A New Hybrid Extragradient Iterative Method for Approximating the Common Solutions of a System of Variational Inequalities, a Mixed Equilibrium Problem, and a Hierarchical Fixed Point Problem

Abstract

We suggest and analyze an iterative scheme for finding the approximate element of the common set of solutions of a system of variational inequalities, a mixed equilibrium problem, and a hierarchical fixed point problem in a real Hilbert space. Strong convergence of the proposed method is proved under some conditions. The results presented in this paper extend and improve some well-known results in the literature.

1. Introduction

Let 𝐻 be a real Hilbert space, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻. We consider the system of variational inequalities of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥0;∀𝑥∈𝐶,𝜇₁𝜇>0,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0;∀𝑥∈𝐶,𝜇₂>0, (1) where 𝐵_𝑖∶𝐶→𝐶 is a nonlinear mapping for each 𝑖=1,2. The solution set of (1) is denoted by 𝑆^∗.

If 𝐵₁=𝐵₂=𝐵, then the problem (1) reduces finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥0;∀𝑥∈𝐶,𝜇₁𝜇>0,₂𝐵𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0;∀𝑥∈𝐶,𝜇₂>0, (2) which has been introduced and studied by Verma [1, 2].

If 𝑥^∗=𝑦^∗ and 𝜇₁=𝜇₂, then the problem (2) collapses to the classical variational inequality finding 𝑥^∗∈𝐶, such that 𝐵𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶 (3) is called the classical variational inequality problem, which was introduced by Stampacchia [3] in 1964. For the recent applications, numerical techniques, and physical formulation, see [1–33]. We now have a variety of techniques to suggest and analyze various iterative algorithms for solving the system of variational inequalities (1); see [1, 2, 7, 8, 12, 14, 24, 28, 30].

We introduce the following definitions which are useful in the following analysis.

Definition 1. The mapping 𝑇∶𝐶→𝐻 is said to be (a)monotone, if 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶; (4)(b)strongly monotone, if there exists an 𝛼>0 such that 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝛼𝑥−𝑦²,∀𝑥,𝑦∈𝐶; (5)(c)𝛼-inverse strongly monotone, if there exists an 𝛼>0 such that 𝑇𝑥−𝑇𝑦,𝑥−𝑦≥𝛼𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶; (6)(d)nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶; (7)(e)𝑘-Lipschitz continuous, if there exists a constant 𝑘>0 such that 𝑇𝑥−𝑇𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶; (8)(f)contraction on 𝐶, if there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑇𝑦≤𝑘𝑥−𝑦,∀𝑥,𝑦∈𝐶. (9)It is easy to observe that every 𝛼-inverse strongly monotone 𝑇 is monotone and Lipschitz continuous. A mapping 𝑇∶𝐶→𝐻 is called 𝑘-strict pseudocontraction, if there exists a constant 0≤𝑘<1 such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (10) The fixed-point problem for the mapping 𝑇 is to find 𝑥∈𝐶 such that 𝑇𝑥=𝑥. (11) We denote by 𝐹(𝑇) the set of solutions of (11). It is well-known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings; then 𝐹(𝑇) is closed and convex and 𝑃_𝐹(𝑇) is well defined (see [33]).
The mixed equilibrium problem, denoted by MEP, is to find 𝑥∈𝐶 such that 𝐹𝑥,𝑦+⟨𝐷𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶, (12) where 𝐹∶𝐶×𝐶→ℝ is bifunction and 𝐷∶𝐶→𝐻 is a nonlinear mapping. This problem was introduced and studied by Moudafi and Théra [21] and Moudafi [22]. The set of solutions of (12) is denoted by MEP(𝐹)∶=𝑥∈𝐶∶𝐹𝑥,𝑦+𝐷𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (13)If 𝐷=0, then it is reduced to the equilibrium problem is to find 𝑥∈𝐶 such that𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (14) The solution set of (14) is denoted by EP(𝐹). Numerous problems in physics, optimization, and economics reduce to find a solution of (14); see [9, 13, 25, 26]. In 1997, Flåm and Antipin [10] introduced an iterative scheme of finding the best approximation to the initial data when EP(𝐹) is nonempty. Recently, Plubtieng and Punpaeng [25] introduced an iterative method for finding the common element of the set 𝐹(𝑇)∩Ω^∗∩EP(𝐹).
Let 𝑆∶𝐶→𝐻 be a nonexpansive mapping. The following problem is called a hierarchical fixed point problem: Find 𝑥∈𝐹(𝑇) such that 𝑥−𝑆𝑥,𝑦−𝑥≥0,∀𝑦∈𝐹(𝑇). (15) It is known that the hierarchical fixed-point problem (15) links with some monotone variational inequalities and convex programming problems; see [11, 31]. Various methods have been proposed to solve the hierarchical fixed point problem; see Moudafi [23], Maingé and Moudafi in [17], Marino and Xu in [19], and Cianciaruso et al. [6]. Very recently, Yao et al. [31] introduced the following strong convergence iterative algorithm to solve the problem (15): 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥0, (16) where 𝑓∶𝐶→𝐻 is a contraction mapping and {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1). Under some certain restrictions on parameters, Yao et al. proved that the sequence {𝑥_𝑛} generated by (16) converges strongly to 𝑧∈𝐹(𝑇), which is the unique solution of the following variational inequality: 𝐼−𝑓𝑧,𝑦−𝑧≥0,∀𝑦∈𝐹(𝑇). (17) By changing the restrictions on parameters, the authors obtained another result on the iterative scheme (16); the sequence {𝑥_𝑛} generated by (16) converges strongly to a point 𝑧∈𝐹(𝑇), which is the unique solution of the following variational inequality: 1𝜏𝐼−𝑓𝑧+(𝐼−𝑆)𝑧,𝑦−𝑧≥0,∀𝑦∈𝐹(𝑇). (18)
Let 𝑆∶𝐶→𝐻 be a nonexpansive mapping and {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 a countable family of nonexpansive mappings. Very recently, Gu et al. [11] introduced the following iterative algorithm: 𝑦_𝑛=𝑃_𝐶𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑇_𝑖𝑦_𝑛,∀𝑛≥1, (19) where 𝛼₀=1, {𝛼_𝑛} is a strictly decreasing sequence in (0,1), and {𝛽_𝑛} is a sequence in (0,1). Under some certain conditions on parameters, Gu et al. proved that the sequence {𝑥_𝑛} generated by (19) converges strongly to 𝑧∈⋂^∞_𝑖=1𝐹(𝑇_𝑖), which is unique solution of one of the variational inequalities (17) and (18).
In this paper, motivated by the work of Yao et al. [31] and Gu et al. [11] and by the recent work going in this direction, we give an iterative method for finding the approximate element of the common set of solutions of (1), (12), and (15) for a strictly pseudocontraction mapping in real Hilbert space. We establish a strong convergence theorem based on this method. The presented method improves and generalizes many known results for solving system of variational inequality problems, mixed equilibrium problems, and hierarchical fixed point problems; see, for example [6, 11, 17, 31] and relevant references cited therein.

2. Preliminaries

In this section, we list some fundamental lemmas that are useful in the consequent analysis. The first lemma provides some basic properties of projection onto 𝐶.

Lemma 2. Let 𝑃_𝐶 denote the projection of 𝐻 onto 𝐶. Then, one has the following inequalities: 𝑧−𝑃_𝐶[𝑧],𝑃_𝐶[𝑧]−𝑣≥0,∀𝑧∈𝐻,𝑣∈𝐶;𝑢−𝑣,𝑃_𝐶[𝑢]−𝑃_𝐶[𝑣]≥𝑃_𝐶[𝑢]−𝑃_𝐶[𝑣]²,𝑃∀𝑢,𝑣∈𝐻;_𝐶[𝑢]−𝑃_𝐶[𝑣]≤‖𝑢−𝑣‖,∀𝑢,𝑣∈𝐻;𝑢−𝑃_𝐶[𝑧]²≤‖𝑧−𝑢‖²−𝑧−𝑃_𝐶[𝑧]²,∀𝑧∈𝐻,𝑢∈𝐶. (20)(21)(22)

Lemma 3 (see [7]). For any (𝑥^∗,𝑦^∗)∈𝐶×𝐶,(𝑥^∗,𝑦^∗) is a solution of (1) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄∶𝐶→𝐶 de fined by 𝑄(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (23) where 𝑦^∗=𝑃_𝐶[𝑥^∗−𝜇₂𝐵₂𝑥^∗],𝜇_𝑖∈(0,2𝜃_𝑖), and 𝐵_𝑖∶𝐶→𝐶 is 𝜃_𝑖-inverse strongly monotone mappings for each 𝑖=1,2.

Lemma 4 (see [5]). Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the following assumptions. (i)𝐹(𝑥,𝑥)=0,∀𝑥∈𝐶. (ii)𝐹 is monotone; that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0,∀𝑥,𝑦∈𝐶.(iii)For each 𝑥,𝑦,𝑧∈𝐶,lim_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦).(iv)For each 𝑥∈𝐶,𝑦→𝐹(𝑥,𝑦) is convex and lower semicontinuous. Let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (24)

Lemma 5 (see [10]). Assume that 𝐹∶𝐶×𝐶→ℝ satisfies assumptions (𝑖)--(𝑖𝑣) of Lemma 4. For 𝑟>0 and 𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (25) Then, the following hold. (i)𝑇_𝑟 is single valued.(ii)𝑇_𝑟 is firmly nonexpansive; that is, 𝑇_𝑟𝑥−𝑇_𝑟𝑦²≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻. (26)(iii)𝐹(𝑇_𝑟)=EP(𝐹). (iv)EP(𝐹) is closed and convex.

Lemma 6 (see [32]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. If 𝑇∶𝐶→𝐶 is a 𝑘-strict pseudocontraction, then (i)the mapping 𝐼−𝑇 is demiclosed at 0; that is, if {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 0, then (𝐼−𝑇)𝑥=0;(ii)the set 𝐹(𝑇) of 𝑇 is closed and convex so that the projection 𝑃_𝐹(𝑇) is well defined.

Lemma 7 (see [29]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (27) where {𝛾_𝑛} is a sequence in (0,1) and 𝛿_𝑛 is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞.Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 8 (see [4]). Let 𝐶 be a closed convex subset of 𝐻. Let {𝑥_𝑛} be a bounded sequence in 𝐻. Assume that (i)the weak 𝑤-limit set 𝑤_𝑤(𝑥_𝑛)⊂𝐶 where 𝑤_𝑤(𝑥_𝑛)={𝑥∶𝑥_𝑛𝑖⇀𝑥},(ii)for each 𝑧∈𝐶,lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists. Then, {𝑥_𝑛} is weakly convergent to a point in 𝐶.

Lemma 9 (see [33]). Let 𝐻 be a Hilbert space, 𝐶 a closed and convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a 𝑘-strict pseudocontraction mapping. Define a mapping 𝑉∶𝐶→𝐻 by 𝑉𝑥=𝜆𝑥+(1−𝜆)𝑇𝑥,  𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈𝐶. Then, as 𝑘≤𝜆<1,𝑉 is a nonexpansive mapping such that 𝐹(𝑉)=𝐹(𝑇).

Lemma 10 (see [11]). Let 𝐻 be a Hilbert space, 𝐶 a closed and convex subset of 𝐻, and 𝑇∶𝐶→𝐶 a nonexpansive mapping such that 𝐹(𝑇)≠∅. Then, ‖𝑇𝑥−𝑥‖²≤2𝑥−𝑇𝑥,𝑥−𝑥^′,∀𝑥^′∈𝐹(𝑇),∀𝑥∈𝐶. (28)

3. The Proposed Method and Some Properties

In this section, we suggest and analyze our method for finding the common solutions of the system of variational inequality problem (1), the mixed equilibrium problem (12), and the hierarchical fixed point problem (15).

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐷,𝐵_𝑖∶𝐶→𝐻 be 𝜂,𝜃_𝑖-inverse strongly monotone mappings for each 𝑖=1,2, respectively. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the assumptions (i)--(iv) of Lemma 4, 𝑆∶𝐶→𝐻 a nonexpansive mapping, and {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 a countable family of 𝑘_𝑖-strict pseudocontraction mappings such that 𝑆^∗∩MEP(𝐹)∩𝐹(𝑇)≠∅, where 𝐹(𝑇)∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)=⋂^∞_𝑖=1𝐹(𝑉_𝑖). Let 𝑓 be a 𝜌-contraction mapping.

Algorithm 11. For a given 𝑥₀∈𝐶 arbitrarily, let the iterative sequences {𝑢_𝑛},{𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be generated by 𝐹𝑢_𝑛,𝑦+𝐷𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶;_𝑛=𝑃_𝐶𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛;𝑦_𝑛=𝑃_𝐶𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛;𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛,∀𝑛≥0, (29) where 𝑉_𝑖=𝑘_𝑖𝐼+(1−𝑘_𝑖)𝑇_𝑖,0≤𝑘_𝑖<1,𝜇_𝑖∈(0,2𝜃_𝑖) for each 𝑖=1,2,{𝑟_𝑛}⊂(0,2𝜂),𝛼₀=1,{𝛼_𝑛} is a strictly decreasing sequence in (0,1), and {𝛽_𝑛} is a sequence in (0,1) satisfying the following conditions: (a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0, (c)∑^∞_𝑛=1|𝛼_𝑛−1−𝛼_𝑛|<∞ and ∑^∞_𝑛=1|𝛽_𝑛−1−𝛽_𝑛|<∞,(d)liminf_𝑛→∞𝑟_𝑛<limsup_𝑛→∞𝑟_𝑛<2𝜂 and ∑^∞_𝑛=1|𝑟_𝑛−1−𝑟_𝑛|<∞.

Lemma 12. Let 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇). Then, {𝑥_𝑛},{𝑢_𝑛},{𝑧_𝑛}, and {𝑦_𝑛} are bounded.

Proof. First, we show that the mapping (𝐼−𝑟_𝑛𝐷) is nonexpansive. For any 𝑥,𝑦∈𝐶, 𝐼−𝑟_𝑛𝐷𝑥−𝐼−𝑟_𝑛𝐷𝑦²=𝑥−𝑦−𝑟_𝑛𝐷𝑥−𝐷𝑦²=𝑥−𝑦²−2𝑟_𝑛𝑥−𝑦,𝐷𝑥−𝐷𝑦+𝑟²_𝑛𝐷𝑥−𝐷𝑦²≤𝑥−𝑦²−𝑟_𝑛2𝜂−𝑟_𝑛𝐷𝑥−𝐷𝑦²≤𝑥−𝑦². (30) Similarly, we can show that the mapping (𝐼−𝜇_𝑖𝐵_𝑖) is nonexpansive for each 𝑖=1,2. It follows from Lemma 5 that 𝑢_𝑛=𝑇_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛). Let 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇); we have 𝑥^∗=𝑇_𝑟𝑛(𝑥^∗−𝑟_𝑛𝐷𝑥^∗), and it follows that 𝑢_𝑛−𝑥^∗2=𝑇_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗−𝑟_𝑛𝐷𝑥^∗2≤𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐷𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑟_𝑛2𝜂−𝑟_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗2≤𝑥_𝑛−𝑥^∗2. (31) Let 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇); we have 𝑥^∗=𝑃_𝐶𝑦^∗−𝜇₁𝐵₁𝑦^∗, (32) where 𝑦^∗=𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗. (33) Setting 𝑣_𝑛=𝑃_𝐶[𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛]. Since 𝐵₂ is 𝜃₂-inverse strongly monotone mapping, it follows that 𝑣_𝑛−𝑦^∗2=𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗2≤𝑢_𝑛−𝑢^∗−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝜇₂2𝜃−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2≤𝑥_𝑛−𝑥^∗2. (34) Since 𝐵_𝑖 is 𝜃_𝑖-inverse strongly monotone mappings for each 𝑖=1,2, we get 𝑧_𝑛−𝑥^∗2=𝑃_𝐶𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗2≤𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−(𝑃_𝐶[𝑥^∗−𝜇₂𝐵₂𝑥^∗]−𝜇₁𝐵₁𝑃_𝐶[𝑥^∗−𝜇₂𝐵₂𝑥^∗])²=𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁(𝐵₁𝑃_𝐶[𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛]−𝐵₁𝑃_𝐶[𝑥^∗−𝜇₂𝐵₂𝑥^∗])²≤𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗2−𝜇₁2𝜃₁−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝐵₁𝑃_𝐶[𝑥^∗−𝜇₂𝐵₂𝑥^∗]²≤𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑥^∗−𝜇₂𝐵₂𝑥^∗2−𝜇₁2𝜃₁−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝐵₁𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗2≤𝑢_𝑛−𝑥^∗2−𝜇₂2𝜃₂−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2−𝜇₁2𝜃₁−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2≤𝑢_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2. (35) Next, we prove that the sequence {𝑥_𝑛} is bounded; without loss of generality we can assume that 𝛽_𝑛≤𝛼_𝑛 for all 𝑛≥1. From (29), we have 𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝛼_𝑛𝑥^∗−^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑉_𝑖𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑥^∗+𝛽_𝑛𝑆𝑥^∗−𝑥^∗+1−𝛽_𝑛𝑧_𝑛−𝑥^∗≤𝛼_𝑛𝜌𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑆𝑥^∗−𝑥^∗+1−𝛽_𝑛𝑥_𝑛−𝑥^∗=1−𝛼_𝑛×𝑥1−𝜌_𝑛−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥^∗−𝑥^∗≤1−𝛼_𝑛×𝑥1−𝜌_𝑛−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+𝛽_𝑛𝑆𝑥^∗−𝑥^∗≤1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗+𝑆𝑥^∗−𝑥^∗=1−𝛼_𝑛1−𝜌𝑥_𝑛−𝑥^∗+𝛼_𝑛1−𝜌×𝑓𝑥1−𝜌_∗−𝑥^∗+𝑆𝑥^∗−𝑥^∗≤max𝑥_𝑛−𝑥^∗,1𝑓𝑥1−𝜌^∗−𝑥^∗+𝑆𝑥^∗−𝑥^∗. (36) By induction on 𝑛, we obtain ‖𝑥_𝑛−𝑥^∗‖≤max{‖𝑥₀−𝑥^∗‖,(1/(1−𝜌))(‖𝑓(𝑥^∗)−𝑥^∗‖+‖𝑆𝑥^∗−𝑥^∗‖)}, for 𝑛≥0 and 𝑥₀∈𝐶. Hence, {𝑥_𝑛} is bounded and, consequently, we deduce that {𝑢_𝑛},{𝑧_𝑛},{𝑣_𝑛}, and {𝑦_𝑛} are bounded.

Lemma 13. Let 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇) and {𝑥_𝑛} be the sequence generated by Algorithm 11. Then one has(a)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (b)The weak 𝑤-limit set 𝑤_𝑤(𝑥_𝑛)⊂𝐹(𝑇),(𝑤_𝑤(𝑥_𝑛)={𝑥∶𝑥_𝑛𝑖⇀𝑥}).

Proof. Since 𝑢_𝑛=𝑇_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛) and 𝑢_𝑛−1=𝑇_𝑟𝑛−1(𝑥_𝑛−1−𝑟_𝑛−1𝐷𝑥_𝑛−1), we have 𝐹𝑢_𝑛,𝑦+𝐷𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝐹𝑢≥0,∀𝑦∈𝐶,_𝑛−1,𝑦+𝐷𝑥_𝑛−1,𝑦−𝑢_𝑛−1+1𝑟_𝑛−1𝑦−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1≥0,∀𝑦∈𝐶. (37)(38) Take 𝑦=𝑢_𝑛−1 in (37) and 𝑦=𝑢_𝑛 in (38), we get 𝐹𝑢_𝑛,𝑢_𝑛−1+𝐷𝑥_𝑛,𝑢_𝑛−1−𝑢_𝑛+1𝑟_𝑛𝑢_𝑛−1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝐹𝑢≥0,_𝑛−1,𝑢_𝑛+𝐷𝑥_𝑛−1,𝑢_𝑛−𝑢_𝑛−1+1𝑟_𝑛−1𝑢_𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1≥0. (39) Adding (39) and using the monotonicity of 𝐹, we have 𝐷𝑥_𝑛−1−𝐷𝑥_𝑛,𝑢_𝑛−𝑢_𝑛−1+𝑢_𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1𝑟_𝑛−1−𝑢_𝑛−𝑥_𝑛𝑟_𝑛≥0, (40) which implies that 0≤𝑢_𝑛−𝑢_𝑛−1,𝑟_𝑛𝐷𝑥_𝑛−1−𝐷𝑥_𝑛+𝑟_𝑛𝑟_𝑛−1𝑢_𝑛−1−𝑥_𝑛−1−𝑢_𝑛−𝑥_𝑛=𝑢_𝑛−1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛−1+𝑟1−^𝑛𝑟_𝑛−1𝑢_𝑛−1+𝑥_𝑛−1−𝑟_𝑛𝐷𝑥_𝑛−1−𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛𝑟_𝑛−1𝑥_𝑛−1=𝑢_𝑛−1−𝑢_𝑛,𝑟1−^𝑛𝑟_𝑛−1𝑢_𝑛−1+𝑥_𝑛−1−𝑟_𝑛𝐷𝑥_𝑛−1−𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛𝑟_𝑛−1𝑥_𝑛−1−𝑢_𝑛−𝑢_𝑛−1²=𝑢_𝑛−1−𝑢_𝑛,𝑟1−^𝑛𝑟_𝑛−1𝑢_𝑛−1−𝑥_𝑛−1+𝑥_𝑛−1−𝑟_𝑛𝐷𝑥_𝑛−1−𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑢_𝑛−𝑢_𝑛−1²≤𝑢_𝑛−1−𝑢_𝑛×𝑟1−^𝑛𝑟_𝑛−1𝑢_𝑛−1−𝑥_𝑛−1+𝑥_𝑛−1−𝑟_𝑛𝐷𝑥_𝑛−1−𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑢_𝑛−𝑢_𝑛−1²≤𝑢_𝑛−1−𝑢_𝑛×𝑟1−^𝑛𝑟_𝑛−1𝑢_𝑛−1−𝑥_𝑛−1+𝑥_𝑛−1−𝑥_𝑛−𝑢_𝑛−𝑢_𝑛−1² (41) and then 𝑢_𝑛−1−𝑢_𝑛≤𝑟1−^𝑛𝑟_𝑛−1𝑢_𝑛−1−𝑥_𝑛−1+𝑥_𝑛−1−𝑥_𝑛. (42) Without loss of generality, let us assume that there exists a real number 𝜇 such that 𝑟_𝑛>𝜇>0, for all positive integers 𝑛. Then, we get 𝑢_𝑛−1−𝑢_𝑛≤𝑥_𝑛−1−𝑥_𝑛+1𝜇𝑟_𝑛−1−𝑟_𝑛𝑢_𝑛−1−𝑥_𝑛−1. (43) Next, we estimate 𝑧_𝑛−𝑧_𝑛−1²=𝑃_𝐶𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1²≤𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1²=𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝐵₁𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1²≤𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1²−𝜇₁2𝜃₁−𝜇₁×𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝐵₁𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1²≤𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−1²≤𝑢_𝑛−𝑢_𝑛−1−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑢_𝑛−1²≤𝑢_𝑛−𝑢_𝑛−1²−𝜇₂2𝜃₂−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑢_𝑛−1²≤𝑢_𝑛−𝑢_𝑛−1². (44) It follows from (43) and (44) that 𝑧_𝑛−𝑧_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+1𝜇𝑟_𝑛−1−𝑟_𝑛𝑢_𝑛−1−𝑥_𝑛−1. (45) From (29) and the previous inequality, we get 𝑦_𝑛−𝑦_𝑛−1≤𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+1−𝛽_𝑛−1𝑧_𝑛−1=𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+1−𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝑧_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑧_𝑛−1≤𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛×𝑥_𝑛−1−𝑥_𝑛+1𝜇𝑟_𝑛−1−𝑟_𝑛𝑢_𝑛−1−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑧_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1𝜇𝑟_𝑛−1−𝑟_𝑛𝑢_𝑛−1−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑧_𝑛−1. (46) Next, we estimate 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+^𝑛−1_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑉_𝑖𝑦_𝑛−1+𝛼_𝑛−1−𝛼_𝑛𝑉_𝑛𝑦_𝑛−1≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑉_𝑖𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝑉_𝑛𝑦_𝑛−1≤𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝑉_𝑛𝑦_𝑛−1=𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝑉_𝑛𝑦_𝑛−1. (47) From (46) and (47), we have 𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛×𝑥_𝑛−𝑥_𝑛−1+1𝜇𝑟_𝑛−1−𝑟_𝑛𝑢_𝑛−1−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑧_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝑉_𝑛𝑦_𝑛−1≤1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+1𝜇𝑟_𝑛−1−𝑟_𝑛×𝑢_𝑛−1−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+𝑧_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝑉_𝑛𝑦_𝑛−1≤1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1×1+𝑀𝜇𝑟_𝑛−1−𝑟_𝑛+𝛽_𝑛−𝛽_𝑛−1+𝛼_𝑛−𝛼_𝑛−1, (48) where 𝑀=maxsup_𝑛≥1𝑢_𝑛−1−𝑥_𝑛−1,sup_𝑛≥1𝑆𝑥_𝑛−1+𝑧_𝑛−1,sup_𝑛≥1𝑓𝑥_𝑛−1+𝑉_𝑛𝑦_𝑛−1. (49) It follows by conditions (a)--(d) of Algorithm 11 and Lemma 7 that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (50) Next, we show that lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0. Since 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇) and 𝛼_𝑛+∑^𝑛_𝑖=1(𝛼_𝑖−1−𝛼_𝑖)=1, by using (31) and (35), we obtain 𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝛼_𝑛𝑥^∗−^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑉_𝑖𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑧_𝑛−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛1−𝛽_𝑛×𝑢_𝑛−𝑥^∗2−𝜇₂2𝜃₂−𝜇₂×𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2−𝜇₁2𝜃₁−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛1−𝛽_𝑛×𝑥_𝑛−𝑥^∗2−𝑟_𝑛2𝜂−𝑟_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗2−𝜇₂2𝜃₂−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2−𝜇₁2𝜃₁−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛1−𝛽_𝑛𝑟_𝑛2𝜂−𝑟_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗2+𝜇₂2𝜃₂−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2+𝜇₁2𝜃₁−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2. (51) Then, from the previous inequality, we get 1−𝛼_𝑛1−𝛽_𝑛𝑟_𝑛2𝜂−𝑟_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗2+𝜇₂2𝜃₂−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2+𝜇₁2𝜃₁−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛. (52) Since liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜂,lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0,𝛼_𝑛→0, and 𝛽_𝑛→0, we obtain lim_𝑛→∞‖𝐵₂𝑢_𝑛−𝐵₂𝑥^∗‖=0,lim_𝑛→∞‖𝐵₁𝑣_𝑛−𝐵₁𝑦^∗‖=0, and lim_𝑛→∞‖𝐷𝑥_𝑛−𝐷𝑥^∗‖=0.
Since 𝑇_𝑟𝑛 is firmly nonexpansive, we have 𝑢_𝑛−𝑥^∗2=𝑇_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑇_𝑟𝑛𝑥^∗−𝑟_𝑛𝐷𝑥^∗2≤𝑢_𝑛−𝑥^∗,𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐷𝑥^∗=12𝑢_𝑛−𝑥^∗2+𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐷𝑥^∗2−𝑢_𝑛−𝑥^∗−𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛−𝑥^∗−𝑟_𝑛𝐷𝑥^∗2. (53) Hence, 𝑢_𝑛−𝑥^∗2≤(𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛)−(𝑥^∗−𝑟_𝑛𝐷𝑥^∗)²−𝑢_𝑛−𝑥_𝑛+𝑟_𝑛(𝐷𝑥_𝑛−𝐷𝑥^∗)²≤𝑥_𝑛−𝑥^∗2−𝑢_𝑛−𝑥_𝑛+𝑟_𝑛(𝐷𝑥_𝑛−𝐷𝑥^∗)²≤𝑥_𝑛−𝑥^∗2−𝑢_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗. (54) From (51), (35), and the previous inequality, we have 𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑧_𝑛−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑢_𝑛−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛×𝑥_𝑛−𝑥^∗2−𝑢_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−1−𝛼_𝑛1−𝛽_𝑛𝑢_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗. (55) Hence, 1−𝛼_𝑛1−𝛽_𝑛𝑢_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗. (56) Since lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0,𝛼_𝑛→0,𝛽_𝑛→0, and lim_𝑛→∞‖𝐷𝑥_𝑛−𝐷𝑥^∗‖=0, we obtain lim_𝑛→∞𝑢_𝑛−𝑥_𝑛=0. (57) From (21), we get 𝑣_𝑛−𝑦^∗2=𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑃_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗2≤𝑣_𝑛−𝑦^∗,𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑥^∗−𝜇₂𝐵₂𝑥^∗=12𝑣_𝑛−𝑦^∗2+𝑢_𝑛−𝑥^∗−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2−𝑢_𝑛−𝑥^∗−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗−𝑣_𝑛−𝑦^∗2≤12𝑣_𝑛−𝑦^∗2+𝑢_𝑛−𝑥^∗2−𝜇₂2𝜃₂−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2−𝑢_𝑛−𝑥^∗−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗−(𝑣_𝑛−𝑦^∗)²≤12𝑣_𝑛−𝑦^∗2+𝑢_𝑛−𝑥^∗2−𝑢_𝑛−𝑣_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗−𝑥^∗−𝑦^∗2≤12𝑣_𝑛−𝑦^∗2+𝑢_𝑛−𝑥^∗2−‖𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗‖²+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗,𝐵₂𝑢_𝑛−𝐵₂𝑥^∗−𝜇²₂𝐵₂𝑢_𝑛−𝐵₂𝑥^∗2≤12𝑣_𝑛−𝑦^∗2+𝑢_𝑛−𝑥^∗2−𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗2+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗𝐵₂𝑢_𝑛−𝐵₂𝑥^∗. (58) Hence, 𝑣_𝑛−𝑦^∗2≤𝑢_𝑛−𝑥^∗2−𝑢_𝑛−𝑣_𝑛−(𝑥^∗−𝑦^∗)²+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗𝐵₂𝑢_𝑛−𝐵₂𝑥^∗≤𝑥_𝑛−𝑥^∗2−𝑢_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗−𝑢_𝑛−𝑣_𝑛−(𝑥^∗−𝑦^∗)²+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗𝐵₂𝑢_𝑛−𝐵₂𝑥^∗, (59) where the last inequality follows from (54). On the other hand, from (29) and (21), we obtain 𝑧_𝑛−𝑥^∗2=𝑃_𝐶[𝑣_𝑛−𝜇₁𝐵₁𝑣_𝑛]−𝑃_𝐶[𝑦^∗−𝜇₁𝐵₁𝑦^∗]²≤𝑧_𝑛−𝑥^∗,𝑣_𝑛−𝜇₁𝐵₁𝑣_𝑛−𝑦^∗−𝜇₁𝐵₁𝑦^∗=12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑦^∗−𝜇₁(𝐵₁𝑣_𝑛−𝐵₁𝑦^∗)²−𝑣_𝑛−𝑦^∗−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗−𝑧_𝑛−𝑥^∗2=12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑦^∗2−2𝜇₁𝑣_𝑛−𝑦^∗,𝐵₁𝑣_𝑛−𝐵₁𝑦^∗+𝜇²₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2−𝑣_𝑛−𝑦^∗−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗−𝑧_𝑛−𝑥^∗2≤12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑦^∗2−𝜇₁2𝜃₁−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗2−𝑣_𝑛−𝑦^∗−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗−𝑧_𝑛−𝑥^∗2≤12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑦^∗2−𝑣_𝑛−𝑧_𝑛−𝜇₁𝐵₁𝑣_𝑛−𝐵₁𝑦^∗+𝑥^∗−𝑦^∗2≤12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑦^∗2−𝑣_𝑛−𝑧_𝑛+(𝑥^∗−𝑦^∗)²+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗,𝐵₁𝑣_𝑛−𝐵₁𝑦^∗≤12𝑧_𝑛−𝑥^∗2+𝑣_𝑛−𝑦^∗2−𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗2+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗×𝐵₁𝑣_𝑛−𝐵₁𝑦^∗. (60) Hence, 𝑧_𝑛−𝑥^∗2≤𝑣_𝑛−𝑦^∗2−𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗2+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗𝐵₁𝑣_𝑛−𝐵₁𝑦^∗≤𝑥_𝑛−𝑥^∗2−𝑢_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗−𝑢_𝑛−𝑣_𝑛−(𝑥^∗−𝑦^∗)²+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗𝐵₂𝑢_𝑛−𝐵₂𝑥^∗−𝑣_𝑛−𝑧_𝑛+(𝑥^∗−𝑦^∗)²+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗𝐵₁𝑣_𝑛−𝐵₁𝑦^∗, (61) where the last inequality follows from (59). From (51) and the previous inequality, we have 𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑥^∗2+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑧_𝑛−𝑥^∗2≤𝛼_𝑛𝑓𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑥_𝑛−𝑥^∗2−𝑢_𝑛−𝑥_𝑛²+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗+1−𝛽_𝑛−𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗2+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗×𝐵₂𝑢_𝑛−𝐵₂𝑥^∗+1−𝛽_𝑛−𝑣_𝑛−𝑧_𝑛+(𝑥^∗−𝑦^∗)²+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗×𝐵₁𝑣_𝑛−𝐵₁𝑦^∗≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛×𝐷𝑥_𝑛−𝐷𝑥^∗+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗×𝐵₂𝑢_𝑛−𝐵₂𝑥^∗+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗×𝐵₁𝑣_𝑛−𝐵₁𝑦^∗−1−𝛼_𝑛1−𝛽_𝑛×𝑢_𝑛−𝑥_𝑛²+𝑢_𝑛−𝑣_𝑛−(𝑥^∗−𝑦^∗)²+𝑣_𝑛−𝑧_𝑛+(𝑥^∗−𝑦^∗)². (62) Hence, 1−𝛼_𝑛1−𝛽_𝑛×𝑢_𝑛−𝑥_𝑛²+𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗2+𝑣_𝑛−𝑧_𝑛+(𝑥^∗−𝑦^∗)²≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗𝐵₂𝑢_𝑛−𝐵₂𝑥^∗+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗𝐵₁𝑣_𝑛−𝐵₁𝑦^∗≤𝛼_𝑛𝑓(𝑥_𝑛)−𝑥^∗2+𝛽_𝑛𝑆𝑥_𝑛−𝑥^∗2+𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+2𝑟_𝑛𝑢_𝑛−𝑥_𝑛𝐷𝑥_𝑛−𝐷𝑥^∗+2𝜇₂𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗𝐵₂𝑢_𝑛−𝐵₂𝑥^∗+2𝜇₁𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗𝐵₁𝑣_𝑛−𝐵₁𝑦^∗. (63) Since lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0,𝛼_𝑛→0,𝛽_𝑛→0, and lim_𝑛→∞‖𝐷𝑥_𝑛−𝐷𝑥^∗‖=0, lim_𝑛→∞‖𝐵₂𝑢_𝑛−𝐵₂𝑥^∗‖=0, lim_𝑛→∞‖𝐵₁𝑣_𝑛−𝐵₁𝑦^∗‖=0, we obtain lim_𝑛→∞𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗=0,lim_𝑛→∞𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗=0. (64) Since 𝑢_𝑛−𝑧_𝑛≤𝑢_𝑛−𝑣_𝑛−𝑥^∗−𝑦^∗+𝑣_𝑛−𝑧_𝑛+𝑥^∗−𝑦^∗, (65) we get lim_𝑛→∞𝑢_𝑛−𝑧_𝑛=0. (66) It follows from (57) and (66) that lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (67) Now, let 𝑧∈𝐹(𝑇)∩𝑆^∗∩MEP(𝐹); since for each 𝑖≥1,𝑉_𝑖𝑥_𝑛∈𝐶, and 𝛼_𝑛+∑^𝑛_𝑖=1(𝛼_𝑖−1−𝛼_𝑖)=1, we have ∑^𝑛_𝑖=1(𝛼_𝑖−1−𝛼_𝑖)𝑉_𝑖𝑥_𝑛+𝛼_𝑛𝑧∈𝐶. And ^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑥_𝑛−𝑉_𝑖𝑥_𝑛=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛+1−𝛼_𝑛𝑥_𝑛−^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑥_𝑛+𝛼_𝑛𝑧+𝛼_𝑛𝑧−𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛+𝛼_𝑛𝑧−𝑥_𝑛+1−𝑃_𝐶^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑥_𝑛+𝛼_𝑛𝑧+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1. (68) It follows that ^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑥_𝑛−𝑉_𝑖𝑥_𝑛,𝑥_𝑛−𝑥^∗=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑃_𝐶^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑥_𝑛+𝛼_𝑛𝑧,𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1,𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1,𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑧+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑉_𝑖𝑥_𝑛×𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑧𝑥_𝑛−𝑥^∗+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗=𝛼_𝑛𝑓𝑥_𝑛−𝑧𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑦_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑧𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛×𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑧𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗. (69) From Lemma 10 and the previous inequality, we get 12^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑥_𝑛−𝑉_𝑖𝑥_𝑛²≤^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑥_𝑛−𝑉_𝑖𝑥_𝑛,𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑓𝑥_𝑛−𝑧𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗+1−𝛼_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑧−𝑥_𝑛+1𝑥_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑥^∗. (70) Since lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0,𝛼_𝑛→0,𝛽_𝑛→0, and lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0, we obtain lim^𝑛_{𝑛→∞𝑖=1}𝛼_𝑖−1−𝛼_𝑖𝑥_𝑛−𝑉_𝑖𝑥_𝑛²=0. (71) Since (𝛼_𝑖−1−𝛼_𝑖)‖𝑥_𝑛−𝑉_𝑖𝑥_𝑛‖²≤∑^𝑛_𝑖=1(𝛼_𝑖−1−𝛼_𝑖)‖𝑥_𝑛−𝑉_𝑖𝑥_𝑛‖² and {𝛼_𝑛} is strictly decreasing, we have lim_𝑛→∞𝑥_𝑛−𝑉_𝑖𝑥_𝑛=0. (72) Hence, we obtain lim_𝑛→∞𝑥_𝑛−𝑇_𝑖𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑉_𝑖𝑥_𝑛1−𝑘_𝑖=0,∀𝑖≥1. (73) Since {𝑥_𝑛} is bounded, without loss of generality, we can assume that 𝑥_𝑛⇀𝑤∈𝐶. It follows from Lemma 6 that 𝑤∈𝐹(𝑇). Therefore, 𝑤_𝑤(𝑥_𝑛)⊂𝐹(𝑇).

Theorem 14. The sequence {𝑥_𝑛} generated by Algorithm 11 converges strongly to 𝑧=𝑃_𝑆^∗_{∩MEP(𝐹)∩𝐹(𝑇)}𝑓(𝑧), which is the unique solution of the variational inequality 𝐼−𝑓𝑧,𝑥−𝑧≥0,∀𝑥∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇). (74)

Proof. Since {𝑥_𝑛} is bounded 𝑥_𝑛⇀𝑤 and from Lemma 13, we have 𝑤∈𝐹(𝑇). Next, we show that 𝑤∈MEP(𝐹). Since 𝑢_𝑛=𝑇_𝑟𝑛(𝑥_𝑛−𝑟_𝑛𝐷𝑥_𝑛), we have 𝐹𝑢_𝑛,𝑦+𝐷𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (75) It follows from monotonicity of 𝐹 that 𝐷𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶,𝐷𝑥_𝑛𝑘,𝑦−𝑢_𝑛𝑘+𝑦−𝑢_𝑛𝑘,𝑢_𝑛𝑘−𝑥_𝑛𝑘𝑟_𝑛𝑘≥𝐹𝑦,𝑢_𝑛𝑘,∀𝑦∈𝐶. (76) Since lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0 and 𝑥_𝑛⇀𝑤, it easy to observe that 𝑢_𝑛𝑘→𝑤. For any 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤; we have 𝑦_𝑡∈𝐶. Then from (76), we obtain 𝐷𝑦_𝑡,𝑦_𝑡−𝑢_𝑛𝑘≥𝐷𝑦_𝑡,𝑦_𝑡−𝑢_𝑛𝑘−𝐷𝑥_𝑛𝑘,𝑦_𝑡−𝑢_𝑛𝑘−𝑦_𝑡−𝑢_𝑛𝑘,𝑢_𝑛𝑘−𝑥_𝑛𝑘𝑟_𝑛𝑘+𝐹𝑦_𝑡,𝑢_𝑛𝑘=𝐷𝑦_𝑡−𝐷𝑢_𝑛𝑘,𝑦_𝑡−𝑢_𝑛𝑘+𝐷𝑢_𝑛𝑘−𝐷𝑥_𝑛𝑘,𝑦_𝑡−𝑢_𝑛𝑘−𝑦_𝑡−𝑢_𝑛𝑘,𝑢_𝑛𝑘−𝑥_𝑛𝑘𝑟_𝑛𝑘+𝐹𝑦_𝑡,𝑢_𝑛𝑘. (77) Since 𝐷 is Lipschitz continuous and lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0, we obtain lim_𝑘→∞‖𝐷𝑢_𝑛𝑘−𝐷𝑥_𝑛𝑘‖=0. From the monotonicity of 𝐷 and 𝑢_𝑛𝑘→𝑤, it follows from (77) that 𝐷𝑦_𝑡,𝑦_𝑡−𝑤≥𝐹𝑦_𝑡,𝑤. (78) Hence, from assumptions (i)--(iv) of Lemma 4 and (78), we have 0=𝐹𝑦_𝑡,𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+(1−𝑡)𝐹𝑦_𝑡𝑦,𝑤≤𝑡𝐹_𝑡,𝑦+(1−𝑡)𝐷𝑦_𝑡,𝑦_𝑡𝑦−𝑤≤𝑡𝐹_𝑡,𝑦+(1−𝑡)𝑡𝐷𝑦_𝑡,𝑦−𝑤 (79) which implies that 𝐹(𝑦_𝑡,𝑦)+(1−𝑡)⟨𝐷𝑦_𝑡,𝑦−𝑤⟩≥0. Letting 𝑡→0₊, we have 𝐹𝑤,𝑦+𝐷𝑤,𝑦−𝑤≥0,∀𝑦∈𝐶, (80) which implies that 𝑤∈MEP(𝐹). Next, we show that 𝑤∈𝑆^∗. Since lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0 and there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑤, it easy to observe that 𝑧_𝑛𝑘→𝑤. For any 𝑥,𝑦∈𝐶, using (23), we have 𝑄(𝑥)−𝑄𝑦²=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝑃_𝐶𝑃_𝐶𝑦−𝜇₂𝐵₂𝑦−𝜇₁𝐵₁𝑃_𝐶𝑦−𝜇₂𝐵₂𝑦²≤𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝑃_𝐶𝑦−𝜇₂𝐵₂𝑦−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝐵₁𝑃_𝐶𝑦−𝜇₂𝐵₂𝑦²≤𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝑃_𝐶𝑦−𝜇₂𝐵₂𝑦²−𝜇₁2𝜃₁−𝜇₁𝑃_𝐶[𝑥−𝜇₂𝐵₂𝑥]−𝑃_𝐶[𝑦−𝜇₂𝐵₂𝑦]²≤𝑃_𝐶[𝑥−𝜇₂𝐵₂𝑥]−𝑃_𝐶[𝑦−𝜇₂𝐵₂𝑦]²≤𝑥−𝜇₂𝐵₂𝑥−𝑦−𝜇₂𝐵₂𝑦²≤𝑥−𝑦²−𝜇₂2𝜃₂−𝜇₂𝐵₂𝑥−𝐵₂𝑦²≤𝑥−𝑦². (81) This implies that 𝑄∶𝐶→𝐶 is nonexpansive. On the other hand, 𝑧_𝑛−𝑄𝑧_𝑛²=𝑃_𝐶𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝑄𝑧_𝑛²=𝑄𝑢_𝑛−𝑄𝑧_𝑛²≤𝑢_𝑛−𝑧_𝑛². (82) Since lim_𝑛→∞‖𝑢_𝑛−𝑧_𝑛‖=0 (see (66)), we have lim_𝑛→∞‖𝑧_𝑛−𝑄(𝑧_𝑛)‖=0. It follows from Lemma 6 that 𝑤=𝑄(𝑤), which implies from Lemma 3 that 𝑤∈𝑆^∗.
Thus, we have 𝑤∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇). (83) Next, we claim that limsup_𝑛→∞⟨𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧⟩≤0 where 𝑧=𝑃_𝑆^∗_{∩MEP(𝐹)∩𝐹(𝑇)}𝑓(𝑧).
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑥_𝑛−𝑧=limsup_𝑘→∞𝑓(𝑧)−𝑧,𝑥_𝑛𝑘=𝑓(𝑧)−𝑧−𝑧,𝑤−𝑧≤0. (84) Next, we show that 𝑥_𝑛→𝑧.
One has𝑥_𝑛+1−𝑧²=𝑥_𝑛+1−𝛼_𝑛𝑓𝑥_𝑛−^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛,𝑥_𝑛+1+𝛼−𝑧_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑧,𝑥_𝑛+1≤𝛼−𝑧_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑧,𝑥_𝑛+1−𝑧=𝛼_𝑛𝑓𝑥_𝑛−𝑓(𝑧),𝑥_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1+−𝑧_𝑛𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑧,𝑥_𝑛+1−𝑧≤𝛼_𝑛𝑓𝑥_𝑛−𝑓(𝑧)𝑥_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1+−𝑧_𝑛𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑧𝑥_𝑛+1−𝑧≤𝛼_𝑛𝜌𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1+−𝑧_𝑛𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑧𝑥_𝑛+1−𝑧≤𝛼_𝑛𝜌𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧+1−𝛼_𝑛×𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑧+𝛽_𝑛‖𝑆𝑧−𝑧‖+1−𝛽_𝑛𝑧_𝑛×𝑥−𝑧_𝑛+1−𝑧≤𝛼_𝑛𝜌𝑥_𝑛−𝑧𝑥_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1×−𝑧1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧+𝛽_𝑛‖‖+𝑆𝑧−𝑧1−𝛽_𝑛𝑥_𝑛−𝑧𝑥_𝑛+1≤−𝑧1−𝛼_𝑛1−𝜌𝑥_𝑛×𝑥−𝑧_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1+−𝑧1−𝛼_𝑛𝛽_𝑛‖𝑆𝑧−𝑧‖𝑥_𝑛+1≤−𝑧1−𝛼_𝑛1−𝜌2𝑥_𝑛−𝑧²+𝑥_𝑛+1−𝑧²+𝛼_𝑛𝑓(𝑧)−𝑧,𝑥_𝑛+1+−𝑧1−𝛼_𝑛𝛽_𝑛‖𝑆𝑧−𝑧‖𝑥_𝑛+1−𝑧 (85) which implies that 𝑥_𝑛+1−𝑧²≤1−2𝛼_𝑛1−𝜌1+𝛼_𝑛𝑥1−𝜌^𝑛−𝑧²+2𝛼_𝑛1+𝛼_𝑛𝑓(𝑧)1−𝜌−𝑧,𝑥_𝑛+1+2−𝑧1−𝛼_𝑛𝛽_𝑛1+𝛼_𝑛‖𝑥1−𝜌𝑆𝑧−𝑧‖_𝑛+1≤−𝑧1−2𝛼_𝑛1−𝜌1+𝛼_𝑛𝑥1−𝜌^𝑛−𝑧²+2𝛼_𝑛1−𝜌1+𝛼_𝑛11−𝜌𝑓(𝑧)1−𝜌−𝑧,𝑥_𝑛+1+−𝑧1−𝛼_𝑛𝛽_𝑛𝛼_𝑛‖‖×𝑥1−𝜌𝑆𝑧−𝑧_𝑛+1−𝑧. (86) Let 𝛾_𝑛=2𝛼_𝑛(1−𝜌)/(1+𝛼_𝑛(1−𝜌)) and 𝛿_𝑛=(2𝛼_𝑛(1−𝜌)/(1+𝛼_𝑛(1−𝜌))){(1/(1−𝜌))⟨𝑓(𝑧)−𝑧,𝑥_𝑛+1−𝑧⟩+((1−𝛼_𝑛)𝛽_𝑛/(𝛼_𝑛(1−𝜌)))‖𝑆𝑧−𝑧‖‖𝑥_𝑛+1−𝑧‖}.
Since ^∞_𝑛=1𝛼_𝑛=∞,1+𝛼_𝑛1−𝜌≤2,limsup_𝑛→∞1𝑓(𝑧)1−𝜌−𝑧,𝑥_𝑛+1+−𝑧1−𝛼_𝑛𝛽_𝑛𝛼_𝑛‖𝑥1−𝜌𝑆𝑧−𝑧‖_𝑛+1−𝑧≤0. (87) It follows that ^∞_𝑛=1𝛾_𝑛=∞limsup_𝑛→∞𝛿_𝑛𝛾_𝑛lim≤0. (88) Thus, all the conditions of Lemma 7 are satisfied. Hence, we deduce that 𝑥_𝑛→𝑧.
Since 𝑃_𝑆^∗_{∩MEP(𝐹)∩𝐹(𝑇)}𝑓 is a contraction, there exists a unique 𝑧∈𝐶 such that 𝑧=𝑃_𝑆^∗_{∩MEP(𝐹)∩𝐹(𝑇)}𝑓(𝑧). From (20), it follows that 𝑧 is the unique solution of the problem (74). This completes the proof.

Theorem 15. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐷,𝐵_𝑖∶𝐶→𝐻 be 𝜂,𝜃_𝑖-inverse strongly monotone mappings for each 𝑖=1,2, respectively. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the assumptions (𝑖)--(𝑖𝑣) of Lemma 4, 𝑆∶𝐶→𝐻 a nonexpansive mapping, and {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 a countable family of 𝑘_𝑖-strict pseudocontraction mappings such that 𝑆^∗∩MEP(𝐹)∩𝐹(𝑇)≠∅, where 𝐹(𝑇)∶=⋂^∞_𝑖=1𝐹(𝑇_𝑖)=⋂^∞_𝑖=1𝐹(𝑉_𝑖). Let 𝑓 be a 𝜌-contraction mapping. For a given 𝑥₀∈𝐶 arbitrarily, let the iterative sequences {𝑢_𝑛},{𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be generated by 𝐹𝑢_𝑛,𝑦+𝐷𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶;_𝑛=𝑃_𝐶𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑢_𝑛−𝜇₂𝐵₂𝑢_𝑛;𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑧_𝑛;𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛,∀𝑛≥0, (89) where 𝑉_𝑖=𝑘_𝑖𝐼+(1−𝑘_𝑖)𝑇_𝑖,0≤𝑘_𝑖<1,𝜇_𝑖∈(0,2𝜃_𝑖) for each 𝑖=1,2,{𝑟_𝑛}⊂(0,2𝜂),𝛼₀=1, {𝛼_𝑛} is a strictly decreasing sequence in (0,1), and {𝛽_𝑛} is a sequence in (0,1) satisfying the following conditions: (a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞), (c)∑^∞_𝑛=1(𝛼_𝑛−1−𝛼_𝑛)<∞ and ∑^∞_𝑛=1|𝛽_𝑛−1−𝛽_𝑛|<∞,(d)lim_𝑛→∞((1/𝜇)|𝑟_𝑛−1−𝑟_𝑛|+|𝛼_𝑛−1−𝛼_𝑛|+|𝛽_𝑛−1−𝛽_𝑛|/𝛼_𝑛𝛽_𝑛)=0, (e)there exists a constant 𝐾>0 such that (1/𝛼_𝑛)|(1/𝛽_𝑛)−(1/𝛽_𝑛−1)|≤𝐾,(f)liminf_𝑛→∞𝑟_𝑛<limsup_𝑛→∞𝑟_𝑛<2𝜂 and ∑^∞_𝑛=1|𝑟_𝑛−1−𝑟_𝑛|<∞. Then, sequence {𝑥_𝑛} generated by Algorithm (89) converges strongly to 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇), which is the unique solution of the variational inequality 1𝜏𝐼−𝑓𝑥^∗+(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇). (90)

Proof. From lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞), without loss of generality, we can assume that 𝛽_𝑛≤(1+𝜏)𝛼_𝑛 for all 𝑛≥1. Hence 𝛽_𝑛→0. By similar argument as that lemmas 12 and 13, we can deduce that {𝑥_𝑛} is bounded, lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0,lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0 (see (67)), and ‖(𝐼−𝑉_𝑖)𝑥_𝑛‖→0. Then, we have 𝑦_𝑛−𝑥_𝑛≤𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛⟶0as𝑛⟶∞. (91) It follows that, for all 𝑖≥1, 𝑦_𝑛−𝑉_𝑖𝑥_𝑛≤𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑉_𝑖𝑥_𝑛⟶0as𝑛⟶∞. (92) From (91) and (92), we have 𝑦_𝑛−𝑉_𝑖𝑦_𝑛≤𝑦_𝑛−𝑉_𝑖𝑥_𝑛+𝑉_𝑖𝑥_𝑛−𝑉_𝑖𝑦_𝑛≤𝑦_𝑛−𝑉_𝑖𝑥_𝑛+𝑦_𝑛−𝑥_𝑛⟶0as𝑛⟶∞. (93) Set 𝑤_𝑛=𝛼_𝑛𝑓(𝑥_𝑛)+∑^𝑛_𝑖=1(𝛼_𝑖−1−𝛼_𝑖)𝑉_𝑖𝑦_𝑛. From (47) and (48), we obtain 𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛≤𝑤_𝑛−𝑤_𝑛−1𝛽_𝑛≤1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_𝑛𝑟+𝑀1/𝜇_𝑛−𝑟_𝑛−1𝛽_𝑛+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛=1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_𝑛−1+1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−11𝛽_𝑛−1𝛽_𝑛−1𝑟+𝑀1/𝜇_𝑛−𝑟_𝑛−1𝛽_𝑛+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛≤1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_𝑛−1+𝑥_𝑛−𝑥_𝑛−11𝛽_𝑛−1𝛽_𝑛−1𝑟+𝑀1/𝜇_𝑛−𝑟_𝑛−1𝛽_𝑛+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛≤1−1−𝜌𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1𝛽_𝑛−1+𝛼_𝑛𝐾𝑥_𝑛−𝑥_𝑛−1𝑟+𝑀1/𝜇_𝑛−𝑟_𝑛−1𝛽_𝑛+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛≤1−1−𝜌𝛼_𝑛𝑤_𝑛−1−𝑤_𝑛−2𝛽_𝑛−1+𝛼_𝑛𝐾𝑥_𝑛−𝑥_𝑛−1𝑟+𝑀1/𝜇_𝑛−𝑟_𝑛−1𝛽_𝑛+𝛽_𝑛−𝛽_𝑛−1𝛽_𝑛+𝛼_𝑛−𝛼_𝑛−1𝛽_𝑛. (94) Let 𝛾_𝑛=(1−𝜌)𝛼_𝑛 and 𝛿_𝑛=𝛼_𝑛𝐾‖𝑥_𝑛−𝑥_𝑛−1‖+𝑀((1/𝜇)|𝑟_𝑛−𝑟_𝑛−1|/𝛽_𝑛+|𝛽_𝑛−𝛽_𝑛−1|/𝛽_𝑛+|𝛼_𝑛−𝛼_𝑛−1|/𝛽_𝑛). From conditions (a) and (d), we have ^∞_𝑛=1𝛾_𝑛=∞,lim_𝑛→∞𝛿_𝑛𝛾_𝑛=0. (95) By Lemma 7, we obtain lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛𝛽_𝑛=0,lim_𝑛→∞𝑤_𝑛+1−𝑤_𝑛𝛽_𝑛=lim_𝑛→∞𝑤_𝑛+1−𝑤_𝑛𝛼_𝑛=0. (96) From (89), we have 𝑥_𝑛+1=𝑃_𝐶𝑤_𝑛−𝑤_𝑛+𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑦_𝑛+1−𝛼_𝑛𝑦_𝑛. (97) Hence, it follows that 𝑥_𝑛−𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑥_𝑛−𝑃_𝐶𝑤_𝑛−𝑤_𝑛+𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑉_𝑖𝑦_𝑛−𝑦_𝑛+1−𝛼_𝑛𝑦_𝑛=1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛+𝑤_𝑛−𝑃_𝐶𝑤_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑉_𝑖𝑦_𝑛+𝛼_𝑛𝑥_𝑛−𝑓𝑥_𝑛 (98) and hence 𝑥_𝑛−𝑥_𝑛+11−𝛼_𝑛𝛽_𝑛=𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝛽_𝑛𝑥_𝑛−𝑧_𝑛+11−𝛼_𝑛𝛽_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛+11−𝛼_𝑛𝛽_𝑛^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑉_𝑖𝑦_𝑛+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑓𝑥_𝑛. (99) Let 𝑣_𝑛=(𝑥_𝑛−𝑥_𝑛+1)/(1−𝛼_𝑛)𝛽_𝑛. For any 𝑧∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇), we have 𝑣_𝑛,𝑥_𝑛=1−𝑧1−𝛼_𝑛𝛽_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1+𝛼−𝑧_𝑛1−𝛼_𝑛𝛽_𝑛𝐼−𝑓𝑥_𝑛,𝑥_𝑛+𝑥−𝑧_𝑛−𝑆𝑥_𝑛,𝑥_𝑛+−𝑧1−𝛽_𝑛𝛽_𝑛𝑥_𝑛−𝑧_𝑛,𝑥_𝑛+1−𝑧1−𝛼_𝑛𝛽_𝑛^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑧. (100) Since 𝑆 is nonexpansive mapping, 𝑓 is 𝜌-contraction mapping and 𝑉_𝑖 is 𝑘_𝑖-strict pseudocontraction mapping. Then, (𝐼−𝑆) and (𝐼−𝑉_𝑖) are monotones, and 𝑓 is strongly monotone with coefficient (1−𝜌). We can deduce that 𝑥_𝑛−𝑆𝑥_𝑛,𝑥_𝑛=)𝑥−𝑧(𝐼−𝑆_𝑛−(𝐼−𝑆)𝑧,𝑥_𝑛+)−𝑧(𝐼−𝑆𝑧,𝑥_𝑛≥)−𝑧(𝐼−𝑆𝑧,𝑥_𝑛,𝑥−𝑧𝐼−𝑓_𝑛,𝑥_𝑛=𝑥−𝑧𝐼−𝑓_𝑛−𝐼−𝑓𝑧,𝑥_𝑛+−𝑧𝐼−𝑓𝑧,𝑥_𝑛−𝑧≥1−𝜌𝑥_𝑛−𝑧²+𝐼−𝑓𝑧,𝑥_𝑛−𝑧,𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛=−𝑧𝐼−𝑉_𝑖𝑦_𝑛−𝐼−𝑉_𝑖𝑧,𝑥_𝑛−𝑦_𝑛+𝐼−𝑉_𝑖𝑦_𝑛−𝐼−𝑉_𝑖𝑧,𝑦_𝑛≥−𝑧𝐼−𝑉_𝑖𝑦_𝑛−𝐼−𝑉_𝑖𝑧,𝑥_𝑛−𝑦_𝑛=𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑦_𝑛=𝐼−𝑉_𝑖𝑦_𝑛,𝛽_𝑛𝑥_𝑛−𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧_𝑛. (101) From (20), we get 𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1=𝑤−𝑧_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛+𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛≥𝑤−𝑧_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛. (102) Then, from (100)–(102), we have 𝑣_𝑛,𝑥_𝑛≥1−𝑧1−𝛼_𝑛𝛽_𝑛×𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛+𝛼_𝑛1−𝛼_𝑛𝛽_𝑛𝐼−𝑓𝑧,𝑥_𝑛+)−𝑧(𝐼−𝑆𝑧,𝑥_𝑛+−𝑧1−𝛽_𝑛𝛽_𝑛𝑥_𝑛−𝑧_𝑛,𝑥_𝑛+−𝑧1−𝛽_𝑛1−𝛼_𝑛𝛽_𝑛^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖×𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑧_𝑛+11−𝛼_𝑛^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖×𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑆𝑥_𝑛+1−𝜌𝛼_𝑛1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧². (103) Then, we obtain 𝑥_𝑛−𝑧²≤11−𝜌𝛼_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛𝑤_𝑛−1−𝑤_𝑛−11−𝜌𝐼−𝑓𝑧,𝑥_𝑛+−𝑧1−𝛼_𝑛𝛽_𝑛1−𝜌𝛼_𝑛𝑣_𝑛,𝑥_𝑛−)−𝑧(𝐼−𝑆𝑧,𝑥_𝑛−−𝑧1−𝛽_𝑛1−𝛼_𝑛1−𝜌𝛼_𝑛𝑥_𝑛−𝑧_𝑛,𝑥_𝑛−−𝑧1−𝛽_𝑛1−𝜌𝛼_𝑛^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑧_𝑛−𝛽_𝑛1−𝜌𝛼_𝑛^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖×𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑆𝑥_𝑛≤𝑤_𝑛−1−𝑤_𝑛1−𝜌𝛼_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛−11−𝜌𝐼−𝑓𝑧,𝑥_𝑛+−𝑧1−𝛼_𝑛𝛽_𝑛1−𝜌𝛼_𝑛×𝑣_𝑛,𝑥_𝑛−𝑧−(𝐼−𝑆)𝑧,𝑥_𝑛+1−𝑧1−𝜌1−𝛽_𝑛𝛽_𝑛𝛽_𝑛𝛼_𝑛𝑥_𝑛−𝑧_𝑛𝑥_𝑛+1−𝑧1−𝜌1−𝛽_𝑛𝛽_𝑛𝛽_𝑛𝛼_𝑛×^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝐼−𝑉_𝑖𝑦_𝑛𝑥_𝑛−𝑧_𝑛−𝛽_𝑛1−𝜌𝛼_𝑛×^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝐼−𝑉_𝑖𝑦_𝑛,𝑥_𝑛−𝑆𝑥_𝑛. (104) By condition (e) of Theorem 15, there exists a constant 𝑁>0 such that ((1−𝛽_𝑛)/𝛽_𝑛)≤𝑁. Since lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0, 𝑣_𝑛→0,(𝐼−𝑉_𝑖)𝑦_𝑛→0, and ‖𝑤_𝑛−1−𝑤_𝑛‖/𝛼_𝑛→0 as 𝑛→∞, then every weak cluster point of {𝑥_𝑛} is also a strong cluster point. Since {𝑥_𝑛} is bounded, by Lemma 13 there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} converging to a point 𝑥^∗∈𝐹(𝑇); in similar argument as that Theorem 14 we can show that 𝑥^∗∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇).
From (100)–(102), it follows that, for any 𝑧∈𝑆^∗∩MEP(𝐹)∩𝐹(𝑇), 𝐼−𝑓𝑥_𝑛𝑘,𝑥_𝑛𝑘=−𝑧1−𝛼_𝑛𝑘𝛽_𝑛𝑘𝛼_𝑛𝑘𝑣_𝑛𝑘,𝑥_𝑛𝑘−1−𝑧𝛼_𝑛𝑘×𝑤_𝑛𝑘−𝑃_𝐶𝑤_𝑛𝑘,𝑃_𝐶𝑤_𝑛𝑘−1−−𝑧1−𝛼_𝑛𝑘𝛽_𝑛𝑘𝛼_𝑛𝑘𝑥_𝑛𝑘−𝑆𝑥_𝑛𝑘,𝑥_𝑛𝑘−−𝑧1−𝛼_𝑛𝑘1−𝛽_𝑛𝑘𝛼_𝑛𝑘𝑥_𝑛𝑘−𝑧_𝑛𝑘,𝑥_𝑛𝑘−1−𝑧𝛼_𝑛𝑘^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑦_𝑛𝑘−𝑉_𝑖𝑦_𝑛𝑘,𝑥_𝑛𝑘≤−𝑧1−𝛼_𝑛𝑘𝛽_𝑛𝑘𝛼_𝑛𝑘𝑣_𝑛𝑘,𝑥_𝑛𝑘+1−𝑧𝛼_𝑛𝑘𝑤_𝑛𝑘−𝑃_𝐶𝑤_𝑛𝑘×𝑤_𝑛𝑘−1−𝑤_𝑛𝑘−1−𝛼_𝑛𝑘𝛽_𝑛𝑘𝛼_𝑛𝑘×𝑥_𝑛𝑘−𝑆𝑥_𝑛𝑘,𝑥_𝑛𝑘+−𝑧1−𝛽_𝑛𝑘𝛽_𝑛𝑘𝛽_𝑛𝑘𝛼_𝑛𝑘𝑥_𝑛𝑘−𝑧_𝑛𝑘×𝑥_𝑛𝑘−𝑧+1−𝛽_𝑛𝑘𝛽_𝑛𝑘𝛽_𝑛𝑘𝛼_𝑛𝑘×^𝑛𝑘_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝐼−𝑉_𝑖𝑦_𝑛𝑘𝑥_𝑛𝑘−𝑧_𝑛𝑘−𝛽_𝑛𝑘𝛼_𝑛𝑘^𝑛𝑘_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝐼−𝑉_𝑖𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑆𝑥_𝑛𝑘. (105) Since lim_𝑛→∞‖𝑥_𝑛−𝑧_𝑛‖=0,  𝑣_𝑛→0,   (𝐼−𝑉_𝑖)𝑦_𝑛→0, and ‖𝑤_𝑛−1−𝑤_𝑛‖/𝛼_𝑛→0; letting 𝑘→∞ in (105), we obtain 𝐼−𝑓𝑥^∗,𝑥^∗−𝑧≤−𝜏𝑥^∗−𝑆𝑥^∗,𝑥^∗−𝑧; (106) that is, 1𝜏𝐼−𝑓𝑥^∗+(𝐼−𝑆)𝑥^∗,𝑧−𝑥^∗≥0. (107) In the following, we show that (90) has unique solution. Assume that 𝑥^′ is another solution. Then, we have 𝐼−𝑓𝑥^′,𝑥^′−𝑥^∗≤−𝜏𝑥^′−𝑆𝑥^′,𝑥^′−𝑥^∗,𝐼−𝑓𝑥^∗,𝑥^∗−𝑥^′≤−𝜏𝑥^∗−𝑆𝑥^∗,𝑥^∗−𝑥^′. (108) Adding (108), we get 1−𝜌𝑥^′−𝑥^∗2≤𝐼−𝑓𝑥^′−𝐼−𝑓𝑥^∗,𝑥^′−𝑥^∗≤−𝜏⟨(𝐼−𝑆)𝑥^′−(𝐼−𝑆)𝑥^∗,𝑥^′−𝑥^∗⟩≤0. (109) Then, 𝑥^′=𝑥^∗. Since (90) has unique solution, it follows that 𝑤_𝑤(𝑥_𝑛)={𝑥^∗}. Since every weak cluster point of {𝑥_𝑛} is also a strong cluster point, we conclude that {𝑥_𝑛}→𝑥^∗. This completes the proof.

4. Applications

In this section, we obtain the following results by using a special case of the proposed method. The first result can be viewed as extension and improvement of the method of Gu et al. [11] for finding the approximate element of the common set of solutions of a generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.

Corollary 16. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐷∶𝐶→𝐻 be 𝜂-inverse strongly monotone mappings, respectively. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the assumptions (𝑖)--(𝑖𝑣) of Lemma 4, 𝑆∶𝐶→𝐻 a nonexpansive mapping, and {𝑇_𝑖}^∞_𝑖=1∶𝐶→𝐶 a countable family of 𝑘_𝑖-strict pseudocontraction mappings such that 𝐹(𝑇)∩MEP(𝐹)≠∅, where 𝐹(𝑇)=∩^∞_𝑖=1𝐹(𝑇_𝑖). Let 𝑓 be a 𝜌-contraction mapping. For a given 𝑥₀∈𝐶 arbitrarily, let the iterative sequences {𝑢_𝑛},{𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be generated by 𝐹𝑢_𝑛,𝑦+⟨𝐷𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶;_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑢_𝑛;𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+^𝑛_𝑖=1𝛼_𝑖−1−𝛼_𝑖𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (110) where 𝛼₀=1, {𝛼_𝑛} is a strictly decreasing sequence in (0,1), and {𝛽_𝑛} is a sequence in (0,1) satisfying the following conditions: (a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞), (c)∑^∞_𝑛=1(𝛼_𝑛−1−𝛼_𝑛)<∞ and ∑^∞_𝑛=1|𝛽_𝑛−1−𝛽_𝑛|<∞,(d)lim_𝑛→∞((1/𝜇)(|𝑟_𝑛−𝑟_𝑛−1|+|𝛼_𝑛−1−𝛼_𝑛|+|𝛽_𝑛−1−𝛽_𝑛|)/(𝛼_𝑛𝛽_𝑛))=0, (e)there exists a constant 𝐾>0 such that (1/𝛼_𝑛)|(1/𝛽_𝑛)−(1/𝛽_𝑛−1)|≤𝐾,(f)liminf_𝑛→∞𝑟_𝑛>0 and ∑^∞_𝑛=1|𝑟_𝑛−1−𝑟_𝑛|<∞. Then, sequence {𝑥_𝑛} generated by Algorithm (110) converges strongly to 𝑥^∗∈MEP(𝐹)∩𝐹(𝑇), which is the unique solution of the variational inequality 1𝜏𝐼−𝑓𝑥^∗+(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗(𝐹)(𝑇).≥0,∀𝑥∈MEP∩𝐹 (111)

Proof. Putting 𝐵₁=𝐵₂=0 and 𝑘_𝑖=0,forall𝑖≥1 in Theorem 15, then conclusion of Corollary 16 is obtained.

The following result can be viewed as extension and improvement of the method of Yao et al. [31] for finding the approximate element of the common set of solutions of a generalized equilibrium problem and a hierarchical fixed point problem in a real Hilbert space.

Corollary 17. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐷∶𝐶→𝐻 be 𝜂-inverse strongly monotone mappings, respectively. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the assumptions (𝑖)--(𝑖𝑣) of Lemma 4, 𝑆∶𝐶→𝐻 a nonexpansive mapping, and 𝑇∶𝐶→𝐶 a countable family of 𝑘-strict pseudocontraction mappings such that 𝐹(𝑇)∩MEP(𝐹)≠∅. Let 𝑓 be a 𝜌-contraction mapping. For a given 𝑥₀∈𝐶 arbitrarily, let the iterative sequences {𝑢_𝑛},{𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} be generated by 𝐹𝑢_𝑛,𝑦+⟨𝐷𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦⟩≥0,∀𝑦∈𝐶;_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑢_𝑛;𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥0, (112) where 𝛼₀=1, {𝛼_𝑛} is a strictly decreasing sequence in (0,1) and {𝛽_𝑛} is a sequence in (0,1) satisfying the following conditions: (a)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(b)lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞), (c)∑^∞_𝑛=1(𝛼_𝑛−1−𝛼_𝑛)<∞ and ∑^∞_𝑛=1|𝛽_𝑛−1−𝛽_𝑛|<∞,(d)lim_𝑛→∞((1/𝜇)|𝑟_𝑛−𝑟_𝑛−1|+|𝛼_𝑛−1−𝛼_𝑛|+|𝛽_𝑛−1−𝛽_𝑛|)/𝛼_𝑛𝛽_𝑛=0, (e)there exists a constant 𝐾>0 such that (1/𝛼_𝑛)|(1/𝛽_𝑛)−(1/𝛽_𝑛−1)|≤𝐾,(f)liminf_𝑛→∞𝑟_𝑛>0 and ∑^∞_𝑛=1|𝑟_𝑛−1−𝑟_𝑛|<∞. Then, sequence {𝑥_𝑛} generated by Algorithm (112) converges strongly to 𝑥^∗∈MEP(𝐹)∩𝐹(𝑇), which is the unique solution of the variational inequality 1𝜏𝐼−𝑓𝑥^∗+(𝐼−𝑆)𝑥^∗,𝑥−𝑥^∗(𝐹)(𝑇).≥0,∀𝑥∈MEP∩𝐹 (113)

Proof. Putting 𝐵₁=𝐵₂=0,  𝑘_𝑖=0, and 𝑇_𝑖=𝑇forall𝑖≥1 in Theorem 15, then conclusion of Corollary 17 is obtained.

5. Conclusions

In this paper, we suggest and analyze an iterative method for finding the approximate element of the common set of solutions of (1), (12), and (15) for a strictly pseudocontraction mapping in real Hilbert space, which can be viewed as a refinement and improvement of some existing methods for solving a system of variational inequality problem, a mixed equilibrium problem, and a hierarchical fixed point problem. It is easy to verify that Algorithm 11 includes some existing methods (e.g., [6, 11, 17, 31]) as special cases. Therefore, the new algorithm is expected to be widely applicable.