A Hybrid Iterative Scheme for a Maximal Monotone Operator and Two Countable Families of Relatively Quasi-Nonexpansive Mappings for Generalized Mixed Equilibrium and Variational Inequality Problems

Abstract

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an 𝛼-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

1. Introduction

Let 𝐸 be a Banach space with norm ‖⋅‖, 𝐶 a nonempty closed convex subset of 𝐸, and let 𝐸^∗ denote the dual of 𝐸. Let 𝜃∶𝐶×𝐶→ℝ be a bifunction, 𝜑∶𝐶→ℝ be a real-valued function, and 𝐵∶𝐶→𝐸^∗ a mapping. The generalized mixed equilibrium problem, is to find 𝑥∈𝐶 such that 𝜃(𝑥,𝑦)+⟨𝐵𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.1) The set of solutions to (1.1) is denoted by GMEP(𝜃,𝐵,𝜑), that is, GMEP(𝜃,𝐵,𝜑)={𝑥∈𝐶∶𝜃(𝑥,𝑦)+⟨𝐵𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶}. (1.2) If 𝐵≡0, the problem (1.1) reduces into the mixed equilibrium problem for 𝜃, denoted by MEP(𝜃,𝜑), which is to find 𝑥∈𝐶 such that𝜃(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.3) If 𝜃≡0, the problem (1.1) reduces into the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐵,𝜑), which is to find 𝑥∈𝐶 such that⟨𝐵𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.4) If 𝐵≡0 and 𝜑≡0 the problem (1.1) reduces into the equilibrium problem for 𝜃, denoted by EP(𝜃), which is to find 𝑥∈𝐶 such that𝜃(𝑥,𝑦)≥0,∀𝑦∈𝐶. (1.5) If 𝜃≡0, the problem (1.3) reduces into the minimize problem, denoted by Argmin(𝜑), is to find 𝑥∈𝐶 such that𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.6) The above formulation (1.4) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem and optimization problem, which can also be written in the form of an EP(𝜃). In other words, the EP(𝜃) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP(𝜃); see, for example, [1, 2] and references therein. Some solution methods have been proposed to solve the EP(𝜃); see, for example, [1, 3–11] and references therein.

A Banach space 𝐸 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. Let 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. Then a Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡 (1.7) exists for each 𝑥,𝑦∈𝑆(𝐸). It is also said to be uniformly smooth if the limit exists uniformly for 𝑥,𝑦∈𝑆(𝐸). Let 𝐸 be a Banach space. The modulus of convexity of 𝐸 is the function 𝛿∶[0,2]→[0,1] defined by 𝛿(𝜀)=inf{1−‖𝑥+𝑦2‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀}. (1.8) A Banach space 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀∈(0,2]. Let 𝑝 be a fixed real number with 𝑝≥2. A Banach space 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿(𝜀)≥𝑐𝜀^𝑝 for all 𝜀∈[0,2]; see [12, 13] for more details. Observe that every 𝑝-uniformly convex is uniformly convex. One should note that no Banach space is 𝑝-uniformly convex for 1<𝑝<2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each 𝑝>1, the generalized duality mapping 𝐽_𝑝∶𝐸→2^𝐸∗ is defined by 𝐽_𝑝(𝑥)={𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖^𝑝,‖𝑥^∗‖=‖𝑥‖^𝑝−1} for all 𝑥∈𝐸. In particular, 𝐽=𝐽₂ is called the normalized duality mapping. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping.

A set valued mapping 𝑇∶𝐸⇉𝐸^∗ with graph 𝐺(𝑇)={(𝑥,𝑥^∗)∶𝑥^∗∈𝑇𝑥}, domain 𝐷(𝑇)={𝑥∈𝐸∶𝑇𝑥≠∅}, and rang 𝑅(𝑇)=∪{𝑇𝑥∶𝑥∈𝐷(𝑇)}. 𝑇 is said to be monotone if ⟨𝑥^∗−𝑦^∗,𝑥−𝑦⟩≥0, whenever 𝑥^∗∈𝑇𝑥,𝑦^∗∈𝑇𝑦. A monotone operator 𝑇 is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator on the same space. We know that if 𝑇 is maximal monotone, then the solution set 𝑇⁻¹0={𝑥∈𝐷(𝑇)∶0∈𝑇𝑥} is closed and convex. It is knows that 𝑇 is a maximal monotone if and only if 𝑅(𝐽+𝑟𝑇)=𝐸^∗ for all 𝑟>0 when 𝐸 is a reflexive, strictly convex and smooth Banach space (see Rockafellar [14]).

Let 𝐸 be a smooth, strictly convex and reflexive Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸 and let 𝑇∶𝐸⇉𝐸^∗ be a monotone operator satisfying 𝐷(𝑇)⊂𝐶⊂𝐽⁻¹(⋂_𝑟>0𝑅(𝐽+𝑟𝑇)). Then we define the resolvent 𝑇 by 𝐽_𝑟𝑥={𝑧∈𝐷(𝑇)∶𝐽𝑥∈𝐽𝑧+𝑟𝑇𝑧}, for all 𝑥∈𝐸. In other words, 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. 𝐽_𝑟 is a single-valued mapping from 𝐸 to 𝐷(𝑇). Also, we know that 𝑇⁻¹0=𝐹(𝐽_𝑟) for all 𝑟>0, where 𝐹(𝐽_𝑟) is the set of all fixed points of 𝐽_𝑟. We can define, for 𝑟>0, the Yosida approximation of 𝑇 by 𝑇_𝑟𝑥=(𝐽𝑥−𝐽𝐽_𝑟𝑥)/𝑟 for all 𝑥∈𝐸. We know that 𝑇_𝑟𝑥∈𝑇(𝐽_𝑟𝑥) for all 𝑟>0 and 𝑥∈𝐸.

It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸. We consider the problem of finding:𝑣∈𝐸suchthat0∈𝑇𝑣, (1.9) where 𝑇 is an operator from 𝐸 into 𝐸^∗. Such 𝑣∈𝐸 is called a zero point of 𝑇. Such a problem contains numerous problems in economics, optimization and physics. When 𝑇 is a maximal monotone operator, a well-know method for solving (1.9) in a Hilbert space 𝐻 is the proximal point algorithm: 𝑥₁=𝑥∈𝐻 and,𝑥_𝑛+1=𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,3,…, (1.10) where {𝑟_𝑛}⊂(0,∞) and 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝑇)⁻¹, then Rockafellar [15] proved that the sequence {𝑥_𝑛} converges weakly to an element of 𝑇⁻¹0. Let 𝐸 be a real Banach space and let 𝐶 be a nonempty closed convex subset of 𝐸 and 𝐴∶𝐶→𝐸^∗ be an operator. The classical variational inequality problem for an operator 𝐴 is to find 𝑥∈𝐶 such that⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶. (1.11) The set of solution of (1.11) is denote by VI(𝐴,𝐶). Recall that let 𝐴∶𝐶→𝐸^∗ be a mapping. Then 𝐴 is called (i)monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0,∀𝑥,𝑦∈𝐶, (1.12)(ii)𝛼-inverse-strongly monotone if there exists a constant 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶. (1.13)

The class of inverse-strongly monotone mappings has been studied by many researchers to approximating a common fixed point; see [6, 7, 16, 17] for more details.

Let 𝐶 be a closed convex subset of 𝐸, a mapping 𝑆∶𝐶→𝐶 is said to be nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶. A point 𝑥∈𝐶 is a fixed point of 𝑆 provided 𝑆𝑥=𝑥. Denote by 𝐹(𝑆) the set of fixed points of 𝑆; that is, 𝐹(𝑆)={𝑥∈𝐶∶𝑆𝑥=𝑥}.

Consider the functional defined by𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,for𝑥,𝑦∈𝐸. (1.14) Recall that a point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑆 [18] if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑆𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑆 will be denoted by 𝐹(𝑆). A mapping 𝑆 from 𝐶 into itself is said to be relatively nonexpansive [19–21] if 𝐹(𝑆)=𝐹(𝑆) and 𝜙(𝑝,𝑆𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑆). The asymptotic behavior of a relatively nonexpansive mapping was studied in [22–24]. 𝑆 is said to be 𝜙-nonexpansive, if 𝜙(𝑆𝑥,𝑆𝑦)≤𝜙(𝑥,𝑦) for 𝑥,𝑦∈𝐶. 𝑆 is said to be relatively quasi-nonexpansive (or quasi-𝜙-nonexpansive) if 𝐹(𝑆)≠∅ and 𝜙(𝑝,𝑆𝑥)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶 and 𝑝∈𝐹(𝑆). We note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [22–26] which requires the strong restriction: 𝐹(𝑆)=𝐹(𝑆).

Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑃_𝐶∶𝐻→𝐶 is the metric projection of 𝐻 onto 𝐶, then 𝑃_𝐶 is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [27] recently introduced a generalized projection Π_𝐶 from 𝐸 in to 𝐶 as follows:Π_𝐶(𝑥)=argmin_𝑦∈𝐶𝜙(𝑦,𝑥),∀𝑥∈𝐸. (1.15) It is obvious from the definition of function 𝜙 that(‖𝑦‖−‖𝑥‖)²≤𝜙(𝑦,𝑥)≤(‖𝑦‖+‖𝑥‖)²,∀𝑥,𝑦∈𝐸. (1.16) If 𝐸 is a Hilbert space, then 𝜙(𝑦,𝑥)=‖𝑦−𝑥‖² and Π_𝐶 becomes the metric projection of 𝐸 onto 𝐶. Let Π_𝐶 be the generalized projection from a smooth, strictly convex and reflexive Banach space 𝐸 onto a nonempty closed convex subset 𝐶 of 𝐸. Then, Π_𝐶 is a closed relatively quasi-nonexpansive mapping from 𝐸 onto 𝐶 with 𝐹(Π_𝐶)=𝐶. On the author hand, the generalized projection Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the functional 𝜙(𝑥,𝑦), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙(𝑥,𝑥)=inf_𝑦∈𝐶𝜙(𝑦,𝑥). (1.17) The existence and uniqueness of the operator Π_𝐶 follows from the properties of the functional 𝜙(𝑦,𝑥) and strict monotonicity of the mapping 𝐽 (see, e.g., [27–31]).

Remark 1.1. If 𝐸 is a reflexive, strictly convex and smooth Banach space, then for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦. It is sufficient to show that if 𝜙(𝑥,𝑦)=0 then 𝑥=𝑦. From (1.14), we have ‖𝑥‖=‖𝑦‖. This implies that ⟨𝑥,𝐽𝑦⟩=‖𝑥‖²=‖𝐽𝑦‖². From the definition of 𝐽, one has 𝐽𝑥=𝐽𝑦. Therefore, we have 𝑥=𝑦; see [29, 31] for more details.

In 2004, Matsushita and Takahashi [32] introduced the following iteration: a sequence {𝑥_𝑛} defined by𝑥_𝑛+1=Π_𝐶𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛), (1.18) where the initial guess element 𝑥₀∈𝐶 is arbitrary, {𝛼_𝑛} is a real sequence in [0,1], 𝑇 is a relatively nonexpansive mapping and Π_𝐶 denotes the generalized projection from 𝐸 onto a closed convex subset 𝐶 of 𝐸. They proved that the sequence {𝑥_𝑛} converges weakly to a fixed point of 𝑇.

In 2005, Matsushita and Takahashi [25] proposed the following hybrid iteration method (it is also called the 𝐶𝑄 method) with generalized projection for relatively nonexpansive mapping 𝑇 in a Banach space 𝐸:𝑥₀𝑦∈𝐶,chosenarbitrarily,_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛𝐶),_𝑛={𝑧∈𝐶∶𝜙(𝑧,𝑦_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑄)},_𝑛={𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛𝑥⟩≥0},_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.19)

They proved that {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇) is the generalized projection from 𝐶 onto 𝐹(𝑇). In 2008, Iiduka and Takahashi [33] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator 𝐴 in a 2-uniformly convex and uniformly smooth Banach space 𝐸: 𝑥₁=𝑥∈𝐶 and𝑥_𝑛+1=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛), (1.20) for every 𝑛=1,2,3,…, where Π_𝐶 is the generalized metric projection from 𝐸 onto 𝐶, 𝐽 is the duality mapping from 𝐸 into 𝐸^∗ and {𝜆_𝑛} is a sequence of positive real numbers. They proved that the sequence {𝑥_𝑛} generated by (1.20) converges weakly to some element of VI(𝐴,𝐶).

Recently, Takahashi and Zembayashi [34, 35], studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2008, Cholamjiak [36], proved the following iteration𝑧_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝜃(𝑢_𝑛1,𝑦)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀, (1.21) where 𝐽 is the duality mapping on 𝐸. Assume that {𝛼_𝑛},{𝛽_𝑛} and {𝛾_𝑛} are sequences in [0,1] such that 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, liminf_𝑛→∞𝛼_𝑛𝛽_𝑛>0 and liminf_𝑛→∞𝛼_𝑛𝛾_𝑛>0. Then {𝑥_𝑛} converges strongly to 𝑞=Π_𝐹𝑥₀, where 𝐹∶=𝐹(𝑇)∩𝐹(𝑆)∩EP(𝜃)∩VI(𝐴,𝐶). In 2009, Wei et al. [37] proved the following iteration for two relatively nonexpansive mappings in a Banach space 𝐸:𝑥₀∈𝐶,𝐽𝑧_𝑛=𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑇𝑥_𝑛,𝐽𝑢_𝑛=𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑆𝑧_𝑛,𝐻_𝑛={𝑣∈𝐶∶𝜙(𝑣,𝑢_𝑛)≤𝛽_𝑛𝜙(𝑣,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑣,𝑧_𝑛)≤𝜙(𝑣,𝑥_𝑛𝑊)},_𝑛={𝑧∈𝐶∶⟨𝑧−𝑥_𝑛,𝐽𝑥₀−𝐽𝑥_𝑛𝑥⟩≤0},_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀, (1.22) if {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1) such that 𝛼_𝑛≤1−𝛿₁ and 𝛽_𝑛≤1−𝛿₂ for some 𝛿₁,𝛿₂∈(0,1), then {𝑥_𝑛} generated by (1.22) converges strongly to a point Π_{ℱ∶=𝐹(𝑇)∩𝐹(𝑆)}𝑥₀. Where the mapping Π_ℱ of 𝐸 onto ℱ is the generalized projection operator. Inoue et al. [38] proved strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping by using the hybrid method. After that, Klin-eam et al. [2], extend Inoue et al. [38] to obtain the strong convergence theorem for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of two relatively nonexpansive mappings in a Banach space by using a new hybrid method.

On the other hand, Nakajo et al. [39] introduced the following condition. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, let {𝑆_𝑛} be a family of mappings of 𝐶 into itself with 𝐹∶=⋂^∞_𝑛=1𝐹(𝑆_𝑛)≠∅ and 𝜔_𝑤(𝑧_𝑛) denotes the set of all weak subsequential limits of a bounded sequence {𝑧_𝑛} in 𝐶. {𝑆_𝑛} is said to satisfy the NST-condition if for every bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖=0impliesthat𝜔_𝑤(𝑧_𝑛)⊂𝐹. (1.23)

Recall that a mapping 𝑆∶𝐶→𝐶 is closed if for each {𝑥_𝑛} in 𝐶, if 𝑥_𝑛→𝑥 and 𝑆𝑥_𝑛→𝑦, then 𝑆𝑥=𝑦. Let {𝑆_𝑛} be a family of mappings of 𝐶 in to itself with 𝐹∶=⋂^∞_𝑛=1𝐹(𝑆_𝑛)≠∅, {𝑆_𝑛} is said to satisfy the (∗)-condition if for each bounded sequence {𝑧_𝑛} in 𝐶,lim_𝑛→∞‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖=0,𝑧_𝑛→𝑧imply𝑧∈𝐹. (1.24) It follows directly from the definitions above that if {𝑆_𝑛} satisfies NST-condition, then {𝑆_𝑛} satisfies (∗)-condition. If 𝑆_𝑛≡𝑆 and 𝑆 is closed, then {𝑆_𝑛} satisfies (∗)-condition (see [40] for more details).

In this paper, we introduce a new hybrid projection method for finding a common solution of the set of common fixed points of two countable families of relatively quasi nonexpansive mappings, the set of the variational inequality for an 𝛼-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in a real uniformly smooth and 2-uniformly convex Banach space.

2. Preliminaries

We also need the following lemmas for the proof of our main results.

Lemma 2.1 (Xu [41]). If 𝐸 be a 2-uniformly convex Banach space and 0<𝑐≤1, then for all 𝑥,𝑦∈𝐸, one has 2‖𝑥−𝑦‖≤𝑐²‖𝐽𝑥−𝐽𝑦‖, (2.1) where 𝐽 is the normalized duality mapping of 𝐸.

The best constant 1/𝑐 in lemma is called the 2-uniformly convex constant of 𝐸.

Lemma 2.2 (Chidume [42, Corollary 4.17, pages 36-37]). If 𝐸 be a p-uniformly convex Banach space and let 𝑝 be a given real number with 𝑝≥2, then for all 𝑥,𝑦∈𝐸, 𝑗_𝑥∈𝐽_𝑝𝑥 and 𝑗_𝑦∈𝐽_𝑝𝑦⟨𝑥−𝑦,𝑗_𝑥−𝑗_𝑦𝑐⟩≥^𝑝2^𝑝−2𝑝‖𝑥−𝑦‖^𝑝, (2.2) where 𝐽_𝑝 is the generalized duality mapping of 𝐸 and 1/𝑐 is the p-uniformly convexity constant of 𝐸.

Lemma 2.3 (Kamimura and Takahashi [30]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then ‖𝑥_𝑛−𝑦_𝑛‖→0.

Lemma 2.4 (Alber [27]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥∈𝐸. Then 𝑥₀=Π_𝐶𝑥 if and only if ⟨𝑥₀−𝑦,𝐽𝑥−𝐽𝑥₀⟩≥0,∀𝑦∈𝐶. (2.3)

Lemma 2.5 (Alber [27]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸 and let 𝑥∈𝐸. Then 𝜙(𝑦,Π_𝐶𝑥)+𝜙(Π_𝐶𝑥,𝑥)≤𝜙(𝑦,𝑥),∀𝑦∈𝐶. (2.4)

Lemma 2.6 (Qin et al. [9]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space, and 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑆∶𝐶→𝐶 be a relatively quasi-nonexpansive mapping. Then 𝐹(𝑆) is a closed convex subset of 𝐶.

Let 𝐸 be a reflexive, strictly convex, smooth Banach space and 𝐽 the duality mapping from 𝐸 into 𝐸^∗. Then 𝐽⁻¹ is also single valued, one-to-one, surjective, and it is the duality mapping from 𝐸^∗ into 𝐸. We make use of the following mapping 𝑉 studied in Alber [27]𝑉(𝑥,𝑥^∗)=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥^∗‖², (2.5) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗, that is, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹𝑥^∗).

Lemma 2.7 (Kohsaka and Takahashi [43, Lemma 3.2]). Let 𝐸 be a reflexive, strictly convex smooth Banach space and let 𝑉 be as in (2.5). Then 𝑉(𝑥,𝑥^∗)+2⟨𝐽⁻¹𝑥^∗−𝑥,𝑦^∗⟩≤𝑉(𝑥,𝑥^∗+𝑦^∗), (2.6) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Lemma 2.8 (Kohsaka and Takahashi [44]). Let 𝐸 be a smooth, strictly convex and reflexive Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸 and let 𝑇∶𝐸⇉𝐸^∗ be a monotone operator satisfying 𝐷(𝑇)⊂𝐶⊂𝐽⁻¹(⋂_𝑟>0𝑅(𝐽+𝑟𝑇)). Let 𝑟>0, let 𝐽_𝑟 and 𝑇_𝑟 be the resolvent and the Yosida approximation of 𝐴, respectively. Then the following hold: (i)𝜙(𝑢,𝐽_𝑟𝑥)+𝜙(𝐽_𝑟𝑥,𝑥)≤𝜙(𝑢,𝑥), for all 𝑥∈𝐶,𝑢∈𝑇⁻¹0;(ii)(𝐽_𝑟𝑥,𝑇_𝑟𝑥)∈𝑇, for all 𝑥∈𝐶;(iii)𝐹(𝐽_𝑟)=𝑇⁻¹0.

Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸 and let 𝐴 be an inverse-strongly monotone mapping of 𝐶 into 𝐸^∗ which is said to be hemicontinuous if for all 𝑥,𝑦∈𝐶, the mapping 𝐹 of [0,1] into 𝐸^∗, defined by 𝐹(𝑡)=𝐴(𝑡𝑥+(1−𝑡)𝑦), is continuous with respect to the weak∗ topology of 𝐸^∗. We define by 𝑁_𝐶(𝑣)the normal cone for 𝐶 at a point 𝑣∈𝐶, that is,𝑁_𝐶(𝑣)={𝑥^∗∈𝐸^∗∶⟨𝑣−𝑦,𝑥^∗⟩≥0,∀𝑦∈𝐶}. (2.7)

Lemma 2.9 (Rockafellar [14]). Let 𝐶 be a nonempty, closed convex subset of a Banach space 𝐸 and 𝐴 a monotone, hemicontinuous operator of 𝐶 into 𝐸^∗. Let 𝑈∶𝐸⇉𝐸^∗ be an operator defined as follows: 𝑈𝑣=𝐴𝑣+𝑁_𝐶(𝑣),𝑣∈𝐶,∅,otherwise. (2.8) Then 𝑈 is maximal monotone and 𝑈⁻¹0=VI(𝐴,𝐶).

For solving the equilibrium problem for a bifunction 𝜃∶𝐶×𝐶→ℝ, let us assume that 𝜃 satisfies the following conditions: (A1)𝜃(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝜃 is monotone, that is, 𝜃(𝑥,𝑦)+𝜃(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶; (A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0𝜃(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝜃(𝑥,𝑦); (2.9)(A4)for each 𝑥∈𝐶, 𝑦↦𝜃(𝑥,𝑦) is convex and lower semicontinuous.

For example, let 𝐵 be a continuous and monotone operator of 𝐶 into 𝐸^∗ and define 𝜃(𝑥,𝑦)=⟨𝐵𝑥,𝑦−𝑥⟩,∀𝑥,𝑦∈𝐶. (2.10) Then, 𝜃 satisfies (A1)–(A4).

The following result is in Takahashi and Zembayashi ([34, 35]).

Lemma 2.10 (see [34, 35]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space 𝐸, let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), let 𝑟>0 and let 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 1𝜃(𝑧,𝑦)+𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶. (2.11)

Motivated by Combettes and Hirstoaga [4] in a Hilbert space and Takahashi and Zembayashi [34] in a Banach space, Zhang [45] obtain the following lemma.

Lemma 2.11 (Zhang [45, Lemma 1.5]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space 𝐸. Let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜑∶𝐶→ℝ is convex and lower semicontinuous and 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, then there exists 𝑢∈𝐶 such that 1𝜃(𝑢,𝑦)+⟨𝐵𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)+𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶. (2.12) Define a mapping 𝐾_𝑟∶𝐶→𝐶 as follows: 𝐾_𝑟1(𝑥)={𝑢∈𝐶∶𝜃(𝑢,𝑦)+⟨𝐵𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)+𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶} (2.13) for all 𝑥∈𝐶. Then the followings hold: (1)𝐾_𝑟 is single-valued; (2)𝐾_𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦∈𝐸,⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝐾_𝑟𝑥−𝐽𝐾_𝑟𝑦⟩≤⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(3)𝐹(𝐾_𝑟)=𝐹(𝐾_𝑟)=GMEP(𝜃,𝐵,𝜑);(4)GMEP(𝜃,𝐵,𝜑) is closed and convex; (5)𝜙(𝑝,𝐾_𝑟𝑧)+𝜙(𝐾_𝑟𝑧,𝑧)≤𝜙(𝑝,𝑧), for all 𝑝∈𝐹(𝐾_𝑟) and 𝑧∈𝐸.

3. Main Results

In this section, by using the (∗)-condition, we prove the new convergence theorems for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of two countable families of relatively quasi-nonexpansive mappings, zeros of maximal monotone operators and the solution set of variational inequalities for an 𝛼-inverse strongly monotone mapping in a 2-uniformly convex and uniformly smooth Banach space.

Theorem 3.1. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑇∶𝐸⇉𝐸^∗ be a maximal monotone operator satisfying 𝐷(𝑇)⊂𝐶 and let 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0, where 𝐽 is the duality mapping on 𝐸. Let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–( A4), and let 𝜑∶𝐶→ℝ be a proper lower semicontinuous and convex function. Let 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐸^∗ satisfy ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖, for all 𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅ and let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping. Let 𝑆_𝑛,𝑇_𝑛∶𝐶→𝐶 be two families of relatively quasi-nonexpansive mappings with satisfy the (∗)-condition such that Θ∶=(^∞_𝑛=1𝐹(𝑆_𝑛))∩(^∞_𝑛=1𝐹(𝑇_𝑛))∩𝑇⁻¹0∩GMEP(𝜃,𝐵,𝜑)∩VI(𝐴,𝐶)≠∅. (3.1) For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇_𝑛𝐽_𝑟𝑛𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆_𝑛𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝜃(𝑢_𝑛,𝑦)+⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑(𝑢_𝑛1)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.2) where {𝛼_𝑛},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑑,∞) for some 𝑑>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. If liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Proof. We split the proof into seven steps.Step 1. We first show that 𝐶_𝑛+1 is closed and convex for each 𝑛≥1.
By Lemma 2.6, we know that (⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛)) is closed and convex. We also know that if 𝑇⁻¹0 and VI(𝐴,𝐶) are closed and convex. From Lemma 2.11 (4), we have GMEP(𝜃,𝐵,𝜑) is closed and convex. Hence Θ∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩𝑇⁻¹0∩GMEP(𝜃,𝐵,𝜑)∩VI(𝐴,𝐶) is a nonempty, closed and convex subset of 𝐶. Consequently, Π_Θ𝑥₀ is well defined.
Next, we prove that 𝐶_𝑛 is closed and convex for each 𝑛≥1. It is obvious that 𝐶₁=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for each 𝑛∈ℕ. Since for any 𝑧∈𝐶_𝑛, we know 𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑥_𝑛) is equivalent to 2⟨𝑧,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛‖²−‖𝑢_𝑛‖². (3.3) So, 𝐶_𝑛+1 is closed and convex.Step 2. We show that Θ⊂𝐶_𝑛 for all 𝑛≥1.
Next, we show by induction that Θ⊂𝐶_𝑛 for all 𝑛∈ℕ. Indeed, let 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛 and 𝑣_𝑛=𝐽_𝑟𝑛𝑤_𝑛 for all 𝑛≥1. On the other hand, from Lemma 2.11 one has 𝐾_𝑟𝑛 is relatively quasi-nonexpansive mapping and Θ⊂𝐶₁=𝐶. Suppose that Θ⊂𝐶_𝑛 for some 𝑛≥1. Let 𝑞∈Θ⊂𝐶_𝑛. Since 𝑆_𝑛 is relatively quasi-nonexpansive mapping, we have𝜙(𝑞,𝑦_𝑛)=𝜙(𝑞,𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆_𝑛𝑧_𝑛))=‖𝑞‖²−2⟨𝑞,𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆_𝑛𝑧_𝑛⟩+‖𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆_𝑛𝑧_𝑛‖²≤‖𝑞‖²−2𝛼_𝑛⟨𝑞,𝐽𝑥_𝑛⟩−2(1−𝛼_𝑛)⟨𝑞,𝐽𝑆_𝑛𝑧_𝑛⟩+𝛼_𝑛‖𝑥_𝑛‖²+(1−𝛼_𝑛)‖𝑆_𝑛𝑧_𝑛‖²=𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑞,𝑆_𝑛𝑧_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑞,𝑧_𝑛), (3.4) by nonexpansiveness of 𝐽_𝑟𝑛 (see [31, Theorem 4.6.3, page 130]) and 𝑇_𝑛 is relatively quasi-nonexpansive mappings, we also have 𝜙(𝑞,𝑧_𝑛)=𝜙(𝑞,𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇_𝑛𝑣_𝑛))=‖𝑞‖²−2⟨𝑞,𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇_𝑛𝑣_𝑛⟩+‖𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇_𝑛𝑣_𝑛‖²≤‖𝑞‖²−2𝛽_𝑛⟨𝑞,𝐽𝑥_𝑛⟩−2(1−𝛽_𝑛)⟨𝑞,𝐽𝑇_𝑛𝑣_𝑛⟩+𝛽_𝑛‖𝑥_𝑛‖²+(1−𝛽_𝑛)‖𝑇_𝑛𝑣_𝑛‖²=𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑇_𝑛𝑣_𝑛)≤𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑣_𝑛)=𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝐽_𝑟𝑛𝑤_𝑛)≤𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑤_𝑛). (3.5) So, it follows that 𝜙(𝑞,𝑢_𝑛)=𝜙(𝑞,𝐾_𝑟𝑛𝑦_𝑛)≤𝜙(𝑞,𝑦_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑞,𝑧_𝑛),≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)[𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑤_𝑛)]. (3.6) It follows from Lemmas 2.5 and 2.7, that 𝜙(𝑞,𝑤_𝑛)=𝜙(𝑞,Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛))≤𝜙(𝑞,𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛))=𝑉(𝑞,𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)≤𝑉(𝑞,(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)+𝜆_𝑛𝐴𝑥_𝑛)−2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑞,𝜆_𝑛𝐴𝑥_𝑛⟩=𝑉(𝑞,𝐽𝑥_𝑛)−2𝜆_𝑛⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑞,𝐴𝑥_𝑛⟩=𝜙(𝑞,𝑥_𝑛)−2𝜆_𝑛⟨𝑥_𝑛−𝑞,𝐴𝑥_𝑛⟩+2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩. (3.7) Thus, since 𝑞∈VI(𝐴,𝐶) and 𝐴 is 𝛼-inverse-strongly monotone, we have −2𝜆_𝑛⟨𝑥_𝑛−𝑞,𝐴𝑥_𝑛⟩=−2𝜆_𝑛⟨𝑥_𝑛−𝑞,𝐴𝑥_𝑛−𝐴𝑞⟩−2𝜆_𝑛⟨𝑥_𝑛−𝑞,𝐴𝑞⟩≤−2𝜆_𝑛⟨𝑥_𝑛−𝑞,𝐴𝑥_𝑛−𝐴𝑞⟩=−2𝛼𝜆_𝑛‖𝐴𝑥_𝑛−𝐴𝑞‖². (3.8) By Lemma 2.1 and the fact that ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖ for all 𝑦∈𝐶 and 𝑢∈Θ, we obtain 2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩=2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝐽⁻¹(𝐽𝑥_𝑛),−𝜆_𝑛𝐴𝑥_𝑛⟩≤2‖𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝐽⁻¹(𝐽𝑥_𝑛)‖‖𝜆_𝑛𝐴𝑥_𝑛‖≤4𝑐²‖𝐽𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝐽𝐽⁻¹(𝐽𝑥_𝑛)‖‖𝜆_𝑛𝐴𝑥_𝑛‖=4𝑐²‖𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝐽𝑥_𝑛‖‖𝜆_𝑛𝐴𝑥_𝑛‖=4𝑐²‖𝜆_𝑛𝐴𝑥_𝑛‖²=4𝑐²𝜆²_𝑛‖𝐴𝑥_𝑛‖²≤4𝑐²𝜆²_𝑛‖𝐴𝑥_𝑛−𝐴𝑞‖². (3.9) Substituting (3.8) and (3.9) into (3.7), we have 𝜙(𝑞,𝑤_𝑛)≤𝜙(𝑞,𝑥_𝑛)−2𝛼𝜆_𝑛‖𝐴𝑥_𝑛−𝐴𝑞‖²+4𝑐²𝜆²_𝑛‖𝐴𝑥_𝑛−𝐴𝑞‖²=𝜙(𝑞,𝑥_𝑛)+2𝜆_𝑛(2𝑐²𝜆_𝑛−𝛼)‖𝐴𝑥_𝑛−𝐴𝑞‖²≤𝜙(𝑞,𝑥_𝑛). (3.10) Substituting (3.10) into (3.6), we get 𝜙(𝑞,𝑢_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)[𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑥_𝑛)]=𝜙(𝑞,𝑥_𝑛). (3.11)
This shows that 𝑞∈𝐶_𝑛+1 which implies that Θ⊂𝐶_𝑛+1 and hence, Θ⊂𝐶_𝑛 for all 𝑛≥1. This implies that the sequence {𝑥_𝑛} is well defined.
Step 3. We prove that {𝑥_𝑛} is bounded.
Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀ and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀⊂𝐶_𝑛+1⊂𝐶_𝑛, we have𝜙(𝑥_𝑛,𝑥₀)≤𝜙(𝑥_𝑛+1,𝑥₀),∀𝑛≥1. (3.12) By Lemma 2.5, we get 𝜙(𝑥_𝑛,𝑥₀)=𝜙(Π_𝐶𝑛(𝑥₀),𝑥₀)≤𝜙(𝑝,𝑥₀)−𝜙(𝑝,𝑥_𝑛)≤𝜙(𝑝,𝑥₀),∀𝑝∈Θ. (3.13) From (3.12) and (3.13), then {𝜙(𝑥_𝑛,𝑥₀)} are nondecreasing and bounded. So, we obtain that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. In particular, by (1.16), the sequence {(‖𝑥_𝑛‖−‖𝑥₀‖)²} is bounded. This implies {𝑥_𝑛} is also bounded. So, we have {𝑢_𝑛}, {𝑧_𝑛}, {𝑦_𝑛} and {𝑤_𝑛} are bounded.Step 4. We show that {𝑥_𝑛} is a Cauchy sequence in 𝐶. Since 𝑥_𝑚=Π_𝐶𝑚𝑥₀∈𝐶_𝑚⊂𝐶_𝑛, for 𝑚>𝑛, by Lemma 2.5, we have 𝜙(𝑥_𝑚,𝑥_𝑛)=𝜙(𝑥_𝑚,Π_𝐶𝑛𝑥₀)≤𝜙(𝑥_𝑚,𝑥₀)−𝜙(Π_𝐶𝑛𝑥₀,𝑥₀)=𝜙(𝑥_𝑚,𝑥₀)−𝜙(𝑥_𝑛,𝑥₀). (3.14)
Taking 𝑚,𝑛→∞, we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0. From Lemma 2.3, we get ‖𝑥_𝑛−𝑥_𝑚‖→0. Hence {𝑥_𝑛} is a Cauchy sequence and by the completeness of 𝐸 and the closedness of 𝐶, we can assume that there exists 𝑝∈𝐶 such that 𝑥_𝑛→𝑝∈𝐶 as 𝑛→∞.Step 5. We show that ‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖→0, as 𝑛→∞. We taking 𝑚=𝑛+1 in Step 4, we also have lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑥_𝑛)=0. (3.15) From Lemma 2.3, that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.16) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we have lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖=0. (3.17) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛 and the definition of 𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑢_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛)∀𝑛∈ℕ. (3.18) By (3.15), we obtain lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑢_𝑛)=0. (3.19) Again applying Lemma 2.3, we get lim_𝑛→∞‖𝑥_𝑛+1−𝑢_𝑛‖=0. (3.20) From ‖𝑢_𝑛−𝑥_𝑛‖=‖𝑢_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛‖≤‖𝑢_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖ (3.21) It follows that lim_𝑛→∞‖𝑢_𝑛−𝑥_𝑛‖=0. (3.22) Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, we also have lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖=0. (3.23)Step 6. We will show that 𝑥_𝑛→𝑝∈Θ, where Θ∶=(^∞_𝑛=1𝐹(𝑇_𝑛))∩(^∞_𝑛=1𝐹(𝑆_𝑛))∩GMEP(𝜃,𝐵,𝜑)∩VI(𝐴,𝐶)∩𝑇⁻¹0 (3.24)(a)We show that 𝑥_𝑛→𝑝∈(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑆_𝑛)). From definition of 𝐶_𝑛+1, for any 𝑧∈𝐶_𝑛, we have 𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛). (3.25) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, we get 𝜙(𝑥_𝑛+1,𝑧_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛). It follows from (3.15), that lim_𝑛→∞𝜙(𝑥_𝑛+1,𝑧_𝑛)=0 (3.26) again from Lemma 2.3, that lim_𝑛→∞‖𝑥_𝑛+1−𝑧_𝑛‖=0 (3.27) it follows that since 𝐽 is uniformly norm-to-norm continuous, we also have lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑧_𝑛‖=0. (3.28) Since ‖𝑧_𝑛−𝑥_𝑛‖≤‖𝑧_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑥_𝑛‖ (3.29) from (3.16) and (3.27), we also have lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0. (3.30) Since 𝐽 is uniformly norm-to-norm continuous, we obtain lim_𝑛→∞‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖=0. (3.31) From (3.4), (3.5) and (3.10), we get 𝜙(𝑝,𝑦_𝑛)≤𝜙(𝑝,𝑥_𝑛). By Lemma 2.11 (5) and 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛, we observe that 𝜙(𝑢_𝑛,𝑦_𝑛)=𝜙(𝐾_𝑟𝑛𝑦_𝑛,𝑦_𝑛)≤𝜙(𝑝,𝑦_𝑛)−𝜙(𝑝,𝐾_𝑟𝑛𝑦_𝑛)≤𝜙(𝑝,𝑥_𝑛)−𝜙(𝑝,𝐾_𝑟𝑛𝑦_𝑛)=𝜙(𝑝,𝑥_𝑛)−𝜙(𝑝,𝑢_𝑛)=‖𝑝‖²−2⟨𝑝,𝐽𝑥_𝑛⟩+‖𝑥_𝑛‖²−(‖𝑝‖²−2⟨𝑝,𝐽𝑢_𝑛⟩+‖𝑢_𝑛‖²)=‖𝑥_𝑛‖²−‖𝑢_𝑛‖²−2⟨𝑝,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛−𝑢_𝑛‖(‖𝑥_𝑛‖+‖𝑢_𝑛‖)+2‖𝑝‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖. (3.32) Since {𝑥_𝑛}, {𝑦_𝑛} and {𝑢_𝑛} are bounded, it follows from (3.22), (3.23), and Lemma 2.3, we also have lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0. (3.33) Since 𝐽 is uniformly norm-to-norm continuous, we have lim_𝑛→∞‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖=0. (3.34) By using the triangle inequality, we obtain ‖𝑥_𝑛+1−𝑦_𝑛‖=‖𝑥_𝑛+1−𝑢_𝑛+𝑢_𝑛−𝑦_𝑛‖≤‖𝑥_𝑛+1−𝑢_𝑛‖+‖𝑢_𝑛−𝑦_𝑛‖. (3.35) By (3.20) and (3.33), we get lim_𝑛→∞‖𝑥_𝑛+1−𝑦_𝑛‖=0. (3.36) Since 𝐽 is uniformly norm-to-norm continuous, we obtain lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖=0. (3.37) Since ‖𝑦_𝑛−𝑧_𝑛‖≤‖𝑦_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑧_𝑛‖. (3.38) From (3.27) and (3.36), we have lim_𝑛→∞‖𝑦_𝑛−𝑧_𝑛‖=0. (3.39) Since 𝐽 is uniformly norm-to-norm continuous, we also have lim_𝑛→∞‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖=0. (3.40) From (3.2), we get ‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖=‖𝛼_𝑛(𝐽𝑥_𝑛−𝐽𝑧_𝑛)+(1−𝛼_𝑛)(𝐽𝑆_𝑛𝑧_𝑛−𝐽𝑧_𝑛)‖=‖(1−𝛼_𝑛)(𝐽𝑆_𝑛𝑧_𝑛−𝐽𝑧_𝑛)−𝛼_𝑛(𝐽𝑧_𝑛−𝐽𝑥_𝑛)‖≥(1−𝛼_𝑛)‖𝐽𝑆_𝑛𝑧_𝑛−𝐽𝑧_𝑛‖−𝛼_𝑛‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖, (3.41) and hence (1−𝛼_𝑛)‖𝐽𝑆_𝑛𝑧_𝑛−𝐽𝑧_𝑛‖≤‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖+𝛼_𝑛‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖, (3.42) it follows that ‖𝐽𝑆_𝑛𝑧_𝑛−𝐽𝑧_𝑛1‖≤1−𝛼_𝑛(‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖+𝛼_𝑛‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖). (3.43) Since liminf_𝑛→∞(1−𝛼_𝑛)>0, (3.31) and (3.40), one has lim_𝑛→∞‖𝐽𝑆_𝑛𝑧_𝑛−𝐽𝑧_𝑛‖=0. Since 𝐽⁻¹ is uniformly norm-to-norm continuous, we get lim_𝑛→∞‖𝑆_𝑛𝑧_𝑛−𝑧_𝑛‖=0. (3.44) Since ‖𝑥_𝑛−𝑧_𝑛‖→0 and 𝑥_𝑛→𝑝, then we get 𝑧_𝑛→𝑝, hence it follows from (∗)-condition, that 𝑝∈⋂^∞_𝑛=1𝐹(𝑆_𝑛).
Since 𝑣_𝑛=𝐽_𝑟𝑛𝑤_𝑛, we compute ‖𝐽𝑥_𝑛+1−𝐽𝑧_𝑛‖=‖𝐽𝑥_𝑛+1−(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇_𝑛𝑣_𝑛)‖=‖𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑥_𝑛+1−(1−𝛽_𝑛)𝐽𝑇_𝑛𝑣_𝑛‖=‖𝛽_𝑛(𝐽𝑥_𝑛+1−𝐽𝑥_𝑛)+(1−𝛽_𝑛)(𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑣_𝑛)‖=‖(1−𝛽_𝑛)(𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑣_𝑛)−𝛽_𝑛(𝐽𝑥_𝑛−𝐽𝑥_𝑛+1)‖≥(1−𝛽_𝑛)‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑣_𝑛‖−𝛽_𝑛‖𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖ (3.45) and hence ‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑣_𝑛1‖≤1−𝛽_𝑛(‖𝐽𝑥_𝑛+1−𝐽𝑧_𝑛‖+𝛽_𝑛‖𝐽𝑥_𝑛−𝐽𝑥_𝑛+1‖). (3.46) From (3.17), (3.28) and liminf_𝑛→∞(1−𝛽_𝑛)>0, we obtain that lim_𝑛→∞‖𝐽𝑥_𝑛+1−𝐽𝑇_𝑛𝑣_𝑛‖=0. (3.47) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖𝑥_𝑛+1−𝑇_𝑛𝑣_𝑛‖=0. (3.48) Using the triangle inequality, we have ‖𝑥_𝑛−𝑇_𝑛𝑣_𝑛‖=‖𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑛𝑣_𝑛‖≤‖𝑥_𝑛−𝑥_𝑛+1‖+‖𝑥_𝑛+1−𝑇_𝑛𝑣_𝑛‖. (3.49) From (3.16) and (3.48), we have lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑣_𝑛‖=0. (3.50) On the other hand, for 𝑞∈Θ, we note that 𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑢_𝑛)=‖𝑥_𝑛‖²−‖𝑢_𝑛‖²−2⟨𝑞,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤‖𝑥_𝑛−𝑢_𝑛‖(‖𝑥_𝑛‖+‖𝑢_𝑛‖)+2‖𝑞‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖. (3.51) Since {𝑥_𝑛} and {𝑢_𝑛} are bounded, it follows from ‖𝑥_𝑛−𝑢_𝑛‖→0 and ‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖→0, that 𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑢_𝑛)→0. (3.52) Furthermore, from (3.4), (3.5), (3.6) and (3.10), that 𝜙(𝑞,𝑢_𝑛)≤𝜙(𝑞,𝑦_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝜙(𝑞,𝑧_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)[𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛽_𝑛)𝜙(𝑞,𝑤_𝑛)]=𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)(1−𝛽_𝑛)𝜙(𝑞,𝑤_𝑛)≤𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)(1−𝛽_𝑛)[𝜙(𝑞,𝑥_𝑛)−2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖²]=𝛼_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)𝛽_𝑛𝜙(𝑞,𝑥_𝑛)+(1−𝛼_𝑛)(1−𝛽_𝑛)𝜙(𝑞,𝑥_𝑛)−(1−𝛼_𝑛)(1−𝛽_𝑛)2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖²=𝜙(𝑞,𝑥_𝑛)−(1−𝛼_𝑛)(1−𝛽_𝑛)2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖², (3.53) and hence 2𝑎(𝛼−2𝑏𝑐²)‖𝐴𝑥_𝑛−𝐴𝑞‖²≤2𝜆_𝑛2(𝛼−𝑐²𝜆_𝑛)‖𝐴𝑥_𝑛−𝐴𝑞‖²≤1(1−𝛼_𝑛)(1−𝛽_𝑛)(𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑢_𝑛)). (3.54) From (3.52), liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, obtain that lim_𝑛→∞‖𝐴𝑥_𝑛−𝐴𝑞‖=0. (3.55) From Lemmas 2.5, 2.7, and (3.9), we compute 𝜙(𝑥_𝑛,𝑤_𝑛)=𝜙(𝑥_𝑛,Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛))≤𝜙(𝑥_𝑛,𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛))=𝑉(𝑥_𝑛,𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)≤𝑉(𝑥_𝑛,(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)+𝜆_𝑛𝐴𝑥_𝑛)−2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,𝜆_𝑛𝐴𝑥_𝑛⟩=𝜙(𝑥_𝑛,𝑥_𝑛)+2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩=2⟨𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)−𝑥_𝑛,−𝜆_𝑛𝐴𝑥_𝑛⟩≤4𝜆²_𝑛𝑐²‖𝐴𝑥_𝑛−𝐴𝑞‖²≤4𝑏²𝑐²‖𝐴𝑥_𝑛−𝐴𝑞‖². (3.56)
Applying Lemma 2.3 and (3.55) it follows thatlim_𝑛→∞‖𝑥_𝑛−𝑤_𝑛‖=0. (3.57) Since 𝐽 is uniformly norm-to-norm continuous, we also have lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖=0. (3.58) Again by the triangle inequality, we get ‖𝑤_𝑛−𝑇_𝑛𝑣_𝑛‖=‖𝑤_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇_𝑛𝑣_𝑛‖≤‖𝑤_𝑛−𝑥_𝑛‖+‖𝑥_𝑛−𝑇_𝑛𝑣_𝑛‖. (3.59) From (3.50) and (3.57), we have lim_𝑛→∞‖𝑤_𝑛−𝑇_𝑛𝑣_𝑛‖=0. (3.60) From (3.5), we have 𝜙(𝑞,𝑣_𝑛)≥(1/(1−𝛽_𝑛))(𝜙(𝑞,𝑧_𝑛)−𝛽_𝑛𝜙(𝑞,𝑥_𝑛)), it follows from Lemma 2.8 and (3.10), we note that 𝜙(𝑣_𝑛,𝑤_𝑛)=𝜙(𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛)≤𝜙(𝑞,𝑤_𝑛)−𝜙(𝑞,𝐽_𝑟𝑛𝑤_𝑛)=𝜙(𝑞,𝑤_𝑛)−𝜙(𝑞,𝑣_𝑛)≤𝜙(𝑞,𝑤_𝑛1)−1−𝛽_𝑛(𝜙(𝑞,𝑧_𝑛)−𝛽_𝑛𝜙(𝑞,𝑥_𝑛))≤𝜙(𝑞,𝑥_𝑛1)−1−𝛽_𝑛(𝜙(𝑞,𝑧_𝑛)−𝛽_𝑛𝜙(𝑞,𝑥_𝑛=1))1−𝛽_𝑛(𝜙(𝑞,𝑥_𝑛)−𝜙(𝑞,𝑧_𝑛=1))1−𝛽_𝑛(‖𝑥_𝑛‖²−‖𝑧_𝑛‖²−2⟨𝑞,𝐽𝑥_𝑛−𝐽𝑧_𝑛≤1⟩)1−𝛽_𝑛(‖𝑥_𝑛‖²−‖𝑧_𝑛‖²+2|⟨𝑞,𝐽𝑥_𝑛−𝐽𝑧_𝑛≤1⟩|)1−𝛽_𝑛(|‖𝑥_𝑛‖−‖𝑧_𝑛‖|(‖𝑥_𝑛‖+‖𝑧_𝑛‖)+2‖𝑞‖‖𝐽𝑥_𝑛−𝐽𝑧_𝑛≤1‖)1−𝛽_𝑛(‖𝑥_𝑛−𝑧_𝑛‖(‖𝑥_𝑛‖+‖𝑧_𝑛‖)+2‖𝑞‖‖𝐽𝑥_𝑛−𝐽𝑧_𝑛‖). (3.61) It follows from liminf_𝑛→∞(1−𝛽_𝑛)>0, (3.30) and (3.31), we get lim_𝑛→∞𝜙(𝑣_𝑛,𝑤_𝑛)=0. (3.62) From Lemma 2.3, it follows that lim_𝑛→∞‖𝑣_𝑛−𝑤_𝑛‖=0. (3.63) By using the triangle inequality, we get ‖𝑣_𝑛−𝑇_𝑛𝑣_𝑛‖=‖𝑣_𝑛−𝑤_𝑛+𝑤_𝑛−𝑇_𝑛𝑣_𝑛‖≤‖𝑣_𝑛−𝑤_𝑛‖+‖𝑤_𝑛−𝑇_𝑛𝑣_𝑛‖. (3.64) From (3.60) and (3.63), we have lim_𝑛→∞‖𝑣_𝑛−𝑇_𝑛𝑣_𝑛‖=0. (3.65) Since ‖𝑥_𝑛−𝑣_𝑛‖≤‖𝑥_𝑛−𝑤_𝑛‖+‖𝑤_𝑛−𝑣_𝑛‖, (3.57) and (3.63), then lim_𝑛→∞‖𝑥_𝑛−𝑣_𝑛‖=0. (3.66) From (3.66) and since 𝑥_𝑛→𝑝, then 𝑣_𝑛→𝑝. By (3.60) it follows from (∗)-condition, that 𝑝∈⋂^∞_𝑛=1𝐹(𝑇_𝑛). Hence 𝑝∈(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛)).(b)We show that 𝑥_𝑛→𝑝∈GMEP(𝜃,𝐵,𝜑). Indeed, it follows from (A2), that ⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑(𝑢_𝑛1)+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥−𝜃(𝑢_𝑛,𝑦)≥𝜃(𝑦,𝑢_𝑛),∀𝑦∈𝐶, (3.67) and hence ⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑(𝑢_𝑛)+⟨𝑦−𝑢_𝑛,(𝐽𝑢_𝑛−𝐽𝑦_𝑛)𝑟_𝑛⟩≥𝜃(𝑦,𝑢_𝑛),∀𝑦∈𝐶. (3.68) For 𝑡 with 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝. Then, we get that 𝑦_𝑡∈𝐶. From (3.68), it follows that ⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩≥⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩−⟨𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩−𝜑(𝑦_𝑡)+𝜑(𝑢_𝑛)−⟨𝑦_𝑡−𝑢_𝑛,(𝐽𝑢_𝑛−𝐽𝑦_𝑛)𝑟_𝑛⟩+𝜃(𝑦_𝑡,𝑢_𝑛)=⟨𝐵𝑦_𝑡−𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩−𝜑(𝑦_𝑡)+𝜑(𝑢_𝑛)−⟨𝑦_𝑡−𝑢_𝑛,(𝐽𝑢_𝑛−𝐽𝑦_𝑛)𝑟_𝑛⟩+𝜃(𝑦_𝑡,𝑢_𝑛),∀𝑦_𝑡∈𝐶. (3.69) By the fact that 𝑦_𝑛,𝑢_𝑛→𝑝 as 𝑛→∞, and ‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖/𝑟_𝑛→0 as 𝑛→∞. Since 𝐵 is monotone, we know that ⟨𝐵𝑦_𝑡−𝐵𝑢_𝑛,𝑦_𝑡−𝑢_𝑛⟩≥0. Thus, it follows from (A4) that 𝜃(𝑦_𝑡,𝑝)−𝜑(𝑦_𝑡)+𝜑(𝑝)≤liminf_𝑛→∞𝜃(𝑦_𝑡,𝑢_𝑛)−𝜑(𝑦_𝑡)+𝜑(𝑢_𝑛)≤lim_𝑛→∞⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑢_𝑛⟩=⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑝⟩. (3.70)
By the conditions (A1), (A4) and convexity of 𝜑, we have 0=𝜃(𝑦_𝑡,𝑦_𝑡)+𝜑(𝑦_𝑡)−𝜑(𝑦_𝑡)≤𝑡𝜃(𝑦_𝑡,𝑦)+(1−𝑡)𝜃(𝑦_𝑡,𝑝)+𝑡𝜑(𝑦)+(1−𝑡)𝜑(𝑝)−𝜑(𝑦_𝑡)=𝑡[𝜃(𝑦_𝑡,𝑦)+𝜑(𝑦)−𝜑(𝑦_𝑡)]+(1−𝑡)[𝜃(𝑦_𝑡,𝑝)+𝜑(𝑝)−𝜑(𝑦_𝑡)]≤𝑡[𝜃(𝑦_𝑡,𝑦)+𝜑(𝑦)−𝜑(𝑦_𝑡)]+(1−𝑡)[⟨𝐵𝑦_𝑡,𝑦_𝑡−𝑝⟩]=𝑡[𝜃(𝑦_𝑡,𝑦)+𝜑(𝑦)−𝜑(𝑦_𝑡)]+(1−𝑡)𝑡[⟨𝐵𝑦_𝑡,𝑦−𝑝⟩] (3.71) and hence 0≤𝜃(𝑦_𝑡,𝑦)+𝜑(𝑦)−𝜑(𝑦_𝑡)+(1−𝑡)⟨𝐵𝑦_𝑡,𝑦−𝑝⟩. (3.72) From (A3) and the weakly lower semicontinuity of 𝜑, letting 𝑡→0, we also have 𝜃(𝑝,𝑦)+⟨𝐵𝑝,𝑦−𝑝⟩+𝜑(𝑦)−𝜑(𝑝)≥0,∀𝑦∈𝐶. (3.73) This implies that 𝑝∈GMEP(𝜃,𝐵,𝜑).(c)We show that 𝑥_𝑛→𝑝∈VI(𝐴,𝐶). Indeed, define a set-valued 𝑈∶𝐸⇉𝐸^∗ by Lemma 2.9, 𝑈 is maximal monotone and 𝑈⁻¹0=VI(𝐴,𝐶). Let (𝑣,𝑤)∈𝐺(𝑈). Since 𝑤∈𝑈𝑣=𝐴𝑣+𝑁_𝐶(𝑣), we get 𝑤−𝐴𝑣∈𝑁_𝐶(𝑣). From 𝑤_𝑛∈𝐶, we have ⟨𝑣−𝑤_𝑛,𝑤−𝐴𝑣⟩≥0. (3.74) On the other hand, since 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛). Then by Lemma 2.4, we have ⟨𝑣−𝑤_𝑛,𝐽𝑤_𝑛−(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛)⟩≥0, (3.75) and thus ⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐴𝑥_𝑛⟩≤0. (3.76) It follows from (3.74) and (3.76), that ⟨𝑣−𝑤_𝑛,𝑤⟩≥⟨𝑣−𝑤_𝑛,𝐴𝑣⟩≥⟨𝑣−𝑤_𝑛,𝐴𝑣⟩+⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐴𝑥_𝑛⟩=⟨𝑣−𝑤_𝑛,𝐴𝑣−𝐴𝑥_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛⟩=⟨𝑣−𝑤_𝑛,𝐴𝑣−𝐴𝑤_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐴𝑤_𝑛−𝐴𝑥_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐽𝑥_𝑛−𝐽𝑤_𝑛𝜆_𝑛⟩≥−‖𝑣−𝑤_𝑛‖‖𝑤_𝑛−𝑥_𝑛‖𝛼−‖𝑣−𝑤_𝑛‖‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖𝑎≥−𝑀(‖𝑤_𝑛−𝑥_𝑛‖𝛼+‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖𝑎), (3.77) where 𝑀=sup_𝑛≥1‖𝑣−𝑤_𝑛‖. Take the limit as 𝑛→∞, (3.57) and (3.58), we obtain ⟨𝑣−𝑝,𝑤⟩≥0. By the maximality of 𝑈, we have 𝑝∈𝑈⁻¹0 and hence 𝑝∈VI(𝐴,𝐶).(d)We show that 𝑥_𝑛→𝑝∈𝑇⁻¹0. Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, from (3.63), we get lim_𝑛→∞‖𝐽𝑤_𝑛−𝐽𝑣_𝑛‖=0. (3.78) From 𝑟_𝑛≥𝑑, we have lim_𝑛→∞1𝑟_𝑛‖𝐽𝑤_𝑛−𝐽𝑣_𝑛‖=0. (3.79) Since 𝐽_𝑟𝑛𝑤_𝑛=𝑣_𝑛, therefore, lim_𝑛→∞‖𝑇_𝑟𝑛𝑤_𝑛‖=lim_𝑛→∞1𝑟_𝑛‖𝐽𝑤_𝑛−𝐽𝐽_𝑟𝑛𝑤_𝑛‖=lim_𝑛→∞1𝑟_𝑛‖𝐽𝑤_𝑛−𝐽𝑣_𝑛‖=0. (3.80) For (𝑤,𝑤^∗)∈𝐺(𝑇), from the monotonicity of 𝑇, we have ⟨𝑤−𝑣_𝑛,𝑤^∗−𝑇_𝑟𝑛𝑤_𝑛⟩≥0 for all 𝑛≥0. Letting 𝑛→∞, we get ⟨𝑤−𝑝,𝑤^∗⟩≥0. From the maximality of 𝑇, we have 𝑝∈𝑇⁻¹0. Hence, from (a), (b), (c) and (d), we obtain 𝑝∈Θ.Step 7. we show that 𝑝=Π_Θ𝑥₀.
From 𝑥_𝑛=Π_𝐶𝑛𝑥₀, we have ⟨𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛−𝑧⟩≥0, for all 𝑧∈𝐶_𝑛. Since Θ⊂𝐶_𝑛, we also have ⟨𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛−𝑦⟩≥0,∀𝑦∈Θ. (3.81) Taking limit 𝑛→∞, we obtain ⟨𝐽𝑥₀−𝐽𝑝,𝑝−𝑦⟩≥0,∀𝑦∈Θ. (3.82) By Lemma 2.4, we can conclude that 𝑝=Π_Θ𝑥₀ and 𝑥_𝑛→𝑝 as 𝑛→∞. This completes the proof.

Theorem 3.2. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑇∶𝐸⇉𝐸^∗ be a maximal monotone operator satisfying 𝐷(𝑇)⊂𝐶 and let 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0, where 𝐽 is the duality mapping on 𝐸. Let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝜑∶𝐶→ℝ be a proper lower semicontinuous and convex function. Let 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐸^∗ satisfy ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖, for all 𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅ and let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping. Let 𝑆_𝑛,𝑇_𝑛∶𝐶→𝐶 be two families of relatively quasi-nonexpansive mappings with satisfy the NST-condition such that Θ∶=(⋂^∞_𝑛=1𝐹(𝑆_𝑛))∩(⋂^∞_𝑛=1𝐹(𝑇_𝑛))∩𝑇⁻¹0∩GMEP(𝜃,𝐵,𝜑)∩VI(𝐴,𝐶)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} by (3.2) where {𝛼_𝑛},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑑,∞) for some 𝑑>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. If liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Proof. If {𝑇_𝑛},{𝑆_𝑛} satisfy NST-condition, then {𝑇_𝑛},{𝑆_𝑛} satisfy (∗)-condition.

Setting 𝑆_𝑛≡𝑆 and 𝑇_𝑛≡𝑇 in Theorem 3.1, then we obtain the following result.

Corollary 3.3. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝑇∶𝐸⇉𝐸^∗ be a maximal monotone operator satisfying 𝐷(𝑇)⊂𝐶 and let 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for all 𝑟>0. Let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝜑∶𝐶→ℝ be a proper lower semicontinuous and convex function. Let 𝐴 be an 𝛼-inverse-strongly monotone mapping of 𝐶 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖, for all 𝑦∈𝐶 and 𝑢∈VI(𝐴,𝐶)≠∅ and let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping. Let 𝑇,𝑆∶𝐶→𝐶 be two closed relatively quasi-nonexpansive mappings such that Θ∶=𝐹(𝑆)∩𝐹(𝑇)∩𝑇⁻¹0∩GMEP(𝜃,𝐵,𝜑)∩VI(𝐴,𝐶)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝐽_𝑟𝑛𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝜃(𝑢_𝑛,𝑦)+𝜑(𝑦)−𝜑(𝑢_𝑛)+⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.83) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑑,∞) for some 𝑑>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. If liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Next, we consider the problem of finding a zero point of an inverse-strongly monotone operator of 𝐸 into 𝐸^∗. Assume that 𝐴 satisfies the conditions: (C1)𝐴 is 𝛼-inverse-strongly monotone, (C2)𝐴⁻¹0={𝑢∈𝐸∶𝐴𝑢=0}≠∅.

Hence, setting 𝑇𝑥≡0, for all 𝑥∈𝐶 in Corollary 3.3, then 𝐽_𝑟=𝐼, we also have the following result.

Corollary 3.4. Let 𝐸 be a 2-uniformly convex and uniformly smooth Banach space. Let 𝜃 be a bifunction from 𝐸×𝐸 to ℝ satisfying (A1)–(A4), and let 𝜑∶𝐸→ℝ be a proper lower semicontinuous and convex function. Let 𝐴 be an operator of 𝐸 into 𝐸^∗ satisfying (C1) and (C2), and let 𝐵∶𝐸→𝐸^∗ be a continuous and monotone mapping. Let 𝑇,𝑆∶𝐸→𝐸 be two closed relatively quasi-nonexpansive mappings such that Θ∶=𝐹(𝑆)∩𝐹(𝑇)∩GMEP(𝜃,𝐵,𝜑)∩𝐴⁻¹0≠∅. (3.84) For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐸1𝑥₀ and 𝐸₁=𝐸, we define the sequence {𝑥_𝑛} as follows: 𝑢^{𝑤𝑛=Π𝐶𝐽−1(𝐽𝑥𝑛−𝜆𝑛𝐴𝑥𝑛𝑧),𝑛=𝐽−1(𝛽𝑛𝐽𝑥𝑛+(1−𝛽𝑛)𝐽𝑇𝑤𝑛𝑦),𝑛=𝐽−1(𝛼𝑛𝐽𝑥𝑛+(1−𝛼𝑛)𝐽𝑆𝑧𝑛),}_𝑛∈𝐶suchthat𝜃(𝑢_𝑛,𝑦)+𝜑(𝑦)−𝜑(𝑢_𝑛)+⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐸,_{𝐸𝑛+1={𝑧∈𝐸𝑛∶𝜙(𝑧,𝑢𝑛)≤𝜙(𝑧,𝑧𝑛)≤𝜙(𝑧,𝑥𝑛𝑥)},𝑛+1=Π𝐸𝑛+1𝑥0,∀𝑛≥1,} (3.52) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑑,∞) for some 𝑑>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. If liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Proof. Setting 𝑇𝑥≡0, for all 𝑥∈𝐶, then, 𝐷(𝑇)=𝐸 and hence 𝐶=𝐸 in Corollary 3.3, we also get Π_𝐸=𝐼. We also have VI(𝐴,𝐶)=VI(𝐴,𝐸)={𝑥∈𝐸∶𝐴𝑥=0}≠∅ and then the condition ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖ holds for all 𝑦∈𝐸 and 𝑢∈𝐴⁻¹0. So, we obtain the result.

Setting 𝐴≡0 in Corollary 3.4, then we get 𝑤_𝑛=𝑥_𝑛. Hence we obtain the following Corollary.

Corollary 3.5. Let 𝐶 be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝜃 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), and let 𝜑∶𝐶→ℝ be a proper lower semicontinuous and convex function. Let 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping. Let 𝑇,𝑆∶𝐶→𝐶 be two closed relatively quasi-nonexpansive mappings such that Θ∶=𝐹(𝑆)∩𝐹(𝑇)∩GMEP(𝜃,𝐵,𝜑)≠∅. (3.86) For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶, we define the sequence {𝑥_𝑛} as follows: 𝑧_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝑥_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐶suchthat𝜃(𝑢_𝑛,𝑦)+𝜑(𝑦)−𝜑(𝑢_𝑛)+⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,∀𝑦∈𝐶,_𝑛+1={𝑧∈𝐶_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥1, (3.87) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛},{𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑑,∞) for some 𝑑>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. If liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Remark 3.6. Theorem 3.1, Corollaries 3.4 and 3.5 improve and extend the corresponding results in Cholamjiak [36], Wei et al. [37] and Saewan et al. [26].

4. Application to Complementarity Problem

Let 𝐾 be a nonempty, closed convex cone in 𝐸. We define the polar 𝐾^∗ of 𝐾 as follows:𝐾^∗={𝑦^∗∈𝐸^∗∶⟨𝑥,𝑦^∗⟩≥0,∀𝑥∈𝐾}. (4.1) If 𝐴∶𝐾→𝐸^∗ is an operator, then an element 𝑢∈𝐾 is called a solution of the complementarity problem ([31]) if𝐴𝑢∈𝐾^∗,⟨𝑢,𝐴𝑢⟩=0. (4.2) The set of solutions of the complementarity problem is denoted by 𝐶(𝐴,𝐾).

Theorem 4.1. Let 𝐾 be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space 𝐸. Let 𝐴∶𝐸⇉𝐸^∗ be a maximal monotone operator satisfying 𝐷(𝐴)⊂𝐾 and let 𝐽_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽 for all 𝑟>0. Let 𝜃 be a bifunction from 𝐾×𝐾 to ℝ satisfying (A1)–(A4), and let 𝜑∶𝐾→ℝ be a proper lower semicontinuous and convex function. Let 𝐴∶𝐾→𝐸^∗ be an 𝛼-inverse-strongly monotone mapping of 𝐸 into 𝐸^∗ satisfying ‖𝐴𝑦‖≤‖𝐴𝑦−𝐴𝑢‖, for all 𝑦∈𝐾 and 𝑢∈𝐶(𝐴,𝐾)≠∅ and let 𝐵∶𝐾→𝐸^∗ be a continuous and monotone mapping. Let 𝑆,𝑇∶𝐾→𝐾 be two closed relatively quasi-nonexpansive mappings such that Θ∶=𝐹(𝑆)∩𝐹(𝑇)∩𝐴⁻¹0∩GMEP(𝜃,𝐵,𝜑)∩𝐶(𝐴,𝐾)≠∅. For an initial point 𝑥₀∈𝐸 with 𝑥₁=Π_𝐾1𝑥₀ and 𝐾₁=𝐾, we define the sequence {𝑥_𝑛} as follows: 𝑤_𝑛=Π_𝐾𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛𝑧),_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑇𝐽_𝑟𝑛𝑤_𝑛𝑦),_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑆𝑧_𝑛𝑢),_𝑛∈𝐾suchthat𝜃(𝑢_𝑛,𝑦)+𝜑(𝑦)−𝜑(𝑢_𝑛)+⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩𝐾≥0,∀𝑦∈𝐾,_𝑛+1={𝑧∈𝐾_𝑛∶𝜙(𝑧,𝑢_𝑛)≤𝜙(𝑧,𝑧_𝑛)≤𝜙(𝑧,𝑥_𝑛𝑥)},_𝑛+1=Π_𝐾𝑛+1𝑥₀,∀𝑛≥1, (4.3) where 𝐽 is the duality mapping on 𝐸, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[𝑑,∞) for some 𝑑>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, where 1/𝑐 is the 2-uniformly convexity constant of 𝐸. If liminf_𝑛→∞(1−𝛼_𝑛)>0 and liminf_𝑛→∞(1−𝛽_𝑛)>0, then {𝑥_𝑛} converges strongly to 𝑝∈Θ, where 𝑝=Π_Θ𝑥₀.

Proof. As in the proof of Takahashi in [31, Lemma 7.11], we get that VI(𝐴,𝐾)=𝐶(𝐴,𝐾). So, we obtain the result.

Acknowledgments

The authors are grateful for Professor Douglas Robert Anderson and the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work. This paper was financially supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand and the King Mongkut’s Dimond scholarship for Ph.D. program at King Mongkut’s University of Technology Thonburi (NRU KMUTT, under CSEC project) during its preparation. The second author was supported by the Higher Education Commission and the Thailand Research Fund under Grant MRG5380044.