An Iterative Method for Solving a System of Mixed Equilibrium Problems, System of Quasivariational Inclusions, and Fixed Point Problems of Nonexpansive Semigroups with Application to Optimization Problems

Abstract

We introduce a general implicit iterative scheme base on viscosity approximation method with a ϕ-strongly pseudocontractive mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator.

1. Introduction

Throughout this paper we denoted by ℕ and ℝ⁺ the set of all positive integers and all positive real numbers, respectively. We always assume that 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively, 𝐶 is a nonempty closed convex subset of 𝐻. Let 𝜑∶𝐶→ℝ be a real-valued function and Θ∶𝐶×𝐶→ℝ be an equilibrium bifunction. The mixed equilibrium problem (for short, MEP) is to find 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗≥0,∀𝑦∈𝐶. (1.1) The set of solutions of (1.1) is denoted by MEP(Θ,𝜑), that is,MEPΘ,𝜑=𝑥^∗∈𝐶∶Θ𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗≥0,∀𝑦∈𝐶. (1.2)

In particular, if 𝜑=0, this problem reduces to the equilibrium problem, that is, to find 𝑥^∗∈𝐶 such thatΘ𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (1.3) The set of solution of (1.3) is denoted by EP(Θ).

Mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [1–6]). Some methods have been proposed to solve the equilibrium problem, see, for instance, [7–21].

Let 𝐴 be a strongly positive bounded linear operator on 𝐻, that is, there exists a constant 𝛾>0 such that⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻. (1.4)

Recall that, a mapping 𝑓∶𝐻→𝐻 is said to be contractive if there exists a constant 𝛼∈(0,1) such that𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐻. (1.5) A mapping 𝑇∶𝐻→𝐻 is said to be (i)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻, (1.6)(ii)pseudocontractive if 𝑇𝑥−𝑇𝑦,𝑥−𝑦≤𝑥−𝑦²,∀𝑥,𝑦∈𝐻, (1.7)(iii)𝜙-strongly pseudocontractive if there exists a continuous and strictly increasing function 𝜙∶ℝ⁺→ℝ⁺ with 𝜙(0)=0 such that 𝑇𝑥−𝑇𝑦,𝑥−𝑦≤𝑥−𝑦²−𝜙𝑥−𝑦𝑥−𝑦,∀𝑥,𝑦∈𝐻. (1.8)

It is obvious that pseudocontractive mapping is more general than 𝜙-strongly pseudocontractive mapping. If 𝜙(𝑡)=𝛼𝑡 with 0<𝛼<1, then 𝜙-strongly pseudocontractive mapping reduces to 𝛽-strongly pseudocontractive mapping with 1−𝛼=𝛽∈(0,1), which is more general than contractive mapping.

Definition 1.1. A one-parameter family mapping 𝒮={T(t)∶t∈ℝ⁺} from C into itself is said to be a nonexpansive semigroup on C if it satisfies the following conditions: (i)𝑇(0)𝑥=𝑥𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈𝐶,(ii)𝑇(𝑠+𝑡)=𝑇(𝑠)∘𝑇(𝑡) for all 𝑠,𝑡∈ℝ⁺,(iii) for each 𝑥∈𝐶 the mapping 𝑡↦𝑇(𝑡)𝑥 is continuous,(iv)‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and 𝑡∈ℝ⁺.

Remark 1.2. We denote by F(𝒮) the set of all common fixed points of 𝒮, that is, 𝐹(𝒮)∶=⋂_𝑡∈ℝ+𝐹(𝑇(𝑡))={𝑥∈𝐶∶𝑇(𝑡)𝑥=𝑥}.

Recall the following definitions of a nonlinear mapping 𝐵∶𝐶→𝐻, the following are mentioned.

Definition 1.3. The nonlinear mapping 𝐵∶𝐶→𝐻 is said to be (i)monotone if 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶, (1.9)(ii)𝛽-strongly monotone if there exists a constant 𝛽>0 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐶, (1.10)(iii)𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that 𝐵𝑥−𝐵𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶, (1.11)(iv)𝜈-inverse-strongly monotone if there exists a constant 𝜈>0 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥𝜈𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶, (1.12)(v)relaxed (𝑐,𝑑)-cocoercive if there exists a constants 𝑐,𝑑>0 such that 𝐵𝑥−𝐵𝑦,𝑥−𝑦≥(−𝑐)𝐵𝑥−𝐵𝑦²+𝑑𝑑𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (1.13) The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities, inclusions, and related optimization problems.

Definition 1.4. Let 𝑀∶𝐻→2^𝐻 be a multivalued maximal monotone mapping. The single-valued mapping 𝐽_(𝑀,𝜌)∶𝐻→𝐻, defined by 𝐽_(𝑀,𝜌)(𝑢)=𝐼+𝜌𝑀⁻¹(𝑢),∀𝑢∈𝐻, (1.14) is called resolvent operator associated with 𝑀, where 𝜌 is any positive number and 𝐼 is the identity mapping.
Next, we consider a system of quasivariational inclusions problem is to find (𝑥^∗,𝑦^∗)∈𝐻×𝐻 such that 0∈𝑥^∗−𝑦^∗+𝜌₁𝐵₁𝑦^∗+𝑀₁𝑥^∗,0∈𝑦^∗−𝑥^∗+𝜌₂𝐵₂𝑥^∗+𝑀₂𝑦^∗, (1.15) where 𝐵_𝑖∶𝐻→𝐻 and 𝑀_𝑖∶𝐻→2^𝐻 are nonlinear mappings for each 𝑖=1,2.
As special cases of the problem (1.15), we have the following results. (1)If 𝐵₁=𝐵₂=𝐵 and 𝑀₁=𝑀₂=𝑀, then the problem (1.15) is reduces to the following. Find (𝑥^∗,𝑦^∗)∈𝐻×𝐻 such that 0∈𝑥^∗−𝑦^∗+𝜌₁𝐵𝑦^∗+𝑀𝑥^∗,0∈𝑦^∗−𝑥^∗+𝜌₂𝐵𝑥^∗+𝑀𝑦^∗. (1.16)(2)Further, if 𝑥^∗=𝑦^∗ in problem (1.16), then the problem (1.16) is reduces to the following. Find 𝑥^∗∈𝐻 such that 0∈𝐵𝑥^∗+𝑀𝑥^∗. (1.17)The problem (1.17) is called variational inclusion problem. We denote by VI(𝐻,𝐵,𝑀) the set of solutions of the variational inclusion problem (1.17). Next, we consider two special cases of the problem (1.17). (1)𝑀=𝜕𝜙∶𝐻→2^𝐻, where 𝜙∶𝐻→ℝ∪{+∞} is a proper convex lower semicontinuous function and 𝜕𝜙 is the subdifferential of 𝜙 then the quasivariational inclusion problem (1.17) is equivalent to finding 𝑥^∗∈𝐻 such that ⟨𝐵𝑥^∗,𝑥−𝑥^∗⟩+𝜙(𝑥)−𝜙(𝑥^∗)≥0, for all 𝑥∈𝐻, which is said to be the mixed quasivariational inequality.(2)If 𝑀=𝜕𝛿_𝐶, where 𝐶 is a nonempty closed convex subset of 𝐻, and 𝛿_𝐶∶𝐻→[0,∞) is the indicator function of 𝐶, that is, 𝛿_𝐶(𝑥)=0,𝑥∈𝐶,+∞,𝑥∉𝐶, (1.18) then the quasivariational inclusion problem (1.17) is equivalent to the classical variational inequality problem denoted by VI(𝐶,𝐵) which is to find 𝑥^∗∈𝐶 such that 𝐵𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (1.19)This problem is called Hartman-Stampacchia variational inequality problem (see e.g., [22–24]).
It is known that problem (1.17) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity, variational inequalities, optimal control, mathematical economics, equilibria, and game theory. Also various types of variational inclusions problems have been extended and generalized (see [25–40] and the references therein).
On the other hand, the following optimization problem has been studied extensively by many authors: min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (1.20) where Ω=⋂^∞_𝑛=1𝐶_𝑛,𝐶₁,𝐶₂,… are infinitely many closed convex subsets of 𝐻 such that ⋂^∞_𝑛=1𝐶_𝑛≠∅, 𝑢∈𝐻, 𝜇≥0 is a real number, 𝐴 is a strongly positive linear bounded operator on 𝐻 and ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻). This kind of optimization problem has been studied extensively by many authors (see, e.g. [41–44]) when Ω=⋂^∞_𝑛=1𝐶_𝑛 and ℎ(𝑥)=⟨𝑥,𝑏⟩, where 𝑏 is a given point in 𝐻.
Li et al. [45] introduced two steps of iterative procedures for the approximation of common fixed point of a nonexpansive semigroup 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} on a nonempty closed convex subset 𝐶 in a Hilbert space. Recently, Liu et al. [46] introduced a hybrid iterative scheme for finding a common element of the set of solutions of system of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inverse-strongly monotone mappings. Very recently, Hao [47] introduced a general iterative method for finding a common element of solution set of quasivariational inclusion problems and the set of common fixed points of an infinite family of nonexpansive mappings.
In this paper, motivated and inspired by Li et al. [45], Liu et al. [46], and Hao [47], we introduce a general implicit iterative algorithm base on viscosity approximation methods with a 𝜙-strongly pseudocontractive mapping which is more general than a contraction mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions (1.15) with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. We prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator. The results obtained in this paper extend and improve several recent results in this area.

2. Preliminaries

In the sequel, we use 𝑥_𝑛⇀𝑥 and 𝑥_𝑛→𝑥 to denote the weak convergence and strong convergence of the sequence {𝑥_𝑛} in 𝐻, respectively.

This collects some results that will be used in the proofs of our main results.

Proposition 2.1 (see [21]). (𝑖) The resolvent operator 𝐽_(𝑀,𝜌) associated with 𝑀 is single-valued and nonexpansive for all 𝜌>0, that is, 𝐽_(𝑀,𝜌)(𝑥)−𝐽_(𝑀,𝜌)𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐻,∀𝜌>0. (2.1)(𝑖𝑖) The resolvent operator 𝐽_(𝑀,𝜌) is 1-inverse-strongly monotone, that is, 𝐽_(𝑀,𝜌)(𝑥)−𝐽_(𝑀,𝜌)𝑦²≤𝑥−𝑦,𝐽_(𝑀,𝜌)(𝑥)−𝐽_(𝑀,𝜌)𝑦,∀𝑥,𝑦∈𝐻. (2.2) Obviously, this immediately implies that 𝑥−𝑦−𝐽_(𝑀,𝜌)(𝑥)−𝐽_(𝑀,𝜌)𝑦²≤𝑥−𝑦²−𝐽_(𝑀,𝜌)(𝑥)−𝐽_(𝑀,𝜌)𝑦²,∀𝑥,𝑦∈𝐻. (2.3)

For solving the equilibrium problem for bifunction Θ∶𝐻×𝐻→ℝ, let us assume that satisfies the following conditions:(H1)Θ(𝑥,𝑥)=0 for all 𝑥∈𝐻;(H2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐻;(H3)for each 𝑦∈𝐻, 𝑥↦Θ(𝑥,𝑦) is concave and upper semicontinuous;(H4) for each 𝑦∈𝐻, 𝑥↦Θ(𝑥,𝑦) is convex;(H5)for each 𝑦∈𝐻, 𝑥↦Θ(𝑥,𝑦) is lower semicontinuous.

Definition 2.2. A map 𝜂∶𝐻×𝐻→𝐻 is called Lipschitz continuous, if there exists a constant 𝐿>0 such that 𝜂𝑥,𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐻. (2.4) A differentiable function 𝐾∶𝐻→ℝ on a convex set 𝐻 is called (i)𝜂—convex [7] if𝐾𝑦−𝐾(𝑥)≥𝐾^′(𝑥),𝜂𝑦,𝑥,∀𝑥,𝑦∈𝐻, (2.5) where 𝐾′(𝑥) is the Fréchet differentiable of 𝐾 at 𝑥,(i)𝜂—strongly convex [7] if there exists a constant 𝜈>0 such that𝐾𝑦−𝐾(𝑥)−𝐾^′(𝑥)≥𝜈,𝜂𝑥,𝑦2𝑥−𝑦²,∀𝑥,𝑦∈𝐻. (2.6) Let Θ∶𝐻×𝐻→ℝ be an equilibrium bifunction satisfying the conditions (H1)–(H5). Let 𝑟 be any given positive number. For a given point 𝑥∈𝐻, consider the following auxiliary problem for MEP (for short, MEP (𝑥,𝑦)) to find 𝑦∈𝐻 such that Θ(𝑧)𝑦+1𝑦,𝑧+𝜑−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝜂𝑧,𝑦≥0,∀𝑧∈𝐻, (2.7) where 𝜂∶𝐻×𝐻→𝐻 is a mapping, and 𝐾′(𝑥) is the Fréchet derivative of a functional 𝐾∶𝐻→ℝ at 𝑥. Let 𝑉_𝑟^(Θ,𝜑)∶𝐻→𝐻 be the mapping such that for each 𝑥∈𝐻, 𝑉_𝑟^(Θ,𝜑)(𝑥) is the set of solutions of MEP (𝑥,𝑦), that is, 𝑉_𝑟^(Θ,𝜑)(𝑥)=(𝑧)𝑦+1𝑦∈𝐻∶Θ𝑦,𝑧+𝜑−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝜂𝑧,𝑦≥0,∀𝑧∈𝐻,∀𝑥∈𝐻. (2.8) Then the following conclusion holds.

Proposition 2.3 (see [7]). Let 𝐻 be a real Hilbert space, 𝜑∶𝐻→ℝ be a a lower semicontinuous and convex functional. Let Θ∶𝐻×𝐻→ℝ be an equilibrium bifunction satisfying conditions (H1)–(H5). Assume that (i)𝜂∶𝐻×𝐻→ℝ is Lipschitz continuous with constant 𝜎>0 such that(a)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐻;(b)𝜂(⋅,⋅) is affine in the first variable;(c)for each fixed 𝑦∈𝐻, 𝑥↦𝜂(𝑦,𝑥) is continuous from the weak topology to the weak topology;(ii)𝐾∶𝐻→ℝ is 𝜂-strongly convex with constant 𝜇>0, and its derivative 𝐾′ is continuous from the weak topology to the strong topology;(iii) for each 𝑥∈𝐻, there exists a bounded subset 𝐷_𝑥⊂𝐻 and 𝑧_𝑥∈𝐻 such that for all 𝑦∉𝐷_𝑥,Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝜂𝑧_𝑥,𝑦<0. (2.9) Then the following hold: (i)𝑉_𝑟^(Θ,𝜑) is single valued;(ii)𝐹(𝑉_𝑟^(Θ,𝜑))=MEP(Θ,𝜑);(iii)MEP(Θ,𝜑) is closed and convex.

Lemma 2.4 (see [48]). Let 𝐶 be a nonempty bounded closed and convex subset of a real Hilbert space 𝐻. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup on 𝐶, then for all ℎ>0, lim_𝑡→∞sup_𝑥∈𝐶1𝑡^𝑡₀𝑇(𝑠)(ℎ)1𝑥𝑑𝑠−𝑇𝑡^𝑡₀𝑇(𝑠)𝑥𝑑𝑠=0. (2.10)

Lemma 2.5 (see [49]). Let 𝑋 be a uniformly convex Banach space, 𝐶 be a nonempty closed and convex subset of 𝑋, and 𝑇∶𝐶→𝑋 be a nonexpansive mapping. Then 𝐼−𝑇 is demiclosed at zero.

Lemma 2.6 (see [50]). Assume that 𝐴 is a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 2.7 (see [51]). Let 𝑋 be a Banach space and 𝑓∶𝑋→𝑋 be a 𝜙-strongly pseudocontractive and continuous mapping. Then 𝑓 has a unique fixed point in 𝑋.

Lemma 2.8. In a real Hilbert space H, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝐻. (2.11)

The following lemma can be found in [52, 53] (see also Lemma  2.2 in [54]).

Lemma 2.9. Let C be a nonempty closed and convex subset of a real Hilbert space H and g∶C→ℝ∪{+∞} be a proper lower semicontinuous differentiable convex function differentiable convex function. If x^∗ is a solution to the minimization problem 𝑔𝑥^∗=inf_𝑥∈𝐶𝑔(𝑥), (2.12) then 𝑔^′(𝑥),𝑥−𝑥^∗≥0,𝑥∈𝐶. (2.13) In particular, if 𝑥^∗ solves the optimization problem min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (2.14) then 𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥^∗,𝑥−𝑥^∗≤0,𝑥∈𝐶, (2.15) where ℎ is a potential function for 𝛾𝑓.

The following lemmas can be found in ([55, 56]). For the sake of the completeness, one includes its proof in a Hilbert space version. Without loss of generality, one assumes that 𝑐,𝑑∈(0,1) and 𝐿_𝐵∈[1,∞).

Lemma 2.10. Let 𝐻 be a real Hilbert space, 𝐵∶𝐻→𝐻 be an 𝐿_𝐵-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping. Then, one has 𝐼−𝜌𝐵𝑥−𝐼−𝜌𝐵𝑦²≤1+2𝜌𝑐𝐿²_𝐵−2𝜌𝑑+𝜌²𝐿²_𝐵𝑥−𝑦², (2.16) where 𝜌>0. In particular, if 0<𝜌≤2(𝑑−𝑐𝐿²_𝐵)/𝐿²_𝐵, then 𝐼−𝜌𝐵 is nonexpansive.

Proof. For all 𝑥,𝑦∈𝐻, we have (𝐼−𝜌𝐵)𝑥−(𝐼−𝜌𝐵)𝑦²=(𝑥−𝑦)−(𝜌𝐵𝑥−𝜌𝐵𝑦)²=𝑥−𝑦²−2𝜌𝐵𝑥−𝐵𝑦,𝑥−𝑦+𝜌²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²−2𝜌−𝑐𝐵𝑥−𝐵𝑦²+𝑑𝑥−𝑦²+𝜌²𝐵𝑥−𝐵𝑦²=𝑥−𝑦²−2𝜌𝑑𝑥−𝑦²+2𝜌𝑐𝐵𝑥−𝐵𝑦²+𝜌²𝐵𝑥−𝐵𝑦²≤1+2𝜌𝑐𝐿²_𝐵−2𝜌𝑑+𝜌²𝐿²_𝐵𝑥−𝑦². (2.17) It is clear that, if 0<𝜌≤2(𝑑−𝑐𝐿²_𝐵)/𝐿²_𝐵, then 𝐼−𝜌𝐵 is nonexpansive. This completes the proof.

Lemma 2.11. Let 𝐻 be a real Hilbert space, 𝑀_𝑖∶𝐻→2^𝐻 be the a maximal monotone mapping and 𝐵_𝑖∶𝐻→𝐻 be an 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping for all 𝑖=1,2. Let 𝑄∶𝐻→𝐻 be a mapping defined by 𝑄𝑥∶=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥,∀𝑥∈𝐻. (2.18) If 0<𝜌_𝑖≤2(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐿²_𝑖 for all 𝑖=1,2, then 𝑄∶𝐻→𝐻 is nonexpansive.

Proof. By Lemma 2.10, we know that (𝐼−𝜌₂𝐵₂) and (𝐼−𝜌₁𝐵₁) are nonexpansive, for all 𝑥,𝑦∈𝐻, we have 𝑄𝑥−𝑄𝑦=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑦−𝜌₂𝐵₂𝑦−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑦−𝜌₂𝐵₂𝑦≤𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝐽_{(𝑀2,𝜌2)}𝑦−𝜌₂𝐵₂𝑦−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑦−𝜌₂𝐵₂𝑦=𝐽_{(𝑀2,𝜌2)}𝐼−𝜌₂𝐵₂𝐼−𝜌₁𝐵₁𝑥−𝐽_{(𝑀2,𝜌2)}𝐼−𝜌₂𝐵₂𝐼−𝜌₁𝐵₁𝑦≤𝐼−𝜌₂𝐵₂𝐼−𝜌₁𝐵₁𝑥−𝐼−𝜌₂𝐵₂𝐼−𝜌₁𝐵₁𝑦≤𝐼−𝜌₂𝐵₂𝑥−𝐼−𝜌₂𝐵₂𝑦≤𝑥−𝑦, (2.19) which implies that 𝑄 is nonexpansive. This completes the proof.

Lemma 2.12. For all (𝑥^∗,𝑦^∗)∈𝐻×𝐻, where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑥^∗−𝜌₂𝐵₂𝑥^∗), (𝑥^∗,𝑦^∗) is a solution of the problem (1.15) if and only if 𝑥^∗ is a fixed point of the mapping 𝑄∶𝐻→𝐻 defined as in Lemma 2.11.

Proof. Let (𝑥^∗,𝑦^∗)∈𝐻×𝐻 be a solution of the problem (1.15). Then, we have 𝑦^∗−𝜌₁𝐵₁𝑦^∗∈𝐼+𝜌₁𝑀₁𝑥^∗,𝑥^∗−𝜌₂𝐵₂𝑥^∗∈𝐼+𝜌₂𝑀₂𝑦^∗, (2.20) which implies that 𝑥^∗=𝐽_{(𝑀1,𝜌1)}𝑦^∗−𝜌₁𝐵₁𝑦^∗,𝑦^∗=𝐽_{(𝑀2,𝜌2)}𝑥^∗−𝜌₂𝐵₂𝑥^∗. (2.21) We can deduce that (2.21) is equivalent to 𝑥^∗=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑥^∗−𝜌₂𝐵₂𝑥^∗−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥^∗−𝜌₂𝐵₂𝑥^∗. (2.22) This completes the proof.

3. Main Results

Now, in this section, we prove our main results of this article. Before proving the main result we need the following lemma.

Lemma 3.1. Let 𝐻 be a real Hilbert space. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐻 into itself. Then 𝐼−𝜎_𝑡(⋅) is monotone, where 𝜎_𝑡(𝑥)∶=(1/𝑡)∫^𝑡₀𝑇(𝑠)𝑥𝑑𝑠 for all 𝑥∈𝐻 and 𝑡≥0.

Proof. For all 𝑥,𝑦∈𝐻, we have 𝑥−𝑦,𝐼−𝜎_𝑡(⋅)𝑥−𝐼−𝜎_𝑡(⋅)𝑦=1𝑥−𝑦,𝐼−𝑡^𝑡₀𝑇(𝑠)1𝑑𝑠𝑥−𝐼−𝑡^𝑡₀𝑇(𝑠)𝑦=𝑑𝑠𝑥−𝑦²−1𝑥−𝑦,𝑡^𝑡₀𝑇(𝑠)1𝑥𝑑𝑠−𝑡^𝑡₀𝑇(𝑠)≥𝑦𝑑𝑠𝑥−𝑦²−1𝑥−𝑦𝑡^𝑡₀𝑇(𝑠)(𝑠)𝑦≥𝑥−𝑇𝑑𝑠𝑥−𝑦²−𝑥−𝑦²=0, (3.1) which implies that 𝐼−𝜎_𝑡(⋅) is monotone. This completes the proof.

Theorem 3.2. Let 𝐻 be a real Hilbert space. Let 𝜑_𝑖∶𝐻→ℝ(𝑖=1,2,…,𝑁) be a finite family of lower semicontinuous and convex function, Θ_𝑖∶𝐻×𝐻→ℝ(𝑖=1,2,…,𝑁) be a finite family of bifunctions satisfying (H1)–(H5), and 𝜂_𝑖∶𝐻×𝐻→𝐻 be a finite family of Lipschitz continuous mappings with a constant 𝜎_𝑖(𝑖=1,2,…,𝑁). Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐻 into itself, 𝐵_𝑖∶𝐻→𝐻(𝑖=1,2) be an 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping with 𝜌_𝑖∈(0,2(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐿²_𝑖] for all 𝑖=1,2 and 𝑀_𝑖∶𝐻→2^𝐻(𝑖=1,2) be a maximal monotone mapping. Assume that Ω∶=𝐹(𝒮)∩⋂^𝑁_𝑘=1MEP (Θ_𝑘,𝜑_𝑘)∩𝐹(𝑄)≠∅, where 𝑄 is defined as in Lemma 2.11. Let 𝑓∶𝐻→𝐻 be a 𝜙-strongly pseudocontractive mapping with lim_𝑡→+∞𝜙(𝑡)=+∞ and 𝐴 be a strongly positive linear bounded operator on 𝐻 with a coefficient 𝛾>0. Let 𝜇>0 and 𝛾>0 be two constants such that 0<𝛾<1+𝜇𝛾. Let {𝑟_𝑖,𝑛}(𝑖=1,2,…,𝑁) be a finite family of positive real sequence such that liminf_𝑛→∞𝑟_𝑖,𝑛>0, {𝛼_𝑛} and {𝛽_𝑛} be two sequences in [0,1], and {𝑡_𝑛} be a positive real divergent sequence. For any fixed 𝑢∈𝐻, let {𝑥_𝑛} be the sequence defined by Θ₁𝑢_𝑛⁽¹⁾,𝑥+𝜑₁(𝑥)−𝜑₁𝑢_𝑛⁽¹⁾+1𝑟_1,𝑛𝐾^′₁𝑢_𝑛⁽¹⁾−𝐾^′₁𝑥_𝑛,𝜂₁𝑥,𝑢_𝑛⁽¹⁾Θ≥0,∀𝑥∈𝐻,₂𝑢_𝑛⁽²⁾,𝑥+𝜑₂(𝑥)−𝜑₂𝑢_𝑛⁽²⁾+1𝑟_2,𝑛𝐾^′₂𝑢_𝑛⁽²⁾−𝐾^′₂𝑢_𝑛⁽¹⁾,𝜂₁𝑥,𝑢_𝑛⁽²⁾⋮Θ≥0,∀𝑥∈𝐻,_𝑁𝑢_𝑛^(𝑁),𝑥+𝜑_𝑁(𝑥)−𝜑_𝑁𝑢_𝑛^(𝑁)+1𝑟_𝑁,𝑛𝐾^′_𝑁𝑢_𝑛^(𝑁)−𝐾^′_𝑁𝑢_𝑛^(𝑁−1),𝜂_𝑁𝑥,𝑢_𝑛^(𝑁)𝑦≥0,∀𝑥∈𝐻,_𝑛=𝐽_{(𝑀2,𝜌2)}𝑢_𝑛^(𝑁)−𝜌₂𝐵₂𝑢_𝑛^(𝑁),𝑥_𝑛=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛𝑑𝑠, (3.2) where 𝑢_𝑛⁽¹⁾=𝑉^(Θ1,𝜑1)_𝑟1,𝑛𝑥_𝑛,𝑢_𝑛^(𝑖)=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}𝑢_𝑛^(𝑖−1)=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}𝑉^{(Θ𝑖−1,𝜑𝑖−1)}_{𝑟𝑖−1,𝑛}𝑢_𝑛^(𝑖−2)=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}⋅⋅⋅𝑉^(Θ2,𝜑2)_𝑟2,𝑛𝑢_𝑛⁽¹⁾=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}⋯𝑉^(Θ2,𝜑2)_𝑟2,𝑛𝑉^(Θ1,𝜑1)_𝑟1,𝑛𝑥_𝑛,𝑖=2,3,…,𝑁, (3.3) and 𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}∶𝐻→𝐻, 𝑖=1,2,…,𝑁 is the mapping defined by (2.8). Assume the following. (i)𝜂_𝑖∶𝐻×𝐻→𝐻 is Lipschitz continuous with constant 𝜎_𝑖>0(𝑖=1,2,…,𝑁) such that(a)𝜂_𝑖(𝑥,𝑦)+𝜂_𝑖(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐻,(b)𝜂_𝑖(⋅,⋅) is affine in the first variable,(c)for each fixed 𝑦∈𝐻, 𝑥↦𝜂_𝑖(𝑦,𝑥) is sequentially continuous from the weak topology to the weak topology.(ii)𝐾_𝑖∶𝐻→ℝ is 𝜂_𝑖-strongly convex with constant 𝜇_𝑖>0, and its derivative 𝐾^′_𝑖 is not only continuous from the weak topology to the strong topology but also Lipschitz continuous with constant 𝜈_𝑖 such that 𝜇_𝑖≥𝜎_𝑖𝜈_𝑖.(iii) For all 𝑖=1,2,...,𝑁 and for all 𝑥∈𝐻, there exists a bounded subset 𝐷_𝑥⊂𝐻 and 𝑧_𝑥∈𝐻 such that for all 𝑦∉𝐷_𝑥,Θ_𝑖𝑦,𝑧_𝑛+𝜑_𝑖𝑧_𝑥−𝜑_𝑖𝑦+1𝑟_𝑖,𝑛⟨𝐾^′_𝑖𝑦−𝐾^′_𝑖(𝑥),𝜂_𝑖𝑧_𝑥,𝑦⟩<0. (3.4) If the following conditions are satisfied: (C1)lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, then the sequence {𝑥_𝑛} defined by (3.2) converges strongly to 𝑥^∗∈Ω∶=𝐹(𝒮)∩⋂^𝑁_𝑘=1MEP(Θ_𝑘,𝜑_𝑘)∩𝐹(𝑄), provided 𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛} is firmly nonexpansive, where 𝑥^∗ is the unique solution of the variational inequality 𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥^∗,𝑧−𝑥^∗≤0,∀𝑧∈Ω, (3.5) or, equivalently, 𝑥^∗ is the unique solution of the optimization problem min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (3.6) where ℎ is a potential function for 𝛾𝑓 and (𝑥^∗,𝑦^∗) is the solution of the problem (1.15), where 𝑦^∗=𝐽_{(𝑀2,𝜌2)}(𝑥^∗−𝜌₂𝐵₂𝑥^∗).

Proof. By 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 𝐻, by (1.4), we have ‖𝐴‖=sup{|⟨𝐴𝑢,𝑢⟩|∶𝑢∈𝐻,‖𝑢‖=1}. (3.7) 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. (3.8) This shows 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∈𝐻,‖𝑢‖=1=sup1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝐴𝑢,𝑢⟩∶𝑢∈𝐻,‖𝑢‖=1≤1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾. (3.9) In fact, by the assumption that for all 𝑘∈{1,2,…,𝑁}, 𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛} is nonexpansive. Setting 𝑉^𝑘_𝑛∶=𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}⋅⋅⋅𝑉^(Θ2,𝜑2)_𝑟2,𝑛𝑉^(Θ1,𝜑1)_𝑟1,𝑛 for 𝑘∈{1,2,…,𝑁} and 𝑉⁰_𝑛∶=𝐼. Define a mapping 𝑊_𝑛∶𝐻→𝐻 by 𝑊_𝑛1𝑥∶=𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑄𝑉^𝑁_𝑛𝑥𝑑𝑠,∀𝑥∈𝐻. (3.10) Hence, by Lemma 2.11 and nonexpansiveness (semigroup) of 𝑇(𝑠) and 𝑉^𝑘_𝑛, for all 𝑥,𝑦∈𝐶, we have 𝑊_𝑛𝑥−𝑊_𝑛𝑦=1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑄𝑉^𝑁_𝑛1𝑥𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑄𝑉^𝑁_𝑛≤𝑦𝑑𝑠𝑄𝑉^𝑁_𝑛𝑥−𝑄𝑉^𝑁_𝑛𝑦≤𝑉^𝑁_𝑛𝑥−𝑉^𝑁_𝑛𝑦≤𝑥−𝑦, (3.11) which implies that 𝑊_𝑛 is nonexpansive.
First, we show that {𝑥_𝑛} defined by (3.2) is well defined. Define a mapping 𝑇^𝑓_𝑛∶𝐻→𝐻 by 𝑇^𝑓_𝑛𝑥∶=𝛼_𝑛𝑢+𝛾𝑓(𝑥)+𝛽_𝑛𝑥+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝑊_𝑛𝑥,∀𝑥∈𝐻. (3.12)
Indeed, by Lemma 2.6, and from (3.11), for all 𝑥,𝑦∈𝐻, we have 𝑇^𝑓_𝑛𝑥−𝑇^𝑓_𝑛𝑦,𝑥−𝑦=𝛼_𝑛𝛾𝑓(𝑥)−𝑓𝑦,𝑥−𝑦+𝛽_𝑛+𝑥−𝑦,𝑥−𝑦1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝐴𝑊𝑥−𝑊𝑦,𝑥−𝑦≤𝛼_𝑛𝛾𝑥−𝑦²−𝜙‖𝑥−𝑦‖𝑥−𝑦+𝛽_𝑛𝑥−𝑦²+1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝑥−𝑦²=1+𝛼_𝑛𝛾−1+𝜇𝛾‖𝑥−𝑦‖²−𝛼_𝑛𝛾𝜙‖𝑥−𝑦‖‖𝑥−𝑦‖≤‖𝑥−𝑦‖²−𝛼_𝑛𝛾𝜙‖𝑥−𝑦‖‖𝑥−𝑦‖. (3.13) This shows that 𝑇^𝑓_𝑛 is a 𝜙-strongly pseudocontractive and strongly continuous. It follows from Lemma 2.7 that 𝑇^𝑓_𝑛 has a unique fixed point 𝑥_𝑛∈𝐻, that is, {𝑥_𝑛} defined by (3.2) is well defined.
Next, we show that uniqueness of the solution of the variational inequality (3.5). Suppose that 𝑥,𝑥^∗∈Ω satisfy (3.5), then 𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥^∗,𝑥−𝑥^∗≤0,𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥,𝑥^∗−𝑥≤0. (3.14) Adding up (3.14), we have 0≥𝐼+𝜇𝐴𝑥^∗−𝐼+𝜇𝐴𝑥−𝛾𝑓𝑥^∗−𝑓(𝑥),𝑥^∗=𝑥−𝑥𝐼+𝜇𝐴^∗−𝐼+𝜇𝐴𝑥,𝑥^∗−𝑥−𝛾𝑓𝑥^∗−𝑓(𝑥),𝑥^∗≥−𝑥1+𝜇𝛾𝑥^∗−𝑥²−𝛾𝑥^∗−𝑥²+𝜙𝑥^∗−𝑥𝑥^∗=−𝑥1+𝜇𝛾−𝛾𝑥^∗−𝑥²+𝜙𝑥^∗−𝑥𝑥^∗−𝑥. (3.15) It follows that 1+𝜇𝛾−𝛾𝑥^∗−𝑥+𝜙𝑥^∗−𝑥≤0, (3.16) which is a contradiction. Hence, 𝑥=𝑥^∗ and the uniqueness is proved.
Next, we show that {𝑥_𝑛} is bounded. Taking 𝑥∈Ω, it follows from Lemma 2.11 that 𝑥=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥. (3.17) Putting 𝑦=𝐽_{(𝑀2,𝜌2)}(𝑥−𝜌₂𝐵₂𝑥), we have 𝑥=𝐽_{(𝑀1,𝜌1)}(𝑦−𝜌₁𝐵₁𝑦). Setting 𝑧_𝑛∶=𝑉^𝑁_𝑛𝑥_𝑛 and 𝑣_𝑛∶=𝐽_{(𝑀1,𝜌1)}(𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛), then 𝑥_𝑛=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠. (3.18) Since for all 𝑘∈{1,2,…,𝑁}, 𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛} is nonexpansive, we also have that 𝑉^𝑁_𝑛 is nonexpansive and 𝑥=𝑉^𝑁_𝑛𝑥, then 𝑧_𝑛−𝑥=𝑉^𝑁_𝑛𝑥_𝑛−𝑉^𝑁_𝑛𝑥≤𝑥_𝑛−𝑥,∀𝑛∈ℕ. (3.19) By nonexpansiveness of 𝐽_{(𝑀𝑖,𝜌𝑖)} and 𝐼−𝜌_𝑖𝐵_𝑖  (𝑖=1,2), we have 𝑣_𝑛−𝑥=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦≤𝑦_𝑛−𝑦=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥≤𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑥−𝜌₂𝐵₂𝑥≤𝑧_𝑛−𝑥≤𝑥_𝑛−𝑥. (3.20) It follows from (3.20) that 𝑥_𝑛−𝑥²=𝑥_𝑛−𝑥,𝑥_𝑛−𝑥=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−𝑥,𝑥_𝑛−𝑥+1−𝛽_𝑛𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛−𝑥𝑑𝑠,𝑥_𝑛−𝑥=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑥),𝑥_𝑛−𝑥+𝛼_𝑛𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−𝑥,𝑥_𝑛−𝑥+1−𝛽_𝑛𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛−𝑥𝑑𝑠,𝑥_𝑛−𝑥≤𝛼_𝑛𝛾𝑥_𝑛−𝑥²−𝜙𝑥_𝑛−𝑥𝑥_𝑛−𝑥+𝛼_𝑛𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥𝛽_𝑛𝑥_𝑛−𝑥²+1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝑥_𝑛−𝑥², (3.21) and, so 𝛾𝜙𝑥_𝑛−𝑥𝑥_𝑛−𝑥+1−𝛾+𝜇𝛾𝑥_𝑛−𝑥²≤𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥. (3.22) It follows that 𝛾𝜙𝑥_𝑛−𝑥𝑥_𝑛−𝑥≤𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥≤𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥𝑥_𝑛−𝑥. (3.23) Hence 𝑥_𝑛−𝑥≤𝜙⁻¹𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥𝛾, (3.24) which implies that {𝑥_𝑛} is bounded, so are {𝑧_𝑛}, {𝑦_𝑛}, and {𝑣_𝑛}. Since 𝑓 is 𝜙-strongly pseudocontractive, we have 𝑓𝑥_𝑛−𝑓(𝑥),𝑥_𝑛−𝑥≤𝑥_𝑛−𝑥²−𝜙𝑥_𝑛−𝑥𝑥_𝑛−𝑥≤𝑥_𝑛−𝑥². (3.25) Thus {𝑓(𝑥_𝑛)} is bounded.
Next, we show that lim_𝑛→∞‖𝑥_𝑛−𝑇(ℎ)𝑥_𝑛‖=0, for all ℎ≥0. From (3.18), we observe that 𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠≤𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛−1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠+𝛽_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠. (3.26) It follows that 𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛≤𝛼𝑑𝑠^𝑛1−𝛽_𝑛𝑢+𝛾𝑓𝑥_𝑛−1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠. (3.27) By the conditions (C1) and (C2), we obtain lim_𝑛→∞𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠=0. (3.28) Let 𝐵={𝜔∈𝐻∶‖𝜔−𝑥‖≤𝜙⁻¹(‖𝑢+𝛾𝑓(𝑥)−(𝐼+𝜇𝐴)𝑥‖/𝛾)}, then 𝐵 is nonempty bounded closed and convex subset of 𝐻, which is 𝑇(ℎ)-invariant for all ℎ≥0 and contains {𝑥_𝑛}. It follows from Lemma 2.4 that lim_𝑛→∞1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛(ℎ)1𝑑𝑠−𝑇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠=0. (3.29) On the other hand, we note that 𝑥_𝑛−𝑇(ℎ)𝑥_𝑛≤𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛+1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛(ℎ)1𝑑𝑠−𝑇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛+𝑇(ℎ)1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑇(ℎ)𝑥_𝑛≤2𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛+1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛(ℎ)1𝑑𝑠−𝑇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠. (3.30) From (3.28) and (3.29), for all ℎ≥0, we have lim_𝑛→∞𝑥_𝑛−𝑇(ℎ)𝑥_𝑛=0. (3.31)
Next, we show that lim_𝑛→∞‖𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛‖=0 for all 𝑘∈{1,2,…,𝑁}. Since 𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛} is firmly nonexpansive for all 𝑘∈{1,2,…,𝑁} and 𝑉^𝑘_𝑛=𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝑉^{(Θ𝑘−1,𝜑𝑘−1)}_{𝑟𝑘−1,𝑛}⋅⋅⋅𝑉^(Θ1,𝜑1)_𝑟1,𝑛 for 𝑘∈{1,2,…,𝑁}, hence for 𝑥∈Ω, we have 𝑉^𝑘_𝑛𝑥_𝑛−𝑥²=𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝑉_𝑛^𝑘−1𝑥_𝑛−𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝑥²≤𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝑉_𝑛^𝑘−1𝑥_𝑛−𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝑥,𝑉_𝑛^𝑘−1𝑥_𝑛−𝑥=𝑉^𝑘_𝑛𝑥_𝑛−𝑥,𝑉_𝑛^𝑘−1𝑥_𝑛−𝑥=12𝑉^𝑘_𝑛𝑥_𝑛−𝑥²+𝑉_𝑛^𝑘−1𝑥_𝑛−𝑥²−𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛². (3.32) It follows that 𝑉^𝑘_𝑛𝑥_𝑛−𝑥²≤𝑥_𝑛−𝑥²−𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛². (3.33) Now, by Lemma 2.8, we have 𝑥_𝑛−𝑥²=𝛼_𝑛𝑢+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥+𝛽_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛+𝑑𝑠𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑥²≤𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑥+𝛽_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠²+2𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥≤𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑥+𝛽_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠²+2𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝑣_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠²+2𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥𝑥_𝑛−𝑥=1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑣_𝑛−𝑥²+𝑐_𝑛, (3.34) where 𝑐_𝑛∶=𝛽²_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠²+21−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝛽_𝑛𝑣_𝑛−𝑥𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠+2𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛−𝐼+𝜇𝐴𝑥𝑥_𝑛−𝑥. (3.35) From the condition (C1) and (3.28), we have lim_𝑛→∞𝑐_𝑛=0. (3.36) From (3.20), we observe that 𝑣_𝑛−𝑥≤𝑦_𝑛−𝑥≤𝑉^𝑁_𝑛𝑥_𝑛−𝑥≤𝑉^𝑘_𝑛𝑥_𝑛−𝑥,∀𝑘∈{1,2,…,𝑁}. (3.37) Substituting (3.33) into (3.34), we have 𝑥_𝑛−𝑥²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑥_𝑛−𝑥²−𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛²+𝑐_𝑛=1−2𝛼_𝑛1+𝜇𝛾+𝛼²_𝑛1+𝜇𝛾²𝑥_𝑛−𝑥²−1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛²+𝑐_𝑛, (3.38) which in turn implies that 1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛²≤1+𝛼²_𝑛1+𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+𝑐_𝑛. (3.39) From the condition (C1) and from (3.36), we obtain that lim_𝑛→∞𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛=0. (3.40) On the other hand, we observe that 𝑥_𝑛−𝑧_𝑛=𝑥_𝑛−𝑉^𝑁_𝑛𝑥_𝑛=^𝑁_𝑘=1𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛≤^𝑁_𝑘=1𝑉^𝑘_𝑛𝑥_𝑛−𝑉_𝑛^𝑘−1𝑥_𝑛. (3.41) Then, we have lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (3.42) Moreover, we observe that 𝑧_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠≤𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠, (3.43) and hence lim_𝑛→∞𝑧_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠=0. (3.44) Next, we show that for all 𝑥,𝑦∈Ω, lim_𝑛→∞‖𝐵₁𝑦_𝑛−𝐵₁𝑦‖=0 and lim_𝑛→∞‖𝐵₂𝑧_𝑛−𝐵₂𝑥‖=0. By the cocoercivity of the mapping 𝐵₁, we have 𝑣_𝑛−𝑥²=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦²≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦²=𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦²=𝑦_𝑛−𝑦²−2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦,𝑦_𝑛−𝑦+𝜌²₁𝐵₁𝑦_𝑛−𝐵₁𝑦²≤𝑥_𝑛−𝑥²−2𝜌₁−𝑐₁𝐵₁𝑦_𝑛−𝐵₁𝑦²+𝑑₁𝑦_𝑛−𝑦²+𝜌²₁𝐵₁𝑦_𝑛−𝐵₁𝑦²≤𝑥_𝑛−𝑥²+2𝜌₁𝑐₁+𝜌²₁−2𝜌₁𝑑₁𝐿²₁𝐵₁𝑦_𝑛−𝐵₁𝑦². (3.45) Similarly, we have 𝑦_𝑛−𝑦²=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥²≤𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑥−𝜌₂𝐵₂𝑥²=𝑧_𝑛−𝑥−𝜌₂𝐵₂𝑧_𝑛−𝐵₂𝑥²=𝑧_𝑛−𝑥²−2𝜌₂𝐵₂𝑧_𝑛−𝐵₂𝑥,𝑧_𝑛−𝑥+𝜌²₂𝐵₂𝑧_𝑛−𝐵₂𝑥²≤𝑥_𝑛−𝑥²−2𝜌₂−𝑐₂𝐵₂𝑧_𝑛−𝐵₂𝑥²+𝑑₂𝑧_𝑛−𝑥²+𝜌²₂𝐵₂𝑧_𝑛−𝐵₂𝑥²≤𝑥_𝑛−𝑥²+2𝜌₂𝑐₂+𝜌²₂−2𝜌₂𝑑₂𝐿²₂𝐵₂𝑧_𝑛−𝐵₂𝑥². (3.46) Substituting (3.45) into (3.34), we have 𝑥_𝑛−𝑥²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑥_𝑛−𝑥²+2𝜌₁𝑐₁+𝜌²₁−2𝜌₁𝑑₁𝐿²₁𝐵₁𝑦_𝑛−𝐵₁𝑦²+𝑐_𝑛. (3.47) Again, from (3.34), we obtain 𝑥_𝑛−𝑥²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑣_𝑛−𝑥²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑦_𝑛−𝑦²+𝑐_𝑛≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑥_𝑛−𝑥²+2𝜌₂𝑐₂+𝜌²₂−2𝜌₂𝑑₂𝐿²₂𝐵₂𝑧_𝑛−𝐵₂𝑥²+𝑐_𝑛. (3.48) Therefore, from (3.47) and (3.48), we obtain 1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²−2𝜌₁𝑐₁−𝜌²₁+2𝜌₁𝑑₁𝐿²₁𝐵₁𝑦_𝑛−𝐵₁𝑦²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+𝑐_𝑛≤1+𝛼²_𝑛1+𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+𝑐_𝑛,1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²−2𝜌₂𝑐₂−𝜌²₂+2𝜌₂𝑑₂𝐿²₂𝐵₂𝑧_𝑛−𝐵₂𝑥²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+𝑐_𝑛≤1+𝛼²_𝑛1+𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+𝑐_𝑛. (3.49) From the condition (𝐶1) and from (3.36), we obtain that lim_𝑛→∞𝐵₁𝑦_𝑛−𝐵₁𝑦=0,lim_𝑛→∞𝐵₂𝑧_𝑛−𝐵₂𝑥=0. (3.50) Next, we show that lim_𝑛→∞‖𝑧_𝑛−𝑣_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑣_𝑛‖=0. By the nonexpansiveness of 𝐼−𝜌₂𝐵₂, we have 𝑦_𝑛−𝑦²=𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝐽_{(𝑀2,𝜌2)}𝑥−𝜌₂𝐵₂𝑥²≤𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑥−𝜌₂𝐵₂𝑥,𝑦_𝑛−𝑦=12𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑥−𝜌₂𝐵₂𝑥²+𝑦_𝑛−𝑦²−𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑥−𝜌₂𝐵₂𝑥−𝑦_𝑛−𝑦²≤12𝑧_𝑛−𝑥²+𝑦_𝑛−𝑦²−𝑧_𝑛−𝑦_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝐵₂𝑥−𝑥−𝑦²=12𝑧_𝑛−𝑥²+𝑦_𝑛−𝑦²−𝑧_𝑛−𝑦_𝑛−𝑥−𝑦²+2𝜌₂𝑧_𝑛−𝑦_𝑛−𝑥−𝑦,𝐵₂𝑧_𝑛−𝐵₂𝑥−𝜌²₂𝐵₂𝑧_𝑛−𝐵₂𝑥². (3.51) So, we obtain 𝑦_𝑛−𝑦²≤𝑧_𝑛−𝑥²−𝑧_𝑛−𝑦_𝑛−𝑥−𝑦²+2𝜌₂𝑧_𝑛−𝑦_𝑛−𝑥−𝑦,𝐵₂𝑧_𝑛−𝐵₂𝑥. (3.52) Substituting (3.52) into (3.34), we have 𝑥_𝑛−𝑥²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑧_𝑛−𝑥²−𝑧_𝑛−𝑦_𝑛−𝑥−𝑦²+2𝜌₂𝑧_𝑛−𝑦_𝑛−𝑥−𝑦,𝐵₂𝑧_𝑛−𝐵₂𝑥+𝑐_𝑛, (3.53) which in turn implies that 1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑧_𝑛−𝑦_𝑛−𝑥−𝑦²≤1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²2𝜌₂𝑧_𝑛−𝑦_𝑛−𝑥−𝑦×𝐵₂𝑧_𝑛−𝐵₂𝑥+𝑐_𝑛≤1+𝛼²_𝑛1+𝜇𝛾²𝑥_𝑛−𝑥²−𝑥_𝑛−𝑥²+1−𝛼_𝑛−𝛼_𝑛𝜇𝛾²2𝜌₂𝑧_𝑛−𝑦_𝑛−𝑥−𝑦×𝐵₂𝑧_𝑛−𝐵₂𝑥+𝑐_𝑛. (3.54) From the condition (𝐶1) and from (3.36), (3.50), we obtain that lim_𝑛→∞𝑧_𝑛−𝑦_𝑛−𝑥−𝑦=0. (3.55) On the other hand, by Lemma 2.8 and from (2.3), we have 𝑦_𝑛−𝑣_𝑛+𝑥−𝑦²=𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦−𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦+𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦²≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦−𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦²+2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦,𝑦_𝑛−𝑣_𝑛+𝑥−𝑦≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦²−𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦²+2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦𝑦_𝑛−𝑣_𝑛+𝑥−𝑦≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝐽_{(𝑀1,𝜌1)}𝑦−𝜌₁𝐵₁𝑦𝑑𝑠²+2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦𝑦_𝑛−𝑣_𝑛+𝑥−𝑦=𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦²−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥𝑑𝑠²+2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦𝑦_𝑛−𝑣_𝑛+𝑥−𝑦≤𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑥×𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦+1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑥+2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦𝑦_𝑛−𝑣_𝑛+𝑥−𝑦=𝑧_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠+𝑥−𝑦−𝑧_𝑛−𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦×𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑦−𝜌₁𝐵₁𝑦+1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠−𝑥+2𝜌₁𝐵₁𝑦_𝑛−𝐵₁𝑦𝑦_𝑛−𝑣_𝑛+𝑥−𝑦. (3.56) From (3.44), (3.50), and (3.55), we obtain that lim_𝑛→∞𝑦_𝑛−𝑣_𝑛+𝑥−𝑦=0. (3.57) In fact, since 𝑧_𝑛−𝑣_𝑛≤𝑧_𝑛−𝑦_𝑛−𝑥−𝑦+𝑦_𝑛−𝑣_𝑛+𝑥−𝑦, (3.58) so lim_𝑛→∞𝑧_𝑛−𝑣_𝑛=0. (3.59) And since 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑧_𝑛+𝑧_𝑛−𝑣_𝑛, (3.60) and so lim_𝑛→∞𝑥_𝑛−𝑣_𝑛=0. (3.61)
Next, we show that 𝑥∈Ω∶=𝐹(𝒮)∩⋂^𝑁_𝑘=1MEP(Θ_𝑘,𝜑_𝑘)∩𝐹(𝑄).(i)We first show that 𝑥∈𝐹(𝒮). Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥∈𝐻 as 𝑗→∞. From (3.31) and Lemma 2.5, we obtain that 𝑥∈𝐹(𝒮).(ii)Now, we show that 𝑥∈⋂^𝑁_𝑘=1MEP (Θ_𝑘,𝜑_𝑘). Since 𝑉^𝑘_𝑛=𝑉^{(Θ𝑘,𝜑𝑘)}_{𝑟𝑘,𝑛}𝑉_𝑛^𝑘−1 for 𝑘∈{1,2,…,𝑁}. Hence, for all 𝑥∈𝐻 and for all 𝑘∈{1,2,…,𝑁}, we obtainΘ_𝑘𝑉^𝑘_𝑛𝑥_𝑛,𝑥+𝜑_𝑘(𝑥)−𝜑_𝑘𝑉^𝑘_𝑛𝑥_𝑛+1𝑟_𝑘,𝑛𝐾^′_𝑘𝑉^𝑘_𝑛𝑥_𝑛−𝐾^′_𝑘𝑉_𝑛^𝑘−1𝑥_𝑛,𝜂_𝑘𝑥,𝑉^𝑘_𝑛𝑥_𝑛≥0. (3.62) And hence 𝐾^′_𝑘𝑉^𝑘_𝑛𝑥_𝑛𝑗−𝐾^′_𝑘𝑉_𝑛^𝑘−1𝑥_𝑛𝑗𝑟_𝑘,𝑛,𝜂_𝑘𝑥,𝑉^𝑘_𝑛𝑥_𝑛𝑗≥−Θ_𝑘𝑉^𝑘_𝑛𝑥_𝑛𝑗,𝑥−𝜑_𝑘(𝑥)+𝜑_𝑘𝑉^𝑘_𝑛𝑥_𝑛𝑗,∀𝑥∈𝐻. (3.63)
By the assumptions that 𝜑_𝑘 is lower semicontinuous and by conditions (H4), (H5), the mapping 𝑥↦(−Θ_𝑘(𝑥,𝑦)) is lower semicontinuous. So, they are weakly lower semicontinuous. Since 𝑥_𝑛𝑗⇀𝑥, we have that 𝑉^𝑘_𝑛𝑥_𝑛⇀𝑥 for all 𝑘∈{1,2,…,𝑁} and from (i)(c), (ii), (3.40). Now, taking lower limit as 𝑗→∞ in (3.63), we obtain that Θ_𝑘(𝑥,𝑥)+𝜑_𝑘(𝑥)−𝜑_𝑘(𝑥)≥0,∀𝑥∈𝐻,∀𝑘∈{1,2,…,𝑁}. (3.64) Therefore 𝑥∈⋂^𝑁_𝑘=1MEP(Θ_𝑘,𝜑_𝑘).
(iii) Now, we show that 𝑥∈𝐹(𝑄), where 𝑄 is defined as in Lemma 2.11. Since 𝑄 is nonexpansive. Then, we have 𝑣_𝑛−𝑄𝑣_𝑛=𝐽_{(𝑀1,𝜌1)}𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛−𝑄𝑣_𝑛=𝐽_{(𝑀1,𝜌1)}𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝜌₁𝐵₁𝐽_{(𝑀2,𝜌2)}𝑧_𝑛−𝜌₂𝐵₂𝑧_𝑛−𝑄𝑣_𝑛=𝑄𝑧_𝑛−𝑄𝑣_𝑛≤𝑧_𝑛−𝑣_𝑛. (3.65) From (3.59), we have lim_𝑛→∞‖𝑣_𝑛−𝑄𝑣_𝑛‖=0. Since 𝑥_𝑛𝑗⇀𝑥 and from (3.61), we also have 𝑣_𝑛𝑗⇀𝑥. Hence, we obtain by Lemma 2.5 that 𝑥∈𝐹(𝑄).
Next, we show that {𝑥_𝑛} is sequentially compact, namely, there is a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} that converges strongly to 𝑥∈Ω as 𝑗→∞. From (3.23), replacing 𝑥 by 𝑥 to obtain 𝛾𝜙𝑥_𝑛−𝑥𝑥_𝑛−𝑥≤𝑢+𝛾𝑓(𝑥)−𝐼+𝜇𝐴𝑥,𝑥_𝑛−𝑥. (3.66) Now, replacing 𝑛 with 𝑛_𝑗 in (3.66) and letting 𝑗→∞, since 𝑥_𝑛𝑗⇀𝑥, we obtain that 𝑥_𝑛𝑗→𝑥 as 𝑗→∞.
Next, we show that 𝑥 is the unique solution of the variational inequality (3.5). Since 𝑥_𝑛=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠, (3.67) we derive that −𝑢+𝐼+𝜇𝐴−𝛾𝑓𝑥_𝑛=−1−𝛽_𝑛𝛼_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛+𝑥𝑑𝑠𝐼+𝜇𝐴_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠. (3.68) For all 𝑧∈Ω, it follows that −𝑢+𝐼+𝜇𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑧=−1−𝛽_𝑛𝛼_𝑛1𝐼−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑄𝑉^𝑁_𝑛𝑑𝑠𝑥_𝑛−1𝐼−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑄𝑉^𝑁_𝑛𝑑𝑠𝑧,𝑥_𝑛+𝑥−𝑧𝐼+𝜇𝐴_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑣_𝑛𝑑𝑠,𝑥_𝑛−𝑧. (3.69)
By Lemma 2.10, we obtain that 𝐼−𝜎_𝑡(⋅) is monotone, where 𝜎_𝑡(⋅)∶=(1/𝑡)∫^𝑡₀𝑇(𝑠)𝑑𝑠. Then, from (3.69), we have −𝑢+𝐼+𝜇𝐴−𝛾𝑓𝑥_𝑛,𝑥_𝑛−𝑧≤𝐼+𝜇𝐴𝑥_𝑛−𝜎_𝑡𝑣_𝑛,𝑥_𝑛−𝑧. (3.70) Now, replacing 𝑛 by 𝑛_𝑗 in (3.70) and letting 𝑗→∞ and 𝑥_𝑛𝑗→𝑥, we notice that 𝑥_𝑛𝑗−1𝑡_𝑛𝑗^𝑡𝑛𝑗₀𝑇(𝑠)𝑣_𝑛𝑗𝑑𝑠→0. (3.71) Then, we have 𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥,𝑧−𝑥≤0,∀𝑧∈Ω. (3.72) That is, 𝑥 is the solution of variational inequality (3.5).
Finally, we show that {𝑥_𝑛} converges strongly to 𝑥∈Ω. Suppose that there exists another subsequence 𝑥_𝑛𝑘→𝑥 as 𝑘→∞. We note Lemma 2.5 that 𝑥∈Ω is the solution of the variational inequality (3.5). Hence 𝑥=𝑥=𝑥^∗ by uniqueness. In summary, we have shown that {𝑥_𝑛} is sequentially compact and each cluster point of the sequence {𝑥_𝑛} is equal to 𝑥^∗. Then, we conclude that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. This completes the proof.

From Theorem 3.2, we can deduce the following result.

Theorem 3.3. Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝜑_𝑖∶𝐶→ℝ(𝑖=1,2,…,𝑁) be a finite family of lower semicontinuous and convex function, Θ_𝑖∶𝐶×𝐶→ℝ(𝑖=1,2,…,𝑁) be a finite family of bifunctions satisfying (H1)–(H5), 𝜂_𝑖∶𝐶×𝐻→𝐻 be a finite family of Lipschitz continuous mappings with a constant 𝜎_𝑖(𝑖=1,2,…,𝑁). Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐶 into 𝐻 and 𝐵_𝑖∶𝐻→𝐻(𝑖=1,2) be an 𝐿_𝑖-Lipschitzian and relaxed (𝑐_𝑖,𝑑_𝑖)-cocoercive mapping with 𝜌_𝑖∈(0,2(𝑑_𝑖−𝑐_𝑖𝐿²_𝑖)/𝐿²_𝑖] for all 𝑖=1,2. Let 𝑄∶𝐶→𝐶 be a mapping defined by 𝑄𝑥∶=𝑃_𝐶𝑃_𝐶𝑥−𝜌₂𝐵₂𝑥−𝜌₁𝐵₁𝑃_𝐶𝑥−𝜌₂𝐵₂𝑥. (3.73) Assume that Ω∶=𝐹(𝒮)∩⋂^𝑁_𝑘=1MEP(Θ_𝑘,𝜑_𝑘)∩𝐹(𝑄)≠∅. Let 𝑓∶𝐻→𝐻 be a 𝜙-strongly pseudocontractive mapping with lim_𝑡→+∞𝜙(𝑡)=+∞ and 𝐴 be a strongly positive linear bounded operator on 𝐻 with a coefficient 𝛾>0. Let 𝜇>0 and 𝛾>0 be two constants such that 0<𝛾<1+𝜇𝛾. Let {𝑟_𝑖,𝑛}(𝑖=1,2,…,𝑁) be a finite family of positive real sequence such that liminf_𝑛→∞𝑟_𝑖,𝑛>0, {𝛼_𝑛} and {𝛽_𝑛} be two sequences in [0,1], and {𝑡_𝑛} be a positive real divergent sequence. For any fixed 𝑢∈𝐻, let {𝑥_𝑛} be the sequence defined by Θ₁𝑢_𝑛⁽¹⁾,𝑥+𝜑₁(𝑥)−𝜑₁𝑢_𝑛⁽¹⁾+1𝑟_1,𝑛𝐾^′₁𝑢_𝑛⁽¹⁾−𝐾^′₁𝑥_𝑛,𝜂₁𝑥,𝑢_𝑛⁽¹⁾Θ≥0,∀𝑥∈𝐶,₂𝑢_𝑛⁽²⁾,𝑥+𝜑₂(𝑥)−𝜑₂𝑢_𝑛⁽²⁾+1𝑟_2,𝑛𝐾^′₂𝑢_𝑛⁽²⁾−𝐾^′₂𝑢_𝑛⁽¹⁾,𝜂₁𝑥,𝑢_𝑛⁽²⁾⋮Θ≥0,∀𝑥∈𝐶,_𝑁𝑢_𝑛^(𝑁),𝑥+𝜑_𝑁(𝑥)−𝜑_𝑁𝑢_𝑛^(𝑁)+1𝑟_𝑁,𝑛𝐾^′_𝑁𝑢_𝑛^(𝑁)−𝐾^′_𝑁𝑢_𝑛^(𝑁−1),𝜂_𝑁𝑥,𝑢_𝑛^(𝑁)𝑦≥0,∀𝑥∈𝐶,_𝑛=𝑃_𝐶𝑢_𝑛^(𝑁)−𝜌₂𝐵₂𝑢_𝑛^(𝑁),𝑥_𝑛=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛1𝐼+𝜇𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑃_𝐶𝑦_𝑛−𝜌₁𝐵₁𝑦_𝑛𝑑𝑠, (3.74) where 𝑢_𝑛⁽¹⁾=𝑉^(Θ1,𝜑1)_𝑟1,𝑛𝑥_𝑛,𝑢_𝑛^(𝑖)=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}𝑢_𝑛^(𝑖−1)=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}𝑉^{(Θ𝑖−1,𝜑𝑖−1)}_{𝑟𝑖−1,𝑛}𝑢_𝑛^(𝑖−2)=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}⋯𝑉^(Θ2,𝜑2)_𝑟2,𝑛𝑢_𝑛⁽¹⁾=𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}⋯𝑉^(Θ2,𝜑2)_𝑟2,𝑛𝑉^(Θ1,𝜑1)_𝑟1,𝑛𝑥_𝑛,𝑖=2,3,…,𝑁, (3.75) and 𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛}∶𝐶→𝐶, 𝑖=1,2,…,𝑁 is the mapping defined by (2.8). Assume the following.(i)𝜂_𝑖∶𝐶×𝐶→ℝ is Lipschitz continuous with constant 𝜎_𝑖>0(𝑖=1,2,…,𝑁) such that(a)𝜂_𝑖(𝑥,𝑦)+𝜂_𝑖(𝑦,𝑥)=0 for all 𝑥,𝑦∈𝐶,(b)𝜂_𝑖(⋅,⋅) is affine in the first variable,(c)for each fixed 𝑦∈𝐶, 𝑥↦𝜂_𝑖(𝑦,𝑥) is sequentially continuous from the weak topology to the weak topology.(ii)𝐾_𝑖∶𝐶→ℝ is 𝜂_𝑖—strongly convex with constant 𝜇_𝑖>0, and its derivative 𝐾^′_𝑖 is not only continuous from the weak topology to the strong topology but also Lipschitz continuous with constant 𝜈_𝑖 such that 𝜇_𝑖≥𝜎_𝑖𝜈_𝑖.(iii)For all 𝑖=1,2,…,𝑁 and for all 𝑥∈𝐶, there exists a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that for all 𝑦∉𝐷_𝑥,Θ_𝑖𝑦,𝑧_𝑛+𝜑_𝑖𝑧_𝑥−𝜑_𝑖𝑦+1𝑟_𝑖,𝑛𝐾^′_𝑖𝑦−𝐾^′_𝑖(𝑥),𝜂_𝑖𝑧_𝑥,𝑦<0. (3.76) If the following conditions are satisfied:(C1)lim_𝑛→∞𝛼_𝑛=0,(C2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, then the sequence {𝑥_𝑛} defined by (3.74) converges strongly to 𝑥^∗∈Ω∶=𝐹(𝒮)∩⋂^𝑁_𝑘=1MEP(Θ_𝑘,𝜑_𝑘)∩𝐹(𝑄), provided 𝑉^{(Θ𝑖,𝜑𝑖)}_{𝑟𝑖,𝑛} is firmly nonexpansive, where 𝑥^∗ is the unique solution of the variational inequality 𝑢+𝛾𝑓−𝐼+𝜇𝐴𝑥^∗,𝑧−𝑥^∗≤0,∀𝑧∈Ω, (3.77) or, equivalently, 𝑥^∗ is the unique solution of the optimization problem min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥), (3.78) where ℎ is a potential function for 𝛾𝑓 and (𝑥^∗,𝑦^∗) is the solution of general system of variational inequality problem 𝜌₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜌≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶. (3.79)

Proof. From Theorem 3.2, taking 𝑀₁=𝑀₂=𝜕𝛿_𝐶, where 𝐶 is a nonempty closed convex subset of 𝐻 and 𝛿_𝐶∶𝐻→[0,+∞) is the indicator function of 𝐶, then, we have 𝐽_{(𝑀1,𝜌1)}=𝐽_{(𝑀2,𝜌2)}=𝑃_𝐶 and the quasivariational inclusion problem (1.17) is equivalent to the classical variational inequality (1.19). Thus, we can get the desired conclusion immediately.

Acknowledgments

This paper was supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand (under Project no. RG-1-54-02-1). P. Sunthrayuth was partially supported by the Centre of Excellence in Mathematics for the Ph.D. Program at KMUTT. Moreover, the authors are greatly indebted to the reviewers for their extremely constructive comments and valuable suggestions leading to the revised version.