Iterative Algorithms for Systems of Generalized Equilibrium Problems with the Constraints of Variational Inclusion and Fixed Point Problems

Abstract

We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, 𝐶 be a nonempty closed convex subset of 𝐻 and 𝑃_𝐶 be the metric projection of 𝐻 onto 𝐶. Let 𝑆∶𝐶→𝐻 be a nonlinear mapping on 𝐶. We denote by Fix(𝑆) the set of fixed points of 𝑆 and by 𝐑 the set of all real numbers. A mapping 𝑉 is called strongly positive on 𝐻 if there exists a constant 𝛾∈(0,1] such that ⟨𝑉𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻.(1) A mapping 𝑆∶𝐶→𝐻 is called 𝐿-Lipschitz continuous if there exists a constant 𝐿≥0 such that 𝑆𝑥−𝑆𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶.(2) In particular, if 𝐿=1 then 𝑆 is called a nonexpansive mapping; if 𝐿∈[0,1) then 𝐴 is called a contraction.

Let 𝐴∶𝐶→𝐻 be a nonlinear mapping on 𝐶. We consider the following variational inequality problem (VIP) [1] which is to find a point 𝑥∈𝐶 such that 𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶.(3) The solution set of VIP (3) is denoted by VI(𝐶,𝐴).

Let 𝜑∶𝐶→𝐑 be a real-valued function, 𝐴∶𝐻→𝐻 be a nonlinear mapping and Θ∶𝐶×𝐶→𝐑 be a bifunction. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding 𝑥∈𝐶 such that Θ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)+𝐴𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶.(4) We denote the set of solutions of GMEP (4) by GMEP(Θ,𝜑,𝐴). The system of equilibrium problems or generalized equilibrium problems is a tool to study Nash eequilibrium problems, see for example [3–8]. In fact, the GMEP (4) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see for example, [9–15]. Here we also consider a system of two generalized equilibrium problem that could be usefull to study the Two players game problem, see [16].

Throughout this paper, it is assumed as in [2] that Θ∶𝐶×𝐶→𝐑 is a bifunction satisfying conditions (H1)–(H4) and 𝜑∶𝐶→𝐑 is a lower semicontinuous and convex function with restriction (H5), where(H1)Θ(𝑥,𝑥)=0 for all 𝑥∈𝐶;(H2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(H3)Θ is upper-hemicontinuous, that is, for each 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡→0+Θ𝑡𝑧+(1−𝑡)𝑥,𝑦≤Θ𝑥,𝑦;(5)(H4)Θ(𝑥,⋅) is convex and lower semicontinuous for each 𝑥∈𝐶;(H5)for each 𝑥∈𝐻 and 𝑟>0, there exists a bounded subset 𝐷_𝑥⊂𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, Θ𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0.(6)

Given a positive number 𝑟>0. Let 𝑆_𝑟^(Θ,𝜑)∶𝐻→𝐶 is the solution set of the auxiliary mixed equilibrium problem, that is, for each 𝑥∈𝐻, 𝑆_𝑟^(Θ,𝜑)(𝑥)(𝑧)𝑦+1∶=𝑦∈𝐶∶Θ𝑦,𝑧+𝜑−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝑧−𝑦≥0,∀𝑧∈𝐶.(7) In particular, whenever 𝐾(𝑥)=(1/2)‖𝑥‖²,∀𝑥∈𝐻, 𝑆_𝑟^(Θ,𝜑) is rewritten as 𝑇_𝑟^(Θ,𝜑).

Let Θ₁,Θ₂∶𝐶×𝐶→𝐑 be two bifunctions, and 𝐴₁,𝐴₂∶𝐶→𝐻 be two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that Θ₁𝑥^∗,𝑥+𝐴₁𝑦^∗,𝑥−𝑥^∗+1𝜈₁𝑥^∗−𝑦^∗,𝑥−𝑥^∗Θ≥0,∀𝑥∈𝐶,₂𝑦^∗,𝑦+𝐴₂𝑥^∗,𝑦−𝑦^∗+1𝜈₂𝑦^∗−𝑥^∗,𝑦−𝑦^∗≥0,∀𝑦∈𝐶,(8) where 𝜈₁>0 and 𝜈₂>0 are two constants. It is introduced and studied in [17]. Whenever Θ₁≡Θ₂≡0, the SGEP reduces to a system of variational inequalities, which is considered and studied in [18]. It is worth to mention that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.

In 2010, Ceng and Yao [17] transformed the SGEP into a fixed point problem in the following way.

Proposition CY (see [17]). Let Θ₁,Θ₂∶𝐶×𝐶→𝐑 be two bifunctions satisfying conditions (H1)–(H4) and let 𝐴_𝑘∶𝐶→𝐻 be 𝜁_𝑘-inverse-strongly monotone for 𝑘=1,2. Let 𝜈_𝑘∈(0,2𝜁_𝑘) for 𝑘=1,2. Then, (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of 𝑆𝐺𝐸𝑃 (8) if and only if 𝑥^∗ is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺=𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂) where 𝑦^∗=𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑥^∗. Here, we denote the fixed point set of 𝐺 by 𝑆𝐺𝐸𝑃(𝐺).

Let {𝑇_𝑛}^∞_𝑛=1 be an infinite family of nonexpansive mappings on 𝐻 and {𝜆_𝑛}^∞_𝑛=1 be a sequence of nonnegative numbers in [0,1]. For any 𝑛≥1, define a mapping 𝑊_𝑛 on 𝐻 as follows: 𝑈_𝑛,𝑛+1𝑈=𝐼,_𝑛,𝑛=𝜆_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝑈𝐼,_{𝑛,𝑛−1}=𝜆_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜆_𝑛−1⋯𝑈𝐼,_𝑛,𝑘=𝜆_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝑈𝐼,_{𝑛,𝑘−1}=𝜆_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜆_𝑘−1⋯𝑈𝐼,_𝑛,2=𝜆₂𝑇₂𝑈_𝑛,3+1−𝜆₂𝑊𝐼,_𝑛=𝑈_𝑛,1=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝐼.(9) Such a mapping 𝑊_𝑛 is called the 𝑊-mapping generated by 𝑇_𝑛,𝑇_𝑛−1,…,𝑇₁ and 𝜆_𝑛,𝜆_𝑛−1,…,𝜆₁.

In 2011, for the case where 𝐶=𝐻, Yao et al. [14] proposed the following hybrid iterative algorithm Θ𝑦_𝑛,𝑧+𝜑(𝑧)−𝜑𝑦_𝑛+1𝑟𝐾^′𝑦_𝑛−𝐾^′𝑥_𝑛,𝑧−𝑦_𝑛𝑥≥0,𝑧∈𝐻,_𝑛+1=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑊_𝑛𝑦_𝑛,∀𝑛≥1,(10) where 𝑓∶𝐻→𝐻 be a contraction, 𝐾∶𝐻→𝐑 is differentiable and strongly convex, {𝛼_𝑛},{𝛽_𝑛}⊂(0,1) and 𝑥₀,𝑢∈𝐻 are given, for finding a common element of the set MEP(Θ,𝜑) and the fixed point set ∩^∞_𝑛=1Fix(𝑇_𝑛) of an infinite family of nonexpansive mappings {𝑇_𝑛}^∞_𝑛=1 on 𝐻. They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (10) to a point 𝑥^∗∈∩^∞_𝑛=1Fix(𝑇_𝑛)∩MEP(Θ,𝜑) under some appropriate conditions. This point 𝑥^∗ also solves the following optimization problem: min_{𝑥∈∩∞𝑛=1Fix(𝑇𝑛)∩MEP(Θ,𝜑)}𝜇2⟨⟩+1𝑉𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥),(OP0) where ℎ∶𝐻→𝐑 is the potential function of 𝛾𝑓.

Let 𝑓∶𝐻→𝐻 be a contraction and 𝑉 be a strongly positive bounded linear operator on 𝐻. Assume that 𝜑∶𝐻→𝐑 is a lower semicontinuous and convex functional, that Θ,Θ₁,Θ₂∶𝐻×𝐻→𝐑 satisfy conditions (H1)–(H4), and that 𝐴,𝐴₁,𝐴₂∶𝐻→𝐻 are inverse-strongly monotone. Let the mapping 𝐺 be defined as in Proposition CY. Very recently, Ceng et al. [11] introduced the following hybrid extragradient-like iterative algorithm 𝑧_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑊_𝑛𝐺𝑧_𝑛,∀𝑛≥0,(11) for finding a common solution of GMEP (4), SGEP (8) and the fixed point problem of an infinite family of nonexpansive mappings {𝑇_𝑛}^∞_𝑛=1 on 𝐻, where {𝑟_𝑛}⊂(0,∞),{𝛼_𝑛},{𝛽_𝑛}⊂(0,1),𝜈_𝑘∈(0,2𝜁_𝑘),𝑘=1,2, and 𝑥₀,𝑢∈𝐻 are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (11) to a point 𝑥^∗∈∩^∞_𝑛=1Fix(𝑇_𝑛)∩GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺) under some suitable conditions. This point 𝑥^∗ also solves the following optimization problem: min_{𝑥∈∩∞𝑛=1Fix(𝑇𝑛)∩GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺)}𝜇2⟨⟩+1𝑉𝑥,𝑥2‖𝑥−𝑢‖²(𝑥),()−ℎOP1 where ℎ∶𝐻→𝐑 is the potential function of 𝛾𝑓.

On the other hand, let 𝐵 be a single-valued mapping of 𝐶 into 𝐻 and 𝑅 be a set-valued mapping with 𝐷(𝑅)=𝐶. Consider the following variational inclusion: find a point 𝑥∈𝐶 such that 0∈𝐵𝑥+𝑅𝑥.(12) We denote by 𝐼(𝐵,𝑅) the solution set of the variational inclusion (12). In particular, if 𝐵=𝑅=0, then 𝐼(𝐵,𝑅)=𝐶. If 𝐵=0, then problem (12) becomes the inclusion problem introduced by Rockafellar [19]. It is known that problem (12) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, and so forth. Let a set-valued mapping 𝑅∶𝐷(𝑅)⊂𝐻→2^𝐻 be maximal monotone. We define the resolvent operator 𝐽_𝑅,𝜆∶𝐻→𝐷(𝑅) associated with 𝑅 and 𝜆 as follows: 𝐽_𝑅,𝜆=(𝐼+𝜆𝑅)⁻¹,∀𝑥∈𝐻,(13) where 𝜆 is a positive number.

In this paper, we will introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems. Such solution also solves an optimization problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement and generalization of the previously known results in this area.

2. Preliminaries

Throughout this paper, we assume that 𝐻 is a real Hilbert space whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 to indicate that the sequence {𝑥_𝑛} converges strongly to 𝑥. Moreover, we use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛}, that is, 𝜔_𝑤𝑥_𝑛∶=𝑥∈𝐻∶𝑥_𝑛𝑖𝑥⇀𝑥forsomesubsequence_𝑛𝑖of𝑥_𝑛.(14)

Definition 1. A mapping 𝐴∶𝐶→𝐻 is called
(i) monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶,(15)
(ii) 𝜂-strongly monotone if there exists a constant 𝜂>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝜂𝑥−𝑦²,∀𝑥,𝑦∈𝐶,(16)
(iii) 𝜁-inverse-strongly monotone if there exists a constant 𝜁>0 such that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝜁𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶.(17)

It is easy to see that the projection 𝑃_𝐶 is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields.

Definition 2. A differentiable function 𝐾∶𝐻→𝐑 is called:
(i) convex, if 𝐾𝑦−𝐾(𝑥)≥𝐾^′(𝑥),𝑦−𝑥,∀𝑥,𝑦∈𝐻,(18) where 𝐾^′(𝑥) is the Frechet derivative of 𝐾 at 𝑥;
(ii) strongly convex, if there exists a constant 𝜎>0 such that 𝐾𝑦−𝐾(𝑥)−𝐾^′(𝑥)≥𝜎,𝑦−𝑥2𝑥−𝑦²,∀𝑥,𝑦∈𝐻.(19)

It is easy to see that if 𝐾∶𝐻→𝐑 is a differentiable strongly convex function with constant 𝜎>0 then 𝐾^′∶𝐻→𝐻 is strongly monotone with constant 𝜎>0.

The metric (or nearest point) projection from 𝐻 onto 𝐶 is the mapping 𝑃_𝐶∶𝐻→𝐶 which assigns to each point 𝑥∈𝐻 the unique point 𝑃_𝐶𝑥∈𝐶 satisfying the property 𝑥−𝑃_𝐶𝑥=inf_𝑦∈𝐶𝑥−𝑦=∶𝑑(𝑥,𝐶).(20)

Some important properties of projections are gathered in the following proposition.

Proposition 3. For given 𝑥∈𝐻 and 𝑧∈𝐶:(i)𝑧=𝑃_𝐶𝑥⇔⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶;(ii)𝑧=𝑃_𝐶𝑥⇔‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖², ∀𝑦∈𝐶;(iii)⟨𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖², ∀𝑦∈𝐻. (This implies that 𝑃_𝐶 is nonexpansive and monotone.)

By using the technique of [20], we can readily obtain the following elementary result.

Proposition 4 (see [11, Lemma 1 and Proposition 1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex function. Let Θ∶𝐶×𝐶→𝐑 be a bifunction satisfying the conditions (H1)–(H4). Assume that (i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻 and 𝑟>0, there exists a bounded subset 𝐷_𝑥⊂𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, Θ𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝐾^′(𝑧)−𝐾^′(𝑥),𝑦_𝑥−𝑧<0.(21) Then the following hold:(a) for each 𝑥∈𝐻,𝑆_𝑟^(Θ,𝜑)(𝑥)≠∅;(b) 𝑆_𝑟^(Θ,𝜑) is single-valued;(c) 𝑆_𝑟^(Θ,𝜑) is nonexpansive if 𝐾^′ is Lipschitz continuous with constant 𝜈>0 and𝐾^′𝑥₁−𝐾^′𝑥₂,𝑢₁−𝑢₂≥𝐾^′𝑢₁−𝐾^′𝑢₂,𝑢₁−𝑢₂,∀𝑥₁,𝑥₂∈𝐻×𝐻,(22) where 𝑢_𝑖=𝑆_𝑟^(Θ,𝜑)(𝑥_𝑖) for 𝑖=1,2;(d) for all 𝑠,𝑡>0 and 𝑥∈𝐻𝐾^′𝑆_𝑠^(Θ,𝜑)𝑥−𝐾^′𝑆_𝑡^(Θ,𝜑)𝑥,𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥≤𝑠−𝑡𝑠𝐾^′𝑆_𝑠^(Θ,𝜑)𝑥−𝐾^′(𝑥),𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥;(23)(e) 𝐹𝑖𝑥(𝑆_𝑟^(Θ,𝜑))=𝑀𝐸𝑃(Θ,𝜑);(f) 𝑀𝐸𝑃(Θ,𝜑) is closed and convex.
In particular, whenever Θ∶𝐶×𝐶→𝐑 is a bifunction satisfying the conditions (H1)–(H4) and 𝐾(𝑥)=(1/2)‖𝑥‖²,∀𝑥∈𝐻, then that is, for any 𝑥,𝑦∈𝐻, 𝑆_𝑟^(Θ,𝜑)𝑥−𝑆_𝑟^(Θ,𝜑)𝑦²≤𝑆_𝑟^(Θ,𝜑)𝑥−𝑆_𝑟^(Θ,𝜑)𝑦,𝑥−𝑦(24) ( 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive) and𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥≤|𝑠−𝑡|𝑠𝑆_𝑠^(Θ,𝜑)𝑥−𝑥,∀𝑠,𝑡>0,𝑥∈𝐻.(25) In this case, 𝑆_𝑟^(Θ,𝜑) is rewritten as 𝑇_𝑟^(Θ,𝜑). If, in addition, 𝜑≡0, then 𝑇_𝑟^(Θ,𝜑) is rewritten as 𝑇^Θ_𝑟.

Remark 5. Suppose 𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and 𝐾^′∶𝐻→𝐻 is Lipschitz continuous with constant 𝜈>0. Then 𝐾^′∶𝐻→𝐻 is 𝜎-strongly monotone and 𝜈-Lipschitz continuous with positive constants 𝜎,𝜈>0. Utilizing Proposition 4 (d) we can show that for all 𝑠,𝑡>0 and 𝑥∈𝐻, 𝑆_𝑠^(Θ,𝜑)𝑥−𝑆_𝑡^(Θ,𝜑)𝑥≤|𝑠−𝑡|𝑠⋅𝜈𝜎𝑆_𝑠^(Θ,𝜑)𝑥−𝑥.(26)

We need some facts and tools in a real Hilbert space 𝐻 which are listed as lemmas below.

Lemma 6. Let 𝑋 be a real inner product space. Then there holds the following inequality 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦,∀𝑥,𝑦∈𝑋.(27)

Lemma 7. Let 𝐻 be a real Hilbert space. Then the following hold:(a)‖𝑥−𝑦‖²=‖𝑥‖²−‖𝑦‖²−2⟨𝑥−𝑦,𝑦⟩ for all 𝑥,𝑦∈𝐻;(b)‖𝜆𝑥+𝜇𝑦‖²=𝜆‖𝑥‖²+𝜇‖𝑦‖²−𝜆𝜇‖𝑥−𝑦‖² for all 𝑥,𝑦∈𝐻 and 𝜆,𝜇∈[0,1] with 𝜆+𝜇=1;(c)If {𝑥_𝑛} is a sequence in 𝐻 such that 𝑥_𝑛⇀𝑥, it follows that limsup_𝑛→∞𝑥_𝑛−𝑦²=limsup_𝑛→∞𝑥_𝑛−𝑥²+𝑥−𝑦²,∀𝑦∈𝐻.(28)We have the following crucial lemmas concerning the 𝑊-mappings defined by (9).

Lemma 8 (see [21, Lemma 3.2]). Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive self-mappings on 𝐻 such that ∩^∞_𝑛=1Fix(𝑇_𝑛)≠∅ and let {𝜆_𝑛} be a sequence in (0,𝑏] for some 𝑏∈(0,1). Then, for every 𝑥∈𝐻 and 𝑘≥1 the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists, where 𝑈_𝑛,𝑘 is defined by (9).

Remark 9 (see [22, Remark 3.1]). It can be known from Lemma 8 that if 𝐷 is a nonempty bounded subset of 𝐻, then for 𝜖>0 there exists 𝑛₀≥𝑘 such that for all 𝑛>𝑛₀sup_𝑥∈𝐷𝑈_𝑛,𝑘𝑥−𝑈_𝑘𝑥≤𝜖.(29)

Remark 10 (see [22, Remark 3.2]). Utilizing Lemma 8, we define a mapping 𝑊∶𝐻→𝐻 as follows: 𝑊𝑥=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥,∀𝑥∈𝐻.(30) Such a 𝑊 is called the 𝑊-mapping generated by 𝑇₁,𝑇₂,… and 𝜆₁,𝜆₂,…. Since 𝑊_𝑛 is nonexpansive, 𝑊∶𝐻→𝐻 is also nonexpansive. Indeed, observe that for each 𝑥,𝑦∈𝐻𝑊𝑥−𝑊𝑦=lim_𝑛→∞𝑊_𝑛𝑥−𝑊_𝑛𝑦≤𝑥−𝑦.(31) If {𝑥_𝑛} is a bounded sequence in 𝐻, then we put 𝐷={𝑥_𝑛∶𝑛≥1}. Hence, it is clear from Remark 5 that for an arbitrary 𝜖>0 there exists 𝑁₀≥1 such that for all 𝑛>𝑁₀𝑊_𝑛𝑥_𝑛−𝑊𝑥_𝑛=𝑈_𝑛,1𝑥_𝑛−𝑈₁𝑥_𝑛≤sup_𝑥∈𝐷𝑈_𝑛,1𝑥−𝑈₁𝑥≤𝜖.(32) This implies that lim_𝑛→∞𝑊_𝑛𝑥_𝑛−𝑊𝑥_𝑛=0.(33)

Lemma 11 (see [21, Lemma 3.3]). Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive self-mappings on 𝐻 such that ∩^∞_𝑛=1Fix(𝑇_𝑛)≠∅, and let {𝜆_𝑛} be a sequence in (0,𝑏] for some 𝑏∈(0,1). Then, Fix(𝑊)=∩^∞_𝑛=1Fix(𝑇_𝑛).

Lemma 12 (see [23, Demiclosedness principle]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇 be a nonexpansive self-mapping on 𝐶. Then 𝐼−𝑇 is demiclosed. That is, whenever {𝑥_𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑇)𝑥_𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑇)𝑥=𝑦. Here 𝐼 is the identity operator of 𝐻.

Lemma 13. Let 𝐴∶𝐶→𝐻 be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3 (i)) implies 𝑢∈VI(𝐶,𝐴)⟺𝑢=𝑃_𝐶(𝑢−𝜆𝐴𝑢),𝜆>0.(34)

Lemma 14 (see [24]). Let {𝑥_𝑛} and {𝑤_𝑛} be bounded sequences in a real Banach space 𝑋 and {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑤_𝑛,∀𝑛≥0,limsup_𝑛→∞𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0.(35) Then, lim_𝑛→∞‖𝑤_𝑛−𝑥_𝑛‖=0.

Lemma 15 (see [25]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝜎_𝑛𝛾_𝑛,∀𝑛≥1,(36) where {𝛾_𝑛} is a sequence in [0,1] and {𝜎_𝑛} is a real sequence such that(i)∑^∞_𝑛=1𝛾_𝑛=∞;(ii)limsup_𝑛→∞𝜎_𝑛≤0 or ∑^∞_𝑛=1|𝜎_𝑛𝛾_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

Recall that a set-valued mapping 𝑇∶𝐷(𝑇)⊂𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐷(𝑇),𝑓∈𝑇𝑥 and 𝑔∈𝑇𝑦 imply 𝑓−𝑔,𝑥−𝑦≥0.(37) A set-valued mapping 𝑇 is called maximal monotone if 𝑇 is monotone and (𝐼+𝜆𝑇)𝐷(𝑇)=𝐻 for each 𝜆>0, where 𝐼 is the identity mapping of 𝐻. We denote by 𝐺(𝑇) the graph of 𝑇. It is known that a monotone mapping 𝑇 is maximal if and only if, for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑓−𝑔,𝑥−𝑦⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑇) implies 𝑓∈𝑇𝑥. Next we provide an example to illustrate the concept of maximal monotone mapping.

Let 𝐴∶𝐶→𝐻 be a monotone, 𝑘-Lipschitz-continuous mapping and let 𝑁_𝐶𝑣 be the normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐶}.(38) Define 𝑇𝑣=𝐴𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶.(39) Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐴); see [19].

Assume that 𝑅∶𝐷(𝑅)⊂𝐻→2^𝐻 is a maximal monotone mapping. Let 𝜆>0. In terms of Huang [26] (see also [27]), there holds the following property for the resolvent operator 𝐽_𝑅,𝜆∶𝐻→𝐷(𝑅).

Lemma 16. 𝐽_𝑅,𝜆 is single-valued and firmly nonexpansive, that is, 𝐽_𝑅,𝜆𝑥−𝐽_𝑅,𝜆𝑦,𝑥−𝑦≥𝐽_𝑅,𝜆𝑥−𝐽_𝑅,𝜆𝑦²,∀𝑥,𝑦∈𝐻.(40) Consequently, 𝐽_𝑅,𝜆 is nonexpansive and monotone.

Lemma 17 (see [28]). Let 𝑅 be a maximal monotone mapping with 𝐷(𝑅)=𝐶. Then for any given 𝜆>0,𝑢∈𝐶 is a solution of problem (12) if and only if 𝑢∈𝐶 satisfies 𝑢=𝐽_𝑅,𝜆(𝑢−𝜆𝐵𝑢).(41)

Lemma 18 (see [27]). Let 𝑅 be a maximal monotone mapping with 𝐷(𝑅)=𝐶 and let 𝐵∶𝐶→𝐻 be a strongly monotone, continuous and single-valued mapping. Then for each 𝑧∈𝐻, the equation 𝑧∈(𝐵+𝜆𝑅)𝑥 has a unique solution 𝑥_𝜆 for 𝜆>0.

Lemma 19 (see [28]). Let 𝑅 be a maximal monotone mapping with 𝐷(𝑅)=𝐶 and 𝐵∶𝐶→𝐻 be a monotone, continuous and single-valued mapping. Then (𝐼+𝜆(𝑅+𝐵))𝐶=𝐻 for each 𝜆>0. In this case, 𝑅+𝐵 is maximal monotone.

Lemma 20 (see [29]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and 𝑔∶𝐶→𝐑∪+∞ be a proper lower semicontinuous differentiable convex function. If 𝑥^∗ is a solution the minimization problem 𝑔𝑥^∗=inf_𝑥∈𝐶𝑔(𝑥),(42) then, 𝑔^′(𝑥),𝑥−𝑥^∗≥0,∀𝑥∈𝐶.(43) In particular, if 𝑥^∗ solves (OP), then 𝑢+𝛾𝑓−𝐼+𝜇𝑉𝑥^∗,𝑥−𝑥^∗≤0.(44)

3. Main Results

In this section, we introduce and analyze an iterative algorithm by hybrid extragradient-like viscosity method for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.

Theorem 21. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑁 be an integer. Let Θ,Θ₁,Θ₂ be three bifunctions from 𝐶×𝐶 to 𝐑 satisfying (H1)–(H4) and 𝜑∶𝐶→𝐑 be a lower semicontinuous and convex functional. Let 𝑅_𝑖∶𝐶→2^𝐻 be a maximal monotone mapping and let 𝐴,𝐴_𝑘∶𝐻→𝐻 and 𝐵_𝑖∶𝐶→𝐻 be 𝜁-inverse strongly monotone, 𝜁_𝑘-inverse strongly monotone and 𝜂_𝑖-inverse strongly monotone, respectively, where 𝑘∈{1,2} and 𝑖∈{1,2,…,𝑁}. Let {𝑇_𝑛}^∞_𝑛=1 be a sequence of nonexpansive mappings on 𝐻 and {𝜆_𝑛} be a sequence in (0,𝑏] for some 𝑏∈(0,1). Let 𝑉 be a 𝛾-strongly positive bounded linear operator and 𝑓∶𝐻→𝐻 be an 𝑙-Lipschitzian mapping with 𝛾𝑙<(1+𝜇)𝛾. Let 𝑊_𝑛 be the 𝑊-mapping defined by (9). Assume that Ω∶=∩^∞_𝑛=1Fix(𝑇_𝑛)∩GMEP(Θ,𝜑,𝐴)∩SGEP(𝐺)∩∩^𝑁_𝑖=1𝐼(𝐵_𝑖,𝑅_𝑖)≠∅ where 𝐺 is defined as in Proposition CY. Let {𝛼_𝑛},{𝛽_𝑛} and {𝛿_𝑛} be three sequences in [0,1]. Assume that:(i)𝐾∶𝐻→𝐑 is strongly convex with constant 𝜎>0 and its derivative 𝐾^′ is Lipschitz continuous with constant 𝜈>0 such that the function 𝑥↦⟨𝑦−𝑥,𝐾^′(𝑥)⟩ is weakly upper semicontinuous for each 𝑦∈𝐻;(ii)for each 𝑥∈𝐻, there exist a bounded subset 𝐷_𝑥⊂𝐶 and 𝑧_𝑥∈𝐶 such that for any 𝑦∉𝐷_𝑥, Θ𝑦,𝑧_𝑥+𝜑𝑧_𝑥𝑦+1−𝜑𝑟𝐾^′𝑦−𝐾^′(𝑥),𝑧_𝑥−𝑦<0;(45)(iii)lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞, 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1 and 0<liminf_𝑛→∞𝛿_𝑛≤limsup_𝑛→∞𝛿_𝑛<1;(iv)𝜈_𝑘∈(0,2𝜁_𝑘),𝑘∈{1,2},𝜇_𝑖∈(0,2𝜂_𝑖),𝑖∈{1,2,…,𝑁}, and {𝑟_𝑛}⊂[0,2𝜁] satisfies 0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁;(46)(v)lim_𝑛→∞(|𝛿_𝑛+1−𝛿_𝑛|+|𝑟_𝑛+1−𝑟_𝑛|)=0.
Given 𝑥₁∈𝐻 arbitrarily, then the sequence {𝑥_𝑛} generated iteratively by 𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛,𝑧_𝑛=𝐽_{𝑅𝑁,𝜇𝑁}𝐼−𝜇_𝑁𝐵_𝑁𝐽_{𝑅𝑁−1,𝜇𝑁−1}×𝐼−𝜇_𝑁−1𝐵_𝑁−1⋯𝐽_𝑅1,𝜇1𝐼−𝜇₁𝐵₁𝑢_𝑛,𝑦_𝑛=𝛿_𝑛𝐺𝑧_𝑛+1−𝛿_𝑛𝑊_𝑛𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑊_𝑛𝑦_𝑛,∀𝑛≥1,(47) converges strongly to 𝑥^∗∈Ω which solves the following optimization problem provided 𝑆_𝑟^(Θ,𝜑) is firmly nonexpansive: min_𝑥∈Ω𝜇2⟨⟩+1𝐴𝑥,𝑥2‖𝑥−𝑢‖²−ℎ(𝑥),(OP2) where ℎ is the potential function of 𝛾𝑓.

Proof. Since lim_𝑛→∞𝛼_𝑛=0 and 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1, we may assume, without loss of generality, that 𝛼_𝑛≤(1−𝛽_𝑛)(1+𝜇‖𝑉‖)⁻¹. Since 𝑉 is a 𝛾-strongly positive bounded linear operator on 𝐻, we know that ‖𝑉‖=sup{⟨𝑉𝑢,𝑢⟩∶𝑢∈𝐻,‖𝑢‖=1}.(48) Observe that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑢,𝑢=1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝑉𝑢,𝑢⟩≥1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇‖𝑉‖≥0(49) that is, (1−𝛽_𝑛)𝐼−𝛼_𝑛(𝐼+𝜇𝑉) is positive. It follows that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉=sup1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑢,𝑢∶𝑢∈𝐻,‖𝑢‖=1=sup1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇⟨𝑉𝑢,𝑢⟩∶𝑢∈𝐻,‖𝑢‖=1≤1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾.(50) Put Λ^𝑖=𝐽_{𝑅𝑖,𝜇𝑖}𝐼−𝜇_𝑖𝐵_𝑖𝐽_{𝑅𝑖−1,𝜇𝑖−1}×𝐼−𝜇_𝑖−1𝐵_𝑖−1⋯𝐽_𝑅1,𝜇1𝐼−𝜇₁𝐵₁(51) for all 𝑖∈{1,2,…,𝑁}, and Λ⁰=𝐼, where 𝐼 is the identity mapping on 𝐻. Then we have that 𝑧_𝑛=Λ^𝑁𝑢_𝑛.
We divide the rest of the proof into several steps.
Step 1. We show that {𝑥_𝑛} is bounded. Indeed, take 𝑝∈Ω arbitrarily. Since 𝑝=𝑆_𝑟^(Θ,𝜑)_𝑛(𝑝−𝑟_𝑛𝐴𝑝), 𝐴 is 𝜁-inverse strongly monotone and 0≤𝑟_𝑛≤2𝜁, we have, for any 𝑛≥1, 𝑢_𝑛−𝑝²=𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑝²≤𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑝²=𝑥_𝑛−𝑝−𝑟_𝑛𝐴𝑥_𝑛−𝐴𝑝²=𝑥_𝑛−𝑝²−2𝑟_𝑛𝑥_𝑛−𝑝,𝐴𝑥_𝑛−𝐴𝑝+𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝²−2𝑟_𝑛𝜁𝐴𝑥_𝑛−𝐴𝑝²+𝑟²_𝑛𝐴𝑥_𝑛−𝐴𝑝²=𝑥_𝑛−𝑝²+𝑟_𝑛𝑟_𝑛−2𝜁𝐴𝑥_𝑛−𝐴𝑝²≤𝑥_𝑛−𝑝².(52) Since 𝑝=𝐽_{𝑅𝑖,𝜇𝑖}(𝐼−𝜇_𝑖𝐵_𝑖)𝑝,Λ^𝑖𝑝=𝑝 and 𝐵_𝑖 is 𝜂_𝑖-inverse strongly monotone, where 𝜇_𝑖∈(0,2𝜂_𝑖),𝑖∈{1,2,…,𝑁}, by Lemma 16 we deduce that for each 𝑛≥1𝑧_𝑛−𝑝²=𝐽_{𝑅𝑁,𝜇𝑁}𝐼−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛−𝐽_{𝑅𝑁,𝜇𝑁}𝐼−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑝²≤𝐼−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛−𝐼−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑝²=Λ^𝑁−1𝑢_𝑛−Λ^𝑁−1𝑝−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛−𝐵_𝑁Λ^𝑁−1𝑝²≤Λ^𝑁−1𝑢_𝑛−Λ^𝑁−1𝑝²+𝜇_𝑁𝜇_𝑁−2𝜂_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛−𝐵_𝑁Λ^𝑁−1𝑝²≤Λ^𝑁−1𝑢_𝑛−Λ^𝑁−1𝑝²⋯≤Λ⁰𝑢_𝑛−Λ⁰𝑝²=𝑢_𝑛−𝑝².(53) Combining (52) and (53), we have 𝑧_𝑛−𝑝≤𝑥_𝑛−𝑝.(54) Since 𝑝=𝐺𝑝=𝑇^Θ1_𝜈1(𝐼−𝜈₁𝐴₁)𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝, 𝐴_𝑘 is 𝜁_𝑘-inverse-strongly monotone for 𝑘=1,2, and 0≤𝜈_𝑘≤2𝜁_𝑘 for 𝑘=1,2, we deduce that, for any 𝑛≥1, 𝐺𝑧_𝑛−𝑝²=𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑧_𝑛−𝑇^Θ1_𝜈1𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝²≤𝐼−𝜈₁𝐴₁𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑧_𝑛−(𝐼−𝜈₁𝐴₁)𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²=𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑧_𝑛−𝑇^Θ2_𝜈2𝐼−𝜈₂𝐴₂𝑝−𝜈₁[𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑧_𝑛−𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝]²≤𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑧_𝑛−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²+𝜈₁𝜈₁−2𝜁₁×𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑧_𝑛−𝐴₁𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²≤𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑧_𝑛−𝑇^Θ2_𝜈2(𝐼−𝜈₂𝐴₂)𝑝²≤(𝐼−𝜈₂𝐴₂)𝑧_𝑛−(𝐼−𝜈₂𝐴₂)𝑝²=(𝑧_𝑛−𝑝)−𝜈₂(𝐴₂𝑧_𝑛−𝐴₂𝑝)²≤𝑧_𝑛−𝑝²+𝜈₂𝜈₂−2𝜁₂𝐴₂𝑧_𝑛−𝐴₂𝑝²≤𝑧_𝑛−𝑝².(55) (This shows that 𝐺 is nonexpansive.) Thus, from (54), we get 𝑦_𝑛−𝑝=𝛿_𝑛𝐺𝑧_𝑛−𝑝+1−𝛿_𝑛𝑊_𝑛𝑧_𝑛−𝑝≤𝛿_𝑛𝐺𝑧_𝑛−𝑝+1−𝛿_𝑛𝑊_𝑛𝑧_𝑛−𝑝≤𝛿_𝑛𝑧_𝑛−𝑝+1−𝛿_𝑛𝑧_𝑛=𝑧−𝑝_𝑛−𝑝≤𝑥_𝑛−𝑝.(56) Set 𝑉=𝐼+𝜇𝑉. Then from (47) we have 𝑥_𝑛+1=𝛼−𝑝_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐼+𝜇𝑉𝑊_𝑛𝑦_𝑛=𝛼−𝑝_𝑛𝑢+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑉𝑝+𝛽_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝑉𝑊_𝑛𝑦_𝑛≤−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝑉𝑊_𝑛𝑦_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛‖𝑢‖+𝛼_𝑛𝛾𝑓𝑥_𝑛−≤𝑉𝑝1−𝛽_𝑛−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝑦_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛‖𝑢‖+𝛼_𝑛𝛾𝑓𝑥_𝑛−≤𝑉𝑝1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝑥_𝑛−𝑝+𝛼_𝑛‖𝑢‖+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑝+𝛾𝑓𝑝−≤𝑉𝑝1−𝛼_𝑛−𝛼_𝑛𝜇𝛾𝑥_𝑛−𝑝+𝛼_𝑛‖𝑢‖+𝛼_𝑛𝛾𝑙𝑥_𝑛−𝑝+𝛾𝑓𝑝−≤𝑉𝑝1−1+𝜇𝛾−𝛾𝑙𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑝−+‖𝑢‖=𝑉𝑝1−1+𝜇𝛾−𝛾𝑙𝛼_𝑛𝑥_𝑛−𝑝+1+𝜇𝛾−𝛾𝑙𝛼_𝑛×𝛾𝑓𝑝−𝑉𝑝+‖𝑢‖1+𝜇𝑥𝛾−𝛾𝑙≤max_𝑛,𝑝−−𝑝𝛾𝑓𝑉𝑝+‖𝑢‖1+𝜇.𝛾−𝛾𝑙(57) By induction, we get 𝑥_𝑛−𝑝≤max𝑥₀,𝑝−−𝑝𝛾𝑓𝑉𝑝+‖𝑢‖1+𝜇.𝛾−𝛾𝑙(58) Therefore, {𝑥_𝑛} is bounded and so are the sequences {𝑢_𝑛},{𝑧_𝑛},{𝑦_𝑛},{𝑓(𝑥_𝑛)} and {𝑊_𝑛𝑦_𝑛}.
Step 2. We show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
Indeed, define 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑤_𝑛,∀𝑛≥1.(59) Then from the definition of 𝑤_𝑛, we obtain 𝑤_𝑛+1−𝑤_𝑛=𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+11−𝛽_𝑛+1−𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛1−𝛽_𝑛=𝛼_𝑛+1𝑢+𝛾𝑓𝑥_𝑛+1+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝑉𝑊_𝑛+1𝑦_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝑢+𝛾𝑓𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝑉𝑊_𝑛𝑦_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝑢+𝛾𝑓𝑥_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝑢+𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛𝑦_𝑛+𝛼_𝑛1−𝛽_𝑛𝑉𝑊_𝑛𝑦_𝑛−𝛼_𝑛+11−𝛽_𝑛+1𝑉𝑊_𝑛+1𝑦_𝑛+1=𝛼_𝑛+11−𝛽_𝑛+1𝑢+𝛾𝑓𝑥_𝑛+1−𝑉𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑉𝑊_𝑛𝑦_𝑛−𝑢−𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛+1−𝑊_𝑛+1𝑦_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛.(60) It follows that 𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1‖𝑢‖+𝛾𝑓𝑥_𝑛+1+𝑉𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑉𝑊_𝑛𝑦_𝑛+‖𝑢‖+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+1‖𝑢‖+𝛾𝑓𝑥_𝑛+1+𝑉𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑉𝑊_𝑛𝑦_𝑛+‖𝑢‖+𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛+𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛.(61) From (9), since 𝑊_𝑛,𝑇_𝑛 and 𝑈_𝑛,𝑖 are all nonexpansive, we have 𝑊_𝑛+1𝑧_𝑛−𝑊_𝑛𝑧_𝑛=𝜆₁𝑇₁𝑈_𝑛+1,2𝑧_𝑛−𝜆₁𝑇₁𝑈_𝑛,2𝑧_𝑛≤𝜆₁𝑈_𝑛+1,2𝑧_𝑛−𝑈_𝑛,2𝑧_𝑛=𝜆₁𝜆₂𝑇₂𝑈_𝑛+1,3𝑧_𝑛−𝜆₂𝑇₂𝑈_𝑛,3𝑧_𝑛≤𝜆₁𝜆₂𝑈_𝑛+1,3𝑧_𝑛−𝑈_𝑛,3𝑧_𝑛⋯≤𝜆₁𝜆₂⋯𝜆_𝑛𝑈_{𝑛+1,𝑛+1}𝑧_𝑛−𝑈_𝑛,𝑛+1𝑧_𝑛≤𝑀^𝑛_𝑖=1𝜆_𝑖,𝑊(62)_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛=𝜆₁𝑇₁𝑈_𝑛+1,2𝑦_𝑛−𝜆₁𝑇₁𝑈_𝑛,2𝑦_𝑛≤𝜆₁𝑈_𝑛+1,2𝑦_𝑛−𝑈_𝑛,2𝑦_𝑛=𝜆₁𝜆₂𝑇₂𝑈_𝑛+1,3𝑦_𝑛−𝜆₂𝑇₂𝑈_𝑛,3𝑦_𝑛≤𝜆₁𝜆₂𝑈_𝑛+1,3𝑦_𝑛−𝑈_𝑛,3𝑦_𝑛⋯≤𝜆₁𝜆₂⋯𝜆_𝑛𝑈_{𝑛+1,𝑛+1}𝑦_𝑛−𝑈_𝑛,𝑛+1𝑦_𝑛≤𝑀^𝑛_𝑖=1𝜆_𝑖,(62)^′ where 𝑀 is a constant such that sup_𝑛≥1𝑈_{𝑛+1,𝑛+1}𝑧_𝑛+𝑈_𝑛,𝑛+1𝑧_𝑛≤𝑀,sup_𝑛≥1𝑈_{𝑛+1,𝑛+1}𝑦_𝑛+𝑈_𝑛,𝑛+1𝑦_𝑛≤𝑀.(63) On the other hand, we estimate ‖𝑦_𝑛+1−𝑦_𝑛‖. Taking into account that 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1,0<liminf_𝑛→∞𝛿_𝑛≤limsup_𝑛→∞𝛿_𝑛<1 and 0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝜁, we may assume, without loss of generality, that {𝑟_𝑛}⊂[𝑐,𝑑]⊂(0,2𝜁) and {𝛽_𝑛},{𝛿_𝑛}⊂[𝑐,̂𝑑]⊂(0,1). Utilizing Remark 5 and Lemma 16, we have 𝑧_𝑛+1−𝑧_𝑛²=𝐽_{𝑅𝑁,𝜇𝑁}𝐼−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛+1−𝐽_{𝑅𝑁,𝜇𝑁}𝐼−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛²≤(𝐼−𝜇_𝑁𝐵_𝑁)Λ^𝑁−1𝑢_𝑛+1−(𝐼−𝜇_𝑁𝐵_𝑁)Λ^𝑁−1𝑢_𝑛²=Λ^𝑁−1𝑢_𝑛+1−Λ^𝑁−1𝑢_𝑛−𝜇_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛+1−𝐵_𝑁Λ^𝑁−1𝑢_𝑛²≤Λ^𝑁−1𝑢_𝑛+1−Λ^𝑁−1𝑢_𝑛²+𝜇_𝑁𝜇_𝑁−2𝜂_𝑁𝐵_𝑁Λ^𝑁−1𝑢_𝑛+1−𝐵_𝑁Λ^𝑁−1𝑢_𝑛²≤Λ^𝑁−1𝑢_𝑛+1−Λ^𝑁−1𝑢_𝑛²⋯≤Λ⁰𝑢_𝑛+1−Λ⁰𝑢_𝑛²=𝑢_𝑛+1−𝑢_𝑛²,(64)𝐼−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝐼−𝑟_𝑛𝐴𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝐴𝑥_𝑛+𝑟_𝑛−𝑟_𝑛+1𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝐴𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐴𝑥_𝑛,𝑢(65)_𝑛+1−𝑢_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛=𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛+𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛≤𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛+𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛≤𝐼−𝑟_𝑛+1𝐴𝑥_𝑛+1−𝐼−𝑟_𝑛𝐴𝑥_𝑛+𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐴𝑥_𝑛+𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝑆_𝑟^(Θ,𝜑)_𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝐴𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑟_𝑛+1⋅𝜈𝜎𝑆_𝑟^(Θ,𝜑)_𝑛+1𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛×𝐴𝑥_𝑛+𝜈𝑆𝑐𝜎^{𝑟(Θ,𝜑)𝑛+1}𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝑟_𝑛𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑀₁,(66) where sup_𝑛≥1{‖𝐴𝑥_𝑛‖+(𝜈/𝑐𝜎)‖𝑆_𝑟^(Θ,𝜑)_𝑛+1(𝐼−𝑟_𝑛𝐴)𝑥_𝑛−(𝐼−𝑟_𝑛𝐴)𝑥_𝑛‖}≤𝑀₁ for some 𝑀₁>0.
Note that 𝑦_𝑛+1−𝑦_𝑛=𝛿_𝑛𝐺𝑧_𝑛+1−𝐺𝑧_𝑛+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+1−𝛿_𝑛𝑊_𝑛+1𝑧_𝑛+1−𝑊_𝑛𝑧_𝑛.(67) Since 𝐺 is nonexpansive, from (62), (64) and (66) it follows that 𝑦_𝑛+1−𝑦_𝑛≤𝛿_𝑛𝐺𝑧_𝑛+1−𝐺𝑧_𝑛+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+1−𝛿_𝑛𝑊_𝑛+1𝑧_𝑛+1−𝑊_𝑛𝑧_𝑛≤𝛿_𝑛𝑧_𝑛+1−𝑧_𝑛+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+1−𝛿_𝑛𝑊_𝑛+1𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+𝑊_𝑛+1𝑧_𝑛−𝑊_𝑛𝑧_𝑛≤𝛿_𝑛𝑧_𝑛+1−𝑧_𝑛+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+1−𝛿_𝑛𝑧_𝑛+1−𝑧_𝑛+𝑀^𝑛_𝑖=1𝜆_𝑖≤𝑧_𝑛+1−𝑧_𝑛+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+𝑀^𝑛_𝑖=1𝜆_𝑖≤𝑢_𝑛+1−𝑢_𝑛+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+𝑀^𝑛_𝑖=1𝜆_𝑖≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑀₁+𝛿_𝑛+1−𝛿_𝑛𝐺𝑧_𝑛+1−𝑊_𝑛+1𝑧_𝑛+1+𝑀^𝑛_𝑖=1𝜆_𝑖.(68) Utilizing (61), (62)^′ and (68), we have 𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1‖𝑢‖+𝛾𝑓𝑥_𝑛+1+𝑉𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑉𝑊_𝑛𝑦_𝑛+‖𝑢‖+𝛾𝑓𝑥_𝑛+𝑊_𝑛+1𝑦_𝑛−𝑊_𝑛𝑦_𝑛+𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1‖𝑢‖+𝛾𝑓𝑥_𝑛+1+𝑉𝑊_𝑛+1𝑦_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑉𝑊_𝑛𝑦_𝑛+‖𝑢‖+𝛾𝑓𝑥_𝑛+𝑀^𝑛_𝑖=1𝜆_𝑖+