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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 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: YonghongΒ Yao

Copyright Β© 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-πœ™-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-πœ™-nonexpansive mappings in the framework of Banach spaces.

Let 𝐸 be a Banach space with norm β€–β‹…β€– and 𝐢 a nonempty closed convex subset of 𝐸 and let πΈβˆ— denote the dual of 𝐸.

A mapping π‘†βˆΆπΆβ†’πΆ is said to be(1)nonexpansive [1] if ‖𝑆π‘₯βˆ’π‘†π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπΆ,(2)relatively nonexpansive [10–12] if 𝐹(𝑆)=𝐹(𝑆) and πœ™(𝑝,𝑆π‘₯)β‰€πœ™(𝑝,π‘₯) for all π‘₯∈𝐢 and π‘βˆˆπΉ(𝑆), where the functional πœ™ defined by (2.6). The asymptotic behavior of a relatively nonexpansive mapping was studied in [13, 14],(3)πœ™-nonexpansive, if πœ™(𝑆π‘₯,𝑆𝑦)β‰€πœ™(π‘₯,𝑦) for π‘₯,π‘¦βˆˆπΆ,(4)quasi- πœ™-nonexpansive if 𝐹(𝑆)β‰ βˆ… and πœ™(𝑝,𝑆π‘₯)β‰€πœ™(𝑝,π‘₯) for π‘₯∈𝐢 and π‘βˆˆπΉ(𝑆).

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

A point 𝑝 in 𝐢 is said to be an asymptotic fixed point of 𝑆 [16] if 𝐢 contains a sequence {π‘₯𝑛} which converges weakly to 𝑝 such that limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘†π‘₯𝑛‖=0. The set of asymptotic fixed points of 𝑆 will be denoted by 𝐹(𝑆).

Let 𝐡 be an operator from 𝐢 into πΈβˆ—, and 𝐡 is said to be 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼 such that ⟨π‘₯βˆ’π‘¦,𝐡π‘₯βˆ’π΅π‘¦βŸ©β‰₯𝛼‖𝐡π‘₯βˆ’π΅π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.1)

If an operator 𝐡 is an 𝛼-inverse-strongly monotone, then we can said that 𝐡 is Lipschitz continuous; that is, ‖𝐡π‘₯βˆ’π΅π‘¦β€–β‰€(1/𝛼)β€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπΆ.

Let π‘“βˆΆπΆΓ—πΆβ†’β„ be a bifunction, πœ‘βˆΆπΆβ†’β„ a real-valued function, and π΅βˆΆπΆβ†’πΈβˆ— be a nonlinear mapping. The generalized mixed equilibrium problem is to find π‘₯∈𝐢 such that𝑓(π‘₯,𝑦)+⟨𝐡π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.2)

We denote Ξ© as the set of solutions to (1.2) that is,Ξ©={π‘₯βˆˆπΆβˆΆπ‘“(π‘₯,𝑦)+⟨𝐡π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ}.(1.3)

If 𝐡≑0, the problem (1.2) reduced into the mixed equilibrium problem for 𝑓, denoted by MEP(𝑓,πœ‘), is to find π‘₯∈𝐢 such that𝑓(π‘₯,𝑦)+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.4)

If 𝑓≑0, the problem (1.2) reduced into the mixed variational inequality of Browder type, denoted by VI(𝐢,𝐡,πœ‘), is to find π‘₯∈𝐢 such that⟨𝐡π‘₯,π‘¦βˆ’π‘₯⟩+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.5)

If 𝐡≑0 and πœ‘β‰‘0 the problem (1.2), reduced into the equilibrium problem for 𝐹, denoted by EP(𝑓), is to find π‘₯∈𝐢 such that𝑓(π‘₯,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΆ.(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 𝐡 be a monotone operator of 𝐢 into πΈβˆ—. The so-called variational inequality problem is to find a point π‘₯∈𝐢 such that⟨𝐡π‘₯,π‘¦βˆ’π‘₯⟩β‰₯0βˆ€y∈𝐢.(1.7) The set of solutions of the variational inequality problem is denoted by VI(𝐢,𝐡).

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 𝐢 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 π‘₯1=π‘₯∈𝐢 andπ‘₯𝑛+1=𝛼𝑛π‘₯+1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘›π‘₯𝑛,(1.8) for all π‘›βˆˆβ„•, where 𝐢 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 {π‘₯𝑛} 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 𝐢,limπ‘›β†’βˆžβ€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π‘§π‘›βˆ’π‘§π‘›+1β€–β€–=0impliesπœ”π‘€ξ€·π‘§π‘›ξ€ΈβŠ‚πΉ.(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 GΓ’teaux differentiable.

In Hilbert space 𝐻, Iiduka et al. [33] introduced an iterative scheme and proved that the sequence {π‘₯𝑛} generated by the following algorithm: π‘₯1=π‘₯∈𝐢, and π‘₯𝑛+1=𝑃𝐢π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛,(1.10) where 𝑃𝐢 is the metric projection of 𝐻 onto 𝐢 and {πœ†π‘›} is a sequence of positive real numbers, converges weakly to some element of VI(𝐢,𝐡).

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 𝐡∢π‘₯1=π‘₯∈𝐢, andπ‘₯𝑛+1=Ξ πΆπ½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛,(1.11) for every 𝑛=1,2,3,…, where Π𝐢 is the generalized metric projection from 𝐸 onto 𝐢, 𝐽 is the duality mapping from 𝐸 into πΈβˆ—, and {πœ†π‘›} is a sequence of positive real numbers. They proved that the sequence {π‘₯𝑛} generated by (1.11) converges weakly to some element of VI(𝐢,𝐡).

In 1974, Rockafellar interested in the following problem of finding:π‘£βˆˆπΈsuchthat0∈𝐴(𝑣),(1.12) where 𝐴 is an operator from 𝐸 into πΈβˆ—. Such π‘£βˆˆπΈ 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: π‘₯1=π‘₯∈𝐻 andπ‘₯𝑛+1=π½π‘Ÿπ‘›π‘₯𝑛,𝑛=1,2,3,…,(1.13) where {π‘Ÿπ‘›}βŠ‚(0,∞), 𝐴 is a maximal monotone and π½π‘Ÿπ‘›=(𝐼+π‘Ÿπ‘›π΄)βˆ’1. He proved that the sequence {π‘₯𝑛} converges weakly to an element of π΄βˆ’1(0).

In 2004, Kamimura et al. [35] considered the algorithm (1.14) in a uniformly smooth and uniformly convex Banach space 𝐸; namely,π‘₯𝑛+1=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘₯𝑛,𝑛=1,2,3,….(1.14) They proved that the algorithm {π‘₯𝑛} generated by (1.14) converges weakly to some element of π΄βˆ’10.

In 2008, Li and Song [36] established a strong convergence theorem in a Banach space. They introduced the following algorithm: π‘₯1=π‘₯∈𝐸 and 𝑦𝑛=π½βˆ’1𝛽𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘₯𝑛,π‘₯𝑛+1=π½βˆ’1𝛼𝑛𝐽π‘₯+1βˆ’π›Όπ‘›ξ€Έπ½ξ€·π‘¦π‘›.ξ€Έξ€Έ(1.15) Under the suitable conditions of the coefficient sequences {𝛼𝑛}, {𝛽𝑛}, and {π‘Ÿπ‘›}, they proved that the sequence {π‘₯𝑛} generated by the above scheme converges strongly to Π𝐢π‘₯, where Π𝐢 is the generalized projection from 𝐸 onto 𝐢.

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-πœ™-nonexpansive mappings in a real uniformly convex and uniformly smooth Banach space by the following manner: π‘₯0𝑦=π‘₯∈𝐢,𝑛=π½βˆ’1𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½π‘§π‘›ξ€Έ,𝑧𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇π‘₯𝑛+𝛾𝑛𝐽𝑆π‘₯𝑛,𝑒𝑛=πΎπ‘Ÿπ‘›π‘¦π‘›,𝐢𝑛=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛,𝑄𝑛=ξ€½π‘§βˆˆπΆβˆΆβŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯βˆ’π½π‘₯𝑛,π‘₯⟩β‰₯0𝑛+1=Ξ πΆπ‘›βˆ©π‘„π‘›π‘₯.(1.16) They proved that {π‘₯𝑛} converges strongly to π‘βˆˆπΉ(𝑇)∩𝐹(𝑆)∩Ω, where π‘βˆˆΞ πΉ(𝑇)∩𝐹(𝑆)∩Ωπ‘₯.

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-πœ™-nonexpansive mappings and improve the iterative scheme to be more general as shown in the following: π‘₯1=π‘₯∈𝐢, 𝑀𝑛=Ξ πΆπ½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛,𝑧𝑛=π½βˆ’1𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘€π‘›,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇𝑛π‘₯𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛,π‘’π‘›ξ€·π‘’βˆˆπΆsuchthat𝐹𝑛,𝑦+βŸ¨π‘Œπ‘’π‘›,π‘¦βˆ’π‘’π‘›ξ€·π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘π‘›ξ€Έ+1π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯.(1.17) By the new iterative scheme, we will prove the strong convergence theorems of the sequence {π‘₯𝑛} which could be converged to the point Ξ (βˆ©βˆžπ‘›=1𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=1𝐹(𝑆𝑛))∩Ω∩VI(𝐢,𝐡)βˆ©π΄βˆ’1(0)π‘₯. Furthermore, we propose the new better appropriate conditions of the coefficient sequences {𝛼𝑛},{𝛽𝑛},{𝛾𝑛}, and {π‘Ÿπ‘›}. 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 β€–β‹…β€–, and 𝐢 a nonempty closed convex subset of 𝐸, and let πΈβˆ— denote the dual of 𝐸. We write π‘₯𝑛⇀π‘₯ to indicate that the sequence {π‘₯𝑛} converges weakly to π‘₯ and π‘₯𝑛→π‘₯ implies that the sequence {π‘₯𝑛} converges strongly to π‘₯.

Let π‘ˆ={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any π‘₯,π‘¦βˆˆπ‘ˆ,β€–β€–β€–π‘₯≠𝑦impliesπ‘₯+𝑦2β€–β€–β€–<1.(2.1)

It is also said to be uniformly convex if for each πœ–βˆˆ(0,2], there exists 𝛿>0 such that for any π‘₯,π‘¦βˆˆπ‘ˆβ€–β€–β€–β€–π‘₯βˆ’π‘¦β€–β‰₯πœ–impliesπ‘₯+𝑦2β€–β€–β€–<1βˆ’π›Ώ.(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 π›ΏβˆΆ[0,2]β†’[0,1] defined by‖‖‖𝛿(πœ€)=inf1βˆ’π‘₯+𝑦2‖‖‖.∢π‘₯,π‘¦βˆˆπΈ,β€–π‘₯β€–=‖𝑦‖=1,β€–π‘₯βˆ’π‘¦β€–β‰₯πœ€(2.3)

Furthermore, it is said to be smooth, provided that lim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(2.4) exists for all π‘₯,π‘¦βˆˆπ‘ˆ. It is also said to be uniformly smooth if the limit is attained uniformly for π‘₯,π‘¦βˆˆπΈ.

Let 𝑝 be a fixed real number with 𝑝β‰₯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 π½π‘βˆΆπΈβ†’2πΈβˆ— is defined by𝐽𝑝π‘₯(π‘₯)=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘₯βˆ—βŸ©=β€–π‘₯‖𝑝,β€–π‘₯βˆ—β€–=β€–π‘₯β€–π‘βˆ’1ξ€Ύ,(2.5) for all π‘₯∈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 ⟨π‘₯βˆ’π‘¦,π‘₯βˆ—βˆ’π‘¦βˆ—βŸ©>0 holds for all (π‘₯,π‘₯βˆ—),(𝑦,π‘¦βˆ—)∈𝐽 with π‘₯≠𝑦,(3)if 𝐸 is reflexive, then 𝐽 is surjective,(4)if 𝐸 is uniformly convex, then it is reflexive,(5)if πΈβˆ— 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 πΈβˆ— is said to be weakly sequentially continuous [44] if π‘₯𝑛⇀π‘₯ implies 𝐽π‘₯π‘›β‡€βˆ—π½π‘₯, where β‡€βˆ— implies the weak* convergence.

Let 𝐸 be a smooth, strictly convex and reflexive Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸. Throughout this paper, we denote by πœ™ the function defined byπœ™(π‘₯,𝑦)=β€–π‘₯β€–2βˆ’2⟨π‘₯,π½π‘¦βŸ©+‖𝑦‖2,forπ‘₯,π‘¦βˆˆπΈ.(2.6)

Remark 2.1. We know the following: for each π‘₯,𝑦,π‘§βˆˆπΈ,(i)(β€–π‘₯β€–βˆ’β€–π‘¦β€–)2β‰€πœ™(π‘₯,𝑦)≀(β€–π‘₯β€–+‖𝑦‖)2,(ii)πœ™(π‘₯,𝑦)=πœ™(π‘₯,𝑧)+πœ™(𝑧,𝑦)+2⟨π‘₯βˆ’π‘§,π½π‘§βˆ’π½π‘¦βŸ©,(iii)πœ™(π‘₯,𝑦)=β€–π‘₯βˆ’π‘¦β€–2 in a real Hilbert space.

The generalized projection, introduced by Alber [45], Ξ πΆβˆΆπΈβ†’πΆ is a map that assigns to an arbitrary point π‘₯∈𝐸 the minimum point of the function πœ™(π‘₯,𝑦); that is, Π𝐢π‘₯=π‘₯, where π‘₯ is the solution to the minimization problemπœ™ξ€·ξ€Έπ‘₯,π‘₯=infπ‘¦βˆˆπΆπœ™(𝑦,π‘₯),(2.7) existence and uniqueness of the operator Π𝐢 follows from the properties of the functional πœ™(π‘₯,𝑦) and strict monotonicity of the mapping 𝐽.

If 𝐸 is a reflexive, strictly convex and smooth Banach space, then for π‘₯,π‘¦βˆˆπΈ, πœ™(π‘₯,𝑦)=0, if and only if π‘₯=𝑦. It is sufficient to show that if πœ™(π‘₯,𝑦)=0, then π‘₯=𝑦. From Remark 2.1 (i), we have β€–π‘₯β€–=‖𝑦‖. This implies that ⟨π‘₯,π½π‘¦βŸ©=β€–π‘₯β€–2=‖𝐽𝑦‖2. From the definition of 𝐽, one has 𝐽π‘₯=𝐽𝑦. Therefore, we have π‘₯=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 π‘₯,π‘¦βˆˆπΈ, one has 2β€–π‘₯βˆ’π‘¦β€–β‰€π‘2‖𝐽π‘₯βˆ’π½π‘¦β€–,(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 𝑝β‰₯2, then for all π‘₯,π‘¦βˆˆπΈ,𝐽π‘₯βˆˆπ½π‘(π‘₯) and π½π‘¦βˆˆπ½π‘(𝑦)π‘βŸ¨π‘₯βˆ’π‘¦,𝐽π‘₯βˆ’π½π‘¦βŸ©β‰₯𝑝2π‘βˆ’2𝑝‖π‘₯βˆ’π‘¦β€–π‘,(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 π‘Ÿ>0, there exists a strictly increasing, continuous and convex function π‘”βˆΆ[0,∞)β†’[0,∞) such that 𝑔(0)=0 and β€–β€–πœ†π‘₯+(1βˆ’πœ†π‘¦)2β‰€πœ†β€–π‘₯β€–2+(1βˆ’πœ†)‖𝑦‖2βˆ’πœ†(1βˆ’πœ†)𝑔(β€–π‘₯βˆ’π‘¦β€–),(2.10) for all π‘₯,π‘¦βˆˆ{π‘§βˆˆπΈβˆΆβ€–π‘§β€–β‰€π‘Ÿ} and πœ†βˆˆ[0,1].

Lemma 2.5 (Kamimura and Takahashi [50]). Let 𝐸 be a uniformly convex and smooth real Banach space and {π‘₯𝑛},{𝑦𝑛} two sequences of 𝐸. If πœ™(π‘₯𝑛,𝑦𝑛)β†’0 and either {π‘₯𝑛} or {𝑦𝑛} is bounded, then β€–π‘₯π‘›βˆ’π‘¦π‘›β€–β†’0.

Lemma 2.6 (Alber [45]). Let 𝐢 be a nonempty closed convex subset of a smooth Banach space 𝐸 and π‘₯∈𝐸. Then, π‘₯0=Π𝐢π‘₯ if and only if ⟨π‘₯0βˆ’π‘¦,𝐽π‘₯βˆ’π½π‘₯0⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.11)

Lemma 2.7 (Alber [45]). Let 𝐸 be a reflexive, strictly convex and smooth Banach space and 𝐢 a nonempty closed convex subset of 𝐸 and let π‘₯∈𝐸. Then, πœ™ξ€·π‘¦,Π𝐢π‘₯ξ€Έξ€·Ξ +πœ™πΆξ€Έπ‘₯,π‘₯β‰€πœ™(𝑦,π‘₯),βˆ€π‘¦βˆˆπΆ.(2.12)

Let 𝐸 be a strictly convex, smooth, and reflexive Banach space and 𝐽 the duality mapping from 𝐸 into πΈβˆ—. Then, π½βˆ’1 is also single-valued, one-to-one, and surjective, and it is the duality mapping from πΈβˆ— into 𝐸. Define a function π‘‰βˆΆπΈΓ—πΈβˆ—β†’β„ as follows (see [51]):𝑉π‘₯,π‘₯βˆ—ξ€Έ=β€–π‘₯β€–2βˆ’2⟨π‘₯,π‘₯βˆ—βŸ©+β€–π‘₯βˆ—β€–2,(2.13) for all π‘₯∈𝐸 and π‘₯βˆ—βˆˆπΈβˆ—. Then, it is obvious that 𝑉(π‘₯,π‘₯βˆ—)=πœ™(π‘₯,π½βˆ’1(π‘₯βˆ—)) and 𝑉(π‘₯,𝐽(𝑦))=πœ™(π‘₯,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π‘₯,π‘₯βˆ—ξ€Έξ«π½+2βˆ’1ξ€·π‘₯βˆ—ξ€Έβˆ’π‘₯,π‘¦βˆ—ξ¬ξ€·β‰€π‘‰π‘₯,π‘₯βˆ—+π‘¦βˆ—ξ€Έ,(2.14) for all π‘₯∈𝐸 and π‘₯βˆ—,π‘¦βˆ—βˆˆπΈβˆ—.

For solving the generalized mixed equilibrium problem, let us assume that the bifunction πΉβˆΆπΆΓ—πΆβ†’β„ and πœ‘βˆΆπΆβ†’β„ is convex and lower semicontinuous, satisfying the following conditions:(A1)𝐹(π‘₯,π‘₯)=0 for all π‘₯∈𝐢,(A2)𝐹 is monotone, that is, 𝐹(π‘₯,𝑦)+𝐹(𝑦,π‘₯)≀0 for all π‘₯,π‘¦βˆˆπΆ,(A3) for each π‘₯,𝑦,π‘§βˆˆπΆ, limsup𝑑↓0𝐹(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)≀𝐹(π‘₯,𝑦),(2.15)(A4) for each π‘₯∈𝐢, 𝑦↦𝐹(π‘₯,𝑦) is convex and lower semicontinuous.

Lemma 2.9 (Blum and Oettli [17]). Let 𝐢 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸 and 𝐹 a bifunction of 𝐢×𝐢 into ℝ satisfying (A1)–(A4). Let π‘Ÿ>0 and π‘₯∈𝐸. Then, there exists π‘§βˆˆπΆ such that 1𝐹(𝑧,𝑦)+π‘ŸβŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0βˆ€π‘¦βˆˆπΆ.(2.16)

Lemma 2.10 (Takahashi and Zembayashi [52]). Let 𝐢 be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸 and 𝐹 a bifunction from 𝐢×𝐢 to ℝ satisfying (A1)–(A4). For all π‘Ÿ>0 and π‘₯∈𝐸, define a mapping π‘‡π‘ŸβˆΆπΈβ†’πΆ as follows: π‘‡π‘Ÿξ‚†1π‘₯=π‘§βˆˆπΆβˆΆπΉ(𝑧,𝑦)+π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘§,π½π‘§βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ,(2.17) for all π‘₯∈𝐸. Then, the following hold:(1)π‘‡π‘Ÿ is single-valued,(2)π‘‡π‘Ÿ is a firmly nonexpansive-type mapping, that is, for all π‘₯,π‘¦βˆˆπΈ, βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π½π‘‡π‘Ÿπ‘₯βˆ’π½π‘‡π‘Ÿπ‘¦βŸ©β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,𝐽π‘₯βˆ’π½π‘¦βŸ©,(2.18)(3)𝐹(π‘‡π‘Ÿ)=𝐸𝑃(𝐹), (4)𝐸𝑃(𝐹) is closed and convex.

Lemma 2.11 (Takahashi and Zembayashi [52]). Let 𝐢 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, and 𝐹 a bifunction from 𝐢×𝐢 to ℝ satisfying (A1)–(A4) and let π‘Ÿ>0. Then, for π‘₯∈𝐸 and π‘žβˆˆπΉ(π‘‡π‘Ÿ), πœ™ξ€·π‘ž,π‘‡π‘Ÿπ‘₯𝑇+πœ™π‘Ÿξ€Έπ‘₯,π‘₯β‰€πœ™(π‘ž,π‘₯).(2.19)

Lemma 2.12 (Zhang [53]). Let 𝐢 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let π΅βˆΆπΆβ†’πΈβˆ— be a continuous and monotone mapping, πœ‘βˆΆπΆβ†’β„ convex and lower semi-continuous, and 𝐹 a bifunction from 𝐢×𝐢 to ℝ satisfying (A1)–(A4). For π‘Ÿ>0 and π‘₯∈𝐸, then there exists π‘’βˆˆπΆ such that 1𝐹(𝑒,𝑦)+βŸ¨π΅π‘’,π‘¦βˆ’π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘(𝑒)+π‘ŸβŸ¨π‘¦βˆ’π‘’,π½π‘’βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.20) Define a mapping πΎπ‘ŸβˆΆπΆβ†’πΆ as follows: πΎπ‘Ÿξ‚†1(π‘₯)=π‘’βˆˆπΆβˆΆπΉ(𝑒,𝑦)+βŸ¨π΅π‘’,π‘¦βˆ’π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘(𝑒)+π‘Ÿξ‚‡,βŸ¨π‘¦βˆ’π‘’,π½π‘’βˆ’π½π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(2.21) for all π‘₯∈𝐸. Then, the following hold:(i)πΎπ‘Ÿ is single-valued,(ii)πΎπ‘Ÿ is firmly nonexpansive, that is, for all π‘₯,π‘¦βˆˆπΈ, βŸ¨πΎπ‘Ÿπ‘₯βˆ’πΎπ‘Ÿπ‘¦,π½πΎπ‘Ÿπ‘₯βˆ’π½πΎπ‘Ÿπ‘¦βŸ©β‰€βŸ¨πΎπ‘Ÿπ‘₯βˆ’πΎπ‘Ÿπ‘¦,𝐽π‘₯βˆ’π½π‘¦βŸ©,(iii)𝐹(πΎπ‘Ÿ)=Ξ©, (iv)Ξ© is closed and convex,(v)πœ™(𝑝,πΎπ‘Ÿπ‘§)+πœ™(πΎπ‘Ÿπ‘§,𝑧)β‰€πœ™(𝑝,𝑧) for all π‘βˆˆπΉ(πΎπ‘Ÿ), π‘§βˆˆπΈ.

It follows from Lemma 2.10 that the mapping πΎπ‘ŸβˆΆπΆβ†’πΆ defined by (2.21) is a relatively nonexpansive mapping. Thus, it is quasi-πœ™-nonexpansive.

Let 𝐸 be a reflexive, strictly convex and smooth Banach space. Let 𝐢 be a closed convex subset of 𝐸. Because πœ™(π‘₯,𝑦) is strictly convex and coercive in the first variable, we know that the minimization problem infπ‘¦βˆˆπΆπœ™(π‘₯,𝑦) has a unique solution. The operator Π𝐢π‘₯∢=argminπ‘¦βˆˆπΆπœ™(π‘₯,𝑦) is said to be the generalized projection of π‘₯ on 𝐢.

Let 𝐴 be a set-valued mapping from 𝐸 to πΈβˆ— with graph 𝐺(𝐴)={(π‘₯,π‘₯βˆ—)∢π‘₯βˆ—βˆˆπ΄π‘₯}, domain 𝐷(𝐴)={π‘₯∈𝐸∢𝐴(π‘₯)β‰ βˆ…}, and range 𝑅(𝐴)={π‘₯βˆ—βˆˆπΈβˆ—βˆΆπ‘₯βˆ—βˆˆπ΄(π‘₯),π‘₯∈𝐷(𝐴)}. We denote a set-valued operator 𝐴 from 𝐸 to πΈβˆ— by π΄βŠ‚πΈΓ—πΈβˆ—. 𝐴 is said to be monotone if ⟨π‘₯βˆ’π‘¦,π‘₯βˆ—βˆ’π‘¦βˆ—βŸ©β‰₯0 for all (π‘₯,π‘₯βˆ—),(𝑦,π‘¦βˆ—)∈𝐴. A monotone operator π΄βŠ‚πΈΓ—πΈβˆ— 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 π΄βˆ’10={π‘§βˆˆπ·(𝐴)∢0βˆˆπ΄π‘§} 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+π‘Ÿπ΄)=πΈβˆ— for all π‘Ÿ>0.

Define the resolvent of 𝐴 by π½π‘Ÿπ‘₯=π‘₯π‘Ÿ. In other words, π½π‘Ÿ=(𝐽+π‘Ÿπ΄)βˆ’1𝐽 for all π‘Ÿ>0. π½π‘Ÿ is a single-valued mapping from 𝐸 to 𝐷(𝐴). Also, π΄βˆ’1(0)=𝐹(π½π‘Ÿ) for all π‘Ÿ>0, where 𝐹(π½π‘Ÿ) is the set of all fixed points of π½π‘Ÿ. Define, for π‘Ÿ>0, the Yosida approximation of 𝐴 by π΄π‘Ÿ=(π½βˆ’π½π½π‘Ÿ)/π‘Ÿ. We know that π΄π‘Ÿπ‘₯∈𝐴(π½π‘Ÿπ‘₯) for all π‘Ÿ>0 and π‘₯∈𝐸.

Lemma 2.13 (Kohsaka and Takahashi [51, Lemma  3.1]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let π΄βŠ‚πΈΓ—πΈβˆ— be a maximal monotone operator with π΄βˆ’10β‰ βˆ…, π‘Ÿ>0, and π½π‘Ÿ=(𝐽+π‘Ÿπ‘‡)βˆ’1𝐽. Then, πœ™ξ€·π‘₯,π½π‘Ÿπ‘¦ξ€Έξ€·π½+πœ™π‘Ÿξ€Έπ‘¦,π‘¦β‰€πœ™(π‘₯,𝑦)(2.22) for all π‘₯βˆˆπ΄βˆ’10 and π‘¦βˆˆπΈ.

Let 𝐡 be an inverse-strongly monotone mapping of 𝐢 into πΈβˆ— which is said to be hemicontinuous if for all π‘₯,π‘¦βˆˆπΆ, the mapping 𝐹 of [0,1] into πΈβˆ—, defined by 𝐹(𝑑)=𝐡(𝑑π‘₯+(1βˆ’π‘‘)𝑦), is continuous with respect to the weak* topology of πΈβˆ—. We define by 𝑁𝐢(𝑣) the normal cone for 𝐢 at a point π‘£βˆˆπΆ; that is,𝑁𝐢π‘₯(𝑣)=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘£βˆ’π‘¦,π‘₯βˆ—ξ€ΎβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.23)

Theorem 2.14. (Rockafellar [54]). Let 𝐢 be a nonempty, closed convex subset of a Banach space 𝐸 and 𝐡 a monotone, hemicontinuous operator of 𝐢 into πΈβˆ—. Let π‘‡βŠ‚πΈΓ—πΈβˆ— be an operator defined as follows: 𝑇𝑣=𝐡𝑣+𝑁𝐢(𝑣),π‘£βˆˆπΆ,βˆ…,π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’.(2.24) Then 𝑇, is maximal monotone and π‘‡βˆ’10=VI(𝐢,𝐡).

Lemma 2.15 (Tan and Xu [55]). Let {π‘Žπ‘›} and {𝑏𝑛} be two sequences of nonnegative real numbers satisfying π‘Žπ‘›+1β‰€π‘Žπ‘›+𝑏𝑛,βˆ€π‘›β‰₯0.(2.25) If βˆ‘βˆžπ‘›=1𝑏𝑛<∞, then limπ‘›β†’βˆžπ‘Žπ‘› 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-πœ™-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 𝐢 a nonempty closed convex subset of 𝐸. Let 𝐹 be a bifunction from 𝐢×𝐢 to ℝ satisfying (A1)–(A4), let πœ‘βˆΆπΆβ†’β„ be a proper lower semicontinuous and convex function, and let π΄βˆΆπΈβ†’πΈβˆ— be a maximal monotone operator satisfying 𝐷(𝐴)βŠ‚πΈ. Let π½π‘Ÿ=(𝐽+π‘Ÿπ΄)βˆ’1𝐽 for π‘Ÿ>0, let 𝐡 be an 𝛼-inverse-strongly monotone operator of 𝐸 into πΈβˆ—, and let π‘ŒβˆΆπΈβ†’πΈβˆ— be a continuous and monotone mapping. Let {𝑇𝑛} and {𝑆𝑛} be two families of quasi-πœ™-nonexpansive mappings of 𝐸 into itself satisfies the NSTβˆ—-condition, with Θ∢=(βˆ©βˆžπ‘›=1𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=1𝐹(𝑆𝑛))βˆ©Ξ©βˆ©π‘‰πΌ(𝐢,𝐡)βˆ©π΄βˆ’1(0)β‰ βˆ… and β€–π΅π‘¦β€–β‰€β€–π΅π‘¦βˆ’π΅π‘’β€– for all π‘¦βˆˆπΆ and π‘’βˆˆΞ˜. Let {π‘₯𝑛} be a sequence generated by π‘₯1=π‘₯∈𝐸, and 𝑀𝑛=Ξ πΆπ½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛,𝑧𝑛=π½βˆ’1𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘€π‘›,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇𝑛π‘₯𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛,π‘’π‘›ξ€·π‘’βˆˆπΆsuchthat𝐹𝑛,𝑦+βŸ¨π‘Œπ‘’π‘›,π‘¦βˆ’π‘’π‘›ξ€·π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘π‘›ξ€Έ+1π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›πΆβŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ,𝑛+1=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯,(3.1) for all π‘›βˆˆβ„•. If the coefficient sequence {𝛼𝑛},{𝛽𝑛}, {𝛾𝑛}, and {𝛿𝑛}βŠ‚[0,1], {π‘Ÿπ‘›}βŠ‚(0,∞) satisfy 𝛼𝑛+𝛽𝑛+𝛾𝑛=1, liminfπ‘›β†’βˆžπ›Όπ‘›π›½π‘›>0, liminfπ‘›β†’βˆžπ›Όπ‘›π›Ύπ‘›>0, liminfπ‘›β†’βˆžπ›Ύπ‘›(1βˆ’π›Ώπ‘›)>0, liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0 and {πœ†π‘›}βŠ‚[π‘Ž,𝑏] for some π‘Ž,𝑏 with 0<π‘Ž<𝑏<𝑐2𝛼/2, 1/𝑐 is the 2-uniformly convexity constant of 𝐸. Then the sequence {π‘₯𝑛} converges strongly to ΠΘπ‘₯.

Proof. We first show that {π‘₯𝑛} is bounded. Let π‘βˆˆΞ˜βˆΆ=(βˆ©βˆžπ‘›=1𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=1𝐹(𝑆𝑛))∩Ω∩VI(𝐢,𝐡)βˆ©π΄βˆ’1(0), and let 𝐻𝑒𝑛𝑒,𝑦=𝐹𝑛,𝑦+βŸ¨π‘Œπ‘’π‘›,π‘¦βˆ’π‘’π‘›ξ€·π‘’βŸ©+πœ‘(𝑦)βˆ’πœ‘π‘›ξ€Έ,𝐾,π‘¦βˆˆπΆπ‘Ÿπ‘›=ξ‚»ξ€·π‘’π‘’βˆˆπΆβˆΆπ»π‘›ξ€Έ+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›ξ‚Ό.⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(3.2) Put 𝑣𝑛=π½βˆ’1(𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛) and 𝑒𝑛=πΎπ‘Ÿπ‘›π‘¦π‘›.
With its relatively nonexpansiveness of π½π‘Ÿπ‘› and by Lemma 2.8, the convexity of the function 𝑉 in the second variable, we have πœ™ξ€·π‘,𝑀𝑛=πœ™π‘,Ξ πΆπ‘£π‘›ξ€Έξ€·β‰€πœ™π‘,𝑣𝑛=πœ™π‘,π½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛≀𝑉𝑝,𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛+πœ†π‘›π΅π‘₯π‘›ξ€Έξ«π½βˆ’2βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯π‘›ξ€Έβˆ’π‘,πœ†π‘›π΅π‘₯𝑛=𝑉𝑝,𝐽π‘₯π‘›ξ€Έβˆ’2πœ†π‘›βŸ¨π‘£π‘›βˆ’π‘,𝐡π‘₯π‘›βŸ©ξ€·=πœ™π‘,π‘₯π‘›ξ€Έβˆ’2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘,𝐡π‘₯π‘›βŸ©+2βŸ¨π‘£π‘›βˆ’π‘₯𝑛,βˆ’πœ†π‘›π΅π‘₯π‘›βŸ©.(3.3) Since π‘βˆˆVI(𝐢,𝐡) and 𝐡 is 𝛼-inverse-strongly monotone, we consider βˆ’2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘,𝐡π‘₯π‘›βŸ©=βˆ’2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘,𝐡π‘₯π‘›βˆ’π΅π‘βŸ©βˆ’2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘,π΅π‘βŸ©β‰€βˆ’2π›Όπœ†π‘›β€–β€–π΅π‘₯π‘›β€–β€–βˆ’π΅π‘2.(3.4) Therefore, by Lemma 2.2, we obtain 2βŸ¨π‘£π‘›βˆ’π‘₯𝑛,βˆ’πœ†π‘›π΅π‘₯π‘›ξ«π½βŸ©=2βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯π‘›ξ€Έβˆ’π½βˆ’1𝐽π‘₯𝑛,βˆ’πœ†π‘›π΅π‘₯𝑛‖‖𝐽≀2βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯π‘›ξ€Έβˆ’π½βˆ’1𝐽π‘₯π‘›ξ€Έβ€–β€–β€–β€–πœ†π‘›π΅π‘₯𝑛‖‖≀4𝑐2‖‖𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–β€–β€–πœ†π‘›π΅π‘₯𝑛‖‖=4𝑐2πœ†2𝑛‖‖𝐡π‘₯𝑛‖‖2≀4𝑐2πœ†2𝑛‖‖𝐡π‘₯π‘›β€–β€–βˆ’π΅π‘2.(3.5) We can rewrite (3.3), which yield that πœ™ξ€·π‘,π‘€π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έβˆ’2π›Όπœ†π‘›β€–β€–π΅π‘₯π‘›β€–β€–βˆ’π΅π‘2+4𝑐2πœ†2𝑛‖‖𝐡π‘₯π‘›β€–β€–βˆ’π΅π‘2ξ€·β‰€πœ™π‘,π‘₯𝑛+2πœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Όπ΅π‘₯π‘›β€–β€–βˆ’π΅π‘2ξ€·β‰€πœ™π‘,π‘₯𝑛.(3.6) Apply the Lemma 2.8, Lemma 2.13 and (3.6), we consider πœ™ξ€·π‘,𝑧𝑛=πœ™π‘,π½βˆ’1𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘€π‘›ξ€·ξ€Έξ€Έξ€Έ=𝑉𝑝,𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘€π‘›ξ€Έξ€Έβ‰€π›Ώπ‘›π‘‰ξ€·ξ€·π‘₯𝑝,𝐽𝑛+ξ€·ξ€Έξ€Έ1βˆ’π›Ώπ‘›ξ€Έπ‘‰ξ€·ξ€·π½π‘,π½π‘Ÿπ‘›π‘€π‘›ξ€Έξ€Έ=π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘,π½π‘Ÿπ‘›π‘€π‘›ξ€Έβ‰€π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘€π‘›ξ€Έξ€·π½βˆ’πœ™π‘Ÿπ‘›π‘€π‘›,π‘€π‘›ξ€Έξ€Έβ‰€π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘,π‘€π‘›ξ€Έβ‰€π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛=πœ™π‘,π‘₯𝑛,(3.7) hence, we obtain πœ™ξ€·π‘,𝑦𝑛=πœ™π‘,π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇𝑛π‘₯𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛=‖𝑝‖2βˆ’2π›Όπ‘›βŸ¨π‘,𝐽π‘₯π‘›βŸ©βˆ’2π›½π‘›βŸ¨π‘,𝐽𝑇𝑛π‘₯π‘›βŸ©βˆ’2π›Ύπ‘›βŸ¨π‘,π½π‘†π‘›π‘§π‘›βŸ©+‖‖𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇𝑛π‘₯𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛‖‖2≀‖𝑝‖2βˆ’2π›Όπ‘›βŸ¨π‘,𝐽π‘₯π‘›βŸ©βˆ’2π›½π‘›βŸ¨π‘,𝐽𝑇𝑛π‘₯π‘›βŸ©βˆ’2π›Ύπ‘›βŸ¨π‘,π½π‘†π‘›π‘§π‘›βŸ©+𝛼𝑛‖‖𝐽π‘₯𝑛‖‖2+𝛽𝑛‖‖𝐽𝑇𝑛π‘₯𝑛‖‖2+𝛾𝑛‖‖𝐽𝑆𝑛𝑧𝑛‖‖2=π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,𝑇𝑛π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘†π‘›π‘§π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘§π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘₯𝑛=πœ™π‘,π‘₯𝑛.(3.8) By (3.1), again, πœ™ξ€·π‘,𝑒𝑛=πœ™π‘,πΎπ‘Ÿπ‘›π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘,π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯𝑛.(3.9) This shows that π‘βˆˆπΆπ‘›+1. Consequently, Ξ˜βŠ‚πΆπ‘›, for all π‘›βˆˆβ„•.
Next, we show that limπ‘›β†’βˆžπœ™(π‘₯𝑛,π‘₯0) exists. Since π‘₯𝑛=Π𝐢𝑛π‘₯, it follows from Lemma 2.7 that πœ™ξ€·π‘₯𝑛,π‘₯β‰€πœ™(𝑝,π‘₯)βˆ’πœ™π‘,π‘₯π‘›ξ€Έβ‰€πœ™(𝑝,π‘₯),(3.10) for each π‘βˆˆΞ˜βŠ‚πΆπ‘›. Then, πœ™(π‘₯𝑛,π‘₯) is bounded. It implies that {π‘₯𝑛} is bounded and {𝑦𝑛}, {𝑧𝑛}, {𝑀𝑛}, and {π½π‘Ÿπ‘›π‘€π‘›} are also bounded.
From π‘₯𝑛=Π𝐢𝑛π‘₯ and π‘₯𝑛+1βˆˆπΆπ‘›+1βŠ‚πΆπ‘›, we have πœ™ξ€·π‘₯𝑛π‘₯,π‘₯β‰€πœ™π‘›+1ξ€Έ,π‘₯,βˆ€π‘›βˆˆβ„•.(3.11) Therefore, {πœ™(π‘₯𝑛,π‘₯)} is nondecreasing. It follows that the limit of {πœ™(π‘₯𝑛,π‘₯)} exists, and from Lemma 2.7, we have πœ™ξ€·π‘₯𝑛+1,π‘₯𝑛π‘₯=πœ™π‘›+1,Π𝐢𝑛π‘₯ξ€Έξ€·π‘₯β‰€πœ™π‘›+1ξ€Έξ€·Ξ ,π‘₯βˆ’πœ™πΆπ‘›ξ€Έξ€·π‘₯π‘₯,π‘₯=πœ™π‘›+1ξ€Έξ€·π‘₯,π‘₯βˆ’πœ™π‘›ξ€Έ,,π‘₯(3.12) for all π‘›βˆˆβ„•. Thus, we have limπ‘›β†’βˆžπœ™ξ€·π‘₯𝑛+1,π‘₯𝑛=0.(3.13) Since π‘₯𝑛+1=Π𝐢𝑛+1π‘₯βˆˆπΆπ‘›+1, it follows from the definition of 𝐢𝑛+1 that πœ™ξ€·π‘₯𝑛+1,𝑒𝑛π‘₯β‰€πœ™π‘›+1,π‘₯π‘›ξ€ΈβŸΆ0.(3.14) By Lemma 2.5, (3.13), and (3.14), we note that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘’π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–=0.(3.15) Since 𝐽 is uniformly norm-to-norm continuous on the bounded set, we obtain limπ‘›β†’βˆžβ€–β€–π½π‘₯𝑛+1βˆ’π½π‘’π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π½π‘₯𝑛+1βˆ’π½π‘₯𝑛‖‖=limπ‘›β†’βˆžβ€–β€–π½π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–=0.(3.16) Since π‘₯π‘š=Ξ πΆπ‘šβŠ‚πΆπ‘› for any positive integer π‘šβ‰₯𝑛, it follows from Lemma 2.7 that πœ™ξ€·π‘₯π‘š,π‘₯𝑛π‘₯=πœ™π‘š,Π𝐢𝑛π‘₯𝑛π‘₯β‰€πœ™π‘šξ€Έξ€·Ξ ,π‘₯βˆ’πœ™πΆπ‘›π‘₯𝑛π‘₯,π‘₯=πœ™π‘šξ€Έξ€·π‘₯,π‘₯βˆ’πœ™π‘›ξ€Έ.,π‘₯(3.17) Taking π‘š,π‘›β†’βˆž, we have πœ™(π‘₯π‘š,π‘₯𝑛)β†’0 as π‘›β†’βˆž. It follows from Lemma 2.5, that β€–π‘₯π‘šβˆ’π‘₯𝑛‖→0 as π‘š,π‘›β†’βˆž. Hence, {π‘₯𝑛} is a Cauchy sequence. Since 𝐸 is a Banach space and 𝐢 is closed and convex, we can assume that π‘₯π‘›β†’π‘’βˆˆπΆ as π‘›β†’βˆž.
Next, we show that π‘’βˆˆ(βˆ©βˆžπ‘›=1𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=1𝐹(𝑆𝑛)).
Since 𝐸 is a uniformly smooth Banach space, we know that πΈβˆ— is a uniformly convex Banach space. Let π‘Ÿ=supπ‘›βˆˆβ„•{β€–π‘₯𝑛‖,‖𝑇𝑛π‘₯𝑛‖,‖𝑆𝑛𝑧𝑛‖}. From Lemma 2.4, we have πœ™ξ€·π‘,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘,𝑦𝑛=πœ™π‘,π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇𝑛π‘₯𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛=‖𝑝‖2βˆ’2π›Όπ‘›βŸ¨π‘,𝐽π‘₯π‘›βŸ©βˆ’2π›½π‘›βŸ¨π‘,𝐽𝑇𝑛π‘₯π‘›βŸ©βˆ’2π›Ύπ‘›βŸ¨π‘,π½π‘†π‘›π‘§π‘›βŸ©+‖‖𝛼𝑛𝐽π‘₯𝑛+𝛽𝑛𝐽𝑇𝑛π‘₯𝑛+𝛾𝑛𝐽𝑆𝑛𝑧𝑛‖‖2≀‖𝑝‖2βˆ’2π›Όπ‘›βŸ¨π‘,𝐽π‘₯π‘›βŸ©βˆ’2π›½π‘›βŸ¨π‘,𝐽𝑇𝑛π‘₯π‘›βŸ©βˆ’2π›Ύπ‘›βŸ¨π‘,π½π‘†π‘›π‘§π‘›βŸ©+𝛼𝑛‖‖𝐽π‘₯𝑛‖‖2+𝛽𝑛‖‖𝐽𝑇𝑛π‘₯𝑛‖‖2+𝛾𝑛‖‖𝐽𝑆𝑛𝑧𝑛‖‖2βˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯𝑛‖‖=π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,𝑇𝑛π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘†π‘›π‘§π‘›ξ€Έβˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘§π‘›ξ€Έβˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘₯π‘›ξ€Έβˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯𝑛‖‖=πœ™π‘,π‘₯π‘›ξ€Έβˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯𝑛‖‖.(3.18) This implies that 𝛼𝑛𝛽𝑛𝑔‖‖𝐽𝑇𝑛π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛.(3.19) On the other hand, we have πœ™ξ€·π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛=β€–β€–π‘₯𝑛‖‖2βˆ’β€–β€–π‘’π‘›β€–β€–2βˆ’2βŸ¨π‘,𝐽π‘₯π‘›βˆ’π½π‘’π‘›βŸ©=β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–ξ€·β€–β€–π‘₯𝑛‖‖+‖‖𝑒𝑛‖‖‖‖+2‖𝑝‖𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–.(3.20) Noticing (3.15) and (3.16), we obtain πœ™ξ€·π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,π‘’π‘›ξ€ΈβŸΆ0,asπ‘›βŸΆβˆž.(3.21) Since liminfπ‘›β†’βˆžπ›Όπ‘›π›½π‘›>0 and (3.21), it follows from (3.19) that 𝑔‖‖𝐽𝑇𝑛π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€ΈβŸΆ0,asπ‘›βŸΆβˆž.(3.22) It follows from the property of 𝑔 that ‖‖𝐽𝑇𝑛π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–βŸΆ0,asπ‘›βŸΆβˆž.(3.23) Since π½βˆ’1 is uniformly norm-to-norm continuous on bounded sets, we see that limπ‘›β†’βˆžβ€–β€–π‘‡π‘›π‘₯π‘›βˆ’π‘₯𝑛‖‖=0.(3.24) Similarly, using the condition limsupπ‘›β†’βˆžπ›Όπ‘›π›Ύπ‘›>0, one can obtain limπ‘›β†’βˆžβ€–β€–π‘†π‘›π‘§π‘›βˆ’π‘₯𝑛‖‖=0.(3.25) By (3.6), (3.8), and (3.18), we have πœ™ξ€·π‘,π‘’π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘§π‘›ξ€Έβˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›ξ€Ίπ›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘,π‘€π‘›ξ€Έξ€»βˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘,π‘€π‘›ξ€Έβˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έξ‚ƒπœ™ξ€·π‘,π‘₯𝑛+2πœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Όπ΅π‘₯π‘›β€–β€–βˆ’π΅π‘2ξ‚„βˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›π›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛+𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έ2πœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Όπ΅π‘₯π‘›β€–β€–βˆ’π΅π‘2βˆ’π›Όπ‘›π›½π‘›π‘”ξ€·β€–β€–π½π‘‡π‘›π‘₯π‘›βˆ’π½π‘₯π‘›β€–β€–ξ€Έξ€·β‰€πœ™π‘,π‘₯𝑛+𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έ2πœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Όπ΅π‘₯π‘›β€–β€–βˆ’π΅π‘2.(3.26) This implies that 2πœ†π‘›ξ‚€2π›Όβˆ’π‘2πœ†π‘›ξ‚β€–β€–π΅π‘₯π‘›β€–β€–βˆ’π΅π‘2≀1𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έξ€Ίπœ™ξ€·π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛.ξ€Έξ€»(3.27) By assumption, liminfπ‘›β†’βˆžπ›Ύπ‘›(1βˆ’π›Ώπ‘›)>0 and (3.21), we get that limπ‘›β†’βˆžβ€–β€–π΅π‘₯π‘›β€–β€–βˆ’π΅π‘=0.(3.28) From Lemma 2.7, Lemma 2.8, and (3.5), we have πœ™ξ€·π‘₯𝑛,𝑀𝑛π‘₯=πœ™π‘›,Π𝐢𝑣𝑛π‘₯β‰€πœ™π‘›,𝑣𝑛π‘₯=πœ™π‘›,π½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛π‘₯ξ€Έξ€Έ=𝑉𝑛,𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛π‘₯≀𝑉𝑛,𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛+πœ†π‘›π΅π‘₯π‘›ξ€Έξ«π½βˆ’2βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯π‘›ξ€Έβˆ’π‘₯𝑛,πœ†π‘›π΅π‘₯𝑛π‘₯=𝑉𝑛,𝐽π‘₯π‘›ξ€Έβˆ’2πœ†π‘›βŸ¨π‘£π‘›βˆ’π‘₯𝑛,𝐡π‘₯π‘›βŸ©ξ€·π‘₯=πœ™π‘›,π‘₯𝑛+2βŸ¨π‘£π‘›βˆ’π‘₯𝑛,πœ†π‘›π΅π‘₯π‘›βŸ©β‰€4𝑐2πœ†2𝑛‖‖𝐡π‘₯π‘›β€–β€–βˆ’π΅π‘2.(3.29) By Lemma 2.8 and Lemma 2.13, we have πœ™ξ€·π‘₯𝑛,𝑧𝑛π‘₯=πœ™π‘›,π½βˆ’1𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘€π‘›ξ€·π‘₯ξ€Έξ€Έξ€Έ=𝑉𝑛,𝛿𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘€π‘›ξ€Έξ€Έβ‰€π›Ώπ‘›π‘‰ξ€·π‘₯𝑛,𝐽π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘‰ξ€·π‘₯𝑛𝐽,π½π‘Ÿπ‘›π‘€π‘›ξ€Έξ€Έ=π›Ώπ‘›πœ™ξ€·π‘₯𝑛,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘₯𝑛,π½π‘Ÿπ‘›π‘€π‘›ξ€Έ=π›Ώπ‘›πœ™ξ€·π‘₯𝑛,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›πœ™ξ€·π‘₯𝑛,π‘€π‘›ξ€Έξ€·π½βˆ’πœ™π‘Ÿπ‘›π‘€π‘›,𝑀𝑛=ξ€·ξ€Έξ€Έ1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π‘₯𝑛,𝑀𝑛≀1βˆ’π›Ώπ‘›ξ€Έ4𝑐2πœ†2𝑛‖‖𝐡π‘₯π‘›β€–β€–βˆ’π΅π‘2.(3.30) From Lemma 2.5 and (3.28), we obtain limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘€π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘§π‘›β€–β€–=0.(3.31) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that limπ‘›β†’βˆžβ€–β€–π½π‘₯π‘›βˆ’π½π‘€π‘›β€–β€–=limπ‘›β†’βˆžβ€–β€–π½π‘₯π‘›βˆ’π½π‘§π‘›β€–β€–=0.(3.32) Since π‘₯𝑛→𝑒 as π‘›β†’βˆž, 𝑧𝑛→𝑒 as π‘›β†’βˆž. Combining (3.15), (3.25), and (3.28), we also obtain β€–β€–π‘†π‘›π‘§π‘›βˆ’π‘§π‘›β€–β€–β‰€β€–β€–π‘†π‘›π‘§π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘§π‘›β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(3.33) By (3.15) and (3.31), we obtain that ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–β‰€β€–β€–π‘§π‘›+1βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘§π‘›β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(3.34) By (3.15), (3.24), (3.33), and (3.34), and {𝑇𝑛}, {𝑆𝑛} satisfies the NST*-condition and π‘₯𝑛→𝑝, then we have π‘βˆˆ(βˆ©βˆžπ‘›=1𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=1𝐹(𝑆𝑛)).
Since {π‘₯𝑛} is bounded, there exists a subsequence {π‘₯𝑛𝑖} of {π‘₯𝑛} such that π‘₯π‘›π‘–β‡€π‘’βˆˆπΆ. It follows from (3.31) that we have 𝑀𝑛𝑖⇀𝑒 as π‘–β†’βˆž. Next, we show that π‘’βˆˆπ΄βˆ’10.
By (3.6), (3.8), and (3.9), we obtain πœ™ξ€·π‘,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘,π‘¦π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›πœ™ξ€·π‘,π‘§π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›ξ€Ίπ›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘€π‘›ξ€Έξ€·π½βˆ’πœ™π‘Ÿπ‘›π‘€π‘›,π‘€π‘›ξ€Έξ€Έξ€»β‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯𝑛+π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+π›Ύπ‘›ξ€Ίπ›Ώπ‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Ώπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘₯π‘›ξ€Έξ€·π½βˆ’πœ™π‘Ÿπ‘›π‘€π‘›,π‘€π‘›ξ€·ξ€Έξ€Έξ€»β‰€πœ™π‘,π‘₯π‘›ξ€Έβˆ’π›Ύπ‘›ξ€·1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π½π‘Ÿπ‘›π‘€π‘›,𝑀𝑛.(3.35) This implies that 𝛾𝑛1βˆ’π›Ώπ‘›ξ€Έπœ™ξ€·π½π‘Ÿπ‘›π‘€π‘›,π‘€π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛.(3.36) By (3.21), we have limπ‘›β†’βˆžβ€–β€–π½π‘Ÿπ‘›π‘€π‘›βˆ’π‘€π‘›β€–β€–=0.(3.37) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we note that limπ‘›β†’βˆžβ€–β€–π½π½π‘Ÿπ‘›π‘€π‘›βˆ’π½π‘€π‘›β€–β€–=0.(3.38) Indeed, since liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0, it follows from (3.38) that limπ‘›β†’βˆžβ€–β€–π΄π‘Ÿπ‘›π‘€π‘›β€–β€–=limπ‘›β†’βˆž1π‘Ÿπ‘›β€–β€–π½π‘€π‘›ξ€·π½βˆ’π½π‘Ÿπ‘›π‘€π‘›ξ€Έβ€–β€–=0.(3.39) If (𝑀,π‘€βˆ—)∈𝐴, then it holds from the monotonicity of 𝐴 that ξ‚¬π‘€βˆ’π‘€π‘›π‘–,π‘€βˆ—βˆ’π΄π‘Ÿπ‘›π‘–π‘€π‘›π‘–ξ‚­β‰₯0,(3.40) for all π‘–βˆˆβ„•. Letting π‘–β†’βˆž, we get βŸ¨π‘€βˆ’π‘’,π‘€βˆ—βŸ©β‰₯0. Then, the maximality of 𝐴 implies π‘’βˆˆπ΄βˆ’10.
Next, we show that π‘’βˆˆVI(𝐢,𝐡). Let πΎβŠ‚πΈΓ—πΈβˆ— be an operator as follows: 𝐾𝑣=𝐡𝑣+𝑁𝐢(𝑣),π‘£βˆˆπΆ,βˆ…,otherwise.(3.41) By Theorem 2.14, 𝐾 is maximal monotone and πΎβˆ’10=VI(𝐢,𝐡).
Let (𝑣,𝑀)∈𝐺(𝐾). Since π‘€βˆˆπΎπ‘£=𝐡𝑣+𝑁𝐢(𝑣), we get π‘€βˆ’π΅π‘£βˆˆπ‘πΆ(𝑣). From π‘€π‘›βˆˆπΆ, we have βŸ¨π‘£βˆ’π‘€π‘›,π‘€βˆ’πΎπ‘£βŸ©β‰₯0.(3.42) On the other hand, since 𝑀𝑛=Ξ πΆπ½βˆ’1(𝐽π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛), then by Lemma 2.6, we have ξ«π‘£βˆ’π‘€π‘›,π½π‘€π‘›βˆ’ξ€·π½π‘₯π‘›βˆ’πœ†π‘›π΅π‘₯𝑛β‰₯0,(3.43) thus ξƒ‘π‘£βˆ’π‘€π‘›,𝐽π‘₯π‘›βˆ’π½π‘€π‘›πœ†π‘›βˆ’π΅π‘₯𝑛≀0.(3.44) It follows from (3.42) and (3.44) that βŸ¨π‘£βˆ’π‘€π‘›,π‘€βŸ©β‰₯βŸ¨π‘£βˆ’π‘€π‘›,π΅π‘£βŸ©β‰₯βŸ¨π‘£βˆ’π‘€π‘›ξƒ‘,π΅π‘£βŸ©+π‘£βˆ’π‘€π‘›,𝐽π‘₯π‘›βˆ’π½π‘€π‘›πœ†π‘›βˆ’π΅π‘₯𝑛=βŸ¨π‘£βˆ’π‘€π‘›,π΅π‘£βˆ’π΅π‘₯π‘›ξƒ‘βŸ©+π‘£βˆ’π‘€π‘›,𝐽π‘₯π‘›βˆ’π½π‘€π‘›πœ†π‘›ξƒ’=βŸ¨π‘£βˆ’π‘€π‘›,π΅π‘£βˆ’π΅π‘€π‘›βŸ©+βŸ¨π‘£βˆ’π‘€π‘›,π΅π‘€π‘›βˆ’π΅π‘₯π‘›ξƒ‘βŸ©+π‘£βˆ’π‘€π‘›,𝐽π‘₯π‘›βˆ’π½π‘€π‘›πœ†π‘›ξƒ’β€–β€–β‰₯βˆ’π‘£βˆ’π‘€π‘›β€–β€–β€–β€–π‘€π‘›βˆ’π‘₯π‘›β€–β€–π›Όβˆ’β€–β€–π‘£βˆ’π‘€π‘›β€–β€–β€–β€–π½π‘₯π‘›βˆ’π½π‘€π‘›β€–β€–π‘Žξ‚΅β€–β€–π‘€β‰₯βˆ’π‘€π‘›βˆ’π‘₯𝑛‖‖𝛼+‖‖𝐽π‘₯π‘›βˆ’π½π‘€π‘›β€–β€–π‘Žξ‚Ά,(3.45) where 𝑀=sup𝑛β‰₯1{β€–π‘£βˆ’π‘€π‘›β€–}. From (3.31) and (3.32), we obtain βŸ¨π‘£βˆ’π‘’,π‘€βŸ©β‰₯0. By the maximality of 𝐾, we have π‘’βˆˆπΎβˆ’10 and hence π‘’βˆˆVI(𝐢,𝐡).
Next, we show that π‘’βˆˆΞ©. From 𝑒𝑛=πΎπ‘Ÿπ‘›π‘¦π‘› and Lemma 2.12, we obtain πœ™ξ€·π‘’π‘›,𝑦𝑛𝐾=πœ™π‘Ÿπ‘›π‘¦π‘›,π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘¦π‘›ξ€Έξ€·βˆ’πœ™π‘’,πΎπ‘Ÿπ‘›π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,πΎπ‘Ÿπ‘›π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,𝑒𝑛.(3.46) On the other hand, we have πœ™ξ€·π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,𝑒𝑛=β€–β€–π‘₯𝑛‖‖2βˆ’β€–β€–π‘’π‘›β€–β€–2βˆ’2βŸ¨π‘’,𝐽π‘₯π‘›βˆ’π½π‘’π‘›βŸ©=β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–ξ€·β€–β€–π‘₯𝑛‖‖+‖‖𝑒𝑛‖‖‖‖+2‖𝑒‖𝐽π‘₯π‘›βˆ’π½π‘’π‘›β€–β€–.(3.47) Noticing (3.15) and (3.16), we obtain πœ™ξ€·π‘’,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘’,π‘’π‘›ξ€ΈβŸΆ0,asπ‘›βŸΆβˆž.(3.48) It follows that πœ™ξ€·π‘’π‘›,π‘¦π‘›ξ€ΈβŸΆ0,asπ‘›βŸΆβˆž.(3.49) By Lemma 2.5, we have limπ‘›β†’βˆžβ€–β€–π‘’π‘›βˆ’π‘¦π‘›β€–β€–=0.(3.50) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we get limπ‘›β†’βˆžβ€–β€–π½π‘’π‘›βˆ’π½π‘¦π‘›β€–β€–=0.(3.51) From the assumption liminfπ‘›β†’βˆžπ‘Ÿπ‘›>π‘Ž, we get limπ‘›β†’βˆžβ€–β€–π½π‘’π‘›βˆ’π½π‘¦π‘›β€–β€–π‘Ÿπ‘›=0.(3.52) Noticing that 𝑒𝑛=πΎπ‘Ÿπ‘›π‘¦π‘›, we have 𝐻𝑒𝑛+1,π‘¦π‘Ÿπ‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.53) Hence, 𝐻𝑒𝑛𝑖+1,π‘¦π‘Ÿπ‘›π‘–ξ«π‘¦βˆ’π‘’π‘›π‘–,π½π‘’π‘›π‘–βˆ’π½π‘¦π‘›π‘–ξ¬β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.54) From the (A2), we note that β€–β€–π‘¦βˆ’π‘’π‘›π‘–β€–β€–β€–β€–π½π‘’π‘›π‘–βˆ’π½π‘¦π‘›π‘–β€–β€–π‘Ÿπ‘›π‘–β‰₯1π‘Ÿπ‘›π‘–ξ«π‘¦βˆ’π‘’π‘›π‘–,π½π‘’π‘›π‘–βˆ’π½π‘¦π‘›π‘–ξ¬ξ€·π‘’β‰₯βˆ’π»π‘›π‘–ξ€Έξ€·,𝑦β‰₯𝐻𝑦,𝑒𝑛𝑖,βˆ€π‘¦βˆˆπΆ.(3.55) Taking the limit as π‘›β†’βˆž in the above inequality, and from (A4) and 𝑒𝑛𝑖⇀𝑒, we have 𝐻(𝑦,𝑒)≀0,forallπ‘¦βˆˆπΆ. For 0<𝑑<1 and π‘¦βˆˆπΆ, define 𝑦𝑑=𝑑𝑦+(1βˆ’π‘‘)𝑒. Noticing that 𝑦,π‘’βˆˆπΆ, we obtain π‘¦π‘‘βˆˆπΆ, which yields that 𝐻(𝑦𝑑,𝑒)≀0. It follows from (A1) that 𝑦0=𝐻𝑑,𝑦𝑑𝑦≀𝑑𝐻𝑑+𝑦,𝑦(1βˆ’π‘‘)𝐻𝑑𝑦,𝑒≀𝑑𝐻𝑑,𝑦.(3.56) That is, 𝐻(𝑦𝑑,𝑦)β‰₯0.
Let 𝑑↓0, from (A3), we obtain 𝐻(𝑒,𝑦)β‰₯0,forallπ‘¦βˆˆπΆ. This implies that π‘’βˆˆΞ©. Hence, π‘’βˆˆΞ˜βˆΆ=(βˆ©βˆžπ‘›=1𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=1𝐹(𝑆𝑛))βˆ©Ξ©βˆ©π‘‰πΌ(𝐢,𝐡)βˆ©π΄βˆ’1(0).
Finally, we show that 𝑒=ΠΘπ‘₯. Indeed, from π‘₯𝑛=Π𝐢𝑛π‘₯ and Lemma 2.6, we have ⟨𝐽π‘₯βˆ’π½π‘₯𝑛,π‘₯𝑛