜏−𝛾𝑙𝑠_𝑛𝑥_𝑛−𝑥_𝑛−1𝛼_𝑛−1+𝜏−𝛾𝑙𝑠_𝑛⋅𝑀₄×1𝜏−𝛾𝑙𝑠_𝑛1𝛼_𝑛−1𝛼_𝑛−1+^𝑁_𝑖=1𝜆_𝑖,𝑛−𝜆_{𝑖,𝑛−1}𝑠_𝑛𝛼_𝑛+^𝑀_𝑘=1𝑟_𝑘,𝑛−𝑟_{𝑘,𝑛−1}𝑠_𝑛𝛼_𝑛+1𝑠_𝑛𝛼1−^𝑛−1𝛼_𝑛+1𝛼_𝑛𝑠1−^𝑛−1𝑠_𝑛+𝑏^𝑛−1𝑠_𝑛𝛼_𝑛, (80) where 𝑀₃+‖𝑥_𝑛−𝑥_𝑛−1‖≤𝑀₄, for all 𝑛>1 for some 𝑀₄>0. From (H2), (H3), (H5), and (H6), it follows that ∑^∞_𝑛=1(𝜏−𝛾𝑙)𝑠_𝑛=∞ and lim_𝑛→∞𝑀₄1𝜏−𝛾𝑙𝑠_𝑛1𝛼_𝑛−1𝛼_𝑛−1+^𝑁_𝑖=1𝜆_𝑖,𝑛−𝜆_{𝑖,𝑛−1}𝑠_𝑛𝛼_𝑛+^𝑀_𝑘=1𝑟_𝑘,𝑛−𝑟_{𝑘,𝑛−1}𝑠_𝑛𝛼_𝑛+1𝑠_𝑛𝛼1−^𝑛−1𝛼_𝑛+1𝛼_𝑛𝑠1−^𝑛−1𝑠_𝑛+𝑏^𝑛−1𝑠_𝑛𝛼_𝑛=0. (81) Thus, applying Lemma 16 to (80), we immediately conclude that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛𝛼_𝑛=0. (82) So, from lim_𝑛→∞𝛼_𝑛=0 it follows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (83)
Step 3. We prove that 𝜔_𝑤(𝑥_𝑛)⊂Ω provided lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.
Indeed, first of all, let us show that ‖𝑦_𝑛−𝑃_𝐶(𝐼−(2/𝐿)∇𝑓)𝑦_𝑛‖→0, ‖𝑥_𝑛−𝑢_𝑛‖→0, ‖𝑥_𝑛−𝑣_𝑛‖→0, ‖𝑣_𝑛−𝐺𝑣_𝑛‖→0, and ‖𝑣_𝑛−𝑊𝑣_𝑛‖→0 as 𝑛→∞. As a matter of fact, utilizing Lemmas 7 and 15 we obtain from (54), (61), and (62) that 𝑦_𝑛−𝑝²=𝛼_𝑛𝛾𝑆𝑣_𝑛−𝑆𝑝+𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑝+𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝²≤𝛼_𝑛𝛾𝑆𝑣_𝑛+−𝑆𝑝𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤𝛼−𝑝_𝑛𝛾𝑆𝑣_𝑛+−𝑆𝑝(𝐼−𝛼_𝑛𝜇𝐹)𝑊_𝑛𝐺𝑣_𝑛−(𝐼−𝛼_𝑛𝜇𝐹)𝑝²+2𝛼_𝑛𝛾𝑉−𝜇𝐹𝑝,𝑦_𝑛≤𝛼−𝑝_𝑛𝛾𝑣_𝑛−𝑝+1−𝛼_𝑛𝜏𝐺𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤𝛼−𝑝_𝑛𝜏𝑣_𝑛−𝑝+1−𝛼_𝑛𝜏𝐺𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏𝐺𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛=𝑣−𝑝_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛−𝑝. (84)
Note that 𝑥_𝑛+1=𝑠_𝑛𝛾𝑉𝑥_𝑛+(𝐼−𝑠_𝑛𝜇𝐹)𝑇_𝑛𝑦_𝑛. Hence we have 𝑥_𝑛+1−𝑦_𝑛=𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛+𝑇_𝑛𝑦_𝑛−𝑦_𝑛, (85) which yields 𝑇_𝑛𝑦_𝑛−𝑦_𝑛≤𝑥_𝑛+1−𝑦_𝑛−𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛≤𝑥_𝑛+1−𝑦_𝑛+𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛+𝑠_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑦_𝑛. (86) Since lim_𝑛→∞𝑠_𝑛=0 and lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0, from the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, we obtain lim_𝑛→∞𝑦_𝑛−𝑇_𝑛𝑦_𝑛=0. (87) It is clear that 𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝑦_𝑛−𝑦_𝑛=𝑠_𝑛𝑦_𝑛+1−𝑠_𝑛𝑇_𝑛𝑦_𝑛−𝑦_𝑛=1−𝑠_𝑛𝑇_𝑛𝑦_𝑛−𝑦_𝑛≤𝑇_𝑛𝑦_𝑛−𝑦_𝑛, (88) where 𝑠_𝑛=((2−𝜃_𝑛𝐿)/4)∈(0,1/2) for each 𝜃_𝑛∈(0,2/𝐿). Hence we have 𝑃_𝐶2𝐼−𝐿∇𝑓𝑦_𝑛−𝑦_𝑛≤𝑃_𝐶2𝐼−𝐿∇𝑓𝑦_𝑛−𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝑦_𝑛+𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝑦_𝑛−𝑦_𝑛≤2𝐼−𝐿∇𝑓𝑦_𝑛−𝐼−𝜃_𝑛∇𝑓𝑦_𝑛+𝑃_𝐶𝐼−𝜃_𝑛∇𝑓𝑦_𝑛−𝑦_𝑛≤2𝐿−𝜃_𝑛∇𝑓𝑦_𝑛+𝑇_𝑛𝑦_𝑛−𝑦_𝑛. (89) From the boundedness of {𝑦_𝑛},𝑠_𝑛→0  (⇔𝜃_𝑛→2/𝐿) and ‖𝑇_𝑛𝑦_𝑛−𝑦_𝑛‖→0 (due to (87)), it follows that lim_𝑛→∞𝑦_𝑛−𝑃_𝐶2𝐼−𝐿∇𝑓𝑦_𝑛=0. (90) Also, from (30) it follows that for all 𝑖∈{1,2,…,𝑁} and 𝑘∈{1,2,…,𝑀}𝑣_𝑛−𝑝²=Λ^𝑁_𝑛𝑢_𝑛−𝑝²≤Λ^𝑖_𝑛𝑢_𝑛−𝑝²=𝑃_𝐶(𝐼−𝜆_𝑖,𝑛𝐴_𝑖)Λ_𝑛^𝑖−1𝑢_𝑛−𝑃_𝐶(𝐼−𝜆_𝑖,𝑛𝐴_𝑖)𝑝²≤𝐼−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐼−𝜆_𝑖,𝑛𝐴_𝑖𝑝²≤Λ_𝑛^𝑖−1𝑢_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²≤𝑢_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²≤𝑥_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²,𝑢_𝑛−𝑝²=Δ^𝑀_𝑛𝑥_𝑛−𝑝²≤Δ^𝑘_𝑛𝑥_𝑛−𝑝²=𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝐼−𝑟_𝑘,𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝐼−𝑟_𝑘,𝑛𝐵_𝑘𝑝²≤(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)Δ_𝑛^𝑘−1𝑥_𝑛−(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)𝑝²≤Δ_𝑛^𝑘−1𝑥_𝑛−𝑝²+𝑟_𝑘,𝑛𝑟_𝑘,𝑛−2𝜇_𝑘𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝²≤𝑥_𝑛−𝑝²+𝑟_𝑘,𝑛𝑟_𝑘,𝑛−2𝜇_𝑘𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝². (91) So, from (84) and (91) it follows that 𝑦_𝑛−𝑝²≤𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤𝑢−𝑝_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝²+𝑟_𝑘,𝑛𝑟_𝑘,𝑛−2𝜇_𝑘𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝, (92) which hence leads to 𝑟_𝑘,𝑛2𝜇_𝑘−𝑟_𝑘,𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝²+𝜆_𝑖,𝑛2𝜂_𝑖−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑦_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝. (93) Since lim_𝑛→∞𝛼_𝑛=0,{𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖) and {𝑟_𝑘,𝑛}⊂[𝑒_𝑘,𝑓_𝑘]⊂(0,2𝜇_𝑘) for all 𝑖∈{1,2,…,𝑁} and 𝑘∈{1,2,…,𝑀}, by the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 and the boundedness of {𝑥_𝑛}, {𝑦_𝑛}, we conclude immediately that lim_𝑛→∞𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝=0,lim_𝑛→∞𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝=0, (94) for all 𝑖∈{1,2,…,𝑁} and 𝑘∈{1,2,…,𝑀}.
Furthermore, by Proposition 6(ii) we obtain that for each 𝑘∈{1,2,…,𝑀}Δ^𝑘_𝑛𝑥_𝑛−𝑝²=𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)Δ_𝑛^𝑘−1𝑥_𝑛−𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)𝑝²≤𝐼−𝑟_𝑘,𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐼−𝑟_𝑘,𝑛𝐵_𝑘𝑝,Δ^𝑘_𝑛𝑥_𝑛=1−𝑝2(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)Δ_𝑛^𝑘−1𝑥_𝑛−(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)𝑝²+Δ^𝑘_𝑛𝑥_𝑛−𝑝²−𝐼−𝑟_𝑘,𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐼−𝑟_𝑘,𝑛𝐵_𝑘𝑝−Δ^𝑘_𝑛𝑥_𝑛−𝑝²≤12Δ_𝑛^𝑘−1𝑥_𝑛−𝑝²+Δ^𝑘_𝑛𝑥_𝑛−𝑝²−Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛−𝑟_𝑘,𝑛(𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝)², (95) which implies that Δ^𝑘_𝑛𝑥_𝑛−𝑝²≤Δ_𝑛^𝑘−1𝑥_𝑛−𝑝²−Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛−𝑟_𝑘,𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝²=Δ_𝑛^𝑘−1𝑥_𝑛−𝑝²−Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛²−𝑟²_𝑘,𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘p²+2𝑟_𝑘,𝑛Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝≤Δ_𝑛^𝑘−1𝑥_𝑛−𝑝²−Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛²+2𝑟_𝑘,𝑛Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝≤𝑥_𝑛−𝑝²−Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛²+2𝑟_𝑘,𝑛Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝. (96) Also, by Proposition 1(iii), we obtain that for each 𝑖∈{1,2,…,𝑁}Λ^𝑖_𝑛𝑢_𝑛−𝑝²=𝑃_𝐶𝐼−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝑃_𝐶𝐼−𝜆_𝑖,𝑛𝐴_𝑖𝑝²≤𝐼−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐼−𝜆_𝑖,𝑛𝐴_𝑖𝑝,Λ^𝑖_𝑛𝑢_𝑛=1−𝑝2𝐼−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐼−𝜆_𝑖,𝑛𝐴_𝑖𝑝²+Λ^𝑖_𝑛𝑢_𝑛−𝑝²−𝐼−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐼−𝜆_𝑖,𝑛𝐴_𝑖𝑝−Λ^𝑖_𝑛𝑢_𝑛−𝑝²≤12Λ_𝑛^𝑖−1𝑢_𝑛−𝑝²+Λ^𝑖_𝑛𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²≤12𝑢_𝑛−𝑝²+Λ^𝑖_𝑛𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝², (97) which implies Λ^𝑖_𝑛𝑢_𝑛−𝑝²≤𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛−𝜆_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²=𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²−𝜆²_𝑖,𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛,𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝≤𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝. (98) Thus, from (84), (96), and (98), we have 𝑦_𝑛−𝑝²≤𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤Λ−𝑝_𝑖𝑛𝑢_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇F𝑝,𝑦_𝑛≤𝑢−𝑝_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤Δ−𝑝_𝑘𝑛𝑥_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝²−Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛²+2𝑟_𝑘,𝑛Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝, (99) which yields Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛²+Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝑟_𝑘,𝑛Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑦_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+2𝑟_𝑘,𝑛Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝐵_𝑘𝑝+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐴_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐴_𝑖𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝. (100) Since lim_𝑛→∞𝛼_𝑛=0, {𝑥_𝑛}, {𝑦_𝑛}, and {𝑢_𝑛} are bounded. For all 𝑖∈{1,2,…,𝑁} and 𝑘∈{1,2,…,𝑀}, we have {𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖) and {𝑟_𝑘,𝑛}⊂[𝑒_𝑘,𝑓_𝑘]⊂(0,2𝜇_𝑘), then by (94) and the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, we conclude immediately that lim_𝑛→∞Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛,lim_𝑛→∞Δ_𝑛^𝑘−1𝑥_𝑛−Δ^𝑘_𝑛𝑥_𝑛=0, (101) for all 𝑖∈{1,2,…,𝑁} and 𝑘∈{1,2,…,𝑀}. Note that 𝑥_𝑛−𝑢_𝑛=Δ⁰_𝑛𝑥_𝑛−Δ^𝑀_𝑛𝑥_𝑛≤Δ⁰_𝑛𝑥_𝑛−Δ¹_𝑛𝑥_𝑛+Δ¹_𝑛𝑥_𝑛−Δ²_𝑛𝑥_𝑛+Δ+⋯_𝑛𝑀−1𝑥_𝑛−Δ^𝑀_𝑛𝑥_𝑛,𝑢_𝑛−𝑣_𝑛=Λ⁰_𝑛𝑢_𝑛−Λ^𝑁_𝑛𝑢_𝑛≤Λ⁰_𝑛𝑢_𝑛−Λ¹_𝑛𝑢_𝑛+Λ¹_𝑛𝑢_𝑛−Λ²_𝑛𝑢_𝑛+Λ+⋯_𝑛𝑁−1𝑢_𝑛−Λ^𝑁_𝑛𝑢_𝑛. (102) Thus, from (101) we have lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0,lim_𝑛→∞𝑢_𝑛−𝑣_𝑛=0. (103) It is easy to see that as 𝑛→∞𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑣_𝑛⟶0. (104)
On the other hand, for simplicity, we write 𝑝=𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑝, 𝑣_𝑛=𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑣_𝑛, and 𝑤_𝑛=𝐺𝑣_𝑛=𝑃_𝐶(𝐼−𝜈₁𝐹₁)𝑣_𝑛 for all 𝑛≥1. Then 𝑝=𝐺𝑝=𝑃_𝐶𝐼−𝜈₁𝐹₁𝑝=𝑃_𝐶𝐼−𝜈₁𝐹₁𝑃_𝐶𝐼−𝜈₂𝐹₂𝑝. (105)
We now show that lim_𝑛→∞‖𝐺𝑣_𝑛−𝑣_𝑛‖=0; that is, lim_𝑛→∞‖𝑤_𝑛−𝑣_𝑛‖=0. As a matter of fact, for 𝑝∈Ω, it follows from (61), (62), and (84) that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏𝐺𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏𝑤_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝑣_𝑛−𝑝²+𝜈₁𝜈₁−2𝜁₁𝐹₁𝑣_𝑛−𝐹₁𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝑣_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐹₂𝑣_𝑛−𝐹₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐹₁𝑣_𝑛−𝐹₁𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛=𝑣−𝑝_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝜈₂𝜈₂−2𝜁₂𝐹₂𝑣_𝑛−𝐹₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐹₁𝑣_𝑛−𝐹₁𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝜈₂𝜈₂−2𝜁₂𝐹₂𝑣_𝑛−𝐹₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐹₁𝑣_𝑛−𝐹₁𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝, (106) which immediately yields 1−𝛼_𝑛𝜏𝜈₂2𝜁₂−𝜈₂𝐹₂𝑣_𝑛−𝐹₂𝑝²+𝜈₁2𝜁₁−𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑦_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝. (107) Since lim_𝑛→∞𝛼_𝑛=0, 𝜈_𝑗∈(0,2𝜁_𝑗) for 𝑗=1,2 and {𝑥_𝑛} and {𝑦_𝑛} are bounded, by the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, we get lim_𝑛→∞𝐹₂𝑣_𝑛−𝐹₂𝑝=0,lim_𝑛→∞𝐹₁𝑣_𝑛−𝐹₁𝑝=0. (108) Also, in terms of the firm nonexpansivity of 𝑃_𝐶 and the 𝜁_𝑗-inverse-strong monotonicity of 𝐹_𝑗 for 𝑗=1,2, we obtain from 𝜈_𝑗∈(0,2𝜁_𝑗), 𝑗=1,2 and (67) that 𝑣_𝑛−𝑝²=𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑣_𝑛−𝑃_𝐶(𝐼−𝜈₂𝐹₂)𝑝²≤𝐼−𝜈₂𝐹₂𝑣_𝑛−𝐼−𝜈₂𝐹₂𝑝,𝑣_𝑛−𝑝=12(𝐼−𝜈₂𝐹₂)𝑣_𝑛−(𝐼−𝜈₂𝐹₂)𝑝²+𝑣_𝑛−𝑝²−𝐼−𝜈₂𝐹₂𝑣_𝑛−𝐼−𝜈₂𝐹₂𝑝−𝑣_𝑛−𝑝²≤12𝑣_𝑛−𝑝²+𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−𝜈₂(𝐹₂𝑣_𝑛−𝐹₂𝑝)−(𝑝−𝑝)²=12𝑣_𝑛−𝑝²+𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝,𝐹₂𝑣_𝑛−𝐹₂𝑝−𝜈²₂𝐹₂𝑣_𝑛−𝐹₂𝑝²,𝑤_𝑛−𝑝²=𝑃_𝐶(𝐼−𝜈₁𝐹₁)𝑣_𝑛−𝑃_𝐶(𝐼−𝜈₁𝐹₁)𝑝²≤𝐼−𝜈₁𝐹₁𝑣_𝑛−𝐼−𝜈₁𝐹₁𝑝,𝑤_𝑛=1−𝑝2(𝐼−𝜈₁𝐹₁)𝑣_𝑛−(𝐼−𝜈₁𝐹₁)𝑝²+𝑤_𝑛−𝑝²−𝐼−𝜈₁𝐹₁𝑣_𝑛−𝐼−𝜈₁𝐹₁𝑝−𝑤_𝑛−𝑝²≤12𝑣_𝑛−𝑝²+𝑤_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝,𝑣_𝑛−𝑤_𝑛+𝑝−𝑝−𝜈²₁𝐹₁𝑣_𝑛−𝐹₁𝑝²≤12𝑣_𝑛−𝑝²+𝑤_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝,𝑣_𝑛−𝑤_𝑛+𝑝−𝑝. (109) Thus, we have 𝑣_𝑛−𝑝²≤𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝,𝐹₂𝑣_𝑛−𝐹₂𝑝−𝜈²₂𝐹₂𝑣_𝑛−𝐹₂𝑝²,𝑤_𝑛−𝑝²≤𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝. (110)(111) Consequently, from (61), (106), and (110) it follows that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝑣_𝑛−𝑝²+𝜈₁𝜈₁−2𝜁₁𝐹₁𝑣_𝑛−𝐹₁𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏𝑣_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝑣_𝑛−𝑝²−𝑣_𝑛−𝑣_𝑛−𝑝−𝑝²+2𝜈₂⟨𝑣_𝑛−𝑣_𝑛−𝑝−𝑝,𝐹₂𝑣_𝑛−𝐹₂𝑝⟩−𝜈²₂𝐹₂𝑣_𝑛−𝐹₂𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐹₂𝑣_𝑛−𝐹₂𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑣−𝑝_𝑛−𝑝²−1−𝛼_𝑛𝜏(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐹₂𝑣_𝑛−𝐹₂𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝛼_𝑛𝜏(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐹₂𝑣_𝑛−𝐹₂𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝, (112) which hence leads to 1−𝛼_𝑛𝜏(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐹₂𝑣_𝑛−𝐹₂𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑦_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐹₂𝑣_𝑛−𝐹₂𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝. (113) Since lim_𝑛→∞𝛼_𝑛=0, {𝑥_𝑛}, {𝑦_𝑛}, {𝑣_𝑛}, and {𝑣_𝑛} are bounded sequences, by the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, we conclude from (108) that lim_𝑛→∞𝑣_𝑛−𝑣_𝑛−𝑝−𝑝=0. (114) Furthermore, from (62), (106), and (111) it follows that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏𝑤_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝≤𝛼_𝑛𝜏𝑣_𝑛−𝑝²+1−𝛼_𝑛𝜏×𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑣−𝑝_𝑛−𝑝²−1−𝛼_𝑛𝜏(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝛼_𝑛𝜏(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝, (115) which hence yields 1−𝛼_𝑛𝜏(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛≤𝑥−𝑝_𝑛−𝑦_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+2𝜈₁𝐹₁𝑣_𝑛−𝐹₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝𝑦_𝑛−𝑝. (116) Since lim_𝑛→∞𝛼_𝑛=0 and {𝑥_𝑛}, {𝑦_𝑛}, {𝑤_𝑛}, and {𝑣_𝑛} are bounded sequences, by the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, we conclude from (108) that lim_𝑛→∞𝑣_𝑛−𝑤_𝑛+𝑝−𝑝=0. (117) Note that 𝑣_𝑛−𝑤_𝑛≤𝑣_𝑛−𝑣_𝑛−𝑝+𝑣𝑝−_𝑛−𝑤_𝑛+𝑝−𝑝. (118) Hence from (114) and (117) we get lim_𝑛→∞𝑣_𝑛−𝐺𝑣_𝑛=lim_𝑛→∞𝑣_𝑛−𝑤_𝑛=0. (119) Also, observe that 𝑦_𝑛=𝛼_𝑛𝛾𝑆𝑣_𝑛+𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛. (120) Hence we get 𝑦_𝑛−𝑊_𝑛𝐺𝑣_𝑛=𝛼_𝑛𝛾𝑆𝑣_𝑛−𝜇𝐹𝑊_𝑛𝐺𝑣_𝑛. (121) So, from lim_𝑛→∞𝛼_𝑛=0 and the boundedness of {𝑣_𝑛} we deduce that lim_𝑛→∞𝑦_𝑛−𝑊_𝑛𝐺𝑣_𝑛=0. (122) In addition, it is readily found that 𝑊_𝑛𝑣_𝑛−𝑣_𝑛≤𝑊_𝑛𝑣_𝑛−𝑊_𝑛𝐺𝑣_𝑛+𝑊_𝑛𝐺𝑣_𝑛−𝑣_𝑛≤𝑣_𝑛−𝐺𝑣_𝑛+𝑊_𝑛𝐺𝑣_𝑛−𝑣_𝑛≤𝑣_𝑛−𝐺𝑣_𝑛+𝑊_𝑛𝐺𝑣_𝑛−𝑦_𝑛+𝑦_𝑛−𝑣_𝑛≤𝑣_𝑛−𝐺𝑣_𝑛+𝑊_𝑛𝐺𝑣_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑣_𝑛. (123) Thus, by the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, from (103) and (119)–(123) we have lim_𝑛→∞𝑊_𝑛𝑣_𝑛−𝑣_𝑛=0. (124) Taking into account that ‖𝑣_𝑛−𝑊𝑣_𝑛‖≤‖𝑣_𝑛−𝑊_𝑛𝑣_𝑛‖+‖𝑊_𝑛𝑣_𝑛−𝑊𝑣_𝑛‖, we obtain from ‖𝑣_𝑛−𝑊_𝑛𝑣_𝑛‖→0 and Remark 12 that lim_𝑛→∞𝑣_𝑛−𝑊𝑣_𝑛=0. (125) Next, let us show that 𝜔_𝑤(𝑥_𝑛)⊂Ω. In fact, since 𝐻 is reflexive and {𝑥_𝑛} is bounded, there exists at least a weak convergence subsequence of {𝑥_𝑛}. Hence it is known that 𝜔_𝑤(𝑥_𝑛)≠0/. Now, take an arbitrary 𝑤∈𝜔_𝑤(𝑥_𝑛). Then there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑤. From (101) and (103) and the assumption lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, we have that 𝑦_𝑛𝑖⇀𝑤, 𝑢_𝑛𝑖⇀𝑤, 𝑣_𝑛𝑖⇀𝑤, Δ^𝑘_𝑛𝑖𝑥_𝑛𝑖⇀𝑤, and Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖⇀𝑤, where 𝑘∈{1,2,…,𝑀} and 𝑚∈{1,2,…,𝑁}. Utilizing Lemma 9, we deduce from 𝑥_𝑛𝑖⇀𝑤, 𝑣_𝑛𝑖⇀𝑤, (90), (119), and (125) that 𝑤∈Fix(𝑃_𝐶(𝐼−(2/𝐿)∇𝑓))=VI(𝐶,∇𝑓)=Γ, 𝑤∈GSVI(𝐺), and 𝑤∈Fix(𝑊)=∩^∞_𝑛=1Fix(𝑆_𝑛) (due to Lemma 13). Thus, we get 𝑤∈∩^∞_𝑛=1Fix(𝑆_𝑛)∩GSVI(𝐺)∩Γ. Next we prove that 𝑤∈∩^𝑁_𝑚=1VI(𝐶,𝐴_𝑚). Let 𝑇_𝑚𝐴𝑣=^𝑚𝑣+𝑁_𝐶𝑣,𝑣∈𝐶,0/,𝑣∉𝐶, (126) where 𝑚∈{1,2,…,𝑁}. Let (𝑣,𝑢)∈𝐺(𝑇_𝑚). Since 𝑢−𝐴_𝑚𝑣∈𝑁_𝐶𝑣 and Λ^𝑚_𝑛𝑢_𝑛∈𝐶, we have 𝑣−Λ^𝑚_𝑛𝑢_𝑛,𝑢−𝐴_𝑚𝑣≥0. (127) On the other hand, from Λ^𝑚_𝑛𝑢_𝑛=𝑃_𝐶(𝐼−𝜆_𝑚,𝑛𝐴_𝑚)Λ_𝑛^𝑚−1𝑢_𝑛 and 𝑣∈𝐶, we have 𝑣−Λ^𝑚_𝑛𝑢_𝑛,Λ^𝑚_𝑛𝑢_𝑛−Λ_𝑛^𝑚−1𝑢_𝑛−𝜆_𝑚,𝑛𝐴_𝑚Λ_𝑛^𝑚−1𝑢_𝑛≥0, (128) and hence 𝑣−Λ^𝑚_𝑛𝑢_𝑛,Λ^𝑚_𝑛𝑢_𝑛−Λ_𝑛^𝑚−1𝑢_𝑛𝜆_𝑚,𝑛+𝐴_𝑚Λ_𝑛^𝑚−1𝑢_𝑛≥0. (129) Therefore we have 𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖≥,𝑢𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,𝐴_𝑚𝑣≥𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,𝐴_𝑚𝑣−𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖−Λ_𝑛^𝑚−1_𝑖𝑢_𝑛𝑖𝜆_{𝑚,𝑛𝑖}+𝐴_𝑚Λ_𝑛^𝑚−1_𝑖𝑢_𝑛𝑖=𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,𝐴_𝑚𝑣−𝐴_𝑚Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖+𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,𝐴_𝑚Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖−𝐴_𝑚Λ_𝑛^𝑚−1_𝑖𝑢_𝑛𝑖−𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖−Λ_𝑛^𝑚−1_𝑖𝑢_𝑛𝑖𝜆_{𝑚,𝑛𝑖}≥𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,𝐴_𝑚Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖−𝐴_𝑚Λ_𝑛^𝑚−1_𝑖𝑢_𝑛𝑖−𝑣−Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖,Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖−Λ_𝑛^𝑚−1_𝑖𝑢_𝑛𝑖𝜆_{𝑚,𝑛𝑖}. (130) From (101) and since 𝐴_𝑚 is Lipschitz continuous, we obtain that lim_𝑛→∞‖𝐴_𝑚Λ^𝑚_𝑛𝑢_𝑛−𝐴_𝑚Λ_𝑛^𝑚−1𝑢_𝑛‖=0. From Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖⇀𝑤, {𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖), for all 𝑖∈{1,2,…,𝑁} and (101), we have ⟨𝑣−𝑤,𝑢⟩≥0. (131) Since 𝑇_𝑚 is maximal monotone, we have 𝑤∈𝑇_𝑚⁻¹0 and hence 𝑤∈VI(𝐶,𝐴_𝑚), 𝑚=1,2,…,𝑁, which implies 𝑤∈∩^𝑁_𝑚=1VI(𝐶,𝐴_𝑚). Next we prove that 𝑤∈∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘). Since Δ^𝑘_𝑛𝑥_𝑛=𝑇^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}(𝐼−𝑟_𝑘,𝑛𝐵_𝑘)Δ_𝑛^𝑘−1𝑥_𝑛, 𝑛≥1, 𝑘∈{1,2,…,𝑀}, we have Θ_𝑘Δ^𝑘_𝑛𝑥_𝑛,𝑦+𝜑_𝑘𝑦−𝜑_𝑘Δ^𝑘_𝑛𝑥_𝑛+𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛,𝑦−Δ^𝑘_𝑛𝑥_𝑛+1𝑟_𝑘,𝑛𝑦−Δ^𝑘_𝑛𝑥_𝑛,Δ^𝑘_𝑛𝑥_𝑛−Δ_𝑛^𝑘−1𝑥_𝑛≥0. (132) By (A2), we have 𝜑_𝑘𝑦−𝜑_𝑘Δ^𝑘_𝑛𝑥_𝑛+𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛,𝑦−Δ^𝑘_𝑛𝑥_𝑛+1𝑟_𝑘,𝑛𝑦−Δ^𝑘_𝑛𝑥_𝑛,Δ^𝑘_𝑛𝑥_𝑛−Δ_𝑛^𝑘−1𝑥_𝑛≥Θ_𝑘𝑦,Δ^𝑘_𝑛𝑥_𝑛. (133) Let 𝑧_𝑡=𝑡𝑦+(1−𝑡)𝑤 for all 𝑡∈(0,1] and 𝑦∈𝐶. This implies that 𝑧_𝑡∈𝐶. Then, we have 𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘𝑧_𝑡≥𝜑_𝑘Δ^𝑘_𝑛𝑥_𝑛−𝜑_𝑘𝑧_𝑡+𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘𝑧_𝑡−𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,Δ^𝑘_𝑛𝑥_𝑛−Δ_𝑛^𝑘−1𝑥_𝑛𝑟_𝑘,𝑛+Θ_𝑘𝑧_𝑡,Δ^𝑘_𝑛𝑥_𝑛=𝜑_𝑘Δ^𝑘_𝑛𝑥_𝑛−𝜑_𝑘𝑧_𝑡+𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘𝑧_𝑡−𝐵_𝑘Δ^𝑘_𝑛𝑥_𝑛+𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘Δ^𝑘_𝑛𝑥_𝑛−𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛−𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,Δ^𝑘_𝑛𝑥_𝑛−Δ_𝑛^𝑘−1𝑥_𝑛𝑟_𝑘,𝑛+Θ_𝑘𝑧_𝑡,Δ^𝑘_𝑛𝑥_𝑛. (134) By (101), we have ‖𝐵_𝑘Δ^𝑘_𝑛𝑥_𝑛−𝐵_𝑘Δ_𝑛^𝑘−1𝑥_𝑛‖→0 as 𝑛→∞. Furthermore, by the monotonicity of 𝐵_𝑘, we obtain ⟨𝑧_𝑡−Δ^𝑘_𝑛𝑥_𝑛,𝐵_𝑘𝑧_𝑡−𝐵_𝑘Δ^𝑘_𝑛𝑥_𝑛⟩≥0. Then, by (A4) we obtain 𝑧_𝑡−𝑤,𝐵_𝑘𝑧_𝑡≥𝜑_𝑘(𝑤)−𝜑_𝑘𝑧_𝑡+Θ_𝑘𝑧_𝑡,𝑤. (135) Utilizing (A1), (A4), and (135), we obtain 0=Θ_𝑘𝑧_𝑡,𝑧_𝑡+𝜑_𝑘𝑧_𝑡−𝜑_𝑘𝑧_𝑡≤𝑡Θ_𝑘𝑧_𝑡,𝑦+(1−𝑡)Θ_𝑘𝑧_𝑡,𝑤+𝑡𝜑_𝑘𝑦+(1−𝑡)𝜑_𝑘(𝑤)−𝜑_𝑘𝑧_𝑡≤𝑡Θ_𝑘𝑧_𝑡,𝑦+𝜑_𝑘𝑦−𝜑_𝑘𝑧_𝑡+(1−𝑡)𝑧_𝑡−𝑤,𝐵_𝑘𝑧_𝑡=𝑡Θ_𝑘𝑧_𝑡,𝑦+𝜑_𝑘𝑦−𝜑_𝑘𝑧_𝑡+(1−𝑡)𝑡𝑦−𝑤,𝐵_𝑘𝑧_𝑡, (136) and hence 0≤Θ_𝑘𝑧_𝑡,𝑦+𝜑_𝑘𝑦−𝜑_𝑘𝑧_𝑡+(1−𝑡)𝑦−𝑤,𝐵_𝑘𝑧_𝑡. (137) Letting 𝑡→0, we have, for each 𝑦∈𝐶, 0≤Θ_𝑘𝑤,𝑦+𝜑_𝑘𝑦−𝜑_𝑘(𝑤)+⟨𝑦−𝑤,𝐵_𝑘𝑤⟩. (138) This implies that 𝑤∈GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘) and hence 𝑤∈∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘). Consequently, 𝑤∈(∩^∞_𝑛=1Fix(𝑆_𝑛))∩(∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘))∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))∩GSVI(𝐺)∩Γ=∶Ω. (This shows that 𝜔_𝑤(𝑥_𝑛)⊂Ω.)
Step 4. We prove that 𝜔_𝑤(𝑥_𝑛)⊂Ξ provided that ‖𝑥_𝑛−𝑦_𝑛‖=𝑜(𝛼_𝑛) additionally.
Indeed, let 𝑤∈𝜔_𝑤(𝑥_𝑛) be the same as mentioned in Step 3. Then we get 𝑥_𝑛𝑖⇀𝑤. In addition, from (84) we have that for every 𝑝∈Ω𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝²+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛−𝑝, (139) which immediately implies that 2𝛾𝑆−𝜇𝐹𝑝,𝑝−𝑦_𝑛≤1𝛼_𝑛𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑦_𝑛𝛼_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝. (140) This together with ‖𝑥_𝑛−𝑦_𝑛‖=𝑜(𝛼_𝑛) leads to limsup_𝑛→∞𝛾𝑆−𝜇𝐹𝑝,𝑝−𝑦_𝑛≤0. (141) Observe that limsup_𝑛→∞𝛾𝑆−𝜇𝐹𝑝,𝑝−𝑥_𝑛=limsup_𝑛→∞𝛾𝑆−𝜇𝐹𝑝,𝑦_𝑛−𝑥_𝑛+𝛾𝑆−𝜇𝐹𝑝,𝑝−𝑦_𝑛=limsup_𝑛→∞𝛾𝑆−𝜇𝐹𝑝,𝑝−𝑦_𝑛≤0. (142) So, it follows from 𝑥_𝑛𝑖⇀𝑤 that 𝛾𝑆−𝜇𝐹𝑝,𝑝−𝑤≤0,∀𝑝∈Ω. (143) Also, note that 0<𝛾≤𝜏 and 𝜇𝜂≥𝜏⟺𝜇𝜂≥1−1−𝜇2𝜂−𝜇𝜅²⟺1−𝜇2𝜂−𝜇𝜅²≥1−𝜇𝜂⟺1−2𝜇𝜂+𝜇²𝜅²≥1−2𝜇𝜂+𝜇²𝜂²⟺𝜅²≥𝜂²⟺𝜅≥𝜂. (144) It is clear that ≥𝜇𝐹−𝛾𝑆𝑥−𝜇𝐹−𝛾𝑆𝑦,𝑥−𝑦𝜇𝜂−𝛾𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (145) Hence, it follows from 0<𝛾≤𝜏≤𝜇𝜂 that 𝜇𝐹−𝛾𝑆 is monotone. Since 𝑤∈𝜔_𝑤(𝑥_𝑛)⊂Ω, by Minty’s lemma [39] we have 𝛾𝑆−𝜇𝐹𝑤,𝑝−𝑤≤0,∀𝑝∈Ω; (146) that is, 𝑤∈Ξ. Therefore, 𝜔_𝑤(𝑥_𝑛)⊂Ξ. This completes the proof.

Theorem 20. Assume that there hold all the conditions in Theorem 19. Then we have the following.(i){𝑥_𝑛} converges strongly to a point 𝑥^∗∈Ω provided that lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, which is a unique solution of the VIP: ⟨(𝛾𝑉−𝜇𝐹)𝑥^∗,𝑝−𝑥^∗⟩≤0, for all 𝑝∈Ω; equivalently, 𝑃_Ω𝐼−𝜇𝐹−𝛾𝑉𝑥^∗=𝑥^∗; (147)(ii){𝑥_𝑛} converges strongly to a unique solution of THVI (24) provided that ‖𝑥_𝑛−𝑦_𝑛‖=𝑜(𝛼_𝑛) additionally.

Proof . Observe that ≥𝜇𝐹−𝛾𝑉𝑥−𝜇𝐹−𝛾𝑉𝑦,𝑥−𝑦𝜇𝜂−𝛾𝑙𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (148) 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 𝛾𝑉−𝜇𝐹𝑥^∗,𝑝−𝑥^∗≤0,∀𝑝∈Ω. (149) Equivalently, 𝑥^∗=𝑃_Ω(𝐼−(𝜇𝐹−𝛾𝑉))𝑥^∗. Now, let us show that limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗≤0. (150) Since {𝑥_𝑛} is bounded, we may assume, without loss of generality, that there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑤 and limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛𝑖−𝑥^∗=𝛾𝑉−𝜇𝐹𝑥^∗,𝑤−𝑥^∗. (151) In terms of Theorem 19(ii), we know that 𝑤∈𝜔_𝑤(𝑥_𝑛)⊂Ω. So, from (149) it follows that limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗=𝛾𝑉−𝜇𝐹𝑥^∗,𝑤−𝑥^∗≤0. (152)
Next, let us show that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0. In fact, put 𝑝=𝑥^∗ in (84). Then from (54) we get 𝑥_𝑛+1−𝑥^∗2=𝑠_𝑛𝛾𝑉𝑥_𝑛−𝑉𝑥^∗+𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑦_𝑛−𝐼−𝑠_𝑛𝜇𝐹𝑥^∗+𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗2≤𝑠_𝑛𝛾(𝑉𝑥_𝑛−𝑉𝑥^∗)+(𝐼−𝑠_𝑛𝜇𝐹)𝑇_𝑛𝑦_𝑛−(𝐼−𝑠_𝑛𝜇𝐹)𝑥^∗)²+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑠_𝑛𝛾𝑉𝑥_𝑛−𝑉𝑥^∗+(𝐼−𝑠_𝑛𝜇𝐹)𝑇_𝑛𝑦_𝑛−(𝐼−𝑠_𝑛𝜇𝐹)𝑥^∗2+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑠_𝑛𝛾𝑙𝑥_𝑛−𝑥^∗+1−𝑠_𝑛𝜏𝑦_𝑛−𝑥^∗2+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝑠_𝑛𝜏𝛾𝑙𝜏𝑥_𝑛−𝑥^∗+1−𝑠_𝑛𝜏𝑦_𝑛−𝑥^∗2+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑠_𝑛𝜏𝛾𝑙²𝜏²𝑥_𝑛−𝑥^∗2+1−𝑠_𝑛𝜏𝑦_𝑛−𝑥^∗2+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑠_𝑛𝛾𝑙²𝜏𝑥_𝑛−𝑥^∗2+1−𝑠_𝑛𝜏×𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑦_𝑛−𝑥^∗+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗=1−𝑠_𝑛𝜏²−𝛾𝑙²𝜏𝑥_𝑛−𝑥^∗2+21−𝑠_𝑛𝜏𝛼_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑦_𝑛−𝑥^∗+2𝑠_𝑛𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗=1−𝑠_𝑛𝜏²−𝛾𝑙²𝜏𝑥_𝑛−𝑥^∗2+𝑠_𝑛𝜏²−𝛾𝑙²𝜏⋅𝜏𝜏²−𝛾𝑙²×21−𝑠_𝑛𝜏𝛼_𝑛𝑠_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑦_𝑛−𝑥^∗+2𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗. (153) Since ∑^∞_𝑛=1𝑠_𝑛=∞, lim_𝑛→∞(𝛼_𝑛/𝑠_𝑛)=0, and limsup_𝑛→∞⟨(𝛾𝑉−𝜇𝐹)𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩≤0 (due to (152)), we deduce that ∑^∞_𝑛=1𝑠_𝑛((𝜏²−(𝛾𝑙)²)/𝜏)=∞ and limsup_𝑛→∞𝜏𝜏²−𝛾𝑙²×21−𝑠_𝑛𝜏𝛼_𝑛𝑠_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑦_𝑛−𝑥^∗+2𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤0. (154) Therefore, applying Lemma 16 to (153) we infer that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0.
On the other hand, let us suppose that ‖𝑥_𝑛−𝑦_𝑛‖=𝑜(𝛼_𝑛). Then by Theorem 19(iii) we know that 𝜔_𝑤(𝑥_𝑛)⊂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 𝛾𝑉𝑥^∗−𝜇𝐹𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈Ξ. (155) Since the sequence {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛𝑖−𝑥^∗. (156) Also, since 𝐻 is reflexive and {𝑥_𝑛} is bounded, without loss of generality we may assume that 𝑥_𝑛𝑖⇀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 limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛾𝑉−𝜇𝐹𝑥^∗,𝑥−𝑥^∗≤0. (157) Repeating the same argument as in (153) we immediately conclude that 𝑥_𝑛+1−𝑥^∗2≤1−𝑠_𝑛𝜏²−𝛾𝑙²𝜏𝑥_𝑛−𝑥^∗2+𝑠_𝑛𝜏²−𝛾𝑙²𝜏⋅𝜏𝜏²−𝛾𝑙²×21−𝑠_𝑛𝜏𝛼_𝑛𝑠_𝑛𝛾𝑆−𝜇𝐹𝑥^∗,𝑦_𝑛−𝑥^∗+2𝛾𝑉−𝜇𝐹𝑥^∗,𝑥_𝑛+1−𝑥^∗. (158) Repeating the same arguments as above, we can readily see that lim_𝑛→∞‖𝑥_𝑛−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, {𝑥_𝑛} is bounded, Fix(𝑇)∩int𝐶≠0/ and ‖𝑥−𝑇𝑥‖≥𝑘Dist(𝑥,Fix(𝑇)), for all 𝑥∈𝐶 for some 𝑘>0, the sequence {𝑥_𝑛} generated by (54) converges strongly to a point 𝑥^∗∈(∩^∞_𝑛=1Fix(𝑆_𝑛))∩(∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘))∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))∩GSVI(𝐺)∩Γ=∶Ω, which is a unique solution of the VIP: ⟨𝛾𝑉𝑥^∗−𝜇𝐹𝑥^∗,𝑥−𝑥^∗⟩≤0, for all 𝑥∈Ω; equivalently, 𝑃_Ω(𝐼−(𝜇𝐹−𝛾𝑉))𝑥^∗=𝑥^∗ (see Theorem 20(i)).

Remark 22. Our Theorems 19 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 𝑥^∗=𝑃_(∩^∞_{𝑛=1Fix(𝑆𝑛))∩(∩}^𝑀_{𝑘=1GMEP(Θ𝑘,𝜑𝑘,𝐵𝑘))∩(∩}^𝑁_{𝑖=1VI(𝐶,𝐴𝑖))∩GSVI(𝐺)∩Γ}×𝐼−𝜇𝐹−𝛾𝑆𝑥^∗ (159) is more general than the problem of finding a point 𝑥∈𝐶 satisfying 𝑥=𝑃_Fix(𝑇)𝑆𝑥 in [33] and than the problem of finding a point 𝑥^∗∈(∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘))∩(∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖))∩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=1Fix(𝑆_𝑛), ∩^𝑀_𝑘=1GMEP(Θ_𝑘,𝜑_𝑘,𝐵_𝑘), ∩^𝑁_𝑖=1VI(𝐶,𝐴_𝑖), GSVI(𝐺), and Γ. In addition, it also drops the crucial requirements that Fix(𝑇)∩int𝐶≠0/ 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 Theorems 19 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 {𝑆_𝑛}^∞_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 𝜔_𝑤(𝑥_𝑛)⊂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 {𝑆_𝑛}^∞_𝑛=1, but the problem in [23, Theorem 21] involves no HFPP for nonexpansive mappings.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This work was supported partly by the National Science Council of the Republic of China. This research was partially supported by a Grant from NSC.