A New Iterative Algorithm for the Set of Fixed-Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem

Abstract

We introduce a new iterative scheme and a new mapping generated by infinite family of nonexpansive mappings and infinite real number. By using both of these ideas, we obtain strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems of infinite family of nonexpansive mappings. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

1. Introduction

Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐴∶𝐶→𝐻 be a nonlinear mapping and let 𝐹∶𝐶×𝐶→ℝ be a bifunction. A mapping 𝑇 of 𝐻 into itself is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖forall𝑥,𝑦∈𝐻. We denote by 𝐹(𝑇) the set of fixed points of 𝑇 (i.e., 𝐹(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}). Goebel and Kirk [1] showed that 𝐹(𝑇) is always closed convex, and also nonempty provided 𝑇 has a bounded trajectory.

A bounded linear operator 𝐴 on 𝐻 is called strongly positive with coefficient 𝛾 if there is a constant 𝛾>0 with the property⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖². (1.1)

The equilibrium problem for 𝐹 is to find 𝑥∈𝐶, such that𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (1.2) The set of solutions of (1.2) is denoted by EP(𝐹). Many problems in physics, optimization, and economics are seeking some elements of EP(𝐹), see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme for finding the best approximation to the initial data when EP(𝐹) is nonempty and proved a strong convergence theorem.

The variational inequality problem is to find a point 𝑢∈𝐶, such that⟨𝑣−𝑢,𝐴𝑢⟩≥0,∀𝑣∈𝐶. (1.3) The set of solutions of the variational inequality is denoted by VI(𝐶,𝐴). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of (1.2) and (1.3).

A mapping 𝐴 of 𝐶 into 𝐻 is called inverse-strongly monotone, see [5], if there exists a positive real number 𝛼, such that𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛼𝐴𝑥−𝐴𝑦² (1.4) for all 𝑥,𝑦∈𝐶.

The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed-point sets of a family of nonexpansive mapping (see [6, 7]).

The problem of finding a common element of EP(𝐹) and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed-point problem of nonexpansive mappings, see [8–10].

In 2007, S. Takahashi and W. Takahashi [10] introduced a general iterative method for finding a common element of EP(𝐹,𝐴) and 𝐹(𝑇). They defined {𝑥_𝑛} in the following way:𝑢,𝑥₁𝐹𝑧∈𝐶,arbitrarily;_𝑛+1,𝑦𝜆_𝑛⟨𝑦−𝑧_𝑛,𝑧_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑆𝑧_𝑛,∀𝑛∈ℕ, (1.5) where {𝛽_𝑛}⊂[0,1], and proved strong convergence of the scheme (1.5) to 𝑧∈𝐹(𝑇)⋂EP(𝐹), where 𝑧=𝑃_{⋂𝐹(𝑆)EP(𝐹)}𝑓(𝑧) in the framework of a Hilbert space, under some suitable conditions on {𝛽_𝑛}, {𝜆_𝑛} and bifunction 𝐹.

In this paper, by motive of (1.5), we prove strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems by using a new mapping generated by infinite family of nonexpansive mapping and infinite real number. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

2. Preliminaries

In this section, we collect and give some useful lemmas that will be used for our main result in the next section.

Let 𝐶 be closed convex subset of a real Hilbert space 𝐻, and let 𝑃_𝐶 be the metric projection of 𝐻 onto 𝐶, that is, for 𝑥∈𝐻, 𝑃_𝐶𝑥 satisfies the property‖𝑥−𝑃_𝐶𝑥‖=min_𝑦∈𝐶‖𝑥−𝑦‖. (2.1) The following characterizes the projection 𝑃_𝐶.

Lemma 2.1 (see [11]). Given 𝑥∈𝐻 and 𝑦∈𝐶, then 𝑃_𝐶𝑥=𝑦 if and only if there holds the inequality 𝑥−𝑦,𝑦−𝑧≥0,∀𝑧∈𝐶. (2.2)

Lemma 2.2 (see [12]). Let 𝐸 be a uniformly convex Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping, then 𝐼−𝑆 is demiclosed at zero.

Lemma 2.3 (see [13]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1=1−𝛼_𝑛𝑠_𝑛+𝛿_𝑛,∀𝑛≥0, (2.3) where {𝛼_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence, such that (1)∑^∞_𝑛=1𝛼_𝑛=∞, (2)limsup_𝑛→∞𝛿_𝑛/𝛼_𝑛≤0or∑^∞_𝑛=1|𝛿_𝑛|<∞ then lim_𝑛→∞𝑠_𝑛=0.

For solving the equilibrium problem for a bifunction 𝐹∶𝐶×𝐶→ℝ, let us assume that 𝐹 satisfies the following conditions: (A1)𝐹(𝑥,𝑥)=0,forall𝑥∈𝐶,(A2)𝐹ismonotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶,(A3)for all 𝑥,𝑦,𝑧∈𝐶,lim_𝑡→0+𝐹𝑡𝑧+(1−𝑡)𝑥,𝑦≤𝐹𝑥,𝑦, (2.4)(A4)for all 𝑥∈𝐶,𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

The following lemma appears implicitly in [2].

Lemma 2.4 (see [2]). Let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐻, then there exists 𝑧∈𝐶, such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥, (2.5) for all 𝑥∈𝐶.

Lemma 2.5 (see [3]). Assume that 𝐹∶𝐶×𝐶→ℝ satisfies (A1)–(A4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶, (2.6) for all 𝑧∈𝐻, then the following hold: (1)𝑇_𝑟 is single valued, (2)𝑇_𝑟 is firmly nonexpansive, that is, 𝑇_𝑟(𝑥)−𝑇_𝑟(𝑦)²≤𝑇_𝑟(𝑥)−𝑇_𝑟𝑦,𝑥−𝑦∀𝑥,𝑦∈𝐻, (2.7)(3)𝐹(𝑇_𝑟)=EP(𝐹), (4)EP(𝐹) is closed and convex.

Lemma 2.6 (see [14]). Let 𝐻 be a Hibert space, let 𝐶 be a nonempty closed convex subset of 𝐻, and let 𝐴 be a mapping of 𝐶 into 𝐻. Let 𝑢∈𝐶, then for 𝜆>0, 𝑢=𝑃_𝐶(𝐼−𝜆𝐴)𝑢⟺𝑢∈VI(𝐶,𝐴), (2.8) where 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶.

Definition 2.7. Let 𝐶 be a nonempty convex subset of a real Hilbert space. Let 𝑇_𝑖,𝑖=1,2,… be mappings of 𝐶 into itself. For each 𝑗=1,2,…, let 𝛼_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼 where 𝐼∈[0,1] and 𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1. For every 𝑛∈ℕ, we define the mapping 𝑆_𝑛∶𝐶→𝐶 as follows: 𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝛼^𝑛₁𝑇_𝑛𝑈_𝑛,𝑛+1+𝛼^𝑛₂𝑈_𝑛,𝑛+1+𝛼^𝑛₃𝑈𝐼,_{𝑛,𝑛−1}=𝛼₁^𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+𝛼₂^𝑛−1𝑈_𝑛,𝑛+𝛼₃^𝑛−1⋮𝑈𝐼,_𝑛,𝑘+1=𝛼₁^𝑘+1𝑇_𝑘+1𝑈_𝑛,𝑘+2+𝛼₂^𝑘+1𝑈_𝑛,𝑘+2+𝛼₃^𝑘+1𝑈𝐼,_𝑛,𝑘=𝛼^𝑘₁𝑇_𝑘𝑈_𝑛,𝑘+1+𝛼^𝑘₂𝑈_𝑛,𝑘+1+𝛼^𝑘₃⋮𝑈𝐼,_𝑛,2=𝛼²₁𝑇₂𝑈_𝑛,3+𝛼²₂𝑈_𝑛,3+𝛼²₃𝑆𝐼,_𝑛=𝑈_𝑛,1=𝛼¹₁𝑇₁𝑈_𝑛,2+𝛼¹₂𝑈_𝑛,2+𝛼¹₃𝐼. (2.9) This mapping is called S-mapping generated by 𝑇_𝑛,…,𝑇₁ and 𝛼_𝑛,𝛼_𝑛−1,…,𝛼₁.

Lemma 2.8. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space. Let {𝑇_𝑖}^∞_𝑖=1 be nonexpansive mappings of 𝐶 into itself with ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅, and let 𝛼_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1],𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1,𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃∈(0,1),forall𝑗=1,2,…. For every 𝑛∈ℕ, let 𝑆_𝑛 be 𝑆-mapping generated by 𝑇_𝑛,…,𝑇₁ and 𝛼_𝑛,𝛼_𝑛−1,…,𝛼₁, then for every 𝑥∈𝐶 and 𝑘∈ℕ,lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

Proof. Let 𝑥∈𝐶 and 𝑦∈⋂^∞_𝑖=1𝐹(𝑇_𝑖). Fix 𝑘∈ℕ, then for every 𝑛∈ℕ with 𝑛≥𝑘, we have 𝑈_𝑛+1,𝑘𝑥−𝑈_𝑛,𝑘𝑥²=𝛼^𝑘₁𝑇_𝑘𝑈_{𝑛+1,𝑘+1}𝑥+𝛼^𝑘₂𝑈_{𝑛+1,𝑘+1}𝑥+𝛼^𝑘₃𝑥−𝛼^𝑘₁𝑇_𝑘𝑈_𝑛,𝑘+1𝑥−𝛼^𝑘₂𝑈_𝑛,𝑘+1𝑥−𝛼^𝑘₃𝑥²=𝛼^𝑘₁𝑇_𝑘𝑈_{𝑛+1,𝑘+1}𝑥−𝑇_𝑘𝑈_𝑛,𝑘+1𝑥+𝛼^𝑘₂𝑈_{𝑛+1,𝑘+1}𝑥−𝑈_𝑛,𝑘+1𝑥²≤𝛼^𝑘₁𝑇_𝑘𝑈_{𝑛+1,𝑘+1}𝑥−𝑇_𝑘𝑈_𝑛,𝑘+1𝑥²+𝛼^𝑘₂𝑈_{𝑛+1,𝑘+1}𝑥−𝑈_𝑛,𝑘+1𝑥²≤𝛼^𝑘₁𝑈_{𝑛+1,𝑘+1}𝑥−𝑈_𝑛,𝑘+1𝑥²+𝛼^𝑘₂𝑈_{𝑛+1,𝑘+1}𝑥−𝑈_𝑛,𝑘+1𝑥²≤1−𝛼^𝑘₃𝑈_{𝑛+1,𝑘+1}𝑥−𝑈_𝑛,𝑘+1𝑥²⋮≤^𝑛_𝑗=𝑘1−𝛼^𝑗₃𝑈_{𝑛+1,𝑛+1}𝑥−𝑈_𝑛,𝑛+1𝑥²=^𝑛_𝑗=𝑘1−𝛼^𝑗₃𝛼₁^𝑛+1𝑇_𝑛+1𝑈_{𝑛+1,𝑛+2}𝑥+𝛼₂^𝑛+1𝑈_{𝑛+1,𝑛+2}𝑥+𝛼₃^𝑛+1𝑥−𝑥²=^𝑛_𝑗=𝑘1−𝛼^𝑗₃𝛼₁^𝑛+1𝑇_𝑛+1𝑥+(1−𝛼₁^𝑛+1)𝑥−𝑥²=^𝑛_𝑗=𝑘1−𝛼^𝑗₃𝛼₁^𝑛+1(𝑇_𝑛+1𝑥−𝑥)²≤^𝑛_𝑗=𝑘1−𝛼^𝑗₃‖𝑇_𝑛+1𝑥−𝑦‖+‖𝑦−𝑥‖²≤^𝑛_𝑗=𝑘1−𝛼^𝑗₃‖𝑥−𝑦‖+‖𝑦−𝑥‖²≤^𝑛_𝑗=𝑘1−𝛼^𝑗₃2‖𝑥−𝑦‖²≤𝑏^{𝑛−(𝑘−1)}2‖𝑥−𝑦‖². (2.10) It follows that ‖𝑈_𝑛+1,𝑘𝑥−𝑈_𝑛,𝑘𝑥‖≤𝑏^{(𝑛−(𝑘−1))/2}=𝑏2‖𝑥−𝑦‖_𝑛/2𝑏^(𝑘−1)/2=𝑎2‖𝑥−𝑦‖_𝑛𝑎^𝑘−1𝑀, (2.11) where 𝑎=𝑏^1/2∈(0,1) and 𝑀=2‖𝑥−𝑦‖.
For any 𝑘,𝑛,𝑝∈ℕ,𝑝>0,𝑛≥𝑘, we have ‖𝑈_{𝑛+𝑝,𝑘}𝑥−𝑈_𝑛,𝑘𝑥‖≤‖𝑈_{𝑛+𝑝,𝑘}𝑥−𝑈_{𝑛+𝑝−1,𝑘}𝑥‖+‖𝑈_{𝑛+𝑝−1,𝑘}𝑥−𝑈_{𝑛+𝑝−2,𝑘}𝑥‖+⋯+‖𝑈_𝑛+1,𝑘𝑥−𝑈_𝑛,𝑘=𝑥‖_{𝑛+𝑝−1𝑗=𝑛}‖𝑈_𝑗+1,𝑘𝑥−𝑈_𝑗,𝑘≤𝑥‖_{𝑛+𝑝−1𝑗=𝑛}𝑎^𝑗𝑎^𝑘−1𝑀≤𝑎^𝑛(1−𝑎)𝑎^𝑘−1𝑀. (2.12) Since 𝑎∈(0,1), we have lim_𝑛→∞𝑎^𝑛=0. From (2.12), we have that {𝑈_𝑛,𝑘𝑥} is a Cauchy sequence. Hence, lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists.

For every 𝑘∈ℕ and 𝑥∈𝐶, we define mapping 𝑈_∞,𝑘 and 𝑆∶𝐶→𝐶 as follows:lim_𝑛→∞𝑈_𝑛,𝑘𝑥=𝑈_∞,𝑘𝑥,lim_𝑛→∞𝑆_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥=𝑆𝑥. (2.13) Such a mapping 𝑆 is called 𝑆-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,… and 𝛼_𝑛,𝛼_𝑛−1,….

Remark 2.9. For each 𝑛∈ℕ,𝑆_𝑛 is nonexpansive and lim_𝑛→∞sup_𝑥∈𝐷‖𝑆_𝑛𝑥−𝑆𝑥‖=0 for every bounded subset 𝐷 of 𝐶. To show this, let 𝑥,𝑦∈𝐶 and 𝐷 be a bounded subset of 𝐶, then we have 𝑆_𝑛𝑥−𝑆_𝑛𝑦²=𝛼¹₁(𝑇₁𝑈_𝑛,2𝑥−𝑇₁𝑈_𝑛,2𝑦)+𝛼¹₂(𝑈_𝑛,2𝑥−𝑈_𝑛,2𝑦)+𝛼¹₃𝑥−𝑦²≤𝛼¹₁𝑇₁𝑈_𝑛,2𝑥−𝑇₁𝑈_𝑛,2𝑦²+𝛼¹₂𝑈_𝑛,2𝑥−𝑈_𝑛,2𝑦²+𝛼¹₃𝑥−𝑦²≤𝛼¹₁𝑈_𝑛,2𝑥−𝑈_𝑛,2𝑦²+𝛼¹₂𝑈_𝑛,2𝑥−𝑈_𝑛,2𝑦²+𝛼¹₃𝑥−𝑦²=1−𝛼¹₃𝑈_𝑛,2𝑥−𝑈_𝑛,2𝑦²+𝛼¹₃𝑥−𝑦²≤1−𝛼¹₃1−𝛼²₃𝑈_𝑛,3𝑥−𝑈_𝑛,3𝑦²+𝛼²₃𝑥−𝑦²+𝛼¹₃𝑥−𝑦²=1−𝛼¹₃1−𝛼²₃𝑈_𝑛,3𝑥−𝑈_𝑛,3𝑦²+𝛼²₃1−𝛼¹₃𝑥−𝑦²+𝛼¹₃𝑥−𝑦²=²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,3𝑥−𝑈_𝑛,3𝑦²+1−²_𝑗=11−𝛼^𝑗₃𝑥−𝑦²⋮≤²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑛+1𝑥−𝑈_𝑛,𝑛+1𝑦²+1−²_𝑗=11−𝛼^𝑗₃𝑥−𝑦²=𝑥−𝑦². (2.14) Then, we have that 𝑆∶𝐶→𝐶 is also nonexpansive, Indeed, observe that for each 𝑥,𝑦∈𝐶, ‖𝑆𝑥−𝑆𝑦‖=lim_𝑛→∞‖𝑆_𝑛𝑥−𝑆_𝑛𝑦‖≤‖𝑥−𝑦‖. (2.15) By (2.11), we have ‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖=‖𝑈_𝑛+1,1𝑥−𝑈_𝑛,1𝑥‖≤𝑎^𝑛𝑀. (2.16) This implies that for 𝑚>𝑛 and 𝑥∈𝐷, ‖𝑆_𝑚𝑥−𝑆_𝑛𝑥‖≤^𝑚−1_𝑗=𝑛‖𝑆_𝑗+1𝑥−𝑆_𝑗≤𝑥‖_{𝑚−1𝑗=𝑛}𝑎^𝑗𝑀≤𝑎^𝑛1−𝑎𝑀. (2.17) By letting 𝑚→∞, for any 𝑥∈𝐷, we have ‖𝑆𝑥−𝑆_𝑛𝑎𝑥‖≤^𝑛1−𝑎𝑀. (2.18) It follows that lim_𝑛→∞sup_𝑥∈𝐷‖𝑆_𝑛𝑥−𝑆𝑥‖=0. (2.19)

Lemma 2.10. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space. Let {𝑇_𝑖}^∞_𝑖=1 be nonexpansive mappings of 𝐶 into itself with ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅, and let 𝛼_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1],𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1,𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃∈(0,1)forall𝑗=1,2,…. For every 𝑛∈ℕ, let 𝑆_𝑛 and 𝑆 be 𝑆-mappings generated by 𝑇_𝑛,…,𝑇₁ and 𝛼_𝑛,𝛼_𝑛−1,…,𝛼₁ and 𝑇_𝑛,𝑇_𝑛−1,…, and 𝛼_𝑛,𝛼_𝑛−1,…, respectively, then 𝐹(𝑆)=⋂^∞_𝑖=1𝐹(𝑇_𝑖).

Proof. It is easy to see that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)⊆𝐹(𝑆). For every 𝑛,𝑘∈ℕ,with𝑛≥𝑘, let 𝑥₀∈𝐹(𝑆) and 𝑥^∗∈⋂^∞_𝑖=1𝐹(𝑇_𝑖), then we have 𝑆_𝑛𝑥₀−𝑥^∗2=𝛼¹₁(𝑇₁𝑈_𝑛,2𝑥₀−𝑥^∗)+𝛼¹₂𝑈_𝑛,2𝑥₀−𝑥^∗+𝛼¹₃(𝑥₀−𝑥^∗)²≤𝛼¹₁𝑇₁𝑈_𝑛,2𝑥₀−𝑥^∗2+𝛼¹₂𝑈_𝑛,2𝑥₀−𝑥^∗2+𝛼¹₃𝑥₀−𝑥^∗2−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²≤𝛼¹₁𝑈_𝑛,2𝑥₀−𝑥^∗2+𝛼¹₂𝑈_𝑛,2𝑥₀−𝑥^∗2+𝛼¹₃𝑥₀−𝑥^∗2−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²=1−𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥^∗2+𝛼¹₃𝑥₀−𝑥^∗2−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²≤1−𝛼¹₃1−𝛼²₃𝑈_𝑛,3𝑥₀−𝑥^∗2+𝛼²₃𝑥₀−𝑥^∗2−𝛼²₁𝛼²₂𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃𝑈_𝑛,3𝑥₀−𝑥₀²+𝛼¹₃𝑥₀−𝑥^∗2−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²=1−𝛼¹₃1−𝛼²₃𝑈_𝑛,3𝑥₀−𝑥^∗2+𝛼²₃1−𝛼¹₃𝑥₀−𝑥^∗2+𝛼¹₃𝑥₀−𝑥^∗2−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²=²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,3𝑥₀−𝑥^∗2+1−²_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²≤²_𝑗=11−𝛼^𝑗₃1−𝛼³₃𝑈_𝑛,4𝑥₀−𝑥^∗2+𝛼³₃𝑥₀−𝑥^∗2−𝛼³₁𝛼³₂𝑇₃𝑈_𝑛,4𝑥₀−𝑈_𝑛,4𝑥₀²−𝛼³₂𝛼³₃𝑈_𝑛,4𝑥₀−𝑥₀²+1−²_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²=²_𝑗=11−𝛼^𝑗₃1−𝛼³₃𝑈_𝑛,4𝑥₀−𝑥^∗2+𝛼³₃²_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼³₁𝛼³₂²_𝑗=11−𝛼^𝑗₃𝑇₃𝑈_𝑛,4𝑥₀−𝑈_𝑛,4𝑥₀²−𝛼³₂𝛼³₃²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,4𝑥₀−𝑥₀²+1−²_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²=³_𝑗=11−𝛼^𝑗₃𝑈_𝑛,4𝑥₀−𝑥^∗2+1−³_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼³₁𝛼³₂²_𝑗=11−𝛼^𝑗₃𝑇₃𝑈_𝑛,4𝑥₀−𝑈_𝑛,4𝑥₀²−𝛼³₂𝛼³₃²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,4𝑥₀−𝑥₀²−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²⋮≤^𝑘+1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+2𝑥₀−𝑥^∗2+1−^𝑘+1_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼₁^𝑘+1𝛼₂^𝑘𝑘+1_𝑗=11−𝛼^𝑗₃𝑇_𝑘+1𝑈_𝑛,𝑘+2𝑥₀−𝑈_𝑛,𝑘+2𝑥₀²−𝛼₂^𝑘+1𝛼₃^𝑘𝑘+1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+2𝑥₀−𝑥₀²−𝛼^𝑘₁𝛼^𝑘₂^𝑘−1_𝑗=11−𝛼^𝑗₃𝑇_𝑘𝑈_𝑛,𝑘+1𝑥₀−𝑈_𝑛,𝑘+1𝑥₀²−𝛼^𝑘₂𝛼^𝑘₃^𝑘−1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+1𝑥₀−𝑥₀²−𝛼₁^𝑘−1𝛼₂^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑇_𝑘−1𝑈_𝑛,𝑘𝑥₀−𝑈_𝑛,𝑘𝑥₀²−𝛼₂^𝑘−1𝛼₃^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘𝑥₀−𝑥₀²⋮−𝛼³₁𝛼³₂²_𝑗=11−𝛼^𝑗₃𝑇₃𝑈_𝑛,4𝑥₀−𝑈_𝑛,4𝑥₀²−𝛼³₂𝛼³₃²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,4𝑥₀−𝑥₀²−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²⋮≤^𝑛_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑛+1𝑥₀−𝑥^∗2+1−^𝑛_𝑗=11−𝛼^𝑗₃𝑥₀−𝑥^∗2−𝛼^𝑛₁𝛼^𝑛₂^𝑛−1_𝑗=11−𝛼^𝑗₃𝑇_𝑛𝑈_𝑛,𝑛+1𝑥₀−𝑈_𝑛,𝑛+1𝑥₀²−𝛼^𝑛₂𝛼^𝑛₃^𝑛−1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑛+1𝑥₀−𝑥₀²⋮−𝛼₁^𝑘+1𝛼₂^𝑘𝑘+1_𝑗=11−𝛼^𝑗₃𝑇_𝑘+1𝑈_𝑛,𝑘+2𝑥₀−𝑈_𝑛,𝑘+2𝑥₀²−𝛼₂^𝑘+1𝛼₃^𝑘𝑘+1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+2𝑥₀−𝑥₀²−𝛼^𝑘₁𝛼^𝑘₂^𝑘−1_𝑗=11−𝛼^𝑗₃𝑇_𝑘𝑈_𝑛,𝑘+1𝑥₀−𝑈_𝑛,𝑘+1𝑥₀²−𝛼^𝑘₂𝛼^𝑘₃^𝑘−1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+1𝑥₀−𝑥₀²−𝛼₁^𝑘−1𝛼₂^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑇_𝑘−1𝑈_𝑛,𝑘𝑥₀−𝑈_𝑛,𝑘𝑥₀²−𝛼₂^𝑘−1𝛼₃^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘𝑥₀−𝑥₀²⋮−𝛼³₁𝛼³₂²_𝑗=11−𝛼^𝑗₃𝑇₃𝑈_𝑛,4𝑥₀−𝑈_𝑛,4𝑥₀²−𝛼³₂𝛼³₃²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,4𝑥₀−𝑥₀²−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀²=𝑥₀−𝑥^∗2−𝛼^𝑛₁𝛼^𝑛₂^𝑛−1_𝑗=11−𝛼^𝑗₃𝑇_𝑛𝑈_𝑛,𝑛+1𝑥₀−𝑈_𝑛,𝑛+1𝑥₀²⋮−𝛼₁^𝑘+1𝛼₂^𝑘𝑘+1_𝑗=11−𝛼^𝑗₃𝑇_𝑘+1𝑈_𝑛,𝑘+2𝑥₀−𝑈_𝑛,𝑘+2𝑥₀²−𝛼₂^𝑘+1𝛼₃^𝑘𝑘+1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+2𝑥₀−𝑥₀²−𝛼^𝑘₁𝛼^𝑘₂^𝑘−1_𝑗=11−𝛼^𝑗₃𝑇_𝑘𝑈_𝑛,𝑘+1𝑥₀−𝑈_𝑛,𝑘+1𝑥₀²−𝛼^𝑘₂𝛼^𝑘₃^𝑘−1_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘+1𝑥₀−𝑥₀²−𝛼₁^𝑘−1𝛼₂^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑇_𝑘−1𝑈_𝑛,𝑘𝑥₀−𝑈_𝑛,𝑘𝑥₀²−𝛼₂^𝑘−1𝛼₃^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘𝑥₀−𝑥₀²⋮−𝛼³₁𝛼³₂²_𝑗=11−𝛼^𝑗₃𝑇₃𝑈_𝑛,4𝑥₀−𝑈_𝑛,4𝑥₀²−𝛼³₂𝛼³₃²_𝑗=11−𝛼^𝑗₃𝑈_𝑛,4𝑥₀−𝑥₀²−𝛼²₁𝛼²₂1−𝛼¹₃𝑇₂𝑈_𝑛,3𝑥₀−𝑈_𝑛,3𝑥₀²−𝛼²₂𝛼²₃1−𝛼¹₃𝑈_𝑛,3𝑥₀−𝑥₀²−𝛼¹₁𝛼¹₂𝑇₁𝑈_𝑛,2𝑥₀−𝑈_𝑛,2𝑥₀²−𝛼¹₂𝛼¹₃𝑈_𝑛,2𝑥₀−𝑥₀². (2.20) For 𝑘∈ℕ and (2.20), we have 𝛼₂^𝑘−1𝛼₃^{𝑘−1𝑘−2}_𝑗=11−𝛼^𝑗₃𝑈_𝑛,𝑘𝑥₀−𝑥₀²≤𝑥₀−𝑥^∗2−𝑆_𝑛𝑥₀−𝑥^∗2, (2.21) as 𝑛→∞. This implies that 𝑈_∞,𝑘𝑥₀=𝑥₀,forall𝑘∈ℕ.
Again by (2.20), we have 𝛼^𝑘₁𝛼^𝑘₂^𝑘−1_𝑗=11−𝛼^𝑗₃𝑇_𝑘𝑈_𝑛,𝑘+1𝑥₀−𝑈_𝑛,𝑘+1𝑥₀²≤𝑥₀−𝑥^∗2−𝑆_𝑛𝑥₀−𝑥^∗2, (2.22) as 𝑛→∞. Hence, 𝛼^𝑘₁𝛼^𝑘₂^𝑘−1_𝑗=11−𝛼^𝑗₃𝑇_𝑘𝑈_∞,𝑘+1𝑥₀−𝑈_∞,𝑘+1𝑥₀²≤0. (2.23) From 𝑈_∞,𝑘𝑥₀=𝑥₀,forall𝑘∈ℕ, and (2.23), we obtain that 𝑇_𝑘𝑥₀=𝑥₀,forall𝑘∈ℕ. This implies that 𝑥₀∈⋂^∞_𝑖=1𝐹(𝑇_𝑖).

3. Main Result

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let 𝐹 be bifunctions from 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝐴∶𝐶→𝐻 be a 𝛼-inverse-strongly monotone mapping. Let {𝑇_𝑖}^∞_𝑖=1 be infinite family of nonexpansive mappings with 𝔉=⋂^∞_𝑖=1𝐹(𝑇_𝑖)⋂EP(𝐹)⋂VI(𝐶,𝐴), and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1],𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1,𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃∈(0,1)forall𝑗=1,2,…. For every 𝑛∈ℕ, let 𝑆_𝑛 and 𝑆 be 𝑆-mappings generated by 𝑇_𝑛,…,𝑇₁ and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₁ and 𝑇_𝑛,𝑇_𝑛−1,…, and 𝜌_𝑛,𝜌_𝑛−1,…, respectively. Let {𝑥_𝑛},{𝑢_𝑛} be sequences generated by 𝑥₁,𝑢∈𝐶 and 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛,∀𝑛≥1, (3.1) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}∈(0,1), such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,𝛽_𝑛∈[𝑐,𝑑]⊂(0,1)𝑟_𝑛∈[𝑎,𝑏]⊂(0,2𝛼),𝜆⊂(0,2𝛼). Assume that(i)lim_𝑛→∞𝛼_𝑛=0and∑^∞_𝑛=0𝛼_𝑛=∞, (ii)∑^∞_𝑛=1𝛼^𝑛₁<∞, (iii)∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|,∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|,∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|,∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞,then the sequence {𝑥_𝑛},{𝑢_𝑛} converge strongly to 𝑧=𝑃_𝔉𝑢.

Proof. First, we show that (𝐼−𝜆𝐴) is nonexpansive. Let 𝑥,𝑦∈𝐶. Since 𝐴 is 𝛼-inverse-strongly monotone and 𝜆<2𝛼, we have (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦²=𝑥−𝑦−𝜆(𝐴𝑥−𝐴𝑦)²=𝑥−𝑦²−2𝜆𝑥−𝑦,𝐴𝑥−𝐴𝑦+𝜆²𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²−2𝛼𝜆𝐴𝑥−𝐴𝑦²+𝜆²𝐴𝑥−𝐴𝑦²=𝑥−𝑦²+𝜆(𝜆−2𝛼)𝐴𝑥−𝐴𝑦²≤𝑥−𝑦². (3.2) Thus, (𝐼−𝜆𝐴) is nonexpansive. We will divide our proof into 5 steps.Step 1. We shall show that the sequence {𝑥_𝑛} is bounded. Since 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.3) By Lemma 2.5, we have 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and EP(𝐹)=𝐹(𝑇_𝑟𝑛).
Let 𝑧∈𝔉. By nonexpansiveness of 𝐼−𝜆𝐴 and 𝑇_𝑟𝑛, we have ‖𝑥_𝑛+1−𝑧‖=‖𝛼_𝑛𝑢+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧‖=‖𝛼_𝑛(𝑢−𝑧)+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑧+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧‖≤𝛼_𝑛‖𝑢−𝑧‖+𝛽_𝑛‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑧‖+𝛾_𝑛‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧‖≤𝛼_𝑛‖𝑢−𝑧‖+𝛽_𝑛‖𝑥_𝑛−𝑧‖+𝛾_𝑛‖𝑢_𝑛−𝑧‖=𝛼_𝑛‖𝑢−𝑧‖+𝛽_𝑛‖𝑥_𝑛−𝑧‖+𝛾_𝑛‖𝑇_𝑟𝑛𝑥_𝑛−𝑧‖≤𝛼_𝑛‖𝑢−𝑧‖+1−𝛼_𝑛‖𝑥_𝑛−𝑧‖≤max‖𝑢−𝑧‖,‖𝑥_𝑛−𝑧‖. (3.4) By induction, we can prove that {𝑥_𝑛} is bounded and so is {𝑢_𝑛}.Step 2. We will show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. By definition of 𝑥_𝑛, we have ‖𝑥_𝑛+1−𝑥_𝑛‖=𝛼_𝑛𝑢+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝛼_𝑛−1𝑢−𝛽_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−1−𝛾_𝑛−1𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1=𝛼_𝑛−𝛼_𝑛−1𝑢+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝛽_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛽_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝛾_𝑛−1𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛+𝛾_𝑛−1𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝛽_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−1−𝛾_𝑛−1𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1=𝛼_𝑛−𝛼_𝑛−1𝑢+𝛽_𝑛−𝛽_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛽_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛+𝛾_𝑛−1𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−1‖+𝛾_𝑛−𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1‖≤𝛼_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+𝛽_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+𝛾_𝑛−𝛾_𝑛−1‖𝑆_𝑛𝑃_C(𝐼−𝜆𝐴)𝑢_𝑛‖+𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1‖+‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1−𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1‖≤𝛼_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+𝛽_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+𝛾_𝑛−𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+𝛾_𝑛−1‖𝑢_𝑛−𝑢_𝑛−1‖+𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1−𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1‖. (3.5) Since 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, by definition of 𝑇_𝑟𝑛, we have 𝐹𝑇_𝑟𝑛𝑥_𝑛+1,𝑦𝑟_𝑛𝑦−𝑇_𝑟𝑛𝑥_𝑛,𝑇_𝑟𝑛𝑥_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.6) Similarly, 𝐹𝑇_𝑟𝑛+1𝑥_𝑛+1+1,𝑦𝑟_𝑛+1𝑦−𝑇_𝑟𝑛+1𝑥_𝑛+1,𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐶. (3.7) From (3.6) and (3.7), we obtain 𝐹𝑇_𝑟𝑛𝑥_𝑛,𝑇_𝑟𝑛+1𝑥_𝑛+1+1𝑟_𝑛𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛,𝑇_𝑟𝑛𝑥_𝑛−𝑥_𝑛𝐹𝑇≥0,_𝑟𝑛+1𝑥_𝑛+1,𝑇_𝑟𝑛𝑥_𝑛+1𝑟_𝑛+1𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛+1𝑥_𝑛+1,𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1≥0. (3.8) By (3.8), we have 1𝑟_𝑛𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛,𝑇_𝑟𝑛𝑥_𝑛−𝑥_𝑛+1𝑟_𝑛+1𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛+1𝑥_𝑛+1,𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1≥0. (3.9) It follows that 𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛+1𝑥_𝑛+1,𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛−𝑥_𝑛𝑟_𝑛≥0. (3.10) This implies that 0≤𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛,𝑇_𝑟𝑛𝑥_𝑛−𝑇_𝑟𝑛+1𝑥_𝑛+1+𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1. (3.11) It follows that 𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛²≤𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛,𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1=𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1≤‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛‖𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1≤‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛𝑟‖+1−^𝑛𝑟_𝑛+1‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖=‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛1‖+𝑟_𝑛+1𝑟_𝑛+1−𝑟_𝑛‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖≤‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑇_𝑟𝑛𝑥_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛1‖+𝑎𝑟_𝑛+1−𝑟_𝑛‖𝑇_𝑟𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖. (3.12) It follows that ‖𝑢_𝑛+1−𝑢_𝑛‖≤‖𝑥_𝑛+1−𝑥_𝑛1‖+𝑎𝑟_𝑛+1−𝑟_𝑛‖𝑢_𝑛+1−𝑥_𝑛+1‖. (3.13) Putting 𝑦_𝑛=𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛, then {𝑦_𝑛} is bounded. By definition of 𝑆_𝑛,forall𝑛∈ℕ, we have ‖𝑆_𝑛𝑦_𝑛−1−𝑆_𝑛−1𝑦_𝑛−1‖=‖𝑈_𝑛,1𝑦_𝑛−1−𝑈_𝑛−1,1𝑦_𝑛−1‖=‖𝛼¹₁𝑇₁𝑈_𝑛,2𝑦_𝑛−1+𝛼¹₂𝑈_𝑛,2𝑦_𝑛−1+𝛼¹₃𝑦_𝑛−1−𝛼¹₁𝑇₁𝑈_𝑛−1,2𝑦_𝑛−1−𝛼¹₂𝑈_𝑛−1,2𝑦_𝑛−1−𝛼¹₃𝑦_𝑛−1‖≤1−𝛼¹₃‖𝑈_𝑛,2𝑦_𝑛−1−𝑈_𝑛−1,2𝑦_𝑛−1‖≤1−𝛼¹₃1−𝛼²₃‖𝑈_𝑛,3𝑦_𝑛−1−𝑈_𝑛−1,3𝑦_𝑛−1‖=²_𝑗=11−𝛼^𝑗₃‖𝑈_𝑛,3𝑦_𝑛−1−𝑈_𝑛−1,3𝑦_𝑛−1‖⋮≤^𝑛−1_𝑗=11−𝛼^𝑗₃‖𝑈_𝑛,𝑛𝑦_𝑛−1−𝑈_{𝑛−1,𝑛}𝑦_𝑛−1‖≤‖𝑈_𝑛,𝑛𝑦_𝑛−1−𝑦_𝑛−1‖=‖𝛼^𝑛₁𝑇_𝑛𝑦_𝑛−1+1−𝛼^𝑛₁𝑦_𝑛−1−𝑦_𝑛−1‖=𝛼^𝑛₁‖𝑇_𝑛𝑦_𝑛−1−𝑦_𝑛−1‖≤𝛼^𝑛₁2‖𝑦_𝑛−1−𝑧‖. (3.14) Substituting (3.13) and (3.14) into (3.5), we have ‖𝑥_𝑛+1−𝑥_𝑛‖≤𝛼_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+𝛽_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+𝛾_𝑛−𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+𝛾_𝑛−1‖𝑢_𝑛−𝑢_𝑛−1‖+𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1−𝑆_𝑛−1𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−1‖≤𝛼_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+𝛽_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+𝛾_𝑛−𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+𝛾_𝑛−1‖𝑥_𝑛+1−𝑥_𝑛1‖+𝑎𝑟_𝑛+1−𝑟_𝑛‖𝑢_𝑛+1−𝑥_𝑛+1‖+2𝛾_𝑛−1𝛼^𝑛₁‖𝑦_𝑛−1≤𝛼−𝑧‖_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+1−𝛼_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+𝛾_𝑛−𝛾_𝑛−1‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+1𝑎𝑟_𝑛+1−𝑟_𝑛‖𝑢_𝑛+1−𝑥_𝑛+1‖+2𝛾_𝑛−1𝛼^𝑛₁‖𝑦_𝑛−1≤−𝑧‖1−𝛼_𝑛−1‖𝑥_𝑛−𝑥_𝑛−1‖+𝛽_𝑛−𝛽_𝑛−1𝑀₁+𝛼_𝑛−𝛼_𝑛−1𝑀₁+𝛾_𝑛−𝛾_𝑛−1𝑀₁+1𝑎𝑟_𝑛+1−𝑟_𝑛𝑀₁+2𝛼^𝑛₁𝑀₁, (3.15) where 𝑀₁=max_𝑛∈ℕ{‖𝑢‖,‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖,‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖,‖𝑢_𝑛−𝑥_𝑛‖,‖𝑦_𝑛−𝑧‖}. By (3.15), Lemma 2.3, and conditions (i)–(iii), we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.16)Step 3. We shall show that lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0.
Let 𝑣∈𝔉. Since 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛 and 𝑇_𝑟𝑛 is firmly nonexpansive, we have 𝑣−𝑇_𝑟𝑛𝑥_𝑛²=𝑇_𝑟𝑛𝑣−𝑇_𝑟𝑛𝑥_𝑛²≤𝑇_𝑟𝑛𝑣−𝑇_𝑟𝑛𝑥_𝑛,𝑣−𝑥_𝑛=12𝑇_𝑟𝑛𝑥_𝑛−𝑣²+𝑥_𝑛−𝑣²−𝑇_𝑟𝑛𝑥_𝑛−𝑥_𝑛². (3.17) Hence, 𝑢_𝑛−𝑣²≤𝑥_𝑛−𝑣²−𝑢_𝑛−𝑥_𝑛². (3.18) By (3.18), we have 𝑥_𝑛+1−𝑣²=𝛼_𝑛(𝑢−𝑣)+𝛽_𝑛(𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑣)+𝛾_𝑛(𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑣)²≤𝛼_𝑛‖𝑢−𝑣‖²+𝛽_𝑛𝑥_𝑛−𝑣²+𝛾_𝑛𝑢_𝑛−𝑣²≤𝛼_𝑛‖𝑢−𝑣‖²+𝛽_𝑛𝑥_𝑛−𝑣²+𝛾_𝑛𝑥_𝑛−𝑣²−𝑢_𝑛−𝑥_𝑛²≤𝛼_𝑛‖𝑢−𝑣‖²+𝑥_𝑛−𝑣²−𝛾_𝑛𝑢_𝑛−𝑥_𝑛². (3.19) it implies that 𝛾_𝑛𝑢_𝑛−𝑥_𝑛²≤𝛼_𝑛‖𝑢−𝑣‖²+𝑥_𝑛−𝑣²−𝑥_𝑛+1−𝑣²≤𝛼_𝑛‖𝑢−𝑣‖²+‖𝑥_𝑛−𝑣‖−‖𝑥_𝑛+1−𝑣‖‖𝑥_𝑛−𝑣‖+‖𝑥_𝑛+1−𝑣‖≤𝛼_𝑛‖𝑢−𝑣‖²+‖𝑥_𝑛−𝑥_𝑛+1‖‖𝑥_𝑛−𝑣‖+‖𝑥_𝑛+1−𝑣‖. (3.20) By (3.16) and condition (i), we have lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0. (3.21) Let 𝑧∈𝔉 and by nonexpansiveness of 𝐼−𝜆𝐴, we have 𝑥_𝑛+1−𝑧²≤𝛼_𝑛‖𝑢−𝑧‖²+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑧²+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧²−𝛽_𝑛𝛾_𝑛‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖≤𝛼_𝑛‖𝑢−𝑧‖²+1−𝛼_𝑛𝑥_𝑛−𝑧²−𝛽_𝑛𝛾_𝑛‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖≤𝛼_𝑛‖𝑢−𝑧‖²+𝑥_𝑛−𝑧²−𝛽_𝑛𝛾_𝑛‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖. (3.22) It implies that 𝛽_𝑛𝛾_𝑛‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖≤𝛼_𝑛‖𝑢−𝑧‖²+𝑥_𝑛−𝑧²−𝑥_𝑛+1−𝑧²=𝛼_𝑛‖𝑢−𝑧‖²+‖𝑥_𝑛−𝑧‖−‖𝑥_𝑛+1−𝑧‖‖𝑥_𝑛−𝑧‖+‖𝑥_𝑛+1−𝑧‖=𝛼_𝑛‖𝑢−𝑧‖²+‖𝑥_𝑛−𝑥_𝑛+1‖‖𝑥_𝑛−𝑧‖+‖𝑥_𝑛+1−𝑧‖. (3.23) By (3.16) and condition (i), we have lim_𝑛→∞‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖=0. (3.24) Since ‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖≤‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖≤‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+‖𝑢_𝑛−𝑥_𝑛‖, (3.25) by (3.24) and (3.21), we have lim_𝑛→∞‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖=0. (3.26) Since ‖𝑥_𝑛−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+𝛼_𝑛‖𝑢−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖+𝛾_𝑛‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖, (3.27) by (3.24), (3.16), and condition (i), we have lim_𝑛→∞‖𝑥_𝑛−𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖=0. (3.28) Since ‖𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+𝛼_𝑛‖𝑢−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+𝛽_𝑛‖𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖, (3.29) again by (3.24), (3.16), and condition (i), we have lim_𝑛→∞‖𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖=0. (3.30) Since ‖𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖≤‖𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+‖𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖≤‖𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛‖+‖𝑢_𝑛−𝑥_𝑛‖, (3.31) by (3.21) and (3.30), we have lim_𝑛→∞‖𝑥_𝑛−𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛‖=0. (3.32)Step 4. Putting 𝑧₀=𝑃_𝔉𝑢, we will show that limsup_𝑛→∞𝑢−𝑧₀,𝑥_𝑛−𝑧₀≤0. (3.33) To show this inequality, take a subsequence {𝑥_𝑛𝑚} of {𝑥_𝑛}, such that limsup_𝑛→∞𝑢−𝑧₀,𝑥_𝑛−𝑧₀=limsup_𝑚→∞𝑢−𝑧₀,𝑥_𝑛𝑚−𝑧₀. (3.34) Without loss of generality, we may assume that 𝑥_𝑛𝑚⇀𝜔as𝑚→∞ where 𝜔∈𝐶. By nonexpansiveness of 𝑃_𝐶(𝐼−𝜆𝐴), (3.28), and Lemma 2.2, we have 𝜔∈𝐹(𝑃_𝐶(𝐼−𝜆𝐴)). By Lemma 2.6, we obtain that 𝜔∈VI(𝐶,𝐴). Since ‖𝑢_𝑛𝑚−𝑥_𝑛𝑚‖→0 as 𝑚→∞, we have 𝑢_𝑛𝑚⇀𝜔as𝑚→∞. Since 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.35) By (A2), we have 1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶. (3.36) In particular, 𝑦−𝑢_𝑛𝑚,1𝑟_𝑛𝑚𝑢_𝑛𝑚−𝑥_𝑛𝑚≥𝐹𝑦,𝑢_𝑛𝑚. (3.37) By condition (A4), 𝐹(𝑦,⋅) is lower semicontinuous and convex, and thus weakly semicontinuous. By (3.21) imply that (1/𝑟_𝑛𝑚)(𝑢_𝑛𝑚−𝑥_𝑛𝑚)→0 in norm. Therefore, letting 𝑚→∞ in (3.37), we have 𝐹𝑦,𝜔≤lim_𝑚→∞𝐹𝑦,𝑢_𝑛𝑚≤0,∀𝑦∈𝐶. (3.38) Replacing 𝑦 with 𝑦_𝑡∶=𝑡𝑦+(1−𝑡)𝜔, 𝑡∈(0,1], we have 𝑦_𝑡∈𝐶, and using (A1), (A4), and (3.38), we obtain 0=𝐹𝑦_𝑡,𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+(1−𝑡)𝐹𝑦_𝑡,𝜔≤𝑡𝐹𝑦_𝑡,𝑦. (3.39) Hence, 𝐹(𝑡𝑦+(1−𝑡)𝜔,𝑦)≥0,forall𝑡∈(0,1] and for all 𝑦∈𝐶. Letting 𝑡→0⁺ and using assumption (A3), we can conclude that 𝐹𝜔,𝑦≥0,𝑦∈𝐶. (3.40) Therefore, 𝜔∈EP(𝐹).
We will show that 𝜔∈⋂^∞_𝑖=1𝐹(𝑇_𝑖). By Lemma 2.10, we have 𝐹(𝑆)=⋂^∞_𝑖=1𝐹(𝑇_𝑖). Assume that 𝜔≠𝑆𝜔. Using Opial 𝑠^′ property, (3.32), 𝜔∈𝐹(𝑃_𝐶(𝐼−𝜆𝐴)), and Remark 2.9, we have liminf_𝑚→∞‖𝑥_𝑛𝑚−𝜔‖<liminf_𝑚→∞‖𝑥_𝑛𝑚−𝑆𝜔‖≤liminf_𝑚→∞‖𝑥_𝑛𝑚−𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛𝑚‖+‖𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛𝑚−𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝜔‖+‖𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝜔−𝑆𝜔‖=liminf_𝑚→∞‖𝑥_𝑛𝑚−𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛𝑚‖+‖𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛𝑚−𝑆_𝑛𝑚𝑃_𝐶(𝐼−𝜆𝐴)𝜔‖+‖𝑆_𝑛𝑚𝜔−𝑆𝜔‖≤liminf_𝑚→∞‖𝑥_𝑛𝑚−𝜔‖. (3.41) This is a contradiction, then 𝜔∈⋂^∞_𝑖=1𝐹(𝑇_𝑖). Hence, 𝜔∈𝔉.
Since 𝑥_𝑛𝑚⇀𝜔 and 𝜔∈𝔉, we have limsup_𝑛→∞⟨𝑢−𝑧₀,𝑥_𝑛−𝑧₀⟩=limsup_𝑚→∞⟨𝑢−𝑧₀,𝑥_𝑛𝑚−𝑧₀⟩=⟨𝑢−𝑧₀,𝜔−𝑧₀⟩≤0. (3.42)Step 5. Finally, we show that {𝑥_𝑛} and {𝑢_𝑛} converse strongly to 𝑧₀=𝑃_𝔉𝑢. Putting 𝑧₀=𝑃_𝔉𝑢, by nonexpansiveness of 𝑃_𝐶(𝐼−𝜆𝐴), 𝑆_𝑛, and 𝑇_𝑟𝑛, we have 𝑥_𝑛+1−𝑧₀²=𝛼_𝑛𝑢−𝑧₀+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑧₀+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧₀²≤𝛽_𝑛(𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑧₀)+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧₀²+2𝛼_𝑛𝑢−𝑧₀,𝑥_𝑛+1−𝑧₀≤𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛−𝑧₀²+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛−𝑧₀²+2𝛼_𝑛𝑢−𝑧₀,𝑥_𝑛+1−𝑧₀≤𝛽_𝑛𝑥_𝑛−𝑧₀²+𝛾_𝑛𝑇_𝑟𝑛𝑥_𝑛−𝑧₀²+2𝛼_𝑛𝑢−𝑧₀,𝑥_𝑛+1−𝑧₀≤1−𝛼_𝑛‖𝑥_𝑛−𝑧₀‖²+2𝛼_𝑛𝑢−𝑧₀,𝑥_𝑛+1−𝑧₀. (3.43)

From Step 4 and Lemma 2.3, we obtain that {𝑥_𝑛} converse strongly to 𝑧₀=𝑃_𝔉𝑢. By using (3.21), we have {𝑢_𝑛} converse strongly to 𝑧₀=𝑃_𝔉𝑢.

4. Application

Using our main theorem (Theorem 3.1), we obtain the following strong convergence theorems involving infinite family of 𝜅-strict pseudocontractions.

To prove strong convergence theorem in this section, we need definition and lemma as follows.

Definition 4.1. A mapping 𝑇∶𝐶→𝐶 is said to be a 𝜅-strongly pseudocontraction mapping, if there exists 𝜅∈[0,1), such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜅(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (4.1)

Lemma 4.2 (see [15]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 a 𝜅-strict pseudocontraction. Define 𝑆∶𝐶→𝐶 by 𝑆𝑥=𝛼𝑥+(1−𝛼)𝑇𝑥 for each 𝑥∈𝐶. Then, as 𝛼∈[𝜅,1)𝑆 is nonexpansive, such that F(𝑆)=𝐹(𝑇).

Theorem 4.3. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Let 𝐹 be bifunctions from 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝐴∶𝐶→𝐻 be a 𝛼-inverse-strongly monotone mapping. Let {𝑇_𝑖}^∞_𝑖=0 be infinite family of 𝜅_𝑖-pseudocontractions mappings with 𝔉=⋂^∞_𝑖=1𝐹(𝑇_𝑖)⋂EP(𝐹)⋂VI(𝐶,𝐴). Define a mapping 𝑇_𝜅𝑖 by 𝑇_𝜅𝑖=𝜅_𝑖𝑥+(1−𝜅_𝑖)𝑇_𝑖𝑥, forall𝑥∈𝐶,𝑖∈ℕ, and let 𝜌_𝑗=(𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃)∈𝐼×𝐼×𝐼, where 𝐼=[0,1],𝛼^𝑗₁+𝛼^𝑗₂+𝛼^𝑗₃=1,𝛼^𝑗₁+𝛼^𝑗₂≤𝑏<1, and 𝛼^𝑗₁,𝛼^𝑗₂,𝛼^𝑗₃∈(0,1)forall𝑗=1,2,…. For every 𝑛∈ℕ, let 𝑆_𝑛 and 𝑆 be 𝑆-mappings generated by 𝑇_𝜅𝑛,…,𝑇_𝜅1 and 𝜌_𝑛,𝜌_𝑛−1,…,𝜌₁ and 𝑇_𝜅𝑛,𝑇_𝜅𝑛−1,…, and 𝜌_𝑛,𝜌_𝑛−1,…, respectively. Let {𝑥_𝑛},{𝑢_𝑛} be sequences generated by 𝑥₁,𝑢∈𝐶 and 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑥_𝑛+𝛾_𝑛𝑆_𝑛𝑃_𝐶(𝐼−𝜆𝐴)𝑢_𝑛,∀𝑛≥1, (4.2) where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}∈(0,1), such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,𝛽_𝑛∈[𝑐,d]⊂(0,1)𝑟_𝑛∈[𝑎,𝑏]⊂(0,2𝛼),𝜆⊂(0,2𝛼). Assume that(i)lim_𝑛→∞𝛼_𝑛=0and∑^∞_𝑛=0𝛼_𝑛=∞, (ii)∑^∞_𝑛=1𝛼^𝑛₁<∞, (iii)∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|,∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛|,∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|,∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛|<∞,
then the sequence {𝑥_𝑛},{𝑢_𝑛} converges strongly to 𝑧=𝑃_𝔉𝑢.

Proof. For every 𝑖∈ℕ, by Lemma 4.2, we have that 𝑇_𝜅𝑖 is nonexpansive mappings. From Theorem 3.1, we could have the desired conclusion.