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

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 Θ∶𝐶×𝐶→𝑅 be a bifunction and 𝜑∶𝐶→𝑅 a real-valued function. The generalized mixed equilibrium problem (GMEP) of finding 𝑥∈𝐶 is such that Θ(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)+⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(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 𝑥∈𝐶 such that Θ(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶,(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 𝑥∈𝐶 such that Θ(𝑥,𝑦)+⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶,(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 𝑥∈𝐶 such that Θ(𝑥,𝑦)≥0,∀𝑦∈𝐶.(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 𝑥∈𝐶 such that ⟨𝐴𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(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, 3–7].

The normalized duality mapping from 𝐸 to 2^𝐸∗ is defined by 𝐽𝑥=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²=‖𝑓‖²,𝑥∈𝐸,(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 𝐽⁻¹ is single valued, one to one, surjective, and it is the duality mapping from 𝐸^∗ into 𝐸 and thus 𝐽𝐽⁻¹=𝐼_𝐸^∗ and 𝐽⁻¹𝐽=𝐼_𝐸 (see [8]).

On the other hand, let 𝑊∶𝐸⇉𝐸^∗ be a set-valued mapping. The problem of finding 𝑣∈𝐸 satisfying 0∈𝑊𝑣 contains numerous problems in economics, optimization, and physics. Such 𝑣∈𝐸 is called a zero point of 𝑊.

A set-valued mapping 𝑊∶𝐸⇉𝐸^∗ with graph 𝐺(𝑊)={(𝑥,𝑥^∗)∶𝑥^∗∈𝑊𝑥}, domain 𝐷(𝑊)={𝑥∈𝐸∶𝑊𝑥≠∅}, and range 𝑅(𝑊)=∪{𝑊𝑥∶𝑥∈𝐷(𝑊)} is said to be monotone if ⟨𝑥−𝑦,𝑥^∗−𝑦^∗⟩≥0 for all 𝑥^∗∈𝑊𝑥,𝑦^∗∈𝑊𝑦. 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 𝑊∶𝐸⇉𝐸^∗ be a monotone operator satisfying 𝐷(𝑊)⊂𝐶⊂𝐽⁻¹(∩_𝑟>0𝑅(𝐽+𝑟𝑊)). Then the resolvent of 𝑊 defined by 𝐽_𝑟=(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 𝑊_𝑟𝑥=(𝐽𝑥−𝐽𝐽_𝑟𝑥)/𝑟 for all 𝑥∈𝐸.

A mapping 𝐴∶𝐶→𝐸^∗ is said to be monotone if, for each 𝑥,𝑦∈𝐶, ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥0.(1.7)𝐴 is said to be 𝛾-inverse strongly monotone if there exists a positive real number 𝛾>0 such that ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝛾‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶.(1.8) If 𝐴 is 𝛾-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛾, that is, ‖𝐴𝑥−𝐴𝑦‖≤1𝛾‖𝑥−𝑦‖,𝑥,𝑦∈𝐶.(1.9)

Let 𝐸 be a smooth Banach space. The function 𝜙∶𝐸×𝐸→𝑅 defined by 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸,(1.10) is studied by Alber [10], Kamimura and Takahashi [11], and Reich [12]. It follows from the definition of the function 𝜙 that (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,∀𝑥,𝑦∈𝐸.(1.11) Observe that, in a Hilbert space 𝐻, 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖²,forall𝑥,𝑦∈𝐻.

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 𝑥∈𝐸. Then there exists a unique element 𝑥₀∈𝐶 such that 𝜙(𝑥₀,𝑥)=min{𝜙(𝑧,𝑥)∶𝑧∈𝐶}.

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 Π_𝐶∶𝐸→𝐶 that assigns to an arbitrary point 𝑥∈𝐸, the minimum point of the functional 𝜙(𝑥,𝑦), that is, Π_𝐶𝑥=𝑥₀ due to Lemma 1.1, where 𝑥₀ is the solution to the minimization problem 𝜙(𝑥₀,𝑥)=min{𝜙(𝑧,𝑥)∶𝑧∈𝐶}.

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 {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑇𝑥_𝑛−𝑥_𝑛‖=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 ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and relatively nonexpansive (see [13, 14]) if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(𝑇). 𝑇 is said to be 𝜙-nonexpansive if 𝜙(𝑇𝑥,𝑇𝑦)≤𝜙(𝑥,𝑦) for all 𝑥,𝑦∈𝐶. 𝑇 is said to be relatively quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶 and 𝑝∈𝐹(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 0∈𝑊𝑣 in a Hilbert space 𝐻 is the proximal point algorithm (see [15]): 𝑥₁=𝑥∈𝐻 and 𝑥_𝑛+1=𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,…,(1.12) where {𝑟_𝑛}⊂(0,∞) and 𝐽_𝑟=(𝐼+𝑟𝑊)⁻¹ for all 𝑟>0 is the resolvent operator for 𝑊, then Rockafellar proved that the sequence {𝑥_𝑛} converges weakly to an element of 𝑊⁻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: 𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛𝐽𝑥₁+1−𝛽_𝑛𝐽𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,…,(1.13) and Kamimura et al. [17] considered Algorithm (1.14) in a uniformly smooth and uniformly convex Banach space: 𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝐽_𝑟𝑛𝑥_𝑛,𝑛=1,2,….(1.14) They showed that Algorithm (1.13) converges strongly and Algorithm (1.14) converges weakly provided that the sequences {𝛽_𝑛}, {𝑟_𝑛} of real numbers are chosen appropriately.

Recently, Saewan and Kumam [18] proposed the following iterative scheme: for an initial 𝑥₀∈𝐸 with 𝑥₁=Π_𝐶1𝑥₀ and 𝐶₁=𝐶𝜔𝑧_𝑛𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇_𝑛𝐽_𝑟𝑛𝜔_𝑛,𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑆_𝑛𝑧_𝑛,𝐶=_𝑛+1𝐾_𝑟𝑛𝑦_𝑛,𝑥=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑧_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₀,𝑛≥0,(1.15) where 𝐾_𝑟 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 𝑥₀∈𝐸 be chosen arbitrarily, 𝑦𝑥̃_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝐽_𝑟𝑛𝑣_𝑛_𝑛,𝑢=𝐽⁻¹𝛼^𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝐽𝑥̃_𝑛+1−𝛽̃_𝑛𝐽𝐽_𝑟𝑛𝑥̃_𝑛_𝑛,𝐻_𝑛=𝐾_𝑟𝑛𝑦_𝑛,𝑊=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛≤𝛼_𝑛+1−𝛼_𝑛𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑧,𝑥_𝑛_𝑛,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛≥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 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, let {𝑇_𝑛} be a family of mappings of 𝐶 into itself with 𝐹=∩^∞𝑛=1𝐹(𝑇_𝑛)≠∅, and 𝜔_𝑤(𝑧_𝑛) denotes the set of all weak subsequential limits of a bounded sequence {𝑧_𝑛} in 𝐶. {𝑇_𝑛} is said to satisfy the NST-condition if, for every bounded sequence {𝑧_𝑛} in 𝐶, lim_𝑛→∞‖‖𝑧_𝑛−𝑆_𝑛𝑧_𝑛‖‖=0impliesthat𝜔_𝑤𝑧_𝑛⊂𝐹.(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 𝐸≥2. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,+∞)→[0,+∞) defined by 𝜌_𝐸(𝜏)∶=sup‖𝑥+𝑦‖+‖𝑥−𝑦‖2−1∶‖𝑥‖=1,‖𝑦‖=𝜏.(2.1) The space 𝐸 is said to be smooth if 𝜌_𝐸(𝜏)>0,forall𝜏>0, and 𝐸 is called uniformly smooth if and only if lim_𝑡→0⁺(𝜌_𝐸(𝑡)/𝑡)=0.

The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜀)∶=inf1−‖‖‖𝑥+𝑦2‖‖‖∶‖𝑥‖=‖𝑦‖=1;𝜀=‖𝑥−𝑦‖.(2.2)𝐸 is called uniformly convex if and only if 𝛿_𝐸(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 𝛿_𝐸(𝜀)≥𝑐𝜀^𝑝 for every 𝜀∈(0,2]. Observe that every 𝑝-uniformly convex is uniformly convex. It is well known (see, e.g., [7]) that 𝐿_𝑝𝑙_𝑝or𝑊^𝑝𝑚is𝑝2--uunniiffoorrmmllyyccoonnvveexxiiff𝑝1≥<2𝑝;≤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 𝑥,𝑦∈𝐸, one has ‖𝑥−𝑦‖≤2𝑐²‖𝐽𝑥−𝐽𝑦‖,(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 𝐶 a nonempty closed convex subset. Then the following conclusions hold: (1)𝜙(𝑦,Π_𝐶𝑥)+𝜙(Π_𝐶𝑥,𝑥)≤𝜙(𝑦,𝑥),∀𝑥∈𝐸,𝑦∈𝐶; (2)suppose 𝑥∈𝐸 and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥⟺⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩≥0,∀𝑦∈𝐶.(2.5)

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

Lemma 2.4 (see [25]). Let 𝐸 be a real smooth Banach space, and let 𝐴∶𝐸⇉𝐸^∗ be a maximal monotone mapping, then 𝐴⁻¹(0) is a closed and convex subset of 𝐸.

We denote by 𝑁_𝐶(𝑣) the normal cone for 𝐶 at a point 𝑣∈𝐶, that is, 𝑁_𝐶(𝑣)∶={𝑥^∗∈𝐸^∗∶⟨𝑣−𝑦,𝑥^∗⟩≥0forall𝑦∈𝐶}. 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 𝑇⊂𝐸×𝐸^∗ be an operator defined as follows: 𝑇𝑣=𝐴𝑣+𝑁^𝐶∅,(𝑣),𝑣𝑣∉∈𝐶𝐶.,(2.6) Then 𝑇 is maximal monotone and 𝑇⁻¹0=VI(𝐶,𝐴).

We make use of the function 𝑉∶𝐸×𝐸^∗→𝑅 defined by 𝑉𝑥,𝑥^∗=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥‖²,∀𝑥∈𝐸,𝑥^∗∈𝐸^∗,(2.7) studied by Alber [10]; that is, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹𝑥^∗) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗. 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 𝑉𝑥,𝑥^∗+2𝐽⁻¹𝑥^∗−𝑥,𝑦^∗≤𝑉𝑥,𝑥^∗+𝑦^∗,(2.8) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Lemma 2.7 (see [26]). Let 𝐸 be a uniformly convex Banach space, and let 𝐵_𝑟(0)={𝑥∈𝐸∶‖𝑥‖≤𝑟} 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+𝛾𝑧‖²≤𝜆‖𝑥‖²+𝜇‖𝑦‖²+𝛾‖𝑧‖²−𝜆𝜇𝑔(‖𝑥−𝑦‖),(2.9) for all 𝑥,𝑦,𝑧∈𝐵_𝑟(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 𝑥∈𝐶;(A2)Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶;(A3)for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡→0Θ(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤Θ(𝑥,𝑦);(2.10)(A4)for each 𝑥∈𝐶,𝑦↦Θ(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 𝐵∶𝐶→𝐸^∗ be a continuous and monotone mapping, let 𝜑∶𝐶→𝑅 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 𝑥∈𝐸, then there exists 𝑢∈𝐶 such that Θ(𝑢,𝑦)+⟨𝐵𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.11) Define a mapping 𝐾_𝑟∶𝐸→𝐶 as follows: 𝐾_𝑟(𝑥)=𝑢∈𝐶∶Θ(𝑢,𝑦)+⟨𝐵𝑢,𝑦−𝑢⟩+𝜑(𝑦)−𝜑(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶,(∗) for all 𝑥∈𝐸. Then, the following conclusions hold:(1)𝐾_𝑟 is single valued;(2)𝐾_𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑦∈𝐸, ⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝐾_𝑟𝑥−𝐽𝐾_𝑟𝑦⟩≤⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝑥−𝐽𝑦⟩;(3)𝐹(𝐾_𝑟)=GMEP;(4)GMEP is closed and convex;(5)𝜙(𝑝,𝐾_𝑟𝑧)+𝜙(𝐾_𝑟𝑧,𝑧)≤𝜙(𝑝,𝑧),∀𝑝∈𝐹(𝐾_𝑟),𝑧∈𝐸.

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 𝑊∶𝐸⇉𝐸^∗ be a monotone operator satisfying 𝐷(𝑊)⊂𝐶⊂𝐽⁻¹(∩_𝑟>0𝑅(𝐽+𝑟𝑊)). Let 𝑟>0, and let 𝐽_𝑟 and 𝑊_𝑟 be the resolvent and the Yosida approximation of 𝑊, respectively. Then the following hold:(i)𝜙(𝑢,𝐽_𝑟𝑥)+𝜙(𝐽_𝑟𝑥,𝑥)≤𝜙(𝑢,𝑥), for all 𝑥∈𝐶,𝑢∈𝑊⁻¹0;(ii)(𝐽_𝑟𝑥,𝑊_𝑟𝑥)∈𝐺(𝑊), for all 𝑥∈𝐶;(iii)𝐹(𝐽_𝑟)=𝑊⁻¹0.

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 𝑆∶𝐶→𝐶 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,𝑇∶𝐸⇉𝐸^∗ be two maximal monotone operators satisfying 𝐷(𝑇),𝐷(𝑇)⊂𝐶. We denote the resolvent operators of 𝑇 and 𝑇 by 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 and 𝐽_𝑟=(𝐽+𝑟𝑇)⁻¹𝐽 for each 𝑟>0, respectively. For each 𝑟>0, the Yosida approximations of 𝑇 and 𝑇 are defined by 𝐴_𝑟=(𝐽−𝐽𝐽_𝑟)/𝑟 and 𝐴_𝑟=(𝐽−𝐽𝐽_𝑟)/𝑟, respectively. It is known that 𝐴_𝑟𝑥∈𝑇𝐽_𝑟𝑥,𝐴_𝑟𝑥∈𝑇𝐽_𝑟𝑥,foreach𝑟>0,𝑥∈𝐸.(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 ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖ for all 𝑥∈𝐶 and 𝑝∈𝑉𝐼(𝐶,𝐴). Let 𝐵∶𝐶→𝐸^∗ be a monotone continuous mapping, and let {𝑇_𝑛},{𝑆_𝑛} be two countable families of relatively quasi-nonexpansive mappings from 𝐶 into itself satisfying NST-conditions such that Ω∶=(∩^∞𝑛=0𝐹(𝑇_𝑛))∩(∩^∞𝑛=0𝐹(𝑆_𝑛))∩VI(𝐶,𝐴)∩𝐺𝑀𝐸𝑃∩𝑇⁻¹0∩𝑇⁻¹0≠∅. Suppose that 0<𝑎<𝜆_𝑛<𝑏<(𝑐²𝛾)/2, where 𝑐 is the constant in (2.4). Let {𝑡_𝑛}⊂[𝑐^∗,+∞) for some 𝑐^∗>0 and {𝑟_𝑛}⊂(0,+∞) satisfy liminf_𝑛→∞𝑟_𝑛>0. Let {𝑥_𝑛} be the sequence generated by 𝑥𝑣₀∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛𝑧_𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑦=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛_𝑛,𝑢=𝐽⁻¹𝛼^𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝐽𝑧_𝑛+1−𝛽̃_𝑛𝐽𝑇_𝑛𝐽_𝑟𝑛𝑧_𝑛_𝑛,𝐻_𝑛=𝐾_𝑡𝑛𝑦_𝑛,𝑊=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛≤𝛼_𝑛+1−𝛼_𝑛𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑧,𝑥_𝑛_𝑛,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛≥0,(3.2) where 𝐽 is the normalized duality mapping, {𝛼_𝑛},{𝛽_𝑛},{𝛼_𝑛}, and {𝛽̃_𝑛} are four sequences in [0,1], and 𝐾_𝑟 is defined by (*). The following conditions hold:(i)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛼_𝑛=0; (ii)liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0; (iii)liminf_𝑛→∞𝛽̃_𝑛(1−𝛽̃_𝑛)>0.

Then {𝑥_𝑛} converges strongly to Π_Ω𝑥₀.

Proof. We have the following steps.Step 1. First we prove that 𝐻_𝑛 and 𝑊_𝑛 are both closed and convex and Ω⊂𝐻_𝑛∩𝑊_𝑛,forall𝑛≥0.
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 𝑊_𝑛 is closed and convex for each 𝑛≥0. Let 𝛾_𝑛=𝛼_𝑛+(1−𝛼_𝑛)𝛼_𝑛,𝑤_𝑛=𝐽_𝑟𝑛𝑣_𝑛,𝑧_𝑛̃=𝐽_𝑟𝑛𝑧_𝑛. For any 𝑧∈𝐶, 𝜙𝑧,𝑢_𝑛≤𝛾_𝑛𝜙𝑧,𝑥₀+1−𝛾_𝑛𝜙𝑧,𝑥_𝑛(3.3) is equivalent to −2⟨𝑧,𝐽𝑢_𝑛⟩+‖‖𝑢_𝑛‖‖²≤−2𝛾_𝑛⟨𝑧,𝐽𝑥₀⟩+𝛾_𝑛‖‖𝑥₀‖‖²+1−𝛾_𝑛−2⟨𝑧,𝐽𝑥_𝑛⟩+‖‖𝑥_𝑛‖‖²,(3.4) which implies that 𝐻_𝑛 is closed and convex for each 𝑛≥0. Next, we prove that Ω⊂𝐻_𝑛∩𝑊_𝑛,forall𝑛≥0. For any given 𝑝∈Ω, by Lemma 2.9 we have 𝜙𝜙𝑝𝑝,,𝑧𝑤_𝑛̃_𝑛=𝜙𝑝,𝐽_𝑟𝑛𝑣_𝑛≤𝜙𝑝,𝑣_𝑛,=𝜙𝑝,𝐽_𝑟𝑛𝑧_𝑛≤𝜙𝑝,𝑧^𝑛.(3.5) Since 𝑇_𝑛,𝑆_𝑛 are relatively quasi-nonexpansive, from the definition of 𝜙(𝑥,𝑦), the convexity of ‖⋅‖², and (3.5), we have 𝜙𝑝,𝑦_𝑛==𝜙‖𝑝‖𝑝₂,𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝐽𝑧_𝑛+1−𝛽̃_𝑛𝐽𝑇_𝑛𝑧_𝑛̃+‖‖𝛼_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥₀⟩−21−𝛼_𝑛𝛽̃_𝑛⟨𝑝,𝐽𝑧_𝑛⟩−21−𝛼_𝑛1−𝛽̃_𝑛⟨𝑝,𝐽𝑇_𝑛𝑧_𝑛̃⟩𝐽𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝐽𝑧_𝑛+1−𝛽̃_𝑛𝐽𝑇_𝑛𝑧_𝑛̃‖‖²≤+‖𝑝𝛼‖²_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥₀⟩−21−𝛼_𝑛𝛽̃_𝑛⟨𝑝,𝐽𝑧_𝑛⟩−21−𝛼_𝑛1−𝛽̃_𝑛⟨𝑝,𝐽𝑇_𝑛𝑧_𝑛̃⟩=𝛼_𝑛‖‖𝑥₀‖‖²+1−𝛼_𝑛𝛽̃_𝑛‖‖𝑧_𝑛‖‖²+1−𝛽̃_𝑛‖‖𝑇_𝑛𝑧_𝑛̃‖‖²≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝜙𝑝,𝑧_𝑛+1−𝛼_𝑛1−𝛽̃_𝑛𝜙𝑝,𝑇_𝑛𝑧_𝑛̃≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝜙𝑝,𝑧_𝑛+1−𝛼_𝑛1−𝛽̃_𝑛𝜙𝑝,𝑧_𝑛̃𝜙𝑝,𝑧_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑧_𝑛,(3.6)=≤𝜙‖𝑝‖𝑝₂,𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆_𝑛𝑤_𝑛+𝛼_𝑛−2𝛼_𝑛⟨𝑝,𝐽𝑥₀⟩−21−𝛼_𝑛𝛽_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−21−𝛼_𝑛1−𝛽_𝑛⟨𝑝,𝐽𝑆_𝑛𝑤_𝑛⟩=𝛼_𝑛‖‖𝑥₀‖‖²+1−𝛼_𝑛𝛽_𝑛‖‖𝑥_𝑛‖‖²+1−𝛽_𝑛‖‖𝑆_𝑛𝑤_𝑛‖‖²≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝛽_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝜙𝑝,𝑆_𝑛𝑤_𝑛≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝛽_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝜙𝑝,𝑤_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝛽_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝜙𝑝,𝑣_𝑛.(3.7) Moreover, it follows from Lemmas 2.2 and 2.6 that 𝜙𝑝,𝑣_𝑛≤≤𝜙𝑉𝑝𝑝,,𝐽𝐽𝑥⁻¹_𝑛𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛≤𝑉𝑝,𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛=𝜙𝑝,𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛+𝜆_𝑛𝐴𝑥_𝑛−2𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑝,𝜆_𝑛𝐴𝑥_𝑛−2𝜆_𝑛−2𝜆_𝑛⟨𝑥_𝑛−𝑝,𝐴𝑥_𝑛−𝐴𝑝⟩−2𝜆_𝑛⟨𝑥_𝑛−𝑝,𝐴𝑝⟩𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑥_𝑛,𝐴𝑥_𝑛.(3.8) Since 𝑝∈VI(𝐶,𝐴), 𝐴 is 𝛾-inverse strongly monotone, from the above inequality, Lemma 2.1, and the fact that ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖ for all 𝑥∈𝐶 and 𝑝∈VI(𝐶,𝐴), we obtain 𝜙𝑝,𝑣_𝑛≤≤𝜙𝜙𝑝𝑝,,𝑥𝑥_𝑛𝑛−2𝜆_𝑛𝛾‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²+2𝜆_𝑛‖‖𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑥_𝑛‖‖‖‖𝐴𝑥_𝑛‖‖−2𝜆_𝑛𝛾‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²+4𝑐²𝜆^2𝑛‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²=𝜙𝑝,𝑥_𝑛+2𝜆_𝑛2𝑐²𝜆_𝑛−𝛾‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²≤𝜙𝑝,𝑥_𝑛.(3.9) From (3.7)-(3.9), we have 𝜙𝑝,𝑧_𝑛≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛.(3.10) So from (3.6) and (3.10), we have 𝜙𝑝,𝑦_𝑛≤𝛼_𝑛+1−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛.(3.11) By Lemma 2.8(5) and (3.11), we have 𝜙𝑝,𝑢_𝑛=≤𝜙𝛼_𝑛𝑝,𝐾_𝑡𝑛𝑦_𝑛≤𝜙𝑝,𝑦_𝑛+1−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛.(3.12) Therefore, 𝑝∈𝐻_𝑛; that is Ω⊂𝐻_𝑛 for each 𝑛≥0.
Next we prove that Ω⊂𝐻_𝑛∩𝑊_𝑛 for each 𝑛≥0 by induction. For 𝑛=0,𝑊₀=𝐶, thus Ω⊂𝐻₀∩𝑊₀. Suppose that Ω⊂𝐻_𝑛∩𝑊_𝑛 for some 𝑛≥1. Since 𝑥_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀, by Lemma 2.2, for any 𝑞∈𝐻_𝑛∩𝑊_𝑛, we have𝑥_𝑛+1−𝑞,𝐽𝑥₀−𝐽𝑥_𝑛+1≥0.(3.13) Since Ω⊂𝐻_𝑛∩𝑊_𝑛, for any 𝑝∈Ω, we have 𝑥_𝑛+1−𝑝,𝐽𝑥₀−𝐽𝑥_𝑛+1≥0,(3.14) which implies that 𝑝∈𝑊_𝑛+1, that is, Ω⊂𝐻_𝑛∩𝑊_𝑛 for each 𝑛≥0.Step 2. Next we prove that ‖𝑤_𝑛−𝑆_𝑛𝑤_𝑛‖→0,‖𝑧_𝑛̃−𝑇_𝑛𝑧_𝑛̃‖→0(𝑛→∞).
Similar to the proof of Step  3 in [19, Theorem  3.1], we have that {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing and bounded and 𝜙(𝑥_𝑛+1,𝑥_𝑛)→0(𝑛→∞). So {𝑥_𝑛} is bounded, then, by (3.5)–(3.12), we obtain that {𝑢_𝑛},{𝑧_𝑛},{𝑣_𝑛},{𝑤_𝑛}, {𝑧_𝑛̃}, and {𝑦_𝑛} are all bounded. Furthermore, {𝑆_𝑛𝑤_𝑛} and {𝑇_𝑛𝑧_𝑛̃} are both bounded. By Lemma 2.3, we have‖‖𝑥_𝑛−𝑥_𝑛+1‖‖⟶0(𝑛⟶∞).(3.15) Since 𝑥_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀∈𝐻_𝑛 and by condition (i), we have 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝛼_𝑛+1−𝛼_𝑛𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛⟶0(𝑛⟶∞),(3.16) which together with Lemma 2.3 implies that ‖𝑥_𝑛+1−𝑢_𝑛‖→0(𝑛→∞). So we obtain ‖‖𝑥_𝑛−𝑢_𝑛‖‖⟶0(𝑛⟶∞).(3.17) Let 𝑎_𝑛=𝐽⁻¹(𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛),𝑏_𝑛=𝐽⁻¹(𝛽̃_𝑛𝐽𝑧_𝑛+(1−𝛽̃_𝑛)𝐽𝑇_𝑛𝐽_𝑟𝑛𝑧_𝑛). It follows from the boundedness of {𝑥_𝑛} that {𝑆_𝑛𝑤_𝑛} and {𝑇_𝑛𝑧_𝑛̃} are bounded. Let 𝑟=sup{‖𝑥_𝑛‖,‖𝑧_𝑛‖,‖𝑇_𝑛𝑧_𝑛̃‖,‖𝑆_𝑛𝑤_𝑛‖}. By Lemma 2.7, (3.5), and (3.9), for any 𝑝∈Ω, we obtain 𝜙𝑝,𝑎_𝑛==‖𝜙𝑝‖𝑝₂,𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛−2𝑝,𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛+‖‖𝛽_𝑛𝐽𝑥_𝑛+(1−𝛽_𝑛)𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛‖‖²≤+‖𝑝𝛽‖²_𝑛−2𝛽_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−21−𝛽_𝑛𝑝,𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛≤𝛽_𝑛‖‖𝑥_𝑛‖‖²+1−𝛽_𝑛‖‖𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛‖‖²−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+1−𝛽_𝑛𝜙𝑝,𝑆_𝑛𝑤_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖≤𝜙𝜙𝑝,𝑥𝑝,_𝑛𝑥_𝑛+1−𝛽_𝑛𝜙𝑝,𝑤_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖𝜙𝑝,𝑏_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖,(3.18)=≤𝜙𝛽̃𝑝,𝐽⁻¹𝛽̃_𝑛𝐽𝑧_𝑛+1−𝛽̃_𝑛𝐽𝑇_𝑛𝑧_𝑛̃_𝑛≤𝛽̃_𝑛𝜙𝑝,𝑧_𝑛+1−𝛽̃_𝑛𝜙𝑝,𝑇_𝑛𝑧_𝑛̃−𝛽̃_𝑛1−𝛽̃_𝑛𝑔‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖≤𝜙𝜙𝑝,𝑧𝑝_𝑛,𝑧_𝑛+1−𝛽̃_𝑛𝜙𝑝,𝑧_𝑛̃−𝛽̃_𝑛1−𝛽̃_𝑛𝑔‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖−𝛽̃_𝑛1−𝛽̃_𝑛𝑔‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖.(3.19) It follows from (3.10), (3.12), and (3.19) that 𝜙𝑝,𝑢_𝑛≤=𝜙𝜙𝑝𝑝,,𝐽𝑦_𝑛₋₁≤𝛼_𝑛𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑏_𝑛≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑏_𝑛≤𝛼𝜙_𝑛𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑧_𝑛−𝛽̃_𝑛1−𝛽̃_𝑛𝑔‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖−1+−𝛼𝛼_𝑛𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛𝛽̃_𝑛1−𝛽̃_𝑛𝑔‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖.(3.20) From (3.17) and (3.20), we have 1≤−𝛼𝛼_𝑛_𝑛𝛽̃_𝑛1−𝛽̃_𝑛𝑔‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖≤𝛼_𝑛+𝛼_𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀+𝛼_𝑛𝛼_𝑛−𝛼_𝑛−𝛼_𝑛𝜙𝑝,𝑥_𝑛+𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛+2𝑝+‖‖𝐽𝛼𝑢_𝑛𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛−𝐽𝑥_𝑛‖‖+‖‖𝑥_𝑛‖‖−‖‖𝑢_𝑛‖‖‖𝑥_𝑛‖+‖𝑢_𝑛‖⟶0(𝑛⟶∞),(3.21) which together with condition (iii) implies that ‖‖𝐽𝑧_𝑛−𝐽𝑇_𝑛𝑧_𝑛̃‖‖⟶0(𝑛⟶∞).(3.22) Since 𝐽 is uniformly continuous on bounded sets, then ‖‖𝑧_𝑛−𝑇_𝑛𝑧_𝑛̃‖‖⟶0(𝑛⟶∞).(3.23) By (3.18) and (3.19), we have 𝜙𝑝,𝑏_𝑛=≤𝜙𝜙𝑝𝑝,,𝐽𝑧_𝑛₋₁≤𝛼_𝑛𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑎_𝑛≤𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑎_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖.(3.24) It follows from (3.20) and (3.24) thatss 𝜙𝑝,𝑢_𝑛≤≤𝛼𝛼_𝑛𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑏_𝑛+𝜙1−𝑝,𝛼𝑥_0𝑛=𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖−1+−𝛼𝛼_𝑛𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖.(3.25) So by (3.17) and (3.25), we obtain 1≤−𝛼𝛼_𝑛_𝑛𝛽_𝑛1−𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖≤𝛼_𝑛+𝛼_𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛+𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛+‖‖𝑥+_𝑛𝛼_𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛+2𝑝‖‖𝐽𝑢_𝑛−𝐽𝑥_𝑛‖‖‖‖−‖‖𝑢_𝑛‖‖‖‖𝑥_𝑛‖‖+‖‖𝑢_𝑛‖‖⟶0(𝑛⟶∞),(3.26) which together with conditions (i) and (ii) implies that ‖‖𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑤_𝑛‖‖⟶0(𝑛⟶∞).(3.27) Since 𝐽 is uniformly continuous on bounded sets, then ‖‖𝑥_𝑛−𝑆_𝑛𝑤_𝑛‖‖⟶0(𝑛⟶∞).(3.28) It follows from (3.2) that 𝐽𝑧_𝑛=𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆_𝑛𝐽_𝑟𝑛𝑣_𝑛.(3.29) Therefore, from (3.27) and condition (i), we have ‖‖𝐽𝑧_𝑛−𝐽𝑥_𝑛‖‖≤𝛼_𝑛‖‖𝐽𝑥₀−𝐽𝑥_𝑛‖‖+1−𝛼_𝑛1−𝛽_𝑛‖‖𝐽𝑆_𝑛𝑤_𝑛−𝐽𝑥_𝑛‖‖⟶0(𝑛⟶∞).(3.30) Thus, ‖‖𝑧_𝑛−𝑥_𝑛‖‖⟶0(𝑛→∞).(3.31) From (3.7) and (3.9), we have 𝜙𝑝,𝑧_𝑛≤+𝛼_𝑛𝜙1−𝑝,𝛼𝑥_0𝑛+1−𝛼_𝑛𝛽_𝑛𝜙𝑝,𝑥_𝑛1−𝛽_𝑛𝜙𝑝,𝑥_𝑛+2𝜆_𝑛2𝑐²=𝛼_𝑛𝜆_𝑛−𝛾‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²𝜙𝑝,𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛+21−𝛼_𝑛1−𝛽_𝑛𝜆_𝑛2𝑐²𝜆_𝑛−𝛾‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²,(3.32) which together with (3.31) and condition (i) implies that 21−𝛼_𝑛1−𝛽_𝑛𝜆_𝑛𝛾−2𝑐²𝜆_𝑛‖‖𝐴𝑥^𝑛−𝐴𝑝‖‖²≤𝛼_𝑛𝜙𝑝,𝑥₀−𝜙𝑝,𝑥_𝑛+𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛⟶0(𝑛⟶∞).(3.33) Hence, ‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖⟶0(𝑛⟶∞).(3.34) From Lemmas 2.1, 2.2, and 2.6, (3.34), and the fact that ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖ for all 𝑥∈𝐶 and 𝑝∈VI(𝐶,𝐴), we have 𝜙𝑥_𝑛,𝑣_𝑛=≤𝜙𝜙𝑥𝑥_𝑛𝑛,Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛=𝑉𝑥_𝑛,𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛≤𝑉𝑥_𝑛,𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛=𝜙𝑥_𝑛,𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛+𝜆_𝑛𝐴𝑥_𝑛−2𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑥_𝑛,𝜆_𝑛𝐴𝑥_𝑛≤2𝜆_2𝑛,𝑥_𝑛+2𝜆_𝑛‖‖𝐽⁻¹𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝐽⁻¹𝐽𝑥_𝑛‖‖‖‖𝐴𝑥_𝑛‖‖2𝑐²‖‖𝐴𝑥_𝑛‖‖²≤2𝜆^2𝑛2𝑐²‖‖𝐴𝑥_𝑛−𝐴𝑝‖‖²⟶0(𝑛⟶∞).(3.35) This implies that ‖‖𝑥_𝑛−𝑣_𝑛‖‖⟶0(𝑛⟶∞).(3.36) Combining (3.28) and (3.36), we have ‖‖𝑣_𝑛−𝑆_𝑛𝑤_𝑛‖‖⟶0(𝑛⟶∞).(3.37) It follows from (3.6), (3.7), and (3.12) that 𝜙𝑝,𝑤_𝑛≥𝜙𝑝,𝑢^𝑛1−𝛼_𝑛1−𝛼_𝑛1−𝛽_𝑛−𝛼^𝑛+𝛼_𝑛−𝛼_𝑛𝛼_𝑛𝜙𝑝,𝑥₀−1−𝛼_𝑛1−𝛼_𝑛1−𝛼_𝑛1−𝛽_𝑛1−𝛼_𝑛𝛽_𝑛𝜙𝑝,𝑥_𝑛1−𝛼_𝑛1−𝛼_𝑛1−𝛽_𝑛.(3.38) By Lemma 2.9, (3.9), and (3.38), we have 𝜙𝑤_𝑛,𝑣_𝑛=≤𝜙𝜙𝐽𝑝,_𝑟𝑛𝑥𝑣_𝑛𝑛,𝑣_𝑛≤𝜙𝑝,𝑣_𝑛−𝜙𝑝,𝑤_𝑛≤𝜙𝑝,𝑥−_𝑛𝜙𝑝,𝑤_𝑛−𝜙𝑝,𝑢_𝑛1−𝛼_𝑛1−𝛼_𝑛1−𝛽_𝑛+𝛼^𝑛+𝛼_𝑛−𝛼_𝑛𝛼_𝑛1−𝛼_𝑛1−𝛼_𝑛1−𝛽_𝑛𝜙𝑝,𝑥⁰−𝜙𝑝,𝑥_𝑛.(3.39) So, by (3.17) and conditions (i) and (ii), we have 𝜙𝑤_𝑛,𝑣_𝑛⟶0(𝑛⟶∞),(3.40) which implies that ‖‖𝑤_𝑛−𝑣_𝑛‖‖⟶0(𝑛⟶∞).(3.41) Combining (3.37) and (3.41), we obtain ‖‖𝑤_𝑛−𝑆_𝑛𝑤_𝑛‖‖⟶0(𝑛⟶∞).(3.42) It follows from (3.6), (3.10), and (3.12) that 𝜙𝑝,𝑢_𝑛≤≤𝜙𝛼_𝑛𝑝,𝑦_𝑛≤𝛼𝜙_𝑛𝑝,𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝜙𝑝,𝑧_𝑛+1−𝛼_𝑛1−𝛽̃_𝑛𝜙𝑝,𝑧_𝑛̃+1+−𝛼1_𝑛−𝛼_𝑛𝛽̃_𝑛𝛼_𝑛𝜙𝑝,𝑥₀1−𝛼_𝑛𝛽̃_𝑛𝜙𝑝,𝑥_𝑛+1−𝛼_𝑛1−𝛽̃_𝑛𝜙𝑝,𝑧_𝑛̃,(3.43) which implies that 𝜙𝑝,𝑧_𝑛̃≥𝜙𝑝,𝑢^𝑛1−𝛼_𝑛1−𝛽̃_𝑛−𝛼^𝑛+1−𝛼_𝑛𝛽̃_𝑛𝛼_𝑛𝜙𝑝,𝑥₀1−𝛼_𝑛1−𝛽̃_𝑛−1−𝛼^𝑛1−𝛼_𝑛𝛽̃_𝑛𝜙𝑝,𝑥_𝑛1−𝛼_𝑛1−𝛽̃_𝑛.(3.44) Combining the above inequality, (3.10), and Lemma 2.9, we have 𝜙𝑧_𝑛̃,𝑧_𝑛≤=𝜙𝜙𝑝𝐽,𝑧^𝑟_𝑛𝑧_𝑛𝑛,𝑧_𝑛≤𝛼_𝑛−𝜙𝑝,𝑧_𝑛̃≤𝜙𝜙𝑝,𝑝𝑥,_𝑛𝑥₀+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛̃−𝜙𝑝,𝑢_𝑛1−𝛼_𝑛1−𝛽̃_𝑛+𝛼^𝑛+𝛼_𝑛−𝛼_𝑛𝛼_𝑛1−𝛼_𝑛1−𝛽̃_𝑛𝜙𝑝,𝑥⁰−𝜙𝑝,𝑥_𝑛.(3.45) By conditions (i) and (iii), (3.17), and (3.45), we have 𝜙𝑧_𝑛̃,𝑧_𝑛⟶0(𝑛⟶∞),(3.46) which implies that ‖‖𝑧_𝑛̃−𝑧_𝑛‖‖⟶0(𝑛⟶∞).(3.47) It follows from (3.23) and (3.47) that ‖‖𝑧_𝑛̃−𝑇_𝑛𝑧_𝑛̃‖‖⟶0(𝑛⟶∞).(3.48)Step 3. Now we show that 𝜔_𝑤({𝑥_𝑛})⊂Ω∶=(∩^∞𝑛=0𝐹(𝑇_𝑛))∩(∩^∞𝑛=0𝐹(𝑆_𝑛))∩VI(𝐶,𝐴)∩GMEP∩𝑇⁻¹0∩𝑇⁻¹0, where 𝜔_𝑤𝑥_𝑛=𝑥^∗∈𝐶∶𝑥_𝑛𝑘⇀𝑥^∗forsomesequence𝑛_𝑘⊂{𝑛}with𝑛_𝑘↑∞.(3.49)
Indeed, since {𝑥_𝑛} is bounded and 𝑋 is reflexive, we know that 𝜔_𝑤({𝑥_𝑛})≠∅. For any arbitrary 𝑥^∗∈𝜔_𝑤({𝑥_𝑛}), there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘⇀𝑥^∗. From (3.36), we have 𝑣_𝑛𝑘⇀𝑥^∗. Since {𝑇_𝑛},{𝑆_𝑛} satisfy NST-conditions, from (3.42) and (3.48) we have 𝑥^∗∈(∩^∞𝑛=0𝐹(𝑇_𝑛))∩(∩^∞𝑛=0𝐹(𝑆_𝑛)).
Let 𝑆⊂𝐸×𝐸^∗ be an operator as follows:𝑆𝑣=𝐴𝑣+𝑁^𝐶∅,(𝑣),𝑣𝑣∈∉𝐶𝐶.,(3.50) By Lemma 2.5, 𝑆 is maximal monotone and 𝑆⁻¹(0)=VI(𝐶,𝐴). Let (𝑣,𝑤)∈𝐺(𝑆). Since 𝑤∈𝑆𝑣=𝐴𝑣+𝑁_𝐶(𝑣), we have 𝑤−𝐴𝑣∈𝑁_𝐶(𝑣). Moreover, 𝑣_𝑛∈𝐶 implies that ⟨𝑣−𝑣_𝑛,𝑤−𝐴𝑣⟩≥0.(3.51)
On the other hand, it follows from 𝑣_𝑛=Π_𝐶𝐽⁻¹(𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛) and Lemma 2.2 that𝑣−𝑣_𝑛,𝐽𝑣_𝑛−𝐽𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛≥0,(3.52) and hence 𝑣−𝑣^𝑛,𝐽𝑥^𝑛−𝐽𝑣_𝑛𝜆_𝑛−𝐴𝑥_𝑛≤0.(3.53) So, from (3.51), (3.53), and 𝐴 being 1/𝛾-Lipschitz continuous, we obtain ⟨𝑣−𝑣_𝑛,𝑤⟩≥≥⟨⟨𝑣𝑣−−𝑣𝑣_𝑛𝑛,𝐴𝑣⟩,𝐴𝑣⟩+𝑣−𝑣_𝑛,𝐽𝑥^𝑛−𝐽𝑣_𝑛𝜆_𝑛−𝐴𝑥_𝑛=𝑣−𝑣_𝑛,𝐴𝑣−𝐴𝑥_𝑛+𝐽𝑥^𝑛−𝐽𝑣_𝑛≥⟨𝑣−𝑣_𝑛𝜆_𝑛,𝐴𝑣−𝐴𝑣_𝑛⟩+⟨𝑣−𝑣_𝑛,𝐴𝑣_𝑛−𝐴𝑥_𝑛⟩+𝑣−𝑣_𝑛,𝐽𝑥^𝑛−𝐽𝑣_𝑛≥−‖‖𝑣−𝑣_𝑛𝜆_𝑛‖‖‖‖𝐴𝑣_𝑛−𝐴𝑥_𝑛‖‖−‖‖𝑣−𝑣_𝑛‖‖‖‖‖𝐽𝑥^𝑛−𝐽𝑣_𝑛≥−1𝑎‖‖‖𝛾‖‖𝑣−𝑣_𝑛‖‖‖‖𝑣_𝑛−𝑥_𝑛‖‖−‖‖𝑣−𝑣^𝑛‖‖‖‖‖𝐽𝑥^𝑛−𝐽𝑣_𝑛𝑎‖‖‖.(3.54) Since 𝐽 is uniformly continuous on bounded sets, by (3.36) and replacing 𝑛 by 𝑛_𝑘 in (3.54), as 𝑘→∞ we have ⟨𝑣−𝑥^∗,𝑤⟩≥0. Thus, 𝑥^∗∈𝑆⁻¹(0), and hence 𝑥^∗∈VI(𝐶,𝐴).
Next we show that 𝑥^∗∈GMEP=𝐹(𝐾_𝑟). Let 𝐻(𝑢_𝑛,𝑦)=Θ(𝑢_𝑛,𝑦)+⟨𝐵𝑢_𝑛,𝑦−𝑢_𝑛⟩+𝜑(𝑦)−𝜑(𝑢_𝑛), forall𝑦∈𝐶. It follows from (3.2) that𝐽𝑦_𝑛=𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝐽𝑧_𝑛+1−𝛽̃_𝑛𝐽𝑇_𝑛𝐽_𝑟𝑛𝑧_𝑛,(3.55) hence, from (3.22) and condition (i), we have ‖‖𝐽𝑦_𝑛−𝐽𝑧_𝑛‖‖≤𝛼_𝑛‖‖𝐽𝑥₀−𝐽𝑧_𝑛‖‖+1−𝛼_𝑛1−𝛽̃_𝑛‖‖𝐽𝑇_𝑛𝐽_𝑟𝑛𝑧_𝑛−𝐽𝑧_𝑛‖‖⟶0(𝑛⟶∞),(3.56) which implies that ‖‖𝑦_𝑛−𝑧_𝑛‖‖⟶0(𝑛⟶∞).(3.57) From (3.17), (3.31), and (3.57), we obtain ‖‖𝑦_𝑛−𝑢_𝑛‖‖⟶0(𝑛⟶∞).(3.58) Thus, 𝑢_𝑛𝑘⇀𝑥^∗,𝑦_𝑛𝑘⇀𝑥^∗(𝑘→∞). Since 𝐽 is uniformly continuous on bounded sets, from (3.58) we have lim_𝑘→∞‖𝐽𝑢_𝑛𝑘−𝐽𝑦_𝑛𝑘‖=0. Therefore, it follows from 𝑡_𝑛𝑘≥𝑐^∗ that ‖𝐽𝑢_𝑛𝑘−𝐽𝑦_𝑛𝑘‖/𝑡_𝑛𝑘→0(𝑛→∞). Since 𝑢_𝑛=𝐾_𝑡𝑛𝑦_𝑛, we have 𝐻𝑢_𝑛,𝑦+1𝑡_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥0,∀𝑦∈𝐶.(3.59) Combining the above inequality, (A2) and (A4), we get ‖‖𝑦−𝑢_𝑛‖‖‖‖𝐽𝑢^𝑛−𝐽𝑦_𝑛‖‖𝑡_𝑛≥1𝑡_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛⟩≥−𝐻𝑢_𝑛,𝑦≥𝐻𝑦,𝑢_𝑛,∀𝑦∈𝐶.(3.60) Replacing 𝑛 by 𝑛_𝑘 and taking the limit as 𝑘→∞ in the above inequality and by (A4), we have 𝐻(𝑦,𝑥^∗)≤0,forall𝑦∈𝐶. For any 𝑡∈(0,1) and 𝑦∈𝐶, define 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑥^∗∈𝐶. So 𝐻(𝑦_𝑡,𝑥^∗)≤0. From (A1) and (A4), we have 0=𝐻𝑦_𝑡,𝑦_𝑡≤𝑡𝐻𝑦_𝑡,𝑦+(1−𝑡)𝐻𝑦_𝑡,𝑥^∗≤𝑡𝐻𝑦_𝑡,𝑦,(3.61) that is, 𝐻(𝑦_𝑡,𝑦)≥0. Thus, from (A3), let 𝑡→0, we have 𝐻(𝑥^∗,𝑦)≥0,forall𝑦∈𝐶, which implies that 𝑥^∗∈GMEP.
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 𝑥^∗∈𝑇⁻¹0∩𝑇⁻¹0. Therefore, 𝜔_𝑤({𝑥_𝑛})⊂Ω.Step 4. Finally we prove that {𝑥_𝑛} converges strongly to 𝑥̃=Π_Ω𝑥₀.
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 𝐶 a nonempty, closed, and convex subset of 𝐸. Let 𝐵∶𝐶→𝐸^∗ be a monotone continuous mapping. Let {𝑇_𝑛},{𝑆_𝑛} be two countable families of relatively quasi-nonexpansive mappings from 𝐶 into itself satisfying NST-conditions such that Ω∶=(∩^∞𝑛=0𝐹(𝑇_𝑛))∩(∩^∞𝑛=0𝐹(𝑆_𝑛))∩𝐺𝑀𝐸𝑃∩𝑇⁻¹0∩𝑇⁻¹0≠∅. Let {𝑡_𝑛}⊂[𝑐^∗,+∞) for some 𝑐^∗>0 and {𝑟_𝑛}⊂(0,+∞) satisfy liminf_𝑛→∞𝑟_𝑛>0. Let {𝑥_𝑛} be the sequence generated by 𝑧𝑥₀∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛𝑦=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆_𝑛𝐽_𝑟𝑛𝑥_𝑛_𝑛,𝑢=𝐽⁻¹𝛼^𝑛𝐽𝑥₀+1−𝛼_𝑛𝛽̃_𝑛𝐽𝑧_𝑛+1−𝛽̃_𝑛𝐽𝑇_𝑛𝐽_𝑟𝑛𝑧_𝑛_𝑛,𝐻_𝑛=𝐾_𝑡𝑛𝑦_𝑛,𝑊=𝑧∈𝐶∶𝜙𝑧,𝑢_𝑛≤𝛼_𝑛+1−𝛼_𝑛𝛼_𝑛𝜙𝑧,𝑥₀+1−𝛼_𝑛1−𝛼_𝑛𝜙𝑧,𝑥_𝑛_𝑛,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛≥0,(3.62) where 𝐽 is the normalized duality mapping, {𝛼_𝑛},{𝛽_𝑛},{𝛼_𝑛}, and {𝛽̃_𝑛} are four sequences in [0,1], and 𝐾_𝑟 is defined by (*). The following conditions hold:(i)li