Iterative Schemes for Convex Minimization Problems with Constraints

Abstract

We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.

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 𝑉 is called strongly positive on 𝐻 if there exists a constant 𝛾>0 such that ⟨𝑉𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1) 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. (2) 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. (3) The solution set of VIP (3) is denoted by VI(𝐶,𝐴).

Let 𝜑∶𝐶→𝐑 be a real-valued function, 𝐴∶𝐻→𝐻 a nonlinear mapping, and Θ∶𝐶×𝐶→𝐑 a bifunction. The generalized mixed equilibrium problem (GMEP) introduced in [1] is to find 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).

We assume as in [1] that Θ∶𝐶×𝐶→𝐑 is a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 is a lower semicontinuous and convex function with restriction (H5), where(H1)Θ(𝑥,𝑥)=0 for all 𝑥∈𝐶;(H2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(H3)Θ is upper-hemicontinuous, that is, for each 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡→0+Θ𝑡𝑧+(1−𝑡)𝑥,𝑦≤Θ𝑥,𝑦; (5)(H4)Θ(𝑥,⋅) is convex and lower semicontinuous for each 𝑥∈𝐶;(H5)for each 𝑥∈𝐻 and 𝑟>0, there exist 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. (6)

Given a positive number, 𝑟>0. Let 𝑆_𝑟^(Θ,𝜑)∶𝐻→𝐶 be the solution set of the auxiliary mixed equilibrium problem; that is, for each 𝑥∈𝐻, 𝑆_𝑟^(Θ,𝜑)(𝑥)(𝑧)𝑦+1∶=𝑦∈𝐶∶Θ𝑦,𝑧+𝜑−𝜑𝑟⟨𝐾^′𝑦−𝐾^′(𝑥),𝑧−𝑦⟩≥0,∀𝑧∈𝐶. (7) In particular, whenever 𝐾(𝑥)=(1/2)‖𝑥‖², for  all 𝑥∈𝐻,𝑆_𝑟^(Θ,𝜑)(𝑥) is rewritten as 𝑇_𝑟^(Θ,𝜑)(𝑥).

Let Θ₁,Θ₂∶𝐶×𝐶→𝐑 be two bifunctions and 𝐴₁,𝐴₂∶𝐶→𝐻 two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP) [2]: find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that Θ₁𝑥^∗,𝑥+𝐴₁𝑦^∗,𝑥−𝑥^∗+1𝜈₁𝑥^∗−𝑦^∗,𝑥−𝑥^∗Θ≥0,∀𝑥∈𝐶,₂𝑦^∗,𝑦+𝐴₂𝑥^∗,𝑦−𝑦^∗+1𝜈₂𝑦^∗−𝑥^∗,𝑦−𝑦^∗≥0,∀𝑦∈𝐶, (8) where 𝜈₁>0 and 𝜈₂>0 are two constants.

In 2010, Ceng and Yao [2] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [2]). Let Θ₁,Θ₂∶𝐶×𝐶→𝐑 be two bifunctions satisfying conditions (H1)–(H4) and let 𝐴_𝑘∶𝐶→𝐻 be 𝜁_𝑘-inverse-strongly monotone for 𝑘=1,2. Let 𝜈_𝑘∈(0,2𝜁_𝑘) for 𝑘=1,2. Then, (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of SGEP (8) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺=𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂) where 𝑦^∗=𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑥^∗. Here, we denote the fixed point set of 𝐺 by 𝑆𝐺𝐸𝑃(𝐺).

Let {𝑇_𝑛}^∞_𝑛=1 be an infinite family of nonexpansive mappings on 𝐻 and {𝜆_𝑛}^∞_𝑛=1 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_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜆_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜆_𝑛−1⋮𝑈𝐼,_𝑛,𝑘=𝜆_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜆_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜆_𝑘−1⋮𝑈𝐼,_𝑛,2=𝜆₂𝑇₂𝑈_𝑛,3+1−𝜆₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝐼. (9) Such a mapping 𝑊_𝑛 is called the 𝑊-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝜆_𝑛,𝜆_𝑛−1,…,𝜆₁.

Let 𝑄∶𝐻→𝐻 be a contraction and 𝑉 a strongly positive bounded linear operator on 𝐻. Assume that 𝜑∶𝐻→𝐑 is a lower semicontinuous and convex functional, that Θ,Θ₁,Θ₂∶𝐻×𝐻→𝐑 satisfy conditions (H1)–(H4), and that 𝐴,𝐴₁,𝐴₂∶𝐻→𝐻 are inverse-strongly monotone. Let the mapping 𝐺 be defined as in Proposition CY. Very recently, Ceng et al. [3] introduced the following hybrid extragradient-like iterative scheme: 𝑧_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛾𝑄𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑊_𝑛𝐺𝑧_𝑛,∀𝑛≥0, (10) for finding a common solution of GMEP (4), SGEP (8), and the fixed point problem of an infinite family of nonexpansive mappings {𝑇_𝑛}^∞_𝑛=1 on 𝐻, where {𝑟_𝑛}⊂(0,∞),{𝛼_𝑛}, {𝛽_𝑛}⊂(0,1), 𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2, and 𝑥₀,𝑢∈𝐻 are given. The authors proved the strong convergence of the sequence {𝑥_𝑛} to a point 𝑥^∗∈Ω∶=∩^∞_𝑛=1Fix(𝑇_𝑛)∩GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺) under some suitable conditions. This point 𝑥^∗ also solves the following optimization problem: min_𝑥∈Ω𝜇2⟨⟩+1𝑉𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (OP1) where ℎ∶𝐻→𝐑 is the potential function of 𝛾𝑄.

Let 𝐵 be a single-valued mapping of 𝐶 into 𝐻 and 𝑅 a multivalued mapping with 𝐷(𝑅)=𝐶. Consider the following variational inclusion: find a point 𝑥∈𝐶 such that 0∈𝐵𝑥+𝑅𝑥. (11) We denote by 𝐼(𝐵,𝑅) the solution set of the variational inclusion (11). In 1998, Huang [4] studied problem (11) in the case where 𝑅 is maximal monotone and 𝐵 is strongly monotone and Lipschitz continuous with 𝐷(𝑅)=𝐶=𝐻.

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 𝐶: minimize𝑓(𝑥)∶𝑥∈𝐶. (12) We denote by Γ the set of minimizers of CMP (12).

Very recently, Ceng and Al-Homidan [5] introduced an implicit iterative algorithm for finding a common solution of the CMP (12), finitely many GMEPs and finitely many variational inclusions, and derived its strong convergence under appropriate conditions.

Algorithm CA (see [5, Theorem  18]). 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 (H1)–(H4) and 𝜑_𝑘∶𝐶→𝐑∪{+∞} a proper lower semicontinuous and convex function, where 𝑘∈{1,2,…,𝑀}. Let 𝑅_𝑖∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴_𝑘∶𝐻→𝐻 and 𝐵_𝑖∶𝐶→𝐻 be 𝜇_𝑘-inverse strongly monotone and 𝜂_𝑖-inverse strongly monotone, respectively, where 𝑘∈{1,2,…,𝑀},  𝑖∈{1,2,…,𝑁}. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Let {𝑥_𝑛} be a sequence generated by 𝑢_𝑛=𝑇^{(Θ𝑀,𝜑𝑀)}_{𝑟𝑀,𝑛}𝐼−𝑟_𝑀,𝑛𝐴_𝑀𝑇^{(Θ𝑀−1,𝜑𝑀−1)}_{𝑟𝑀−1,𝑛}×𝐼−𝑟_{𝑀−1,𝑛}𝐴_𝑀−1⋯𝑇^(Θ1,𝜑1)_𝑟1,𝑛𝐼−𝑟_1,𝑛𝐴₁𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅𝑁,𝜆𝑁,𝑛}𝐼−𝜆_𝑁,𝑛𝐵_𝑁𝐽_{𝑅𝑁−1,𝜆𝑁−1,𝑛}×𝐼−𝜆_{𝑁−1,𝑛}𝐵_𝑁−1⋯𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁𝑢_𝑛,𝑥_𝑛=𝑠_𝑛𝛾𝑄𝑥_𝑛+𝐼−𝑠_𝑛𝜇𝐹𝑇_𝑛𝑣_𝑛,∀𝑛≥1, (13) where 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛 (here 𝑇_𝑛 is nonexpansive, 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿))), and the following conditions hold:(i)𝑠_𝑛∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)), lim_𝑛→∞𝑠_𝑛=0 (⇔lim_𝑛→∞𝜆_𝑛=(2/𝐿));(ii){𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  for  all  𝑖∈{1,2,…,𝑁};(iii){𝑟_𝑘,𝑛}⊂[𝑒_𝑘,𝑓_𝑘]⊂(0,2𝜇_𝑘),  for  all  𝑘∈{1,2,…,𝑀}.
Motivated and inspired by the above facts, we first introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. The iterative algorithm is based on Korpelevich's extragradient method, hybrid steepest-descent method in [6], viscosity approximation method, averaged mapping approach to the GPA in [7], and strongly positive bounded linear operator technique. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints. We derive its strong convergence under mild assumptions.

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_𝑛𝑖⇀𝑥forsomesubsequence𝑥_𝑛𝑖of𝑥_𝑛. (14)

Recall that a mapping 𝐴∶𝐶→𝐻 is called(i)monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶; (15)(ii)𝜂-strongly monotone if there exists a constant 𝜂>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶; (16)(iii)𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (17)

It is obvious that if 𝐴 is 𝛼-inverse-strongly monotone, then 𝐴 is monotone and (1/𝛼)-Lipschitz continuous.

The metric (or nearest point) 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). (18) 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.

If 𝐴 is an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐻, then it is obvious that 𝐴 is (1/𝛼)-Lipschitz continuous. If 𝜆≤2𝛼, then it is easy to see that 𝐼−𝜆𝐴 is a nonexpansive mapping from 𝐶 to 𝐻.

Definition 2. A mapping 𝑇∶𝐻→𝐻 is said to be(a)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻; (19)(b)firmly nonexpansive if 2𝑇−𝐼 is nonexpansive, or equivalently, if 𝑇 is 1-inverse strongly monotone (1-ism), 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐻; (20) alternatively, 𝑇 is firmly nonexpansive if and only if 𝑇 can be expressed as 1𝑇=2(𝐼+𝑆), (21) 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. It is also easy to see that a projection 𝑃_𝐶 is 1-ism.

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, (22) 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 [8]). 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 [8]). 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, then 𝐹𝑖𝑥𝑇^𝑁_𝑖=1𝑖=𝐹𝑖𝑥𝑇₁⋅⋅⋅𝑇_𝑁. (23) The notation 𝐹𝑖𝑥(𝑇) denotes the set of all fixed points of the mapping 𝑇; that is, 𝐹𝑖𝑥(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}.
By using the technique in [9], we can readily obtain the following elementary result.

Proposition 6 (see [3, Lemma 1 and Proposition 1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex function. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying the conditions (H1)–(H4). Assume that(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii) 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−𝜑𝑟⟨𝐾^′(𝑧)−𝐾^′(𝑥),𝑦_𝑥−𝑧⟩<0. (24) Then the following hold:(a) for each 𝑥∈𝐻, 𝑆_𝑟^(Θ,𝜑)(𝑥)≠∅;(b)𝑆_𝑟^(Θ,𝜑) is single-valued;(c)𝑆_𝑟^(Θ,𝜑) is nonexpansive if 𝐾^′ is Lipschitz continuous with constant 𝜈>0 and⟨𝐾^′𝑥₁−𝐾^′𝑥₂,𝑢₁−𝑢₂⟩≤⟨𝐾^′𝑢₁−𝐾^′𝑢₂,𝑢₁−𝑢₂⟩,∀𝑥₁,𝑥₂∈𝐻×𝐻, (25) where 𝑢_𝑖=𝑆_𝑟^(Θ,𝜑)(𝑥_𝑖) for 𝑖=1,2;(d) for all 𝑠,𝑡>0 and 𝑥∈𝐻𝐾^′𝑆_𝑠^(Θ,𝜑)𝑥−𝐾^′𝑆_𝑡^(Θ,𝜑)𝑥,𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥≤𝑠−𝑡𝑠𝐾^′𝑆_𝑠^(Θ,𝜑)𝑥−𝐾^′(𝑥),𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥; (26)(e)𝐹𝑖𝑥(𝑆_𝑟^(Θ,𝜑))=𝑀𝐸𝑃(Θ,𝜑);(f)𝑀𝐸𝑃(Θ,𝜑) is closed and convex.

Remark 7. In Proposition 4, whenever Θ∶𝐶×𝐶→𝐑 is a bifunction satisfying the conditions (H1)–(H4) and 𝐾(𝑥)=(1/2)‖𝑥‖²,  for  all  𝑥∈𝐻, we have for any 𝑥,𝑦∈𝐻, 𝑆_𝑟^(Θ,𝜑)𝑥−𝑆_𝑟^(Θ,𝜑)𝑦²≤𝑆_𝑟^(Θ,𝜑)𝑥−𝑆_𝑟^(Θ,𝜑)𝑦,𝑥−𝑦 (27) (𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive) and 𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥≤|𝑠−𝑡|𝑠𝑆_𝑠^(Θ,𝜑)𝑥−𝑥,∀𝑠,𝑡>0,𝑥∈𝐻. (28) In this case, 𝑆_𝑟^(Θ,𝜑) is rewritten as 𝑇_𝑟^(Θ,𝜑). If, in addition, 𝜑≡0, then 𝑇_𝑟^(Θ,𝜑) is rewritten as 𝑇^Θ_𝑟; see [2, Lemma 2.1] for more details.

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

Lemma 8. Let 𝑋 be a real inner product space. Then there holds the following inequality: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝑋. (29)

Lemma 9. 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 𝑢∈𝑉𝐼(𝐶,𝐴)⟺𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢),𝜆>0. (30)

Lemma 10 (see [10, 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.

Lemma 11 (see [11, Lemma 3.2]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive self-mappings on 𝐶 such that ∩^∞_𝑛=1𝐹𝑖𝑥(𝑇_𝑛)≠∅ 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 (9).

Lemma 12 (see [11, Lemma 3.3]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive self-mappings on 𝐶 such that ∩^∞_𝑛=1𝐹𝑖𝑥(𝑇_𝑛)≠∅, and let {𝜆_𝑛} be a sequence in (0,𝑏] for some 𝑏∈(0,1). Then, 𝐹𝑖𝑥(𝑊)=∩^∞_𝑛=1𝐹𝑖𝑥(𝑇_𝑛).

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

Lemma 13. Let 𝑉∶𝐻→𝐻 be a 𝛾-strongly positive bounded linear operator with constant 𝛾>1. Then for 𝛾−1>0, (𝑉−𝐼)𝑥−(𝑉−𝐼)𝑦,𝑥−𝑦≥𝛾−1𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (31) That is, 𝑉−𝐼 is strongly monotone with constant 𝛾−1.

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, (32) 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 conditions 𝐹𝑥−𝐹𝑦≤𝜅𝑥−𝑦,⟨𝐹𝑥−𝐹𝑦,𝑥−𝑦⟩≥𝜂𝑥−𝑦² (33) for all 𝑥,𝑦∈𝐻.

Lemma 14 (see [12, Lemma  3.1]). 𝑇^𝜆 is a contraction provided 0<𝜇<(2𝜂/𝜅²); that is, 𝑇^𝜆𝑥−𝑇^𝜆𝑦≤(1−𝜆𝜏)𝑥−𝑦,∀𝑥,𝑦∈𝐶, (34) where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²)∈(0,1].

We observe that since 𝐹 is 𝜅-Lipschitzian and 𝜂-strongly monotone on 𝐻, we get 0<𝜂≤𝜅. Hence, whenever 0<𝜇<(2𝜂/𝜅²), we have 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²)∈(0,1]. Also in Lemma 14, put 𝐹=(1/2)𝐼 and 𝜇=2. Then we know that 𝜅=𝜂=(1/2),0<𝜇=2<(2𝜂/𝜅²)=4 and 𝜏=1.

Recall that a set-valued mapping 𝑅∶𝐷(𝑅)⊂𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐷(𝑅), 𝑓∈𝑅(𝑥) and 𝑔∈𝑅(𝑦) imply ⟨𝑓−𝑔,𝑥−𝑦⟩≥0. (35) A set-valued mapping 𝑅 is called maximal monotone if 𝑅 is monotone and (𝐼+𝜆𝑅)𝐷(𝑅)=𝐻 for each 𝜆>0, where 𝐼 is the identity mapping of 𝐻. We denote by 𝐺(𝑅) the graph of 𝑅. It is known that a monotone mapping 𝑅 is maximal if and only if, for (𝑥,𝑓)∈𝐻×𝐻, ⟨𝑓−𝑔,𝑥−𝑦⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑅) implies 𝑓∈𝑅(𝑥).

Let 𝐴∶𝐶→𝐻 be a monotone, 𝑘-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}. (36) Define 𝑇𝑣=𝐴𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (37) It is well known that 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈𝑉𝐼(𝐶,𝐴).

Assume that 𝑅∶𝐷(𝑅)⊂𝐻→2^𝐻 is a maximal monotone mapping. Then, for 𝜆>0, associated with 𝑅, the resolvent operator 𝐽_𝑅,𝜆 can be defined as 𝐽_𝑅,𝜆𝑥=(𝐼+𝜆𝑅)⁻¹𝑥,∀𝑥∈𝐻. (38) From Huang [4] (see also [13]), there holds the following property for the resolvent operator 𝐽_𝑅,𝜆∶𝐻→𝐷(𝑅).

Lemma 15. 𝐽_𝑅,𝜆 is single-valued and firmly nonexpansive; that is, ⟨𝐽_𝑅,𝜆𝑥−𝐽_𝑅,𝜆𝑦,𝑥−𝑦⟩≥𝐽_𝑅,𝜆𝑥−𝐽_𝑅,𝜆𝑦²,∀𝑥,𝑦∈𝐻. (39) Consequently, 𝐽_𝑅,𝜆 is nonexpansive and monotone.

Lemma 16 (see [14]). Let 𝑅 be a maximal monotone mapping with 𝐷(𝑅)=𝐶. Then for any given 𝜆>0, 𝑢∈𝐶 is a solution of problem (11) if and only if 𝑢∈𝐶 satisfies 𝑢=𝐽_𝑅,𝜆(𝑢−𝜆𝐵𝑢). (40)

Lemma 17 (see [13]). Let 𝑅 be a maximal monotone mapping with 𝐷(𝑅)=𝐶 and let 𝐵∶𝐶→𝐻 be a strongly monotone, continuous, and single-valued mapping. Then for each 𝑧∈𝐻, the equation 𝑧∈(𝐵+𝜆𝑅)𝑥 has a unique solution 𝑥_𝜆 for 𝜆>0.

Lemma 18 (see [14]). Let 𝑅 be a maximal monotone mapping with 𝐷(𝑅)=𝐶 and 𝐵∶𝐶→𝐻 a monotone, continuous, and single-valued mapping. Then (𝐼+𝜆(𝑅+𝐵))𝐶=𝐻 for each 𝜆>0. In this case, 𝑅+𝐵 is maximal monotone.

3. Convex Minimization Problems with Constraints

In this section, we will introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We po prove strong convergence theorem for the iterative algorithm under suitable conditions.

Theorem 19. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑁 be an integer. Let 𝑓∶𝐶→𝐑 be a convex functional with 𝐿-Lipschitz continuous gradient ∇𝑓. Let Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex functional. Let 𝑅_𝑖∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴∶𝐻→𝐻 and let 𝐴_𝑘,𝐵_𝑖∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone, and 𝜂_𝑖-inverse-strongly monotone, respectively, for 𝑘=1,2 and 𝑖=1,2,…,𝑁. Let 𝑉 be a 𝛾-strongly positive bounded linear operator with 𝛾>1. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=𝐺𝑀𝐸𝑃(Θ,𝜑,𝐴)∩𝑆𝐺𝐸𝑃(𝐺)∩∩^𝑁_𝑖=1𝐼(𝐵_𝑖,𝑅_𝑖)∩Γ≠∅ where 𝐺 is defined as in Proposition CY. Let {𝑟_𝑛} be a sequence in (0,2𝜁] and {𝛼_𝑛} a sequence in (0,1] such that lim_𝑛→∞𝛼_𝑛=0. Let {𝑥_𝑛} be the sequence generated by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅𝑁,𝜆𝑁,𝑛}𝐼−𝜆_𝑁,𝑛𝐵_𝑁𝐽_{𝑅𝑁−1,𝜆𝑁−1,𝑛}×𝐼−𝜆_{𝑁−1,𝑛}𝐵_𝑁−1⋯𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁𝑢_𝑛,𝑥_𝑛=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛,∀𝑛≥1, (41) where 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛 (here 𝑇_𝑛 is nonexpansive, 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿))). Suppose that the following conditions hold.(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that, for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟⟨𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦⟩<0; (42)(iii)𝑠_𝑛∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)), lim_𝑛→∞𝑠_𝑛=0  (⇔lim_𝑛→∞𝜆_𝑛=(2/𝐿));(iv)𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2 and {𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  𝑖=1,2,…,𝑁;(v)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁. Assume that 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive. Then {𝑥_𝑛} converges strongly as 𝜆_𝑛→(2/𝐿)(⇔𝑠_𝑛→0) to a point 𝑞∈Ω, which is a unique solution in Ω to the VIP: ⟨(𝐼−𝑉)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (43)

Proof. First of all, let us show that the sequence {𝑥_𝑛} is well defined. Indeed, since ∇𝑓 is 𝐿-Lipschitzian, it follows that ∇𝑓 is 1/𝐿-ism. By Proposition 4(ii) we know that, for 𝜆>0,𝜆∇𝑓 is (1/𝜆𝐿)-ism. So by Proposition 4(iii) we deduce that 𝐼−𝜆∇𝑓 is (𝜆𝐿/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−𝜆∇𝑓) is ((2+𝜆𝐿)/4)-averaged for 𝜆∈(0,(2/𝐿)). Hence we obtain that, for each 𝑛≥1, 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓) is ((2+𝜆_𝑛𝐿)/4)-averaged for each 𝜆_𝑛∈(0,(2/𝐿)). Therefore, we can write 𝑃_𝐶𝐼−𝜆_𝑛∇𝑓=2−𝜆_𝑛𝐿4𝐼+2+𝜆_𝑛𝐿4𝑇_𝑛=𝑠_𝑛𝐼+1−𝑠_𝑛𝑇_𝑛, (44) where 𝑇_𝑛 is nonexpansive and 𝑠_𝑛∶=𝑠_𝑛(𝜆_𝑛)=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)). It is clear that 𝜆_𝑛⟶2𝐿⟺𝑠_𝑛⟶0. (45) As lim_𝑛→∞𝑠_𝑛=0 and 0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁, we may assume, without loss of generality, that {𝑟_𝑛}⊂[𝑎,𝑎]⊂(0,2𝜁) and 𝑠_𝑛‖𝑉‖≤1 for all 𝑛≥1. Since 𝑉 is a 𝛾-strongly positive bounded linear operator on 𝐻, we know that ‖𝑉‖=sup{⟨𝑉𝑢,𝑢⟩∶𝑢∈𝐻,‖𝑢‖=1}≥𝛾>1. (46) Taking into account that 𝑠_𝑛‖𝑉‖≤1 for all 𝑛≥1, we have 𝐼−𝑠_𝑛𝑉𝑢,𝑢=1−𝑠_𝑛⟨𝑉𝑢,𝑢⟩≥1−𝑠_𝑛‖𝑉‖≥0, (47) that is, (1−𝛽_𝑛)𝐼−𝑠_𝑛𝑉 is positive. It follows that 𝐼−𝑠_𝑛𝑉=sup⟨𝐼−𝑠_𝑛𝑉𝑢,𝑢⟩∶𝑢∈𝐻,‖𝑢‖=1=sup1−𝑠_𝑛⟨𝑉𝑢,𝑢⟩∶𝑢∈𝐻,‖𝑢‖=1≤1−𝑠_𝑛𝛾. (48) Put Λ^𝑖_𝑛=𝐽_{𝑅𝑖,𝜆𝑖,𝑛}𝐼−𝜆_𝑖,𝑛𝐵_𝑖𝐽_{𝑅𝑖−1,𝜆𝑖−1,𝑛}×𝐼−𝜆_{𝑖−1,𝑛}𝐵_𝑖−1⋯𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁ (49) for all 𝑖∈{1,2,…,𝑁}, and Λ⁰_𝑛=𝐼, where 𝐼 is the identity mapping on 𝐻. Then we have 𝑣_𝑛=Λ^𝑁_𝑛𝑢_𝑛.
Consider the following mapping 𝐺_𝑛 on 𝐻 defined by 𝐺_𝑛𝑥=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥+𝑠_𝑛𝑇_𝑛𝑥−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥−𝛾𝑄𝑥,∀𝑥∈𝐻,𝑛≥1, (50) where 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)). Since 𝐴_𝑘 is 𝜁_𝑘-inverse-strongly monotone with 0≤𝜈_𝑘≤2𝜁_𝑘 for 𝑘=1,2, we deduce that. for any 𝑥,𝑦∈𝐻, 𝐺𝑥−𝐺𝑦²=𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑥−𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑦²≤𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑥−𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑦²=𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑥−𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑦−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑥−𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑦²≤𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑥−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑦²+𝜈₁𝜈₁−2𝜁₁×𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑥−𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑦²≤𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑥−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑦²≤(𝐼−𝜈₂𝐴₂)𝑥−(𝐼−𝜈₂𝐴₂)𝑦²=(𝑥−𝑦)−𝜈₂(𝐴₂𝑥−𝐴₂𝑦)²≤𝑥−𝑦²+𝜈₂𝜈₂−2𝜁₂𝐴₂𝑥−𝐴₂𝑦²≤𝑥−𝑦². (51) By the nonexpansivity of 𝑆_𝑟^(Θ,𝜑) and Lemma 14 we obtain from (41), (48), and (51) that for all 𝑥,𝑦∈𝐶𝐺_𝑛𝑥−𝐺_𝑛𝑦≤𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−𝑇_𝑛𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑦+𝑠_𝑛𝛼_𝑛𝛾≤𝑄𝑥−𝑄𝑦1−𝑠_𝑛𝛾×𝑇_𝑛𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−𝑇_𝑛𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾×𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾×Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾×𝐼−𝜆_𝑁,𝑛𝐵_𝑁Λ_𝑛^𝑁−1𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−𝐼−𝜆_𝑁,𝑛𝐵_𝑁Λ_𝑛^𝑁−1𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾×Λ_𝑛^𝑁−1𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−Λ_𝑛^𝑁−1𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛⋮≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾Λ⁰_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−Λ⁰_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛=𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾𝐼−𝑟_𝑛𝐴𝑥−𝐼−𝑟_𝑛𝐴𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛾𝑥−𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛≤𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝑥−𝑦+𝑠_𝑛1−𝛼_𝑛𝜏𝑥−𝑦+𝑠_𝑛𝛼_𝑛=𝛾𝑙𝑥−𝑦1−𝑠_𝑛𝛼_𝑛𝜏−𝛾𝑙𝑥−𝑦. (52) Since 0<1−𝑠_𝑛𝛼_𝑛(𝜏−𝛾𝑙)<1, 𝐺_𝑛∶𝐻→𝐻 is a contraction. Therefore, by the Banach contraction principle, 𝐺_𝑛 has a unique fixed point 𝑥_𝑛∈𝐻, which uniquely solves the fixed point equation 𝑥_𝑛=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛. (53) This shows that the sequence {𝑥_𝑛} is defined well.
Note that (𝑉−𝐼)𝑥−(𝑉−𝐼)𝑦,𝑥−𝑦≥𝛾−1𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (54) That is, 𝑉−𝐼 is (𝛾−1)-strongly monotone for 𝛾−1>0. Moreover, it is clear that 𝑉−𝐼 is Lipschitz continuous. So the VIP (43) has only one solution in Ω. Below we use 𝑞∈Ω to denote the unique solution of the VIP (43).
Now, let us show that {𝑥_𝑛} is bounded. In fact, take 𝑝∈Ω arbitrarily. Then from the nonexpansivity of 𝑆_𝑟^(Θ,𝜑) and {𝑟_𝑛}⊂[𝑎,𝑎]⊂(0,2𝜁) we have 𝑢_𝑛−𝑝²=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑝²≤(𝐼−𝑟_𝑛𝐴)𝑥_𝑛−(𝐼−𝑟_𝑛𝐴)𝑝²=(𝑥_𝑛−𝑝)−𝑟_𝑛(𝐴𝑥_𝑛−𝐴𝑝)²=𝑥_𝑛−𝑝²−2𝑟_𝑛⟨𝑥_𝑛−𝑝,𝐴𝑥_𝑛−𝐴𝑝⟩+𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝²−2𝑟_𝑛𝜁𝐴𝑥_𝑛−𝐴𝑝²+𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑝²=𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝜁𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝². (55) Since 𝑝=𝐽_{𝑅𝑖,𝜆𝑖,𝑛}(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)𝑝, Λ^𝑖_𝑛𝑝=𝑝 and 𝐵_𝑖 is 𝜂_𝑖-inverse strongly monotone, where 𝜆_𝑖,𝑛∈(0,2𝜂_𝑖),  𝑖∈{1,2,…,𝑁}, by Lemma 15 we deduce that for each 𝑛≥1𝑣_𝑛−𝑝²=Λ^𝑁_𝑛𝑢_𝑛−𝑝²≤Λ^𝑖_𝑛𝑢_𝑛−𝑝²=𝐽_{𝑅𝑖,𝜆𝑖,𝑛}(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)Λ_𝑛^𝑖−1𝑢_𝑛−𝐽_{𝑅𝑖,𝜆𝑖,𝑛}(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)𝑝²≤(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)Λ_𝑛^𝑖−1𝑢_𝑛−(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)𝑝²≤Λ_𝑛^𝑖−1𝑢_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖×𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²≤𝑢_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²≤𝑢_𝑛−𝑝². (56) Combining (55) and (56), we have 𝑣_𝑛−𝑝≤𝑥_𝑛−𝑝. (57) Since 𝑝=𝐺𝑝=𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝,𝐴_𝑘 is 𝜁_𝑘-inverse-strongly monotone for 𝑘=1,2, and 0≤𝜈_𝑘≤2𝜁_𝑘 for 𝑘=1,2, we deduce that, for any 𝑛≥1, 𝐺𝑣_𝑛−𝑝²=𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑣_𝑛−𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝²≤𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑣_𝑛−(𝐼−𝜈₁𝐴₁)𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²=𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑣_𝑛−𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑣_𝑛−𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝²≤𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑣_𝑛−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²+𝜈₁𝜈₁−2𝜁₁×𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑣_𝑛−𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²≤𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑣_𝑛−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²≤(𝐼−𝜈₂𝐴₂)𝑣_𝑛−(𝐼−𝜈₂𝐴₂)𝑝²=(𝑣_𝑛−𝑝)−𝜈₂(𝐴₂𝑣_𝑛−𝐴₂𝑝)²≤𝑣_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐴₂𝑣_𝑛−𝐴₂𝑝²≤𝑣_𝑛−𝑝². (58) Thus, utilizing Lemmas 8 and 14, from (41), (43), (55), (56), and (58) we have 𝑥_𝑛−𝑝²=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛−𝑝+(𝐼−𝑠_𝑛𝑉)𝑝−(1−𝑠_𝑛)𝑝²=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝜇𝐹𝑝+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑝+𝑠_𝑛(𝐼−𝑉)𝑝²=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝑄𝑝+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑝+𝑠_𝑛[(𝐼−𝑉)𝑝+𝛼_𝑛(𝛾𝑄−𝜇𝐹)𝑝]²≤𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝑄𝑝+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−(𝐼−𝛼_𝑛𝜇𝐹)𝑇_𝑛𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝑄𝑝+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑝+1−𝛼_𝑛𝜏𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝+𝑠_𝑛1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛=−𝑝⟩1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝+𝑠_𝑛𝛾1−𝛼_𝑛𝜏−𝛾𝑙𝛾𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝²+𝑠_𝑛𝛾1−𝛼_𝑛𝜏−𝛾𝑙²𝛾²𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝²+𝑠_𝑛1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝑣_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾𝑢_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛=−𝑝⟩1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (59) which implies that 𝑥_𝑛≤2−𝑝)𝑝+𝑝.𝛾−1(𝐼−𝑉𝛾𝑄−𝜇𝐹 (60) Hence {𝑥_𝑛} is bounded. So, according to (55) and (57) we know that {𝑢_𝑛},{𝑣_𝑛},{𝐺𝑣_𝑛}, {𝑇_𝑛𝐺𝑣_𝑛},{𝑇_𝑛𝑥_𝑛},{𝑄𝑥_𝑛}, and {𝐹𝑇_𝑛𝑥_𝑛} are bounded.
Next let us show that ‖𝑢_𝑛−𝑥_𝑛‖→0, ‖𝑣_𝑛−𝑢_𝑛‖→0, ‖𝑣_𝑛−𝐺𝑣_𝑛‖→0 and ‖𝑣_𝑛−𝑃_𝐶(𝐼−(2/𝐿)∇𝑓)𝑣_𝑛‖→0 as 𝑛→∞. Indeed, combining (55) and (59), we obtain 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝑢_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝜁𝐴𝑥_𝑛−𝐴𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑝²+1−𝑠_𝑛𝛾𝑟_𝑛𝑟_𝑛−2𝜁𝐴𝑥_𝑛−𝐴𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²+1−𝑠_𝑛𝛾𝑟_𝑛𝑟_𝑛−2𝜁𝐴𝑥_𝑛−𝐴𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (61) which immediately yields 1−𝑠_𝑛𝛾𝑎(2𝜁−𝑎)𝐴𝑥_𝑛−𝐴𝑝²≤1−𝑠_𝑛𝛾𝑟_𝑛2𝜁−𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝. (62) From lim_𝑛→∞𝑠_𝑛=0 and the boundedness of {𝑥_𝑛} we conclude that lim_𝑛→∞𝐴𝑥_𝑛−𝐴𝑝=0. (63) Furthermore, from the firm nonexpansivity of 𝑆_𝑟^(Θ,𝜑)_𝑛, we have 𝑢_𝑛−𝑝²=𝑆_𝑟^(Θ,𝜑)_𝑛(𝐼−𝑟_𝑛𝐴)𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛(𝐼−𝑟_𝑛𝐴)𝑝²≤𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑝,𝑢_𝑛=1−𝑝2(𝐼−𝑟_𝑛𝐴)𝑥_𝑛−(𝐼−𝑟_𝑛𝐴)𝑝²+𝑢_𝑛−𝑝²−𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑝−𝑢_𝑛−𝑝²≤12𝑥_𝑛−𝑝²+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛−𝑟_𝑛(𝐴𝑥_𝑛−𝐴𝑝)²=12𝑥_𝑛−𝑝²+𝑢_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛⟨𝐴𝑥_𝑛−𝐴𝑝,𝑥_𝑛−𝑢_𝑛⟩−𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑝², (64) which leads to 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝𝑥_𝑛−𝑢_𝑛. (65) From (59) and (65), we get 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝑢_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾×𝑥_𝑛−𝑝²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝𝑥_𝑛−𝑢_𝑛+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑝²−1−𝑠_𝑛𝛾𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝𝑥_𝑛−𝑢_𝑛+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝑠_𝑛𝛾𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝𝑥_𝑛−𝑢_𝑛+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (66) which hence implies that 1−𝑠_𝑛𝛾𝑥_𝑛−𝑢_𝑛²≤2𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝𝑥_𝑛−𝑢_𝑛+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛𝑎−𝑝≤2𝐴𝑥_𝑛−𝐴𝑝𝑥_𝑛−𝑢_𝑛+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝. (67) Since lim_𝑛→∞𝑠_𝑛=0 and {𝑥_𝑛} and {𝑢_𝑛} are bounded sequences, it follows from (63) that lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0. (68)
Next we show that lim_𝑛→∞‖𝐵_𝑖Λ^𝑖_𝑛𝑢_𝑛−𝐵_𝑖𝑝‖=0,  𝑖=1,2,…,𝑁. As a matter of fact, combining (55), (56), and (59), we have 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝑣_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝑢_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖×𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑥_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛=−𝑝1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑝²+1−𝑠_𝑛𝛾𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²+1−𝑠_𝑛𝛾𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖×𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (69) which leads to 1−𝑠_𝑛𝛾𝜆_𝑖,𝑛2𝜂_𝑖−𝜆_𝑖,𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²≤2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝. (70) Since lim_𝑛→∞𝑠_𝑛=0,{𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  𝑖=1,2,…,𝑁, and {𝑥_𝑛} is bounded, it follows that lim_𝑛→∞𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝=0. (71) Also, by Lemma 15, we obtain that for each 𝑖∈{1,2,…,𝑁}Λ^𝑖_𝑛𝑢_𝑛−𝑝²=𝐽_{𝑅𝑖,𝜆𝑖,𝑛}(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)Λ_𝑛^𝑖−1𝑢_𝑛−𝐽_{𝑅𝑖,𝜆𝑖,𝑛}(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)𝑝²≤𝐼−𝜆_𝑖,𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐼−𝜆_𝑖,𝑛𝐵_𝑖𝑝,Λ^𝑖_𝑛𝑢_𝑛=1−𝑝2(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)Λ_𝑛^𝑖−1𝑢_𝑛−(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)𝑝²+Λ^𝑖_𝑛𝑢_𝑛−𝑝²−(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)Λ_𝑛^𝑖−1𝑢_𝑛−(𝐼−𝜆_𝑖,𝑛𝐵_𝑖)𝑝−(Λ^𝑖_𝑛𝑢_𝑛−𝑝)²≤12Λ_𝑛^𝑖−1𝑢_𝑛−𝑝²+Λ^𝑖_𝑛𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛−𝜆_𝑖,𝑛(𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝)²≤12𝑢_𝑛−𝑝²+Λ^𝑖_𝑛𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛−𝜆_𝑖,𝑛(𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝)², (72) which yields Λ^𝑖_𝑛𝑢_𝑛−𝑝²≤𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛−𝜆_𝑖,𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²=𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²−𝜆²_𝑖,𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛,𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝≤𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝. (73) Thus, utilizing Lemma 8, from (41), (55), (56), (59), and (73) we have 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝑣_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾Λ^𝑖_𝑛𝑢_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝑠_𝑛𝛾×𝑢_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝑥_𝑛−𝑝²−Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝑝×𝑥𝛾𝑄−𝜇𝐹_𝑛≤−𝑝1−𝑠_𝑛×𝑥𝛾−1_𝑛−𝑝²−1−𝑠_𝑛𝛾Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝑠_𝑛𝛾Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²+2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (74) which leads to 1−𝑠_𝑛𝛾Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛²≤2𝜆_𝑖,𝑛Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝. (75) Since lim_𝑛→∞𝑠_𝑛=0,{𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  𝑖=1,2,…,𝑁 and {𝑥_𝑛} and {𝑢_𝑛} are bounded sequences, it follows from (71) that lim_𝑛→∞Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛=0. (76) From (76) we get 𝑢_𝑛−𝑣_𝑛=Λ⁰_𝑛𝑢_𝑛−Λ^𝑁_𝑛𝑢_𝑛≤Λ⁰_𝑛𝑢_𝑛−Λ¹_𝑛𝑢_𝑛+Λ¹_𝑛𝑢_𝑛−Λ²_𝑛𝑢_𝑛+⋯+Λ_𝑛^𝑁−1𝑢_𝑛−Λ^𝑁_𝑛𝑢_𝑛⟶0as𝑛⟶∞. (77) Taking into account that ‖𝑥_𝑛−𝑣_𝑛‖≤‖𝑥_𝑛−𝑢_𝑛‖+‖𝑢_𝑛−𝑣_𝑛‖, we conclude from (68) and (77) that lim_𝑛→∞𝑥_𝑛−𝑣_𝑛=0. (78)
On the other hand, for simplicity, we write 𝑝=𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝, 𝑣_𝑛=𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑣_𝑛 and 𝑤_𝑛=𝐺𝑣_𝑛=𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑣_𝑛 for all 𝑛≥1. Then 𝑝=𝐺𝑝=𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑝=𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝. (79)
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 (56), (57), and (59) that 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝐺𝑣_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛=−𝑝1−𝑠_𝑛𝛾𝑤_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝑤_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑣_𝑛−𝑝²+𝜈₁𝜈₁−2𝜁₁𝐴₁𝑣_𝑛−𝐴₁𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛)𝑝+𝑝×𝑥(𝐼−𝑉𝛾𝑄−𝜇𝐹_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑣_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐴₂𝑣_𝑛−𝐴₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐴₁𝑣_𝑛−𝐴₁𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛)𝑝+𝑝×𝑥(𝐼−𝑉𝛾𝑄−𝜇𝐹_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑥_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐴₂𝑣_𝑛−𝐴₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐴₁𝑣_𝑛−𝐴₁𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛=−𝑝1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑝²+1−𝑠_𝑛𝛾×𝜈₂𝜈₂−2𝜁₂𝐴₂𝑣_𝑛−𝐴₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐴₁𝑣_𝑛−𝐴₁𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²+1−𝑠_𝑛𝛾×𝜈₂𝜈₂−2𝜁₂𝐴₂𝑣_𝑛−𝐴₂𝑝²+𝜈₁𝜈₁−2𝜁₁𝐴₁𝑣_𝑛−𝐴₁𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (80) which immediately implies that 1−𝑠_𝑛𝛾𝜈₂2𝜁₂−𝜈₂𝐴₂𝑣_𝑛−𝐴₂𝑝²+𝜈₁2𝜁₁−𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝²≤2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (81) Since lim_𝑛→∞𝑠_𝑛=0, 𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2 and {𝑥_𝑛} is bounded, it follows that lim_𝑛→∞𝐴₂𝑣_𝑛−𝐴₂𝑝=0,lim_𝑛→∞𝐴₁𝑣_𝑛−𝐴₁𝑝=0. (82) Also, in terms of the firm nonexpansivity of 𝑇^Θ𝑘_𝜈𝑘 and the 𝜁_𝑘-inverse strong monotonicity of 𝐴_𝑘 for 𝑘=1,2, we obtain from 𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2, and (58) that 𝑣_𝑛−𝑝²=𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑣_𝑛−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²≤𝐼−𝜈₂𝐴₂𝑣_𝑛−𝐼−𝜈₂𝐴₂𝑝,𝑣_𝑛−𝑝=12(𝐼−𝜈₂𝐴₂)𝑣_𝑛−(𝐼−𝜈₂𝐴₂)𝑝²+𝑣_𝑛−𝑝²−(𝐼−𝜈₂𝐴₂)𝑣_𝑛−(𝐼−𝜈₂𝐴₂)𝑝−(𝑣_𝑛−𝑝)²≤12𝑣_𝑛−𝑝²+𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−𝜈₂(𝐴₂𝑣_𝑛−𝐴₂𝑝)−(𝑝−𝑝)²=12𝑣_𝑛−𝑝²+𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝,𝐴₂𝑣_𝑛−𝐴₂𝑝−𝜈²₂𝐴₂𝑣_𝑛−𝐴₂𝑝²,𝑤_𝑛−𝑝²=𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑣_𝑛−𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑝²≤𝐼−𝜈₁𝐴₁𝑣_𝑛−𝐼−𝜈₁𝐴₁𝑝,𝑤_𝑛=1−𝑝2(𝐼−𝜈₁𝐴₁)𝑣_𝑛−(𝐼−𝜈₁𝐴₁)𝑝²+𝑤_𝑛−𝑝²−(𝐼−𝜈₁𝐴₁)𝑣_𝑛−(𝐼−𝜈₁𝐴₁)𝑝−(𝑤_𝑛−𝑝)²≤12𝑣_𝑛−𝑝²+𝑤_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝,𝑣_𝑛−𝑤_𝑛+𝑝−𝑝−𝜈²₁𝐴₁𝑣_𝑛−𝐴₁𝑝²≤12𝑣_𝑛−𝑝²+𝑤_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝,𝑣_𝑛−𝑤_𝑛+𝑝−𝑝. (83) Thus, we have 𝑣_𝑛−𝑝²≤𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝,𝐴₂𝑣_𝑛−𝐴₂𝑝−𝜈²₂𝐴₂𝑣_𝑛−𝐴₂𝑝²,𝑤_𝑛−𝑝²≤𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝. (84)(85) Consequently, from (57), (80), and (84) it follows that 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾×𝑣_𝑛−𝑝²+𝜈₁𝜈₁−2𝜁₁𝐴₁𝑣_𝑛−𝐴₁𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑣_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝,𝐴₂𝑣_𝑛−𝐴₂𝑝−𝜈²₂𝐴₂𝑣_𝑛−𝐴₂𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑥_𝑛−𝑝²−(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐴₂𝑣_𝑛−𝐴₂𝑝+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛×𝑥𝛾−1_𝑛−𝑝²−1−𝑠_𝑛𝛾(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐴₂𝑣_𝑛−𝐴₂𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝑠_𝑛𝛾(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²+2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐴₂𝑣_𝑛−𝐴₂𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (86) which hence leads to 1−𝑠_𝑛𝛾(𝑣_𝑛−𝑣_𝑛)−(𝑝−𝑝)²≤2𝜈₂𝑣_𝑛−𝑣_𝑛−𝑝−𝑝𝐴₂𝑣_𝑛−𝐴₂𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝. (87) Since lim_𝑛→∞𝑠_𝑛=0 and {𝑥_𝑛},{𝑣_𝑛}, and {𝑣_𝑛} are bounded sequences, we conclude from (82) that lim_𝑛→∞𝑣_𝑛−𝑣_𝑛−𝑝−𝑝=0. (88) Consequently, from (57), (80), and (85) it follows that 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝑤_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝑣_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛)𝑝+𝑝×𝑥(𝐼−𝑉𝛾𝑄−𝜇𝐹_𝑛≤−𝑝1−𝑠_𝑛𝛾×𝑥_𝑛−𝑝²−(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛)𝑝+𝑝×𝑥(𝐼−𝑉𝛾𝑄−𝜇𝐹_𝑛≤−𝑝1−𝑠_𝑛×𝑥𝛾−1_𝑛−𝑝²−1−𝑠_𝑛𝛾(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²−1−𝑠_𝑛𝛾(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²+2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝, (89) which hence yields 1−𝑠_𝑛𝛾(𝑣_𝑛−𝑤_𝑛)+(𝑝−𝑝)²≤2𝜈₁𝐴₁𝑣_𝑛−𝐴₁𝑝𝑣_𝑛−𝑤_𝑛+𝑝−𝑝+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛾𝑄−𝜇𝐹𝑝𝑥_𝑛−𝑝. (90) Since lim_𝑛→∞𝑠_𝑛=0 and {𝑥_𝑛},{𝑤_𝑛}, and {𝑣_𝑛} are bounded sequences, we conclude from (82) that lim_𝑛→∞𝑣_𝑛−𝑤_𝑛+𝑝−𝑝=0. (91) Note that 𝑣_𝑛−𝑤_𝑛≤𝑣_𝑛−𝑣_𝑛−𝑝−𝑝+𝑣_𝑛−𝑤_𝑛+𝑝−𝑝. (92) Hence from (88) and (91) we get lim_𝑛→∞𝑣_𝑛−𝐺𝑣_𝑛=lim_𝑛→∞𝑣_𝑛−𝑤_𝑛=0. (93) Also, it is clear from (41) that 𝑥_𝑛−𝑇_𝑛𝐺𝑣_𝑛=𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛−𝑉𝑇_𝑛𝐺𝑣_𝑛. (94) So, it follows from lim_𝑛→∞𝑠_𝑛=0 that lim_𝑛→∞𝑥_𝑛−𝑇_𝑛𝐺𝑣_𝑛=0. (95) Observe that 𝑣_𝑛−𝑇_𝑛𝑣_𝑛≤𝑣_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇_𝑛𝐺𝑣_𝑛+𝑇_𝑛𝐺𝑣_𝑛−𝑇_𝑛𝑣_𝑛≤𝑣_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇_𝑛𝐺𝑣_𝑛+𝐺𝑣_𝑛−𝑣_𝑛. (96) Hence, from (78), (93), and (95) we have lim_𝑛→∞𝑣_𝑛−𝑇_𝑛𝑣_𝑛=0. (97) It is easy to see that 𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑣_𝑛−𝑣_𝑛=𝑠_𝑛𝑣_𝑛+1−𝑠_𝑛𝑇_𝑛𝑣_𝑛−𝑣_𝑛=1−𝑠_𝑛𝑇_𝑛𝑣_𝑛−𝑣_𝑛≤𝑇_𝑛𝑣_𝑛−𝑣_𝑛, (98) where 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)). Hence we have 𝑃_𝐶2𝐼−𝐿∇𝑓𝑣_𝑛−𝑣_𝑛≤𝑃_𝐶2𝐼−𝐿∇𝑓𝑣_𝑛−𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑣_𝑛+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑣_𝑛−𝑣_𝑛≤2𝐼−𝐿∇𝑓𝑣_𝑛−𝐼−𝜆_𝑛∇𝑓𝑣_𝑛+𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑣_𝑛−𝑣_𝑛≤2𝐿−𝜆_𝑛∇𝑓𝑣_𝑛+𝑇_𝑛𝑣_𝑛−𝑣_𝑛. (99) From the boundedness of {𝑣_𝑛},𝑠_𝑛→0  (⇔𝜆_𝑛→(2/𝐿)) and ‖𝑇_𝑛𝑣_𝑛−𝑣_𝑛‖→0 (due to (97)), it follows that lim_𝑛→∞𝑣_𝑛−𝑃_𝐶2𝐼−𝐿∇𝑓𝑣_𝑛=0. (100)
Furthermore, we show that 𝜔_𝑤(𝑥_𝑛)⊂Ω. Indeed, since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖}  of  {𝑥_𝑛} which converges weakly to some 𝑤. Note that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0 (due to (68)). Hence 𝑢_𝑛𝑖⇀𝑤. Since 𝐶 is closed and convex, 𝐶 is weakly closed. So, we have 𝑤∈𝐶. From (68), (76), and (78) we have that 𝑢_𝑛𝑖⇀𝑤, 𝑣_𝑛𝑖⇀𝑤 and Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖⇀𝑤, where 𝑚∈{1,2,…,𝑁}. First, we prove that 𝑤∈∩^𝑁_𝑚=1𝐼(𝐵_𝑚,𝑅_𝑚). However, the argument to show that 𝑤 lies in 𝐼(𝐵_𝑚,𝑅_𝑚) is quite standard by using maximality and hence is omitted. Next, we show that 𝑤∈GMEP(Θ,𝜑,𝐴). In fact, from 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛(𝐼−𝑟_𝑛𝐴)𝑥_𝑛, we know that Θ𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+⟨𝐴𝑥_𝑛,𝑦−𝑢_𝑛⟩+1𝑟_𝑛⟨𝐾^′𝑢_𝑛−𝐾^′𝑥_𝑛,𝑦−𝑢_𝑛⟩≥0,∀𝑦∈𝐶. (101) From (H2) it follows that 𝜑𝑦−𝜑𝑢_𝑛+⟨𝐴𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛×𝐾^′𝑢_𝑛−𝐾^′𝑥_𝑛,𝑦−𝑢_𝑛≥Θ𝑦,𝑢_𝑛,∀𝑦∈𝐶. (102) Replacing 𝑛 by 𝑛_𝑖, we have 𝜑𝑦−𝜑𝑢_𝑛𝑖+⟨𝐴𝑥_𝑛𝑖,𝑦−𝑢_𝑛𝑖⟩+𝐾^′𝑢_𝑛𝑖−𝐾^′𝑥_𝑛𝑖𝑟_𝑛𝑖,𝑦−𝑢_𝑛𝑖≥Θ𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (103) Put 𝑢_𝑡=𝑡𝑦+(1−𝑡)𝑤 for all 𝑡∈(0,1] and 𝑦∈𝐶. Then, from (103) we have ⟨𝑢_𝑡−𝑢_𝑛𝑖,𝐴𝑢_𝑡⟩≥⟨𝑢_𝑡−𝑢_𝑛𝑖,𝐴𝑢_𝑡⟩−𝜑𝑢_𝑡+𝜑𝑢_𝑛𝑖−𝑢_𝑡−𝑢_𝑛𝑖,𝐴𝑥_𝑛𝑖−𝐾^′𝑢_𝑛𝑖−𝐾^′𝑥_𝑛𝑖𝑟_𝑛𝑖,𝑢_𝑡−𝑢_𝑛𝑖+Θ𝑢_𝑡,𝑢_𝑛𝑖≥𝑢_𝑡−𝑢_𝑛𝑖,𝐴𝑢_𝑡−𝐴𝑢_𝑛𝑖+⟨𝑢_𝑡−𝑢_𝑛𝑖,𝐴𝑢_𝑛𝑖−𝐴𝑥_𝑛𝑖⟩−𝜑𝑢_𝑡+𝜑𝑢_𝑛𝑖−𝐾^′𝑢_𝑛𝑖−𝐾^′𝑥_𝑛𝑖𝑟_𝑛𝑖,𝑢_𝑡−𝑢_𝑛𝑖+Θ𝑢_𝑡,𝑢_𝑛𝑖. (104) Since ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0 as 𝑖→∞, we deduce from the Lipschitz continuity of 𝐴 and 𝐾^′ that ‖𝐴𝑢_𝑛𝑖−𝐴𝑥_𝑛𝑖‖→0 and ‖𝐾^′(𝑢_𝑛𝑖)−𝐾^′(𝑥_𝑛𝑖)‖→0 as 𝑖→∞. Furthermore, from the monotonicity of 𝐴, we have ⟨𝑢_𝑡−𝑢_𝑛𝑖,𝐴𝑢_𝑡−𝐴𝑢_𝑛𝑖⟩≥0. So, from (H4), the weakly lower semicontinuity of 𝜑,((𝐾^′(𝑢_𝑛𝑖)−𝐾^′(𝑥_𝑛𝑖))/𝑟_𝑛𝑖)→0, and 𝑢_𝑛𝑖⇀𝑤, we have ⟨𝑢_𝑡−𝑤,𝐴𝑢_𝑡⟩≥−𝜑𝑢_𝑡+𝜑(𝑤)+Θ𝑢_𝑡,𝑤,as𝑖⟶∞. (105) From (H1), (H4), and (105) we also have 0=Θ𝑢_𝑡,𝑢_𝑡+𝜑𝑢_𝑡−𝜑𝑢_𝑡≤𝑡Θ𝑢_𝑡,𝑦+(1−𝑡)Θ𝑢_𝑡𝑦+()𝜑(𝑤)𝑢,𝑤+𝑡𝜑1−𝑡−𝜑_𝑡=𝑡Θ𝑢_𝑡,𝑦+𝜑𝑦−𝜑𝑢_𝑡+(1−𝑡)Θ𝑢_𝑡,𝑤+𝜑(𝑤)−𝜑(𝑤)−𝜑𝑢_𝑡≤𝑡Θ𝑢_𝑡,𝑦+𝜑𝑦−𝜑𝑢_𝑡+(1−𝑡)⟨𝑢_𝑡−𝑤,𝐴𝑢_𝑡⟩=𝑡Θ𝑢_𝑡,𝑦+𝜑𝑦−𝜑𝑢_𝑡+(1−𝑡)𝑡⟨𝑦−𝑤,𝐴𝑢_𝑡⟩, (106) and hence 0≤Θ𝑢_𝑡,𝑦+𝜑𝑦−𝜑𝑢_𝑡+(1−𝑡)⟨𝑦−𝑤,𝐴𝑢_𝑡⟩. (107) Letting 𝑡→0, we have, for each 𝑦∈𝐶, 0≤Θ𝑤,𝑦+𝜑𝑦−𝜑(𝑤)+⟨𝐴𝑤,𝑦−𝑤⟩. (108) This implies that 𝑤∈GMEP(Θ,𝜑,𝐴). In addition, let us show that 𝑤∈SGEP(𝐺)∩Γ. As a matter of fact, from (93), (100), 𝑣_𝑛𝑖⇀𝑤, and Lemma 10, we deduce that 𝑤∈SGEP(𝐺) and 𝑤∈Fix(𝑃_𝐶(𝐼−(2/𝐿)∇𝑓))=VI(𝐶,∇𝑓)=Γ. Hence we get 𝑤∈SGEP(𝐺)∩Γ. Therefore, 𝑤∈GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺)∩∩^𝑁_𝑖=1𝐼(𝐵_𝑖,𝑅_𝑖)∩Γ=∶Ω. This shows that 𝜔_𝑤(𝑥_𝑛)⊂Ω.
Finally, we prove that {𝑥_𝑛} converges strongly as 𝜆_𝑛→(2/𝐿) (⇔𝑠_𝑛→0) to 𝑞∈Ω, which is the unique solution in Ω to the VIP (43). In fact, we note that, for 𝑤∈Ω with 𝑥_𝑛𝑖⇀𝑤, 𝑥_𝑛−𝑤=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑤+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝑄𝑤+𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑤+𝑠_𝑛(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤. (109) By (48), (57), and Lemma 14, we obtain that 𝑥_𝑛−𝑤²=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛−𝑤,𝑥_𝑛−𝑤+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝑄𝑤,𝑥_𝑛+−𝑤𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑤,𝑥_𝑛−𝑤+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛≤−𝑤⟩1−𝑠_𝑛𝛾𝑇_𝑛𝐺𝑣_𝑛−𝑤𝑥_𝑛−𝑤+𝑠_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛−𝑄𝑤𝑥_𝑛+−𝑤1−𝛼_𝑛𝜏𝑥_𝑛−𝑤²+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛≤−𝑤⟩1−𝑠_𝑛𝛾𝑣_𝑛−𝑤𝑥_𝑛−𝑤+𝑠_𝑛𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑤²+1−𝛼_𝑛𝜏𝑥_𝑛−𝑤²+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛≤−𝑤⟩1−𝑠_𝑛𝛾𝑥_𝑛−𝑤²+𝑠_𝑛𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑤²+1−𝛼_𝑛𝜏𝑥_𝑛−𝑤²+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛=−𝑤⟩1−𝑠_𝑛𝛾𝑥_𝑛−𝑤²+𝑠_𝑛1−𝛼_𝑛𝜏−𝛾𝑙𝑥_𝑛−𝑤²+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛≤−𝑤⟩1−𝑠_𝑛𝛾𝑥_𝑛−𝑤²+𝑠_𝑛𝑥_𝑛−𝑤²+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛=−𝑤⟩1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑤²+𝑠_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛−𝑤⟩, (110) which hence leads to 𝑥_𝑛−𝑤²≤⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛−𝑤⟩.𝛾−1 (111) In particular, we have 𝑥_𝑛𝑖−𝑤²≤⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝑖𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛𝑖−𝑤⟩.𝛾−1 (112) Since 𝑥_𝑛𝑖⇀𝑤 and lim_𝑛→∞𝛼_𝑛=0, it follows from (112) that 𝑥_𝑛𝑖→𝑤 as 𝑖→∞.
Now we show that 𝑤 solves the VIP (43). As a matter of fact, from (55) and (59) we obtain that, for any 𝑝∈Ω, 𝑥_𝑛−𝑝²≤1−𝑠_𝑛𝛾𝑢_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝1−𝑠_𝑛𝛾𝑥_𝑛−𝑝²+𝑠_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛=−𝑝1−𝑠_𝑛𝛾−1𝑥_𝑛−𝑝²+2𝑠_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛−𝑝⟩, (113) which immediately implies that (𝑉−𝐼)𝑝+𝛼_𝑛𝜇𝐹−𝛾𝑄𝑝,𝑥_𝑛−𝑝≤0. (114) Since 𝛼_𝑛→0 and 𝑥_𝑛𝑖→𝑤, we get ⟨(𝑉−𝐼)𝑝,𝑤−𝑝⟩≤0,∀𝑝∈Ω. (115) By Minty's lemma, 𝑤 is a solution in Ω to the VIP (43). In terms of the uniqueness of solutions of VIP (43), we deduce that 𝑤=𝑞 and 𝑥_𝑛𝑖→𝑞 as 𝑛→∞. So, every weak convergence subsequence of {𝑥_𝑛} converges strongly to the unique solution 𝑞 of VIP (43). Therefore, {𝑥_𝑛} converges strongly to the unique solution 𝑞 of VIP (43). This completes the proof.

Corollary 20. 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 Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and let 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex functional. Let 𝑅_𝑖∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴∶𝐻→𝐻 and 𝐴_𝑘,𝐵_𝑖∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone, and 𝜂_𝑖-inverse-strongly monotone, respectively, for 𝑘=1,2 and 𝑖=1,2. Let 𝑉 be a 𝛾-strongly positive bounded linear operator with 𝛾>1. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=𝐺𝑀𝐸𝑃(Θ,𝜑,𝐴)∩𝑆𝐺𝐸𝑃(𝐺)∩𝐼(𝐵₁,𝑅₁)∩𝐼(𝐵₂,𝑅₂)∩Γ≠∅ where 𝐺 is defined as in Proposition CY. Let {𝑟_𝑛} be a sequence in (0,2𝜁] and {𝛼_𝑛} a sequence in (0,1] such that lim_𝑛→∞𝛼_𝑛=0. Let {𝑥_𝑛} be the sequence generated by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅2,𝜆2,𝑛}𝐼−𝜆_2,𝑛𝐵₂𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁𝑢_𝑛,𝑥_𝑛=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛,∀𝑛≥1, (116) where 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛  (here 𝑇_𝑛 is nonexpansive, 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿))). Suppose that the following conditions hold.(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻.(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟⟨𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦⟩<0. (117)(iii)𝑠_𝑛∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)), lim_𝑛→∞𝑠_𝑛=0  (⇔lim_𝑛→∞𝜆_𝑛=(2/𝐿)).(iv)𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2 and {𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  𝑖=1,2.(v)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁. Assume that 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive. Then {𝑥_𝑛} converges strongly as 𝜆_𝑛→(2/𝐿)   (⇔𝑠_𝑛→0) to a point 𝑞∈Ω, which is a unique solution in Ω to the VIP: ⟨(𝐼−𝑉)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (118)

Corollary 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 Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex functional. Let 𝑅∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴∶𝐻→𝐻 and 𝐴_𝑘,𝐵∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone, and 𝜉-inverse-strongly monotone, respectively, for 𝑘=1,2. Let 𝑉 be a 𝛾-strongly positive bounded linear operator with 𝛾>1. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=𝐺𝑀𝐸𝑃(Θ,𝜑,𝐴)∩𝑆𝐺𝐸𝑃(𝐺)∩𝐼(𝐵,𝑅)∩Γ≠∅ where 𝐺 is defined as in Proposition CY. Let {𝑟_𝑛} be a sequence in (0,2𝜁] and {𝛼_𝑛} be a sequence in (0,1] such that lim_𝑛→∞𝛼_𝑛=0. Let {𝑥_𝑛} be the sequence generated by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅,𝜌𝑛}𝐼−𝜌_𝑛𝐵𝑢_𝑛,𝑥_𝑛=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛,∀𝑛≥1, (119) where 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛  (here 𝑇_𝑛 is nonexpansive, 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿))). Suppose that the following conditions hold(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦<0; (120)(iii)𝑠_𝑛∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)), lim_𝑛→∞𝑠_𝑛=0  (⇔lim_𝑛→∞𝜆_𝑛=(2/𝐿));(iv)𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2 and {𝜌_𝑛}⊂[𝑎,𝑏]⊂(0,2𝜉);(v)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁. Assume that 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive. Then {𝑥_𝑛} converges strongly as 𝜆_𝑛→(2/𝐿) (⇔𝑠_𝑛→0) to a point 𝑞∈Ω, which is a unique solution in Ω to the VIP: ⟨(𝐼−𝑉)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (121)
We next provide one example to illustrate Corollary 21.

Example 22. Let 𝐻=𝐑² with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ which are defined by 𝑥,𝑦=𝑎𝑐+𝑏𝑑,‖𝑥‖=𝑎²+𝑏², (122) for all 𝑥,𝑦∈𝐑² with 𝑥=(𝑎,𝑏) and 𝑦=(𝑐,𝑑). Let 𝐶={(𝑎,𝑎)∶𝑎∈𝐑} and 𝐾(𝑥)=(1/2)‖𝑥‖²,  for  all  𝑥∈𝐑². Clearly, 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻=𝐑². Let 𝑓∶𝐶→𝐑 be a convex functional with 𝐿-Lipschitz continuous gradient ∇𝑓; for instance, putting 𝑆=^2/31/3_1/32/3 and 𝑓(𝑥)=(1/2)⟨𝑆𝑥,𝑥⟩, we obtain that ‖𝑆‖=1 and ∇𝑓=𝑆 with Lipschitz constant 𝐿=4. Put Θ=Θ₁=Θ₂=0 and 𝜑=0. Then Θ,Θ₁,Θ₂ are three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and let 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex functional. Let 𝑅∶𝐶→2^𝐻 be a maximal monotone mapping, for instance, putting 𝑅𝑣=𝑆𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶, (123) where 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,  for  all  𝑢∈𝐶}. It is known that 𝑅 is maximal monotone and 0∈𝑅𝑣 if and only if 𝑣∈𝑉𝐼(𝐶,𝑆). Put 𝐴=^3/52/5_2/53/5, 𝐵=0, 𝐴₁=𝐼−𝑆=^1/3−1/3_−1/31/3, 𝐴₂=𝐼−𝐴=^2/5−2/5_−2/52/5. Then 𝐴∶𝐻→𝐻 and 𝐴_𝑘,𝐵∶𝐶→𝐻 are 𝜁-inverse strongly monotone with 𝜁=(1/2), 𝜁_𝑘-inverse strongly monotone with 𝜁_𝑘=(1/2) and 𝜉-inverse-strongly monotone with 𝜉=(1/2), respectively, for 𝑘=1,2. Also, put 𝜇=2, 𝑄=0, 𝑉=(3/2)𝑆, and 𝐹=(1/2)𝐼, where 𝐼 is the identity mapping of 𝐑². Then 𝑉 is a 𝛾-strongly positive bounded linear operator with 𝛾=(3/2). It is easy to see that Ω∶=GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺)∩𝐼(𝐵,𝑅)∩Γ={0} where 𝐺 is defined as in Proposition CY. Let {𝛼_𝑛}⊂(0,1], {𝑟_𝑛}⊂(0,2𝜁], and {𝜌_𝑛}⊂(0,2𝜉] with 𝜁=𝜉=(1/2); that is, {𝛼_𝑛},{𝜌_𝑛},{𝑟_𝑛}⊂(0,1]. In this case, for any given 𝑥₁∈𝐶, the iterative scheme (119) is equivalent to the following one: 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛=𝑃_𝐶𝐼−𝑟_𝑛𝐴𝑥_𝑛=1−𝑟_𝑛𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅,𝜌𝑛}𝐼−𝜌_𝑛𝐵𝑢_𝑛=𝐽_{𝑅,𝜌𝑛}𝑢_𝑛=𝑃_𝐶𝑢_𝑛−𝜌_𝑛𝑆𝑣_𝑛=𝑢_𝑛−𝜌_𝑛𝑣_𝑛,𝑥_𝑛=𝐼−𝑠_𝑛𝑉𝑇_𝑛𝐺𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛=3𝐼−2𝑠_𝑛𝑆𝑇_𝑛𝑃_𝐶𝐼−𝜈₁𝐴₁𝑃_𝐶𝐼−𝜈₂𝐴₂𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛=3𝐼−2𝑠_𝑛𝑆𝑇_𝑛𝑣_𝑛+𝑠_𝑛𝑇_𝑛𝑥_𝑛−𝛼_𝑛𝑇_𝑛𝑥_𝑛=3𝐼−2𝑠_𝑛𝑆𝑇_𝑛𝑣_𝑛+𝑠_𝑛1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,∀𝑛≥1, (124) where 𝑃_𝐶(𝐼−𝜆_𝑛∇𝑓)=𝑠_𝑛𝐼+(1−𝑠_𝑛)𝑇_𝑛 and 𝑠_𝑛=((2−𝜆_𝑛𝐿)/4)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(2/𝐿)) with 𝐿=4; that is, 𝑠_𝑛=((1−2𝜆_𝑛)/2)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(1/2)).
Next, taking into account 𝑣_𝑛=𝐽_{𝑅,𝜌𝑛}𝑢_𝑛=(𝐼+𝜌_𝑛𝑅)⁻¹𝑢_𝑛, we get 𝑢_𝑛∈𝑣_𝑛+𝜌_𝑛𝑅𝑣_𝑛, which leads to 𝑢_𝑛−𝑣_𝑛𝜌_𝑛∈𝑅𝑣_𝑛=𝑆𝑣_𝑛+𝑁_𝐶𝑣_𝑛. (125) Hence, we have ⟨𝑣_𝑛−𝑢,((𝑢_𝑛−𝑣_𝑛)/𝜌_𝑛)−𝑆𝑣_𝑛⟩≥0,  for  all  𝑢∈𝐶; that is, ⟨𝑢−𝑣_𝑛,𝑢_𝑛−𝜌_𝑛𝑆𝑣_𝑛−𝑣_𝑛⟩≤0,∀𝑢∈𝐶, (126) which immediately implies that 𝑣_𝑛=𝑃_𝐶(𝑢_𝑛−𝜌_𝑛𝑆𝑣_𝑛)=𝑢_𝑛−𝜌_𝑛𝑣_𝑛. Thus, 𝑣_𝑛=11+𝜌_𝑛𝑢_𝑛=1−𝑟_𝑛1+𝜌_𝑛𝑥_𝑛. (127) Also, note that 1−𝜆_𝑛𝑥_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑥_𝑛=𝑠_𝑛𝑥_𝑛+1−𝑠_𝑛𝑇_𝑛𝑥_𝑛. (128) So, we obtain 𝑇_𝑛𝑥_𝑛=1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛𝑥_𝑛. (129) On the other hand, we have 1−𝜆_𝑛𝑣_𝑛=𝑃_𝐶𝐼−𝜆_𝑛∇𝑓𝑣_𝑛=𝑠_𝑛𝑣_𝑛+1−𝑠_𝑛𝑇_𝑛𝑣_𝑛, (130) and hence 𝑇_𝑛𝑣_𝑛=1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛𝑣_𝑛=1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛⋅1−𝑟_𝑛1+𝜌_𝑛𝑥_𝑛, (131) which together with (124) implies that 𝑥_𝑛=3𝐼−2𝑠_𝑛𝑆𝑇_𝑛𝑣_𝑛+𝑠_𝑛1−𝛼_𝑛𝑇_𝑛𝑥_𝑛=3𝐼−2𝑠_𝑛𝑆1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛1−𝑟_𝑛1+𝜌_𝑛𝑥_𝑛+𝑠_𝑛1−𝛼_𝑛1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛𝑥_𝑛=31−2𝑠_𝑛1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛1−𝑟_𝑛1+𝜌_𝑛𝑥_𝑛+𝑠_𝑛1−𝛼_𝑛1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛𝑥_𝑛=31−2𝑠_𝑛1−𝑟_𝑛1+𝜌_𝑛+𝑠_𝑛1−𝛼_𝑛1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛𝑥_𝑛. (132) It is clear that if 𝑠_𝑛=((1−2𝜆_𝑛)/2)=((1−2𝜆_𝑛)/2)∈(0,(1/2)) for each 𝜆_𝑛∈(0,(1/2)), then, for {𝑟_𝑛},{𝛼_𝑛},{𝜌_𝑛}⊂(0,1], we deduce from (132) that 𝑥_𝑛=31−2𝑠_𝑛1−𝑟_𝑛1+𝜌_𝑛+𝑠_𝑛1−𝛼_𝑛1−𝑠_𝑛−𝜆_𝑛1−𝑠_𝑛𝑥_𝑛≤31−2𝑠_𝑛1−𝑟_𝑛1+𝜌_𝑛+𝑠_𝑛1−𝛼_𝑛𝑥_𝑛≤31−2𝑠_𝑛+𝑠_𝑛𝑥_𝑛=11−2𝑠_𝑛𝑥_𝑛. (133) Therefore, 𝑥_𝑛=0 for all 𝑛≥1. There is no doubt that {𝑥_𝑛} converges to the unique element 0 in Ω, which solves the VIP (121).

4. Fixed Point Problems with Constraints

In this section, we will introduce and analyze another implicit iterative algorithm for solving the fixed point problem of infinitely many nonexpansive mappings with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under mild assumptions.

Theorem 23. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑁 be an integer. Let Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex functional. Let 𝑅_𝑖∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴∶𝐻→𝐻 and 𝐴_𝑘,𝐵_𝑖∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone, and 𝜂_𝑖-inverse-strongly monotone, respectively, for 𝑘=1,2 and 𝑖=1,2,…,𝑁. Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive mappings on 𝐻 and {𝜆_𝑛} a sequence in (0,𝑏] for some 𝑏∈(0,1). Let 𝑉 be a 𝛾-strongly positive bounded linear operator with 𝛾>1. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=∩^∞_𝑛=1𝐹𝑖𝑥(𝑇_𝑛)∩𝐺𝑀𝐸𝑃(Θ,𝜑,𝐴)∩𝑆𝐺𝐸𝑃(𝐺)∩∩^𝑁_𝑖=1𝐼(𝐵_𝑖,𝑅_𝑖)≠∅ where 𝐺 is defined as in Proposition CY. Let {𝑟_𝑛} be a sequence in (0,2𝜁] and let {𝛼_𝑛} and {𝜎_𝑛} be sequences in (0,1]. Let {𝑥_𝑛} be the sequence generated by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅𝑁,𝜆𝑁,𝑛}𝐼−𝜆_𝑁,𝑛𝐵_𝑁𝐽_{𝑅𝑁−1,𝜆𝑁−1,𝑛}×𝐼−𝜆_{𝑁−1,𝑛}𝐵_𝑁−1⋯𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁𝑢_𝑛,𝑥_𝑛=𝐼−𝜎_𝑛𝑉𝑊_𝑛𝐺𝑣_𝑛+𝜎_𝑛𝑊_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑊_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛,∀𝑛≥1, (134) where 𝑊_𝑛 is the 𝑊-mapping defined by (9). Suppose that the following conditions hold:(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟⟨𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦⟩<0; (135)(iii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞𝜎_𝑛=0;(iv)𝜈_𝑘∈(0,2𝜁_𝑘),𝑘=1,2 and {𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  𝑖=1,2,…,𝑁;(v)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁. Assume that 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive. Then {𝑥_𝑛} converges strongly to a point 𝑞∈Ω, which is a unique solution in Ω to the VIP: ⟨(𝐼−𝑉)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (136)

Proof. First of all, let us show that the sequence {𝑥_𝑛} is well defined. As lim_𝑛→∞𝜎_𝑛=0 and 0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁, we may assume, without loss of generality, that {𝑟_𝑛}⊂[𝑎,𝑎]⊂(0,2𝜁) and 𝜎_𝑛‖𝑉‖≤1 for all 𝑛≥1. Utilizing the arguments similar to those in the proof of Theorem 19, we get 𝐼−𝜎_𝑛𝑉≤1−𝜎_𝑛𝛾. (137) Put Λ^𝑖_𝑛=𝐽_{𝑅𝑖,𝜆𝑖,𝑛}𝐼−𝜆_𝑖,𝑛𝐵_𝑖𝐽_{𝑅𝑖−1,𝜆𝑖−1,𝑛}×𝐼−𝜆_{𝑖−1,𝑛}𝐵_𝑖−1⋯𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁ (138) for all 𝑖∈{1,2,…,𝑁}, and Λ⁰_𝑛=𝐼, where 𝐼 is the identity mapping on 𝐻. Then we have 𝑣_𝑛=Λ^𝑁_𝑛𝑢_𝑛.
Consider the following mapping 𝐺_𝑛 on 𝐻 defined by 𝐺_𝑛𝑥=𝐼−𝜎_𝑛𝑉𝑇_𝑛𝐺Λ^𝑁_𝑛𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥+𝜎_𝑛𝑇_𝑛𝑥−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥−𝛾𝑄𝑥,∀𝑥∈𝐻,𝑛≥1. (139) Since 𝐴_𝑘 is 𝜁_𝑘-inverse-strongly monotone with 0≤𝜈_𝑘≤2𝜁_𝑘 for 𝑘=1,2, repeating the same arguments as in the proof of Theorem 19 we deduce that 𝐺 is a nonexpansive mapping on 𝐻. Utilizing the arguments similar to those in the proof of Theorem 19, we deduce that, for any 𝑥,𝑦∈𝐻, 𝐺_𝑛𝑥−𝐺_𝑛𝑦²≤1−𝜎_𝑛𝛼_𝑛𝜏−𝛾𝑙𝑥−𝑦. (140) Since 0<1−𝜎_𝑛𝛼_𝑛(𝜏−𝛾𝑙)<1, 𝐺_𝑛∶𝐻→𝐻 is a contraction. Therefore, by the Banach contraction principle, 𝐺_𝑛 has a unique fixed point 𝑥_𝑛∈𝐻, which uniquely solves the fixed point equation 𝑥_𝑛=𝐼−𝜎_𝑛𝑉𝑊_𝑛𝐺𝑣_𝑛+𝜎_𝑛𝑊_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑊_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛. (141) This shows that the sequence {𝑥_𝑛} is defined well.
It is easy to see the VIP (43) has only one solution in Ω. Below we use 𝑞∈Ω to denote the unique solution of the VIP (43).
Now, let us show that {𝑥_𝑛} is bounded. In fact, take 𝑝∈Ω arbitrarily. Repeating the same arguments as in the proof of Theorem 19 we obtain 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝜁𝐴𝑥_𝑛−𝐴𝑝²,𝑣_𝑛−𝑝²≤Λ^𝑖_𝑛𝑢_𝑛−𝑝²≤𝑢_𝑛−𝑝²+𝜆_𝑖,𝑛𝜆_𝑖,𝑛−2𝜂_𝑖𝐵_𝑖Λ_𝑛^𝑖−1𝑢_𝑛−𝐵_𝑖𝑝²,𝐺𝑣_𝑛−𝑝²≤𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑣_𝑛−𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝²+𝜈₁𝜈₁−2𝜁₁×𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑣_𝑛−𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²≤𝑣_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐴₂𝑣_𝑛−𝐴₂𝑝²,𝑥_𝑛−𝑝²≤1−𝜎_𝑛𝛾𝐺𝑣_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝⟩1−𝜎_𝑛𝛾𝑣_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑝²+2𝑠_𝑛⟨(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛=−𝑝⟩1−𝜎_𝑛𝛾−1𝑥_𝑛−𝑝²+2𝜎_𝑛)𝑝+𝑝×𝑥(𝐼−𝑉𝛾𝑄−𝜇𝐹_𝑛−𝑝. (142)(143)(144)(145)
From (145) we conclude that 𝑥_𝑛≤2−𝑝)𝑝+𝑝.𝛾−1(𝐼−𝑉𝛾𝑄−𝜇𝐹 (146) Hence {𝑥_𝑛} is bounded. So, according to (142) and (143) we know that {𝑢_𝑛},{𝑣_𝑛},{𝐺𝑣_𝑛}, {𝑊_𝑛𝐺𝑣_𝑛},{𝑊_𝑛𝑥_𝑛},{𝑄𝑥_𝑛}, and {𝐹𝑊_𝑛𝑥_𝑛} are bounded.
Repeating the arguments similar to those of (68), (76), (78), (93), and (97) in the proof of Theorem 19 we obtain that ‖𝑥_𝑛−𝑢_𝑛‖→0, ‖Λ_𝑛^𝑖−1𝑢_𝑛−Λ^𝑖_𝑛𝑢_𝑛‖→0, ‖𝑥_𝑛−𝑣_𝑛‖→0, ‖𝑣_𝑛−𝐺𝑣_𝑛‖→0 and ‖𝑣_𝑛−𝑊_𝑛𝑣_𝑛‖→0 as 𝑛→∞. In addition, note that 𝑣_𝑛−𝑊𝑣_𝑛≤𝑣_𝑛−𝑊_𝑛𝑣_𝑛+𝑊_𝑛𝑣_𝑛−𝑊𝑣_𝑛. (147) So, from ‖𝑣_𝑛−𝑊_𝑛𝑣_𝑛‖→0 and [15, Remark 3.2] it follows that lim_𝑛→∞𝑣_𝑛−𝑊𝑣_𝑛=0. (148)
Further, we show that 𝜔_𝑤(𝑥_𝑛)⊂Ω. Indeed, since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} which converges weakly to some 𝑤. Note that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. Hence 𝑢_𝑛𝑖⇀𝑤. Since 𝐶 is closed and convex, 𝐶 is weakly closed. So, we have 𝑤∈𝐶. On the other hand, it is easy to see that 𝑢_𝑛𝑖⇀𝑤, 𝑣_𝑛𝑖⇀𝑤, and Λ^𝑚_𝑛𝑖𝑢_𝑛𝑖⇀𝑤, where 𝑚∈{1,2,…,𝑁}. Repeating the same arguments as in the proof of Theorem 19, we obtain that 𝑤∈∩^𝑁_𝑖=1𝐼(𝐵_𝑖,𝑅_𝑖)∩GMEP(Θ,𝜑,𝐴). Next let us show that 𝑤∈SGEP(𝐺)∩∩^∞_𝑛=1Fix(𝑇_𝑛). As a matter of fact, from ‖𝑣_𝑛−𝐺𝑣_𝑛‖→0, ‖𝑣_𝑛−𝑊𝑣_𝑛‖→0, 𝑣_𝑛𝑖⇀𝑤 and Lemma 10, we deduce that 𝑤∈SGEP(𝐺) and 𝑤∈Fix(𝑊)=∩^∞_𝑛=1Fix(𝑇_𝑛) (due to Lemma 12). Hence we get 𝑤∈SGEP(𝐺)∩∩^∞_𝑛=1Fix(𝑇_𝑛). Therefore, 𝑤∈∩^∞_𝑛=1Fix(𝑇_𝑛)∩GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺)∩∩^𝑁_𝑖=1𝐼(𝐵_𝑖,𝑅_𝑖)=∶Ω. This shows that 𝜔_𝑤(𝑥_𝑛)⊂Ω.
Finally, we prove that {𝑥_𝑛} converges strongly to 𝑞∈Ω, which is the unique solution in Ω to the VIP (136). In fact, we note that, for 𝑤∈Ω with 𝑥_𝑛𝑖⇀𝑤, 𝑥_𝑛−𝑤=𝐼−𝜎_𝑛𝑉𝑊_𝑛𝐺𝑣_𝑛−𝑤+𝜎_𝑛𝛼_𝑛𝛾𝑄𝑥_𝑛+−𝑄𝑤𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝜇𝐹𝑊_𝑛𝑤+𝜎_𝑛(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤. (149) Utilizing the arguments similar to those in the proof of Theorem 19, we obtain that 𝑥_𝑛−𝑤²≤1−𝜎_𝑛𝛾−1𝑥_𝑛−𝑤²+𝜎_𝑛⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛−𝑤⟩, (150) which hence leads to 𝑥_𝑛−𝑤²≤⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛−𝑤⟩.𝛾−1 (151) In particular, we have 𝑥_𝑛𝑖−𝑤²≤⟨(𝐼−𝑉)𝑤+𝛼_𝑛𝑖𝛾𝑄−𝜇𝐹𝑤,𝑥_𝑛𝑖−𝑤⟩.𝛾−1 (152) Since 𝑥_𝑛𝑖⇀𝑤 and lim_𝑛→∞𝛼_𝑛=0, it follows from (152) that 𝑥_𝑛𝑖→𝑤 as 𝑖→∞.
Now we show that 𝑤 solves the VIP (136). As a matter of fact, from (142) and (145) we obtain that, for any 𝑝∈Ω, 𝑥_𝑛−𝑝²≤1−𝜎_𝑛𝛾𝑢_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑝²+2𝜎_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤−𝑝1−𝜎_𝑛𝛾𝑥_𝑛−𝑝²+𝜎_𝑛𝑥_𝑛−𝑝²+2𝜎_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛=−𝑝1−𝜎_𝑛𝛾−1𝑥_𝑛−𝑝²+2𝜎_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝²+2𝜎_𝑛(𝐼−𝑉)𝑝+𝛼_𝑛𝛾𝑄−𝜇𝐹𝑝,𝑥_𝑛−𝑝, (153) which immediately implies that ⟨(𝑉−𝐼)𝑝+𝛼_𝑛𝜇𝐹−𝛾𝑄𝑝,𝑥_𝑛−𝑝⟩≤0. (154) Since 𝛼_𝑛→0 and 𝑥_𝑛𝑖→𝑤, we get ⟨(𝑉−𝐼)𝑝,𝑤−𝑝⟩≤0,∀𝑝∈Ω. (155) By Minty's lemma, 𝑤 is a solution in Ω to the VIP (136). In terms of the uniqueness of solutions of VIP (136), we deduce that 𝑤=𝑞 and 𝑥_𝑛𝑖→𝑞 as 𝑛→∞. So, every weak convergence subsequence of {𝑥_𝑛} converges strongly to the unique solution 𝑞 of VIP (136). Therefore, {𝑥_𝑛} converges strongly to the unique solution 𝑞 of VIP (136). This completes the proof.

Corollary 24. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and let 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex functional. Let 𝑅_𝑖∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴,𝐴_𝑘∶𝐻→𝐻 and 𝐵_𝑖∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone, and 𝜂_𝑖-inverse strongly monotone, respectively, for 𝑘=1,2 and 𝑖=1,2. Let {𝑇_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 𝑉 be a 𝛾-strongly positive bounded linear operator with 𝛾>1. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=∩^∞_𝑛=1𝐹𝑖𝑥(𝑇_𝑛)∩𝐺𝑀𝐸𝑃(Θ,𝜑,𝐴)∩𝑆𝐺𝐸𝑃(𝐺)∩𝐼(𝐵₁,𝑅₁)∩𝐼(𝐵₂,𝑅₂)≠∅ where 𝐺 is defined as in Proposition CY. Let {𝑟_𝑛} be a sequence in (0,2𝜁] and {𝛼_𝑛} and {𝜎_𝑛} sequences in (0,1]. Let {𝑥_𝑛} be the sequence generated by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅2,𝜆2,𝑛}𝐼−𝜆_2,𝑛𝐵₂𝐽_{𝑅1,𝜆1,𝑛}𝐼−𝜆_1,𝑛𝐵₁𝑢_𝑛,𝑥_𝑛=𝐼−𝜎_𝑛𝑉𝑊_𝑛𝐺𝑣_𝑛+𝜎_𝑛𝑊_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑊_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛,∀𝑛≥1, (156) where 𝑊_𝑛 is the 𝑊-mapping defined by (9). Suppose that the following conditions hold:(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that, for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟⟨𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦⟩<0; (157)(iii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞𝜎_𝑛=0;(iv)𝜈_𝑘∈(0,2𝜁_𝑘),𝑘=1,2 and {𝜆_𝑖,𝑛}⊂[𝑎_𝑖,𝑏_𝑖]⊂(0,2𝜂_𝑖),  𝑖=1,2;(v)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁. Assume that 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive. Then {𝑥_𝑛} converges strongly to a point 𝑞∈Ω, which is a unique solution in Ω to the VIP: ⟨(𝐼−𝑉)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (158)

Corollary 25. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and 𝜑∶𝐶→𝐑 a lower semicontinuous and convex functional. Let 𝑅∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴,𝐴_𝑘∶𝐻→𝐻 and 𝐵∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone, and 𝜉-inverse strongly monotone, respectively, for 𝑘=1,2. Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive mappings on 𝐻 and {𝜆_𝑛} a sequence in (0,𝑏] for some 𝑏∈(0,1). Let 𝑉 be a 𝛾-strongly positive bounded linear operator with 𝛾>1. Let 𝐹∶𝐻→𝐻 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with positive constants 𝜅,𝜂>0. Let 𝑄∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with constant 𝑙≥0. Let 0<𝜇<(2𝜂/𝜅²) and 0≤𝛾𝑙<𝜏, where 𝜏=1−1−𝜇(2𝜂−𝜇𝜅²). Assume that Ω∶=∩^∞_𝑛=1𝐹𝑖𝑥(𝑇_𝑛)∩𝐺𝑀𝐸𝑃(Θ,𝜑,𝐴)∩𝑆𝐺𝐸𝑃(𝐺)∩𝐼(𝐵,𝑅)≠∅ where 𝐺 is defined as in Proposition CY. Let {𝑟_𝑛} be a sequence in (0,2𝜁] and {𝛼_𝑛} and {𝜎_𝑛} sequences in (0,1]. Let {𝑥_𝑛} be the sequence generated by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑣_𝑛=𝐽_{𝑅,𝜌𝑛}𝐼−𝜌_𝑛𝐵𝑢_𝑛,𝑥_𝑛=𝐼−𝜎_𝑛𝑉𝑊_𝑛𝐺𝑣_𝑛+𝜎_𝑛𝑊_𝑛𝑥_𝑛−𝛼_𝑛𝜇𝐹𝑊_𝑛𝑥_𝑛−𝛾𝑄𝑥_𝑛,∀𝑛≥1, (159) where 𝑊_𝑛 is the 𝑊-mapping defined by (9). Suppose that the following conditions hold:(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that, for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟⟨𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦⟩<0; (160)(iii)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞𝜎_𝑛=0;(iv)𝜈_𝑘∈(0,2𝜁_𝑘),  𝑘=1,2 and {𝜌_𝑛}⊂[𝑎,𝑏]⊂(0,2𝜉);(v)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁. Assume that 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive. Then {𝑥_𝑛} converges strongly to a point 𝑞∈Ω, which is a unique solution in Ω to the VIP: ⟨(𝐼−𝑉)𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (161)

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.