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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 712651, 21 pages
http://dx.doi.org/10.1155/2012/712651
Research Article

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

1Mathematical College, Sichuan University, Chengdu, Sichuan 610064, China
2Department of Mathematics, Shaoxing University, Shaoxing 312000, China

Received 5 August 2011; Accepted 10 October 2011

Academic Editor: Ya PingΒ Fang

Copyright Β© 2012 Yaqin Wang. 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

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, πΈβˆ— the dual space of 𝐸, and 𝐢 a nonempty closed convex subset of 𝐸. Let Ξ˜βˆΆπΆΓ—πΆβ†’π‘… be a bifunction and πœ‘βˆΆπΆβ†’π‘… a real-valued function. The generalized mixed equilibrium problem (GMEP) of finding π‘₯∈𝐢 is such that Θ(π‘₯,𝑦)+πœ‘(𝑦)βˆ’πœ‘(π‘₯)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(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 π‘₯∈𝐢 such that Θ(π‘₯,𝑦)+πœ‘(𝑦)βˆ’πœ‘(π‘₯)β‰₯0,βˆ€π‘¦βˆˆπΆ,(1.2) which was considered by Ceng and Yao [3]. The set of solutions of this problem is denoted by MEP.

If πœ‘=0, then problem (1.1) reduces to the following generalized equilibrium problem of finding π‘₯∈𝐢 such that Θ(π‘₯,𝑦)+⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ,(1.3) which was studied by S. Takahashi and W. Takahashi [4].

If πœ‘=0 and 𝐴=0, then problem (1.1) reduces to the following equilibrium problem of finding π‘₯∈𝐢 such that Θ(π‘₯,𝑦)β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.4) The set of solutions of problem (1.4) is denoted by EP.

If Θ=0,πœ‘=0, then problem (1.1) reduces to the following classical variational inequality problem of finding π‘₯∈𝐢 such that ⟨𝐴π‘₯,π‘¦βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ.(1.5) The set of solutions of problem (1.5) is denoted by VI(𝐢,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, 3–7].

The normalized duality mapping from 𝐸 to 2πΈβˆ— is defined by 𝐽π‘₯=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2=‖𝑓‖2ξ€Ύ,π‘₯∈𝐸,(1.6) where βŸ¨β‹…,β‹…βŸ© 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 π½βˆ’1 is single valued, one to one, surjective, and it is the duality mapping from πΈβˆ— into 𝐸 and thus π½π½βˆ’1=πΌπΈβˆ— and π½βˆ’1𝐽=𝐼𝐸 (see [8]).

On the other hand, let π‘ŠβˆΆπΈβ‡‰πΈβˆ— be a set-valued mapping. The problem of finding π‘£βˆˆπΈ satisfying 0βˆˆπ‘Šπ‘£ contains numerous problems in economics, optimization, and physics. Such π‘£βˆˆπΈ is called a zero point of π‘Š.

A set-valued mapping π‘ŠβˆΆπΈβ‡‰πΈβˆ— with graph 𝐺(π‘Š)={(π‘₯,π‘₯βˆ—)∢π‘₯βˆ—βˆˆπ‘Šπ‘₯}, domain 𝐷(π‘Š)={π‘₯βˆˆπΈβˆΆπ‘Šπ‘₯β‰ βˆ…}, and range 𝑅(π‘Š)=βˆͺ{π‘Šπ‘₯∢π‘₯∈𝐷(π‘Š)} is said to be monotone if ⟨π‘₯βˆ’π‘¦,π‘₯βˆ—βˆ’π‘¦βˆ—βŸ©β‰₯0 for all π‘₯βˆ—βˆˆπ‘Šπ‘₯,π‘¦βˆ—βˆˆπ‘Šπ‘¦. π‘Š is said to be maximal monotone if the graph 𝐺(π‘Š) of π‘Š is not properly contained in the graph of any other monotone operator. It is known that π‘Š is a maximal monotone if and only if 𝑅(𝐽+π‘Ÿπ‘Š)=πΈβˆ— for all π‘Ÿ>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 𝐢 be a nonempty closed convex subset of 𝐸, and let π‘ŠβˆΆπΈβ‡‰πΈβˆ— be a monotone operator satisfying 𝐷(π‘Š)βŠ‚πΆβŠ‚π½βˆ’1(βˆ©π‘Ÿ>0𝑅(𝐽+π‘Ÿπ‘Š)). Then the resolvent of π‘Š defined by π½π‘Ÿ=(𝐽+π‘Ÿπ‘Š)βˆ’1𝐽 is a single-valued mapping from 𝐸 to 𝐷(π‘Š) for all π‘Ÿ>0. For π‘Ÿ>0, the Yosida approximation of π‘Š is defined by π‘Šπ‘Ÿπ‘₯=(𝐽π‘₯βˆ’π½π½π‘Ÿπ‘₯)/π‘Ÿ for all π‘₯∈𝐸.

A mapping π΄βˆΆπΆβ†’πΈβˆ— is said to be monotone if, for each π‘₯,π‘¦βˆˆπΆ, ⟨π‘₯βˆ’π‘¦,𝐴π‘₯βˆ’π΄π‘¦βŸ©β‰₯0.(1.7)𝐴 is said to be 𝛾-inverse strongly monotone if there exists a positive real number 𝛾>0 such that ⟨π‘₯βˆ’π‘¦,𝐴π‘₯βˆ’π΄π‘¦βŸ©β‰₯𝛾‖𝐴π‘₯βˆ’π΄π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.8) If 𝐴 is 𝛾-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛾, that is, ξ‚΅1‖𝐴π‘₯βˆ’π΄π‘¦β€–β‰€π›Ύξ‚Άβ€–π‘₯βˆ’π‘¦β€–,π‘₯,π‘¦βˆˆπΆ.(1.9)

Let 𝐸 be a smooth Banach space. The function πœ™βˆΆπΈΓ—πΈβ†’π‘… defined by πœ™(π‘₯,𝑦)=β€–π‘₯β€–2βˆ’2⟨π‘₯,π½π‘¦βŸ©+‖𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπΈ,(1.10) is studied by Alber [10], Kamimura and Takahashi [11], and Reich [12]. It follows from the definition of the function πœ™ that ()β€–π‘₯β€–βˆ’β€–π‘¦β€–2)β‰€πœ™(π‘₯,𝑦)≀(β€–π‘₯β€–+‖𝑦‖2,βˆ€π‘₯,π‘¦βˆˆπΈ.(1.11) Observe that, in a Hilbert space 𝐻, πœ™(π‘₯,𝑦)=β€–π‘₯βˆ’π‘¦β€–2,forallπ‘₯,π‘¦βˆˆπ».

Lemma 1.1 (see [10]). Let 𝐢 be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space 𝐸, and let π‘₯∈𝐸. Then there exists a unique element π‘₯0∈𝐢 such that πœ™(π‘₯0,π‘₯)=min{πœ™(𝑧,π‘₯)βˆΆπ‘§βˆˆπΆ}.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space and 𝐢 a nonempty closed and convex subset of 𝐸. The generalized projection mapping, introduced by Alber [10], is a mapping Ξ πΆβˆΆπΈβ†’πΆ that assigns to an arbitrary point π‘₯∈𝐸, the minimum point of the functional πœ™(π‘₯,𝑦), that is, Π𝐢π‘₯=π‘₯0 due to Lemma 1.1, where π‘₯0 is the solution to the minimization problem πœ™(π‘₯0,π‘₯)=min{πœ™(𝑧,π‘₯)βˆΆπ‘§βˆˆπΆ}.

Let 𝑇 be a mapping from 𝐢 into itself. 𝐹(𝑇) denotes the set of fixed points of 𝑇. A point 𝑝 in 𝐢 is said to be an asymptotic fixed point of 𝑇 if 𝐢 contains a sequence {π‘₯𝑛} which converges weakly to 𝑝 such that limπ‘›β†’βˆžβ€–π‘‡π‘₯π‘›βˆ’π‘₯𝑛‖=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐢 into itself is called nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€– for all π‘₯,π‘¦βˆˆπΆ and relatively nonexpansive (see [13, 14]) if 𝐹(𝑇)=𝐹(𝑇) and πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯) for all π‘₯∈𝐢 and π‘βˆˆπΉ(𝑇). 𝑇 is said to be πœ™-nonexpansive if πœ™(𝑇π‘₯,𝑇𝑦)β‰€πœ™(π‘₯,𝑦) for all π‘₯,π‘¦βˆˆπΆ. 𝑇 is said to be relatively quasi-nonexpansive if 𝐹(𝑇)β‰ βˆ… and πœ™(𝑝,𝑇π‘₯)β‰€πœ™(𝑝,π‘₯) for all π‘₯∈𝐢 and π‘βˆˆπΉ(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 π‘Š is a maximal monotone operator, a well-known method for solving the equation 0βˆˆπ‘Šπ‘£ in a Hilbert space 𝐻 is the proximal point algorithm (see [15]): π‘₯1=π‘₯∈𝐻 and π‘₯𝑛+1=π½π‘Ÿπ‘›π‘₯𝑛,𝑛=1,2,…,(1.12) where {π‘Ÿπ‘›}βŠ‚(0,∞) and π½π‘Ÿ=(𝐼+π‘Ÿπ‘Š)βˆ’1 for all π‘Ÿ>0 is the resolvent operator for π‘Š, then Rockafellar proved that the sequence {π‘₯𝑛} converges weakly to an element of π‘Šβˆ’10.

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: π‘₯𝑛+1=π½βˆ’1𝛽𝑛𝐽π‘₯1ξ€Έ+ξ€·1βˆ’π›½π‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘₯𝑛,𝑛=1,2,…,(1.13) and Kamimura et al. [17] considered Algorithm (1.14) in a uniformly smooth and uniformly convex Banach space: π‘₯𝑛+1=π½βˆ’1𝛽𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½ξ€·π½π‘Ÿπ‘›π‘₯𝑛,𝑛=1,2,….(1.14) They showed that Algorithm (1.13) converges strongly and Algorithm (1.14) converges weakly provided that the sequences {𝛽𝑛}, {π‘Ÿπ‘›} of real numbers are chosen appropriately.

Recently, Saewan and Kumam [18] proposed the following iterative scheme: for an initial π‘₯0∈𝐸 with π‘₯1=Π𝐢1π‘₯0 and 𝐢1=πΆπœ”π‘›=Ξ πΆπ½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛,𝑧𝑛=π½βˆ’1𝛽𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘‡π‘›π½π‘Ÿπ‘›πœ”π‘›ξ€Έ,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘†π‘›π‘§π‘›ξ€Έ,𝑒𝑛=πΎπ‘Ÿπ‘›π‘¦π‘›,𝐢𝑛+1=ξ€½π‘§βˆˆπΆπ‘›ξ€·βˆΆπœ™π‘§,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘§π‘›ξ€Έξ€·β‰€πœ™π‘§,π‘₯𝑛,π‘₯𝑛+1=Π𝐢𝑛+1π‘₯0,𝑛β‰₯0,(1.15) where πΎπ‘Ÿ 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 π‘₯0∈𝐸 be chosen arbitrarily, Μƒπ‘₯𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’π›Όπ‘›π›½ξ€Έξ€·π‘›π½π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π½π‘Ÿπ‘›π‘£π‘›,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ‚€Μƒπ›½π‘›π½Μƒπ‘₯𝑛+̃𝛽1βˆ’π‘›ξ€Έπ½ξ‚π½π‘Ÿπ‘›Μƒπ‘₯𝑛,𝑒𝑛=πΎπ‘Ÿπ‘›π‘¦π‘›,𝐻𝑛=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,𝑒𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›Όπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯𝑛,π‘Šξ€Έξ€Ύπ‘›=ξ€½π‘§βˆˆπΆβˆΆβŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯0βˆ’π½π‘₯𝑛,π‘₯⟩β‰₯0𝑛+1=Ξ π»π‘›βˆ©π‘Šπ‘›π‘₯0,𝑛β‰₯0.(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, [20–23].

On the other hand, Nakajo et al. [24] introduced the following condition. Let 𝐢 be a nonempty closed convex subset of a Hilbert space 𝐻, let {𝑇𝑛} be a family of mappings of 𝐢 into itself with 𝐹=βˆ©βˆžπ‘›=1𝐹(𝑇𝑛)β‰ βˆ…, and πœ”π‘€(𝑧𝑛) denotes the set of all weak subsequential limits of a bounded sequence {𝑧𝑛} in 𝐢. {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€–=0impliesthatπœ”π‘€ξ€·π‘§π‘›ξ€ΈβŠ‚πΉ.(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 𝐸β‰₯2. The modulus of smoothness of 𝐸 is the function 𝜌𝐸∢[0,+∞)β†’[0,+∞) defined by πœŒπΈξ‚»(𝜏)∢=supβ€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–2ξ‚Όβˆ’1βˆΆβ€–π‘₯β€–=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 lim𝑑→0+(𝜌𝐸(𝑑)/𝑑)=0.

The modulus of convexity of 𝐸 is the function π›ΏπΈβˆΆ(0,2]β†’[0,1] defined by 𝛿𝐸(ξ‚†β€–β€–β€–πœ€)∢=inf1βˆ’π‘₯+𝑦2β€–β€–β€–ξ‚‡βˆΆβ€–π‘₯β€–=‖𝑦‖=1;πœ€=β€–π‘₯βˆ’π‘¦β€–.(2.2)𝐸 is called uniformly convex if and only if 𝛿𝐸(πœ€)>0 for every πœ€βˆˆ(0,2]. Let 𝑝>1, then 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿𝐸(πœ€)β‰₯π‘πœ€π‘ for every πœ€βˆˆ(0,2]. Observe that every 𝑝-uniformly convex is uniformly convex. It is well known (see, e.g., [7]) that 𝐿𝑝𝑙𝑝orπ‘Šπ‘π‘šξ‚»is𝑝-uniformlyconvexif𝑝β‰₯2;2-uniformlyconvexif1<𝑝≀2.(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 π‘₯,π‘¦βˆˆπΈ, one has 2β€–π‘₯βˆ’π‘¦β€–β‰€π‘2‖𝐽π‘₯βˆ’π½π‘¦β€–,(2.4) where 𝐽 is the normalized duality mapping of 𝐸 and 1/𝑐(0<𝑐≀1) 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 𝐢 a nonempty closed convex subset. Then the following conclusions hold: (1)πœ™(𝑦,Π𝐢π‘₯)+πœ™(Π𝐢π‘₯,π‘₯)β‰€πœ™(𝑦,π‘₯),βˆ€π‘₯∈𝐸,π‘¦βˆˆπΆ; (2)suppose π‘₯∈𝐸 and π‘§βˆˆπΆ, then 𝑧=Π𝐢π‘₯βŸΊβŸ¨π‘§βˆ’π‘¦,𝐽π‘₯βˆ’π½π‘§βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(2.5)

Lemma 2.3 (see [11]). Let 𝐸 be a real smooth and uniformly convex Banach space, and let {π‘₯𝑛} and {𝑦𝑛} be two sequences of 𝐸. If either {π‘₯𝑛} or {𝑦𝑛} is bounded and πœ™(π‘₯𝑛,𝑦𝑛)β†’0 as π‘›β†’βˆž, then π‘₯π‘›βˆ’π‘¦π‘›β†’0 as π‘›β†’βˆž.

Lemma 2.4 (see [25]). Let 𝐸 be a real smooth Banach space, and let π΄βˆΆπΈβ‡‰πΈβˆ— be a maximal monotone mapping, then π΄βˆ’1(0) is a closed and convex subset of 𝐸.

We denote by 𝑁𝐢(𝑣) the normal cone for 𝐢 at a point π‘£βˆˆπΆ, that is, 𝑁𝐢(𝑣)∢={π‘₯βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘£βˆ’π‘¦,π‘₯βˆ—βŸ©β‰₯0forallπ‘¦βˆˆπΆ}. In the following, we will use the following Lemma.

Lemma 2.5 (see [15]). Let 𝐢 be a nonempty closed convex subset of a Banach space 𝐸, and let 𝐴 be a monotone and hemicontinuous operator of 𝐢 into πΈβˆ—. Let π‘‡βŠ‚πΈΓ—πΈβˆ— be an operator defined as follows: 𝑇𝑣=𝐴𝑣+𝑁𝐢(𝑣),π‘£βˆˆπΆ,βˆ…,π‘£βˆ‰πΆ.(2.6) Then 𝑇 is maximal monotone and π‘‡βˆ’10=VI(𝐢,𝐴).

We make use of the function π‘‰βˆΆπΈΓ—πΈβˆ—β†’π‘… defined by 𝑉π‘₯,π‘₯βˆ—ξ€Έ=β€–π‘₯β€–2βˆ’2⟨π‘₯,π‘₯βˆ—βŸ©+β€–π‘₯β€–2,βˆ€π‘₯∈𝐸,π‘₯βˆ—βˆˆπΈβˆ—,(2.7) studied by Alber [10]; that is, 𝑉(π‘₯,π‘₯βˆ—)=πœ™(π‘₯,π½βˆ’1π‘₯βˆ—) for all π‘₯∈𝐸 and π‘₯βˆ—βˆˆπΈβˆ—. We know the following lemma.

Lemma 2.6 (see [10]). Let 𝐸 be a reflexive strictly convex and smooth Banach space with πΈβˆ— as its dual. Then 𝑉π‘₯,π‘₯βˆ—ξ€Έξ«π½+2βˆ’1π‘₯βˆ—βˆ’π‘₯,π‘¦βˆ—ξ¬ξ€·β‰€π‘‰π‘₯,π‘₯βˆ—+π‘¦βˆ—ξ€Έ,(2.8) for all π‘₯∈𝐸 and π‘₯βˆ—,π‘¦βˆ—βˆˆπΈβˆ—.

Lemma 2.7 (see [26]). Let 𝐸 be a uniformly convex Banach space, and let π΅π‘Ÿ(0)={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–β‰€π‘Ÿ} be a closed ball of 𝐸. Then there exists a continuous strictly increasing convex function π‘”βˆΆ[0,∞)β†’[0,∞) with 𝑔(0)=0 such that β€–πœ†π‘₯+πœ‡π‘¦+𝛾𝑧‖2β‰€πœ†β€–π‘₯β€–2+πœ‡β€–π‘¦β€–2+𝛾‖𝑧‖2(βˆ’πœ†πœ‡π‘”β€–π‘₯βˆ’π‘¦β€–),(2.9) for all π‘₯,𝑦,π‘§βˆˆπ΅π‘Ÿ(0) and πœ†,πœ‡,π›Ύβˆˆ[0,1] with πœ†+πœ‡+𝛾=1.

For solving the equilibrium problem, let us assume that Θ satisfies the following conditions: (A1)Θ(π‘₯,π‘₯)=0 for all π‘₯∈𝐢;(A2)Θ is monotone, that is, Θ(π‘₯,𝑦)+Θ(𝑦,π‘₯)≀0 for all π‘₯,π‘¦βˆˆπΆ;(A3)for each π‘₯,𝑦,π‘§βˆˆπΆ, lim𝑑→0Θ(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)β‰€Ξ˜(π‘₯,𝑦);(2.10)(A4)for each π‘₯∈𝐢,π‘¦β†¦Ξ˜(π‘₯,𝑦) is convex and lower semicontinuous.

Lemma 2.8 (see [2]). Let 𝐢 be a closed subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let π΅βˆΆπΆβ†’πΈβˆ— be a continuous and monotone mapping, let πœ‘βˆΆπΆβ†’π‘… be a lower semicontinuous and convex function, and let Θ be 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 1Θ(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.(2.11) Define a mapping πΎπ‘ŸβˆΆπΈβ†’πΆ as follows: πΎπ‘Ÿξ‚†1(π‘₯)=π‘’βˆˆπΆβˆΆΞ˜(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,(βˆ—) for all π‘₯∈𝐸. Then, the following conclusions hold:(1)πΎπ‘Ÿ is single valued;(2)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©;(3)𝐹(πΎπ‘Ÿ)=GMEP;(4)GMEP is closed and convex;(5)πœ™(𝑝,πΎπ‘Ÿπ‘§)+πœ™(πΎπ‘Ÿπ‘§,𝑧)β‰€πœ™(𝑝,𝑧),βˆ€π‘βˆˆπΉ(πΎπ‘Ÿ),π‘§βˆˆπΈ.

Lemma 2.9 (see [27]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐢 be a nonempty closed convex subset of 𝐸, and let π‘ŠβˆΆπΈβ‡‰πΈβˆ— be a monotone operator satisfying 𝐷(π‘Š)βŠ‚πΆβŠ‚π½βˆ’1(βˆ©π‘Ÿ>0𝑅(𝐽+π‘Ÿπ‘Š)). Let π‘Ÿ>0, and let π½π‘Ÿ and π‘Šπ‘Ÿ be the resolvent and the Yosida approximation of π‘Š, respectively. Then the following hold:(i)πœ™(𝑒,π½π‘Ÿπ‘₯)+πœ™(π½π‘Ÿπ‘₯,π‘₯)β‰€πœ™(𝑒,π‘₯), for all π‘₯∈𝐢,π‘’βˆˆπ‘Šβˆ’10;(ii)(π½π‘Ÿπ‘₯,π‘Šπ‘Ÿπ‘₯)∈𝐺(π‘Š), for all π‘₯∈𝐢;(iii)𝐹(π½π‘Ÿ)=π‘Šβˆ’10.

Lemma 2.10 (see [28]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space and 𝐢 a nonempty closed convex subset of 𝐸. Let π‘†βˆΆπΆβ†’πΆ be a relatively quasi-nonexpansive mapping. Then 𝐹(𝑆) is a closed convex subset of 𝐢.

3. Strong Convergence Theorems

In this section, let 𝑇,π‘‡βˆΆπΈβ‡‰πΈβˆ— be two maximal monotone operators satisfying 𝐷(𝑇),𝐷(𝑇)βŠ‚πΆ. We denote the resolvent operators of 𝑇 and 𝑇 by π½π‘Ÿ=(𝐽+π‘Ÿπ‘‡)βˆ’1𝐽 and ξ‚π½π‘Ÿξ‚=(𝐽+π‘Ÿπ‘‡)βˆ’1𝐽 for each π‘Ÿ>0, respectively. For each π‘Ÿ>0, the Yosida approximations of 𝑇 and 𝑇 are defined by π΄π‘Ÿ=(π½βˆ’π½π½π‘Ÿ)/π‘Ÿ and ξ‚π΄π‘Ÿξ‚π½=(π½βˆ’π½π‘Ÿ)/π‘Ÿ, respectively. It is known that π΄π‘Ÿξ€·π½π‘₯βˆˆπ‘‡π‘Ÿπ‘₯ξ€Έ,ξ‚π΄π‘Ÿξ‚π‘‡ξ‚€ξ‚π½π‘₯βˆˆπ‘Ÿπ‘₯,foreachπ‘Ÿ>0,π‘₯∈𝐸.(3.1)

Theorem 3.1. Let 𝐸 be a real uniformly smooth and 2-uniformly convex Banach space and 𝐢 a nonempty, closed, and convex subset of 𝐸. Let 𝐴 be a 𝛾-inverse strongly monotone mapping of 𝐢 into πΈβˆ— satisfying ‖𝐴π‘₯‖≀‖𝐴π‘₯βˆ’π΄π‘β€– for all π‘₯∈𝐢 and π‘βˆˆπ‘‰πΌ(𝐢,𝐴). Let π΅βˆΆπΆβ†’πΈβˆ— be a monotone continuous mapping, and let {𝑇𝑛},{𝑆𝑛} be two countable families of relatively quasi-nonexpansive mappings from 𝐢 into itself satisfying NST-conditions such that Ω∢=(βˆ©βˆžπ‘›=0𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=0𝐹(𝑆𝑛))∩VI(𝐢,𝐴)βˆ©πΊπ‘€πΈπ‘ƒβˆ©π‘‡βˆ’1𝑇0βˆ©βˆ’10β‰ βˆ…. Suppose that 0<π‘Ž<πœ†π‘›<𝑏<(𝑐2𝛾)/2, where 𝑐 is the constant in (2.4). Let {𝑑𝑛}βŠ‚[π‘βˆ—,+∞) for some π‘βˆ—>0 and {π‘Ÿπ‘›}βŠ‚(0,+∞) satisfy liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0. Let {π‘₯𝑛} be the sequence generated by π‘₯0π‘£βˆˆπΆ,π‘β„Žπ‘œπ‘ π‘’π‘›π‘Žπ‘Ÿπ‘π‘–π‘‘π‘Ÿπ‘Žπ‘Ÿπ‘–π‘™π‘¦,𝑛=Ξ πΆπ½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛,𝑧𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’π›Όπ‘›π›½ξ€Έξ€·π‘›π½π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ‚€Μƒπ›½π‘›π½π‘§π‘›+̃𝛽1βˆ’π‘›ξ€Έπ½π‘‡π‘›ξ‚π½π‘Ÿπ‘›π‘§π‘›,𝑒𝑛=𝐾𝑑𝑛𝑦𝑛,𝐻𝑛=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,𝑒𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›Όπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯𝑛,π‘Šξ€Έξ€Ύπ‘›=ξ€½π‘§βˆˆπΆβˆΆβŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯0βˆ’π½π‘₯𝑛,π‘₯⟩β‰₯0𝑛+1=Ξ π»π‘›βˆ©π‘Šπ‘›π‘₯0,𝑛β‰₯0,(3.2) where 𝐽 is the normalized duality mapping, {𝛼𝑛},{𝛽𝑛},{𝛼𝑛}, and {̃𝛽𝑛} are four sequences in [0,1], and πΎπ‘Ÿ is defined by (*). The following conditions hold:(i)limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžξ‚π›Όπ‘›=0; (ii)liminfπ‘›β†’βˆžπ›½π‘›(1βˆ’π›½π‘›)>0; (iii)liminfπ‘›β†’βˆžΜƒπ›½π‘›Μƒπ›½(1βˆ’π‘›)>0.

Then {π‘₯𝑛} converges strongly to Ξ Ξ©π‘₯0.

Proof. We have the following steps.Step 1. First we prove that 𝐻𝑛 and π‘Šπ‘› are both closed and convex and Ξ©βŠ‚π»π‘›βˆ©π‘Šπ‘›,forall𝑛β‰₯0.
In fact, it follows from Lemmas 2.4, 2.5, 2.8, and 2.10 that Ξ© is closed and convex. It is obvious that π‘Šπ‘› is closed and convex for each 𝑛β‰₯0. Let 𝛾𝑛=𝛼𝑛+(1βˆ’ξ‚π›Όπ‘›)𝛼𝑛,𝑀𝑛=π½π‘Ÿπ‘›π‘£π‘›,̃𝑧𝑛=ξ‚π½π‘Ÿπ‘›π‘§π‘›. For any π‘§βˆˆπΆ, πœ™ξ€·π‘§,π‘’π‘›ξ€Έβ‰€π›Ύπ‘›πœ™ξ€·π‘§,π‘₯0ξ€Έ+ξ€·1βˆ’π›Ύπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯𝑛(3.3) is equivalent to βˆ’2βŸ¨π‘§,π½π‘’π‘›β€–β€–π‘’βŸ©+𝑛‖‖2β‰€βˆ’2π›Ύπ‘›βŸ¨π‘§,𝐽π‘₯0⟩+𝛾𝑛‖‖π‘₯0β€–β€–2+ξ€·1βˆ’π›Ύπ‘›ξ€Έξ‚€βˆ’2βŸ¨π‘§,𝐽π‘₯𝑛‖‖π‘₯⟩+𝑛‖‖2,(3.4) which implies that 𝐻𝑛 is closed and convex for each 𝑛β‰₯0. Next, we prove that Ξ©βŠ‚π»π‘›βˆ©π‘Šπ‘›,forall𝑛β‰₯0. For any given π‘βˆˆΞ©, by Lemma 2.9 we have πœ™ξ€·π‘,𝑀𝑛=πœ™π‘,π½π‘Ÿπ‘›π‘£π‘›ξ€Έξ€·β‰€πœ™π‘,𝑣𝑛,πœ™ξ€·π‘,̃𝑧𝑛𝐽=πœ™π‘,π‘Ÿπ‘›π‘§π‘›ξ‚ξ€·β‰€πœ™π‘,𝑧𝑛.(3.5) Since 𝑇𝑛,𝑆𝑛 are relatively quasi-nonexpansive, from the definition of πœ™(π‘₯,𝑦), the convexity of β€–β‹…β€–2, and (3.5), we have πœ™ξ€·π‘,𝑦𝑛=πœ™π‘,π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·π‘›π½π‘§π‘›+̃𝛽1βˆ’π‘›ξ€Έπ½π‘‡π‘›Μƒπ‘§π‘›ξ€Έξ€Έξ€Έ=‖𝑝‖2βˆ’2ξ‚π›Όπ‘›βŸ¨π‘,𝐽π‘₯0ξ€·βŸ©βˆ’21βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›βŸ¨π‘,π½π‘§π‘›ξ€·βŸ©βˆ’21βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€ΈβŸ¨π‘,π½π‘‡π‘›Μƒπ‘§π‘›βŸ©+‖‖𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·π‘›π½π‘§π‘›+̃𝛽1βˆ’π‘›ξ€Έπ½π‘‡π‘›Μƒπ‘§π‘›ξ€Έβ€–β€–2≀‖𝑝‖2βˆ’2ξ‚π›Όπ‘›βŸ¨π‘,𝐽π‘₯0ξ€·βŸ©βˆ’21βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›βŸ¨π‘,π½π‘§π‘›ξ€·βŸ©βˆ’21βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€ΈβŸ¨π‘,π½π‘‡π‘›Μƒπ‘§π‘›βŸ©+𝛼𝑛‖‖π‘₯0β€–β€–2+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ‚€Μƒπ›½π‘›β€–β€–π‘§π‘›β€–β€–2+̃𝛽1βˆ’π‘›ξ€Έβ€–β€–π‘‡π‘›Μƒπ‘§π‘›β€–β€–2=ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›πœ™ξ€·π‘,𝑧𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έπœ™ξ€·π‘,π‘‡π‘›Μƒπ‘§π‘›ξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›πœ™ξ€·π‘,𝑧𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έπœ™ξ€·π‘,Μƒπ‘§π‘›ξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπœ™ξ€·π‘,𝑧𝑛,πœ™ξ€·(3.6)𝑝,𝑧𝑛=πœ™π‘,π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’π›Όπ‘›π›½ξ€Έξ€·π‘›π½π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘†π‘›π‘€π‘›ξ€Έξ€Έξ€Έβ‰€β€–π‘β€–2βˆ’2π›Όπ‘›βŸ¨π‘,𝐽π‘₯0ξ€·βŸ©βˆ’21βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨π‘,𝐽π‘₯π‘›ξ€·βŸ©βˆ’21βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€ΈβŸ¨π‘,π½π‘†π‘›π‘€π‘›βŸ©+𝛼𝑛‖‖π‘₯0β€–β€–2+ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚€π›½π‘›β€–β€–π‘₯𝑛‖‖2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘†π‘›π‘€π‘›β€–β€–2=π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έπœ™ξ€·π‘,π‘†π‘›π‘€π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έπœ™ξ€·π‘,π‘€π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έπœ™ξ€·π‘,𝑣𝑛.(3.7) Moreover, it follows from Lemmas 2.2 and 2.6 that πœ™ξ€·π‘,π‘£π‘›ξ€Έξ€·β‰€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≀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=πœ™π‘,π‘₯π‘›ξ€Έβˆ’2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘,𝐴π‘₯π‘›βˆ’π΄π‘βŸ©βˆ’2πœ†π‘›βŸ¨π‘₯π‘›βˆ’π‘,π΄π‘βŸ©βˆ’2πœ†π‘›ξ«π½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’π‘₯𝑛,𝐴π‘₯𝑛.(3.8) Since π‘βˆˆVI(𝐢,𝐴), 𝐴 is 𝛾-inverse strongly monotone, from the above inequality, Lemma 2.1, and the fact that ‖𝐴π‘₯‖≀‖𝐴π‘₯βˆ’π΄π‘β€– for all π‘₯∈𝐢 and π‘βˆˆVI(𝐢,𝐴), we obtain πœ™ξ€·π‘,π‘£π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έβˆ’2πœ†π‘›π›Ύβ€–β€–π΄π‘₯π‘›β€–β€–βˆ’π΄π‘2+2πœ†π‘›β€–β€–π½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’π‘₯𝑛‖‖‖‖𝐴π‘₯π‘›β€–β€–ξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έβˆ’2πœ†π‘›π›Ύβ€–β€–π΄π‘₯π‘›β€–β€–βˆ’π΄π‘2+4𝑐2πœ†2𝑛‖‖𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘2ξ€·=πœ™π‘,π‘₯𝑛+2πœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Ύπ΄π‘₯π‘›β€–β€–βˆ’π΄π‘2ξ€·β‰€πœ™π‘,π‘₯𝑛.(3.9) From (3.7)-(3.9), we have πœ™ξ€·π‘,π‘§π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛.(3.10) So from (3.6) and (3.10), we have πœ™ξ€·π‘,𝑦𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛.(3.11) By Lemma 2.8(5) and (3.11), we have πœ™ξ€·π‘,𝑒𝑛=πœ™π‘,πΎπ‘‘π‘›π‘¦π‘›ξ€Έξ€·β‰€πœ™π‘,𝑦𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛.(3.12) Therefore, π‘βˆˆπ»π‘›; that is Ξ©βŠ‚π»π‘› for each 𝑛β‰₯0.
Next we prove that Ξ©βŠ‚π»π‘›βˆ©π‘Šπ‘› for each 𝑛β‰₯0 by induction. For 𝑛=0,π‘Š0=𝐢, thus Ξ©βŠ‚π»0βˆ©π‘Š0. Suppose that Ξ©βŠ‚π»π‘›βˆ©π‘Šπ‘› for some 𝑛β‰₯1. Since π‘₯𝑛+1=Ξ π»π‘›βˆ©π‘Šπ‘›π‘₯0, by Lemma 2.2, for any π‘žβˆˆπ»π‘›βˆ©π‘Šπ‘›, we haveπ‘₯𝑛+1βˆ’π‘ž,𝐽π‘₯0βˆ’π½π‘₯𝑛+1β‰₯0.(3.13) Since Ξ©βŠ‚π»π‘›βˆ©π‘Šπ‘›, for any π‘βˆˆΞ©, we have π‘₯𝑛+1βˆ’π‘,𝐽π‘₯0βˆ’π½π‘₯𝑛+1β‰₯0,(3.14) which implies that π‘βˆˆπ‘Šπ‘›+1, that is, Ξ©βŠ‚π»π‘›βˆ©π‘Šπ‘› for each 𝑛β‰₯0.
Step 2. Next we prove that β€–π‘€π‘›βˆ’π‘†π‘›π‘€π‘›β€–β†’0,β€–Μƒπ‘§π‘›βˆ’π‘‡π‘›Μƒπ‘§π‘›β€–β†’0(π‘›β†’βˆž).
Similar to the proof of Step  3 in [19, Theorem  3.1], we have that {πœ™(π‘₯𝑛,π‘₯0)} is nondecreasing and bounded and πœ™(π‘₯𝑛+1,π‘₯𝑛)β†’0(π‘›β†’βˆž). So {π‘₯𝑛} is bounded, then, by (3.5)–(3.12), we obtain that {𝑒𝑛},{𝑧𝑛},{𝑣𝑛},{𝑀𝑛}, {̃𝑧𝑛}, and {𝑦𝑛} are all bounded. Furthermore, {𝑆𝑛𝑀𝑛} and {𝑇𝑛̃𝑧𝑛} are both bounded. By Lemma 2.3, we haveβ€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.15) Since π‘₯𝑛+1=Ξ π»π‘›βˆ©π‘Šπ‘›π‘₯0βˆˆπ»π‘› and by condition (i), we have πœ™ξ€·π‘₯𝑛+1,𝑒𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›Όπ‘›ξ€Έπœ™ξ€·π‘₯𝑛+1,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘₯𝑛+1,π‘₯π‘›ξ€ΈβŸΆ0(π‘›βŸΆβˆž),(3.16) which together with Lemma 2.3 implies that β€–π‘₯𝑛+1βˆ’π‘’π‘›β€–β†’0(π‘›β†’βˆž). So we obtain β€–β€–π‘₯π‘›βˆ’π‘’π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.17) Let π‘Žπ‘›=π½βˆ’1(𝛽𝑛𝐽π‘₯𝑛+(1βˆ’π›½π‘›)π½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›),𝑏𝑛=π½βˆ’1(̃𝛽𝑛𝐽𝑧𝑛̃𝛽+(1βˆ’π‘›)π½π‘‡π‘›ξ‚π½π‘Ÿπ‘›π‘§π‘›). It follows from the boundedness of {π‘₯𝑛} that {𝑆𝑛𝑀𝑛} and {𝑇𝑛̃𝑧𝑛} are bounded. Let π‘Ÿ=sup{β€–π‘₯𝑛‖,‖𝑧𝑛‖,‖𝑇𝑛̃𝑧𝑛‖,‖𝑆𝑛𝑀𝑛‖}. By Lemma 2.7, (3.5), and (3.9), for any π‘βˆˆΞ©, we obtain πœ™ξ€·π‘,π‘Žπ‘›ξ€Έξ€·=πœ™π‘,π½βˆ’1𝛽𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›ξ€Έξ€Έ=‖𝑝‖2ξ«βˆ’2𝑝,𝛽𝑛𝐽π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›ξ¬+‖‖𝛽𝑛𝐽π‘₯𝑛+(1βˆ’π›½π‘›)π½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›β€–β€–2≀‖𝑝‖2βˆ’2π›½π‘›βŸ¨π‘,𝐽π‘₯π‘›ξ€·βŸ©βˆ’21βˆ’π›½π‘›ξ€Έξ«π‘,π½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›ξ¬+𝛽𝑛‖‖π‘₯𝑛‖‖2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›β€–β€–2βˆ’π›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–ξ€Έβ‰€π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπœ™ξ€·π‘,π‘†π‘›π‘€π‘›ξ€Έβˆ’π›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–ξ€Έβ‰€π›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπœ™ξ€·π‘,π‘€π‘›ξ€Έβˆ’π›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–ξ€Έξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έβˆ’π›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–ξ€Έ,πœ™ξ€·(3.18)𝑝,𝑏𝑛=πœ™π‘,π½βˆ’1̃𝛽𝑛𝐽𝑧𝑛+̃𝛽1βˆ’π‘›ξ€Έπ½π‘‡π‘›Μƒπ‘§π‘›β‰€Μƒπ›½ξ€Έξ€Έπ‘›πœ™ξ€·π‘,𝑧𝑛+̃𝛽1βˆ’π‘›ξ€Έπœ™ξ€·π‘,π‘‡π‘›Μƒπ‘§π‘›ξ€Έβˆ’Μƒπ›½π‘›ξ€·Μƒπ›½1βˆ’π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–ξ€Έβ‰€Μƒπ›½π‘›πœ™ξ€·π‘,𝑧𝑛+̃𝛽1βˆ’π‘›ξ€Έπœ™ξ€·π‘,Μƒπ‘§π‘›ξ€Έβˆ’Μƒπ›½π‘›ξ€·Μƒπ›½1βˆ’π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–ξ€Έξ€·β‰€πœ™π‘,π‘§π‘›ξ€Έβˆ’Μƒπ›½π‘›ξ€·Μƒπ›½1βˆ’π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–ξ€Έ.(3.19) It follows from (3.10), (3.12), and (3.19) that πœ™ξ€·π‘,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘,𝑦𝑛=πœ™π‘,π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ½π‘π‘›ξ€Έξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘π‘›ξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘§π‘›ξ€Έβˆ’Μƒπ›½π‘›ξ€·Μƒπ›½1βˆ’π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–β‰€ξ€·ξ€Έξ€Έξ‚π›Όπ‘›+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯π‘›ξ€Έβˆ’ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›ξ€·Μƒπ›½1βˆ’π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–ξ€Έ.(3.20) From (3.17) and (3.20), we have ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›ξ€·Μƒπ›½1βˆ’π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–ξ€Έβ‰€ξ€·ξ‚π›Όπ‘›+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·π›Όπ‘›ξ‚π›Όπ‘›βˆ’π›Όπ‘›βˆ’ξ‚π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛+πœ™π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛≀𝛼𝑛+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘₯0ξ€Έξ€·βˆ’πœ™π‘,π‘₯𝑛‖‖+2π‘π½π‘’π‘›βˆ’π½π‘₯𝑛‖‖+ξ€·β€–β€–π‘₯π‘›β€–β€–βˆ’β€–β€–π‘’π‘›β€–β€–ξ€Έξ€·β€–π‘₯𝑛‖+β€–π‘’π‘›β€–ξ€ΈβŸΆ0(π‘›βŸΆβˆž),(3.21) which together with condition (iii) implies that β€–β€–π½π‘§π‘›βˆ’π½π‘‡π‘›Μƒπ‘§π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.22) Since 𝐽 is uniformly continuous on bounded sets, then β€–β€–π‘§π‘›βˆ’π‘‡π‘›Μƒπ‘§π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.23) By (3.18) and (3.19), we have πœ™ξ€·π‘,π‘π‘›ξ€Έξ€·β‰€πœ™π‘,𝑧𝑛=πœ™π‘,π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’π›Όπ‘›ξ€Έπ½π‘Žπ‘›ξ€Έξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘Žπ‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘₯π‘›ξ€Έβˆ’π›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–.ξ€Έξ€Έ(3.24) It follows from (3.20) and (3.24) thatss πœ™ξ€·π‘,π‘’π‘›ξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘π‘›ξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›π›Όξ€Έξ€·π‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘₯π‘›ξ€Έβˆ’π›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–=𝛼𝑛+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯π‘›ξ€Έβˆ’ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–ξ€Έ.(3.25) So by (3.17) and (3.25), we obtain ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›½π‘›ξ€·1βˆ’π›½π‘›ξ€Έπ‘”ξ€·β€–β€–π½π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–ξ€Έβ‰€ξ€·ξ‚π›Όπ‘›+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘₯0ξ€Έξ€·βˆ’πœ™π‘,π‘₯𝑛+ξ€·πœ™ξ€·ξ€Έξ€Έπ‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛≀𝛼𝑛+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›πœ™ξ€·ξ€Έξ€·π‘,π‘₯0ξ€Έξ€·βˆ’πœ™π‘,π‘₯𝑛‖‖+2π‘π½π‘’π‘›βˆ’π½π‘₯𝑛‖‖+ξ€·β€–β€–π‘₯π‘›β€–β€–βˆ’β€–β€–π‘’π‘›β€–β€–β€–β€–π‘₯𝑛‖‖+β€–β€–π‘’π‘›β€–β€–ξ€ΈβŸΆ0(π‘›βŸΆβˆž),(3.26) which together with conditions (i) and (ii) implies that ‖‖𝐽π‘₯π‘›βˆ’π½π‘†π‘›π‘€π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.27) Since 𝐽 is uniformly continuous on bounded sets, then β€–β€–π‘₯π‘›βˆ’π‘†π‘›π‘€π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.28) It follows from (3.2) that 𝐽𝑧𝑛=𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’π›Όπ‘›π›½ξ€Έξ€·π‘›π½π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘†π‘›π½π‘Ÿπ‘›π‘£π‘›ξ€Έ.(3.29) Therefore, from (3.27) and condition (i), we have β€–β€–π½π‘§π‘›βˆ’π½π‘₯𝑛‖‖≀𝛼𝑛‖‖𝐽π‘₯0βˆ’π½π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π½π‘†π‘›π‘€π‘›βˆ’π½π‘₯π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.30) Thus, β€–β€–π‘§π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0(π‘›β†’βˆž).(3.31) From (3.7) and (3.9), we have πœ™ξ€·π‘,π‘§π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έξ‚€πœ™ξ€·π‘,π‘₯𝑛+2πœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Ύπ΄π‘₯π‘›β€–β€–βˆ’π΄π‘2=π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯𝑛+21βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έπœ†π‘›ξ‚€2𝑐2πœ†π‘›ξ‚β€–β€–βˆ’π›Ύπ΄π‘₯π‘›β€–β€–βˆ’π΄π‘2,(3.32) which together with (3.31) and condition (i) implies that 2ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έπœ†π‘›ξ‚€2π›Ύβˆ’π‘2πœ†π‘›ξ‚β€–β€–π΄π‘₯π‘›β€–β€–βˆ’π΄π‘2β‰€π›Όπ‘›ξ€·πœ™ξ€·π‘,π‘₯0ξ€Έξ€·βˆ’πœ™π‘,π‘₯𝑛+πœ™π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,π‘§π‘›ξ€ΈβŸΆ0(π‘›βŸΆβˆž).(3.33) Hence, ‖‖𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘βŸΆ0(π‘›βŸΆβˆž).(3.34) From Lemmas 2.1, 2.2, and 2.6, (3.34), and the fact that ‖𝐴π‘₯‖≀‖𝐴π‘₯βˆ’π΄π‘β€– for all π‘₯∈𝐢 and π‘βˆˆVI(𝐢,𝐴), we have πœ™ξ€·π‘₯𝑛,𝑣𝑛π‘₯=πœ™π‘›,Ξ πΆπ½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛π‘₯ξ€Έξ€Έβ‰€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‘₯=πœ™π‘›,π‘₯𝑛+2πœ†π‘›β€–β€–π½βˆ’1𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’π½βˆ’1𝐽π‘₯𝑛‖‖‖‖𝐴π‘₯𝑛‖‖≀2πœ†2𝑛2𝑐2‖‖𝐴π‘₯𝑛‖‖2≀2πœ†2𝑛2𝑐2‖‖𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘2⟢0(π‘›βŸΆβˆž).(3.35) This implies that β€–β€–π‘₯π‘›βˆ’π‘£π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.36) Combining (3.28) and (3.36), we have β€–β€–π‘£π‘›βˆ’π‘†π‘›π‘€π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.37) It follows from (3.6), (3.7), and (3.12) that πœ™ξ€·π‘,𝑀𝑛β‰₯πœ™ξ€·π‘,𝑒𝑛1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έβˆ’ξ€·ξ‚π›Όπ‘›+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έβˆ’ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έ.(3.38) By Lemma 2.9, (3.9), and (3.38), we have πœ™ξ€·π‘€π‘›,𝑣𝑛𝐽=πœ™π‘Ÿπ‘›π‘£π‘›,π‘£π‘›ξ€Έξ€·β‰€πœ™π‘,π‘£π‘›ξ€Έξ€·βˆ’πœ™π‘,π‘€π‘›ξ€Έξ€·β‰€πœ™π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,π‘€π‘›ξ€Έβ‰€πœ™ξ€·π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έ+𝛼𝑛+π›Όπ‘›βˆ’ξ‚π›Όπ‘›π›Όπ‘›ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’π›½π‘›ξ€Έξ€·πœ™ξ€·π‘,π‘₯0ξ€Έξ€·βˆ’πœ™π‘,π‘₯𝑛.ξ€Έξ€Έ(3.39) So, by (3.17) and conditions (i) and (ii), we have πœ™ξ€·π‘€π‘›,π‘£π‘›ξ€ΈβŸΆ0(π‘›βŸΆβˆž),(3.40) which implies that β€–β€–π‘€π‘›βˆ’π‘£π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.41) Combining (3.37) and (3.41), we obtain β€–β€–π‘€π‘›βˆ’π‘†π‘›π‘€π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.42) It follows from (3.6), (3.10), and (3.12) that πœ™ξ€·π‘,π‘’π‘›ξ€Έξ€·β‰€πœ™π‘,π‘¦π‘›ξ€Έβ‰€ξ‚π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›πœ™ξ€·π‘,𝑧𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έπœ™ξ€·π‘,̃𝑧𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€ΈΜƒπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έπœ™ξ€·π‘,̃𝑧𝑛,(3.43) which implies that πœ™ξ€·π‘,̃𝑧𝑛β‰₯πœ™ξ€·π‘,𝑒𝑛1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έβˆ’ξ€·ξ‚π›Όπ‘›+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€ΈΜƒπ›½π‘›π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯0ξ€Έξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έβˆ’ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€ΈΜƒπ›½π‘›πœ™ξ€·π‘,π‘₯𝑛1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έ.(3.44) Combining the above inequality, (3.10), and Lemma 2.9, we have πœ™ξ€·Μƒπ‘§π‘›,𝑧𝑛𝐽=πœ™π‘Ÿπ‘›π‘§π‘›,π‘§π‘›ξ‚ξ€·β‰€πœ™π‘,π‘§π‘›ξ€Έξ€·βˆ’πœ™π‘,Μƒπ‘§π‘›ξ€Έβ‰€π›Όπ‘›πœ™ξ€·π‘,π‘₯0ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,Μƒπ‘§π‘›ξ€Έβ‰€πœ™ξ€·π‘,π‘₯π‘›ξ€Έξ€·βˆ’πœ™π‘,𝑒𝑛1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έ+𝛼𝑛+ξ‚π›Όπ‘›βˆ’π›Όπ‘›ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έξ€·πœ™ξ€·π‘,π‘₯0ξ€Έξ€·βˆ’πœ™π‘,π‘₯𝑛.ξ€Έξ€Έ(3.45) By conditions (i) and (iii), (3.17), and (3.45), we have πœ™ξ€·Μƒπ‘§π‘›,π‘§π‘›ξ€ΈβŸΆ0(π‘›βŸΆβˆž),(3.46) which implies that β€–β€–Μƒπ‘§π‘›βˆ’π‘§π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.47) It follows from (3.23) and (3.47) that β€–β€–Μƒπ‘§π‘›βˆ’π‘‡π‘›Μƒπ‘§π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.48)
Step 3. Now we show that πœ”π‘€({π‘₯𝑛})βŠ‚Ξ©βˆΆ=(βˆ©βˆžπ‘›=0𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=0𝐹(𝑆𝑛))∩VI(𝐢,𝐴)∩GMEPβˆ©π‘‡βˆ’1𝑇0βˆ©βˆ’10, where πœ”π‘€π‘₯𝑛=ξ€½π‘₯ξ€Ύξ€Έβˆ—βˆˆπΆβˆΆπ‘₯π‘›π‘˜β‡€π‘₯βˆ—ξ€½π‘›forsomesequenceπ‘˜ξ€ΎβŠ‚{𝑛}withπ‘›π‘˜ξ€Ύβ†‘βˆž.(3.49)
Indeed, since {π‘₯𝑛} is bounded and 𝑋 is reflexive, we know that πœ”π‘€({π‘₯𝑛})β‰ βˆ…. For any arbitrary π‘₯βˆ—βˆˆπœ”π‘€({π‘₯𝑛}), there exists a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} such that π‘₯π‘›π‘˜β‡€π‘₯βˆ—. From (3.36), we have π‘£π‘›π‘˜β‡€π‘₯βˆ—. Since {𝑇𝑛},{𝑆𝑛} satisfy NST-conditions, from (3.42) and (3.48) we have π‘₯βˆ—βˆˆ(βˆ©βˆžπ‘›=0𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=0𝐹(𝑆𝑛)).
Let π‘†βŠ‚πΈΓ—πΈβˆ— be an operator as follows:𝑆𝑣=𝐴𝑣+𝑁𝐢(𝑣),π‘£βˆˆπΆ,βˆ…,π‘£βˆ‰πΆ.(3.50) By Lemma 2.5, 𝑆 is maximal monotone and π‘†βˆ’1(0)=VI(𝐢,𝐴). Let (𝑣,𝑀)∈𝐺(𝑆). Since π‘€βˆˆπ‘†π‘£=𝐴𝑣+𝑁𝐢(𝑣), we have π‘€βˆ’π΄π‘£βˆˆπ‘πΆ(𝑣). Moreover, π‘£π‘›βˆˆπΆ implies that βŸ¨π‘£βˆ’π‘£π‘›,π‘€βˆ’π΄π‘£βŸ©β‰₯0.(3.51)
On the other hand, it follows from 𝑣𝑛=Ξ πΆπ½βˆ’1(𝐽π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛) and Lemma 2.2 thatξ«π‘£βˆ’π‘£π‘›,π½π‘£π‘›βˆ’ξ€·π½π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛β‰₯0,(3.52) and hence ξƒ‘π‘£βˆ’π‘£π‘›,𝐽π‘₯π‘›βˆ’π½π‘£π‘›πœ†π‘›βˆ’π΄π‘₯𝑛≀0.(3.53) So, from (3.51), (3.53), and 𝐴 being 1/𝛾-Lipschitz continuous, we obtain 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=ξƒ‘π‘£βˆ’π‘£π‘›,π΄π‘£βˆ’π΄π‘₯𝑛+𝐽π‘₯π‘›βˆ’π½π‘£π‘›πœ†π‘›ξƒ’β‰₯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‘£π‘›βˆ’π΄π‘₯π‘›β€–β€–βˆ’β€–β€–π‘£βˆ’π‘£π‘›β€–β€–β€–β€–β€–π½π‘₯π‘›βˆ’π½π‘£π‘›π‘Žβ€–β€–β€–β‰₯βˆ’1π›Ύβ€–β€–π‘£βˆ’π‘£π‘›β€–β€–β€–β€–π‘£π‘›βˆ’π‘₯π‘›β€–β€–βˆ’β€–β€–π‘£βˆ’π‘£π‘›β€–β€–β€–β€–β€–π½π‘₯π‘›βˆ’π½π‘£π‘›π‘Žβ€–β€–β€–.(3.54) Since 𝐽 is uniformly continuous on bounded sets, by (3.36) and replacing 𝑛 by π‘›π‘˜ in (3.54), as π‘˜β†’βˆž we have βŸ¨π‘£βˆ’π‘₯βˆ—,π‘€βŸ©β‰₯0. Thus, π‘₯βˆ—βˆˆπ‘†βˆ’1(0), and hence π‘₯βˆ—βˆˆVI(𝐢,𝐴).
Next we show that π‘₯βˆ—βˆˆGMEP=𝐹(πΎπ‘Ÿ). Let 𝐻(𝑒𝑛,𝑦)=Θ(𝑒𝑛,𝑦)+βŸ¨π΅π‘’π‘›,π‘¦βˆ’π‘’π‘›βŸ©+πœ‘(𝑦)βˆ’πœ‘(𝑒𝑛), forallπ‘¦βˆˆπΆ. It follows from (3.2) that𝐽𝑦𝑛=𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ‚€Μƒπ›½π‘›π½π‘§π‘›+̃𝛽1βˆ’π‘›ξ€Έπ½π‘‡π‘›ξ‚π½π‘Ÿπ‘›π‘§π‘›ξ‚,(3.55) hence, from (3.22) and condition (i), we have β€–β€–π½π‘¦π‘›βˆ’π½π‘§π‘›β€–β€–β‰€ξ‚π›Όπ‘›β€–β€–π½π‘₯0βˆ’π½π‘§π‘›β€–β€–+ξ€·1βˆ’ξ‚π›Όπ‘›Μƒπ›½ξ€Έξ€·1βˆ’π‘›ξ€Έβ€–β€–π½π‘‡π‘›ξ‚π½π‘Ÿπ‘›π‘§π‘›βˆ’π½π‘§π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž),(3.56) which implies that β€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.57) From (3.17), (3.31), and (3.57), we obtain β€–β€–π‘¦π‘›βˆ’π‘’π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.58) Thus, π‘’π‘›π‘˜β‡€π‘₯βˆ—,π‘¦π‘›π‘˜β‡€π‘₯βˆ—(π‘˜β†’βˆž). Since 𝐽 is uniformly continuous on bounded sets, from (3.58) we have limπ‘˜β†’βˆžβ€–π½π‘’π‘›π‘˜βˆ’π½π‘¦π‘›π‘˜β€–=0. Therefore, it follows from π‘‘π‘›π‘˜β‰₯π‘βˆ— that β€–π½π‘’π‘›π‘˜βˆ’π½π‘¦π‘›π‘˜β€–/π‘‘π‘›π‘˜β†’0(π‘›β†’βˆž). Since 𝑒𝑛=𝐾𝑑𝑛𝑦𝑛, we have 𝐻𝑒𝑛+1,π‘¦π‘‘π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›βŸ©β‰₯0,βˆ€π‘¦βˆˆπΆ.(3.59) Combining the above inequality, (A2) and (A4), we get β€–β€–π‘¦βˆ’π‘’π‘›β€–β€–β€–β€–π½π‘’π‘›βˆ’π½π‘¦π‘›β€–β€–π‘‘π‘›β‰₯1π‘‘π‘›βŸ¨π‘¦βˆ’π‘’π‘›,π½π‘’π‘›βˆ’π½π‘¦π‘›ξ€·π‘’βŸ©β‰₯βˆ’π»π‘›ξ€Έξ€·,𝑦β‰₯𝐻𝑦,𝑒𝑛,βˆ€π‘¦βˆˆπΆ.(3.60) Replacing 𝑛 by π‘›π‘˜ and taking the limit as π‘˜β†’βˆž in the above inequality and by (A4), we have 𝐻(𝑦,π‘₯βˆ—)≀0,forallπ‘¦βˆˆπΆ. For any π‘‘βˆˆ(0,1) and π‘¦βˆˆπΆ, define 𝑦𝑑=𝑑𝑦+(1βˆ’π‘‘)π‘₯βˆ—βˆˆπΆ. So 𝐻(𝑦𝑑,π‘₯βˆ—)≀0. From (A1) and (A4), we have 𝑦0=𝐻𝑑,𝑦𝑑𝑦≀𝑑𝐻𝑑+𝑦,𝑦(1βˆ’π‘‘)𝐻𝑑,π‘₯βˆ—ξ€Έξ€·π‘¦β‰€π‘‘π»π‘‘ξ€Έ,𝑦,(3.61) that is, 𝐻(𝑦𝑑,𝑦)β‰₯0. Thus, from (A3), let 𝑑→0, we have 𝐻(π‘₯βˆ—,𝑦)β‰₯0,forallπ‘¦βˆˆπΆ, which implies that π‘₯βˆ—βˆˆGMEP.
From (3.31), (3.36), (3.41), and (3.47), similar to the proof of Step  5 in Theorem  3.1 of [19], we have π‘₯βˆ—βˆˆπ‘‡βˆ’1𝑇0βˆ©βˆ’10. Therefore, πœ”π‘€({π‘₯𝑛})βŠ‚Ξ©.
Step 4. Finally we prove that {π‘₯𝑛} converges strongly to Μƒπ‘₯=Ξ Ξ©π‘₯0.
It follows from Step  6 in Theorem  3.1 of [19] that we have the conclusion. This completes the proof.

If 𝐴≑0, then we have the following result from Theorem 3.1.

Corollary 3.2. Let 𝐸 be a real uniformly smooth and uniformly convex Banach space and 𝐢 a nonempty, closed, and convex subset of 𝐸. Let π΅βˆΆπΆβ†’πΈβˆ— be a monotone continuous mapping. Let {𝑇𝑛},{𝑆𝑛} be two countable families of relatively quasi-nonexpansive mappings from 𝐢 into itself satisfying NST-conditions such that Ω∢=(βˆ©βˆžπ‘›=0𝐹(𝑇𝑛))∩(βˆ©βˆžπ‘›=0𝐹(𝑆𝑛))βˆ©πΊπ‘€πΈπ‘ƒβˆ©π‘‡βˆ’1𝑇0βˆ©βˆ’10β‰ βˆ…. Let {𝑑𝑛}βŠ‚[π‘βˆ—,+∞) for some π‘βˆ—>0 and {π‘Ÿπ‘›}βŠ‚(0,+∞) satisfy liminfπ‘›β†’βˆžπ‘Ÿπ‘›>0. Let {π‘₯𝑛} be the sequence generated by π‘₯0π‘§βˆˆπΆ,π‘β„Žπ‘œπ‘ π‘’π‘›π‘Žπ‘Ÿπ‘π‘–π‘‘π‘Ÿπ‘Žπ‘Ÿπ‘–π‘™π‘¦,𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’π›Όπ‘›π›½ξ€Έξ€·π‘›π½π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π‘†π‘›π½π‘Ÿπ‘›π‘₯𝑛,𝑦𝑛=π½βˆ’1𝛼𝑛𝐽π‘₯0+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ‚€Μƒπ›½π‘›π½π‘§π‘›+̃𝛽1βˆ’π‘›ξ€Έπ½π‘‡π‘›ξ‚π½π‘Ÿπ‘›π‘§π‘›,𝑒𝑛=𝐾𝑑𝑛𝑦𝑛,𝐻𝑛=ξ€½ξ€·π‘§βˆˆπΆβˆΆπœ™π‘§,𝑒𝑛≀𝛼𝑛+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έπ›Όπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯0ξ€Έ+ξ€·1βˆ’ξ‚π›Όπ‘›ξ€Έξ€·1βˆ’π›Όπ‘›ξ€Έπœ™ξ€·π‘§,π‘₯𝑛,π‘Šξ€Έξ€Ύπ‘›=ξ€½π‘§βˆˆπΆβˆΆβŸ¨π‘₯π‘›βˆ’π‘§,𝐽π‘₯0βˆ’π½π‘₯𝑛,π‘₯⟩β‰₯0𝑛+1=Ξ π»π‘›βˆ©π‘Šπ‘›π‘₯0,𝑛β‰₯0,(3.62) where 𝐽 is the normalized duality mapping, {𝛼𝑛},{𝛽𝑛},{𝛼𝑛}, and {̃𝛽𝑛} are four sequences in [0,1], and πΎπ‘Ÿ is defined by (*). The following conditions hold:(i)limπ‘›β†’βˆžπ›Όπ‘›=limπ‘›β†’βˆžξ‚