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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 692829, 27 pages
http://dx.doi.org/10.1155/2012/692829
Research Article

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

1Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand
3Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thungkru, Bangkok 10140, Thailand

Received 3 November 2011; Accepted 16 November 2011

Academic Editor: Yonghong Yao

Copyright © 2012 Kriengsak Wattanawitoon and Poom Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [19]. 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 𝑆𝐶𝐶 is said to be(1)nonexpansive [1] if 𝑆𝑥𝑆𝑦𝑥𝑦 for all 𝑥,𝑦𝐶,(2)relatively nonexpansive [1012] if 𝐹(𝑆)=𝐹(𝑆) and 𝜙(𝑝,𝑆𝑥)𝜙(𝑝,𝑥) for all 𝑥𝐶 and 𝑝𝐹(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,𝑦𝐶,(4)quasi- 𝜙-nonexpansive if 𝐹(𝑆) and 𝜙(𝑝,𝑆𝑥)𝜙(𝑝,𝑥) for 𝑥𝐶 and 𝑝𝐹(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 {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛𝑥𝑛𝑆𝑥𝑛=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 𝑥𝑦,𝐵𝑥𝐵𝑦𝛼𝐵𝑥𝐵𝑦2,𝑥,𝑦𝐶.(1.1)

If an operator 𝐵 is an 𝛼-inverse-strongly monotone, then we can said that 𝐵 is Lipschitz continuous; that is, 𝐵𝑥𝐵𝑦(1/𝛼)𝑥𝑦 for all 𝑥,𝑦𝐶.

Let 𝑓𝐶×𝐶 be a bifunction, 𝜑𝐶 a real-valued function, and 𝐵𝐶𝐸 be a nonlinear mapping. The generalized mixed equilibrium problem is to find 𝑥𝐶 such that𝑓(𝑥,𝑦)+𝐵𝑥,𝑦𝑥+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶.(1.2)

We denote Ω as the set of solutions to (1.2) that is,Ω={𝑥𝐶𝑓(𝑥,𝑦)+𝐵𝑥,𝑦𝑥+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶}.(1.3)

If 𝐵0, the problem (1.2) reduced into the mixed equilibrium problem for 𝑓, denoted by MEP(𝑓,𝜑), is to find 𝑥𝐶 such that𝑓(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶.(1.4)

If 𝑓0, the problem (1.2) reduced into the mixed variational inequality of Browder type, denoted by VI(𝐶,𝐵,𝜑), is to find 𝑥𝐶 such that𝐵𝑥,𝑦𝑥+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶.(1.5)

If 𝐵0 and 𝜑0 the problem (1.2), reduced into the equilibrium problem for 𝐹, denoted by EP(𝑓), is to find 𝑥𝐶 such that𝑓(𝑥,𝑦)0,𝑦𝐶.(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 [1721] and reference therein. Furthermore, there are many solution methods proposed continuously to solve the EP(𝑓) as shown in [2, 3, 18, 20, 2226] 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 𝑥𝐶 such that𝐵𝑥,𝑦𝑥0y𝐶.(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=𝑥𝐶 and𝑥𝑛+1=𝛼𝑛𝑥+1𝛼𝑛𝑇𝑛𝑥𝑛,(1.8) for all 𝑛, where 𝐶 is a nonempty closed convex subset of a Banach space, {𝛼𝑛} is a sequence in [0,1], and {𝑇𝑛} is a sequence of nonexpansive mappings with some conditions. They proved that {𝑥𝑛} converges strongly to a common fixed point of {𝑇𝑛}.

Recently, Nakajo et al. [32] introduced the more general condition so-called the NST*-condition, and {𝑇𝑛} is said to satisfy the NST*-condition if for every bounded sequence {𝑧𝑛} in 𝐶,lim𝑛𝑧𝑛𝑇𝑛𝑧𝑛=lim𝑛𝑧𝑛𝑧𝑛+1=0implies𝜔𝑤𝑧𝑛𝐹.(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 {𝑥𝑛} generated by the following algorithm: 𝑥1=𝑥𝐶, and 𝑥𝑛+1=𝑃𝐶𝑥𝑛𝜆𝑛𝐵𝑥𝑛,(1.10) where 𝑃𝐶 is the metric projection of 𝐻 onto 𝐶 and {𝜆𝑛} 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 𝐵𝑥1=𝑥𝐶, and𝑥𝑛+1=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛,(1.11) for every 𝑛=1,2,3,, where Π𝐶 is the generalized metric projection from 𝐸 onto 𝐶, 𝐽 is the duality mapping from 𝐸 into 𝐸, and {𝜆𝑛} is a sequence of positive real numbers. They proved that the sequence {𝑥𝑛} generated by (1.11) converges weakly to some element of VI(𝐶,𝐵).

In 1974, Rockafellar interested in the following problem of finding:𝑣𝐸suchthat0𝐴(𝑣),(1.12) where 𝐴 is an operator from 𝐸 into 𝐸. Such 𝑣𝐸 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=𝑥𝐻 and𝑥𝑛+1=𝐽𝑟𝑛𝑥𝑛,𝑛=1,2,3,,(1.13) where {𝑟𝑛}(0,), 𝐴 is a maximal monotone and 𝐽𝑟𝑛=(𝐼+𝑟𝑛𝐴)1. He proved that the sequence {𝑥𝑛} converges weakly to an element of 𝐴1(0).

In 2004, Kamimura et al. [35] considered the algorithm (1.14) in a uniformly smooth and uniformly convex Banach space 𝐸; namely,𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝐽𝑟𝑛𝑥𝑛,𝑛=1,2,3,.(1.14) They proved that the algorithm {𝑥𝑛} generated by (1.14) converges weakly to some element of 𝐴10.

In 2008, Li and Song [36] established a strong convergence theorem in a Banach space. They introduced the following algorithm: 𝑥1=𝑥𝐸 and 𝑦𝑛=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝐽𝑟𝑛𝑥𝑛,𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥+1𝛼𝑛𝐽𝑦𝑛.(1.15) Under the suitable conditions of the coefficient sequences {𝛼𝑛}, {𝛽𝑛}, and {𝑟𝑛}, they proved that the sequence {𝑥𝑛} generated by the above scheme converges strongly to Π𝐶𝑥, where Π𝐶 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: 𝑥0𝑦=𝑥𝐶,𝑛=𝐽1𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝑧𝑛,𝑧𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑥𝑛+𝛾𝑛𝐽𝑆𝑥𝑛,𝑢𝑛=𝐾𝑟𝑛𝑦𝑛,𝐶𝑛=𝑧𝐶𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶𝑥𝑛𝑧,𝐽𝑥𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐶𝑛𝑄𝑛𝑥.(1.16) They proved that {𝑥𝑛} converges strongly to 𝑝𝐹(𝑇)𝐹(𝑆)Ω, where 𝑝Π𝐹(𝑇)𝐹(𝑆)Ω𝑥.

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=𝑥𝐶, 𝑤𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛,𝑧𝑛=𝐽1𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝐽𝑟𝑛𝑤𝑛,𝑦𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑛𝑥𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛,𝑢𝑛𝑢𝐶suchthat𝐹𝑛,𝑦+𝑌𝑢𝑛,𝑦𝑢𝑛𝑢+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥.(1.17) By the new iterative scheme, we will prove the strong convergence theorems of the sequence {𝑥𝑛} which could be converged to the point Π(𝑛=1𝐹(𝑇𝑛))(𝑛=1𝐹(𝑆𝑛))ΩVI(𝐶,𝐵)𝐴1(0)𝑥. Furthermore, we propose the new better appropriate conditions of the coefficient sequences {𝛼𝑛},{𝛽𝑛},{𝛾𝑛}, and {𝑟𝑛}. 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 𝑥𝑛𝑥 to indicate that the sequence {𝑥𝑛} converges weakly to 𝑥 and 𝑥𝑛𝑥 implies that the sequence {𝑥𝑛} converges strongly to 𝑥.

Let 𝑈={𝑥𝐸𝑥=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any 𝑥,𝑦𝑈,𝑥𝑦implies𝑥+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,𝑦𝑈𝑥𝑦𝜖implies𝑥+𝑦2<1𝛿.(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)=inf1𝑥+𝑦2.𝑥,𝑦𝐸,𝑥=𝑦=1,𝑥𝑦𝜀(2.3)

Furthermore, it is said to be smooth, provided that lim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.4) exists for all 𝑥,𝑦𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦𝐸.

Let 𝑝 be a fixed real number with 𝑝2. 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 𝐽𝑝𝐸2𝐸 is defined by𝐽𝑝𝑥(𝑥)=𝐸𝑥,𝑥=𝑥𝑝,𝑥=𝑥𝑝1,(2.5) for all 𝑥𝐸.

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 𝑥𝑦,𝑥𝑦>0 holds for all (𝑥,𝑥),(𝑦,𝑦)𝐽 with 𝑥𝑦,(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 𝑥𝑛𝑥 implies 𝐽𝑥𝑛𝐽𝑥, 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)=𝑥22𝑥,𝐽𝑦+𝑦2,for𝑥,𝑦𝐸.(2.6)

Remark 2.1. We know the following: for each 𝑥,𝑦,𝑧𝐸,(i)(𝑥𝑦)2𝜙(𝑥,𝑦)(𝑥+𝑦)2,(ii)𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2𝑥𝑧,𝐽𝑧𝐽𝑦,(iii)𝜙(𝑥,𝑦)=𝑥𝑦2 in a real Hilbert space.

The generalized projection, introduced by Alber [45], Π𝐶𝐸𝐶 is a map that assigns to an arbitrary point 𝑥𝐸 the minimum point of the function 𝜙(𝑥,𝑦); that is, Π𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem𝜙𝑥,𝑥=inf𝑦𝐶𝜙(𝑦,𝑥),(2.7) existence and uniqueness of the operator Π𝐶 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,𝑦𝐸, 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=𝑥2=𝐽𝑦2. 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,𝑦𝐸, one has 2𝑥𝑦𝑐2𝐽𝑥𝐽𝑦,(2.8) where 𝐽 is the normalized duality mapping of 𝐸 and 0<𝑐1.

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 𝑝2, then for all 𝑥,𝑦𝐸,𝐽𝑥𝐽𝑝(𝑥) and 𝐽𝑦𝐽𝑝(𝑦)𝑐𝑥𝑦,𝐽𝑥𝐽𝑦𝑝2𝑝2𝑝𝑥𝑦𝑝,(2.9) where 𝐽𝑝 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+(1𝜆𝑦)2𝜆𝑥2+(1𝜆)𝑦2𝜆(1𝜆)𝑔(𝑥𝑦),(2.10) for all 𝑥,𝑦{𝑧𝐸𝑧𝑟} and 𝜆[0,1].

Lemma 2.5 (Kamimura and Takahashi [50]). Let 𝐸 be a uniformly convex and smooth real Banach space and {𝑥𝑛},{𝑦𝑛} two sequences of 𝐸. If 𝜙(𝑥𝑛,𝑦𝑛)0 and either {𝑥𝑛} or {𝑦𝑛} is bounded, then 𝑥𝑛𝑦𝑛0.

Lemma 2.6 (Alber [45]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and 𝑥𝐸. Then, 𝑥0=Π𝐶𝑥 if and only if 𝑥0𝑦,𝐽𝑥𝐽𝑥00,𝑦𝐶.(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 𝑥𝐸. Then, 𝜙𝑦,Π𝐶𝑥Π+𝜙𝐶𝑥,𝑥𝜙(𝑦,𝑥),𝑦𝐶.(2.12)

Let 𝐸 be a strictly convex, smooth, and reflexive Banach space and 𝐽 the duality mapping from 𝐸 into 𝐸. Then, 𝐽1 is also single-valued, one-to-one, and surjective, and it is the duality mapping from 𝐸 into 𝐸. Define a function 𝑉𝐸×𝐸 as follows (see [51]):𝑉𝑥,𝑥=𝑥22𝑥,𝑥+𝑥2,(2.13) for all 𝑥𝐸 and 𝑥𝐸. Then, it is obvious that 𝑉(𝑥,𝑥)=𝜙(𝑥,𝐽1(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, 𝑉𝑥,𝑥𝐽+21𝑥𝑥,𝑦𝑉𝑥,𝑥+𝑦,(2.14) for all 𝑥𝐸 and 𝑥,𝑦𝐸.

For solving the generalized mixed equilibrium problem, let us assume that the bifunction 𝐹𝐶×𝐶 and 𝜑𝐶 is convex and lower semicontinuous, satisfying the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥𝐶,(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)0 for all 𝑥,𝑦𝐶,(A3) for each 𝑥,𝑦,𝑧𝐶, limsup𝑡0𝐹(𝑡𝑧+(1𝑡)𝑥,𝑦)𝐹(𝑥,𝑦),(2.15)(A4) for each 𝑥𝐶, 𝑦𝐹(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 satisfying (A1)–(A4). Let 𝑟>0 and 𝑥𝐸. Then, there exists 𝑧𝐶 such that 1𝐹(𝑧,𝑦)+𝑟𝑦𝑧,𝑧𝑥0𝑦𝐶.(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 satisfying (A1)–(A4). For all 𝑟>0 and 𝑥𝐸, define a mapping 𝑇𝑟𝐸𝐶 as follows: 𝑇𝑟1𝑥=𝑧𝐶𝐹(𝑧,𝑦)+𝑟𝑦𝑧,𝐽𝑧𝐽𝑥0,𝑦𝐶,(2.17) for all 𝑥𝐸. Then, the following hold:(1)𝑇𝑟 is single-valued,(2)𝑇𝑟 is a firmly nonexpansive-type mapping, that is, for all 𝑥,𝑦𝐸, 𝑇𝑟𝑥𝑇𝑟𝑦,𝐽𝑇𝑟𝑥𝐽𝑇𝑟𝑦𝑇𝑟𝑥𝑇𝑟𝑦,𝐽𝑥𝐽𝑦,(2.18)(3)𝐹(𝑇𝑟)=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 satisfying (A1)–(A4) and let 𝑟>0. Then, for 𝑥𝐸 and 𝑞𝐹(𝑇𝑟), 𝜙𝑞,𝑇𝑟𝑥𝑇+𝜙𝑟𝑥,𝑥𝜙(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 𝐵𝐶𝐸 be a continuous and monotone mapping, 𝜑𝐶 convex and lower semi-continuous, and 𝐹 a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). For 𝑟>0 and 𝑥𝐸, then there exists 𝑢𝐶 such that 1𝐹(𝑢,𝑦)+𝐵𝑢,𝑦𝑢+𝜑(𝑦)𝜑(𝑢)+𝑟𝑦𝑢,𝐽𝑢𝐽𝑥0,𝑦𝐶.(2.20) Define a mapping 𝐾𝑟𝐶𝐶 as follows: 𝐾𝑟1(𝑥)=𝑢𝐶𝐹(𝑢,𝑦)+𝐵𝑢,𝑦𝑢+𝜑(𝑦)𝜑(𝑢)+𝑟,𝑦𝑢,𝐽𝑢𝐽𝑥0,𝑦𝐶(2.21) for all 𝑥𝐸. Then, the following hold:(i)𝐾𝑟 is single-valued,(ii)𝐾𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦𝐸, 𝐾𝑟𝑥𝐾𝑟𝑦,𝐽𝐾𝑟𝑥𝐽𝐾𝑟𝑦𝐾𝑟𝑥𝐾𝑟𝑦,𝐽𝑥𝐽𝑦,(iii)𝐹(𝐾𝑟)=Ω, (iv)Ω is closed and convex,(v)𝜙(𝑝,𝐾𝑟𝑧)+𝜙(𝐾𝑟𝑧,𝑧)𝜙(𝑝,𝑧) for all 𝑝𝐹(𝐾𝑟), 𝑧𝐸.

It follows from Lemma 2.10 that the mapping 𝐾𝑟𝐶𝐶 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 inf𝑦𝐶𝜙(𝑥,𝑦) has a unique solution. The operator Π𝐶𝑥=argmin𝑦𝐶𝜙(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)𝑥𝐴𝑥}, domain 𝐷(𝐴)={𝑥𝐸𝐴(𝑥)}, and range 𝑅(𝐴)={𝑥𝐸𝑥𝐴(𝑥),𝑥𝐷(𝐴)}. We denote a set-valued operator 𝐴 from 𝐸 to 𝐸 by 𝐴𝐸×𝐸. 𝐴 is said to be monotone if 𝑥𝑦,𝑥𝑦0 for all (𝑥,𝑥),(𝑦,𝑦)𝐴. A monotone operator 𝐴𝐸×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 𝐴10={𝑧𝐷(𝐴)0𝐴𝑧} 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 𝐽𝑟𝑥=𝑥𝑟. In other words, 𝐽𝑟=(𝐽+𝑟𝐴)1𝐽 for all 𝑟>0. 𝐽𝑟 is a single-valued mapping from 𝐸 to 𝐷(𝐴). Also, 𝐴1(0)=𝐹(𝐽𝑟) for all 𝑟>0, where 𝐹(𝐽𝑟) is the set of all fixed points of 𝐽𝑟. Define, for 𝑟>0, the Yosida approximation of 𝐴 by 𝐴𝑟=(𝐽𝐽𝐽𝑟)/𝑟. We know that 𝐴𝑟𝑥𝐴(𝐽𝑟𝑥) for all 𝑟>0 and 𝑥𝐸.

Lemma 2.13 (Kohsaka and Takahashi [51, Lemma  3.1]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐴𝐸×𝐸 be a maximal monotone operator with 𝐴10, 𝑟>0, and 𝐽𝑟=(𝐽+𝑟𝑇)1𝐽. Then, 𝜙𝑥,𝐽𝑟𝑦𝐽+𝜙𝑟𝑦,𝑦𝜙(𝑥,𝑦)(2.22) for all 𝑥𝐴10 and 𝑦𝐸.

Let 𝐵 be an inverse-strongly monotone mapping of 𝐶 into 𝐸 which is said to be hemicontinuous if for all 𝑥,𝑦𝐶, the mapping 𝐹 of [0,1] into 𝐸, defined by 𝐹(𝑡)=𝐵(𝑡𝑥+(1𝑡)𝑦), is continuous with respect to the weak* topology of 𝐸. We define by 𝑁𝐶(𝑣) the normal cone for 𝐶 at a point 𝑣𝐶; that is,𝑁𝐶𝑥(𝑣)=𝐸𝑣𝑦,𝑥0,𝑦𝐶.(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 𝑇𝐸×𝐸 be an operator defined as follows: 𝑇𝑣=𝐵𝑣+𝑁𝐶(𝑣),𝑣𝐶,,𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒.(2.24) Then 𝑇, is maximal monotone and 𝑇10=VI(𝐶,𝐵).

Lemma 2.15 (Tan and Xu [55]). Let {𝑎𝑛} and {𝑏𝑛} be two sequences of nonnegative real numbers satisfying 𝑎𝑛+1𝑎𝑛+𝑏𝑛,𝑛0.(2.25) If 𝑛=1𝑏𝑛<, then lim𝑛𝑎𝑛 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 satisfying (A1)–(A4), let 𝜑𝐶 be a proper lower semicontinuous and convex function, and let 𝐴𝐸𝐸 be a maximal monotone operator satisfying 𝐷(𝐴)𝐸. Let 𝐽𝑟=(𝐽+𝑟𝐴)1𝐽 for 𝑟>0, let 𝐵 be an 𝛼-inverse-strongly monotone operator of 𝐸 into 𝐸, and let 𝑌𝐸𝐸 be a continuous and monotone mapping. Let {𝑇𝑛} and {𝑆𝑛} be two families of quasi-𝜙-nonexpansive mappings of 𝐸 into itself satisfies the NST-condition, with Θ=(𝑛=1𝐹(𝑇𝑛))(𝑛=1𝐹(𝑆𝑛))Ω𝑉𝐼(𝐶,𝐵)𝐴1(0) and 𝐵𝑦𝐵𝑦𝐵𝑢 for all 𝑦𝐶 and 𝑢Θ. Let {𝑥𝑛} be a sequence generated by 𝑥1=𝑥𝐸, and 𝑤𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛,𝑧𝑛=𝐽1𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝐽𝑟𝑛𝑤𝑛,𝑦𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑛𝑥𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛,𝑢𝑛𝑢𝐶suchthat𝐹𝑛,𝑦+𝑌𝑢𝑛,𝑦𝑢𝑛𝑢+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0,𝑦𝐶,𝑛+1=𝑧𝐶𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥,(3.1) for all 𝑛. If the coefficient sequence {𝛼𝑛},{𝛽𝑛}, {𝛾𝑛}, and {𝛿𝑛}[0,1], {𝑟𝑛}(0,) satisfy 𝛼𝑛+𝛽𝑛+𝛾𝑛=1, liminf𝑛𝛼𝑛𝛽𝑛>0, liminf𝑛𝛼𝑛𝛾𝑛>0, liminf𝑛𝛾𝑛(1𝛿𝑛)>0, liminf𝑛𝑟𝑛>0 and {𝜆𝑛}[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 {𝑥𝑛} converges strongly to ΠΘ𝑥.

Proof. We first show that {𝑥𝑛} is bounded. Let 𝑝Θ=(𝑛=1𝐹(𝑇𝑛))(𝑛=1𝐹(𝑆𝑛))ΩVI(𝐶,𝐵)𝐴1(0), and let 𝐻𝑢𝑛𝑢,𝑦=𝐹𝑛,𝑦+𝑌𝑢𝑛,𝑦𝑢𝑛𝑢+𝜑(𝑦)𝜑𝑛,𝐾,𝑦𝐶𝑟𝑛=𝑢𝑢𝐶𝐻𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛.0,𝑦𝐶(3.2) Put 𝑣𝑛=𝐽1(𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛) and 𝑢𝑛=𝐾𝑟𝑛𝑦𝑛.
With its relatively nonexpansiveness of 𝐽𝑟𝑛 and by Lemma 2.8, the convexity of the function 𝑉 in the second variable, we have 𝜙𝑝,𝑤𝑛=𝜙𝑝,Π𝐶𝑣𝑛𝜙𝑝,𝑣𝑛=𝜙𝑝,𝐽1𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝑉𝑝,𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛+𝜆𝑛𝐵𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝑝,𝜆𝑛𝐵𝑥𝑛=𝑉𝑝,𝐽𝑥𝑛2𝜆𝑛𝑣𝑛𝑝,𝐵𝑥𝑛=𝜙𝑝,𝑥𝑛2𝜆𝑛𝑥𝑛𝑝,𝐵𝑥𝑛+2𝑣𝑛𝑥𝑛,𝜆𝑛𝐵𝑥𝑛.(3.3) Since 𝑝VI(𝐶,𝐵) and 𝐵 is 𝛼-inverse-strongly monotone, we consider 2𝜆𝑛𝑥𝑛𝑝,𝐵𝑥𝑛=2𝜆𝑛𝑥𝑛𝑝,𝐵𝑥𝑛𝐵𝑝2𝜆𝑛𝑥𝑛𝑝,𝐵𝑝2𝛼𝜆𝑛𝐵𝑥𝑛𝐵𝑝2.(3.4) Therefore, by Lemma 2.2, we obtain 2𝑣𝑛𝑥𝑛,𝜆𝑛𝐵𝑥𝑛𝐽=21𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝐽1𝐽𝑥𝑛,𝜆𝑛𝐵𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛4𝑐2𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛=4𝑐2𝜆2𝑛𝐵𝑥𝑛24𝑐2𝜆2𝑛𝐵𝑥𝑛𝐵𝑝2.(3.5) We can rewrite (3.3), which yield that 𝜙𝑝,𝑤𝑛𝜙𝑝,𝑥𝑛2𝛼𝜆𝑛𝐵𝑥𝑛𝐵𝑝2+4𝑐2𝜆2𝑛𝐵𝑥𝑛𝐵𝑝2𝜙𝑝,𝑥𝑛+2𝜆𝑛2𝑐2𝜆𝑛𝛼𝐵𝑥𝑛𝐵𝑝2𝜙𝑝,𝑥𝑛.(3.6) Apply the Lemma 2.8, Lemma 2.13 and (3.6), we consider 𝜙𝑝,𝑧𝑛=𝜙𝑝,𝐽1𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝐽𝑟𝑛𝑤𝑛=𝑉𝑝,𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝐽𝑟𝑛𝑤𝑛𝛿𝑛𝑉𝑥𝑝,𝐽𝑛+1𝛿𝑛𝑉𝐽𝑝,𝐽𝑟𝑛𝑤𝑛=𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝐽𝑟𝑛𝑤𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝑤𝑛𝐽𝜙𝑟𝑛𝑤𝑛,𝑤𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝑤𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝑥𝑛=𝜙𝑝,𝑥𝑛,(3.7) hence, we obtain 𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑛𝑥𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛=𝑝22𝛼𝑛𝑝,𝐽𝑥𝑛2𝛽𝑛𝑝,𝐽𝑇𝑛𝑥𝑛2𝛾𝑛𝑝,𝐽𝑆𝑛𝑧𝑛+𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑛𝑥𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛2𝑝22𝛼𝑛𝑝,𝐽𝑥𝑛2𝛽𝑛𝑝,𝐽𝑇𝑛𝑥𝑛2𝛾𝑛𝑝,𝐽𝑆𝑛𝑧𝑛+𝛼𝑛𝐽𝑥𝑛2+𝛽𝑛𝐽𝑇𝑛𝑥𝑛2+𝛾𝑛𝐽𝑆𝑛𝑧𝑛2=𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑇𝑛𝑥𝑛+𝛾𝑛𝜙𝑝,𝑆𝑛𝑧𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑧𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑥𝑛=𝜙𝑝,𝑥𝑛.(3.8) By (3.1), again, 𝜙𝑝,𝑢𝑛=𝜙𝑝,𝐾𝑟𝑛𝑦𝑛𝜙𝑝,𝑦𝑛𝜙𝑝,𝑥𝑛.(3.9) This shows that 𝑝𝐶𝑛+1. Consequently, Θ𝐶𝑛, for all 𝑛.
Next, we show that lim𝑛𝜙(𝑥𝑛,𝑥0) exists. Since 𝑥𝑛=Π𝐶𝑛𝑥, it follows from Lemma 2.7 that 𝜙𝑥𝑛,𝑥𝜙(𝑝,𝑥)𝜙𝑝,𝑥𝑛𝜙(𝑝,𝑥),(3.10) for each 𝑝Θ𝐶𝑛. Then, 𝜙(𝑥𝑛,𝑥) is bounded. It implies that {𝑥𝑛} is bounded and {𝑦𝑛}, {𝑧𝑛}, {𝑤𝑛}, and {𝐽𝑟𝑛𝑤𝑛} are also bounded.
From 𝑥𝑛=Π𝐶𝑛𝑥 and 𝑥𝑛+1𝐶𝑛+1𝐶𝑛, we have 𝜙𝑥𝑛𝑥,𝑥𝜙𝑛+1,𝑥,𝑛.(3.11) Therefore, {𝜙(𝑥𝑛,𝑥)} is nondecreasing. It follows that the limit of {𝜙(𝑥𝑛,𝑥)} exists, and from Lemma 2.7, we have 𝜙𝑥𝑛+1,𝑥𝑛𝑥=𝜙𝑛+1,Π𝐶𝑛𝑥𝑥𝜙𝑛+1Π,𝑥𝜙𝐶𝑛𝑥𝑥,𝑥=𝜙𝑛+1𝑥,𝑥𝜙𝑛,,𝑥(3.12) for all 𝑛. Thus, we have lim𝑛𝜙𝑥𝑛+1,𝑥𝑛=0.(3.13) Since 𝑥𝑛+1=Π𝐶𝑛+1𝑥𝐶𝑛+1, it follows from the definition of 𝐶𝑛+1 that 𝜙𝑥𝑛+1,𝑢𝑛𝑥𝜙𝑛+1,𝑥𝑛0.(3.14) By Lemma 2.5, (3.13), and (3.14), we note that lim𝑛𝑥𝑛+1𝑢𝑛=lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛𝑥𝑛𝑢𝑛=0.(3.15) Since 𝐽 is uniformly norm-to-norm continuous on the bounded set, we obtain lim𝑛𝐽𝑥𝑛+1𝐽𝑢𝑛=lim𝑛𝐽𝑥𝑛+1𝐽𝑥𝑛=lim𝑛𝐽𝑥𝑛𝐽𝑢𝑛=0.(3.16) Since 𝑥𝑚=Π𝐶𝑚𝐶𝑛 for any positive integer 𝑚𝑛, it follows from Lemma 2.7 that 𝜙𝑥𝑚,𝑥𝑛𝑥=𝜙𝑚,Π𝐶𝑛𝑥𝑛𝑥𝜙𝑚Π,𝑥𝜙𝐶𝑛𝑥𝑛𝑥,𝑥=𝜙𝑚𝑥,𝑥𝜙𝑛.,𝑥(3.17) Taking 𝑚,𝑛, we have 𝜙(𝑥𝑚,𝑥𝑛)0 as 𝑛. It follows from Lemma 2.5, that 𝑥𝑚𝑥𝑛0 as 𝑚,𝑛. Hence, {𝑥𝑛} is a Cauchy sequence. Since 𝐸 is a Banach space and 𝐶 is closed and convex, we can assume that 𝑥𝑛𝑢𝐶 as 𝑛.
Next, we show that 𝑢(𝑛=1𝐹(𝑇𝑛))(𝑛=1𝐹(𝑆𝑛)).
Since 𝐸 is a uniformly smooth Banach space, we know that 𝐸 is a uniformly convex Banach space. Let 𝑟=sup𝑛{𝑥𝑛,𝑇𝑛𝑥𝑛,𝑆𝑛𝑧𝑛}. From Lemma 2.4, we have 𝜙𝑝,𝑢𝑛𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑛𝑥𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛=𝑝22𝛼𝑛𝑝,𝐽𝑥𝑛2𝛽𝑛𝑝,𝐽𝑇𝑛𝑥𝑛2𝛾𝑛𝑝,𝐽𝑆𝑛𝑧𝑛+𝛼𝑛𝐽𝑥𝑛+𝛽𝑛𝐽𝑇𝑛𝑥𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛2𝑝22𝛼𝑛𝑝,𝐽𝑥𝑛2𝛽𝑛𝑝,𝐽𝑇𝑛𝑥𝑛2𝛾𝑛𝑝,𝐽𝑆𝑛𝑧𝑛+𝛼𝑛𝐽𝑥𝑛2+𝛽𝑛𝐽𝑇𝑛𝑥𝑛2+𝛾𝑛𝐽𝑆𝑛𝑧𝑛2𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛=𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑇𝑛𝑥𝑛+𝛾𝑛𝜙𝑝,𝑆𝑛𝑧𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑧𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑥𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛=𝜙𝑝,𝑥𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛.(3.18) This implies that 𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛.(3.19) On the other hand, we have 𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛=𝑥𝑛2𝑢𝑛22𝑝,𝐽𝑥𝑛𝐽𝑢𝑛=𝑥𝑛𝑢𝑛𝑥𝑛+𝑢𝑛+2𝑝𝐽𝑥𝑛𝐽𝑢𝑛.(3.20) Noticing (3.15) and (3.16), we obtain 𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛0,as𝑛.(3.21) Since liminf𝑛𝛼𝑛𝛽𝑛>0 and (3.21), it follows from (3.19) that 𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛0,as𝑛.(3.22) It follows from the property of 𝑔 that 𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛0,as𝑛.(3.23) Since 𝐽1 is uniformly norm-to-norm continuous on bounded sets, we see that lim𝑛𝑇𝑛𝑥𝑛𝑥𝑛=0.(3.24) Similarly, using the condition limsup𝑛𝛼𝑛𝛾𝑛>0, one can obtain lim𝑛𝑆𝑛𝑧𝑛𝑥𝑛=0.(3.25) By (3.6), (3.8), and (3.18), we have 𝜙𝑝,𝑢𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑧𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝑤𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛1𝛿𝑛𝜙𝑝,𝑤𝑛𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛1𝛿𝑛𝜙𝑝,𝑥𝑛+2𝜆𝑛2𝑐2𝜆𝑛𝛼𝐵𝑥𝑛𝐵𝑝2𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛1𝛿𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛1𝛿𝑛2𝜆𝑛2𝑐2𝜆𝑛𝛼𝐵𝑥𝑛𝐵𝑝2𝛼𝑛𝛽𝑛𝑔𝐽𝑇𝑛𝑥𝑛𝐽𝑥𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛1𝛿𝑛2𝜆𝑛2𝑐2𝜆𝑛𝛼𝐵𝑥𝑛𝐵𝑝2.(3.26) This implies that 2𝜆𝑛2𝛼𝑐2𝜆𝑛𝐵𝑥𝑛𝐵𝑝21𝛾𝑛1𝛿𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛.(3.27) By assumption, liminf𝑛𝛾𝑛(1𝛿𝑛)>0 and (3.21), we get that lim𝑛𝐵𝑥𝑛𝐵𝑝=0.(3.28) From Lemma 2.7, Lemma 2.8, and (3.5), we have 𝜙𝑥𝑛,𝑤𝑛𝑥=𝜙𝑛,Π𝐶𝑣𝑛𝑥𝜙𝑛,𝑣𝑛𝑥=𝜙𝑛,𝐽1𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝑥=𝑉𝑛,𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝑥𝑉𝑛,𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛+𝜆𝑛𝐵𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛𝑥𝑛,𝜆𝑛𝐵𝑥𝑛𝑥=𝑉𝑛,𝐽𝑥𝑛2𝜆𝑛𝑣𝑛𝑥𝑛,𝐵𝑥𝑛𝑥=𝜙𝑛,𝑥𝑛+2𝑣𝑛𝑥𝑛,𝜆𝑛𝐵𝑥𝑛4𝑐2𝜆2𝑛𝐵𝑥𝑛𝐵𝑝2.(3.29) By Lemma 2.8 and Lemma 2.13, we have 𝜙𝑥𝑛,𝑧𝑛𝑥=𝜙𝑛,𝐽1𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝐽𝑟𝑛𝑤𝑛𝑥=𝑉𝑛,𝛿𝑛𝐽𝑥𝑛+1𝛿𝑛𝐽𝐽𝑟𝑛𝑤𝑛𝛿𝑛𝑉𝑥𝑛,𝐽𝑥𝑛+1𝛿𝑛𝑉𝑥𝑛𝐽,𝐽𝑟𝑛𝑤𝑛=𝛿𝑛𝜙𝑥𝑛,𝑥𝑛+1𝛿𝑛𝜙𝑥𝑛,𝐽𝑟𝑛𝑤𝑛=𝛿𝑛𝜙𝑥𝑛,𝑥𝑛+1𝛿𝑛𝜙𝑥𝑛,𝑤𝑛𝐽𝜙𝑟𝑛𝑤𝑛,𝑤𝑛=1𝛿𝑛𝜙𝑥𝑛,𝑤𝑛1𝛿𝑛4𝑐2𝜆2𝑛𝐵𝑥𝑛𝐵𝑝2.(3.30) From Lemma 2.5 and (3.28), we obtain lim𝑛𝑥𝑛𝑤𝑛=lim𝑛𝑥𝑛𝑧𝑛=0.(3.31) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim𝑛𝐽𝑥𝑛𝐽𝑤𝑛=lim𝑛𝐽𝑥𝑛𝐽𝑧𝑛=0.(3.32) Since 𝑥𝑛𝑢 as 𝑛, 𝑧𝑛𝑢 as 𝑛. Combining (3.15), (3.25), and (3.28), we also obtain 𝑆𝑛𝑧𝑛𝑧𝑛𝑆𝑛𝑧𝑛𝑥𝑛+𝑥𝑛𝑧𝑛0as𝑛.(3.33) By (3.15) and (3.31), we obtain that 𝑧𝑛+1𝑧𝑛𝑧𝑛+1𝑥𝑛+1+𝑥𝑛+1𝑥𝑛+𝑥𝑛𝑧𝑛0as𝑛.(3.34) By (3.15), (3.24), (3.33), and (3.34), and {𝑇𝑛}, {𝑆𝑛} satisfies the NST*-condition and 𝑥𝑛𝑝, then we have 𝑝(𝑛=1𝐹(𝑇𝑛))(𝑛=1𝐹(𝑆𝑛)).
Since {𝑥𝑛} is bounded, there exists a subsequence {𝑥𝑛𝑖} of {𝑥𝑛} such that 𝑥𝑛𝑖𝑢𝐶. It follows from (3.31) that we have 𝑤𝑛𝑖𝑢 as 𝑖. Next, we show that 𝑢𝐴10.
By (3.6), (3.8), and (3.9), we obtain 𝜙𝑝,𝑢𝑛𝜙𝑝,𝑦𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑧𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝑤𝑛𝐽𝜙𝑟𝑛𝑤𝑛,𝑤𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝛿𝑛𝜙𝑝,𝑥𝑛+1𝛿𝑛𝜙𝑝,𝑥𝑛𝐽𝜙𝑟𝑛𝑤𝑛,𝑤𝑛𝜙𝑝,𝑥𝑛𝛾𝑛1𝛿𝑛𝜙𝐽𝑟𝑛𝑤𝑛,𝑤𝑛.(3.35) This implies that 𝛾𝑛1𝛿𝑛𝜙𝐽𝑟𝑛𝑤𝑛,𝑤𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛.(3.36) By (3.21), we have lim𝑛𝐽𝑟𝑛𝑤𝑛𝑤𝑛=0.(3.37) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that lim𝑛𝐽𝐽𝑟𝑛𝑤𝑛𝐽𝑤𝑛=0.(3.38) Indeed, since liminf𝑛𝑟𝑛>0, it follows from (3.38) that lim𝑛𝐴𝑟𝑛𝑤𝑛=lim𝑛1𝑟𝑛𝐽𝑤𝑛𝐽𝐽𝑟𝑛𝑤𝑛=0.(3.39) If (𝑤,𝑤)𝐴, then it holds from the monotonicity of 𝐴 that 𝑤𝑤𝑛𝑖,𝑤𝐴𝑟𝑛𝑖𝑤𝑛𝑖0,(3.40) for all 𝑖. Letting 𝑖, we get 𝑤𝑢,𝑤0. Then, the maximality of 𝐴 implies 𝑢𝐴10.
Next, we show that 𝑢VI(𝐶,𝐵). Let 𝐾𝐸×𝐸 be an operator as follows: 𝐾𝑣=𝐵𝑣+𝑁𝐶(𝑣),𝑣𝐶,,otherwise.(3.41) By Theorem 2.14, 𝐾 is maximal monotone and 𝐾10=VI(𝐶,𝐵).
Let (𝑣,𝑤)𝐺(𝐾). Since 𝑤𝐾𝑣=𝐵𝑣+𝑁𝐶(𝑣), we get 𝑤𝐵𝑣𝑁𝐶(𝑣). From 𝑤𝑛𝐶, we have 𝑣𝑤𝑛,𝑤𝐾𝑣0.(3.42) On the other hand, since 𝑤𝑛=Π𝐶𝐽1(𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛), then by Lemma 2.6, we have 𝑣𝑤𝑛,𝐽𝑤𝑛𝐽𝑥𝑛𝜆𝑛𝐵𝑥𝑛0,(3.43) thus 𝑣𝑤𝑛,𝐽𝑥𝑛𝐽𝑤𝑛𝜆𝑛𝐵𝑥𝑛0.(3.44) It follows from (3.42) and (3.44) that 𝑣𝑤𝑛,𝑤𝑣𝑤𝑛,𝐵𝑣𝑣𝑤𝑛,𝐵𝑣+𝑣𝑤𝑛,𝐽𝑥𝑛𝐽𝑤𝑛𝜆𝑛𝐵𝑥𝑛=𝑣𝑤𝑛,𝐵𝑣𝐵𝑥𝑛+𝑣𝑤𝑛,𝐽𝑥𝑛𝐽𝑤𝑛𝜆𝑛=𝑣𝑤𝑛,𝐵𝑣𝐵𝑤𝑛+𝑣𝑤𝑛,𝐵𝑤𝑛𝐵𝑥𝑛+𝑣𝑤𝑛,𝐽𝑥𝑛𝐽𝑤𝑛𝜆𝑛𝑣𝑤𝑛𝑤𝑛𝑥𝑛𝛼𝑣𝑤𝑛𝐽𝑥𝑛𝐽𝑤𝑛𝑎𝑤𝑀𝑛𝑥𝑛𝛼+𝐽𝑥𝑛𝐽𝑤𝑛𝑎,(3.45) where 𝑀=sup𝑛1{𝑣𝑤𝑛}. From (3.31) and (3.32), we obtain 𝑣𝑢,𝑤0. By the maximality of 𝐾, we have 𝑢𝐾10 and hence 𝑢VI(𝐶,𝐵).
Next, we show that 𝑢Ω. From 𝑢𝑛=𝐾𝑟𝑛𝑦𝑛 and Lemma 2.12, we obtain 𝜙𝑢𝑛,𝑦𝑛𝐾=𝜙𝑟𝑛𝑦𝑛,𝑦𝑛𝜙𝑢,𝑦𝑛𝜙𝑢,𝐾𝑟𝑛𝑦𝑛𝜙𝑢,𝑥𝑛𝜙𝑢,𝐾𝑟𝑛𝑦𝑛𝜙𝑢,𝑥𝑛𝜙𝑢,𝑢𝑛.(3.46) On the other hand, we have 𝜙𝑢,𝑥𝑛𝜙𝑢,𝑢𝑛=𝑥𝑛2𝑢𝑛22𝑢,𝐽𝑥𝑛𝐽𝑢𝑛=𝑥𝑛𝑢𝑛𝑥𝑛+𝑢𝑛+2𝑢𝐽𝑥𝑛𝐽𝑢𝑛.(3.47) Noticing (3.15) and (3.16), we obtain 𝜙𝑢,𝑥𝑛𝜙𝑢,𝑢𝑛0,as𝑛.(3.48) It follows that 𝜙𝑢𝑛,𝑦𝑛0,as𝑛.(3.49) By Lemma 2.5, we have lim𝑛𝑢𝑛𝑦𝑛=0.(3.50) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we get lim𝑛𝐽𝑢𝑛𝐽𝑦𝑛=0.(3.51) From the assumption liminf𝑛𝑟𝑛>𝑎, we get lim𝑛𝐽𝑢𝑛𝐽𝑦𝑛𝑟𝑛=0.(3.52) Noticing that 𝑢𝑛=𝐾𝑟𝑛𝑦𝑛, we have 𝐻𝑢𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛0,𝑦𝐶.(3.53) Hence, 𝐻𝑢𝑛𝑖+1,𝑦𝑟𝑛𝑖𝑦𝑢𝑛𝑖,𝐽𝑢𝑛𝑖𝐽𝑦𝑛𝑖0,𝑦𝐶.(3.54) From the (A2), we note that 𝑦𝑢𝑛𝑖𝐽𝑢𝑛𝑖𝐽𝑦𝑛𝑖𝑟𝑛𝑖1𝑟𝑛𝑖𝑦𝑢𝑛𝑖,𝐽𝑢𝑛𝑖𝐽𝑦𝑛𝑖𝑢𝐻𝑛𝑖,𝑦𝐻𝑦,𝑢𝑛𝑖,𝑦𝐶.(3.55) Taking the limit as 𝑛 in the above inequality, and from (A4) and 𝑢𝑛𝑖𝑢, we have 𝐻(𝑦,𝑢)0,forall𝑦𝐶. For 0<𝑡<1 and 𝑦𝐶, define 𝑦𝑡=𝑡𝑦+(1𝑡)𝑢. Noticing that 𝑦,𝑢𝐶, we obtain 𝑦𝑡𝐶, which yields that 𝐻(𝑦𝑡,𝑢)0. It follows from (A1) that 𝑦0=𝐻𝑡,𝑦𝑡𝑦𝑡𝐻𝑡+𝑦,𝑦(1𝑡)𝐻𝑡𝑦,𝑢𝑡𝐻𝑡,𝑦.(3.56) That is, 𝐻(𝑦𝑡,𝑦)0.
Let 𝑡0, from (A3), we obtain 𝐻(𝑢,𝑦)0,forall𝑦𝐶. This implies that 𝑢Ω. Hence, 𝑢Θ=(𝑛=1𝐹(𝑇𝑛))(𝑛=1𝐹(𝑆𝑛))Ω𝑉𝐼(𝐶,𝐵)𝐴1(0).
Finally, we show that 𝑢=ΠΘ𝑥. Indeed, from 𝑥𝑛=Π𝐶𝑛𝑥 and Lemma 2.6, we have 𝐽𝑥𝐽𝑥𝑛,𝑥𝑛𝑧0,𝑧𝐶𝑛.(3.57) Since Θ𝐶𝑛, we also have