Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 712651 | https://doi.org/10.1155/2012/712651

Yaqin Wang, "A New Hybrid Method for Equilibrium Problems, Variational Inequality Problems, Fixed Point Problems, and Zero of Maximal Monotone Operators", Journal of Applied Mathematics, vol. 2012, Article ID 712651, 21 pages, 2012. https://doi.org/10.1155/2012/712651

A New Hybrid Method for Equilibrium Problems, Variational Inequality Problems, Fixed Point Problems, and Zero of Maximal Monotone Operators

Academic Editor: Ya Ping Fang
Received05 Aug 2011
Accepted10 Oct 2011
Published08 Dec 2011

Abstract

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality, the set of solutions of the generalized mixed equilibrium problem, and zeros of maximal monotone operators in a Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. The results obtained in this paper improve and extend the result of Zeng et al. (2010) and many others.

1. Introduction

In 1994, Blum and Oettli [1] introduced equilibrium problems, which have had a great impact and influence on the development of several branches of pure and applied sciences. It has been shown that the equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, physics, image reconstruction, ecology, transportation, network, elasticity, and optimization.

Let 𝐸 be a real Banach space, 𝐸 the dual space of 𝐸, and 𝐶 a nonempty closed convex subset of 𝐸. Let Θ𝐶×𝐶𝑅 be a bifunction and 𝜑𝐶𝑅 a real-valued function. The generalized mixed equilibrium problem (GMEP) of finding 𝑥𝐶 is such that Θ(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)+𝐴𝑥,𝑦𝑥0,𝑦𝐶.(1.1) Recently, Zhang [2] considered this problem. Here some special cases of problem (1.1) are stated as follows.

If 𝐴=0, then problem (1.1) reduces to the following mixed equilibrium problem of finding 𝑥𝐶 such that Θ(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶,(1.2) which was considered by Ceng and Yao [3]. The set of solutions of this problem is denoted by MEP.

If 𝜑=0, then problem (1.1) reduces to the following generalized equilibrium problem of finding 𝑥𝐶 such that Θ(𝑥,𝑦)+𝐴𝑥,𝑦𝑥0,𝑦𝐶,(1.3) which was studied by S. Takahashi and W. Takahashi [4].

If 𝜑=0 and 𝐴=0, then problem (1.1) reduces to the following equilibrium problem of finding 𝑥𝐶 such that Θ(𝑥,𝑦)0,𝑦𝐶.(1.4) The set of solutions of problem (1.4) is denoted by EP.

If Θ=0,𝜑=0, then problem (1.1) reduces to the following classical variational inequality problem of finding 𝑥𝐶 such that 𝐴𝑥,𝑦𝑥0,𝑦𝐶.(1.5) The set of solutions of problem (1.5) is denoted by VI(𝐶,𝐴).

The problem (1.1) is very general in the sense that it includes, as special cases, numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instance, [1, 37].

The normalized duality mapping from 𝐸 to 2𝐸 is defined by 𝐽𝑥=𝑓𝐸𝑥,𝑓=𝑥2=𝑓2,𝑥𝐸,(1.6) where , denotes the generalized duality pairing. It is well known that if 𝐸 is smooth then 𝐽 is single valued and if 𝐸 is uniformly smooth then 𝐽 is uniformly continuous on bounded subsets of 𝐸. Moreover, if 𝐸 is a reflexive and strictly convex Banach space with a strictly convex dual, then 𝐽1 is single valued, one to one, surjective, and it is the duality mapping from 𝐸 into 𝐸 and thus 𝐽𝐽1=𝐼𝐸 and 𝐽1𝐽=𝐼𝐸 (see [8]).

On the other hand, let 𝑊𝐸𝐸 be a set-valued mapping. The problem of finding 𝑣𝐸 satisfying 0𝑊𝑣 contains numerous problems in economics, optimization, and physics. Such 𝑣𝐸 is called a zero point of 𝑊.

A set-valued mapping 𝑊𝐸𝐸 with graph 𝐺(𝑊)={(𝑥,𝑥)𝑥𝑊𝑥}, domain 𝐷(𝑊)={𝑥𝐸𝑊𝑥}, and range 𝑅(𝑊)={𝑊𝑥𝑥𝐷(𝑊)} is said to be monotone if 𝑥𝑦,𝑥𝑦0 for all 𝑥𝑊𝑥,𝑦𝑊𝑦. 𝑊 is said to be maximal monotone if the graph 𝐺(𝑊) of 𝑊 is not properly contained in the graph of any other monotone operator. It is known that 𝑊 is a maximal monotone if and only if 𝑅(𝐽+𝑟𝑊)=𝐸 for all 𝑟>0 when 𝐸 is a reflexive, strictly convex, and smooth Banach space (see [9]).

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑊𝐸𝐸 be a monotone operator satisfying 𝐷(𝑊)𝐶𝐽1(𝑟>0𝑅(𝐽+𝑟𝑊)). Then the resolvent of 𝑊 defined by 𝐽𝑟=(𝐽+𝑟𝑊)1𝐽 is a single-valued mapping from 𝐸 to 𝐷(𝑊) for all 𝑟>0. For 𝑟>0, the Yosida approximation of 𝑊 is defined by 𝑊𝑟𝑥=(𝐽𝑥𝐽𝐽𝑟𝑥)/𝑟 for all 𝑥𝐸.

A mapping 𝐴𝐶𝐸 is said to be monotone if, for each 𝑥,𝑦𝐶, 𝑥𝑦,𝐴𝑥𝐴𝑦0.(1.7)𝐴 is said to be 𝛾-inverse strongly monotone if there exists a positive real number 𝛾>0 such that 𝑥𝑦,𝐴𝑥𝐴𝑦𝛾𝐴𝑥𝐴𝑦2,𝑥,𝑦𝐶.(1.8) If 𝐴 is 𝛾-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛾, that is, 1𝐴𝑥𝐴𝑦𝛾𝑥𝑦,𝑥,𝑦𝐶.(1.9)

Let 𝐸 be a smooth Banach space. The function 𝜙𝐸×𝐸𝑅 defined by 𝜙(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2,𝑥,𝑦𝐸,(1.10) is studied by Alber [10], Kamimura and Takahashi [11], and Reich [12]. It follows from the definition of the function 𝜙 that ()𝑥𝑦2)𝜙(𝑥,𝑦)(𝑥+𝑦2,𝑥,𝑦𝐸.(1.11) Observe that, in a Hilbert space 𝐻, 𝜙(𝑥,𝑦)=𝑥𝑦2,forall𝑥,𝑦𝐻.

Lemma 1.1 (see [10]). Let 𝐶 be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space 𝐸, and let 𝑥𝐸. Then there exists a unique element 𝑥0𝐶 such that 𝜙(𝑥0,𝑥)=min{𝜙(𝑧,𝑥)𝑧𝐶}.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝐶 a nonempty closed and convex subset of 𝐸. The generalized projection mapping, introduced by Alber [10], is a mapping Π𝐶𝐸𝐶 that assigns to an arbitrary point 𝑥𝐸, the minimum point of the functional 𝜙(𝑥,𝑦), that is, Π𝐶𝑥=𝑥0 due to Lemma 1.1, where 𝑥0 is the solution to the minimization problem 𝜙(𝑥0,𝑥)=min{𝜙(𝑧,𝑥)𝑧𝐶}.

Let 𝑇 be a mapping from 𝐶 into itself. 𝐹(𝑇) denotes the set of fixed points of 𝑇. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛𝑇𝑥𝑛𝑥𝑛=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is called nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦 for all 𝑥,𝑦𝐶 and relatively nonexpansive (see [13, 14]) if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑥𝐶 and 𝑝𝐹(𝑇). 𝑇 is said to be 𝜙-nonexpansive if 𝜙(𝑇𝑥,𝑇𝑦)𝜙(𝑥,𝑦) for all 𝑥,𝑦𝐶. 𝑇 is said to be relatively quasi-nonexpansive if 𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑥𝐶 and 𝑝𝐹(𝑇). Note that the class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the strong restriction: 𝐹(𝑇)=𝐹(𝑇).

When 𝑊 is a maximal monotone operator, a well-known method for solving the equation 0𝑊𝑣 in a Hilbert space 𝐻 is the proximal point algorithm (see [15]): 𝑥1=𝑥𝐻 and 𝑥𝑛+1=𝐽𝑟𝑛𝑥𝑛,𝑛=1,2,,(1.12) where {𝑟𝑛}(0,) and 𝐽𝑟=(𝐼+𝑟𝑊)1 for all 𝑟>0 is the resolvent operator for 𝑊, then Rockafellar proved that the sequence {𝑥𝑛} converges weakly to an element of 𝑊10.

The modifications of the proximal point algorithm for different operators have been investigated by many authors. Kohsaka and Takahashi [16] considered the following Algorithm (1.13) in a smooth and uniformly convex Banach space: 𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥1+1𝛽𝑛𝐽𝐽𝑟𝑛𝑥𝑛,𝑛=1,2,,(1.13) and Kamimura et al. [17] considered Algorithm (1.14) in a uniformly smooth and uniformly convex Banach space: 𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝐽𝑟𝑛𝑥𝑛,𝑛=1,2,.(1.14) They showed that Algorithm (1.13) converges strongly and Algorithm (1.14) converges weakly provided that the sequences {𝛽𝑛}, {𝑟𝑛} of real numbers are chosen appropriately.

Recently, Saewan and Kumam [18] proposed the following iterative scheme: for an initial 𝑥0𝐸 with 𝑥1=Π𝐶1𝑥0 and 𝐶1=𝐶𝜔𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑧𝑛=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑇𝑛𝐽𝑟𝑛𝜔𝑛,𝑦𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑆𝑛𝑧𝑛,𝑢𝑛=𝐾𝑟𝑛𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑧𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛0,(1.15) where 𝐾𝑟 is the same as in Lemma 2.8 and they obtained a strong convergence theorem.

In 2010, Zeng et al. [19] introduced the following hybrid iterative process: let 𝑥0𝐸 be chosen arbitrarily, ̃𝑥𝑛=𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝐽𝑟𝑛𝑣𝑛,𝑦𝑛=𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛̃𝛽𝑛𝐽̃𝑥𝑛+̃𝛽1𝑛𝐽𝐽𝑟𝑛̃𝑥𝑛,𝑢𝑛=𝐾𝑟𝑛𝑦𝑛,𝐻𝑛=𝑧𝐶𝜙𝑧,𝑢𝑛𝛼𝑛+1𝛼𝑛𝛼𝑛𝜙𝑧,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑧,𝑥𝑛,𝑊𝑛=𝑧𝐶𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐻𝑛𝑊𝑛𝑥0,𝑛0.(1.16) Then they proved some strong and weak convergence theorems.

Very recently, for mixed equilibrium problems, variational inequality problems, fixed point problems, and zeros of maximal monotone operators, many authors have studied them and obtained many new results, see, for instance, [2023].

On the other hand, Nakajo et al. [24] introduced the following condition. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, let {𝑇𝑛} be a family of mappings of 𝐶 into itself with 𝐹=𝑛=1𝐹(𝑇𝑛), and 𝜔𝑤(𝑧𝑛) denotes the set of all weak subsequential limits of a bounded sequence {𝑧𝑛} in 𝐶. {𝑇𝑛} is said to satisfy the NST-condition if, for every bounded sequence {𝑧𝑛} in 𝐶, lim𝑛𝑧𝑛𝑆𝑛𝑧𝑛=0impliesthat𝜔𝑤𝑧𝑛𝐹.(1.17)

Motivated and inspired by the above work, the purpose of this paper is to introduce a new hybrid projection iterative scheme which converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings and zero of maximal monotone operators in Banach spaces.

2. Preliminaries

Let 𝐸 be a normed linear space with dim 𝐸2. The modulus of smoothness of 𝐸 is the function 𝜌𝐸[0,+)[0,+) defined by 𝜌𝐸(𝜏)=sup𝑥+𝑦+𝑥𝑦21𝑥=1,𝑦=𝜏.(2.1) The space 𝐸 is said to be smooth if 𝜌𝐸(𝜏)>0,forall𝜏>0, and 𝐸 is called uniformly smooth if and only if lim𝑡0+(𝜌𝐸(𝑡)/𝑡)=0.

The modulus of convexity of 𝐸 is the function 𝛿𝐸(0,2][0,1] defined by 𝛿𝐸(𝜀)=inf1𝑥+𝑦2𝑥=𝑦=1;𝜀=𝑥𝑦.(2.2)𝐸 is called uniformly convex if and only if 𝛿𝐸(𝜀)>0 for every 𝜀(0,2]. Let 𝑝>1, then 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿𝐸(𝜀)𝑐𝜀𝑝 for every 𝜀(0,2]. Observe that every 𝑝-uniformly convex is uniformly convex. It is well known (see, e.g., [7]) that 𝐿𝑝𝑙𝑝or𝑊𝑝𝑚is𝑝-uniformlyconvexif𝑝2;2-uniformlyconvexif1<𝑝2.(2.3)

In what follows, we will make use of the following lemmas.

Lemma 2.1 (see [7]). Let 𝐸 be a 2-uniformly convex and smooth Banach space. Then, for all 𝑥,𝑦𝐸, one has 2𝑥𝑦𝑐2𝐽𝑥𝐽𝑦,(2.4) where 𝐽 is the normalized duality mapping of 𝐸 and 1/𝑐(0<𝑐1) is the 2-uniformly convex constant of 𝐸.

Lemma 2.2 (see [10, 11]). Let 𝐸 be a real smooth, strictly convex, and reflexive Banach space and 𝐶 a nonempty closed convex subset. Then the following conclusions hold: (1)𝜙(𝑦,Π𝐶𝑥)+𝜙(Π𝐶𝑥,𝑥)𝜙(𝑦,𝑥),𝑥𝐸,𝑦𝐶; (2)suppose 𝑥𝐸 and 𝑧𝐶, then 𝑧=Π𝐶𝑥𝑧𝑦,𝐽𝑥𝐽𝑧0,𝑦𝐶.(2.5)

Lemma 2.3 (see [11]). Let 𝐸 be a real smooth and uniformly convex Banach space, and let {𝑥𝑛} and {𝑦𝑛} be two sequences of 𝐸. If either {𝑥𝑛} or {𝑦𝑛} is bounded and 𝜙(𝑥𝑛,𝑦𝑛)0 as 𝑛, then 𝑥𝑛𝑦𝑛0 as 𝑛.

Lemma 2.4 (see [25]). Let 𝐸 be a real smooth Banach space, and let 𝐴𝐸𝐸 be a maximal monotone mapping, then 𝐴1(0) is a closed and convex subset of 𝐸.

We denote by 𝑁𝐶(𝑣) the normal cone for 𝐶 at a point 𝑣𝐶, that is, 𝑁𝐶(𝑣)={𝑥𝐸𝑣𝑦,𝑥0forall𝑦𝐶}. In the following, we will use the following Lemma.

Lemma 2.5 (see [15]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸, and let 𝐴 be a monotone and hemicontinuous operator of 𝐶 into 𝐸. Let 𝑇𝐸×𝐸 be an operator defined as follows: 𝑇𝑣=𝐴𝑣+𝑁𝐶(𝑣),𝑣𝐶,,𝑣𝐶.(2.6) Then 𝑇 is maximal monotone and 𝑇10=VI(𝐶,𝐴).

We make use of the function 𝑉𝐸×𝐸𝑅 defined by 𝑉𝑥,𝑥=𝑥22𝑥,𝑥+𝑥2,𝑥𝐸,𝑥𝐸,(2.7) studied by Alber [10]; that is, 𝑉(𝑥,𝑥)=𝜙(𝑥,𝐽1𝑥) for all 𝑥𝐸 and 𝑥𝐸. We know the following lemma.

Lemma 2.6 (see [10]). Let 𝐸 be a reflexive strictly convex and smooth Banach space with 𝐸 as its dual. Then 𝑉𝑥,𝑥𝐽+21𝑥𝑥,𝑦𝑉𝑥,𝑥+𝑦,(2.8) for all 𝑥𝐸 and 𝑥,𝑦𝐸.

Lemma 2.7 (see [26]). Let 𝐸 be a uniformly convex Banach space, and let 𝐵𝑟(0)={𝑥𝐸𝑥𝑟} be a closed ball of 𝐸. Then there exists a continuous strictly increasing convex function 𝑔[0,)[0,) with 𝑔(0)=0 such that 𝜆𝑥+𝜇𝑦+𝛾𝑧2𝜆𝑥2+𝜇𝑦2+𝛾𝑧2(𝜆𝜇𝑔𝑥𝑦),(2.9) for all 𝑥,𝑦,𝑧𝐵𝑟(0) and 𝜆,𝜇,𝛾[0,1] with 𝜆+𝜇+𝛾=1.

For solving the equilibrium problem, let us assume that Θ satisfies the following conditions: (A1)Θ(𝑥,𝑥)=0 for all 𝑥𝐶;(A2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)0 for all 𝑥,𝑦𝐶;(A3)for each 𝑥,𝑦,𝑧𝐶, lim𝑡0Θ(𝑡𝑧+(1𝑡)𝑥,𝑦)Θ(𝑥,𝑦);(2.10)(A4)for each 𝑥𝐶,𝑦Θ(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.8 (see [2]). Let 𝐶 be a closed subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝐵𝐶𝐸 be a continuous and monotone mapping, let 𝜑𝐶𝑅 be a lower semicontinuous and convex function, and let Θ be a bifunction from 𝐶×𝐶 to 𝑅 satisfying (A1)–(A4). For 𝑟>0 and 𝑥𝐸, then there exists 𝑢𝐶 such that 1Θ(𝑢,𝑦)+𝐵𝑢,𝑦𝑢+𝜑(𝑦)𝜑(𝑢)+𝑟𝑦𝑢,𝐽𝑢𝐽𝑥0,𝑦𝐶.(2.11) Define a mapping 𝐾𝑟𝐸𝐶 as follows: 𝐾𝑟1(𝑥)=𝑢𝐶Θ(𝑢,𝑦)+𝐵𝑢,𝑦𝑢+𝜑(𝑦)𝜑(𝑢)+𝑟𝑦𝑢,𝐽𝑢𝐽𝑥0,𝑦𝐶,() for all 𝑥𝐸. Then, the following conclusions hold:(1)𝐾𝑟 is single valued;(2)𝐾𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦𝐸, 𝐾𝑟𝑥𝐾𝑟𝑦,𝐽𝐾𝑟𝑥𝐽𝐾𝑟𝑦𝐾𝑟𝑥𝐾𝑟𝑦,𝐽𝑥𝐽𝑦;(3)𝐹(𝐾𝑟)=GMEP;(4)GMEP is closed and convex;(5)𝜙(𝑝,𝐾𝑟𝑧)+𝜙(𝐾𝑟𝑧,𝑧)𝜙(𝑝,𝑧),𝑝𝐹(𝐾𝑟),𝑧𝐸.

Lemma 2.9 (see [27]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑊𝐸𝐸 be a monotone operator satisfying 𝐷(𝑊)𝐶𝐽1(𝑟>0𝑅(𝐽+𝑟𝑊)). Let 𝑟>0, and let 𝐽𝑟 and 𝑊𝑟 be the resolvent and the Yosida approximation of 𝑊, respectively. Then the following hold:(i)𝜙(𝑢,𝐽𝑟𝑥)+𝜙(𝐽𝑟𝑥,𝑥)𝜙(𝑢,𝑥), for all 𝑥𝐶,𝑢𝑊10;(ii)(𝐽𝑟𝑥,𝑊𝑟𝑥)𝐺(𝑊), for all 𝑥𝐶;(iii)𝐹(𝐽𝑟)=𝑊10.

Lemma 2.10 (see [28]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝑆𝐶𝐶 be a relatively quasi-nonexpansive mapping. Then 𝐹(𝑆) is a closed convex subset of 𝐶.

3. Strong Convergence Theorems

In this section, let 𝑇,𝑇𝐸𝐸 be two maximal monotone operators satisfying 𝐷(𝑇),𝐷(𝑇)𝐶. We denote the resolvent operators of 𝑇 and 𝑇 by 𝐽𝑟=(𝐽+𝑟𝑇)1𝐽 and 𝐽𝑟=(𝐽+𝑟𝑇)1𝐽 for each 𝑟>0, respectively. For each 𝑟>0, the Yosida approximations of 𝑇 and 𝑇 are defined by 𝐴𝑟=(𝐽𝐽𝐽𝑟)/𝑟 and 𝐴𝑟𝐽=(𝐽𝐽𝑟)/𝑟, respectively. It is known that 𝐴𝑟𝐽𝑥𝑇𝑟𝑥,𝐴𝑟𝑇𝐽𝑥𝑟𝑥,foreach𝑟>0,𝑥𝐸.(3.1)

Theorem 3.1. Let 𝐸 be a real uniformly smooth and 2-uniformly convex Banach space and 𝐶 a nonempty, closed, and convex subset of 𝐸. Let 𝐴 be a 𝛾-inverse strongly monotone mapping of 𝐶 into 𝐸 satisfying 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝𝑉𝐼(𝐶,𝐴). Let 𝐵𝐶𝐸 be a monotone continuous mapping, and let {𝑇𝑛},{𝑆𝑛} be two countable families of relatively quasi-nonexpansive mappings from 𝐶 into itself satisfying NST-conditions such that Ω=(𝑛=0𝐹(𝑇𝑛))(𝑛=0𝐹(𝑆𝑛))VI(𝐶,𝐴)𝐺𝑀𝐸𝑃𝑇1𝑇010. Suppose that 0<𝑎<𝜆𝑛<𝑏<(𝑐2𝛾)/2, where 𝑐 is the constant in (2.4). Let {𝑡𝑛}[𝑐,+) for some 𝑐>0 and {𝑟𝑛}(0,+) satisfy liminf𝑛𝑟𝑛>0. Let {𝑥𝑛} be the sequence generated by 𝑥0𝑣𝐶,𝑐𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑧𝑛=𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛,𝑦𝑛=𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛̃𝛽𝑛𝐽𝑧𝑛+̃𝛽1𝑛𝐽𝑇𝑛𝐽𝑟𝑛𝑧𝑛,𝑢𝑛=𝐾𝑡𝑛𝑦𝑛,𝐻𝑛=𝑧𝐶𝜙𝑧,𝑢𝑛𝛼𝑛+1𝛼𝑛𝛼𝑛𝜙𝑧,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑧,𝑥𝑛,𝑊𝑛=𝑧𝐶𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐻𝑛𝑊𝑛𝑥0,𝑛0,(3.2) where 𝐽 is the normalized duality mapping, {𝛼𝑛},{𝛽𝑛},{𝛼𝑛}, and {̃𝛽𝑛} are four sequences in [0,1], and 𝐾𝑟 is defined by (*). The following conditions hold:(i)lim𝑛𝛼𝑛=lim𝑛𝛼𝑛=0; (ii)liminf𝑛𝛽𝑛(1𝛽𝑛)>0; (iii)liminf𝑛̃𝛽𝑛̃𝛽(1𝑛)>0.

Then {𝑥𝑛} converges strongly to ΠΩ𝑥0.

Proof. We have the following steps.Step 1. First we prove that 𝐻𝑛 and 𝑊𝑛 are both closed and convex and Ω𝐻𝑛𝑊𝑛,forall𝑛0.
In fact, it follows from Lemmas 2.4, 2.5, 2.8, and 2.10 that Ω is closed and convex. It is obvious that 𝑊𝑛 is closed and convex for each 𝑛0. Let 𝛾𝑛=𝛼𝑛+(1𝛼𝑛)𝛼𝑛,𝑤𝑛=𝐽𝑟𝑛𝑣𝑛,̃𝑧𝑛=𝐽𝑟𝑛𝑧𝑛. For any 𝑧𝐶, 𝜙𝑧,𝑢𝑛𝛾𝑛𝜙𝑧,𝑥0+1𝛾𝑛𝜙𝑧,𝑥𝑛(3.3) is equivalent to 2𝑧,𝐽𝑢𝑛𝑢+𝑛22𝛾𝑛𝑧,𝐽𝑥0+𝛾𝑛𝑥02+1𝛾𝑛2𝑧,𝐽𝑥𝑛𝑥+𝑛2,(3.4) which implies that 𝐻𝑛 is closed and convex for each 𝑛0. Next, we prove that Ω𝐻𝑛𝑊𝑛,forall𝑛0. For any given 𝑝Ω, by Lemma 2.9 we have 𝜙𝑝,𝑤𝑛=𝜙𝑝,𝐽𝑟𝑛𝑣𝑛𝜙𝑝,𝑣𝑛,𝜙𝑝,̃𝑧𝑛𝐽=𝜙𝑝,𝑟𝑛𝑧𝑛𝜙𝑝,𝑧𝑛.(3.5) Since 𝑇𝑛,𝑆𝑛 are relatively quasi-nonexpansive, from the definition of 𝜙(𝑥,𝑦), the convexity of 2, and (3.5), we have 𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛̃𝛽𝑛𝐽𝑧𝑛+̃𝛽1𝑛𝐽𝑇𝑛̃𝑧𝑛=𝑝22𝛼𝑛𝑝,𝐽𝑥021𝛼𝑛̃𝛽𝑛𝑝,𝐽𝑧𝑛21𝛼𝑛̃𝛽1𝑛𝑝,𝐽𝑇𝑛̃𝑧𝑛+𝛼𝑛𝐽𝑥0+1𝛼𝑛̃𝛽𝑛𝐽𝑧𝑛+̃𝛽1𝑛𝐽𝑇𝑛̃𝑧𝑛2𝑝22𝛼𝑛𝑝,𝐽𝑥021𝛼𝑛̃𝛽𝑛𝑝,𝐽𝑧𝑛21𝛼𝑛̃𝛽1𝑛𝑝,𝐽𝑇𝑛̃𝑧𝑛+𝛼𝑛𝑥02+1𝛼𝑛̃𝛽𝑛𝑧𝑛2+̃𝛽1𝑛𝑇𝑛̃𝑧𝑛2=𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛̃𝛽𝑛𝜙𝑝,𝑧𝑛+1𝛼𝑛̃𝛽1𝑛𝜙𝑝,𝑇𝑛̃𝑧𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛̃𝛽𝑛𝜙𝑝,𝑧𝑛+1𝛼𝑛̃𝛽1𝑛𝜙𝑝,̃𝑧𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑧𝑛,𝜙(3.6)𝑝,𝑧𝑛=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑆𝑛𝑤𝑛𝑝22𝛼𝑛𝑝,𝐽𝑥021𝛼𝑛𝛽𝑛𝑝,𝐽𝑥𝑛21𝛼𝑛1𝛽𝑛𝑝,𝐽𝑆𝑛𝑤𝑛+𝛼𝑛𝑥02+1𝛼𝑛𝛽𝑛𝑥𝑛2+1𝛽𝑛𝑆𝑛𝑤𝑛2=𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛼𝑛1𝛽𝑛𝜙𝑝,𝑆𝑛𝑤𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛼𝑛1𝛽𝑛𝜙𝑝,𝑤𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛼𝑛1𝛽𝑛𝜙𝑝,𝑣𝑛.(3.7) Moreover, it follows from Lemmas 2.2 and 2.6 that 𝜙𝑝,𝑣𝑛𝜙𝑝,𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑉𝑝,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑉𝑝,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛+𝜆𝑛𝐴𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑝,𝜆𝑛𝐴𝑥𝑛=𝜙𝑝,𝑥𝑛2𝜆𝑛𝑥𝑛𝑝,𝐴𝑥𝑛𝐴𝑝2𝜆𝑛𝑥𝑛𝑝,𝐴𝑝2𝜆𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝐴𝑥𝑛.(3.8) Since 𝑝VI(𝐶,𝐴), 𝐴 is 𝛾-inverse strongly monotone, from the above inequality, Lemma 2.1, and the fact that 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝VI(𝐶,𝐴), we obtain 𝜙𝑝,𝑣𝑛𝜙𝑝,𝑥𝑛2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2+2𝜆𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛𝐴𝑥𝑛𝜙𝑝,𝑥𝑛2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2+4𝑐2𝜆2𝑛𝐴𝑥𝑛𝐴𝑝2=𝜙𝑝,𝑥𝑛+2𝜆𝑛2𝑐2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2𝜙𝑝,𝑥𝑛.(3.9) From (3.7)-(3.9), we have 𝜙𝑝,𝑧𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑥𝑛.(3.10) So from (3.6) and (3.10), we have 𝜙𝑝,𝑦𝑛𝛼𝑛+1𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛.(3.11) By Lemma 2.8(5) and (3.11), we have 𝜙𝑝,𝑢𝑛=𝜙𝑝,𝐾𝑡𝑛𝑦𝑛𝜙𝑝,𝑦𝑛𝛼𝑛+1𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛.(3.12) Therefore, 𝑝𝐻𝑛; that is Ω𝐻𝑛 for each 𝑛0.
Next we prove that Ω𝐻𝑛𝑊𝑛 for each 𝑛0 by induction. For 𝑛=0,𝑊0=𝐶, thus Ω𝐻0𝑊0. Suppose that Ω𝐻𝑛𝑊𝑛 for some 𝑛1. Since 𝑥𝑛+1=Π𝐻𝑛𝑊𝑛𝑥0, by Lemma 2.2, for any 𝑞𝐻𝑛𝑊𝑛, we have𝑥𝑛+1𝑞,𝐽𝑥0𝐽𝑥𝑛+10.(3.13) Since Ω𝐻𝑛𝑊𝑛, for any 𝑝Ω, we have 𝑥𝑛+1𝑝,𝐽𝑥0𝐽𝑥𝑛+10,(3.14) which implies that 𝑝𝑊𝑛+1, that is, Ω𝐻𝑛𝑊𝑛 for each 𝑛0.
Step 2. Next we prove that 𝑤𝑛𝑆𝑛𝑤𝑛0,̃𝑧𝑛𝑇𝑛̃𝑧𝑛0(𝑛).
Similar to the proof of Step  3 in [19, Theorem  3.1], we have that {𝜙(𝑥𝑛,𝑥0)} is nondecreasing and bounded and 𝜙(𝑥𝑛+1,𝑥𝑛)0(𝑛). So {𝑥𝑛} is bounded, then, by (3.5)–(3.12), we obtain that {𝑢𝑛},{𝑧𝑛},{𝑣𝑛},{𝑤𝑛}, {̃𝑧𝑛}, and {𝑦𝑛} are all bounded. Furthermore, {𝑆𝑛𝑤𝑛} and {𝑇𝑛̃𝑧𝑛} are both bounded. By Lemma 2.3, we have𝑥𝑛𝑥𝑛+10(𝑛).(3.15) Since 𝑥𝑛+1=Π𝐻𝑛𝑊𝑛𝑥0𝐻𝑛 and by condition (i), we have 𝜙𝑥𝑛+1,𝑢𝑛𝛼𝑛+1𝛼𝑛𝛼𝑛𝜙𝑥𝑛+1,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑥𝑛+1,𝑥𝑛0(𝑛),(3.16) which together with Lemma 2.3 implies that 𝑥𝑛+1𝑢𝑛0(𝑛). So we obtain 𝑥𝑛𝑢𝑛0(𝑛).(3.17) Let 𝑎𝑛=𝐽1(𝛽𝑛𝐽𝑥𝑛+(1𝛽𝑛)𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛),𝑏𝑛=𝐽1(̃𝛽𝑛𝐽𝑧𝑛̃𝛽+(1𝑛)𝐽𝑇𝑛𝐽𝑟𝑛𝑧𝑛). It follows from the boundedness of {𝑥𝑛} that {𝑆𝑛𝑤𝑛} and {𝑇𝑛̃𝑧𝑛} are bounded. Let 𝑟=sup{𝑥𝑛,𝑧𝑛,𝑇𝑛̃𝑧𝑛,𝑆𝑛𝑤𝑛}. By Lemma 2.7, (3.5), and (3.9), for any 𝑝Ω, we obtain 𝜙𝑝,𝑎𝑛=𝜙𝑝,𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛=𝑝22𝑝,𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛+𝛽𝑛𝐽𝑥𝑛+(1𝛽𝑛)𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛2𝑝22𝛽𝑛𝑝,𝐽𝑥𝑛21𝛽𝑛𝑝,𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛+𝛽𝑛𝑥𝑛2+1𝛽𝑛𝑆𝑛𝐽𝑟𝑛𝑣𝑛2𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛽𝑛𝜙𝑝,𝑆𝑛𝑤𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛽𝑛𝜙𝑝,𝑤𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛𝜙𝑝,𝑥𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛,𝜙(3.18)𝑝,𝑏𝑛=𝜙𝑝,𝐽1̃𝛽𝑛𝐽𝑧𝑛+̃𝛽1𝑛𝐽𝑇𝑛̃𝑧𝑛̃𝛽𝑛𝜙𝑝,𝑧𝑛+̃𝛽1𝑛𝜙𝑝,𝑇𝑛̃𝑧𝑛̃𝛽𝑛̃𝛽1𝑛𝑔𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛̃𝛽𝑛𝜙𝑝,𝑧𝑛+̃𝛽1𝑛𝜙𝑝,̃𝑧𝑛̃𝛽𝑛̃𝛽1𝑛𝑔𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛𝜙𝑝,𝑧𝑛̃𝛽𝑛̃𝛽1𝑛𝑔𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛.(3.19) It follows from (3.10), (3.12), and (3.19) that 𝜙𝑝,𝑢𝑛𝜙𝑝,𝑦𝑛=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛𝐽𝑏𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑏𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑧𝑛̃𝛽𝑛̃𝛽1𝑛𝑔𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛1𝛼𝑛̃𝛽𝑛̃𝛽1𝑛𝑔𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛.(3.20) From (3.17) and (3.20), we have 1𝛼𝑛̃𝛽𝑛̃𝛽1𝑛𝑔𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0+𝛼𝑛𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥𝑛+𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛+2𝑝𝐽𝑢𝑛𝐽𝑥𝑛+𝑥𝑛𝑢𝑛𝑥𝑛+𝑢𝑛0(𝑛),(3.21) which together with condition (iii) implies that 𝐽𝑧𝑛𝐽𝑇𝑛̃𝑧𝑛0(𝑛).(3.22) Since 𝐽 is uniformly continuous on bounded sets, then 𝑧𝑛𝑇𝑛̃𝑧𝑛0(𝑛).(3.23) By (3.18) and (3.19), we have 𝜙𝑝,𝑏𝑛𝜙𝑝,𝑧𝑛=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑥0+1𝛼𝑛𝐽𝑎𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑎𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑥𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛.(3.24) It follows from (3.20) and (3.24) thatss 𝜙𝑝,𝑢𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑏𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑥𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛=𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛1𝛼𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛.(3.25) So by (3.17) and (3.25), we obtain 1𝛼𝑛𝛽𝑛1𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛+𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛+2𝑝𝐽𝑢𝑛𝐽𝑥𝑛+𝑥𝑛𝑢𝑛𝑥𝑛+𝑢𝑛0(𝑛),(3.26) which together with conditions (i) and (ii) implies that 𝐽𝑥𝑛𝐽𝑆𝑛𝑤𝑛0(𝑛).(3.27) Since 𝐽 is uniformly continuous on bounded sets, then 𝑥𝑛𝑆𝑛𝑤𝑛0(𝑛).(3.28) It follows from (3.2) that 𝐽𝑧𝑛=𝛼𝑛𝐽𝑥0+1𝛼𝑛𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑆𝑛𝐽𝑟𝑛𝑣𝑛.(3.29) Therefore, from (3.27) and condition (i), we have 𝐽𝑧𝑛𝐽𝑥𝑛𝛼𝑛𝐽𝑥0𝐽𝑥𝑛+1𝛼𝑛1𝛽𝑛𝐽𝑆𝑛𝑤𝑛𝐽𝑥𝑛0(𝑛).(3.30) Thus, 𝑧𝑛𝑥𝑛0(𝑛).(3.31) From (3.7) and (3.9), we have 𝜙𝑝,𝑧𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛼𝑛1𝛽𝑛𝜙𝑝,𝑥𝑛+2𝜆𝑛2𝑐2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2=𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑥𝑛+21𝛼𝑛1𝛽𝑛𝜆𝑛2𝑐2𝜆𝑛𝛾𝐴𝑥𝑛𝐴𝑝2,(3.32) which together with (3.31) and condition (i) implies that 21𝛼𝑛1𝛽𝑛𝜆𝑛2𝛾𝑐2𝜆𝑛𝐴𝑥𝑛𝐴𝑝2𝛼𝑛𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛+𝜙𝑝,𝑥𝑛𝜙𝑝,𝑧𝑛0(𝑛).(3.33) Hence, 𝐴𝑥𝑛𝐴𝑝0(𝑛).(3.34) From Lemmas 2.1, 2.2, and 2.6, (3.34), and the fact that 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝VI(𝐶,𝐴), we have 𝜙𝑥𝑛,𝑣𝑛𝑥=𝜙𝑛,Π𝐶𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝜙𝑛,𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥=𝑉𝑛,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑉𝑛,𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛+𝜆𝑛𝐴𝑥𝑛𝐽21𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝑥𝑛,𝜆𝑛𝐴𝑥𝑛𝑥=𝜙𝑛,𝑥𝑛+2𝜆𝑛𝐽1𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛𝐽1𝐽𝑥𝑛𝐴𝑥𝑛2𝜆2𝑛2𝑐2𝐴𝑥𝑛22𝜆2𝑛2𝑐2𝐴𝑥𝑛𝐴𝑝20(𝑛).(3.35) This implies that 𝑥𝑛𝑣𝑛0(𝑛).(3.36) Combining (3.28) and (3.36), we have 𝑣𝑛𝑆𝑛𝑤𝑛0(𝑛).(3.37) It follows from (3.6), (3.7), and (3.12) that 𝜙𝑝,𝑤𝑛𝜙𝑝,𝑢𝑛1𝛼𝑛1𝛼𝑛1𝛽𝑛𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛𝜙𝑝,𝑥01𝛼𝑛1𝛼𝑛1𝛽𝑛1𝛼𝑛1𝛼𝑛𝛽𝑛𝜙𝑝,𝑥𝑛1𝛼𝑛1𝛼𝑛1𝛽𝑛.(3.38) By Lemma 2.9, (3.9), and (3.38), we have 𝜙𝑤𝑛,𝑣𝑛𝐽=𝜙𝑟𝑛𝑣𝑛,𝑣𝑛𝜙𝑝,𝑣𝑛𝜙𝑝,𝑤𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑤𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛1𝛼𝑛1𝛼𝑛1𝛽𝑛+𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛1𝛼𝑛1𝛼𝑛1𝛽𝑛𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛.(3.39) So, by (3.17) and conditions (i) and (ii), we have 𝜙𝑤𝑛,𝑣𝑛0(𝑛),(3.40) which implies that 𝑤𝑛𝑣𝑛0(𝑛).(3.41) Combining (3.37) and (3.41), we obtain 𝑤𝑛𝑆𝑛𝑤𝑛0(𝑛).(3.42) It follows from (3.6), (3.10), and (3.12) that 𝜙𝑝,𝑢𝑛𝜙𝑝,𝑦𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛̃𝛽𝑛𝜙𝑝,𝑧𝑛+1𝛼𝑛̃𝛽1𝑛𝜙𝑝,̃𝑧𝑛𝛼𝑛+1𝛼𝑛̃𝛽𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛1𝛼𝑛̃𝛽𝑛𝜙𝑝,𝑥𝑛+1𝛼𝑛̃𝛽1𝑛𝜙𝑝,̃𝑧𝑛,(3.43) which implies that 𝜙𝑝,̃𝑧𝑛𝜙𝑝,𝑢𝑛1𝛼𝑛̃𝛽1𝑛𝛼𝑛+1𝛼𝑛̃𝛽𝑛𝛼𝑛𝜙𝑝,𝑥01𝛼𝑛̃𝛽1𝑛1𝛼𝑛1𝛼𝑛̃𝛽𝑛𝜙𝑝,𝑥𝑛1𝛼𝑛̃𝛽1𝑛.(3.44) Combining the above inequality, (3.10), and Lemma 2.9, we have 𝜙̃𝑧𝑛,𝑧𝑛𝐽=𝜙𝑟𝑛𝑧𝑛,𝑧𝑛𝜙𝑝,𝑧𝑛𝜙𝑝,̃𝑧𝑛𝛼𝑛𝜙𝑝,𝑥0+1𝛼𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,̃𝑧𝑛𝜙𝑝,𝑥𝑛𝜙𝑝,𝑢𝑛1𝛼𝑛̃𝛽1𝑛+𝛼𝑛+𝛼𝑛𝛼𝑛𝛼𝑛1𝛼𝑛̃𝛽1𝑛𝜙𝑝,𝑥0𝜙𝑝,𝑥𝑛.(3.45) By conditions (i) and (iii), (3.17), and (3.45), we have 𝜙̃𝑧𝑛,𝑧𝑛0(𝑛),(3.46) which implies that ̃𝑧𝑛𝑧𝑛0(𝑛).(3.47) It follows from (3.23) and (3.47) that ̃𝑧𝑛𝑇𝑛̃𝑧𝑛0(𝑛).(3.48)
Step 3. Now we show that 𝜔𝑤({𝑥𝑛})Ω=(𝑛=0𝐹(𝑇𝑛))(𝑛=0𝐹(𝑆𝑛))VI(𝐶,𝐴)GMEP𝑇1𝑇010, where 𝜔𝑤𝑥𝑛=𝑥𝐶𝑥𝑛𝑘𝑥𝑛forsomesequence𝑘{𝑛}with𝑛𝑘.(3.49)
Indeed, since {𝑥𝑛} is bounded and 𝑋 is reflexive, we know that 𝜔𝑤({𝑥𝑛}). For any arbitrary 𝑥𝜔𝑤({𝑥𝑛}), there exists a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that 𝑥𝑛𝑘𝑥. From (3.36), we have 𝑣𝑛𝑘𝑥. Since {𝑇𝑛},{𝑆𝑛} satisfy NST-conditions, from (3.42) and (3.48) we have 𝑥(𝑛=0𝐹(𝑇𝑛))(𝑛=0𝐹(𝑆𝑛)).
Let 𝑆𝐸×𝐸 be an operator as follows:𝑆𝑣=𝐴𝑣+𝑁𝐶(𝑣),𝑣𝐶,,𝑣𝐶.(3.50) By Lemma 2.5, 𝑆 is maximal monotone and 𝑆1(0)=VI(𝐶,𝐴). Let (𝑣,𝑤)𝐺(𝑆). Since 𝑤𝑆𝑣=𝐴𝑣+𝑁𝐶(𝑣), we have 𝑤𝐴𝑣𝑁𝐶(𝑣). Moreover, 𝑣𝑛𝐶 implies that 𝑣𝑣𝑛,𝑤𝐴𝑣0.(3.51)
On the other hand, it follows from 𝑣𝑛=Π𝐶𝐽1(𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛) and Lemma 2.2 that𝑣𝑣𝑛,𝐽𝑣𝑛𝐽𝑥𝑛𝜆𝑛𝐴𝑥𝑛0,(3.52) and hence 𝑣𝑣𝑛,𝐽𝑥𝑛𝐽𝑣𝑛𝜆𝑛𝐴𝑥𝑛0.(3.53) So, from (3.51), (3.53), and 𝐴 being 1/𝛾-Lipschitz continuous, we obtain 𝑣𝑣𝑛,𝑤𝑣𝑣𝑛,𝐴𝑣𝑣𝑣𝑛,𝐴𝑣+𝑣𝑣𝑛,𝐽𝑥𝑛𝐽𝑣𝑛𝜆𝑛𝐴𝑥𝑛=𝑣𝑣𝑛,𝐴𝑣𝐴𝑥𝑛+𝐽𝑥𝑛𝐽𝑣𝑛𝜆𝑛𝑣𝑣𝑛,𝐴𝑣𝐴𝑣𝑛+𝑣𝑣𝑛,𝐴𝑣𝑛𝐴𝑥𝑛+𝑣𝑣𝑛,𝐽𝑥𝑛𝐽𝑣𝑛𝜆𝑛𝑣𝑣𝑛𝐴𝑣𝑛𝐴𝑥𝑛𝑣𝑣𝑛𝐽𝑥𝑛𝐽𝑣𝑛𝑎1𝛾𝑣𝑣𝑛𝑣𝑛𝑥𝑛𝑣𝑣𝑛𝐽𝑥𝑛𝐽𝑣𝑛𝑎.(3.54) Since 𝐽 is uniformly continuous on bounded sets, by (3.36) and replacing 𝑛 by 𝑛𝑘 in (3.54), as 𝑘 we have 𝑣𝑥,𝑤0. Thus, 𝑥𝑆1(0), and hence 𝑥VI(𝐶,𝐴).
Next we show that 𝑥GMEP=𝐹(𝐾𝑟). Let 𝐻(𝑢𝑛,𝑦)=Θ(𝑢𝑛,𝑦)+𝐵𝑢𝑛,𝑦𝑢𝑛+𝜑(𝑦)𝜑(𝑢𝑛), forall𝑦𝐶. It follows from (3.2) that𝐽𝑦𝑛=𝛼𝑛𝐽𝑥0+1𝛼𝑛̃𝛽𝑛𝐽𝑧𝑛+̃𝛽1𝑛𝐽𝑇𝑛𝐽𝑟𝑛𝑧𝑛,(3.55) hence, from (3.22) and condition (i), we have 𝐽𝑦𝑛𝐽𝑧𝑛𝛼𝑛𝐽𝑥0𝐽𝑧𝑛+1𝛼𝑛̃𝛽1𝑛𝐽𝑇𝑛𝐽𝑟𝑛𝑧𝑛𝐽𝑧𝑛0(𝑛),(3.56) which implies that