A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory

Abstract

It is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; and the set of fixed points of a continuous pseudocontractive mapping. Moreover, the common element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize, improve, and unify several recently announced results.

1. Introduction

Let 𝐸 be a real normed space with dual 𝐸^∗. The normalized duality mapping 𝐽 from 𝐸 to 2^𝐸∗ is defined by 𝐽𝑥∶=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖²,‖𝑥‖=𝑥^∗, (1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is well known (see, e.g., [1]) that if 𝐸^∗ is strictly convex, then 𝐽 is single-valued and if 𝐸=𝐻 is a Hilbert space, then 𝐽 is the identity mapping. In the sequel, we shall denote the single-valued normalized duality mapping by 𝑗.

A mapping 𝑇 with domain 𝐷(𝑇), and range 𝑅(𝑇), in 𝐸 is called a strict contraction or simply a contraction if and only if there exists 𝛾₀∈[0,1) such that for all 𝑥,𝑦∈𝐷(𝑇), 𝑇(𝑥)−𝑇𝑦≤𝛾₀𝑥−𝑦, (2) and 𝑇 is called nonexpansive if and only if for all 𝑥,𝑦∈𝐷(𝑇), 𝑇𝑥−𝑇𝑦≤𝑥−𝑦. (3) A point 𝑥∈𝐷(𝑇) is called a fixed point of an operator 𝑇 if and only if 𝑇𝑥=𝑥. The set of fixed points of an operator 𝑇 is denoted by Fix(𝑇), that is, Fix(𝑇)∶={𝑥∈𝐷(𝑇)∶𝑇𝑥=𝑥}.

The most important generalization of the class of nonexpansive mappings is, perhaps, the class of pseudocontractive mappings. These mappings are intimately connected with the important class of nonlinear accretive operators. This connection will be made precise in what follows.

A mapping 𝑇 with domain 𝐷(𝑇), and range 𝑅(𝑇), in 𝐸 is called pseudocontractive if and only if for all 𝑥,𝑦∈𝐷(𝑇), the following inequality holds: 𝑥−𝑦≤(1+𝑟)𝑥−𝑦−𝑟𝑇𝑥−𝑇𝑦, (4) for all 𝑟>0. As a consequence of a result of Kato [2], the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows: the mapping 𝑇 is called pseudocontractive if and only if for all 𝑥,𝑦∈𝐷(𝑇), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝑥−𝑦². (5) It now follows trivially from (5) that every nonexpansive mapping is pseudocontractive. We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings. For examples of pseudocontractive mappings which are not nonexpansive, the reader may see [1].

To see the connection between the pseudocontractive mappings and the accretive mappings, we introduce the following definition: a mapping 𝐴 with domain, 𝐷(𝐴), and range, 𝑅(𝐴), in 𝐸 is called accretive if and only if for all 𝑥,𝑦∈𝐷(𝐴), the following inequality is satisfied: 𝑥−𝑦≤𝑥−𝑦+𝑟𝐴𝑥−𝐴𝑦, (6) for all 𝑟>0. Again, as a consequence of Kato [2], it follows that 𝐴 is accretive if and only if for all 𝑥,𝑦∈𝐷(𝐴), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0. (7) The operator 𝐴 is called 𝜂-inverse strongly accretive  if and only if there exists 𝜂∈(0,1) such that for all 𝑥,𝑦∈𝐷(𝐴), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝜂𝐴𝑥−𝐴𝑦². (8) It is easy to see from inequalities (4) and (6) that an operator 𝐴 is accretive if and only if the mapping 𝑇∶=(𝐼−𝐴) is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the mapping theory of accretive operators. For the importance of accretive operators and their connections with evolution equations, the reader may consult any of the references [1, 3].

Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors. It is of interest to note that if 𝐸=𝐻 is a Hilbert space, accretive (𝜂-strongly accretive) operators coincide with the monotone (𝜂-strongly monotone) operators, where an operator 𝐴 with domain, 𝐷(𝐴), and range, 𝑅(𝐴), in 𝐻 is called monotone if and only if for all 𝑥,𝑦∈𝐷(𝐴), we have that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0, (9) and 𝐴 is called 𝜂-inverse strongly monotone if and only if for all 𝑥,𝑦∈𝐷(𝐴), 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝜂𝐴𝑥−𝐴𝑦². (10) A bounded linear operator 𝐴 with domain, 𝐷(𝐴), and range, 𝑅(𝐴), in a Hilbert space 𝐻 is called a strongly positive operator if and only if there exists a constant 𝛾>0 such that for all 𝑥∈𝐻,⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖².

Let 𝐶 be a closed convex nonempty subset of a real Hilbert space 𝐻 with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction and Φ∶𝐶→ℝ∪{+∞} be a proper extended real valued function, where ℝ denotes the set of real numbers. Let Θ∶𝐶→𝐻 be a nonlinear monotone mapping. The generalized mixed equilibrium problem (GMEP) for 𝑓,Φ, and Θ is to find 𝑢^∗∈𝐶 such that 𝑓𝑢^∗,𝑦+Φ𝑦−Φ𝑢^∗+Θ𝑢^∗,𝑦−𝑢^∗≥0∀𝑦∈𝐶. (11) The set of solutions for GMEP (11) is denoted by =𝑦(𝑢)+.GMEP𝑓,Φ,Θ𝑢∈𝐶∶𝑓𝑢,𝑦+Φ−ΦΘ𝑢,𝑦−𝑢≥0∀𝑦∈𝐶 (12) If Φ≡0≡Θ in (11), then (11) reduces to the classical equilibrium problem (EP), that is, the problem of finding 𝑢^∗∈𝐶 such that 𝑓𝑢^∗,𝑦≥0∀𝑦∈𝐶, (13) and GMEP(𝑓,0,0) is denoted by EP(𝑓), where EP𝑓=𝑢∈𝐶∶𝑓𝑢,𝑦≥0∀𝑦∈𝐶. (14) If 𝑓≡0≡Φ in (11), then GMEP (11) reduces to the classical variational inequality problem and GMEP(0,0,Θ) is denoted by VI(Θ,𝐶), where VI(Θ,𝐶)=𝑢∈𝐶∶Θ𝑢,𝑦−𝑢≥0∀𝑦∈𝐶. (15) If 𝑓≡0≡Θ, then GMEP (11) reduces to the following minimization problem: find𝑢^∗∈𝐶suchthatΦ𝑦≥Φ𝑢^∗∀𝑦∈𝐶, (16) and GMEP(0,Φ,0) is denoted by Argmin(Φ), where Argmin(Φ)=𝑢∈𝐶∶Φ(𝑢)≤Φ𝑦∀𝑦∈𝐶. (17) If Θ≡0, then (11) becomes the mixed equilibrium problem (MEP) and GMEP(𝑓,Φ,0) is denoted by MEP(𝑓,Φ), where =𝑦(𝑢).MEP𝑓,Φ𝑢∈𝐶∶𝑓𝑢,𝑦+Φ−Φ≥0∀𝑦∈𝐶 (18) If Φ≡0, then (11) reduces to the generalized equilibrium problem (GEP) and GMEP(𝑓,0,Θ) is denoted by GEP(𝑓,Θ), where =+.GEP𝑓,Θ𝑢∈𝐶∶𝑓𝑢,𝑦Θ𝑢,𝑦−𝑢≥0∀𝑦∈𝐶 (19) If 𝑓≡0, then GMEP (11) reduces to the generalized variational inequality problem (abbreviated GVIP) and GMEP(0,Φ,Θ) is denoted by GVI(Φ,Θ,𝐶), where ()=𝑦(𝑢)+.GVIΦ,Θ,𝐶𝑢∈𝐾∶Φ−ΦΘ𝑢,𝑦−𝑢≥0∀𝑦∈𝐶 (20) The generalized mixed equilibrium problem (GMEP) includes special cases as the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. Furthermore, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the generalized mixed equilibrium problem. In other words, the generalized mixed equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer science, optimization theory, operations research, economics, and countless other fields. For the past 20 years or so, many existence results have been published for various equilibrium problems (see, e.g., [4–6]).

Iterative approximation of fixed points and zeros of nonlinear operators has been studied extensively by many authors to solve nonlinear operator equations as well as variational inequality problems and their generalizations (see, e.g., [7–24]). Most published results on nonexpansive mappings (e.g.) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.

Let 𝐶 be a closed convex nonempty subset of 𝐻, let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑇)≠∅. Given that 𝑢∈𝐶 and a real sequence {𝛼_𝑛}_𝑛≥0 in the interval (0,1), let a sequence {𝑥_𝑛}_𝑛≥0 be defined by 𝑥₀∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0. (21) Under appropriate conditions on the iterative parameter {𝛼_𝑛}_𝑛≥0, it was shown by Bauschke [25], Halpern [26], Lions [27], and Wittmann [28] that {𝑥_𝑛}_𝑛≥0 converges strongly to 𝑃_𝐹(𝑇)𝑢, the projection of 𝑢 to the fixed point set, 𝐹(𝑇), of 𝑇.

Marino and Xu [11], proved that the iteration scheme given by 𝑥₀∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0 (22) converges strongly to a unique solution 𝑥^′∈Fix(𝑇) of the variational inequality 𝛾𝑓−𝐴𝑥^′,𝑦−𝑥^′≤0∀𝑦∈Fix(𝑇), (23) which is the optimality condition for the minimization problem min_{(𝑇)𝑥∈Fix}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑐), (24) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻); provided 𝑓∶𝐻→𝐻 is a strict contraction, 𝑇∶𝐻→𝐻 is nonexpansive and the iterative parameter {𝛼_𝑛}_𝑛≥0 that satisfies appropriate conditions.

In [29], Yamada introduced the following hybrid iterative method: 𝑥₀∈𝐻,𝑥_𝑛+1=𝑇𝑥_𝑛−𝜇𝜆_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0, (25) where 𝑇 is nonexpansive, 𝐴 is 𝐿-Lipschitzian and an 𝜂-strongly monotone operator. He proved that if 𝜇 and {𝜆_𝑛}_𝑛≥1 satisfy appropriate conditions, then (25) converges strongly to a unique solution 𝑥^′∈Fix(𝑇) of the varational inequality ⟨𝐴𝑥^′,𝑦−𝑥^′⟩≥0,forall𝑦∈𝐹(𝑇).

Tian [30] introduced the following iterative method: 𝑥₀∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0, (26) and proved that if 𝑓∶𝐻→𝐻 is a contraction, 𝐴∶𝐻→𝐻 is an 𝜂-strongly monotone mapping, 𝑇∶𝐻→𝐻 a nonexpansive mapping and the parameter {𝛼_𝑛}_𝑛≥1 satisfies appropriate conditions, then the sequence {𝑥_𝑛}_𝑛≥1 converges strongly to a unique solution 𝑥^′∈Fix(𝑇) of the variational inequality 𝛾𝑓−𝜇𝐴𝑥^′,𝑦−𝑥^′≤0∀𝑦∈Fix(𝑇). (27) Let 𝐶 be a convex nonempty subset of a real Banach space. Let {𝑇_𝑗}_𝑗≥1 be a countable infinite family of nonexpansive mappings of 𝐶 into itself and let {𝜆_𝑗}_𝑗≥1 be a sequence of real numbers such that 𝜆_𝑗∈(0,1) for all 𝑗∈ℕ. For all 𝑛∈ℕ, define a mapping 𝑊_𝑛∶𝐶→𝐶 by 𝑈_𝑛,𝑛+1𝑈∶=𝐼,_𝑛,𝑛∶=𝜆_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜆_𝑛𝑈𝐼,_{𝑛,𝑛−1}∶=𝜆_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜆_𝑛−1⋮𝑈𝐼,_𝑛,𝑘∶=𝜆_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜆_𝑘𝑈𝐼,_{𝑛,𝑘−1}∶=𝜆_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜆_𝑘−1⋮𝑈𝐼,_𝑛,2∶=𝜆₂𝑇₂𝑈_𝑛,3+1−𝜆₂𝑊𝐼,_𝑛∶=𝑈_𝑛,1=𝜆₁𝑇₁𝑈_𝑛,2+1−𝜆₁𝐼. (28) The mapping 𝑊_𝑛,𝑛∈ℕ is called the 𝑊-mapping generated by the countable infinite family {𝑇_𝑗}_𝑗≥1 of nonexpansive mappings (see, e.g., [31]).

In [32], Yao et al. studied the following problem. Let 𝐻 be a real Hilbert space. Consider the iterative scheme 𝑥₁∈𝐻,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛𝑥_𝑛, (29) where 𝛾>0 is some constant, 𝑓∶𝐻→𝐻 is a given contractive mapping, and 𝐴 is a strongly positive bounded linear operator on 𝐻. Assuming that 𝐹∶=⋂^∞_𝑗=1Fix(𝑇_𝑗)≠∅, and under appropriate conditions on the iterative parameters, Yao et al. proved that the sequence {𝑥_𝑛}_𝑛≥1 generated by (29) converges strongly to 𝑥^∗∈𝐹, where 𝑥^∗ is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑥^∗,𝑦−𝑥^∗≥0∀𝑦∈𝐹. (30) Colao and Marino [33] proved that if {𝑥_𝑛}_𝑛≥1 and {𝑢_𝑛}_𝑛≥1 are sequences generated by 𝑥₁∈𝐻 and 𝐺𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0𝑦∈𝐻;_𝑛+1=𝜀_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝜀_𝑛𝐴𝑊_𝑛𝑢_𝑛,𝑛≥1, (31) (where 𝜀_𝑛, 𝛽_𝑛, 𝑟_𝑛, and 𝛾 satisfy appropriate conditions), then both {𝑥_𝑛}_𝑛≥1 and {𝑢_𝑛}_𝑛≥1 converge strongly to an element 𝑥^∗∈⋂^∞_𝑗=1Fix(𝑇_𝑗)∩EP(𝐺), which is also a unique solution of the variational inequality 𝐴−𝛾𝑓𝑥^∗,𝑦−𝑥^∗≥0∀𝑦∈^∞_𝑗=1Fix𝑇_𝑗∩EP(𝐺). (32) Let 𝐶 be a nonempty closed and convex subset of real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a continuous pseudocontractive mapping; 𝐴∶𝐶→𝐻 be a continuous monotone mapping; 𝐵∶𝐻→𝐻 an inverse strongly monoyone mapping. For each 𝑛∈ℕ, let 𝑇_𝑟𝑛 and 𝐹_𝑟𝑛 be defined for each 𝑥∈𝐻 by 𝑇_𝑟𝑛−1∶=𝑧∈𝐶∶𝑦−𝑧,𝑇𝑧𝑟_𝑛𝑦−𝑧,1+𝑟_𝑛,𝐹𝑧−𝑥≤0,∀𝑦∈𝐶_𝑟𝑛+1∶=𝑧∈𝐶∶𝑦−𝑧,𝐴𝑧𝑟_𝑛𝑦−𝑧,𝑧−𝑥≤0,∀𝑦∈𝐶. (33) Let {𝑥_𝑛}_𝑛≥1 be a sequence generated iteratively by 𝑥₁∈𝐶, 𝑦_𝑛=𝐹_𝑟𝑛𝑥_𝑛;𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛿_𝑛𝑥_𝑛+1−𝛿_𝑛𝐼−𝛼_𝑛𝐵𝑇_𝑟𝑛𝑦_𝑛,𝑛≥1, (34) (where 𝛼_𝑛, 𝛿_𝑛, 𝑟_𝑛, and 𝛾 satisfy appropriate conditions), then it was recently proved by Chamnarnpan and Kumam [34] that the sequence {𝑥_𝑛}_𝑛≥1 defined by (34) converges strongly to an element 𝑧∈Fix(𝑇)∩VI(𝐶,𝐴), which is a unique solution of the variational inequality 𝐵−𝛾𝑓𝑧,𝑥−𝑧≥0∀𝑥∈Fix(𝑇)∩VI(𝐶,𝐴). (35)

In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; the set of fixed points of a continuous pseudocontractive mapping. Moreover, the element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize improve and unify several recently announced results.

2. Preliminary

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

Lemma 1 (see, e.g., [31]). Let 𝐶 be a closed convex nonempty subset of a strictly convex real Banach space. Let {𝑇_𝑗}_𝑗≥1 be a countable infinite family of nonexpansive mappings of 𝐶 into itself and let {𝜆_𝑗}_𝑗≥1 be a real sequence such that 0<𝜆_𝑗≤𝑏<1 for all 𝑗∈ℕ, for some constant 𝑏∈(0,1). Then, for all 𝑥∈𝐶 and 𝑘∈ℕ,lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists (where 𝑈_𝑛,𝑘 is as defined in (28)).

In particular, for 𝑘=1 in Lemma 1, we define a mapping 𝑊∶𝐶→𝐶 by 𝑊𝑥∶=lim_𝑛→∞𝑈_𝑛,1𝑥=lim_𝑛→∞𝑊_𝑛𝑥. (36)

Lemma 2 (see, e.g., [31]). Let 𝐶 be a closed convex nonempty subset of a strictly convex real Banach space. Let {𝑇_𝑗}_𝑗≥1 be a countable infinite family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑗=1Fix(𝑇_𝑗)≠∅ and let {𝜆_𝑗}_𝑗≥1 be a real sequence such that 0<𝜆_𝑗≤𝑏<1 for all 𝑗∈ℕ, for some constant 𝑏∈(0,1). Then, Fix(𝑊)=⋂^∞_𝑗=1Fix(𝑇_𝑗)=⋂^∞_𝑛=1Fix(𝑊_𝑛).

Lemma 3 (see [33]). Let 𝐶 be a closed convex nonempty subset of a strictly convex real Banach space. Let {𝑇_𝑗}_𝑗≥1 be a countable infinite family of nonexpansive mappings of 𝐶 into itself such that ⋂^∞_𝑗=1Fix(𝑇_𝑗)≠∅ and let {𝜆_𝑗}_𝑗≥1 be a real sequence such that 0<𝜆_𝑗≤𝑏<1 for all 𝑗∈ℕ, for some constant 𝑏∈(0,1). Suppose that 𝑊_𝑛∶𝐶→𝐶 is given by (28) for all 𝑛∈ℕ, then 𝑊_𝑛+1𝑥−𝑊_𝑛𝑥≤2^𝑛_𝑗=1𝜆_𝑗𝑥−𝑝,∀𝑥∈𝐶,𝑝∈^∞_𝑗=1Fix𝑇_𝑗. (37)

Lemma 4. Let 𝐻 be a real Hilbert space, then the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦∀𝑥,𝑦∈𝐻. (38)

Lemma 5 (see, e.g., [35–37]). Let {𝜆_𝑛}_𝑛≥0 be a sequence of nonnegative real numbers satisfying the following condition: 𝜆_𝑛+1≤1−𝛼_𝑛𝜆_𝑛+𝜎_𝑛,𝑛≥0, (39) where {𝛼_𝑛}_𝑛≥0 and {𝜎_𝑛}_𝑛≥0 are sequences of real numbers such that {𝛼_𝑛}_𝑛≥0⊂[0,1],∑^∞_𝑛=0𝛼_𝑛=+∞. Suppose that 𝜎_𝑛=𝑜(𝛼_𝑛),𝑛≥0 (i.e., lim_𝑛→∞(𝜎_𝑛/𝛼_𝑛)=0) or ∑^∞_𝑛=0|𝜎_𝑛|<+∞ or limsup_𝑛→∞(𝜎_𝑛/𝛼_𝑛)≤0, then 𝜆_𝑛→0 as 𝑛→∞.

Lemma 6 (see, e.g., Suzuki [38]). Let {𝑡_𝑛}_𝑛≥0 and {𝑦_𝑛}_𝑛≥0 be two bounded sequences in a real Banach space such that 𝑦_𝑛+1=𝛽_𝑛𝑦_𝑛+(1−𝛽_𝑛)𝑡_𝑛, for all 𝑛≥0, where {𝛽_𝑛}_𝑛≥0 is a real sequence satisfying the condition 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that limsup_𝑛→∞𝑡_𝑛+1−𝑡_𝑛−𝑦_𝑛+1−𝑦_𝑛≤0, (40) then lim_𝑛→∞‖𝑡_𝑛−𝑦_𝑛‖=0.

Lemma 7 (see Zegeye [39]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇^′∶𝐶→𝐻 be a continuous pseudocontractive mapping, then for all 𝑟>0 and 𝑥∈𝐻, there exists 𝑧∈𝐶 such that 𝑦−𝑧,𝑇^′𝑧−1𝑟𝑦−𝑧,(1+𝑟)𝑧−𝑥≤0∀𝑦∈𝐶. (41)

Lemma 8 (see Zegeye [39]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇^′∶𝐶→𝐶 be a continuous pseudocontractive mapping, then for all 𝑟>0 and 𝑥∈𝐻, define a mapping 𝐹_𝑟∶𝐻→𝐶 by 𝐹_𝑟𝑥=𝑧∈𝐶∶𝑦−𝑧,𝑇^′𝑧−1𝑟𝑦−𝑧,(1+𝑟)𝑧−𝑥≤0∀𝑦∈𝐶, (42) then the following hold:(1)𝐹_𝑟 is single-valued; (2)𝐹_𝑟 is firmly nonexpansive type mapping, that is, for all 𝑥,𝑦∈𝐻, 𝐹_𝑟𝑥−𝐹_𝑟𝑦²≤𝐹_𝑟𝑥−𝐹_𝑟𝑦,𝑥−𝑦; (43)(3)Fix(𝐹_𝑟) is closed and convex; and Fix(𝐹_𝑟)=Fix(𝑇^′) for all 𝑟>0.

In the sequel, we shall require that the bifunction 𝑓∶𝐶×𝐶→ℝ satisfies the following conditions:(𝐴1)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶;(𝐴2)𝑓 is monotone, in the sense that 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶;(𝐴3)limsup_𝑡→0+𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦) for all 𝑥,𝑦,𝑧∈𝐶;(𝐴4) the function 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous for all 𝑥∈𝐶.

Lemma 9 (Compare with Lemma 2.4 of [5]). Let 𝐶 be a closed convex nonempty subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying conditions (𝐴1)–(𝐴4);Θ∶𝐶→𝐻 a continuous monotone mapping and Φ∶𝐶→ℝ∪{+∞} a proper lower semicontinuous convex function. Then, for all 𝑟>0 and 𝑥∈𝐻 there exists 𝑢∈𝐶 such that 𝑓𝑦(𝑢)++1𝑢,𝑦+Φ−ΦΘ𝑢,𝑦−𝑢𝑟𝑦−𝑢,𝑢−𝑥≥0∀𝑦∈𝐾. (44) Moreover, if for all 𝑥∈𝐻 we define a mapping 𝐺_𝑟∶𝐻→2^𝐶 by 𝐺_𝑟(𝑥)=𝑦(𝑢)++1𝑢∈𝐶∶𝑓𝑢,𝑦+Φ−ΦΘ𝑢,𝑦−𝑢𝑟,𝑦−𝑢,𝑢−𝑥≥0,∀𝑦∈𝐶 (45) then the following hold: (1)𝐺_𝑟 is single-valued for all 𝑟>0; (2)𝐺_𝑟 is firmly nonexpansive, that is, for all 𝑥,𝑧∈𝐻, 𝐺_𝑟𝑥−𝐺_𝑟𝑧²≤𝐺_𝑟𝑥−𝐺_𝑟𝑧,𝑥−𝑧; (46)(3)Fix(𝐺_𝑟)=GMEP(𝑓,Φ,Θ) for all 𝑟>0; (4)GMEP(𝑓,Φ,Θ) is closed and convex.

Let 𝐶 be a closed convex nonempty subset of a uniformly convex real Banach space 𝐸, where 𝐸 is called uniformly convex if and only if the modulus of convexity 𝛿_𝐸(𝜀)1=inf1−2𝑥+𝑦∶‖𝑥‖≤1,𝑦≤1,𝑥−𝑦≥𝜀 (47) is positive for all 𝜀∈(0,2]. Given a bounded sequence {𝑥_𝑛}_𝑛≥1 in 𝐶, define 𝑟_𝑚𝑦∶=sup𝑥_𝑘−𝑦∶𝑘≥𝑚,𝑦∈𝐸. (48) It is well known (see, e.g., [40]) that there exists unique 𝑐_𝑚∈𝐶 such that 𝑟_𝑚𝑐_𝑚=inf𝑟_𝑚𝑦∶𝑦∈𝐶=𝑟_𝑚∀𝑚∈ℕ. (49) Clearly, 𝑟_𝑚+1≤𝑟_𝑚 and 0≤𝑟_𝑚 for all 𝑚∈ℕ, so that lim_𝑚→∞𝑟_𝑚=inf_𝑚∈ℕ𝑟_𝑚 exists. It is known (see [40]) that if lim_𝑚→∞𝑟_𝑚=inf_𝑚∈ℕ𝑟_𝑚=0, then the sequence {𝑥_𝑛}_𝑛≥1 converges.

Definition 10 (see [40]). If the sequence {𝑐_𝑚}_𝑚≥1 converges, then 𝑐₀=lim_𝑚→∞𝑐_𝑚 is called the asymptotic center of {𝑥_𝑛}_𝑛≥1 (with respect to 𝐶). Equivalently (see, e.g., [22]), 𝑐₀∈𝐶 is called the asymptotic center of {𝑥_𝑛}_𝑛≥1 (with respect to 𝐶) if and only if limsup_𝑛→∞𝑥_𝑛−𝑐₀=inf_𝑧∈𝐶limsup_𝑛→∞𝑥_𝑛−𝑧. (50)

Lemma 11 (see Theorem 1 of Edelstein [40]). Let 𝐶 be a closed convex nonempty subset of a uniformly convex real Banach space 𝐸, Let {𝑥_𝑛}_𝑛≥1 be a bounded sequence in 𝐶 and {𝑐_𝑚}_𝑚≥1 be as in (49), then 𝑐=lim_𝑚→∞𝑐_𝑚 exists. In other words, the asymptotic center 𝑐 of the sequence {𝑥_𝑛}_𝑛≥1 (with respect to 𝐶) exists and is unique.

3. Main Results

Lemma 12. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇^′∶𝐶→𝐶 be a continuous pseudocontractive mapping. For 𝑟>0, let 𝐹_𝑟∶𝐻→𝐶 be the mapping in Lemma 8, then for any 𝑥∈𝐻 and for any 𝑝,𝑞>0, 𝐹_𝑝𝑥−𝐹_𝑞𝑥≤𝑝−𝑞𝑝𝐹_𝑝𝑥+‖𝑥‖. (51)

Proof. Observe that for any 𝑦∈𝐶, 𝑦−𝐹_𝑝𝑥,𝑇^′𝐹_𝑝𝑥−1𝑝𝑦−𝐹_𝑝𝑥,1+𝑝𝐹_𝑝𝑥−𝑥≤0,𝑦−𝐹_𝑞𝑥,𝑇^′𝐹_𝑞𝑥−1𝑞𝑦−𝐹_𝑞𝑥,1+𝑞𝐹_𝑞𝑥−𝑥≤0. (52)(53) So, we obtain (in particular) for 𝑦=𝐹_𝑞𝑥 in (52) and 𝑦=𝐹_𝑝𝑥 in (53) that 𝐹_𝑞𝑥−𝐹_𝑝𝑥,𝑇^′𝐹_𝑝𝑥−1𝑝𝐹_𝑞𝑥−𝐹_𝑝𝑥,1+𝑝𝐹_𝑝𝐹𝑥−𝑥≤0,_𝑝𝑥−𝐹_𝑞𝑥,𝑇^′𝐹_𝑞𝑥−1𝑞𝐹_𝑝𝑥−𝐹_𝑞𝑥,1+𝑞𝐹_𝑞𝑥−𝑥≤0, (54)(55) respectively. Adding (54) and (55) gives −𝐹_𝑞𝑥−𝐹_𝑝𝑥,𝑇^′𝐹_𝑞𝑥−𝑇^′𝐹_𝑝𝑥−𝐹_𝑞𝑥−𝐹_𝑝𝐹𝑥,1+𝑝_𝑝𝑥−𝑥𝑝−1+𝑞𝐹_𝑞𝑥−𝑥𝑞≤0. (56) Using the fact that 𝑇^′ is pseudocontractive, we obtain from (56) that −𝐹_𝑞𝑥−𝐹_𝑝𝑥,𝐹_𝑞𝑥−𝐹_𝑝𝑥−𝐹_𝑞𝑥−𝐹_𝑝𝐹𝑥,1+𝑝_𝑝𝑥−𝑥𝑝−1+𝑞𝐹_𝑞𝑥−𝑥𝑞≤0. (57) This implies that 𝐹_𝑞𝑥−𝐹_𝑝𝑥,𝐹_𝑞𝑥−𝐹_𝑝𝑥+𝐹_𝑞𝑥−𝐹_𝑝𝐹𝑥,1+𝑝_𝑝𝑥−𝑥𝑝−1+𝑞𝐹_𝑞𝑥−𝑥𝑞≥0. (58) Equivalently, we obtain 𝐹_𝑞𝑥−𝐹_𝑝𝐹𝑥,^𝑝𝑥−𝑥𝑝−𝐹_𝑞𝑥−𝑥𝑞≥0. (59) This implies that 0≤𝐹_𝑝𝑥−𝐹_𝑞𝑥,𝐹_𝑞𝑞𝑥−𝑥−𝑝𝐹_𝑝=𝐹𝑥−𝑥_𝑝𝑥−𝐹_𝑞𝑥,𝐹_𝑞𝑥−𝐹_𝑝𝑥+𝐹_𝑝𝑞𝑥−𝑥−𝑝𝐹_𝑝=𝐹𝑥−𝑥_𝑝𝑥−𝐹_𝑞𝑥,𝐹_𝑞𝑥−𝐹_𝑝𝑞𝑥+1−𝑝𝐹_𝑝𝐹𝑥−𝑥≤−_𝑝𝑥−𝐹_𝑞𝑥²+𝑞1−𝑝𝐹_𝑝𝑥−𝐹_𝑞𝑥×𝐹_𝑝𝑥+‖𝑥‖. (60) Thus, 𝐹_𝑝𝑥−𝐹_𝑞𝑥≤𝑝−𝑞𝑝𝐹_𝑝𝑥+‖𝑥‖. (61)

Lemma 13. Let 𝐶 be a closed convex nonempty subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying conditions (𝐴1)–(𝐴4);Θ∶𝐶→𝐻 a monotone mapping and Φ∶𝐶→ℝ∪{+∞} a proper lower semicontinuous convex function. Let 𝑟>0 and let 𝐺_𝑟 be the mapping in Lemma 9, then for all 𝑝,𝑞>0 and for all 𝑥∈𝐻, one has that G_𝑝𝑥−𝐺_𝑞𝑥≤𝑝−𝑞𝑝𝐺_𝑝𝑥+‖𝑥‖. (62)

Proof. We observe that 𝑓𝐺_𝑝𝑥,𝑦+Φ𝑦−Φ𝐺_𝑝𝑥+Θ𝐺_𝑝𝑥,𝑦−𝐺_𝑝𝑥+1𝑝𝑦−𝐺_𝑝𝑥,𝐺_𝑝𝑓𝐺𝑥−𝑥≥0∀𝑦∈𝐻,_𝑞𝑥,𝑦+Φ𝑦−Φ𝐺_𝑞𝑥+Θ𝐺_𝑞𝑥,𝑦−𝐺_𝑞𝑥+1𝑞𝑦−𝐺_𝑞𝑥,𝐺_𝑞𝑥−𝑥≥0∀𝑦∈𝐻. (63)(64) In particular, we obtain from (63) and (64) (resp.) that 𝑓𝐺_𝑝𝑥,𝐺_𝑞𝑥+Φ𝐺_𝑞𝑥−Φ𝐺_𝑝𝑥+Θ𝐺_𝑝𝑥,𝐺_𝑞𝑥−𝐺_𝑝𝑥+1𝑝𝐺_𝑞𝑥−𝐺_𝑝𝑥,𝐺_𝑝𝑓𝐺𝑥−𝑥≥0,_𝑞𝑥,𝐺_𝑝𝑥+Φ𝐺_𝑝𝑥−Φ𝐺_𝑞𝑥+Θ𝐺_𝑞𝑥,𝐺_𝑝𝑥−𝐺_𝑞𝑥+1𝑞𝐺_𝑝𝑥−𝐺_𝑞𝑥,𝐺_𝑞𝑥−𝑥≥0. (65) Adding (65) (and using the monotonicity of 𝑓 and Θ), we get that 0≤𝐺_𝑝𝑥−𝐺_𝑞𝐺𝑥,^𝑞𝑥−𝑥𝑞−𝐺_𝑝𝑥−𝑥𝑝. (66) This implies that 0≤𝐺_𝑝𝑥−𝐺_𝑞𝑥,𝐺_𝑞𝑞𝑥−𝑥−𝑝𝐺_𝑝=𝐺𝑥−𝑥_𝑝𝑥−𝐺_𝑞𝑥,𝐺_𝑞𝑥−𝐺_𝑝𝑥+𝐺_𝑝𝑞𝑥−𝑥−𝑝𝐺_𝑝=𝐺𝑥−𝑥_𝑝𝑥−𝐺_𝑞𝑥,𝐺_𝑞𝑥−𝐺_𝑝𝑞𝑥+1−𝑝𝐺_𝑝𝐺𝑥−𝑥≤−_𝑝𝑥−𝐺_𝑞𝑥²+𝑞1−𝑝×𝐺_𝑝𝑥−𝐺_𝑞𝑥𝐺_𝑝𝑥+‖𝑥‖. (67) Thus, 𝐺_𝑝𝑥−𝐺_𝑞𝑥≤𝑝−𝑞𝑝𝐺_𝑝𝑥+‖𝑥‖. (68)

Lemma 14. Let {𝑎_𝑛}_𝑛≥1 be a bounded sequence of nonnegative real numbers, then limsup_𝑛→∞𝑎²_𝑛=limsup_𝑛→∞𝑎_𝑛². (69)

Proof. It is well known that for any two bounded sequences {𝑢_𝑛}_𝑛≥1 and {𝑣_𝑛}_𝑛≥1 of nonnegative real numbers, we have that limsup_𝑛→∞𝑢_𝑛𝑣_𝑛≤limsup_𝑛→∞𝑢_𝑛×limsup_𝑛→∞𝑣_𝑛. (70) It then follows that for any bounded sequence {𝑎_𝑛}_𝑛≥1 of nonnegative terms, limsup_𝑛→∞𝑎²_𝑛≤limsup_𝑛→∞𝑎_𝑛². (71) To complete our proof, we show that limsup_𝑛→∞𝑎²_𝑛≥limsup_𝑛→∞𝑎_𝑛². (72) Now, let {𝑎_𝑛𝑗}_𝑗≥1 be a subsequence of {𝑎_𝑛}_𝑛≥1 such that limsup_𝑛→∞𝑎_𝑛=lim_𝑗→∞𝑎_𝑛𝑗, then by the continuity of the function 𝑓∶ℝ→ℝ given by 𝑓(𝑡)=𝑡², we obtain that limsup_𝑛→∞𝑎_𝑛²=lim_𝑗→∞𝑎_𝑛𝑗²=lim_𝑗→∞𝑎²_𝑛𝑗≤limsup_𝑛→∞𝑎²_𝑛. (73) The last inequality follows from the fact that limsup_𝑛→∞𝑎²_𝑛 equals sup𝑆, where 𝑆 is the set of all subsequential limits of {𝑎²_𝑛}_𝑛≥1. This completes the proof.

Remark 15. We must note that Lemma 14 holds trivially if the sequence {𝑎_𝑛}_𝑛≥1 was an unbounded sequence of nonnegative real numbers.

Remark 16. In the sequel, we shall make the following assumptions: 𝐻 is a real Hilbert space; 𝑇^′∶𝐻→𝐻 is a continuous pseudocontractive mapping; 𝑇_𝑗∶𝐻→𝐻,𝑗=1,2,3,… is a countable infinite family of nonexpansive mappings; 𝑓∶𝐻×𝐻→ℝ is a bifunction satisfying conditions (𝐴1)–(𝐴4);Φ∶𝐻→ℝ∪{+∞} a proper lower semicontinuous convex function; Θ∶𝐻→𝐻 a continuous monotone mapping; 𝑢∈𝐻 is a fixed vector; 𝐴∶𝐻→𝐻 is a strongly positive bounded linear operator with coefficient 𝛾;𝐵∶𝐻→𝐻is an 𝜂-inverse strongly monotone mapping. The sequences {𝑟_𝑛}_𝑛≥1, {𝛼_𝑛}_𝑛≥1,{𝛽_𝑛}_𝑛≥1, and {𝜆_𝑛}_𝑛≥1 are real sequences such that 𝑟_𝑛>0 for all 𝑛∈ℕ,lim_𝑛→∞𝑟_𝑛=𝑟₀>0;0<𝛼_𝑛<1forall𝑛∈ℕ,lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=+∞;0<𝛽_𝑛<1for all𝑛∈ℕ,0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1;0<𝜆_𝑛≤𝑏<1 for all 𝑛∈ℕ and for some constant 𝑏∈(0,1).𝜉 is a real constant such that 0<𝜉<2𝜂; Ω=Fix(𝑇^′)∩GMEP(𝑓,Φ,Θ)∩𝐵⁻¹(0)∩⋂^∞_𝑗=1Fix(𝑇_𝑗)≠∅. For 𝑟>0, 𝐹_𝑟 and 𝐺_𝑟 are as in Lemmas 8 and 9, respectively.

We shall study the strong convergence of the sequence {𝑥_𝑛}_𝑛≥1 generated iteratively from arbitrary 𝑥₁∈𝐻 by 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑛≥1 (74) to a unique solution 𝑥^′∈Ω of the variational inequality 𝑢−𝐴𝑥^′,𝑥−𝑥^′≤0∀𝑥∈Ω. (75)

Lemma 17. Suppose that the conditions of Remark 16 are satisfied, then {𝑥_𝑛}_𝑛≥1 defined by (74) is bounded.

Proof. Observe that for all 𝑥,𝑦∈𝐻, (𝑥−𝜉𝐵𝑥)−𝑦−𝜉𝐵𝑦²=𝑥−𝑦−𝜉(𝐵𝑥−𝐵𝑦)²=𝑥−𝑦²−2𝜉𝐵𝑥−𝐵𝑦,𝑥−𝑦+𝜉²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²−2𝜉𝜂𝐵𝑥−𝐵𝑦²+𝜉²𝐵𝑥−𝐵𝑦²=𝑥−𝑦²−𝜉2𝜂−𝜉𝐵𝑥−𝐵𝑦². (76) Thus, since 0<𝜉<2𝜂, we get from (76) that (𝑥−𝜉𝐵𝑥)−𝑦−𝜉𝐵𝑦≤𝑥−𝑦∀𝑥,𝑦∈𝐻. (77) It thus follows that for 𝑝∈Ω, (𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²=(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝑝²≤𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝². (78) Next, we may (without loss of generality) assume that 𝛼_𝑛<‖𝐴‖⁻¹(1−𝛽_𝑛) for all 𝑛∈ℕ since lim_𝑛→∞𝛼_𝑛=0. So, for each 𝑥∈𝐻 such that ‖𝑥‖=1, we have 1−𝛽_𝑛𝐼−𝛼_𝑛𝐴=𝑥,𝑥1−𝛽_𝑛−𝛼_𝑛⟨⟩≥𝐴𝑥,𝑥1−𝛽_𝑛−𝛼_𝑛‖𝐴‖>1−𝛽_𝑛−‖𝐴‖⁻¹1−𝛽_𝑛‖𝐴‖=0. (79) Thus, we have that for all 𝑛∈ℕ, the operator (1−𝛽_𝑛)𝐼−𝛼_𝑛𝐴 is a positive bounded linear operator, so that 1−𝛽_𝑛𝐼−𝛼_𝑛𝐴=sup_‖𝑥‖=11−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑥,𝑥=sup_‖𝑥‖=11−𝛽_𝑛−𝛼_𝑛⟨𝐴𝑥,𝑥⟩≤1−𝛽_𝑛−𝛼_𝑛𝛾. (80) Therefore, for 𝑝∈Ω, we obtain from (74) using (78) and (80) that 𝑥_𝑛+1−𝑝=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑢−𝐴𝑝+𝛽_𝑛𝑥_𝑛≤−𝑝1−𝛽_𝑛−𝛼_𝑛𝛾𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛≤𝑢−𝐴𝑝1−𝛼_𝑛𝛾𝑥_𝑛−𝑝+𝛼_𝑛𝑢−𝐴𝑝. (81) It is thus easy to show by mathematical induction (using (81)) that 𝑥_𝑛−𝑝≤max𝑥₁,1−𝑝𝛾𝑢−𝐴𝑝. (82) So, {𝑥_𝑛}_𝑛≥1 is bounded. Consequently, {𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛}_𝑛≥1,  {(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛}_𝑛≥1,  {𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛}_𝑛≥1 and {𝐺_𝑟𝑛𝑥_𝑛}_𝑛≥1 are all bounded.

Lemma 18. Let {𝑥_𝑛}_𝑛≥1 be given by (74). Suppose that conditions of Remark 16 are satisfied, then lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (83)

Proof. Set 𝑡_𝑛∶=(1/(1−𝛽_𝑛))(𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛), then 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑡_𝑛. Thus, 𝑡_𝑛+1−𝑡_𝑛=11−𝛽_𝑛+1𝑥_𝑛+2−𝛽_𝑛+1𝑥_𝑛+1−11−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛=11−𝛽_𝑛+1𝛼_𝑛+1𝑢+1−𝛽_𝑛+1𝐼−𝛼_𝑛+1𝐴×𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛+1𝐺_𝑟𝑛+1𝑥_𝑛+1−11−𝛽_𝑛𝛼_𝑛𝑢+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=𝛼_𝑛+11−𝛽_𝑛+1𝑢−𝐴𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛+1𝐺_𝑟𝑛+1𝑥_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛+1𝐺_𝑟𝑛+1𝑥_𝑛+1−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛. (84)
Thus, for 𝑝∈Ω (using (84), Lemmas 3, 13, and 12), we obtain (for some constant 𝑀₀>0) the following: 𝑡_𝑛+1−𝑡_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑀₀+𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛+1𝐺_𝑟𝑛+1𝑥_𝑛+1−𝑊_𝑛+1(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑀₀+2^𝑛_𝑗=1𝜆_𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝐹−𝑝_𝑟𝑛+1𝐺_𝑟𝑛+1𝑥_𝑛+1−𝐹_𝑟𝑛+1𝐺_𝑟𝑛𝑥_𝑛+𝐹_𝑟𝑛+1𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑀₀+2^𝑛_𝑗=1𝜆_𝑗𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝+𝐺_𝑟𝑛+1𝑥_𝑛+1−𝐺_𝑟𝑛𝑥_𝑛+𝐹_𝑟𝑛+1𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑀₀+2^𝑛_𝑗=1𝜆_𝑗𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝐺−𝑝_𝑟𝑛+1𝑥_𝑛+1−𝐺_𝑟𝑛+1𝑥_𝑛+𝐺_𝑟𝑛+1𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛+𝐹_𝑟𝑛+1𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1+𝛼_𝑛1−𝛽_𝑛𝑀₀+2^𝑛_𝑗=1𝜆_𝑗𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝑟−𝑝^𝑛+1−𝑟_𝑛𝑟_𝑛×𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+2𝐺_𝑟𝑛𝑥_𝑛+𝑥_𝑛. (85) Since 0<𝜆_𝑛<𝑏<1 for all 𝑛∈ℕ and for some constant 𝑏∈(0,1), it is thus easy to see that lim_𝑛→∞∏^𝑛_𝑗=1𝜆_𝑗=0. So, we obtain from (85) that limsup_𝑛→∞𝑡_𝑛+1−𝑡_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (86) Lemma 6 therefore gives lim_𝑛→∞𝑡_𝑛−𝑥_𝑛=0, (87) but 𝑥_𝑛+1−𝑥_𝑛=1−𝛽_𝑛𝑡_𝑛−𝑥_𝑛. (88) Hence, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (89)

Lemma 19. Let {𝑥_𝑛}_𝑛≥1 be given by (74). Suppose conditions of Remark 16 are satisfied, then lim_𝑛→∞𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=lim_𝑛→∞𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=lim_𝑛→∞𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛=0. (90)

Proof. Observe that for some constant 𝑀₁>0, 𝑥_𝑛−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑀₁+𝛽_𝑛𝑥_𝑛−𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛. (91) So, we obtain from (91) that 𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛≤11−𝛽_𝑛𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑀₁. (92) Using (89) and the fact that lim_𝑛→∞𝛼_𝑛=0, we obtain from (92) that lim_𝑛→∞𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛=0. (93) Next, observe that for fixed 𝑝∈Ω and using (2) of Lemma 9, we obtain the following: 𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝐺_𝑟𝑛𝑥_𝑛−𝑝,𝑥_𝑛=1−𝑝2𝐺_𝑟𝑛𝑥_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛², (94) so that 𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛². (95) Furthermore, using (2) of Lemma 8, we obtain 𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝,𝐺_𝑟𝑛𝑥_𝑛=1−𝑝2𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+𝐺_𝑟𝑛𝑥_𝑛−𝑝²−𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛², (96) so that 𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝐺_𝑟𝑛𝑥_𝑛−𝑝²−𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛². (97) Moreover, using the recursion formula (74), Lemma 4, and convexity of ‖⋅‖², we have 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝛽_𝑛𝑥_𝑛−𝑝²+1−𝛽_𝑛×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)×𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1−𝑝≤𝛽_𝑛𝑥_𝑛+1−𝑝²+2𝛽_𝑛𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛²×1−𝛽_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+2𝛼_𝑛𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)×𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1−𝑝. (98) Thus, using (76) and (97), we obtain from (98) the following: 𝑥_𝑛+1−𝑝²≤(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+𝛽_𝑛1−𝛽_𝑛2𝑥_𝑛+1−𝑝+𝑥_𝑛+1−𝑥_𝑛×𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1≤𝐹−𝑝_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²−𝜉2𝜂−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+𝛽_𝑛1−𝛽_𝑛2𝑥_𝑛+1−𝑝+𝑥_𝑛+1−𝑥_𝑛×𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1≤𝑥−𝑝_𝑛−𝑝²−𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²−𝜉2𝜂−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+𝛽_𝑛1−𝛽_𝑛2𝑥_𝑛+1−𝑝+𝑥_𝑛+1−𝑥_𝑛×𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1−𝑝. (99) So, from (99), we have (for some real constant 𝑀₂>0) 𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+𝜉2𝜂−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝛽_𝑛1−𝛽_𝑛×2𝑥_𝑛+1−𝑝+𝑥_𝑛+1−𝑥_𝑛×𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1≤𝛼−𝑝_𝑛+𝑥_𝑛+1−𝑥_𝑛𝑀₂. (100) From (100), we have lim_𝑛→∞𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+𝜉2𝜂−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²=0, (101) and since 𝜉(2𝜂−𝜉)>0, then using (101) and applying sandwich theorem, we obtain lim_𝑛→∞𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=0,lim_𝑛→∞𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=0. (102)(103) Furthermore, observe that 𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛≤𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛. (104) Hence, using (93), (102), (103), and (104), we have lim_𝑛→∞𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=0. (105) Besides, we obtain from the second line of inequality (99) (using (95) and the fact that 𝐹_𝑟𝑛 is firmly nonexpansive; thus nonexpansive) that for some constant 𝑀^′>0, 𝑥_𝑛+1−𝑝²≤𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²−𝜉2𝜂−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+𝛽_𝑛1−𝛽_𝑛2𝑥_𝑛+1−𝑝+𝑥_𝑛+1−𝑥_𝑛×𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1≤𝐺−𝑝_𝑟𝑛𝑥_𝑛−𝑝²+𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑀^′≤𝑥_𝑛−𝑝²−𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛²+𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑀^′. (106) So, we obtain from (106) that for some constant 𝑀^′′>0, 𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑀^′≤𝑥_𝑛+1−𝑥_𝑛𝑀^′′+𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑀^′. (107) Hence, lim_𝑛→∞‖𝑥_𝑛−𝐺_𝑟𝑛𝑥_𝑛‖=0.

Lemma 20. Let {𝑥_𝑛}_𝑛≥1 be given by (74). Suppose that conditions of Remark 16 hold, then lim_𝑛→∞𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=0. (108)

Proof. Let 𝑝∈Ω, then we obtain (𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²=𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²=𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝−𝜉𝐵𝑝²=12𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝−𝜉𝐵𝑝²+(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝜉𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−−()𝐹𝑝−𝜉𝐵𝑝𝐼−𝜉𝐵_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤12𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+2𝜉𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)×𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝜉²𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛². (109) This inequality implies the following: (𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+2𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛×𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝜉²𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²≤𝑥_𝑛−𝑥_𝑛+1²+2𝑥_𝑛+1×𝑥−𝑝_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑝²−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²+2𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛×𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛. (110) But from the first line of inequality (99), we have 𝑥_𝑛+1−𝑝²≤(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+𝛽_𝑛1−𝛽_𝑛2𝑥_𝑛+1−𝑝+𝑥_𝑛+1−𝑥_𝑛×𝑥_𝑛+1−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1−𝑝. (111) So, for some real constant 𝑀₂>0, we obtain from (111) that 𝑥_𝑛+1−𝑝²≤(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑝²+𝛼_𝑛+𝑥_𝑛+1−𝑥_𝑛𝑀₂. (112) Using (112) in (110) and rearranging terms, we obtain (for some real constant 𝑀₃>0) that 𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛²≤𝛼_𝑛+𝑥_𝑛+1−𝑥_𝑛+𝐵𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛𝑀₃. (113) So, (113) implies that lim_𝑛→∞𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=0. (114) Hence, (𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛≤(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛𝑊→as𝑛→∞,_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛≤𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛. (115) The last inequality implies that lim_𝑛→∞𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛=0. (116)

Lemma 21. Let {𝑥_𝑛}_𝑛≥1 be given by (74) and let 𝑥^′∈Ω be the unique solution of the variational inequality (75). Suppose that conditions of Remark 16 hold, then limsup_𝑛→∞𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′≤0. (117)

Proof. Let {𝑥_𝑛𝑗}_𝑗≥1 be a subsequence of {𝑥_𝑛}_𝑛≥1 such that limsup_𝑛→∞𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′=lim_𝑗→∞𝑢−𝐴𝑥^′,𝑥_𝑛𝑗−𝑥^′. (118) Since every real Hilbert space 𝐻 is a uniformly convex real Banach space and {𝑥_𝑛}_𝑛≥1 is a bounded sequence in 𝐻, then we obtain by Lemma 11 that there exists a unique 𝑥^∗∈𝐻 such that 𝑥^∗ is the asymptotic center of the sequence {𝑥_𝑛𝑗}_𝑗≥1. We first show that 𝑥^∗∈Ω. Recall that 𝑟₀=lim_𝑛→∞𝑟_𝑛 and that by Lemma 1, 𝑊𝑥∶=lim_𝑛→∞𝑊_𝑛𝑥 exists for all 𝑥∈𝐻 and by Lemma 2 that Fix(𝑊)=⋂^∞_𝑛=1Fix(𝑊_𝑛)=⋂^∞_𝑗=1Fix(𝑇_𝑗); thus, 𝑥_𝑛𝑗−𝑊𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛼_𝑛𝑗𝑢+𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝛽_𝑛𝑗𝐼−𝛼_𝑛𝑗𝐴×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑊𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑊𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑊_𝑛𝑗𝑥_𝑛𝑗+𝑊_𝑛𝑗𝑥_𝑛𝑗−𝑊_𝑛𝑗𝑥^∗+𝑊_𝑛𝑗𝑥^∗−𝑊𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑊𝑥^∗+1−𝛽_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑊_𝑛𝑗𝑥^∗−𝑊𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (119) Thus, we obtain from (119) that 𝑥_𝑛𝑗−𝑊𝑥^∗≤11−𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑊_𝑛𝑗𝑥^∗−𝑊𝑥^∗+𝛼_𝑛𝑗1−𝛽_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗, (120) so that limsup_𝑗→∞𝑥_𝑛𝑗−𝑊𝑥^∗≤limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗. (121) This implies by Definition 10 (with 𝐶=𝐻) and uniqueness of 𝑥^∗ that 𝑊𝑥^∗=𝑥^∗; that is, 𝑥^∗∈⋂^∞_𝑗=1Fix(𝑇_𝑗).
Next, 𝑥_𝑛𝑗−𝐹_𝑟0𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛼_𝑛𝑗𝑢+𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝛽_𝑛𝑗𝐼−𝛼_𝑛𝑗𝐴×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟0𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟0𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝑥^∗+𝐹_𝑟𝑛𝑗𝑥^∗−𝐹_𝑟0𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟0𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑟_𝑛𝑗−𝑟₀𝑟₀𝐹_𝑟0𝑥^∗+𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗, (122) so that from (122), we obtain 𝑥_𝑛𝑗−𝐹_𝑟0𝑥^∗≤11−𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑟_𝑛𝑗−𝑟₀𝑟₀𝐹_𝑟0𝑥^∗+𝑥^∗×𝛼_𝑛𝑗1−𝛽_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (123) Thus,limsup_𝑛→∞𝑥_𝑛𝑗−𝐹_𝑟0𝑥^∗≤limsup_𝑛→∞𝑥_𝑛𝑗−𝑥^∗. (124) So, by Definition 10, we obtain that 𝐹_𝑟0𝑥^∗=𝑥^∗⇔𝑇^′𝑥^∗=𝑥^∗, that is, 𝑥^∗∈Fix(𝑇^′).
Moreover, 𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛼_𝑛𝑗𝑢+𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝛽_𝑛𝑗𝐼−𝛼_𝑛𝑗𝐴×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐺_𝑟𝑛𝑗𝑥^∗+𝐺_𝑟𝑛𝑗𝑥^∗−𝐺_𝑟0𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑟_𝑛𝑗−𝑟₀𝑟₀𝐺_𝑟0𝑥^∗+𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (125) Thus, we obtain from (125) that 𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗≤11−𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑟_𝑛𝑗−𝑟₀𝑟₀𝐺_𝑟0𝑥^∗+𝑥^∗+𝛼_𝑛𝑗1−𝛽_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (126) Thus, limsup_𝑗→∞𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗≤limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗. (127)
Definition 10 again implies that 𝐺_𝑟0𝑥^∗=𝑥^∗. Hence 𝑥^∗∈GMEP(𝑓,Φ,Θ).
Furthermore, since we have shown that 𝐺_𝑟0𝑥^∗=𝑥^∗=𝐹_𝑟0𝑥^∗, then making use of this fact and Lemma 13, we have 𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛼_𝑛𝑗𝑢+𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝛽_𝑛𝑗𝐼−𝛼_𝑛𝑗𝐴×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐹_𝑟𝑛𝑗𝐺_𝑟0𝑥^∗+𝐹_𝑟𝑛𝑗𝐺_𝑟0𝑥^∗−𝐹_𝑟0𝐺_𝑟0𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+1−𝛽_𝑛𝑗×𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐺_𝑟0𝑥^∗+𝐹_𝑟𝑛𝑗𝐺_𝑟0𝑥^∗−𝐹_𝑟0𝐺_𝑟0𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (128) This implies that 𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+1−𝛽_𝑛𝑗𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−𝐺_𝑟𝑛𝑗𝑥^∗+𝐺_𝑟𝑛𝑗𝑥^∗−𝐺_𝑟0𝑥^∗+𝐹_𝑟𝑛𝑗𝐺_𝑟0𝑥^∗−𝐹_𝑟0𝐺_𝑟0𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗≤𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝛽_𝑛𝑗𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗+1−𝛽_𝑛𝑗𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+𝑟_𝑛𝑗−𝑟₀𝑟₀×𝐹_𝑟0𝐺_𝑟0𝑥^∗+2𝐺_𝑟0𝑥^∗+𝑥^∗+𝛼_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (129) Inequality (129) thus gives 𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗≤11−𝛽_𝑛𝑗𝑥_𝑛𝑗+1−𝑥_𝑛𝑗+𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗−(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥^∗+4𝑥^∗𝑟_𝑛𝑗−𝑟₀𝑟₀+𝛼_𝑛𝑗1−𝛽_𝑛𝑗𝑢−𝐴𝑊_𝑛𝑗(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝑗𝐺_𝑟𝑛𝑗𝑥_𝑛𝑗. (130) Thus, we obtain from (130) that limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗−𝜉𝐵𝑥^∗≤limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗, (131) so that Definition 10 again gives 𝑥^∗−𝜉𝐵𝑥^∗=𝑥^∗ and this implies that 𝑥^∗∈𝐵⁻¹(0). Hence, 𝑥^∗∈Ω.
Now, let 𝜀∈(0,1) and 𝑥∈𝐻 be arbitrary, then, using Lemma 4 we have 𝑥_𝑛𝑗−𝑥^∗−𝜀𝑥−𝑥^∗2≤𝑥_𝑛𝑗−𝑥^∗2−2𝜀𝑥−𝑥^∗,𝑥_𝑛𝑗−𝑥^∗−𝜀𝑥−𝑥^∗=𝑥_𝑛𝑗−𝑥^∗2+2𝜀𝑥−𝑥^∗,𝑥^∗+𝜀𝑥−𝑥^∗−𝑥_𝑛𝑗. (132) Since 𝑥^∗ is the asymptotic center of {𝑥_𝑛𝑗}_𝑗≥1, it is easy to see (using Definition 10 and Lemma 14) that limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗2≤limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗−𝜀𝑥−𝑥^∗2≤limsup_𝑗→∞𝑥_𝑛𝑗−𝑥^∗2+2𝜀limsup_𝑗→∞𝑥−𝑥^∗,𝑥^∗+𝜀𝑥−𝑥^∗−𝑥_𝑛𝑗. (133) Thus, 0≤limsup_𝑗→∞𝑥−𝑥^∗,𝑥^∗+𝜀𝑥−𝑥^∗−𝑥_𝑛𝑗, (134) But, 𝑥−𝑥^∗,𝑥^∗−𝑥_𝑛𝑗=𝑥−𝑥^∗,𝑥^∗+𝜀𝑥−𝑥^∗−𝑥_𝑛𝑗−𝜀𝑥−𝑥^∗2. (135) So, limsup_𝑗→∞𝑥−𝑥^∗,𝑥^∗−𝑥_𝑛𝑗=limsup_𝑗→∞𝑥−𝑥^∗,𝑥^∗+𝜀𝑥−𝑥^∗−𝑥_𝑛𝑗−𝜀𝑥−𝑥^∗2=limsup_𝑗→∞𝑥−𝑥^∗,𝑥^∗+𝜀𝑥−𝑥^∗−𝑥_𝑛𝑗−𝜀𝑥−𝑥^∗2≥−𝜀𝑥−𝑥^∗2. (136) Thus, since 𝜀∈(0,1) is arbitrary, we obtain from (136) that limsup_𝑗→∞𝑥−𝑥^∗,𝑥^∗−𝑥_𝑛𝑗≥0∀𝑥∈𝐻. (137) In particular, for 𝑥=𝑥^∗+𝑢−𝐴𝑥^′∈𝐻 and using the fact that 𝑥^∗∈Ω, we obtain from (137) that 0≤limsup_𝑗→∞𝑢−𝐴𝑥^′,𝑥^∗−𝑥_𝑛𝑗≤𝑢−𝐴𝑥^′,𝑥^∗−𝑥^′+limsup_𝑗→∞𝑢−𝐴𝑥^′,𝑥^′−𝑥_𝑛𝑗=𝑢−𝐴𝑥^′,𝑥^∗−𝑥^′+lim_𝑗→∞𝑢−𝐴𝑥^′,𝑥^′−𝑥_𝑛𝑗≤lim_𝑗→∞𝑢−𝐴𝑥^′,𝑥^′−𝑥_𝑛𝑗. (138) This implies that lim_𝑗→∞𝑢−𝐴𝑥^′,𝑥_𝑛𝑗−𝑥^′≤0. (139) Hence, limsup_𝑛→∞𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′=lim_𝑗→∞𝑢−𝐴𝑥^′,𝑥_𝑛𝑗−𝑥^′≤0. (140)

Theorem 22. Let {𝑥_𝑛}_𝑛≥1 be given by (74). Suppose that the conditins of Remark 16 are satisfied, then {𝑥_𝑛}_𝑛≥1 converges strongly to 𝑥^′∈Ω which is a unique solution of the variational inequality (75).

Proof. From the recursion formula (74) and Lemma 4, we have 𝑥_𝑛+1−𝑥^′2=𝛼_𝑛𝑢+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑥^′2=1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′+𝛽_𝑛𝑥_𝑛−𝑥^′+𝛼_𝑛𝑢−𝐴𝑥^′2≤1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×(𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′)+𝛽_𝑛(𝑥_𝑛−𝑥^′)²+2𝛼_𝑛𝑢−𝐴𝑥^′,𝑥_𝑛+1−𝑥^′=1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×(𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′)+𝛽_𝑛(𝑥_𝑛−𝑥^′)²+2𝛼_𝑛𝑢−𝐴𝑥^′,1−𝛽_𝑛×𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′−𝛼_𝑛𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′+𝛼_𝑛𝑢−𝐴𝑥^′+𝛽_𝑛𝑥_𝑛−𝑥^′=1−𝛽_𝑛𝐼−𝛼_𝑛𝐴×(𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′)+𝛽_𝑛(𝑥_𝑛−𝑥^′)²+2𝛼²_𝑛𝑢−𝐴𝑥^′2+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑥^′,𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′−2𝛼²_𝑛𝑢−𝐴𝑥^′,𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′+2𝛼_𝑛𝛽_𝑛𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′≤1−𝛽_𝑛−𝛼_𝑛𝛾+𝛽_𝑛²𝑥_𝑛−𝑥^′2+2𝛼²_𝑛𝑢−𝐴𝑥^′2+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑥^′,𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛+2𝛼_𝑛1−𝛽_𝑛𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′−2𝛼²_𝑛𝑢−𝐴𝑥^′,𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′+2𝛽_𝑛𝛼_𝑛𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′=1−2𝛼_𝑛𝛾𝑥_𝑛−𝑥^′2+𝛼²_𝑛𝛾²𝑥_𝑛−𝑥^′2+2𝛼²_𝑛𝑢−𝐴𝑥^′2+2𝛼_𝑛1−𝛽_𝑛×𝑢−𝐴𝑥^′,𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛−2𝛼²_𝑛𝑢−𝐴𝑥^′,𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥^′+2𝛼_𝑛𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′≤1−2𝛼_𝑛𝛾𝑥_𝑛−𝑥^′2+𝛼_𝑛𝛼_𝑛+𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛𝑀₄+2𝛼_𝑛𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′, (141) for some real constant 𝑀₄>0. Thus, 𝑥_𝑛+1−𝑥^′2≤1−𝛿_𝑛𝑥_𝑛−𝑥^′2+𝜎_𝑛, (142) where 𝛿_𝑛∶=2𝛼_𝑛𝛾 and 𝜎_𝑛∶=𝛼_𝑛(𝛼_𝑛+‖𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛−𝑥_𝑛‖)𝑀₄+2𝛼_𝑛⟨𝑢−𝐴𝑥^′,𝑥_𝑛−𝑥^′⟩. It is easy to see that limsup_𝑛→∞(𝜎_𝑛/𝛿_𝑛)≤0 (using (93), Lemma 21 and the fact that lim_𝑛→∞𝛼_𝑛=0). Hence, we obtain by Lemma 5 that the sequence {𝑥_𝑛}_𝑛≥1 converges strongly to 𝑥^′∈Ω which is a unique solution of (75).

Remark 23. Let 𝑓∶𝐻→𝐻 be a strict contraction, then our method of proof easily carries over to the iteration method {𝑥_𝑛}_𝑛≥1 given by 𝑥₁∈𝐻, 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴𝑊_𝑛(𝐼−𝜉𝐵)𝐹_𝑟𝑛𝐺_𝑟𝑛𝑥_𝑛, (143) making use of the fact that 𝑓 is a contraction. Thus, we obatin the following theorem.

Theorem 24. Let {𝑥_𝑛}_𝑛≥1 be given by (143). Suppose that the conditions of Remark 16 are satisfied, then {𝑥_𝑛}_𝑛≥1 converges strongly to 𝑥^′∈Ω which is a unique solution of the variational inequality 𝐴−𝛾𝑓𝑥^′,𝑦−𝑥^′≥0,∀𝑦∈Ω⟺𝑥^′=𝑃_Ω𝐼−𝐴+𝛾𝑓𝑥^′, (144) which is the optimality condition for the minimization problem min_𝑥∈Ω(1/2)⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻), and one recalls that Ω=Fix(𝑇^′)∩GMEP(𝑓,Φ,Θ)∩𝐵⁻¹(0)∩⋂^∞_𝑗=1Fix(𝑇_𝑗).

Remark 25. Our theorems extend and unify most of the results that have been proved for the class of nonlinear operators studied in this paper. In particular, it is easy to see that Theorem 24 extends, improves, generalizes, and unifies the corresponding results of Yao et al. [32], Colao and Marino [33], and Chamnarnpan and Kuman [34]. Furthermore, the setting of our theorems corrects the problem of well definedness which might arise in the results of Chamnarnpan and Kuman [34], and our method of proof is of independent interest.