A New General System of Generalized Nonlinear Mixed Composite-Type Equilibria and Fixed Point Problems with an Application to Minimization Problems

Abstract

We introduce a new general system of generalized nonlinear mixed composite-type equilibria and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. Finally, we also give a numerical example which supports our results. The results obtained in this paper extend the recent ones announced by many others.

1. Introduction

Let 𝐻 be a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐶 be a nonempty closed and convex subset of 𝐻. Let 𝜙∶𝐶→ℝ be a real-valued function, where ℝ is the set of real numbers. Let 𝐺,𝐵∶𝐶→𝐻 be two nonlinear mappings and Θ∶𝐻×𝐶×𝐶→ℝ be an equilibrium-like function, that is, Θ(𝑤,𝑢,𝑣)=Θ(𝑤,𝑣,𝑢)=0 for all (𝑤,𝑢,𝑣)∈𝐻×𝐶×𝐶. We consider the following new generalized equilibrium problem: find 𝑥^∗∈𝐶 such that Θ𝐺𝑥^∗,𝑥^∗,𝑦+𝜙𝑦−𝜙𝑥^∗+𝐵𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (1.1) The set of solutions of the problem (1.1) is denoted by GEP(𝐶,𝐺,Θ,𝜙,𝐵). As special cases of the problem (1.1), we have the following results. (1)If 𝐵=0, then problem (1.1) reduces to the following generalized equilibrium problem: find 𝑥^∗∈𝐶 such that Θ𝐺𝑥^∗,𝑥^∗,𝑦+𝜙𝑦−𝜙𝑥^∗≥0,∀𝑦∈𝐶, (1.2) which was considered by Cho et al. [1] for more details. The set of solutions of the problem (1.1) is denoted by GEP(𝐶,𝐺,Θ,𝜙).(2)If 𝐵=0 and Θ(𝑤,𝑢,𝑣)=𝐹(𝑢,𝑣), where 𝐹∶𝐶×𝐶→ℝ is an equilibrium bifunction, then problem (1.1) reduces to the following mixed equilibrium problem: find 𝑥^∗∈𝐶 such that 𝐹𝑥^∗,𝑦+𝜙𝑦−𝜙𝑥^∗≥0,∀𝑦∈𝐶, (1.3) which was considered by Ceng and Yao [2] for more details. The set of solutions of the problem (1.3) is denoted by MEP(𝐹,𝜙).(3)If 𝐵=0, 𝜙=0 and Θ(𝑤,𝑢,𝑣)=𝐹(𝑢,𝑣) where 𝐹∶𝐶×𝐶→ℝ is an equilibrium bifunction, then problem (1.1) reduces to the following equilibrium problem: find 𝑥^∗∈𝐶 such that 𝐹𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (1.4) The set of solutions of problem (1.4) is denoted by EP(𝐹).(4)If Θ=0, 𝜙=0, then problem (1.1) reduces to the following classical variational inequality problem: find 𝑥^∗∈𝐶 such that 𝐵𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (1.5) The set of solutions of the problem (1.5) is denoted by VI(𝐶,𝐵).

In brief, for an appropriate choice of the mapping 𝐺, the function 𝜙, and the convex set 𝐶, one can obtain a number of the various classes of equilibrium problems as special cases. In particular, the equilibrium problems (1.4) which were introduced by Blum and Oettli [3] and Noor and Oettli [4] in 1994 have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. In [3, 4], it has been shown that equilibrium problems include variational inequalities, fixed point, minimax problems, Nash equilibrium problems in noncooperative games, and others as special cases. This means that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Hence collectively, equilibrium problems cover a vast range of applications. Related to the equilibrium problems, we also have the problems of finding the fixed points of the nonlinear mappings, which is the subject of current interest in functional analysis. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of nonlinear mappings (e.g., see [5–18] and the references therein).

Recall the following definitions.

Definition 1.1. The mapping 𝑆∶𝐶→𝐶 is said to be (1)nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.6) (2)𝐿-Lipschitzian if there exists a constant 𝐿>0 such that 𝑆𝑥−𝑆𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.7)(3)𝑘-strict pseudocontraction [19] if there exists a constant 𝑘∈[0,1) such that 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶, (1.8)(4)pseudocontractive if 𝑆𝑥−𝑆𝑦²≤𝑥−𝑦²+(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶. (1.9)
Clearly, the class of strict pseudocontractions falls into the one between classes of nonexpansive mappings and pseudocontractions. It is easy to see that (1.8) is equivalent to 𝑆𝑥−𝑆𝑦,𝑥−𝑦≤𝑥−𝑦²−1−𝑘2(𝐼−𝑆)𝑥−(𝐼−𝑆)𝑦²,∀𝑥,𝑦∈𝐶, (1.10) that is, 𝐼−𝑆 is (1−𝑘)/2-inverse-strongly monotone. From [19], we know that if 𝑆 is a 𝑘-strictly pseudocontractive mapping, then 𝑆 is Lipschitz continuous with constant (3−𝑘)/(1−𝑘), that is, ‖𝑆𝑥−𝑆𝑦‖≤(3−𝑘)/(1−𝑘)‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶.
In this paper, we use Fix(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥} to denote the set of fixed points of 𝑆.

Definition 1.2. A countable family of mapping {𝑇_𝑛}^∞_𝑛=1∶𝐶→𝐶 is called a family of 𝑘-strict pseudocontraction mappings if there exists a constant 𝑘∈[0,1) such that 𝑇_𝑛𝑥−𝑇_𝑛𝑦²≤𝑥−𝑦²+𝑘𝐼−𝑇_𝑛𝑥−𝐼−𝑇_𝑛𝑦²,∀𝑥,𝑦∈𝐶,𝑛≥1. (1.11) On the other hand, let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹₁,𝐹₂,𝐹₃∶𝐶×𝐶→ℝ be three bifunctions and let Ψ₁,Ψ₂,Ψ₃,Φ₁,Φ₂,Φ₃∶𝐶→𝐻 be six nonlinear mappings and let 𝜑₁,𝜑₂,𝜑₃∶𝐶→ℝ be three functions. We consider the following problem of finding (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐶×𝐶×𝐶 such that 𝜇₁𝐹₁𝑥^∗,𝑥+𝜇₁Ψ₁+Φ₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥𝜇₁𝜑₁𝑥^∗−𝜇₁𝜑₁(𝑥)𝜇,∀𝑥∈𝐶,₂𝐹₂𝑦^∗,𝑦+𝜇₂Ψ₂+Φ₂𝑧^∗+𝑦^∗−𝑧^∗,𝑦−𝑦^∗≥𝜇₂𝜑₂𝑦^∗−𝜇₂𝜑₂𝑦𝜇,∀𝑦∈𝐶,₃𝐹₃𝑧^∗,𝑧+𝜇₃Ψ₃+Φ₃𝑥^∗+𝑧^∗−𝑥^∗,𝑧−𝑧^∗≥𝜇₃𝜑₃𝑧^∗−𝜇₃𝜑₃(𝑧),∀𝑧∈𝐶, (1.12) which is called a new general system of generalized nonlinear mixed composite-type equilibria, where 𝜇_𝑖>0 for all 𝑖=1,2,3. Next, we present some special cases of problem (1.12) as follows. (1)If Ψ_𝑖=Ψ, Φ_𝑖=Φ, 𝐹_𝑖=𝐹, and 𝜑_𝑖=𝜑 for all 𝑖=1,2,3, then the problem (1.12) reduces to the following new general system of generalized nonlinear mixed composite-type equilibria: find (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐶×𝐶×𝐶 such that 𝜇₁𝐹𝑥^∗,𝑥+𝜇₁(Ψ+Φ)𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥𝜇₁𝜑𝑥^∗−𝜇₁𝜑(𝑥)𝜇,∀𝑥∈𝐶,₂𝐹𝑦^∗,𝑦+𝜇₂(Ψ+Φ)𝑧^∗+𝑦^∗−𝑧^∗,𝑦−𝑦^∗≥𝜇₂𝜑𝑦^∗−𝜇₂𝜑𝑦𝜇,∀𝑦∈𝐶,₃𝐹𝑧^∗,𝑧+𝜇₃(Ψ+Φ)𝑥^∗+𝑧^∗−𝑥^∗,𝑧−𝑧^∗≥𝜇₃𝜑𝑧^∗−𝜇₃𝜑(𝑧),∀𝑧∈𝐶, (1.13) where 𝜇_𝑖>0 for all 𝑖=1,2.(2)If 𝐹₃=0, Ψ₃=Φ₃=0, 𝜇₃=0, and 𝑧^∗=𝑥^∗, then the problem (1.12) reduces to the following general system of generalized nonlinear mixed composite-type equilibria: find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐹₁𝑥^∗,𝑥+𝜇₁Ψ₁+Φ₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥𝜇₁𝜑₁𝑥^∗−𝜇₁𝜑₁(𝑥)𝜇,∀𝑥∈𝐶,₂𝐹₂𝑦^∗,𝑦+𝜇₂Ψ₂+Φ₂𝑥^∗+𝑦^∗−𝑥^∗,𝑦−𝑦^∗≥𝜇₂𝜑₂𝑦^∗−𝜇₂𝜑₂𝑦,∀𝑦∈𝐶, (1.14) which was introduced and considered by Ceng et al. [20], where 𝜇_𝑖>0 for all 𝑖=1,2.(3)If 𝐹₃=0, Ψ₃=Φ₃=0, 𝜇₃=0, and 𝜑_𝑖=0 for all 𝑖=1,2,3 and 𝑧^∗=𝑥^∗, then the problem (1.12) reduces to the following a general system of generalized equilibria: find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝐹₁𝑥^∗,𝑥+Ψ₁𝑦^∗,𝑥−𝑥^∗+1𝜇₁𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝐹≥0,∀𝑥∈𝐶,₂𝑦^∗,𝑦+Ψ₂𝑥^∗,𝑦−𝑦^∗+1𝜇₂𝑦^∗−𝑥^∗,𝑦−𝑦^∗≥0,∀𝑦∈𝐶, (1.15) which was introduced and considered by Ceng and Yao [21], where 𝜇_𝑖>0 for all 𝑖=1,2.(4)If 𝐹_𝑖=𝐹, Ψ_𝑖=Ψ, and Φ_𝑖=Φ, 𝜑_𝑖=𝜑, for all 𝑖=1,2,3, then the problem (1.12) reduces to the following generalized mixed equilibrium problem with perturbed mapping: find 𝑥^∗∈𝐶 such that 𝐹𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗+(Ψ+Φ)𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶, (1.16) which was introduced and considered by Hu and Ceng [22].(5)If 𝐹_𝑖=0, Φ_𝑖=0, and 𝜑_𝑖=0 for all 𝑖=1,2,3, then the problem (1.12) reduces to the following general system of variational inequalities: find (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐶×𝐶×𝐶 such that 𝜇₁Ψ₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂Ψ₂𝑧^∗+𝑦^∗−𝑧^∗,𝑦−𝑦^∗𝜇≥0,∀𝑦∈𝐶,₃Ψ₃𝑥^∗+𝑧^∗−𝑥^∗,𝑧−𝑧^∗≥0,∀𝑧∈𝐶, (1.17) which was introduced and considered by Kumam et al. [23], where 𝜇_𝑖>0 for all 𝑖=1,2,3. (6)If 𝐹_𝑖=0, Φ_𝑖=0, 𝜑_𝑖=0 for all 𝑖=1,2,3, Ψ₃=0 and 𝑧^∗=𝑥^∗, then the problem (1.12) reduces to the following general system of variational inequalities: find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁Ψ₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂Ψ₂𝑥^∗+𝑦^∗−𝑥^∗,𝑦−𝑦^∗≥0,∀𝑦∈𝐶, (1.18) which was introduced and considered by Ceng et al. [24], where 𝜇_𝑖>0 for all 𝑖=1,2.

In 2010, Cho et al. [1] introduced an iterative method for finding a common element of the set of solutions of generalized equilibrium problems (1.2), the set of solutions for a systems of nonlinear variational inequalities problems (1.18), and the set of fixed points of nonexpansive mappings in Hilbert spaces. Ceng and Yao [21] introduced and considered a relaxed extragradient-like method for finding a common element of the set of solutions of a system of generalized equilibria, the set of fixed points of a strictly pseudocontractive mapping, and the set of solutions of a equilibrium problem in a real Hilbert space and obtained a strong convergence theorem. The result of Ceng and Yao [21] included, as special cases, the corresponding ones of S. Takahashi and W. Takahashi [10], Ceng et al. [24], Peng and Yao [25], and Yao et al. [26].

Motivated and inspired by the works in the literature, we introduce a new general system of generalized nonlinear mixed composite-type equilibria (1.12) and propose a new iterative scheme for finding a common element of the set of solutions of a generalized equilibrium problem, a general system of generalized nonlinear mixed composite-type equilibria, and the set of fixed points of a countable family of strict pseudocontraction mappings. Furthermore, we prove the strong convergence theorem of the purposed iterative scheme in a real Hilbert space. As applications, we apply our results to solve a certain minimization problem related to a strongly positive bounded linear operator. The results presented in this paper extend the recent results of Cho et al. [1], Ceng and Yao [21], Ceng et al.[20], and many authors.

2. Preliminaries

A bounded linear operator 𝐴 is said to be strongly positive, if there exists a constant 𝛾>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (2.1) Recall that, a mapping 𝑓∶𝐶→𝐶 is said to be contractive if there exists a constant 𝛼∈(0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.2) A mapping 𝐴∶𝐶→𝐻 is called 𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛼𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (2.3) Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. 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. (2.4)𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃_𝐶 is nonexpansive (see [27]) and for 𝑥∈𝐻, 𝑧=𝑃_𝐶𝑥⟺𝑥−𝑧,𝑦−𝑧≤0,∀𝑦∈𝐶. (2.5) Let 𝜙∶𝐶→ℝ be a real-valued function, 𝐺∶𝐶→𝐻 be a mapping and Θ∶𝐻×𝐶×𝐶→ℝ be an equilibrium-like function. Let 𝑟 be a positive real number. For all 𝑥∈𝐶, we consider the following problem. Find 𝑦∈𝐶 such that Θ(𝑧)𝑦+1𝐺𝑥,𝑦,𝑧+𝜙−𝜙𝑟𝑦−𝑥,𝑧−𝑦≥0,∀𝑧∈𝐶, (2.6) which is known as the auxiliary generalized equilibrium problem.

Let 𝑇^(𝑟)∶𝐻→𝐶 be the mapping such that, for all 𝑥∈𝐻, 𝑇^(𝑟) is the solution set of the auxiliary problem (2.6), that is, 𝑇^(𝑟)(𝑥)=(𝑧)𝑦+1𝑦∈𝐶∶Θ𝐺𝑥,𝑦,𝑧+𝜙−𝜙𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑧∈𝐶. (2.7) Then, we will assume the Condition (Δ) [28] as follows: (a)𝑇^(𝑟) is single-valued;(b)𝑇^(𝑟) is nonexpansive;(c)Fix(𝑇^(𝑟))=GEP(𝐶,𝐺,Θ,𝜙).

Notice that the examples of showing the sufficient conditions for the existence of the condition (Δ) can be found in [6].

Throughout this paper, we assume that a bifunction 𝐹∶𝐶×𝐶→ℝ and 𝜑∶𝐶→ℝ is a lower semicontinuous and convex function satisfy the following conditions: (H1)𝐹(𝑥,𝑥)=0, ∀𝑥∈𝐶;(H2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0, ∀𝑥,𝑦∈𝐶;(H3)for all 𝑦∈𝐶,  𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous;(H4)for all 𝑥∈𝐶,  𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous;(A1)for all 𝑥∈𝐻 and 𝑟>0, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑦_𝑥∈𝐶 such that for all 𝑧∈𝐶⧵𝐷_𝑥, 𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0; (2.8)(A2)𝐶 is a bounded set.

In order to prove our main results in the next section, we need the following lemmas.

Lemma 2.1 (see [29]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert sapce 𝐻. Let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying condition (𝐻1)–(𝐻4) and let 𝜑∶𝐶→ℝ be a lower semicontinuous and convex function. For 𝑟>0 and 𝑥∈𝐻 define a mapping 𝑇_𝑟^(𝐹,𝜑)∶𝐻→𝐶 follows 𝑇_𝑟^(𝐹,𝜑)(𝑥)=(𝑧)𝑦+1𝑦∈𝐶∶𝐹𝑦,𝑧+𝜑−𝜑𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑧∈𝐶. (2.9) Assume that either (𝐴1) or (𝐴2) holds, then the following statements hold (i)𝑇_𝑟^(𝐹,𝜑)≠∅ for all 𝑥∈𝐻 and 𝑇_𝑟^(𝐹,𝜑) is single-valued;(ii)𝑇_𝑟^(𝐹,𝜑) is firmly nonexpansive, that is, for all 𝑥,𝑦∈𝐻, 𝑇_𝑟^(𝐹,𝜑)𝑥−𝑇_𝑟^(𝐹,𝜑)𝑦²≤𝑇_𝑟^(𝐹,𝜑)𝑥−𝑇_𝑟^(𝐹,𝜑)𝑦,𝑥−𝑦; (2.10)(iii)Fix(𝑇_𝑟^(𝐹,𝜑))=MEP(𝐹,𝜑);(iv)MEP(𝐹,𝜑) is closed and convex.

Remark 2.2. If 𝜑=0, then 𝑇_𝑟^(𝐹,𝜑) is rewritten as 𝑇^𝐹_𝑟 (see [21, Lemma 2.1] for more details).

Lemma 2.3 (see [30]). Let {𝑥_𝑛} and {𝑙_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)𝑙_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞(‖𝑙_𝑛+1−𝑙_𝑛‖−‖𝑥_𝑛+1−𝑥_𝑛‖)≤0. Then, lim_𝑛→∞‖𝑙_𝑛−𝑥_𝑛‖=0.

Lemma 2.4 (see [31]). Let 𝐻 be a real Hilbert space. Then the following inequalities hold. (i)‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖²,  ∀𝑥,𝑦∈𝐻and 𝜆∈[0,1].(ii)‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩,∀𝑥,𝑦∈𝐻.

Definition 2.5 (see [32]). Let {𝑇_𝑛} be a sequence of mappings from a subset 𝐶 of a real Hilbert space 𝐻 into itself. We say that {𝑇_𝑛} satisfies the PT-condition if lim_{𝑘,𝑙→∞}𝜌^𝑘_𝑙=0, (2.11) where 𝜌^𝑘_𝑙=sup_𝑧∈𝐶{‖𝑇_𝑘𝑧−𝑇_𝑙𝑧‖}<∞, for all 𝑘,𝑙∈ℕ.

Lemma 2.6 (see [32]). Suppose that {𝑇_𝑛} satisfies the PT-condition such that (i)for each 𝑥∈𝐶, {𝑇_𝑛} is converse strongly to some point in 𝐶(ii)let the mapping 𝑇∶𝐶→𝐶 defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶. Then, lim_𝑛→∞sup_𝜔∈𝐶‖𝑇𝜔−𝑇_𝑛𝜔‖=0.

Lemma 2.7 (see [33]). Let 𝐶 be a closed and convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛∶𝑛∈ℕ} be a sequence of nonexpansive mappings on 𝐶. Suppose ⋂^∞_𝑛=1Fix(𝑇_𝑛) is nonempty. Let {𝛾_𝑛} be a sequence of positive numbers with ∑^∞_𝑛=1𝛾_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=∑^∞_𝑛=1𝛾_𝑛𝑇_𝑛𝑥 for all 𝑥∈𝐶 is well defined, nonexpansive, and Fix(𝑆)=⋂^∞_𝑛=1Fix(𝑇_𝑛) holds.

Lemma 2.8 (see [19]). Let 𝑇∶𝐶→𝐻 be a 𝑘-strict pseudocontraction. Define 𝑆∶𝐶→𝐻 by 𝑆𝑥=𝛿𝑥+(1−𝛿)𝑇𝑥 for each 𝑥∈𝐶. Then, as 𝛿∈[𝑘,1), S is nonexpansive such that Fix(𝑆)=Fix(𝑇).

Lemma 2.9 (see [34]). Let 𝐶 be a closed and convex subset of a real Hilbert space 𝐻 and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping. then, the mapping 𝐼−𝑆 is demiclosed. That is, if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑧 and (𝐼−𝑆)𝑥_𝑛→𝑦, then (𝐼−𝑆)𝑧=𝑦.

Lemma 2.10 (see [35]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜎_𝑛𝑎_𝑛+𝛿_𝑛, (2.12) where {𝜎_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (i)∑^∞_𝑛=0𝜎_𝑛=∞;(ii)limsup_𝑛→∞(𝛿_𝑛/𝜎_𝑛)≤0 or ∑^∞_𝑛=0|𝛿_𝑛|<∞.
then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.11. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let mappings Ψ,Φ∶𝐶→𝐻 be 𝛽-inverse-strongly monotone and 𝛾-inverse-strongly monotone, respectively. Then, we have 𝐼−𝜇(Ψ+Φ)𝑥−𝐼−𝜇(Ψ+Φ)𝑦²≤𝑥−𝑦²+2𝜇𝜇−𝛽Ψ𝑥−Ψ𝑦²+2𝜇𝜇−𝛾Φ𝑥−Φ𝑦², (2.13) where 𝜇>0. In particular, if 𝜇∈(0,min{𝛽,𝛾}), then 𝐼−𝜇(Ψ+Φ) is nonexpansive.

Proof. From Lemma 2.4(i), for all 𝑥,𝑦∈𝐶, we have 𝐼−𝜇(Ψ+Φ)𝑥−𝐼−𝜇(Ψ+Φ)𝑦²=𝑥−𝑦−𝜇(Ψ+Φ)𝑥−(Ψ+Φ)𝑦²=12+1𝑥−𝑦−2𝜇Ψ𝑥−Ψ𝑦2𝑥−𝑦−2𝜇Φ𝑥−Φ𝑦²≤12𝑥−𝑦−2𝜇Ψ𝑥−Ψ𝑦²+12𝑥−𝑦−2𝜇Φ𝑥−Φ𝑦²=12𝑥−𝑦²−4𝜇𝑥−𝑦,Ψ𝑥−Ψ𝑦+4𝜇²Ψ𝑥−Ψ𝑦²+12𝑥−𝑦²−4𝜇𝑥−𝑦,Φ𝑥−Φ𝑦+4𝜇²Φ𝑥−Φ𝑦²≤12𝑥−𝑦²+4𝜇𝜇−𝛽Ψ𝑥−Ψ𝑦²+12𝑥−𝑦²+4𝜇𝜇−𝛾Φ𝑥−Φ𝑦²=𝑥−𝑦²+2𝜇𝜇−𝛽Ψ𝑥−Ψ𝑦²+2𝜇𝜇−𝛾Φ𝑥−Φ𝑦². (2.14) It is clear that, if 0<𝜇≤min{𝛽,𝛾}, then 𝐼−𝜇(Ψ+Φ) is nonexpansive. This completes the proof.

Lemma 2.12. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let mappings Ψ_𝑖,Φ_𝑖∶𝐶→𝐻(𝑖=1,2,3) be 𝛽_𝑖-inverse-strongly monotone and 𝛾_𝑖-inverse-strongly monotone, respectively. Let 𝑄∶𝐶→𝐶 be the mapping defined by 𝑄𝑥=𝑇^{(𝐹1,𝜑1)}_𝜇1𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥,∀𝑥∈𝐶. (2.15) If 𝜇_𝑖∈(0,min{𝛽_𝑖,𝛾_𝑖})(𝑖=1,2,3), then 𝑄 is nonexpansive.

Proof. From Lemma 2.11, for all 𝑥,𝑦∈𝐶, we have =𝑇𝑄𝑥−𝑄𝑦_{(𝐹1,𝜑1)𝜇1}𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝑇^{(𝐹1,𝜑1)}_𝜇1𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦≤𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥−𝜇₃Ψ₃+Φ₃𝑥−𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑦−𝜇₃Ψ₃+Φ₃𝑦=𝑇^{(𝐹2,𝜑2)}_𝜇2𝐼−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑥−𝑇^{(𝐹2,𝜑2)}_𝜇2𝐼−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑦≤𝐼−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑥−𝐼−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑦≤𝑇^{(𝐹3,𝜑3)}_𝜇3𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑥−𝑇^{(𝐹3,𝜑3)}_𝜇3𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑦≤𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑥−𝐼−𝜇₃Ψ₃+Φ₃𝐼−𝜇₁Ψ₁+Φ₁𝑦≤𝐼−𝜇₁Ψ₁+Φ₁𝑥−𝐼−𝜇₁Ψ₁+Φ₁𝑦≤𝑥−𝑦, (2.16) which implies that 𝑄∶𝐶→𝐶 is nonexpansive. This completes the proof.

Lemma 2.13. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹_𝑖∶𝐶×𝐶→ℝ  (𝑖=1,2,3) be a bifunction satisfying conditions (𝐻1)–(𝐻4) and let Ψ_𝑖,Φ_𝑖∶𝐶→𝐻(𝑖=1,2,3) be a nonlinear mapping. Suppose that 𝜇_𝑖(𝑖=1,2,3) be a real positive number. Let 𝜑_𝑖∶𝐶→ℝ(𝑖=1,2,3) be a lower semicontinuous and convex function. Assume that either condition (𝐴1) or (𝐴2) holds. Then, for (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐶×𝐶×𝐶 is a solution of the problem (1.12) if and only if 𝑥^∗∈Fix(𝑄), 𝑦^∗=𝑇^{(𝐹2,𝜑2)}_𝜇2(𝑧^∗−𝜇₂(Ψ₂+Φ₂)𝑧^∗) and 𝑧^∗=𝑇^{(𝐹3,𝜑3)}_𝜇3(𝑥^∗−𝜇₃(Ψ₃+Φ₃)𝑥^∗), where 𝑄 is the mapping defined as in Lemma 2.12.

Proof. Let (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐶×𝐶×𝐶 be a solution of the problem (1.12). Then, we have 𝜇₁𝐹₁𝑥^∗,𝑥+𝜇₁Ψ₁+Φ₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗≥𝜇₁𝜑₁𝑥^∗−𝜇₁𝜑₁(𝑥)𝜇,∀𝑥∈𝐶,₂𝐹₂𝑦^∗,𝑦+𝜇₂Ψ₂+Φ₂𝑧^∗+𝑦^∗−𝑧^∗,𝑦−𝑦^∗≥𝜇₂𝜑₂𝑦^∗−𝜇₂𝜑₂𝑦𝜇,∀𝑦∈𝐶,₃𝐹₃𝑧^∗,𝑧+𝜇₃Ψ₃+Φ₃𝑥^∗+𝑧^∗−𝑥^∗,𝑧−𝑧^∗≥𝜇₃𝜑₃𝑧^∗−𝜇₃𝜑₃(𝑧),∀𝑧∈𝐶, (2.17)⇔𝑥^∗=𝑇^{(𝐹1,𝜑1)}_𝜇1𝑦^∗−𝜇₁Ψ₁+Φ₁𝑦^∗,𝑦^∗=𝑇^{(𝐹2,𝜑2)}_𝜇2𝑧^∗−𝜇₂Ψ₂+Φ₂𝑧^∗,𝑧^∗=𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗, (2.18)⇔𝑥^∗=𝑇^{(𝐹1,𝜑1)}_𝜇1𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗. (2.19) This completes the proof.

Corollary 2.14 (see [20]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹_𝑖∶𝐶×𝐶→ℝ(𝑖=1,2) be a bifunction satisfying conditions (𝐻1)–(𝐻4) and let Ψ_𝑖,Φ_𝑖∶𝐶→𝐻(𝑖=1,2) be a nonlinear mapping. Suppose that 𝜇_𝑖(𝑖=1,2) be a real positive number. Let 𝜑_𝑖∶𝐶→ℝ(𝑖=1,2) be a lower semicontinuous and convex function. Assume that either condition (𝐴1) or (𝐴2) holds. Then, for (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of the problem (1.14) if and only if 𝑥^∗∈Fix(𝑄), 𝑦^∗=𝑇^{(𝐹2,𝜑2)}_𝜇2(𝑥^∗−𝜇₂(Ψ₂+Φ₂)𝑥^∗), where 𝐺 is the mapping defined by 𝑄𝑥=𝑇^{(𝐹1,𝜑1)}_𝜇1𝑇^{(𝐹2,𝜑2)}_𝜇2𝑥−𝜇₂Ψ₂+Φ₂𝑥−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑥−𝜇₂Ψ₂+Φ₂𝑥,∀𝑥∈𝐶. (2.20)

Corollary 2.15 (see [21]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹_𝑖∶𝐶×𝐶→ℝ(𝑖=1,2) be a bifunction satisfying conditions (𝐻1)–(𝐻4) and let Ψ_𝑖∶𝐶→𝐻(𝑖=1,2) be a nonlinear mapping. Suppose that 𝜇_𝑖(𝑖=1,2) is a real positive number. Assume that either condition (𝐴1) or (𝐴2) holds. Then, for (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of the problem (1.15) if and only if 𝑥^∗∈Fix(𝑄), 𝑦^∗=𝑇^𝐹2_𝜇2(𝑥^∗−𝜇₂Ψ₂𝑥^∗), where 𝑄 is the mapping defined by 𝑄𝑥=𝑇^𝐹1_𝜇1𝑇^𝐹2_𝜇2𝑥−𝜇₂Ψ₂𝑥−𝜇₁Ψ₁𝑇^𝐹2_𝜇2𝑥−𝜇₂Ψ₂𝑥,∀𝑥∈𝐶. (2.21)

Corollary 2.16 (see [23]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. For given (𝑥^∗,𝑦^∗,𝑧^∗)∈𝐶×𝐶×𝐶 is a solution of the problem (1.17) if and only if 𝑥^∗∈Fix(𝑄), 𝑦^∗=𝑃_𝐶(𝑧^∗−𝜇₂Ψ₂𝑧^∗) and 𝑧^∗=𝑃_𝐶(𝑥^∗−𝜇₃Ψ₃𝑥^∗), where 𝑄 is the mapping defined by 𝑄𝑥=𝑃_𝐶[𝑃_𝐶𝑃_𝐶𝑥−𝜇₃Ψ₃𝑥−𝜇₂Ψ₂𝑃_𝐶𝑥−𝜇₃Ψ𝑥−𝜇₁Ψ₁𝑃_𝐶𝑃_𝐶𝑥−𝜇₃Ψ₃𝑥−𝜇₂Ψ₂𝑃_𝐶𝑥−𝜇₃Ψ₃𝑥,∀𝑥∈𝐶. (2.22)

Corollary 2.17 (see [24]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. For given (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of the problem (1.18) if and only if 𝑥^∗∈Fix(𝑄), 𝑦^∗=𝑃_𝐶(𝑥^∗−𝜇₂Ψ₂𝑥^∗), where 𝑄 is the mapping defined by 𝑄𝑥=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂Ψ₂𝑥−𝜇₁Ψ₁𝑃_𝐶𝑥−𝜇₂Ψ₂𝑥,∀𝑥∈𝐶. (2.23)

3. Main Results

We are now in a position to prove the main result of this paper.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻 such that 𝐶±𝐶⊂𝐶. Let 𝜙,𝜑_𝑖∶𝐶→ℝ(𝑖=1,2,3) be lower semicontinuous and convex functionals, Θ∶𝐻×𝐶×𝐶→ℝ be an equilibrium-like function, 𝐺∶𝐶→𝐻 be a mapping, and 𝐹_𝑖∶𝐶×𝐶→ℝ(𝑖=1,2,3) be a bifunction satisfying conditions (𝐻1)–(𝐻4). Assume that either condition (𝐴1) or (𝐴2) holds. Let 𝐵∶𝐶→𝐻 be 𝛽-inverse-strongly monotone, Ψ_𝑖,Φ_𝑖∶𝐶→𝐻(𝑖=1,2,3) be 𝛽_𝑖-inverse-strongly monotone and 𝛾_𝑖-inverse-strongly monotone, respectively. Let {𝑇_𝑛}^∞_𝑛=1∶𝐶→𝐶 be a family of 𝑘-strict pseudocontraction mappings. Define a mapping 𝑆_𝑛𝑥∶=𝛿𝑥+(1−𝛿)𝑇_𝑛𝑥, for all 𝑥∈𝐶, 𝛿∈[𝑘,1) and 𝑛≥1. Assume that the condition (Δ) is satisfied and Ω∶=⋂^∞_𝑛=1Fix(𝑇_𝑛)∩Fix(𝑄)∩GEP(𝐶,𝐺,Θ,𝜙)≠∅, where 𝑄 is defined as in Lemma 2.13. Let 𝜇>0, 𝛾>0, and 𝑟>0 be three constants. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and let 𝐴 be a strongly positive bounded linear operator on 𝐶 with a coefficient 𝛾∈(0,1] such that 0<𝛾<((1+𝜇)𝛾)/𝛼. For 𝑥₁∈𝐶, let the sequence {𝑥_𝑛} defined by Θ𝐺𝑥_𝑛,𝑢_𝑛,𝑦+𝜙𝑦−𝜙𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛−𝑟𝐵𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥_𝑛−𝜇₃Ψ₃+Φ₃𝑥_𝑛,𝑦_𝑛=𝑇^{(𝐹2,𝜑2)}_𝜇2𝑧_𝑛−𝜇₂Ψ₂+Φ₂𝑧_𝑛,𝑣_𝑛=𝑇^{(𝐹1,𝜑1)}_𝜇1𝑦_𝑛−𝜇₁Ψ₁+Φ₁𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝛾₁𝑆_𝑛𝑥_𝑛+𝛾₂𝑢_𝑛+𝛾₃𝑣_𝑛,∀𝑛≥1, (3.1) where 𝛾₁,𝛾₂,𝛾₃∈[0,1] such that 𝛾₁+𝛾₂+𝛾₃=1, 𝜇₁∈(0,min{𝛽₁,𝛾₁}), 𝜇₂∈(0,min{𝛽₂,𝛾₂}), 𝜇₃∈(0,min{𝛽₃,𝛾₃}), 𝑟∈(0,2𝛽) and {𝛼_𝑛},{𝛽_𝑛} are two sequences in [0,1]. Suppose that {𝑇_𝑛} satisfies the PT-condition. Let 𝑇∶𝐶→𝐶 be the mapping defined by 𝑇𝑥=lim_𝑛→∞𝑇_𝑛𝑥 for all 𝑥∈𝐶 and suppose that Fix(𝑇)=⋂^∞_𝑛=1Fix(𝑇_𝑛). Assume the following conditions are satisfied: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.
Then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to 𝑥∈Ω, where 𝑥 is the unique solution of the variational inequality ⟨𝛾𝑓−𝐼+𝜇𝐴𝑥,𝑣−𝑥⟩≤0,∀𝑣∈Ω, (3.2) or equivalently, 𝑥=𝑃_Ω(𝛾𝑓−𝜇𝐴)𝑥, where 𝑃_Ω is a metric projection mapping from 𝐶 onto Ω, and (𝑥,𝑦,𝑧) is a solution of the problem (1.12), where 𝑦=𝑇^{(𝐹2,𝜑2)}_𝜇2(𝑧−𝜇₂(Ψ₂+Φ₂)𝑧) and 𝑧=𝑇^{(𝐹3,𝜑3)}_𝜇3(𝑥−𝜇₃(Ψ₃+Φ₃)𝑥).

Proof. Note that from the conditions (𝐶1) and (𝐶2), we may assume, without loss of generality, that 𝛼_𝑛≤(1−𝛽_𝑛)(1+𝜇‖𝐴‖)⁻¹ for all 𝑛∈ℕ. Since 𝐴 is a linear bounded self-adjoint operator on 𝐶, by (2.2), we have ‖𝐴‖=sup{|⟨𝐴𝑢,𝑢⟩|∶𝑢∈𝐶,‖𝑢‖=1}. (3.3) Observe that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝑢,𝑢=1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝐴𝑢,𝑢⟩≥1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇‖𝐴‖≥0. (3.4) This show that (1−𝛽_𝑛)𝐼−𝛼_𝑛(𝐼+𝜇𝐴) is positive. It follows that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴=sup1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝑢,𝑢∶𝑢∈𝐶,‖𝑢‖=1=sup1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝐴𝑢,𝑢⟩∶𝑢∈𝐶,‖𝑢‖=1≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾. (3.5) First, we show that {𝑥_𝑛} is bounded. Taking 𝑥^∗∈Ω, it follows from Lemma 2.13 that 𝑥^∗=𝑇^{(𝐹1,𝜑1)}_𝜇1𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗−𝜇₁Ψ₁+Φ₁𝑇^{(𝐹2,𝜑2)}_𝜇2𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗−𝜇₂Ψ₂+Φ₂𝑇^{(𝐹3,𝜑3)}_𝜇3𝑥^∗−𝜇₃Ψ₃+Φ₃𝑥^∗. (3.6) Putting 𝑦^∗=𝑇^{(𝐹2,𝜑2)}_𝜇2(𝑧^∗−𝜇₂(Ψ₂+Φ₂)𝑧^∗) and 𝑧^∗=𝑇^{(𝐹3,𝜑3)}_𝜇3(𝑥^∗−𝜇₃(Ψ₃+Φ₃)𝑥^∗), we obtain 𝑥^∗=𝑇^{(𝐹1,𝜑1)}_𝜇1(𝑦^∗−𝜇₁(Ψ₁+Φ₁)𝑦^∗). Notice that 𝑢_𝑛=𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥_𝑛. Since 𝑇^(𝑟) is nonexpansive and 𝐵 is 𝛽-inverse-strongly monotone, we have 𝑢_𝑛−𝑥^∗2=𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥_𝑛−𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥^∗2≤(𝐼−𝑟𝐵)𝑥_𝑛−(𝐼−𝑟𝐵)𝑥^∗2=𝑥_𝑛−𝑥^∗−𝑟𝐵𝑥_𝑛−𝐵𝑥^∗2=𝑥_𝑛−𝑥^∗2−2𝑟𝑥_𝑛−𝑥^∗,𝐵𝑥_𝑛−𝐵𝑥^∗+𝑟²𝐵𝑥_𝑛−𝐵𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝑟𝑟−2𝛽𝐵𝑥_𝑛−𝐵𝑥^∗2≤𝑥_𝑛−𝑥^∗2, (3.7) and hence 𝑢_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.8) We observe that 𝑣_𝑛−𝑥^∗=𝑄𝑥_𝑛−𝑄𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.9) Setting 𝜃_𝑛∶=𝛾₁𝑆_𝑛𝑥_𝑛+𝛾₂𝑢_𝑛+𝛾₃𝑣_𝑛. By Lemma 2.8, we have 𝑆_𝑛 is a nonexpansive mapping such that Fix(𝑆_𝑛)=Fix(𝑇_𝑛) for all 𝑛≥1. Then, we have 𝜃_𝑛−𝑥^∗=𝛾₁𝑆_𝑛𝑥_𝑛−𝑥^∗+𝛾₂𝑢_𝑛−𝑥^∗+𝛾₃𝑣_𝑛−𝑥^∗≤𝛾₁𝑆_𝑛𝑥_𝑛−𝑥^∗+𝛾₂𝑢_𝑛−𝑥^∗+𝛾₃𝑣_𝑛−𝑥^∗≤𝛾₁𝑥_𝑛−𝑥^∗+𝛾₂𝑥_𝑛−𝑥^∗+𝛾₃𝑥_𝑛−𝑥^∗=𝑥_𝑛−𝑥^∗. (3.10) It follows that 𝑥_𝑛+1−𝑥^∗=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝜃_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛼_𝑛𝐼+𝜇𝐴𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾𝜃_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐼+𝜇𝐴𝑥^∗+1−𝛼_𝑛1+𝜇𝛾𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐼+𝜇𝐴𝑥^∗+1−𝛼_𝑛1+𝜇𝛾𝑥_𝑛−𝑥^∗≤1−1+𝜇𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐼+𝜇𝐴𝑥^∗=1−1+𝜇𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑥^∗+1+𝜇𝛾−𝛾𝛼𝛼_𝑛𝛾𝑓𝑥^∗−𝐼+𝜇𝐴𝑥^∗1+𝜇.𝛾−𝛾𝛼 (3.11) By induction, we have 𝑥_𝑛−𝑥^∗≤max𝑥₁−𝑥^∗,𝛾𝑓𝑥^∗−𝐼+𝜇𝐴𝑥^∗1+𝜇𝛾−𝛾𝛼,∀𝑛≥1. (3.12) Hence, {𝑥_𝑛} is bounded, so are {𝑢_𝑛}, {𝑣_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛}. From definition of 𝑆_𝑛 and for all 𝑘,𝑙∈ℕ, it follows that sup_𝜔∈𝐶𝑆_𝑘𝜔−𝑆_𝑙𝜔=𝛾₁sup_𝜔∈𝐶𝑇_𝑘𝜔−𝑇_𝑙𝜔. (3.13) By our assumption, {𝑇_𝑛} satisfies the PT-condition, we obtain that lim_𝑛→∞sup_𝜔∈𝐶𝑆_𝑘𝜔−𝑆_𝑙𝜔=0, (3.14) that is {𝑆_𝑛} satisfies the PT-condition.
Next, we show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Since 𝑢_𝑛=𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥_𝑛, we have 𝑢_𝑛+1−𝑢_𝑛=𝑇^(𝑟)𝑥_𝑛+1−𝑟𝐵𝑥_𝑛+1−𝑇^(𝑟)𝑥_𝑛−𝑟𝐵𝑥_𝑛≤𝑥_𝑛+1−𝑟𝐵𝑥_𝑛+1−𝑥_𝑛−𝑟𝐵𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛,𝑣_𝑛+1−𝑣_𝑛=𝑄𝑥_𝑛+1−𝑄𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (3.15) It follows from (3.15) that 𝜃_𝑛+1−𝜃_𝑛=𝛾₁𝑆_𝑛+1𝑥_𝑛+1+𝛾₂𝑢_𝑛+1+𝛾₃𝑣_𝑛+1−𝛾₁𝑆_𝑛𝑥_𝑛+𝛾₂𝑢_𝑛+𝛾₃𝑣_𝑛=𝛾₁𝑆_𝑛+1𝑥_𝑛+1−𝑆_𝑛𝑥_𝑛+𝛾𝑢_𝑛+1−𝑢_𝑛+𝛾₃𝑣_𝑛+1−𝑣_𝑛≤𝛾₁𝑆_𝑛+1𝑥_𝑛+1−𝑆_𝑛𝑥_𝑛+𝛾₂𝑢_𝑛+1−𝑢_𝑛+𝛾₃𝑣_𝑛+1−𝑣_𝑛≤𝛾₁𝑆_𝑛+1𝑥_𝑛+1−𝑆_𝑛+1𝑥_𝑛+𝛾₁𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛+𝛾₂𝑢_𝑛+1−𝑢_𝑛+𝛾₃𝑣_𝑛+1−𝑣_𝑛≤𝛾₁𝑥_𝑛+1−𝑥_𝑛+𝛾₂𝑥_𝑛+1−𝑥_𝑛+𝛾₃𝑥_𝑛+1−𝑥_𝑛+𝛾₁𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝛾₁𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛. (3.16) Let 𝑥_𝑛+1=(1−𝛽_𝑛)𝑙_𝑛+𝛽_𝑛𝑥_𝑛 for all 𝑛≥1. Then, we have 𝑙_𝑛+1−𝑙_𝑛=𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐼+𝜇𝐴𝜃_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝜃_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐼+𝜇𝐴𝜃_𝑛+11−𝛽_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝜃_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐼+𝜇𝐴𝜃_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝜃_𝑛+𝜃_𝑛+1−𝜃_𝑛. (3.17) Combining (3.16) and (3.17), we have 𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1𝛾𝑓𝑥_𝑛+1−𝐼+𝜇𝐴𝜃_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝜃_𝑛+𝛾₁𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛. (3.18) Since {𝑆_𝑛} satisfies the PT-condition, we can define a mapping 𝑆∶𝐶→𝐶 by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥=lim_𝑛→∞𝛿𝑥+(1−𝛿)𝑇_𝑛𝑥=𝛿𝑥+(1−𝛿)𝑇𝑥,∀𝑥∈𝐶. (3.19) We observe that 𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤𝑆_𝑛+1𝑥_𝑛−𝑆𝑥_𝑛+𝑆𝑥_𝑛−𝑆_𝑛𝑥_𝑛≤sup_𝜔∈𝐶𝑆_𝑛+1𝜔−𝑆𝜔+sup_𝜔∈𝐶𝑆𝜔−𝑆_𝑛𝜔. (3.20) Be Lemma 2.6, we have lim_𝑛→∞𝑆_𝑛+1𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (3.21) Consequently, it follows from the conditions (𝐶1), (𝐶2), and (3.18) that limsup_𝑛→∞𝑙_𝑛+1−𝑙_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.22) Hence, by Lemma 2.3, we obtain that lim_𝑛→∞𝑙_𝑛−𝑥_𝑛=0. (3.23) Consequently, we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑙_𝑛−𝑥_𝑛=0. (3.24) On the other hand, we observe that 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝜃_𝑛−𝑥_𝑛. (3.25) It follows that 1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾𝜃_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥_𝑛. (3.26) From the conditions (𝐶1), (𝐶2), and (3.24), we obtain thatlim_𝑛→∞𝜃_𝑛−𝑥_𝑛=0. (3.27) Next, we show thatlimsup_𝑛→∞𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥≤0. (3.28) To show this, we take a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛𝑗−𝑥. (3.29) Since {𝑥_𝑛𝑗} is bounded, without loss of generality, we can assume that 𝑥_𝑛𝑗⇀𝑣∈𝐶. So, we get limsup_𝑛→∞𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛𝑗−𝑥=𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑣−𝑥≤0. (3.30) Next, we show that 𝑣∈Ω∶=⋂^∞_𝑛=1Fix(𝑇_𝑛)∩Fix(𝑄)∩GEP(𝐶,𝐺,Θ,𝜙). Define a mapping 𝐾_𝑛∶𝐶→𝐶 by 𝐾_𝑛𝑥=𝛾₁𝑆_𝑛𝑥+𝛾₂𝑄𝑥+𝛾₃𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥,∀𝑥∈𝐶. (3.31) From (3.27), we have lim_𝑛→∞𝐾_𝑛𝑥_𝑛−𝐾𝑥_𝑛=0. (3.32) For all 𝑘,𝑙∈ℕ, it follows that sup_𝜔∈𝐶𝐾_𝑘𝜔−𝐾_𝑙𝜔=𝛾₁sup_𝜔∈𝐶𝑆_𝑘𝜔−𝑆_𝑙𝜔. (3.33) Since {𝑆_𝑛} satisfies the PT-condition, we obtain that lim_{𝑘,𝑙→∞}sup_𝜔∈𝐶𝐾_𝑘𝜔−𝐾_𝑙𝜔=0, (3.34) that is {𝐾_𝑛} satisfies the PT-condition. Define a mapping 𝐾∶𝐶→𝐶 by 𝐾𝑥=lim_𝑛→∞𝐾_𝑛𝑥=lim_𝑛→∞𝛾₁𝑆_𝑛𝑥+𝛾₂𝑄𝑥+𝛾₃𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥=𝛾₁𝑆𝑥+𝛾₂𝑄𝑥+𝛾₃𝑇^(𝑟)(𝐼−𝑟𝐵)𝑥,∀𝑥∈𝐶. (3.35) By Lemma 2.6, we obtain that lim_𝑛→∞sup_𝜔∈𝐶𝐾_𝑛𝜔−𝐾𝜔=0. (3.36) From Lemma 2.7, we see that 𝐾 is nonexpansive and Fix(𝐾)=Fix(𝑆)∩Fix(𝑄)∩Fix𝑇^(𝑟)()=𝐼−𝑟𝐵_∞𝑛=1Fix𝑆_𝑛∩Fix(𝑄)∩Fix𝑇^(𝑟)()=𝐼−𝑟𝐵_∞𝑛=1Fix𝑇_𝑛(𝑄)=∩Fix∩GEP𝐶,𝐺,Θ,𝜑_∞𝑛=1Fix𝐾_𝑛. (3.37) Notice that 𝑥_𝑛−𝐾𝑥_𝑛≤𝑥_𝑛−𝐾_𝑛𝑥_𝑛+𝐾_𝑛𝑥_𝑛−𝐾𝑥_𝑛≤𝑥_𝑛−𝐾_𝑛𝑥_𝑛+sup_𝜔∈𝐶𝐾_𝑛𝜔−𝐾𝜔. (3.38) From (3.32) and (3.39), we obtain that lim_𝑛→∞𝑥_𝑛−𝐾𝑥_𝑛=0. (3.39) Thus, by Lemma 2.9, we obtain that 𝑣∈Fix(𝐾)=Ω.
Finally, we show that 𝑥_𝑛→𝑥 as 𝑛→∞. From Lemma 2.4, we compute 𝑥_𝑛+1−𝑥²=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥+𝛽_𝑛𝑥_𝑛−𝑥+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝜃_𝑛−𝑥²≤1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝜃_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾𝜃_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑥),𝑥_𝑛+1−𝑥+2𝛼_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾𝑥_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥+2𝛼_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥≤1−𝛽_𝑛−𝛼_𝑛1+𝜇𝛾𝑥_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥+2𝛼_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥≤1−𝛼_𝑛1+𝜇𝛾²𝑥_𝑛−𝑥²+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥²+𝑥_𝑛+1−𝑥²+2𝛼_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥=1−2𝛼_𝑛1+𝜇𝛾+𝛼²_𝑛1+𝜇𝛾²+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥²+𝛼_𝑛𝛾𝛼𝑥_𝑛+1−𝑥²+2𝛼_𝑛𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥. (3.40) It follows that 𝑥_𝑛+1−𝑥²≤1−2𝛼_𝑛1+𝜇𝛾+𝛼²_𝑛1+𝜇𝛾²+𝛼_𝑛𝛾𝛼1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥²+2𝛼_𝑛1−𝛼_𝑛(𝑥)−𝛾𝛼𝛾𝑓𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥=1−2𝛼_𝑛1+𝜇𝛾+𝛼_𝑛𝛾𝛼1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥²+𝛼²_𝑛1+𝜇𝛾²1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥²+2𝛼_𝑛1−𝛼_𝑛(𝑥)−𝛾𝛼𝛾𝑓𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥≤21−1+𝜇𝛾−𝛾𝛼𝛼_𝑛1−𝛼_𝑛𝑥𝛾𝛼^𝑛−𝑥²+𝛼²_𝑛1+𝜇𝛾²1−𝛼_𝑛𝑀+𝛾𝛼2𝛼_𝑛1−𝛼_𝑛(𝑥)−𝛾𝛼𝛾𝑓𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥, (3.41)
where 𝑀=sup_𝑛≥1{‖𝑥_𝑛−𝑥‖²}. Put 𝜎_𝑛=(2((1+𝜇)𝛾−𝛾𝛼)𝛼_𝑛)/(1−𝛼_𝑛𝛾𝛼) and 𝛿_𝑛=𝛼²_𝑛1+𝜇𝛾²1−𝛼_𝑛𝛾𝛼𝑀+2𝛼_𝑛1−𝛼_𝑛(𝑥)−𝛾𝛼𝛾𝑓𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥. (3.42) Then, the (3.41) reduces to the formula 𝑥_𝑛+1−𝑥²≤1−𝜎_𝑛𝑥_𝑛−𝑥²+𝛿_𝑛. (3.43) It is easily seen that ∑^∞_𝑛=1𝜎_𝑛=∞ and (using (3.30)), we get limsup_𝑛→∞𝛿_𝑛𝜎_𝑛=limsup_𝑛→∞121+𝜇𝛼𝛾−𝛾𝛼^𝑛1+𝜇𝛾²𝑀+2𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛+1−𝑥≤0. (3.44) Hence, by Lemma 2.10, we conclude that 𝑥_𝑛→𝑥 as 𝑛→∞. This completes the proof.

Remark 3.2. Theorem 3.1 improves and generalizes [1, Theorem 2.1] in the following ways. (i)From a one nonexpansive mapping to a countable family of strict pseudocontraction mappings.(ii)From a general system of variational inequalities to a general system of generalized nonlinear mixed composite-type equilibria.(iii)Theorem 3.1 for finding an element 𝑥∈⋂^∞_𝑛=1Fix(𝑇_𝑛)∩Fix(𝑄)∩GEP(𝐶,𝐺,Θ,Φ) (𝑄 is defined as in Lemma 2.13) is more general the one of finding elements of Fix(𝑆)∩Fix(𝐷)∩GEP(𝐶,𝐺,Θ,Φ) (𝐷 is defined as in Lemma 2.17) in [1, Theorem 2.1].
Furthermore, our method of the proof is very different from that in [1, Theorem 2.1] because (3.1) involves the countable family of strict pseudocontraction mappings and strongly positive bounded linear operator.

4. Application to Minimization Problems

Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻 and 𝐴∶𝐻→𝐻 be a strongly positive linear bounded operator with a constant 𝛾>0. In this section, we will utilize the results presented in Section 3 to study the following minimization problem: min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥‖²−ℎ(𝑥), (4.1) where Ω is a nonempty closed and convex subsets of 𝐶, 𝜇≥0 is some constant and ℎ∶𝐶→ℝ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐶), where 𝑓∶𝐶→𝐶 is a contraction mapping with a constant 𝛼∈(0,1). Note that this kind of minimization problems has been studied extensively by many authors (e.g., see [18, 36–39]). We can apply Theorem 3.1 to solve the above minimization problem in the framework of Hilbert spaces as follows.

Theorem 4.1. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻 such that 𝐶±𝐶⊂𝐶. Let 𝑆∶𝐶→𝐶 be a nonexpansive mappings such that Fix(𝑆)≠∅. Let 𝜇>0 and 𝛾>0 be two constants. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator on 𝐶 with a coefficient 𝛾∈(0,1) such that 0<𝛾<((1+𝜇)𝛾)/𝛼. Let {𝑥_𝑛} be a sequence defined by 𝑥₁∈𝐶 and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝑆𝑥_𝑛,∀𝑛≥1, (4.2) where {𝛼_𝑛},{𝛽_𝑛} are two sequences in [0,1]. Assume the following conditions are satisfied: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that Fix(𝑆) is a compact subset of 𝐶. Then the sequence {𝑥_𝑛} defined by (4.2) converges strongly to 𝑥∈Fix(𝑆) which solves the minimization problem (4.1).

Proof. Taking 𝛾₁=1 and 𝛾₂=𝛾₃=0 in Theorem 3.1. Hence, from Theorem 3.1, we know that the sequence {𝑥_𝑛} defined by (4.2) converges strongly to 𝑥∈Fix(𝑆), where 𝑥 is the unique solution of the variational inequality 𝛾𝑓−𝐼+𝜇𝐴𝑥,𝑣−𝑥≤0,∀𝑣∈Fix(𝑆). (4.3) Since 𝑆 is nonexpansive, then Fix(𝑆) is convex. Again by the assumption that Fix(𝑆) is compact, then it is a compact and convex subset of 𝐶, and 𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥‖²−ℎ(𝑥)∶𝐶→ℝ (4.4) is a continuous mapping. By virtue of the well-known Weierstrass's theorem, there exists a point 𝑥∈Fix(𝑆) which is a minimal point of minimization problem (4.1). As is known to all, (4.3) is the optimality necessary condition [18] for the minimization problem (4.1). Therefore we also have 𝛾𝑓−𝐼+𝜇𝐴𝑥,𝑣−𝑥≤0,∀𝑣∈Fix(𝑆). (4.5) Since 𝑥 is the unique solution of (4.3), we have 𝑥=𝑥. This completes the proof.

5. A Numerical Example

In this section, we give a real example in which the conditions satisfy the ones of Theorem 3.1 and some numerical experiment results to explain the main result Theorem 3.1 as follows.

Example 5.1. Let 𝐻=ℝ, 𝐶=[0,1], Θ(𝑥,𝑦,𝑧)=𝜙(𝑥)=𝐵𝑥=0, ∀𝑥,𝑦,𝑧∈𝐶, 𝑟=1, 𝐹_𝑖(𝑥,𝑦)=𝜑_𝑖(𝑥)=Ψ_𝑖𝑥=Φ_𝑖𝑥=0, ∀𝑥,𝑦∈𝐶, 𝜇_𝑖=1(𝑖=1,2,3), 𝐴=𝐼, 𝑓(𝑥)=(1/2)𝑥, and ∀𝑥∈𝐶, with a constant 𝛼=1/2, 𝛼_𝑛=1/𝑛,  𝛽_𝑛=(𝑛+1)/2𝑛, ∀𝑛∈ℕ,  𝛾_𝑖=(1/3)(𝑖=1,2,3), 𝛾=1, and 𝜇=1. For all 𝑛∈ℕ, let 𝑇_𝑛∶𝐶→𝐶 define by 𝑇_𝑛𝑥=𝑛𝑥²/(2𝑛+1), ∀𝑥∈𝐶, we see that, {𝑇_𝑛} is a family of 0-strictly pseudocontractive with ⋂^∞_𝑛=1Fix(𝑇_𝑛)={0}. Then, {𝑥_𝑛} is the sequence defined by 𝑥_𝑛+1=56−2𝑥3𝑛^𝑛+16𝑛−5𝑥2𝑛+1^2𝑛, (5.1) and 𝑥_𝑛→0 as 𝑛→∞, where 0 is the unique solution of the minimization problem min_𝑥∈𝐶34𝑥²+𝑞. (5.2)

Proof. Step 1. We show that 𝑇^(𝑟)(𝑥)=𝑃_𝐶𝑥,∀𝑥∈𝐻, (5.3) where 𝑃_𝐶𝑥𝑥=|𝑥|,𝑥∈𝐻⧵𝐶,𝑥,𝑥∈𝐶. (5.4) Since Θ(𝑥,𝑦,𝑧)=𝜙(𝑥)=𝐵𝑥=0, due to the definition of 𝑇^(𝑟)(𝑥), ∀𝑥∈𝐻 in (2.7), we have 𝑇^(𝑟)(𝑥)=𝑦∈𝐶∶𝑦−𝑧,𝑧−𝑥≥0,∀𝑧∈𝐶. (5.5) Also by the equivalent property (2.5) of the nearest projection 𝑃_𝐶 from 𝐻 to 𝐶, we obtain this conclusion. When we take 𝑥∈𝐶, then 𝑇^(𝑟)(𝑥)=𝑃_𝐶𝑥=𝑥. By the condition (Δ)(𝑐), we have GEP(𝐶,𝐺,Θ,𝜙)=𝐶. In a similar way, for all 𝑖=1,2,3, we can get 𝑇^{(𝐹𝑖,𝜑𝑖)}_𝜇𝑖(𝑥)=𝑃_𝐶𝑥=𝑥,∀𝑥∈𝐶, (5.6) and Fix(𝑄)=𝐶. Hence GEP𝐶,𝐺,Θ,𝜙∩Fix(𝑄)=𝐶. (5.7)
Step 2. We show that {𝑇_𝑛} is satisfies the PT-condition. Since 𝑇_𝑛𝑥=(𝑛𝑥²/(2𝑛+1)), ∀𝑥∈𝐶, and 𝑛∈ℕ. For all 𝑘,𝑙∈ℕ, we have lim_{𝑘,𝑙→∞}𝑘𝑥²−2𝑘+1𝑙𝑥²2𝑙+1≤lim_{𝑘,𝑙→∞}sup_𝜔∈𝐶𝑘𝜔²−2𝑘+1𝑙𝜔²2𝑙+1=lim_{𝑘,𝑙→∞}𝑘−𝑙2𝑘+12𝑙+1sup_𝜔∈𝐶𝜔²=0, (5.8) that is {𝑇_𝑛} satisfies the PT-condition.

Step 3. We show that 𝑥_𝑛+1=56−2𝑥3𝑛^𝑛+16𝑛−5𝑥2𝑛+1^2𝑛,𝑥_𝑛→0,as𝑛→∞, (5.9) where 0 is the unique solution of the minimization problem: min_𝑥∈𝐶34𝑥²+𝑞. (5.10) Due to (5.3) and (5.4), we can obtain a special sequence {𝑥_𝑛} of (3.1) in Theorem 3.1 as follows: 𝑥_𝑛+1=56−2𝑥3𝑛^𝑛+16𝑛−5𝑥2𝑛+1^2𝑛. (5.11) Since ⋂^∞_𝑛=1Fix(𝑇_𝑛)={0}, combining with (5.7), we have Ω∶=GEP𝐶,𝐺,Θ,𝜙∩Fix(𝑄)∩^∞_𝑛=1Fix𝑇_𝑛={0}. (5.12) By Lemma 2.10, we obtain that 𝑥_𝑛→0, where 0 is the unique solution of the minimization problem min_𝑥∈𝐶(3/4)𝑥²+𝑞, where 𝑞 is a constant number.

5.1. Numerical Experiment Results

Next, we show the numerical experiment results using software MATLAB 7.0 and we obtain the results shown in Tables 1 and 2 and Figure 1, which show that the iteration process of the sequence {𝑥_𝑛} as initial points 𝑥₁=1 and 𝑥₁=0.5, respectively.

Table 1 

This table shows the value of sequence {𝑥_𝑛} on each iteration step (initial value 𝑥₁=1).

Table 2 

This table shows the value of sequence {𝑥_𝑛} on each iteration step (initial value 𝑥₁=0.5).

(a)

(b)

Figure 1 

These figures show the iteration comparison chart of different initial values (a)  𝑥1=1 and (b)  𝑥1=0.5, respectively.

Acknowledgment

The authors were supported by the Higher Education Research Promotion and the National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC no. 55000613).