A New Iterative Scheme for Generalized Mixed Equilibrium, Variational Inequality Problems, and a Zero Point of Maximal Monotone Operators

Abstract

The purpose of this paper is to introduce a new iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi-𝜙-nonexpansive mappings in Banach spaces. Moreover, the strong convergence theorems of this method are established under the suitable conditions of the parameter imposed on the algorithm. Finally, we apply our results to finding a zero point of inverse-strongly monotone operators and complementarity problems. Our results presented in this paper improve and extend the recently results by many others.

1. Introduction

Equilibrium problem theory is the most important area of mathematical sciences and widely popular among mathematicians and researchers in other fields due to its applications in a wide class of problems which arise in economics, finance, optimization, network and transportation, image reconstruction, ecology, and many others. It has been improved and extended in many directions. Furthermore, equilibrium problems are related to the problem of finding fixed point of nonexpansive mappings. In this way, they have been extensively studied by many authors; see [1–9]. They introduced new iterative schemes for finding a common element of the set of the solutions of equilibrium problems and the set of fixed points. In this paper, we are interested a new hybrid iterative method for finding a common elements of the set of solutions of generalized mixed equilibrium problems, the set of solutions of variational inequality problems, the zero point of maximal monotone operators, and the set of two countable families of quasi-𝜙-nonexpansive mappings in the framework of Banach spaces.

Let 𝐸 be a Banach space with norm ‖⋅‖ and 𝐶 a nonempty closed convex subset of 𝐸 and let 𝐸^∗ denote the dual of 𝐸.

A mapping 𝑆∶𝐶→𝐶 is said to be(1)nonexpansive [1] if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶,(2)relatively nonexpansive [10–12] if 𝐹(𝑆)=𝐹(𝑆) and 𝜙(𝑝,𝑆𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑆), where the functional 𝜙 defined by (2.6). The asymptotic behavior of a relatively nonexpansive mapping was studied in [13, 14],(3)𝜙-nonexpansive, if 𝜙(𝑆𝑥,𝑆𝑦)≤𝜙(𝑥,𝑦) for 𝑥,𝑦∈𝐶,(4)quasi- 𝜙-nonexpansive if 𝐹(𝑆)≠∅ and 𝜙(𝑝,𝑆𝑥)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶 and 𝑝∈𝐹(𝑆).

In the sequel, we denote 𝐹(𝑇) as the set of fixed points of 𝑆. If 𝐶 is a bounded closed convex set and 𝑆 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑆) is nonempty (see [15]).

A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑆 [16] 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).

Let 𝐵 be an operator from 𝐶 into 𝐸^∗, and 𝐵 is said to be 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼 such that ⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩≥𝛼‖𝐵𝑥−𝐵𝑦‖²,∀𝑥,𝑦∈𝐶.(1.1)

If an operator 𝐵 is an 𝛼-inverse-strongly monotone, then we can said that 𝐵 is Lipschitz continuous; that is, ‖𝐵𝑥−𝐵𝑦‖≤(1/𝛼)‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶.

Let 𝑓∶𝐶×𝐶→ℝ be a bifunction, 𝜑∶𝐶→ℝ a real-valued function, and 𝐵∶𝐶→𝐸^∗ be a nonlinear mapping. The generalized mixed equilibrium problem is to find 𝑥∈𝐶 such that𝑓(𝑥,𝑦)+⟨𝐵𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.2)

We denote Ω as the set of solutions to (1.2) that is,Ω={𝑥∈𝐶∶𝑓(𝑥,𝑦)+⟨𝐵𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶}.(1.3)

If 𝐵≡0, the problem (1.2) reduced into the mixed equilibrium problem for 𝑓, denoted by MEP(𝑓,𝜑), is to find 𝑥∈𝐶 such that𝑓(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.4)

If 𝑓≡0, the problem (1.2) reduced into the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐵,𝜑), is to find 𝑥∈𝐶 such that⟨𝐵𝑥,𝑦−𝑥⟩+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.5)

If 𝐵≡0 and 𝜑≡0 the problem (1.2), reduced into the equilibrium problem for 𝐹, denoted by EP(𝑓), is to find 𝑥∈𝐶 such that𝑓(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.6)

In addition, fixed point problem, optimization problem, and many problems can be written in the form of EP(𝑓). There are the development of researches in this area as seen in many papers which appeared in the literature on the existence of the solutions of EP(𝑓); see, for example [17–21] and reference therein. Furthermore, there are many solution methods proposed continuously to solve the EP(𝑓) as shown in [2, 3, 18, 20, 22–26] and many others.

Next, we let 𝐵 be a monotone operator of 𝐶 into 𝐸^∗. The so-called variational inequality problem is to find a point 𝑥∈𝐶 such that⟨𝐵𝑥,𝑦−𝑥⟩≥0∀y∈𝐶.(1.7) The set of solutions of the variational inequality problem is denoted by VI(𝐶,𝐵).

As we know that the classical variational inequality was first introduced and studied by Stampacchia [27] in 1964. Its solution can be computed by using iterative projection method. There are many results with corresponding to variational inequality; for example, Yao et al. [28] proposed the strong convergence theorem for a system of nonlinear variational inequalities in Banach spaces, and then, they studied the two-step projection methods, and they established the convergence theorem for a system of variational inequality problems in the framework of Banach spaces. Moreover, the important generalized variational inequalities called variational inclusion also have been extensively studied and extended in many different directions. Yao et al. [29] considered the algorithm and proved the strong convergence of common solutions for variational inclusions, mixed equilibrium problems, and fixed point problems.

The one classical way to approximate a fixed point of a nonlinear self mapping 𝑇 on 𝐶 was firstly introduced by Halpern [30], and then, Aoyama et al. [31] extended the mapping in the Halpern-type iterative sequence to be a countable family of nonexpansive mappings. They introduced the following iterative sequence: let 𝑥₁=𝑥∈𝐶 and𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,(1.8) for all 𝑛∈ℕ, where 𝐶 is a nonempty closed convex subset of a Banach space, {𝛼_𝑛} is a sequence in [0,1], and {𝑇_𝑛} is a sequence of nonexpansive mappings with some conditions. They proved that {𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑛}.

Recently, Nakajo et al. [32] introduced the more general condition so-called the NST*-condition, and {𝑇_𝑛} is said to satisfy the NST*-condition if for every bounded sequence {𝑧_𝑛} in 𝐶,lim_𝑛→∞‖‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛‖‖=lim_𝑛→∞‖‖𝑧_𝑛−𝑧_𝑛+1‖‖=0implies𝜔_𝑤𝑧_𝑛⊂𝐹.(1.9) They also prove strong convergence theorems by the hybrid method for families of mappings in a uniformly convex Banach space 𝐸 whose norm is Gâteaux differentiable.

In Hilbert space 𝐻, Iiduka et al. [33] introduced an iterative scheme and proved that the sequence {𝑥_𝑛} generated by the following algorithm: 𝑥₁=𝑥∈𝐶, and 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛,(1.10) where 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶 and {𝜆_𝑛} is a sequence of positive real numbers, converges weakly to some element of VI(𝐶,𝐵).

Later, Iiduka and Takahashi [34] are interested in the similar problem in the framework of Banach spaces, they introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator 𝐵∶𝑥₁=𝑥∈𝐶, and𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛,(1.11) 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 (1.11) converges weakly to some element of VI(𝐶,𝐵).

In 1974, Rockafellar interested in the following problem of finding:𝑣∈𝐸suchthat0∈𝐴(𝑣),(1.12) where 𝐴 is an operator from 𝐸 into 𝐸^∗. Such 𝑣∈𝐸 is called a zero point of 𝐴. He introduced a well-known method, proximal point algorithm, for solving (1.12) in a Hilbert space 𝐻 as shown in the following: 𝑥₁=𝑥∈𝐻 and𝑥_𝑛+1=𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,3,…,(1.13) where {𝑟_𝑛}⊂(0,∞), 𝐴 is a maximal monotone and 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝐴)⁻¹. He proved that the sequence {𝑥_𝑛} converges weakly to an element of 𝐴⁻¹(0).

In 2004, Kamimura et al. [35] considered the algorithm (1.14) in a uniformly smooth and uniformly convex Banach space 𝐸; namely,𝑥_𝑛+1=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,3,….(1.14) They proved that the algorithm {𝑥_𝑛} generated by (1.14) converges weakly to some element of 𝐴⁻¹0.

In 2008, Li and Song [36] established a strong convergence theorem in a Banach space. They introduced the following algorithm: 𝑥₁=𝑥∈𝐸 and 𝑦_𝑛𝑥=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝐽_𝑟𝑛𝑥_𝑛_𝑛+1,=𝐽⁻¹𝛼_𝑛𝐽𝑥+1−𝛼_𝑛𝐽𝑦_𝑛.(1.15) Under the suitable conditions of the coefficient sequences {𝛼_𝑛}, {𝛽_𝑛}, and {𝑟_𝑛}, they proved that the sequence {𝑥_𝑛} generated by the above scheme converges strongly to Π_𝐶𝑥, where Π_𝐶 is the generalized projection from 𝐸 onto 𝐶.

In 2010, Petrot et al. [37] introduced a hybrid projection iterative scheme for approximating a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of two quasi-𝜙-nonexpansive mappings in a real uniformly convex and uniformly smooth Banach space by the following manner: 𝑥𝑦₀=𝑥∈𝐶,_𝑛𝑧_𝑛=𝐽⁻¹𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇𝑥_𝑛+𝛾_𝑛𝐽𝑆𝑥_𝑛,𝐶_𝑛=𝐾_𝑟𝑛𝑦_𝑛,𝑄=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐶𝑛∩𝑄𝑛}𝑥.(1.16) They proved that {𝑥_𝑛} converges strongly to 𝑝∈𝐹(𝑇)∩𝐹(𝑆)∩Ω, where 𝑝∈Π_{𝐹(𝑇)∩𝐹(𝑆)∩Ω}𝑥.

Recently, Klin-eam et al. [38], obtained 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. Saewan and Kumam [39] introduced 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. Wattanawitoon and Kumam [40] proved the strong convergence theorem by using modified hybrid projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of solution of variational inequality operators of an inverse strongly monotone, the zero point of a maximal monotone operator, and the set of fixed point of two relatively quasi-nonexpansive mappings in Banach space.

Motivated and inspired by the ongoing research and the above-mentioned results, we are also interested in generalized mixed equilibrium problem, variational inequality problems, and the zero point of maximal monotone operators. In this paper, we extend the fixed point problems of two relatively quasi-nonexpansive mappings in [40] to the countable families of two quasi-𝜙-nonexpansive mappings and improve the iterative scheme to be more general as shown in the following: 𝑥₁=𝑥∈𝐶, 𝑤𝑧_𝑛𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛,𝑦=𝐽⁻¹𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝐽_𝑟𝑛𝑤_𝑛_𝑛,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑆_𝑛𝑧_𝑛,∈𝐶suchthat𝐹𝑢_𝑛,𝑦++⟨𝑌1𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑𝑢_𝑛𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1𝑥=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥.(1.17) By the new iterative scheme, we will prove the strong convergence theorems of the sequence {𝑥_𝑛} which could be converged to the point Π_{(∩∞𝑛=1𝐹(𝑇𝑛))∩(∩∞𝑛=1𝐹(𝑆𝑛))∩Ω∩VI(𝐶,𝐵)∩𝐴}⁻¹₍₀₎𝑥. Furthermore, we propose the new better appropriate conditions of the coefficient sequences {𝛼_𝑛},{𝛽_𝑛},{𝛾_𝑛}, and {𝑟_𝑛}. Finally, we will apply our result to find a zero point of inverse-strongly monotone operators and complementarity problem in the last section. The results presented in this paper extend and improve the corresponding ones announced by Kamimura et al. [35], Petrot et al. [37], Wattanawitoon and Kumam [40], and some authors in the literature.

2. Preliminaries

In this section, we propose the following preliminaries and lemmas which will be used in our proof.

Throughout this paper, we let 𝐸 be a Banach space with norm ‖⋅‖, and 𝐶 a nonempty closed convex subset of 𝐸, and let 𝐸^∗ denote the dual of 𝐸. We write 𝑥_𝑛⇀𝑥 to indicate that the sequence {𝑥_𝑛} converges weakly to 𝑥 and 𝑥_𝑛→𝑥 implies that the sequence {𝑥_𝑛} converges strongly to 𝑥.

Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any 𝑥,𝑦∈𝑈,𝑥≠𝑦implies‖‖‖𝑥+𝑦2‖‖‖<1.(2.1)

It is also said to be uniformly convex if for each 𝜖∈(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦∈𝑈‖𝑥−𝑦‖≥𝜖implies‖‖‖𝑥+𝑦2‖‖‖<1−𝛿.(2.2) We know that a uniformly convex Banach space is reflexive and strictly convex; see [41, 42] for more details.

The modulus of convexity of 𝐸 is the function 𝛿∶[0,2]→[0,1] defined by𝛿(𝜀)=inf1−‖‖‖𝑥+𝑦2‖‖‖∶𝑥,𝑦∈𝐸,‖𝑥‖=‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀.(2.3)

Furthermore, it is said to be smooth, provided that lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.4) exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝐸.

Let 𝑝 be a fixed real number with 𝑝≥2. Observe that every 𝑝-uniformly convex is uniformly convex. One should note that no a Banach space is 𝑝-uniformly convex for 1<𝑝<2. It is well known that a Hilbert space is 2-uniformly convex and uniformly smooth. For each 𝑝>1, the generalized duality mapping 𝐽_𝑝∶𝐸→2^𝐸∗ is defined by𝐽_𝑝(𝑥)=𝑥^∗∈𝐸^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖^𝑝,‖𝑥^∗‖=‖𝑥‖^𝑝−1,(2.5) 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. 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 know the following (see [43]):(1)if 𝐸 is smooth, then 𝐽 is single-valued,(2)if 𝐸 is strictly convex, then 𝐽 is one-to-one and ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩>0 holds for all (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝐽 with 𝑥≠𝑦,(3)if 𝐸 is reflexive, then 𝐽 is surjective,(4)if 𝐸 is uniformly convex, then it is reflexive,(5)if 𝐸^∗ is uniformly convex, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.

The duality 𝐽 from a smooth Banach space 𝐸 into 𝐸^∗ is said to be weakly sequentially continuous [44] if 𝑥_𝑛⇀𝑥 implies 𝐽𝑥_𝑛⇀^∗𝐽𝑥, where ⇀^∗ implies the weak* convergence.

Let 𝐸 be a smooth, strictly convex and reflexive Banach space, and let 𝐶 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by 𝜙 the function defined by𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,for𝑥,𝑦∈𝐸.(2.6)

Remark 2.1. We know the following: for each 𝑥,𝑦,𝑧∈𝐸,(i)(‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,(ii)𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2⟨𝑥−𝑧,𝐽𝑧−𝐽𝑦⟩,(iii)𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² in a real Hilbert space.

The generalized projection, introduced by Alber [45], Π_𝐶∶𝐸→𝐶 is a map that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the function 𝜙(𝑥,𝑦); that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem𝜙𝑥,𝑥=inf_𝑦∈𝐶𝜙(𝑦,𝑥),(2.7) existence and uniqueness of the operator Π_𝐶 follows from the properties of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽.

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 Remark 2.1 (i), 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 [43, 46] for more details.

Lemma 2.2 (see [47, 48]). If 𝐸 be a 2-uniformly convex Banach space, then for all 𝑥,𝑦∈𝐸, one has ‖𝑥−𝑦‖≤2𝑐²‖𝐽𝑥−𝐽𝑦‖,(2.8) where 𝐽 is the normalized duality mapping of 𝐸 and 0<𝑐≤1.

The best constant 1/𝑐 in the Lemma is called the 2-uniformly convex constant of 𝐸; see [41].

Lemma 2.3 (see [47, 49]). If 𝐸 is a p-uniformly convex Banach space and 𝑝 a given real number with 𝑝≥2, then for all 𝑥,𝑦∈𝐸,𝐽_𝑥∈𝐽_𝑝(𝑥) and 𝐽_𝑦∈𝐽_𝑝(𝑦)⟨𝑥−𝑦,𝐽𝑥−𝐽𝑦⟩≥𝑐^𝑝2^𝑝−2𝑝‖𝑥−𝑦‖^𝑝,(2.9) where 𝐽_𝑝 is the generalized duality mapping of 𝐸 and 1/𝑐 is the p-uniformly convexity constant of 𝐸.

Lemma 2.4 (Xu [48]). Let 𝐸 be a uniformly convex Banach space, then for each 𝑟>0, there exists a strictly increasing, continuous and convex function 𝑔∶[0,∞)→[0,∞) such that 𝑔(0)=0 and ‖𝜆𝑥+(1−𝜆𝑦)‖²≤𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)𝑔(‖𝑥−𝑦‖),(2.10) for all 𝑥,𝑦∈{𝑧∈𝐸∶‖𝑧‖≤𝑟} and 𝜆∈[0,1].

Lemma 2.5 (Kamimura and Takahashi [50]). Let 𝐸 be a uniformly convex and smooth real Banach space and {𝑥_𝑛},{𝑦_𝑛} 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 2.6 (Alber [45]). 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.(2.11)

Lemma 2.7 (Alber [45]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space and 𝐶 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.(2.12)

Let 𝐸 be a strictly convex, smooth, and reflexive Banach space and 𝐽 the duality mapping from 𝐸 into 𝐸^∗. Then, 𝐽⁻¹ is also single-valued, one-to-one, and surjective, and it is the duality mapping from 𝐸^∗ into 𝐸. Define a function 𝑉∶𝐸×𝐸^∗→ℝ as follows (see [51]):𝑉𝑥,𝑥^∗=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥^∗‖²,(2.13) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗. Then, it is obvious that 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹(𝑥^∗)) and 𝑉(𝑥,𝐽(𝑦))=𝜙(𝑥,𝑦).

Lemma 2.8 (Kohsaka and Takahashi [51, Lemma  3.2]). Let 𝐸 be a strictly convex, smooth, and reflexive Banach space and 𝑉 as in (2.13). Then, 𝑉𝑥,𝑥^∗+2𝐽⁻¹𝑥^∗−𝑥,𝑦^∗≤𝑉𝑥,𝑥^∗+𝑦^∗,(2.14) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

For solving the generalized mixed equilibrium problem, let us assume that the bifunction 𝐹∶𝐶×𝐶→ℝ and 𝜑∶𝐶→ℝ is convex and lower semicontinuous, satisfying the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶,(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶,(A3) for each 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦),(2.15)(A4) for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.9 (Blum and Oettli [17]). Let 𝐶 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸 and 𝐹 a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0∀𝑦∈𝐶.(2.16)

Lemma 2.10 (Takahashi and Zembayashi [52]). Let 𝐶 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸 and 𝐹 a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For all 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟𝑥=𝑧∈𝐶∶𝐹(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,(2.17) for all 𝑥∈𝐸. Then, the following hold:(1)𝑇_𝑟 is single-valued,(2)𝑇_𝑟 is a firmly nonexpansive-type mapping, that is, for all 𝑥,𝑦∈𝐸, ⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩,(2.18)(3)𝐹(𝑇_𝑟)=𝐸𝑃(𝐹), (4)𝐸𝑃(𝐹) is closed and convex.

Lemma 2.11 (Takahashi and Zembayashi [52]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, and 𝐹 a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝑟>0. Then, for 𝑥∈𝐸 and 𝑞∈𝐹(𝑇_𝑟), 𝜙𝑞,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙(𝑞,𝑥).(2.19)

Lemma 2.12 (Zhang [53]). Let 𝐶 be a 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 convex and lower semi-continuous, and 𝐹 a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, then there exists 𝑢∈𝐶 such that 𝐹(𝑢,𝑦)+⟨𝐵𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.20) Define a mapping 𝐾_𝑟∶𝐶→𝐶 as follows: 𝐾_𝑟(𝑥)=𝑢∈𝐶∶𝐹(𝑢,𝑦)+⟨𝐵𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶,(2.21) for all 𝑥∈𝐸. Then, the following hold:(i)𝐾_𝑟 is single-valued,(ii)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⟩,(iii)𝐹(𝐾_𝑟)=Ω, (iv)Ω is closed and convex,(v)𝜙(𝑝,𝐾_𝑟𝑧)+𝜙(𝐾_𝑟𝑧,𝑧)≤𝜙(𝑝,𝑧) for all 𝑝∈𝐹(𝐾_𝑟), 𝑧∈𝐸.

It follows from Lemma 2.10 that the mapping 𝐾_𝑟∶𝐶→𝐶 defined by (2.21) is a relatively nonexpansive mapping. Thus, it is quasi-𝜙-nonexpansive.

Let 𝐸 be a reflexive, strictly convex and smooth Banach space. Let 𝐶 be a closed convex subset of 𝐸. Because 𝜙(𝑥,𝑦) is strictly convex and coercive in the first variable, we know that the minimization problem inf_𝑦∈𝐶𝜙(𝑥,𝑦) has a unique solution. The operator Π_𝐶𝑥∶=argmin_𝑦∈𝐶𝜙(𝑥,𝑦) is said to be the generalized projection of 𝑥 on 𝐶.

Let 𝐴 be a set-valued mapping from 𝐸 to 𝐸^∗ with graph 𝐺(𝐴)={(𝑥,𝑥^∗)∶𝑥^∗∈𝐴𝑥}, domain 𝐷(𝐴)={𝑥∈𝐸∶𝐴(𝑥)≠∅}, and range 𝑅(𝐴)={𝑥^∗∈𝐸^∗∶𝑥^∗∈𝐴(𝑥),𝑥∈𝐷(𝐴)}. We denote a set-valued operator 𝐴 from 𝐸 to 𝐸^∗ by 𝐴⊂𝐸×𝐸^∗. 𝐴 is said to be monotone if ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩≥0 for all (𝑥,𝑥^∗),(𝑦,𝑦^∗)∈𝐴. A monotone operator 𝐴⊂𝐸×𝐸^∗ is said to be maximal monotone if it graph is not properly contained in the graph of any other monotone operator. We know that if 𝐴 is maximal monotone, then the solution set 𝐴⁻¹0={𝑧∈𝐷(𝐴)∶0∈𝐴𝑧} is closed and convex.

Let 𝐸 be a reflexive, strictly convex and smooth Banach space, it is known that 𝐴 is maximal monotone if and only if 𝑅(𝐽+𝑟𝐴)=𝐸^∗ for all 𝑟>0.

Define the resolvent of 𝐴 by 𝐽_𝑟𝑥=𝑥_𝑟. In other words, 𝐽_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽 for all 𝑟>0. 𝐽_𝑟 is a single-valued mapping from 𝐸 to 𝐷(𝐴). Also, 𝐴⁻¹(0)=𝐹(𝐽_𝑟) for all 𝑟>0, where 𝐹(𝐽_𝑟) is the set of all fixed points of 𝐽_𝑟. Define, for 𝑟>0, the Yosida approximation of 𝐴 by 𝐴_𝑟=(𝐽−𝐽𝐽_𝑟)/𝑟. We know that 𝐴_𝑟𝑥∈𝐴(𝐽_𝑟𝑥) for all 𝑟>0 and 𝑥∈𝐸.

Lemma 2.13 (Kohsaka and Takahashi [51, Lemma  3.1]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐴⊂𝐸×𝐸^∗ be a maximal monotone operator with 𝐴⁻¹0≠∅, 𝑟>0, and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽. Then, 𝜙𝑥,𝐽_𝑟𝑦+𝜙𝐽_𝑟𝑦,𝑦≤𝜙(𝑥,𝑦)(2.22) for all 𝑥∈𝐴⁻¹0 and 𝑦∈𝐸.

Let 𝐵 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)=𝐵(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.(2.23)

Theorem 2.14. (Rockafellar [54]). Let 𝐶 be a nonempty, closed convex subset of a Banach space 𝐸 and 𝐵 a monotone, hemicontinuous operator of 𝐶 into 𝐸^∗. Let 𝑇⊂𝐸×𝐸^∗ be an operator defined as follows: 𝑇𝑣=𝐵𝑣+𝑁^𝐶∅,(𝑣),𝑜𝑡ℎ𝑣𝑒𝑟∈𝑤𝑖𝐶𝑠,𝑒.(2.24) Then 𝑇, is maximal monotone and 𝑇⁻¹0=VI(𝐶,𝐵).

Lemma 2.15 (Tan and Xu [55]). Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤𝑎_𝑛+𝑏_𝑛,∀𝑛≥0.(2.25) If ∑^∞𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists.

3. The Main Result

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of mixed equilibrium problems, the set of solutions of the variational inequality problem, the zero point of a maximal monotone operator, and the set of two families of quasi-𝜙-nonexpansive mappings in a Banach space by using the shrinking hybrid projection method.

Theorem 3.1. Let 𝐸 be a 2-uniformly convex and uniformly smooth Banach space and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4), let 𝜑∶𝐶→ℝ be a proper lower semicontinuous and convex function, and let 𝐴∶𝐸→𝐸^∗ be a maximal monotone operator satisfying 𝐷(𝐴)⊂𝐸. Let 𝐽_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽 for 𝑟>0, let 𝐵 be an 𝛼-inverse-strongly monotone operator of 𝐸 into 𝐸^∗, and let 𝑌∶𝐸→𝐸^∗ be a continuous and monotone mapping. Let {𝑇_𝑛} and {𝑆_𝑛} be two families of quasi-𝜙-nonexpansive mappings of 𝐸 into itself satisfies the NST∗-condition, with Θ∶=(∩^∞𝑛=1𝐹(𝑇_𝑛))∩(∩^∞𝑛=1𝐹(𝑆_𝑛))∩Ω∩𝑉𝐼(𝐶,𝐵)∩𝐴⁻¹(0)≠∅ and ‖𝐵𝑦‖≤‖𝐵𝑦−𝐵𝑢‖ for all 𝑦∈𝐶 and 𝑢∈Θ. Let {𝑥_𝑛} be a sequence generated by 𝑥₁=𝑥∈𝐸, and 𝑤𝑧_𝑛𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛,𝑦=𝐽⁻¹𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝐽_𝑟𝑛𝑤_𝑛_𝑛,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑆_𝑛𝑧_𝑛,∈𝐶suchthat𝐹𝑢_𝑛,𝑦++⟨𝑌1𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑𝑢_𝑛𝑟_𝑛𝐶⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛+1𝑥=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥,(3.1) for all 𝑛∈ℕ. If the coefficient sequence {𝛼_𝑛},{𝛽_𝑛}, {𝛾_𝑛}, and {𝛿_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,∞) satisfy 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, liminf_𝑛→∞𝛼_𝑛𝛽_𝑛>0, liminf_𝑛→∞𝛼_𝑛𝛾_𝑛>0, liminf_𝑛→∞𝛾_𝑛(1−𝛿_𝑛)>0, liminf_𝑛→∞𝑟_𝑛>0 and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝑐²𝛼/2, 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then the sequence {𝑥_𝑛} converges strongly to Π_Θ𝑥.

Proof. We first show that {𝑥_𝑛} is bounded. Let 𝑝∈Θ∶=(∩^∞𝑛=1𝐹(𝑇_𝑛))∩(∩^∞𝑛=1𝐹(𝑆_𝑛))∩Ω∩VI(𝐶,𝐵)∩𝐴⁻¹(0), and let 𝐾𝐻𝑢_𝑛,𝑦=𝐹𝑢_𝑛,𝑦+⟨𝑌𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑𝑢_𝑛,𝑦∈𝐶_𝑟𝑛,=𝑢∈𝐶∶𝐻𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.2) Put 𝑣_𝑛=𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛) and 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛.
With its relatively nonexpansiveness of 𝐽_𝑟𝑛 and by Lemma 2.8, the convexity of the function 𝑉 in the second variable, we have 𝜙𝑝,𝑤_𝑛=≤𝜙𝜙𝑝𝑝,,Π𝑣_𝐶𝑛𝑣_𝑛≤𝑉𝑝,𝐽𝑥_𝑛=𝜙𝑝,𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛=𝑉𝑝,𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛+𝜆_𝑛𝐵𝑥_𝑛−2𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛−𝑝,𝜆_𝑛𝐵𝑥_𝑛=𝜙𝑝,𝑥_𝑛−2𝜆_𝑛⟨𝑣_𝑛−𝑝,𝐵𝑥_𝑛⟩−2𝜆_𝑛⟨𝑥_𝑛−𝑝,𝐵𝑥_𝑛⟩+2⟨𝑣_𝑛−𝑥_𝑛,−𝜆_𝑛𝐵𝑥_𝑛⟩.(3.3) Since 𝑝∈VI(𝐶,𝐵) and 𝐵 is 𝛼-inverse-strongly monotone, we consider −2𝜆_𝑛⟨𝑥_𝑛−𝑝,𝐵𝑥_𝑛⟩≤=−−22𝜆𝛼_𝑛𝜆⟨𝑥_𝑛𝑛−𝑝,𝐵𝑥_𝑛−𝐵𝑝⟩−2𝜆_𝑛⟨𝑥_𝑛−𝑝,𝐵𝑝⟩‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖².(3.4) Therefore, by Lemma 2.2, we obtain 2⟨𝑣_𝑛−𝑥_𝑛,−𝜆_𝑛𝐵𝑥_𝑛⟩≤=22‖‖𝐽𝐽⁻¹₋₁𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛−𝐽⁻¹𝐽𝑥_𝑛,−𝜆_𝑛𝐵𝑥_𝑛≤4𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛−𝐽⁻¹𝐽𝑥_𝑛‖‖‖‖𝜆_𝑛𝐵𝑥_𝑛‖‖𝑐²=4‖‖𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛−𝐽𝑥_𝑛‖‖‖‖𝜆_𝑛𝐵𝑥_𝑛‖‖𝑐²𝜆^2𝑛‖‖𝐵𝑥_𝑛‖‖²≤4𝑐²𝜆^2𝑛‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖².(3.5) We can rewrite (3.3), which yield that 𝜙𝑝,𝑤_𝑛≤𝜙𝑝,𝑥_𝑛−2𝛼𝜆_𝑛‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖²+4𝑐²𝜆^2𝑛‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖²≤𝜙𝑝,𝑥_𝑛+2𝜆_𝑛2𝑐²𝜆_𝑛−𝛼‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖²≤𝜙𝑝,𝑥_𝑛.(3.6) Apply the Lemma 2.8, Lemma 2.13 and (3.6), we consider 𝜙𝑝,𝑧_𝑛==𝜙𝑉𝑝𝑝,,𝐽𝛿⁻¹_𝑛𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝐽_𝑟𝑛𝑤_𝑛≤𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝐽_𝑟𝑛𝑤_𝑛=𝛿_𝑛𝑉𝑝,𝐽𝑥_𝑛+1−𝛿_𝑛𝑉𝑝,𝐽𝐽_𝑟𝑛𝑤_𝑛≤𝛿_𝑛𝜙𝑝,𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝐽_𝑟𝑛𝑤_𝑛≤𝛿_𝑛𝜙𝑝,𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝑤_𝑛−𝜙𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛≤𝛿_𝑛𝜙𝑝,𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝑤_𝑛=𝜙𝜙𝑝,𝑝𝑥,_𝑛𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝑥_𝑛,(3.7) hence, we obtain 𝜙𝑝,𝑦_𝑛==𝜙‖𝑝‖𝑝₂,𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑆_𝑛𝑧_𝑛+‖‖𝛼_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑝,𝐽𝑇_𝑛𝑥_𝑛⟩−2𝛾_𝑛⟨𝑝,𝐽𝑆_𝑛𝑧_𝑛⟩𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑆_𝑛𝑧_𝑛‖‖²≤+‖𝑝𝛼‖²_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑝,𝐽𝑇_𝑛𝑥_𝑛⟩−2𝛾_𝑛⟨𝑝,𝐽𝑆_𝑛𝑧_𝑛⟩‖‖𝐽𝑥_𝑛‖‖²+𝛽_𝑛‖‖𝐽𝑇_𝑛𝑥_𝑛‖‖²+𝛾_𝑛‖‖𝐽𝑆_𝑛𝑧_𝑛‖‖²=≤𝛼𝛼_𝑛𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑆_𝑛𝑧_𝑛≤𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑧_𝑛=𝜙𝜙𝑝,𝑥𝑝,_𝑛𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑥_𝑛.(3.8) By (3.1), again, 𝜙𝑝,𝑢_𝑛=𝜙𝑝,𝐾_𝑟𝑛𝑦_𝑛≤𝜙𝑝,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛.(3.9) This shows that 𝑝∈𝐶_𝑛+1. Consequently, Θ⊂𝐶_𝑛, for all 𝑛∈ℕ.
Next, we show that lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. Since 𝑥_𝑛=Π_𝐶𝑛𝑥, it follows from Lemma 2.7 that 𝜙𝑥_𝑛,𝑥≤𝜙(𝑝,𝑥)−𝜙𝑝,𝑥_𝑛≤𝜙(𝑝,𝑥),(3.10) for each 𝑝∈Θ⊂𝐶_𝑛. Then, 𝜙(𝑥_𝑛,𝑥) is bounded. It implies that {𝑥_𝑛} is bounded and {𝑦_𝑛}, {𝑧_𝑛}, {𝑤_𝑛}, and {𝐽_𝑟𝑛𝑤_𝑛} are also bounded.
From 𝑥_𝑛=Π_𝐶𝑛𝑥 and 𝑥_𝑛+1∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥≤𝜙𝑥_𝑛+1,𝑥,∀𝑛∈ℕ.(3.11) Therefore, {𝜙(𝑥_𝑛,𝑥)} is nondecreasing. It follows that the limit of {𝜙(𝑥_𝑛,𝑥)} exists, and from Lemma 2.7, we have 𝜙𝑥_𝑛+1,𝑥_𝑛=𝜙𝑥_𝑛+1,Π_𝐶𝑛𝑥≤𝜙𝑥_𝑛+1,𝑥−𝜙Π_𝐶𝑛𝑥,𝑥=𝜙𝑥_𝑛+1,𝑥−𝜙𝑥_𝑛,𝑥,(3.12) for all 𝑛∈ℕ. Thus, we have lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.13) Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥∈𝐶_𝑛+1, it follows from the definition of 𝐶_𝑛+1 that 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛⟶0.(3.14) By Lemma 2.5, (3.13), and (3.14), we note that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑢_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=0.(3.15) Since 𝐽 is uniformly norm-to-norm continuous on the bounded set, we obtain lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑢_𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖=0.(3.16) Since 𝑥_𝑚=Π_𝐶𝑚⊂𝐶_𝑛 for any positive integer 𝑚≥𝑛, it follows from Lemma 2.7 that 𝜙𝑥_𝑚,𝑥_𝑛=𝜙𝑥_𝑚,Π_𝐶𝑛𝑥_𝑛≤=𝜙𝜙𝑥𝑥_𝑚𝑚,𝑥−𝜙Π_𝐶𝑛𝑥_𝑛,𝑥,𝑥−𝜙𝑥_𝑛,𝑥.(3.17) Taking 𝑚,𝑛→∞, we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0 as 𝑛→∞. It follows from Lemma 2.5, that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. Hence, {𝑥_𝑛} is a Cauchy sequence. Since 𝐸 is a Banach space and 𝐶 is closed and convex, we can assume that 𝑥_𝑛→𝑢∈𝐶 as 𝑛→∞.
Next, we show that 𝑢∈(∩^∞𝑛=1𝐹(𝑇_𝑛))∩(∩^∞𝑛=1𝐹(𝑆_𝑛)).
Since 𝐸 is a uniformly smooth Banach space, we know that 𝐸^∗ is a uniformly convex Banach space. Let 𝑟=sup_𝑛∈ℕ{‖𝑥_𝑛‖,‖𝑇_𝑛𝑥_𝑛‖,‖𝑆_𝑛𝑧_𝑛‖}. From Lemma 2.4, we have 𝜙𝑝,𝑢_𝑛≤=𝜙𝜙𝑝𝑝,,𝐽𝑦_𝑛₋₁=‖𝑝‖₂𝛼_𝑛𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑆_𝑛𝑧_𝑛+‖‖𝛼_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑝,𝐽𝑇_𝑛𝑥_𝑛⟩−2𝛾_𝑛⟨𝑝,𝐽𝑆_𝑛𝑧_𝑛⟩𝐽𝑥_𝑛+𝛽_𝑛𝐽𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝐽𝑆_𝑛𝑧_𝑛‖‖²≤+‖𝑝𝛼‖²_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−2𝛽_𝑛⟨𝑝,𝐽𝑇_𝑛𝑥_𝑛⟩−2𝛾_𝑛⟨𝑝,𝐽𝑆_𝑛𝑧_𝑛⟩=𝛼_𝑛‖‖𝐽𝑥_𝑛‖‖²+𝛽_𝑛‖‖𝐽𝑇_𝑛𝑥_𝑛‖‖²+𝛾_𝑛‖‖𝐽𝑆_𝑛𝑧_𝑛‖‖²−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖≤𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑇_𝑛𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑆_𝑛𝑧_𝑛−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖≤𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑧_𝑛−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖=𝜙𝜙𝑝,𝑥𝑝,_𝑛𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑥_𝑛−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖.(3.18) This implies that 𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛.(3.19) On the other hand, we have 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛==‖‖‖‖𝑥𝑥_𝑛𝑛‖‖²−‖‖𝑢_𝑛‖‖²−2⟨𝑝,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩−𝑢_𝑛‖‖‖‖𝑥_𝑛‖‖+‖‖𝑢_𝑛‖‖+2‖𝑝‖‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖.(3.20) Noticing (3.15) and (3.16), we obtain 𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛⟶0,as𝑛⟶∞.(3.21) Since liminf_𝑛→∞𝛼_𝑛𝛽_𝑛>0 and (3.21), it follows from (3.19) that 𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖⟶0,as𝑛⟶∞.(3.22) It follows from the property of 𝑔 that ‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖⟶0,as𝑛⟶∞.(3.23) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we see that lim_𝑛→∞‖‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖‖=0.(3.24) Similarly, using the condition limsup_𝑛→∞𝛼_𝑛𝛾_𝑛>0, one can obtain lim_𝑛→∞‖‖𝑆_𝑛𝑧_𝑛−𝑥_𝑛‖‖=0.(3.25) By (3.6), (3.8), and (3.18), we have 𝜙𝑝,𝑢_𝑛≤≤𝛼𝛼_𝑛𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑧_𝑛−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖+𝛾𝜙_𝑛𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛≤𝛼_𝑛𝛿_𝑛𝜙𝑝,𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝑤_𝑛−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖+𝛾𝜙_𝑛𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝜙𝑝,𝑥_𝑛≤𝛼_𝑛1−𝛿_𝑛𝜙𝑝,𝑤_𝑛−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖+𝛾𝜙_𝑛𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛𝛿_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛1−𝛿_𝑛𝜙𝑝,𝑥_𝑛+2𝜆_𝑛2𝑐²−𝛼_𝑛𝜆_𝑛−𝛼‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖²≤𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖+𝛾𝜙_𝑛𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛1−𝛿_𝑛𝜙𝑝,𝑥_𝑛1−𝛿_𝑛2𝜆_𝑛2𝑐²≤𝜙𝑝,𝑥_𝑛𝜆_𝑛−𝛼‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖²−𝛼_𝑛𝛽_𝑛𝑔‖‖𝐽𝑇_𝑛𝑥_𝑛−𝐽𝑥_𝑛‖‖+𝛾_𝑛1−𝛿_𝑛2𝜆_𝑛2𝑐²𝜆_𝑛−𝛼‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖².(3.26) This implies that 2𝜆_𝑛𝛼−2𝑐²𝜆_𝑛‖‖𝐵𝑥^𝑛−𝐵𝑝‖‖²≤1𝛾_𝑛1−𝛿_𝑛𝜙𝑝,𝑥^𝑛−𝜙𝑝,𝑢_𝑛.(3.27) By assumption, liminf_𝑛→∞𝛾_𝑛(1−𝛿_𝑛)>0 and (3.21), we get that lim_𝑛→∞‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖=0.(3.28) From Lemma 2.7, Lemma 2.8, and (3.5), we have 𝜙𝑥_𝑛,𝑤_𝑛=≤𝜙𝜙𝑥𝑥_𝑛𝑛,Π_𝐶𝑣_𝑛=𝜙𝑥_𝑛,𝑣_𝑛=𝑉𝑥_𝑛,𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛≤𝑉𝑥_𝑛,𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛=𝑉𝑥_𝑛,𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛+𝜆_𝑛𝐵𝑥_𝑛−2𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛−𝑥_𝑛,𝜆_𝑛𝐵𝑥_𝑛=𝜙𝑥_𝑛,𝐽𝑥_𝑛−2𝜆_𝑛⟨𝑣_𝑛−𝑥_𝑛,𝐵𝑥_𝑛⟩≤4,𝑥_𝑛+2⟨𝑣_𝑛−𝑥_𝑛,𝜆_𝑛𝐵𝑥_𝑛⟩𝑐²𝜆^2𝑛‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖².(3.29) By Lemma 2.8 and Lemma 2.13, we have 𝜙𝑥_𝑛,𝑧_𝑛==𝜙𝑉𝑥𝑥_𝑛𝑛,𝐽⁻¹𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝐽_𝑟𝑛𝑤_𝑛≤𝛿_𝑛,𝛿_𝑛𝐽𝑥_𝑛+1−𝛿_𝑛𝐽𝐽_𝑟𝑛𝑤_𝑛=𝛿_𝑛𝑉𝑥_𝑛,𝐽𝑥_𝑛+1−𝛿_𝑛𝑉𝑥_𝑛,𝐽𝐽_𝑟𝑛𝑤_𝑛=𝛿_𝑛𝜙𝑥_𝑛,𝑥_𝑛+1−𝛿_𝑛𝜙𝑥_𝑛,𝐽_𝑟𝑛𝑤_𝑛=1𝜙−𝛿𝑥_𝑛𝑛,𝑥_𝑛+1−𝛿_𝑛𝜙𝑥_𝑛,𝑤_𝑛−𝜙𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛≤1−𝛿_𝑛𝜙𝑥_𝑛,𝑤_𝑛4𝑐²𝜆^2𝑛‖‖𝐵𝑥_𝑛−𝐵𝑝‖‖².(3.30) From Lemma 2.5 and (3.28), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑤_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛−𝑧_𝑛‖‖=0.(3.31) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑤_𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛−𝐽𝑧_𝑛‖‖=0.(3.32) Since 𝑥_𝑛→𝑢 as 𝑛→∞, 𝑧_𝑛→𝑢 as 𝑛→∞. Combining (3.15), (3.25), and (3.28), we also obtain ‖‖𝑆_𝑛𝑧_𝑛−𝑧_𝑛‖‖≤‖‖𝑆_𝑛𝑧_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑧_𝑛‖‖⟶0as𝑛⟶∞.(3.33) By (3.15) and (3.31), we obtain that ‖‖𝑧_𝑛+1−𝑧_𝑛‖‖≤‖‖𝑧_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑧_𝑛‖‖⟶0as𝑛⟶∞.(3.34) By (3.15), (3.24), (3.33), and (3.34), and {𝑇_𝑛}, {𝑆_𝑛} satisfies the NST*-condition and 𝑥_𝑛→𝑝, then we have 𝑝∈(∩^∞𝑛=1𝐹(𝑇_𝑛))∩(∩^∞𝑛=1𝐹(𝑆_𝑛)).
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑢∈𝐶. It follows from (3.31) that we have 𝑤_𝑛𝑖⇀𝑢 as 𝑖→∞. Next, we show that 𝑢∈𝐴⁻¹0.
By (3.6), (3.8), and (3.9), we obtain 𝜙𝑝,𝑢_𝑛≤≤𝜙𝛼_𝑛𝑝,𝑦_𝑛≤𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑧_𝑛≤𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝜙𝑝,𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝑤_𝑛−𝜙𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛≤𝜙𝜙𝑝,𝑥𝑝,_𝑛𝑥_𝑛+𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝜙𝑝,𝑥_𝑛+1−𝛿_𝑛𝜙𝑝,𝑥_𝑛−𝜙𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛−𝛾_𝑛1−𝛿_𝑛𝜙𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛.(3.35) This implies that 𝛾_𝑛1−𝛿_𝑛𝜙𝐽_𝑟𝑛𝑤_𝑛,𝑤_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛.(3.36) By (3.21), we have lim_𝑛→∞‖‖𝐽_𝑟𝑛𝑤_𝑛−𝑤_𝑛‖‖=0.(3.37) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim_𝑛→∞‖‖𝐽𝐽_𝑟𝑛𝑤_𝑛−𝐽𝑤_𝑛‖‖=0.(3.38) Indeed, since liminf_𝑛→∞𝑟_𝑛>0, it follows from (3.38) that lim_𝑛→∞‖‖𝐴_𝑟𝑛𝑤_𝑛‖‖=lim_𝑛→∞1𝑟_𝑛‖‖𝐽𝑤_𝑛−𝐽𝐽_𝑟𝑛𝑤_𝑛‖‖=0.(3.39) If (𝑤,𝑤^∗)∈𝐴, then it holds from the monotonicity of 𝐴 that 𝑤−𝑤^𝑛_𝑖,𝑤^∗−𝐴_𝑟𝑛𝑖𝑤_𝑛𝑖≥0,(3.40) for all 𝑖∈ℕ. Letting 𝑖→∞, we get ⟨𝑤−𝑢,𝑤^∗⟩≥0. Then, the maximality of 𝐴 implies 𝑢∈𝐴⁻¹0.
Next, we show that 𝑢∈VI(𝐶,𝐵). Let 𝐾⊂𝐸×𝐸^∗ be an operator as follows: 𝐾𝑣=𝐵𝑣+𝑁^𝐶∅,(𝑣),othe𝑣rw∈i𝐶se,.(3.41) By Theorem 2.14, 𝐾 is maximal monotone and 𝐾⁻¹0=VI(𝐶,𝐵).
Let (𝑣,𝑤)∈𝐺(𝐾). Since 𝑤∈𝐾𝑣=𝐵𝑣+𝑁_𝐶(𝑣), we get 𝑤−𝐵𝑣∈𝑁_𝐶(𝑣). From 𝑤_𝑛∈𝐶, we have ⟨𝑣−𝑤_𝑛,𝑤−𝐾𝑣⟩≥0.(3.42) On the other hand, since 𝑤_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛), then by Lemma 2.6, we have 𝑣−𝑤_𝑛,𝐽𝑤_𝑛−𝐽𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛≥0,(3.43) thus 𝑣−𝑤^𝑛,𝐽𝑥^𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐵𝑥_𝑛≤0.(3.44) It follows from (3.42) and (3.44) that ⟨𝑣−𝑤_𝑛,𝑤⟩≥≥⟨⟨𝑣𝑣−−𝑤𝑤_𝑛𝑛,𝐵𝑣⟩,𝐵𝑣⟩+𝑣−𝑤_𝑛,𝐽𝑥^𝑛−𝐽𝑤_𝑛𝜆_𝑛−𝐵𝑥_𝑛=⟨𝑣−𝑤_𝑛,𝐵𝑣−𝐵𝑥_𝑛⟩+𝑣−𝑤_𝑛,𝐽𝑥^𝑛−𝐽𝑤_𝑛=⟨𝑣−𝑤_𝑛𝜆_𝑛,𝐵𝑣−𝐵𝑤_𝑛⟩+⟨𝑣−𝑤_𝑛,𝐵𝑤_𝑛−𝐵𝑥_𝑛⟩+𝑣−𝑤_𝑛,𝐽𝑥^𝑛−𝐽𝑤_𝑛≥−‖‖𝑣−𝑤_𝑛𝜆_𝑛‖‖‖‖𝑤^𝑛−𝑥_𝑛‖‖𝛼−‖‖𝑣−𝑤^𝑛‖‖‖‖𝐽𝑥^𝑛−𝐽𝑤_𝑛‖‖≥−𝑀‖‖𝑤_𝑛𝑎−𝑥_𝑛‖‖𝛼+‖‖𝐽𝑥^𝑛−𝐽𝑤_𝑛‖‖𝑎,(3.45) where 𝑀=sup_𝑛≥1{‖𝑣−𝑤_𝑛‖}. From (3.31) and (3.32), we obtain ⟨𝑣−𝑢,𝑤⟩≥0. By the maximality of 𝐾, we have 𝑢∈𝐾⁻¹0 and hence 𝑢∈VI(𝐶,𝐵).
Next, we show that 𝑢∈Ω. From 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛 and Lemma 2.12, we obtain 𝜙𝑢_𝑛,𝑦_𝑛≤=𝜙𝜙𝐾𝑢,𝑦_𝑟𝑛𝑦_𝑛𝑛,𝑦_𝑛≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝐾_𝑟𝑛𝑦_𝑛≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝐾_𝑟𝑛𝑦_𝑛−𝜙𝑢,𝑢_𝑛.(3.46) On the other hand, we have 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛==‖‖‖‖𝑥𝑥_𝑛𝑛‖‖²−‖‖𝑢_𝑛‖‖²−2⟨𝑢,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩−𝑢_𝑛‖‖‖‖𝑥_𝑛‖‖+‖‖𝑢_𝑛‖‖+2‖𝑢‖‖‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖‖.(3.47) Noticing (3.15) and (3.16), we obtain 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛⟶0,as𝑛⟶∞.(3.48) It follows that 𝜙𝑢_𝑛,𝑦_𝑛⟶0,as𝑛⟶∞.(3.49) By Lemma 2.5, we have lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.50) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we get lim_𝑛→∞‖‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖‖=0.(3.51) From the assumption liminf_𝑛→∞𝑟_𝑛>𝑎, we get lim_𝑛→∞‖‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖‖𝑟_𝑛=0.(3.52) Noticing that 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛, we have 𝐻𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.53) Hence, 𝐻𝑢_𝑛𝑖,𝑦+1𝑟_𝑛𝑖𝑦−𝑢_𝑛𝑖,𝐽𝑢_𝑛𝑖−𝐽𝑦_𝑛𝑖≥0,∀𝑦∈𝐶.(3.54) From the (A2), we note that ‖‖𝑦−𝑢_𝑛𝑖‖‖‖‖𝐽𝑢^𝑛𝑖−𝐽𝑦_𝑛𝑖‖‖𝑟_𝑛𝑖≥1𝑟_𝑛𝑖𝑦−𝑢_𝑛𝑖,𝐽𝑢_𝑛𝑖−𝐽𝑦_𝑛𝑖≥−𝐻𝑢_𝑛𝑖,𝑦≥𝐻𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶.(3.55) Taking the limit as 𝑛→∞ in the above inequality, and from (A4) and 𝑢_𝑛𝑖⇀𝑢, we have 𝐻(𝑦,𝑢)≤0,forall𝑦∈𝐶. For 0<𝑡<1 and 𝑦∈𝐶, define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑢. Noticing that 𝑦,𝑢∈𝐶, we obtain 𝑦_𝑡∈𝐶, which yields that 𝐻(𝑦_𝑡,𝑢)≤0. It follows from (A1) that 0=𝐻𝑦_𝑡,𝑦_𝑡≤𝑡𝐻𝑦_𝑡,𝑦+(1−𝑡)𝐻𝑦_𝑡,𝑢≤𝑡𝐻𝑦_𝑡,𝑦.(3.56) That is, 𝐻(𝑦_𝑡,𝑦)≥0.
Let 𝑡↓0, from (A3), we obtain 𝐻(𝑢,𝑦)≥0,fo