Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces

Abstract

We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.

1. Introduction

Throughout this paper, we assume that 𝐻 is a real Hilbert space with inner product and norm denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, ∀𝑥,𝑦∈𝐶. They use 𝐹(𝑇) to denote the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. It is assumed throughout the paper that 𝑇 is a nonexpansive mapping such that 𝐹(𝑇)≠∅. Recall that a self-mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝛼∈[0,1), and 𝑥,𝑦∈𝐶 such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖.

Let 𝜑∶𝐶→ℝ∪{+∞} be a proper extended real-valued function and let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ, where ℝ is the set of real numbers. Ceng and Yao [1] considered the following mixed equilibrium problem for finding 𝑥∈𝐶 such that 𝐹𝑥,𝑦+𝜑𝑦≥𝜑(𝑥),∀𝑦∈𝐶. (1) The set of solutions of (1) is denoted by MEP(𝐹,𝜑). We see that 𝑥 is a solution of problem (1) which implies that 𝑥∈dom𝜑={𝑥∈𝐶∣𝜑(𝑥)<+∞}. If 𝜑≡0, then the mixed equilibrium problem (1) becomes the following equilibrium problem for finding 𝑥∈𝐶 such that 𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (2) The set of solutions of (2) is denoted by EP(𝐹). The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (2). Some methods have been proposed to solve the equilibrium problem (see [2–14]).

Let 𝐵∶𝐶→𝐻 be a mapping. The variational inequality problem, denoted by VI(𝐶,𝐵), is for finding 𝑥∈𝐶 such that 𝐵𝑥,𝑦−𝑥≥0, (3) for all 𝑦∈𝐶. The variational inequality problem has been extensively studied in the literature. See, for example, [15, 16] and the references therein. A mapping 𝐵 of 𝐶 into 𝐻 is called monotone if 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥0, (4) for all 𝑥,𝑦∈𝐶. 𝐵 is called 𝛽-inverse-strongly monotone if there exists a positive real number 𝛽>0 such that for all 𝑥,𝑦∈𝐶𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝛽𝐵𝑥−𝐵𝑦². (5) We consider a system of quasi-variational inclusion for finding (𝑥^∗,𝑦^∗)∈𝐻×𝐻 such that 𝜃∈𝑥^∗−𝑦^∗+𝜌₁𝐵₁𝑦^∗+𝑀₁𝑥^∗,𝜃∈𝑦^∗−𝑥^∗+𝜌₂𝐵₂𝑥^∗+𝑀₂𝑦^∗, (6) where 𝐵_𝑖∶𝐻→𝐻 and 𝑀_𝑖∶𝐻→2^𝐻 are nonlinear mappings for each 𝑖=1,2. The set of solutions of problem (6) is denoted by SQVI(𝐵₁,𝑀₁,𝐵₂,𝑀₂). As special cases of problem (6), we have the following.(1)If 𝐵₁=𝐵₂=𝐵 and 𝑀₁=𝑀₂=𝑀, then problem (6) is reduced to (7) for finding (𝑥^∗,𝑦^∗)∈𝐻×𝐻 such that 𝜃∈𝑥^∗−𝑦^∗+𝜌₁𝐵𝑦^∗+𝑀𝑥^∗,𝜃∈𝑦^∗−𝑥^∗+𝜌₂𝐵𝑥^∗+𝑀𝑦^∗. (7)(2)Further, if 𝑥^∗=𝑦^∗, then problem (7) is reduced to (8) for finding 𝑥^∗∈𝐻 such that 𝜃∈𝐵𝑥^∗+𝑀𝑥^∗, (8) where 𝜃 is the zero vector in 𝐻. The set of solutions of problem (8) is denoted by 𝐼(𝐵,𝑀). A set-valued mapping 𝑀∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻,𝑓∈𝑀(𝑥) and 𝑔∈𝑀(𝑦) imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑀 is maximal if its graph 𝐺(𝑀)∶={(𝑓,𝑥)∈𝐻×𝐻∶𝑓∈𝑀(𝑥)} of 𝑀 is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑀 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for all (𝑦,𝑔)∈𝐺(𝑀) imply 𝑓∈𝑀(𝑥). Let 𝐵 be a monotone mapping of 𝐶 into 𝐻 and let 𝑁_𝐶𝑦 be the normal cone to 𝐶 at 𝑦∈𝐶; that is, 𝑁_𝐶𝑦={𝑤∈𝐻∶⟨𝑢−𝑦,𝑤⟩≤0,∀𝑢∈𝐶}, and define 𝑀𝐵𝑦=𝑦+𝑁_𝐶𝑦,𝑦∈𝐶;∅,𝑦∉𝐶. (9) Then, 𝑀 is the maximal monotone and 𝜃∈𝑀𝑦 if and only if 𝑦∈VI(𝐶,𝐵); see [17].

Let 𝑀∶𝐻→2^𝐻 be a set-valued maximal monotone mapping; then, the single-valued mapping 𝐽_𝑀,𝜆∶𝐻→𝐻 defined by 𝐽_𝑀,𝜆𝑥^∗=(𝐼+𝜆𝑀)⁻¹𝑥^∗,𝑥^∗∈𝐻 (10) is called the resolvent operator associated with 𝑀, where 𝜆 is any positive number and 𝐼 is the identity mapping. The following characterizes the resolvent operator.(R1)The resolvent operator 𝐽_𝑀,𝜆 is single-valued and nonexpansive for all 𝜆>0; that is, 𝐽_𝑀,𝜆(𝑥)−𝐽_𝑀,𝜆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻,∀𝜆>0. (11)(R2)The resolvent operator 𝐽_𝑀,𝜆 is 1-inverse-strongly monotone; see [18]; that is, 𝐽_𝑀,𝜆(𝑥)−𝐽_𝑀,𝜆𝑦²≤𝑥−𝑦,𝐽_𝑀,𝜆(𝑥)−𝐽_𝑀,𝜆𝑦,∀𝑥,𝑦∈𝐻. (12)(R3)The solution of problem (8) is a fixed point of the operator 𝐽_𝑀,𝜆(𝐼−𝜆𝐵) for all 𝜆>0; see also [19]; that is, 𝐼(𝐵,𝑀)=𝐹𝐽_𝑀,𝜆(𝐼−𝜆𝐵),∀𝜆>0. (13)(R4)If 0<𝜆≤2𝛽, then the mapping 𝐽_𝑀,𝜆(𝐼−𝜆𝐵)∶𝐻→𝐻 is nonexpansive.(R5)𝐼(𝐵,𝑀) is closed and convex.

Let 𝐴 be a strongly positive linear bounded operator on 𝐻; that is, there exists a constant 𝛾>0 with property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (14) A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻: min_{𝑥∈𝐹(𝑇)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (15) where 𝐴 is a strongly positive linear bounded operator and ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

In 2007, Plubtieng and Punpaeng [20] proposed the following iterative algorithm: 𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐻,_𝑛+1=𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑇𝑢_𝑛. (16) They proved that if the sequences {𝜖_𝑛} and {𝑟_𝑛} of parameters satisfy appropriate conditions, then the sequences {𝑥_𝑛} and {𝑢_𝑛} both converge to the unique solution 𝑧 of the variational inequality 𝐴−𝛾𝑓𝑧,𝑥−𝑧≥0,∀𝑥∈𝐹(𝑇)∩EP(𝐹), (17) which is the optimality condition for the minimization problem min_{𝑥∈𝐹(𝑇)∩EP(𝐹)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (18) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

In 2009, Peng and Yao [21] introduced an iterative algorithm based on extragradient method which solves the problem for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings, and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. The sequences generated by 𝑣∈𝐶 are 𝑥₁𝐹𝑢=𝑥∈𝐶,_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝑃_𝐶𝑢_𝑛−𝛾_𝑛𝐵𝑢_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑣+𝛽_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝑊_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐵𝑦_𝑛, (19) for all 𝑛≥1, where 𝑊_𝑛 is 𝑊-mapping. They proved the strong convergence theorems under some mild conditions.

In 2010, Qin et al. [22] introduced an iterative method for finding solutions of a generalized equilibrium problem, the set of fixed points of a family of nonexpansive mappings, and the common variational inclusions. The sequences generated by 𝑥₁∈𝐶 and {𝑥_𝑛} are a sequence generated by 𝐹𝑢_𝑛,𝑦+𝐴₃𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴₂𝑢_𝑛,𝑦_𝑛=𝑃_𝐶𝑧_𝑛−𝜂_𝑛𝐴₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑊_𝑛𝑦_𝑛,∀𝑛≥1, (20) where 𝑓 is a contraction and 𝐴_𝑖 is inverse-strongly monotone mappings for 𝑖=1,2,3 and 𝑊_𝑛 is called a 𝑊-mapping generated by 𝑆_𝑛,𝑆_𝑛1,…,𝑆₁ and 𝛾_𝑛,𝛾_𝑛−1,…,𝛾₁. They proved the strong convergence theorems under some mild conditions. Liou [23] introduced an algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of variational inclusion in a real Hilbert space. The sequences generated by 𝑥₀∈𝐶 are 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶1−𝛼_𝑛𝐴𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛, (21) for all 𝑛≥1, where 𝐴 is a strongly positive bounded linear operator and 𝐵,𝑄 are inverse-strongly monotone. They proved the strong convergence theorems under some suitable conditions.

Next, Petrot et al. [24] introduced the new following iterative process for finding the set of solutions of quasi-variational inclusion problem and the set of fixed point of a nonexpansive mapping. The sequence is generated by 𝑥₀𝑥∈𝐻,chosenarbitrary,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑆𝑧_𝑛,𝑧_𝑛=𝐽_𝑀,𝜆𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑦_𝑛=𝐽_𝑀,𝜌𝑥_𝑛−𝜌𝐴𝑥_𝑛, (22) for all 𝑛∈ℕ∪{0}, where {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛} are three sequences in [0,1] and 𝜆∈(0,2𝛼]. They proved that {𝑥_𝑛} generated by (22) converges strongly to 𝑧₀ which is the unique solution in 𝐹(𝑆)∩𝐼(𝐴,𝑀).

In 2011, Jitpeera and Kumam [25] introduced a shrinking projection method for finding the common element of the common fixed points of nonexpansive semigroups, the set of common fixed point for an infinite family, the set of solutions of a system of mixed equilibrium problems, and the set of solution of the variational inclusion problem. Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑣_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝐶₁=𝐶, 𝑥₁=𝑃_𝐶1𝑥₀, 𝑢_𝑛∈𝐶, and 𝑥₀𝑢=𝑥∈𝐶chosenarbitrary,_𝑛=𝐾^𝐹𝑁_{𝑟𝑁,𝑛}𝐾^𝐹𝑁−1_{𝑟𝑁−1,𝑛}𝐾^𝐹𝑁−2_{𝑟𝑁−2,𝑛}⋯𝐾^𝐹2_𝑟2,𝑛𝐾^𝐹1_𝑟1,𝑛𝑥_𝑛,𝑦_𝑛=𝐽_{𝑀2,𝛿𝑛}𝑢_𝑛−𝛿_𝑛𝐵𝑢_𝑛,𝑣_𝑛=𝐽_{𝑀1,𝜆𝑛}𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛,𝑧_𝑛=𝛼_𝑛𝑣_𝑛+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑣_𝑛𝐶𝑑𝑠,_𝑛+1=𝑧∈𝐶_𝑛∶𝑧_𝑛−𝑧²≤𝑥_𝑛−𝑧²−𝛼_𝑛1−𝛼_𝑛×𝑣_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑆(𝑠)𝑊_𝑛𝑣_𝑛𝑑𝑠²,𝑥_𝑛+1=𝑃_𝐶𝑛+1𝑥₀,𝑛∈ℕ, (23) where 𝐾^𝐹𝑘_𝑟𝑘∶𝐶→𝐶, 𝑘=1,2,…,𝑁. We proved the strong convergence theorem under certain appropriate conditions.

In this paper, motivated by the above results, we introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusions, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Then, we prove strong convergence theorems which are connected with [5, 26–29]. Our results extend and improve the corresponding results of Jitpeera and Kumam [25], Liou [23], Plubtieng and Punpaeng [20], Petrot et al. [24], Peng and Yao [21], Qin et al. [22], and some authors.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖ and let 𝐶 be a nonempty closed convex subset of 𝐻. Then, 𝑥−𝑦²=‖𝑥‖²−𝑦²,()𝑦−2𝑥−𝑦,𝑦𝜆𝑥+1−𝜆²=𝜆‖𝑥‖²+(1−𝜆)𝑦²()×−𝜆1−𝜆𝑥−𝑦²,∀𝑥,𝑦∈𝐻,𝜆∈[0,1]. (24) For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (25)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶 and satisfies 𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (26) Moreover, 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and 𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0,𝑥−𝑦²≥𝑥−𝑃_𝐶𝑥²+𝑦−𝑃_𝐶𝑥²,∀𝑥∈𝐻,𝑦∈𝐶. (27)(28) Let 𝐵 be a monotone mapping of 𝐶 into 𝐻. In the context of the variational inequality problem, the characterization of projection (27) implies the following: 𝑢∈VI(𝐶,𝐵)⟺𝑢=𝑃_𝐶(𝑢−𝜆𝐵𝑢),𝜆>0. (29) It is also known that 𝐻 satisfies the Opial condition [30]; that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦 (30) holds for every 𝑦∈𝐻 with 𝑥≠𝑦.

For the infinite family of nonexpansive mappings of 𝑇₁,𝑇₂,…, and sequence {𝜆_𝑖}^∞_𝑖=1 in [0,1), see [31]; we define the mapping 𝑊_𝑛 of 𝐶 into itself as follows: 𝑈_𝑛,0𝑈=𝐼,_𝑛,1=𝜆₁𝑇₁𝑈_𝑛,0+1−𝜆₁𝑈_𝑛,0,𝑈_𝑛,2=𝜆₂𝑇₂𝑈_𝑛,1+1−𝜆₂𝑈_𝑛,1,⋮𝑈_{𝑛,𝑁−1}=𝜆_𝑁−1𝑇_𝑁−1𝑈_{𝑛,𝑁−2}+1−𝜆_𝑁−1𝑈_{𝑛,𝑁−2},𝑊_𝑛=𝑈_𝑛,𝑁=𝜆_𝑁𝑇_𝑁𝑈_{𝑛,𝑁−1}+1−𝜆_𝑁𝑈_{𝑛,𝑁−1}. (31)

Lemma 1 (Shimoji and Takahashi [32]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝒯={𝑇_𝑖}^𝑁_𝑖=1 be a family of infinitely nonexpanxive mappings with 𝐹(𝒯)=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅ and let {𝜆_𝑖} be a real sequence such that 0<𝜆_𝑖≤𝑏<1 for every 𝑖≥1. Then (1)𝑊_𝑛 is nonexpansive and 𝐹(𝑊_𝑛)=⋂^𝑛_𝑖=1𝐹(𝑇_𝑖) for each 𝑛≥1;(2)for each 𝑥∈𝐶 and for each positive integer 𝑘, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists;(3)the mapping 𝑊∶𝐶→𝐶 defined by 𝑊𝑥=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥 is a nonexpansive mapping satisfying 𝐹(𝑊)=𝐹(𝒯) and it is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,…, and 𝜆₁,𝜆₂,…;(4)if 𝐾 is any bounded subset of 𝐶, then lim_𝑛→∞sup_𝑥∈𝐾‖𝑊𝑥−𝑊_𝑛𝑥‖=0.

For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction 𝐹∶𝐶×𝐶→ℝ and a proper extended real-valued function 𝜑∶𝐶→ℝ∪{+∞} satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝐹 is monotone; that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶;(A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦);(A4)for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous;(A5)for each 𝑦∈𝐶, 𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous;(B1)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, 𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥+1𝑟𝑦_𝑥−𝑧,𝑧−𝑥<𝜑(𝑧); (32)(B2)𝐶 is a bounded set.

We need the following lemmas for proving our main results.

Lemma 2 (Peng and Yao [21]). Let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction that satisfies (A1)–(A5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑦+1𝑧∈𝐶∶𝐹𝑧,𝑦+𝜑𝑟𝑦−𝑧,𝑧−𝑥≥𝜑(𝑧),∀𝑦∈𝐶, (33) for all 𝑥∈𝐻. Then, the following hold: (1)for each 𝑥∈𝐻,𝑇_𝑟(𝑥)≠∅;(2)𝑇_𝑟 is single-valued;(3)𝑇_𝑟 is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐻, ‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩;(4)𝐹(𝑇_𝑟)=𝑀𝐸𝑃(𝐹,𝜑);(5)𝑀𝐸𝑃(𝐹,𝜑) is closed and convex.

Lemma 3 (Xu [33]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥0, (34) where {𝛼_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (1)∑^∞_𝑛=1𝛼_𝑛=∞,(2)limsup_𝑛→∞(𝛿_𝑛/𝛼_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞.Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 4 (Suzuki [34]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑦_𝑛+1−𝑦_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 5 (Marino and Xu [35]). Assume 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then, ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 6. For given 𝑥^∗,𝑦^∗∈𝐶×𝐶,(𝑥^∗,𝑦^∗) is a solution of problem (6) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝐽_𝑀1,𝜆𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝜆𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥,∀𝑥∈𝐶, (35) where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥−𝜇𝐸₂𝑥), 𝜆,𝜇 are positive constants, and 𝐸₁,𝐸₂∶𝐶→𝐻 are two mappings.

Proof. 𝜃∈𝑥^∗−𝑦^∗+𝜆𝐸₁𝑦^∗+𝑀₁𝑥^∗,𝜃∈𝑦^∗−𝑥^∗+𝜇𝐸₂𝑥^∗+𝑀₂𝑦^∗ (36)⇔𝑥^∗=𝐽_𝑀1,𝜆𝑦^∗−𝜆𝐸₁𝑦^∗,𝑦^∗=𝐽_𝑀2,𝜇𝑥^∗−𝜇𝐸₂𝑥^∗ (37)⇔𝐺𝑥^∗=𝐽_𝑀1,𝜆𝐽_𝑀2,𝜇𝑥^∗−𝜇𝐸₂𝑥^∗−𝜆𝐸₁𝐽_𝑀2,𝜇𝑥^∗−𝜇𝐸₂𝑥^∗=𝑥^∗. (38) This completes the proof.

Now, we prove the following lemmas which will be applied in the main theorem.

Lemma 7. Let 𝐺∶𝐶→𝐶 be defined as in Lemma 6. If 𝐸₁,𝐸₂∶𝐶→𝐻 is 𝜂₁,𝜂₂-inverse-strongly monotone and 𝜆∈(0,2𝜂₁), and 𝜇∈(0,2𝜂₂), respectively, then 𝐺 is nonexpansive.

Proof. For any 𝑥,𝑦∈𝐶 and 𝜆∈(0,2𝜂₁),  𝜇∈(0,2𝜂₂), we have 𝐺(𝑥)−𝐺𝑦²=𝐽_𝑀1,𝜆𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝜆𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀1,𝜆𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦−𝜆𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²≤𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝜆𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦−𝜆𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²=𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦−𝜆𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²=𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²−2𝜆𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦,𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦+𝜆²𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²≤𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²−2𝜆𝜂₁𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²+𝜆²𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²=𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²+𝜆𝜆−2𝜂₁𝐸₁𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐸₁𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²≤𝐽_𝑀2,𝜇𝑥−𝜇𝐸₂𝑥−𝐽_𝑀2,𝜇𝑦−𝜇𝐸₂𝑦²≤𝑥−𝜇𝐸₂𝑥−𝑦−𝜇𝐸₂𝑦²=𝑥−𝑦−𝜇𝐸₂𝑥−𝐸₂𝑦²=𝑥−𝑦²−2𝜇𝑥−𝑦,𝐸₂𝑥−𝐸₂𝑦+𝜇²𝐸₂𝑥−𝐸₂𝑦²≤𝑥−𝑦²−2𝜂₂𝜇𝐸₂𝑥−𝐸₂𝑦²+𝜇²𝐸₂𝑥−𝐸₂𝑦²=𝑥−𝑦²+𝜇𝜇−2𝜂₂𝐸₂𝑥−𝐸₂𝑦²≤𝑥−𝑦². (39) This shows that 𝐺 is nonexpansive on 𝐶.

3. Main Results

In this section, we show a strong convergence theorem for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of a infinite family of nonexpansive mappings in a real Hilbert space.

Theorem 8. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuos and convex function. Let 𝑇_𝑖∶𝐶→𝐶 be nonexpansive mappings for all 𝑖=1,2,3,…, such that Θ∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)∩𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝑀𝐸𝑃(𝐹,𝜑)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸₁,𝐸₂ be 𝛿,𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝐴 be a strongly positive bounded linear self-adjoint on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼, let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Assume that either 𝐵₁ or 𝐵₂ holds and let 𝑊_𝑛 be the 𝑊-mapping defined by (31). Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛,∀𝑛≥0, (40) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (6).

Proof. Let 𝑥^∗∈Θ; that is 𝑇_𝑟(𝑥^∗−𝑟𝑄𝑥^∗)=𝐽_𝑀1,𝜆[𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐵₂𝑥^∗)−𝜆𝐵₁𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐵₂𝑥^∗)]=𝑇_𝑖(𝑥^∗)=𝑥^∗, 𝑖≥1. Putting 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗), one can see that 𝑥^∗=𝐽_𝑀1,𝜆(𝑦^∗−𝜆𝐵₁𝑦^∗).
We divide our proofs into the following steps: (1)sequences {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} are bounded;(2)lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0;(3)lim_𝑛→∞‖𝑄𝑥_𝑛−𝑄𝑥^∗‖=0, lim_𝑛→∞‖𝐸₁𝑧_𝑛−𝐸₁𝑥^∗‖=0 and lim_𝑛→∞‖𝐸₂𝑢_𝑛−𝐸₂𝑥^∗‖=0;(4)lim_𝑛→∞‖𝑥_𝑛−𝑊𝑥_𝑛‖=0;(5)limsup_𝑛→∞⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0, where 𝑥^∗=𝑃_Θ(𝛾𝑓+𝐼−𝐴)𝑥^∗;(6)lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0.
Step 1. From conditions (C1) and (C2), we may assume that 𝛼_𝑛≤(1−𝛽_𝑛)‖𝐴‖⁻¹. By the same argument as that in [9], we can deduce that (1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴 is positive and ‖(1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴‖≤1−𝛽_𝑛−𝛼_𝑛𝛾. For all 𝑥,𝑦∈𝐶 and 𝑟∈(0,2𝛿). since 𝑄 is a 𝛿-inverse-strongly monotone and 𝐵₁,𝐵₂ are 𝜂₁,𝜂₂-inverse-strongly monotone, we have (𝐼−𝑟𝑄)𝑥−(𝐼−𝑟𝑄)𝑦²=𝑥−𝑦−𝑟𝑄𝑥−𝑄𝑦²=𝑥−𝑦²−2𝑟𝑥−𝑦,𝑄𝑥−𝑄𝑦+𝑟²𝑄𝑥−𝑄𝑦²≤𝑥−𝑦²−2𝑟𝛿𝑄𝑥−𝑄𝑦²+𝑟²𝑄𝑥−𝑄𝑦²=𝑥−𝑦²+𝑟(𝑟−2𝛿)𝑄𝑥−𝑄𝑦²≤𝑥−𝑦². (41) It follows that ‖(𝐼−𝑟𝑄)𝑥−(𝐼−𝑟𝑄)𝑦‖≤‖𝑥−𝑦‖; hence 𝐼−𝑟𝑄 is nonexpansive.
In the same way, we conclude that ‖(𝐼−𝜆𝐸₁)𝑥−(𝐼−𝜆𝐸₁)𝑦‖≤‖𝑥−𝑦‖ and ‖(𝐼−𝜇𝐸₂)𝑥−(𝐼−𝜇𝐸₂)𝑦‖≤‖𝑥−𝑦‖; hence 𝐼−𝜆𝐸₁,𝐼−𝜇𝐸₂ are nonexpansive. Let 𝑦_𝑛=𝐽_𝑀1,𝜆(𝑧_𝑛−𝜆𝐸₁𝑧_𝑛),𝑛≥0. It follows that 𝑦_𝑛−𝑥^∗=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝐽_𝑀1,𝜆𝑦^∗−𝜆𝐸₁𝑦^∗≤𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝑦^∗−𝜆𝐸₁𝑦^∗≤𝑧_𝑛−𝑦^∗,𝑧_𝑛−𝑦^∗=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛−𝐽_𝑀2,𝜇𝑥^∗−𝜇𝐸₂𝑥^∗≤𝑢_𝑛−𝜇𝐸₂𝑢_𝑛−𝑥^∗−𝜇𝐸₂𝑥^∗≤𝑢_𝑛−𝑥^∗. (42) By Lemma 2, we have 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛−𝑟𝑄𝑥_𝑛) for all 𝑛≥0, ∀𝑥,𝑦∈𝐶. Then, for 𝑟∈(0,2𝛿), we obtain 𝑢_𝑛−𝑥^∗2=𝑇_𝑟𝑥_𝑛−𝑟𝑄𝑥_𝑛−𝑇_𝑟𝑥^∗−𝑟𝑄𝑥^∗2≤𝑥_𝑛−𝑟𝑄𝑥_𝑛−𝑥^∗−𝑟𝑄𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑟(𝑟−2𝛿)𝑄𝑥_𝑛−𝑄𝑥^∗2≤𝑥_𝑛−𝑥^∗2. (43) Hence, we have 𝑦_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗. (44) From (40) and (44), we deduce that 𝑥_𝑛+1−𝑥^∗=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛−𝛼_𝑛𝛾𝑦_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗+1−𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗+1−𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗=1−𝛼_𝑛𝛾−𝛾𝛼𝑥_𝑛−𝑥^∗+𝛼_𝑛𝑥𝛾−𝛾𝛼𝛾𝑓^∗−𝐴𝑥^∗𝑥𝛾−𝛾𝛼≤max_𝑛−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗.𝛾−𝛾𝛼 (45) It follows by mathematical induction that 𝑥_𝑛+1−𝑥^∗≤max𝑥₀−𝑥^∗,𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝛾−𝛾𝛼𝑛≥0. (46) Hence, {𝑥_𝑛} is bounded and also {𝑢_𝑛}, {𝑧_𝑛}, {𝑦_𝑛}, {𝑊_𝑛𝑦_𝑛}, {𝐴𝑊_𝑛𝑦_𝑛}, and {𝑓𝑥_𝑛} are all bounded.
Step 2. We show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
Putting 𝑡_𝑛=(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛)=(𝛼_𝑛𝛾𝑓(𝑥_𝑛)+((1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴)𝑊_𝑛𝑦_𝑛)/(1−𝛽_𝑛), we get 𝑥_𝑛+1=(1−𝛽_𝑛)𝑡_𝑛+𝛽_𝑛𝑥_𝑛, 𝑛≥1. We note that 𝑡_𝑛+1−𝑡_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐴𝑊_𝑛+1𝑦_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛−𝛼_𝑛+11−𝛽_𝑛+1𝐴𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑊_𝑛𝑦_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐴𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛+1𝑦_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛. (47) It follows that 𝑡_𝑛+1−𝑡_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐴𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑊_𝑛𝑦_𝑛+𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛+1𝑦_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐴𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑊_𝑛𝑦_𝑛+𝛾𝑓𝑥_𝑛+𝑦_𝑛+1−𝑦_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛. (48) By the definition of 𝑊_𝑛, 𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛=𝜆_𝑛+1,𝑁𝑇_𝑁𝑈_{𝑛+1,𝑁−1}𝑦_𝑛+1−𝜆_𝑛+1,𝑁𝑦_𝑛−𝜆_𝑛,𝑁𝑇_𝑁𝑈_{𝑛,𝑁−1}𝑦_𝑛−1−𝜆_𝑛,𝑁𝑦_𝑛≤𝜆_𝑛+1,𝑁−𝜆_𝑛,𝑁𝑦_𝑛+𝜆_𝑛+1,𝑁𝑇_𝑁𝑈_{𝑛+1,𝑁−1}𝑦_𝑛−𝜆_𝑛,𝑁𝑇_𝑁𝑈_{𝑛,𝑁−1}𝑦_𝑛≤𝜆_𝑛+1,𝑁−𝜆_𝑛,𝑁𝑦_𝑛+𝜆_𝑛+1,𝑁𝑇_𝑁𝑈_{𝑛+1,𝑁−1}𝑦_𝑛−𝑇_𝑁𝑈_{𝑛,𝑁−1}𝑦_𝑛+𝜆_𝑛+1,𝑁−𝜆_𝑛,𝑁𝑇_𝑁𝑈_{𝑛,𝑁−1}𝑦_𝑛≤2𝑀𝜆_𝑛+1,𝑁−𝜆_𝑛,𝑁+𝜆_𝑛+1,𝑁𝑈_{𝑛+1,𝑁−1}𝑦_𝑛−𝑈_{𝑛,𝑁−1}𝑦_𝑛, (49) where 𝑀 is an approximate constant such that 𝑀≥max{sup_𝑛≥1{‖𝑦_𝑛‖},sup_𝑛≥1{‖𝑇_𝑚𝑈_{𝑛,𝑚−1}𝑦_𝑛‖}∣𝑚=1,2,…,𝑁}. Since 0<𝜆_𝑛𝑖≤1 for all 𝑛≥1 and 𝑖=1,2,…,𝑁, we compute 𝑈_{𝑛+1,𝑁−1}𝑦_𝑛−𝑈_{𝑛,𝑁−1}𝑦_𝑛=𝜆_{𝑛+1,𝑁−1}𝑇_𝑁−1𝑈_{𝑛+1,𝑁−2}𝑦_𝑛+1−𝜆_{𝑛+1,𝑁−1}𝑦_𝑛−𝜆_{𝑛,𝑁−1}𝑇_𝑁−1𝑈_{𝑛,𝑁−2}𝑦_𝑛−1−𝜆_{𝑛,𝑁−1}𝑦_𝑛≤𝜆_{𝑛+1,𝑁−1}−𝜆_{𝑛,𝑁−1}𝑦_𝑛+𝜆_{𝑛+1,𝑁−1}𝑇_𝑁−1𝑈_{𝑛+1,𝑁−2}𝑦_𝑛−𝜆_{𝑛,𝑁−1}𝑇_𝑁−1𝑈_{𝑛,𝑁−2}𝑦_𝑛≤𝜆_{𝑛+1,𝑁−1}−𝜆_{𝑛,𝑁−1}𝑦_𝑛+𝜆_{𝑛+1,𝑁−1}𝑇_𝑁−1𝑈_{𝑛+1,𝑁−2}𝑦_𝑛−𝑇_𝑁−1𝑈_{𝑛,𝑁−2}𝑦_𝑛+𝜆_{𝑛+1,𝑁−1}−𝜆_{𝑛,𝑁−1}𝑇_𝑁−1𝑈_{𝑛,𝑁−2}𝑦_𝑛≤2𝑀𝜆_{𝑛+1,𝑁−1}−𝜆_{𝑛,𝑁−1}+𝑈_{𝑛+1,𝑁−2}𝑦_𝑛−𝑈_{𝑛,𝑁−2}𝑦_𝑛. (50) It follows that 𝑈_{𝑛+1,𝑁−1}𝑦_𝑛−𝑈_{𝑛,𝑁−1}𝑦_𝑛≤2𝑀𝜆_{𝑛+1,𝑁−1}−𝜆_{𝑛,𝑁−1}+2𝑀𝜆_{𝑛+1,𝑁−2}−𝜆_{𝑛,𝑁−2}+𝑈_{𝑛+1,𝑁−3}𝑦_𝑛−𝑈_{𝑛,𝑁−3}𝑦_𝑛≤2𝑀^𝑁−1_𝑖=2𝜆_𝑛+1,𝑖−𝜆_𝑛,𝑖+𝑈_𝑛+1,1𝑦_𝑛−𝑈_𝑛,1𝑦_𝑛=2𝑀^𝑁−1_𝑖=2𝜆_𝑛+1,𝑖−𝜆_𝑛,𝑖+𝜆_𝑛+1,1𝑇₁𝑦_𝑛+1−𝜆_𝑛+1,1𝑦_𝑛−𝜆_𝑛,1𝑇₁𝑦_𝑛−1−𝜆_𝑛,1𝑦_𝑛≤2𝑀^𝑁−1_𝑖=1𝜆_𝑛+1,𝑖−𝜆_𝑛,𝑖. (51) Substituting (51) into (49), 𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛≤2𝑀𝜆_𝑛+1,𝑁−𝜆_𝑛,𝑁+2𝜆_𝑛+1,𝑁𝑀^𝑁−1_𝑖=1𝜆_𝑛+1,𝑖−𝜆_𝑛,𝑖≤2𝑀^𝑁_𝑖=1𝜆_𝑛+1,𝑖−𝜆_𝑛,𝑖. (52) We note that 𝑦_𝑛+1−𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛+1−𝜆𝐸₁𝑧_𝑛+1−𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛≤𝑧_𝑛+1−𝜆𝐸₁𝑧_𝑛+1−𝑧_𝑛−𝜆𝐸₁𝑧_𝑛≤𝑧_𝑛+1−𝑧_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛+1−𝜇𝐸₂𝑢_𝑛+1−𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛≤𝑢_𝑛+1−𝜇𝐸₂𝑢_𝑛+1−𝑢_𝑛−𝜇𝐸₂𝑢_𝑛≤𝑢_𝑛+1−𝑢_𝑛=𝑇_𝑟𝑥_𝑛+1−𝑟𝐷𝑥_𝑛+1−𝑇_𝑟𝑥_𝑛−𝑟𝐷𝑥_𝑛≤𝑥_𝑛+1−𝑟𝐷𝑥_𝑛+1−𝑥_𝑛−𝑟𝐷𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (53) Applying (52) and (53) in (48), we get 𝑡_𝑛+1−𝑡_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐴𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝐴𝑊_𝑛𝑦_𝑛+𝛾𝑓𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛+2𝑀^𝑁_𝑖=1𝜆_𝑛+1,𝑖−𝜆_𝑛,𝑖−𝑥_𝑛+1−𝑥_𝑛. (54) By conditions (C1)–(C3), imply that limsup_𝑛→∞𝑡_𝑛+1−𝑡_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (55) Hence, by Lemma 4, we obtain lim_𝑛→∞𝑡_𝑛−𝑥_𝑛=0. (56) It follows that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑡_𝑛−𝑥_𝑛=0. (57) We obtain that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (58)
Step 3. We can rewrite (40) as 𝑥_𝑛+1=𝛼_𝑛(𝛾𝑓(𝑥_𝑛)−𝐴𝑊_𝑛𝑦_𝑛)+𝛽_𝑛(𝑥_𝑛−𝑊_𝑛𝑦_𝑛)+𝑊_𝑛𝑦_𝑛. We observe that 𝑥_𝑛−𝑊_𝑛𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑊_𝑛𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑊_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛; (59) it follows that 𝑥_𝑛−𝑊_𝑛𝑦_𝑛≤11−𝛽_𝑛𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑊_𝑛𝑦_𝑛. (60) By conditions (C1), (C2), and (58), imply that lim_𝑛→∞𝑥_𝑛−𝑊_𝑛𝑦_𝑛=0. (61) From (42) and (43), we get 𝑦_𝑛−𝑥^∗2=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝐽_𝑀1,𝜆𝑥^∗−𝜆𝐸₁𝑥^∗2≤𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝑥^∗−𝜆𝐸₁𝑥^∗2≤𝑧_𝑛−𝑥^∗2+𝜆𝜆−2𝜂₁𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2≤𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛−𝐽_𝑀2,𝜇𝑥^∗−𝜇𝐸₂𝑥^∗2+𝜆𝜆−2𝜂₁𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2≤𝑢_𝑛−𝜇𝐸₂𝑢_𝑛−𝑥^∗−𝜇𝐸₂𝑥^∗2+𝜆𝜆−2𝜂₁𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2≤𝑢_𝑛−𝑥^∗2+𝜇𝜇−2𝜂₂𝐸₂𝑢_𝑛−𝐸₂𝑥^∗2+𝜆𝜆−2𝜂₁𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑟(𝑟−2𝛿)𝑄𝑥_𝑛−𝑄𝑥^∗2+𝜇𝜇−2𝜂₂𝐸₂𝑢_𝑛−𝐸₂𝑥^∗2+𝜆𝜆−2𝜂₁𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2. (62) By (40), we obtain 𝑥_𝑛+1−𝑥^∗2=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛+𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑥^∗2≤𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗=1−𝛼_𝑛𝛾²𝑦_𝑛−𝑥^∗2+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗. (63) Substituting (62) into (63), imply that 𝑥_𝑛+1−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑟(𝑟−2𝛿)𝑄𝑥_𝑛−𝑄𝑥^∗2+𝜇𝜇−2𝜂₂𝐸₂𝑢_𝑛−𝐸₂𝑥^∗2+𝜆𝜆−2𝜂₁𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗. (64) Thus, 𝑟(2𝛿−𝑟)𝑄𝑥_𝑛−𝑄𝑥^∗2+𝜇2𝜂₂−𝜇𝐸₂𝑢_𝑛−𝐸₂𝑥^∗2+𝜆2𝜂₁−𝜆𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗≤𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗. (65) By conditions (C1), (C2), (58), and (61), we deduce immediately that lim_𝑛→∞𝑄𝑥_𝑛−𝑄𝑥^∗=lim_𝑛→∞𝐸₁𝑧_𝑛−𝐸₁𝑥^∗=lim_𝑛→∞𝐸₂𝑢_𝑛−𝐸₂𝑥^∗=0. (66)
Step 4. We show that lim_𝑛→∞‖𝑥_𝑛−𝑊𝑥_𝑛‖=0. Since 𝑇_𝑟 is firmly nonexpansive, we have 𝑢_𝑛−𝑥^∗2=𝑇_𝑟𝑥_𝑛−𝑟𝑄𝑥_𝑛−𝑇_𝑟𝑥^∗−𝑟𝑄𝑥^∗2≤𝑥_𝑛−𝑟𝑄𝑥_𝑛−𝑥^∗−𝑟𝑄𝑥^∗,𝑢_𝑛−𝑥^∗=12𝑥_𝑛−𝑟𝑄𝑥_𝑛−𝑥^∗−𝑟𝑄𝑥^∗2+𝑢_𝑛−𝑥^∗2−12𝑥_𝑛−𝑟𝑄𝑥_𝑛−𝑥^∗−𝑟𝑄𝑥^∗−𝑢_𝑛−𝑥^∗2=12𝑥_𝑛−𝑥^∗2+𝑢_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛−𝑟𝑄𝑥_𝑛−𝑄𝑥^∗2=12𝑥_𝑛−𝑥^∗2+𝑢_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛²+𝑟²𝑄𝑥_𝑛−𝑄𝑥^∗2−2𝑟𝑥_𝑛−𝑢_𝑛,𝑄𝑥_𝑛−𝑄𝑥^∗≤12𝑥_𝑛−𝑥^∗2+𝑢_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛²−𝑟²𝑄𝑥_𝑛−𝑄𝑥^∗2+2𝑟𝑥_𝑛−𝑢_𝑛𝑄𝑥_𝑛−𝑄𝑥^∗, (67) which implies that 𝑢_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛²+2𝑟𝑥_𝑛−𝑢_𝑛𝑄𝑥_𝑛−𝑄𝑥^∗. (68) Since 𝐽_𝑀1,𝜆 is 1-inverse-strongly monotone, we have 𝑦_𝑛−𝑥^∗2=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝐽_𝑀1,𝜆𝑥^∗−𝜆𝐸₁𝑥^∗2≤𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝑥^∗−𝜆𝐸₁𝑥^∗,𝑦_𝑛−𝑥^∗=12𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝑥^∗−𝜆𝐸₁𝑥^∗2+𝑦_𝑛−𝑥^∗2−12𝑧_𝑛−𝜆𝐸₁𝑧_𝑛−𝑥^∗−𝜆𝐸₁𝑥^∗−𝑦_𝑛−𝑥^∗2=12𝑧_𝑛−𝑥^∗2+𝑦_𝑛−𝑥^∗2−𝑧_𝑛−𝑦_𝑛−𝜆𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2=12𝑧_𝑛−𝑥^∗2+𝑦_𝑛−𝑥^∗2−𝑧_𝑛−𝑦_𝑛²+𝜆²𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2−2𝜆𝑧_𝑛−𝑦_𝑛,𝐸₁𝑧_𝑛−𝐸₁𝑥^∗≤12𝑧_𝑛−𝑥^∗2+𝑦_𝑛−𝑥^∗2−𝑧_𝑛−𝑦_𝑛²−𝜆²𝐸₁𝑧_𝑛−𝐸₁𝑥^∗2+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗, (69) which implies that 𝑦_𝑛−𝑥^∗2≤𝑧_𝑛−𝑥^∗2−𝑧_𝑛−𝑦_𝑛²+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗. (70) In the same way with (70), we can get 𝑧_𝑛−𝑥^∗2≤𝑢_𝑛−𝑥^∗2−𝑢_𝑛−𝑧_𝑛²+2𝜇𝑢_𝑛−𝑧_𝑛𝐸₂𝑢_𝑛−𝐸₂𝑥^∗. (71) Substituting (71) into (70), imply that 𝑦_𝑛−𝑥^∗2≤𝑢_𝑛−𝑥^∗2−𝑢_𝑛−𝑧_𝑛²+2𝜇𝑢_𝑛−𝑧_𝑛𝐸₂𝑢_𝑛−𝐸₂𝑥^∗−𝑧_𝑛−𝑦_𝑛²+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗. (72) Again, substituting (68) into (72), we get 𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛²+2𝑟𝑥_𝑛−𝑢_𝑛𝑄𝑥_𝑛−𝑄𝑥^∗−𝑢_𝑛−𝑧_𝑛²+2𝜇𝑢_𝑛−𝑧_𝑛𝐸₂𝑢_𝑛−𝐸₂𝑥^∗−𝑧_𝑛−𝑦_𝑛²+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗. (73) Substituting (73) into (63), imply that 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝛾²𝑥_𝑛−𝑥^∗2−𝑥_𝑛−𝑢_𝑛²+2𝑟𝑥_𝑛−𝑢_𝑛𝑄𝑥_𝑛−𝑄𝑥^∗−𝑢_𝑛−𝑧_𝑛²+2𝜇𝑢_𝑛−𝑧_𝑛𝐸₂𝑢_𝑛−𝐸₂𝑥^∗−𝑧_𝑛−𝑦_𝑛²+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗. (74) Then, we derive 1−𝛼_𝑛𝛾²𝑥_𝑛−𝑢_𝑛²+𝑢_𝑛−𝑧_𝑛²+𝑧_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2+2𝑟𝑥_𝑛−𝑢_𝑛𝑄𝑥_𝑛−𝑄𝑥^∗+2𝜇𝑢_𝑛−𝑧_𝑛𝐸₂𝑢_𝑛−𝐸₂𝑥^∗+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗≤𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+2𝑟𝑥_𝑛−𝑢_𝑛𝑄𝑥_𝑛−𝑄𝑥^∗+2𝜇𝑢_𝑛−𝑧_𝑛𝐸₂𝑢_𝑛−𝐸₂𝑥^∗+2𝜆𝑧_𝑛−𝑦_𝑛𝐸₁𝑧_𝑛−𝐸₁𝑥^∗+𝛽²_𝑛𝑥_𝑛−𝑊_𝑛𝑦_𝑛²+21−𝛼_𝑛𝛾𝛽_𝑛𝑦_𝑛−𝑥^∗𝑥_𝑛−𝑊_𝑛𝑦_𝑛+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥^∗𝑥_𝑛+1−𝑥^∗. (75) By conditions (C1), (C2), (58), (61), and (66), we obtain lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=lim_𝑛→∞𝑢_𝑛−𝑧_𝑛=lim_𝑛→∞𝑧_𝑛−𝑦_𝑛=0. (76) Observe that 𝑊_𝑛𝑦_𝑛−𝑦_𝑛≤𝑊_𝑛𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑧_𝑛+𝑧_𝑛−𝑦_𝑛. (77) By (61) and (76), we have lim_𝑛→∞𝑊_𝑛𝑦_𝑛−𝑦_𝑛=0. (78) Note that 𝑊𝑦_𝑛−𝑦_𝑛≤𝑊𝑦_𝑛−𝑊_𝑛𝑦_𝑛+𝑊_𝑛𝑦_𝑛−𝑦_𝑛. (79) From Lemma 1, we get lim_𝑛→∞𝑊𝑦_𝑛−𝑊_𝑛𝑦_𝑛=0. (80) By (78) and (80), we have lim_𝑛→∞‖𝑊𝑦_𝑛−𝑦_𝑛‖=0. It follows that lim_𝑛→∞‖𝑊𝑥_𝑛−𝑥_𝑛‖=0.
Step 5. We show that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑧,𝑥_𝑛−𝑧⟩≤0, where 𝑧=𝑃_Θ(𝛾𝑓+𝐼−𝐴)𝑧. It is easy to see that 𝑃_Θ(𝛾𝑓+(𝐼−𝐴)) is a contraction of 𝐻 into itself. Indeed, since 0<𝛾<𝛾/𝛼, we have 𝑃_Θ𝛾𝑓+(𝐼−𝐴)𝑥−𝑃_Θ()𝑦≤()()𝑦𝑓(𝑥)𝑦+|+𝛾𝑓+𝐼−𝐴𝛾𝑓+𝐼−𝐴𝑥−𝛾𝑓+𝐼−𝐴≤𝛾−𝑓𝐼−𝐴|𝑥−𝑦≤𝛾𝛼𝑥−𝑦1−𝛾=𝑥−𝑦1−𝛾+𝛾𝛼𝑥−𝑦. (81) Since 𝐻 is complete, there exists a unique fixed point 𝑧∈𝐻 such that 𝑧=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑧). Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛}, such that lim_𝑖→∞𝐴−𝛾𝑓𝑧,𝑧−𝑥_𝑛𝑖=limsup_𝑛→∞𝐴−𝛾𝑓𝑧,𝑧−𝑥_𝑛. (82) Also, since {𝑥_𝑛𝑖} is bounded, there exists a subsequence {𝑥_𝑛𝑖𝑗} of {𝑥_𝑛𝑖} which converges weakly to 𝑤∈𝐶. Without loss of generality, we can assume that 𝑥_𝑛𝑖⇀𝑤. From ‖𝑊𝑥_𝑛−𝑥_𝑛‖→0, we obtain 𝑊𝑥_𝑛𝑖⇀𝑤. Then, by the demiclosed principle of nonexpansive mappings, we obtain 𝑤∈∩^∞_𝑖=1𝐹(𝑇_𝑖).
Next, we show that 𝑤∈MEP(𝐹,𝜑). Since 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛−𝑟𝑄𝑥_𝑛), we obtain 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛≥0,∀𝑦∈𝐶. (83) From (A2), we also have 𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶, (84) and hence, 𝜑𝑦−𝜑𝑢_𝑛𝑖+𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖−𝑟𝑄𝑥_𝑛𝑖𝑟≥𝐹𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (85) For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐻, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. From (85) we have 𝑦_𝑡−𝑢_𝑛𝑖,𝑄𝑦_𝑡≥𝑦_𝑡−𝑢_𝑛𝑖,𝑄𝑦_𝑡−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖−𝑟𝑄𝑥_𝑛𝑖𝑟+𝐹𝑦_𝑡,𝑢_𝑛𝑖=𝑦_𝑡−𝑢_𝑛𝑖,𝑄𝑦_𝑡−𝑄𝑢_𝑛𝑖+𝑦_𝑡−𝑢_𝑛𝑖,𝑄𝑢_𝑛𝑖−𝑄𝑥_𝑛𝑖−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟+𝐹𝑦_𝑡,𝑢_𝑛𝑖. (86) Since ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0, we have ‖𝑄𝑢_𝑛𝑖−𝑄𝑥_𝑛𝑖‖→0. Further, from an inverse-strongly monotonicity of 𝑄, we have ⟨𝑦_𝑡−𝑢_𝑛𝑖,𝑄𝑦_𝑡−𝑄𝑢_𝑛𝑖⟩≥0. So, from (A4), (A5), and the weakly lower semicontinuity of 𝜑,⟨𝑢_𝑛𝑖−𝑥_𝑛𝑖⟩/𝑟→0 and 𝑢_𝑛𝑖→𝑤 weakly, we have 𝑦_𝑡−𝑤,𝑄𝑦_𝑡≥−𝜑𝑦_𝑡+𝜑(𝑤)+𝐹𝑦_𝑡,𝑤. (87) From (A1), (A4), and (87), we also have 0=𝐹𝑦_𝑡,𝑦_𝑡+𝜑𝑦_𝑡−𝜑𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+(1−𝑡)𝐹𝑦_𝑡𝑦+()𝜑(𝑤)𝑦,𝑤+𝑡𝜑1−𝑡−𝜑_𝑡=𝑡𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝐹𝑦_𝑡,𝑤+𝜑(𝑤)−𝜑𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑦_𝑡−𝑤,𝑄𝑦_𝑡=𝑡𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑡𝑦−𝑤,𝑄𝑦_𝑡, (88) and hence, 0≤𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑦−𝑤,𝑄𝑦_𝑡. (89) Letting 𝑡→0, we have, for each 𝑦∈𝐶, 𝐹𝑤,𝑦+𝜑𝑦−𝜑(𝑤)+𝑦−𝑤,𝑄𝑤≥0. (90) This implies that 𝑤∈MEP(𝐹,𝜑).
Lastly, we show that 𝑤∈SQVI(𝐵₁,𝑀₁,𝐵₂,𝑀₂). Since ‖𝑢_𝑛−𝑧_𝑛‖→0 and ‖𝑧_𝑛−𝑦_𝑛‖→0 as 𝑛→∞, we get 𝑢_𝑛−𝑦_𝑛≤𝑢_𝑛−𝑧_𝑛+𝑧_𝑛−𝑦_𝑛, (91) we conclude that ‖𝑢_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. Moreover, by the nonexpansivity of 𝐺 in Lemma 6, we have 𝑦_𝑛−𝐺𝑦_𝑛=𝐽_𝑀1,𝜆𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛−𝜆𝐸₁𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛−𝐺𝑦_𝑛=𝐺𝑢_𝑛−𝐺𝑦_𝑛≤𝑢_𝑛−𝑦_𝑛. (92) Thus, lim_𝑛→∞‖𝑦_𝑛−𝐺(𝑦_𝑛)‖=0. According to Lemma 7, we obtain that 𝑤∈SQVI(𝐵₁,𝑀₁,𝐵₂,𝑀₂). Hence, 𝑤∈Θ. Since 𝑧=𝑃_Θ(𝐼−𝐴+𝛾𝑓)(𝑧), we have limsup_𝑛→∞𝛾𝑓−𝐴𝑧,𝑥_𝑛−𝑧=limsup_𝑖→∞𝛾𝑓−𝐴𝑧,𝑥_𝑛𝑖=−𝑧𝛾𝑓−𝐴𝑧,𝑤−𝑧≤0. (93)
Step 6. We show that {𝑥_𝑛} converges strongly to 𝑧; we compute that 𝑥_𝑛+1−𝑧²=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑧²=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧+𝛽_𝑛𝑥_𝑛+−𝑧1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑧²=𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛−𝑧²+2𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛𝑦_𝑛−𝑧,𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧≤𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛−𝛼_𝑛𝛾𝑦_𝑛−𝑧²+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓𝑥_𝑛−𝐴𝑧+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛−𝑧,𝛾𝑓𝑥_𝑛−𝐴𝑧≤𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+𝛽_𝑛𝑥_𝑛−𝑧+1−𝛽_𝑛−𝛼_𝑛𝛾𝑥_𝑛−𝑧²+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑧)+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛−𝑧,𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑧)+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧≤𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+1−𝛼_𝑛𝛾²𝑥_𝑛−𝑧²+2𝛼_𝑛𝛽_𝑛𝛾𝑥_𝑛−𝑧𝑓𝑥_𝑛−𝑓(𝑧)+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝛾𝑊_𝑛𝑦_𝑛−𝑧𝑓𝑥_𝑛−𝑓(𝑧)+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧≤𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+1−𝛼_𝑛𝛾²𝑥_𝑛−𝑧²+2𝛼_𝑛𝛽_𝑛𝛾𝛼𝑥_𝑛−𝑧²+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝛾𝛼𝑥_𝑛−𝑧²+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧=𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+1−2𝛼_𝑛𝛾+𝛼²_𝑛𝛾²+2𝛼_𝑛𝛾𝛼−2𝛼²_𝑛×𝑥𝛾𝛾𝛼_𝑛−𝑧²+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛(𝑧)≤−𝑧,𝛾𝑓−𝐴𝑧1−𝛼_𝑛2𝛾−𝛼_𝑛𝛾²−2𝛾𝛼+2𝛼_𝑛𝛾𝛾𝛼𝑥_𝑛−𝑧²+𝛼²_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑧²+2𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧+2𝛼_𝑛1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛𝑦_𝑛(𝑧)≤−𝑧,𝛾𝑓−𝐴𝑧1−𝛼_𝑛2𝛾−𝛼_𝑛𝛾²−2𝛾𝛼+2𝛼_𝑛𝛾𝛾𝛼𝑥_𝑛−𝑧²+𝛼_𝑛𝜎_𝑛, (94) where 𝜎_𝑛=𝛼_𝑛‖𝛾𝑓(𝑥_𝑛)−𝐴𝑧‖²+2𝛽_𝑛⟨𝑥_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧⟩+2(1−𝛽_𝑛−𝛼_𝑛𝛾)⟨𝑊_𝑛𝑦_𝑛−𝑧,𝛾𝑓(𝑧)−𝐴𝑧⟩. It is easy to see that limsup_𝑛→∞𝜎_𝑛≤0. Applying Lemma 3 to (94), we conclude that 𝑥_𝑛→𝑧. This completes the proof.

Next, the following example shows that all conditions of Theorem 8 are satisfied.

Example 9. For instance, let 𝛼_𝑛=1/2(𝑛+1), let 𝛽_𝑛=(2𝑛+2)/2(2𝑛), let 𝜆_𝑛=𝑛/(𝑛+1). Then, we will show that the sequences {𝛼_𝑛} satisfy condition (C1). Indeed, we take 𝛼_𝑛=1/2(𝑛+1); then, we have ^∞_𝑛=1𝛼_𝑛=^∞_𝑛=112(𝑛+1)=∞,lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞12(𝑛+1)=0. (95)
We will show that the sequences {𝛽_𝑛} satisfy condition (C2). Indeed, we set 𝛽_𝑛=(2𝑛+2)/2(2𝑛)=(1/2)+(1/2𝑛). It is easy to see that 0<liminf_𝑛→∞𝛽_𝑛<limsup_𝑛→∞𝛽_𝑛<1.
Next, we will show the condition (C3) is satisfied. We take 𝜆_𝑛=𝑛/(𝑛+1); then we compute lim_𝑛→∞𝜆_𝑛−𝜆_𝑛−1=lim_𝑛→∞𝑛−𝑛+1𝑛−1(𝑛−1)+1=lim_𝑛→∞𝑛(𝑛)−(𝑛−1)(𝑛+1)(𝑛+1)𝑛=lim_𝑛→∞𝑛²−𝑛²+1(𝑛+1)𝑛=lim_𝑛→∞1𝑛().𝑛+1 (96) Then, we have lim_𝑛→∞|𝜆_𝑛+1−𝜆_𝑛|=0. The sequence {𝜆_𝑛} satisfies condition (C3).

Using Theorem 8, we obtain the following corollaries.

Corollary 10. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuos and convex function. Let 𝑇_𝑖∶𝐶→𝐶 be nonexpansive mappings for all 𝑖=1,2,3,…, such that Θ∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)∩𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝑀𝐸𝑃(𝐹,𝜑)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸₁,𝐸₂ be 𝛿,𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Assume that either 𝐵₁ or 𝐵₂ holds and let 𝑊_𝑛 be the 𝑊-mapping defined by (31). Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝑊_𝑛𝑦_𝑛,∀𝑛≥0, (97) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝑓+𝐼)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (6).

Proof. Taking 𝛾≡1 and 𝐴≡𝐼 in Theorem 8, we can conclude the desired conclusion easily.

Corollary 11. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuos and convex function. Let 𝑇_𝑖∶𝐶→𝐶 be a nonexpansive mappings for all 𝑖=1,2,3,…, such that Θ∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)∩𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝑀𝐸𝑃(𝐹,𝜑)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝐸₁,𝐸₂ be 𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝐴 be strongly positive bounded linear self-adjoint on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼, let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Assume that either 𝐵₁ or 𝐵₂ holds and let 𝑊_𝑛 be the 𝑊-mapping defined by (31). Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛,∀𝑛≥0, (98) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,∞) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (6).

Proof. Taking 𝑄≡0 in Theorem 8, we can conclude the desired conclusion easily.

Corollary 12. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuos and convex function such that Θ∶=𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝑀𝐸𝑃(𝐹,𝜑)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸₁,𝐸₂ be 𝛿,𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝐴 be a strongly positive bounded linear self-adjoint on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼, let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Assume that either 𝐵₁ or 𝐵₂ holds, let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0, (99) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (7), which is the unique solution of the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Θ, (100)

Proof. Taking 𝑊_𝑛≡𝐼 in Theorem 8, we can conclude the desired conclusion easily.

Corollary 13. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5). Let 𝑇_𝑖∶𝐶→𝐶 be nonexpansive mappings for all 𝑖=1,2,3,…, such that Θ∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)∩𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸₁,𝐸₂ be 𝛿,𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝐴 be a strongly positive bounded linear self-adjoint on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼, let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Assume that either 𝐵₁ or 𝐵₂ holds and let 𝑊_𝑛 be the 𝑊-mapping defined by (31). Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑦_𝑛,∀𝑛≥0, (101) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (6).

Proof. Taking 𝜑≡0 in Theorem 8, we can conclude the desired conclusion easily.

Corollary 14. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) such that Θ∶=𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸₁,𝐸₂ be 𝛿,𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝐴 be a strongly positive bounded linear self-adjoint on 𝐻 with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼, let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Assume that either 𝐵₁ or 𝐵₂ holds, let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0, (102) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (6).

Proof. Taking 𝜑≡0 and 𝑊_𝑛≡𝐼 in Theorem 8, we can conclude the desired conclusion easily.

Corollary 15. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) such that Θ∶=𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸₁,𝐸₂ be 𝛿,𝜂₁,𝜂₂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸₂𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝑦_𝑛,∀𝑛≥0, (103) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂₁), 𝜇∈(0,2𝜂₂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝑓+𝐼)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸₂𝑥^∗) is solution to the problem (6).

Proof. Taking 𝛾≡1, 𝐴≡𝐼, 𝜑≡0, and 𝑊_𝑛≡𝐼 in Theorem 8, we can conclude the desired conclusion easily.

Corollary 16. Let 𝐶 be a nonempty closed convex subset of a real Hilbert Space 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into real numbers ℝ satisfying (A1)–(A5) such that Θ∶=𝑆𝑄𝑉𝐼(𝐵₁,𝑀₁,𝐵₂,𝑀₂)∩𝐸𝑃(𝐹)≠∅. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝛼∈(0,1) and let 𝑄,𝐸 be 𝛿,𝜂-inverse-strongly monotone mapping of 𝐶 into 𝐻. Let 𝑀₁,𝑀₂∶𝐻→2^𝐻 be a maximal monotone mapping. Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢_𝑛} be sequences generated by 𝑥₀∈𝐶, 𝑢_𝑛∈𝐶, and 𝐹𝑢_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝑄𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝐽_𝑀2,𝜇𝑢_𝑛−𝜇𝐸𝑢_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜆𝑧_𝑛−𝜆𝐸𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛−𝛼_𝑛𝑦_𝑛,∀𝑛≥0, (104) where {𝛼_𝑛} and {𝛽_𝑛}⊂(0,1), 𝜆∈(0,2𝜂), 𝜇∈(0,2𝜂), and 𝑟∈(0,2𝛿) satisfy the following conditions:(C1)∑^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, (C3)lim_𝑛→∞|𝜆_𝑛,𝑖−𝜆_{𝑛−1,𝑖}|=0, ∀𝑖=1,2,…,𝑁.Then, {𝑥_𝑛} converges strongly to 𝑥^∗∈Θ, where 𝑥^∗=𝑃_Θ(𝑓+𝐼)(𝑥^∗), 𝑃_Θ is the metric projection of 𝐻 onto Θ and (𝑥^∗,𝑦^∗), where 𝑦^∗=𝐽_𝑀2,𝜇(𝑥^∗−𝜇𝐸𝑥^∗) is solution to the problem (6).

Proof. Taking 𝐸₁=𝐸₂=𝐸 in Corollary 15, we can conclude the desired conclusion easily.

Conflict of Interests

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

Acknowledgments

The first author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under Project: Theoretical and Computational Fixed Points for Optimization Problems, NRU no. 57000621). The second author was supported by the Commission on Higher Education, the Thailand Research Fund, and Rajamangala University of Technology Lanna Chiangrai under Grant no. MRG5680157 during the preparation of this paper.