Hybrid Algorithms for Minimization Problems over the Solutions of Generalized Mixed Equilibrium and Variational Inclusion Problems

Abstract

We introduce a new general hybrid iterative algorithm for finding a common element of the set of solution of fixed point for a nonexpansive mapping, the set of solution of generalized mixed equilibrium problem, and the set of solution of the variational inclusion for a β-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Our results improve and extend the corresponding results of Marino and Xu (2006), Yao and Liou (2010), Tan and Chang (2011), and other authors.

1. Introduction

In the theory of variational inequalities, variational inclusions, and equilibrium problems, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance, and applied sciences.

Equilibrium theory represents an important area of mathematical sciences such as optimization, operations research, game theory, complementarity problems, financial mathematics, and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, fixed point problems, and complementarity problems as special cases; for example, see the references herein. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻 with the inner product ⟨⋅,⋅⟩ and the norm ‖⋅‖. Let 𝐹 be a bifunction of 𝐶×𝐶 into ℛ, where ℛ is the set of real numbers, Φ∶𝐶→𝐻 be a mapping and 𝜑∶𝐶→ℛ be a real-valued function. The generalized mixed equilibrium problem for finding 𝑥∈𝐶 such that𝐹𝑥,𝑦+Φ𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.1) The set of solutions of (1.1) is denoted by GMEP(𝐹,𝜑,Φ), that isGMEP𝐹,𝜑,Φ=𝑥∈𝐶∶𝐹𝑥,𝑦+Φ𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.2) If Φ≡0 and 𝜑≡0, the problem (1.1) is reduced into the equilibrium problem [1] for finding 𝑥∈𝐶 such that𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (1.3) The set of solutions of (1.3) is denoted by EP(𝐹). This problem contains fixed point problems, includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem, please consult [2–4].

If 𝐹≡0 and 𝜑≡0, the problem (1.1) is reduced into the Hartmann-Stampacchia variational inequality [5] for finding 𝑥∈𝐶 such thatΦ𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (1.4) The set of solutions of (1.4) is denoted by VI(𝐶,Φ). The variational inequality has been extensively studied in the literature [6].

If 𝐹≡0 and Φ≡0, the problem (1.1) is reduced into the minimize problem for finding 𝑥∈𝐶 such that𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.5) The set of solutions of (1.5) is denoted by Argmin(𝜑).

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space𝐻:𝜃(𝑥)=12⟨𝐴𝑥,𝑥⟩−𝑥,𝑦,∀𝑥∈𝐹(𝑆), (1.6) where 𝐴 is a linear bounded operator, 𝐹(𝑆) is the fixed point set of a nonexpansive mapping 𝑆, and 𝑦 is a given point in 𝐻 [7].

Recall, a mapping 𝑆∶𝐶→𝐶 is said to be nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦, (1.7) for all 𝑥,𝑦∈𝐶. If 𝐶 is bounded closed convex and 𝑆 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑆) is nonempty [8]. We denote weak convergence and strongly convergence by notations ⇀ and →, respectively. A mapping 𝐴 of 𝐶 into 𝐻 is called monotone if𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0, (1.8) for all 𝑥,𝑦∈𝐶. A mapping 𝐴 of 𝐶 into 𝐻 is called 𝛼- inverse-strongly monotone if there exists a positive real number 𝛼 such that𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦², (1.9) for all 𝑥,𝑦∈𝐶. It is obvious that any 𝛼-inverse-strongly monotone mappings 𝐴 is monotone and Lipschitz continuous mapping. A linear bounded operator 𝐴 is strongly positive if there exists a constant 𝛾>0 with the property⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖², (1.10) for all 𝑥∈𝐻. A self-mapping 𝑓∶𝐶→𝐶 is a contraction on 𝐶 if there exists a constant 𝛼∈(0,1) such that𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦, (1.11) for all 𝑥,𝑦∈𝐶. We use Π_𝐶 to denote the collection of all contraction on C. Note that each 𝑓∈Π_𝐶 has a unique fixed point in 𝐶.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. Let 𝐵∶𝐻→𝐻 be a single-valued nonlinear mapping and 𝑀∶𝐻→2^𝐻 be a set-valued mapping. The variational inclusion problem is to find 𝑥∈𝐻 such that𝜃∈𝐵(𝑥)+𝑀(𝑥), (1.12) where 𝜃 is the zero vector in 𝐻. The set of solutions of problem (1.12) is denoted by 𝐼(𝐵,𝑀). The variational inclusion has been extensively studied in the literature. See, for example, [9–12] and the reference therein.

A set-valued mapping 𝑀∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻,𝑓∈𝑀(𝑥) and 𝑔∈𝑀(𝑦) imply ⟨𝑥−𝑦,𝑓−𝑔⟩≥0. A monotone mapping 𝑀 is maximal if its graph 𝐺(𝑀)∶={(𝑓,𝑥)∈𝐻×𝐻∶𝑓∈𝑀(𝑥)} of 𝑀 is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑀 is maximal if and only if for (𝑥,𝑓)∈𝐻×𝐻,⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for all (𝑦,𝑔)∈𝐺(𝑀) imply 𝑓∈𝑀(𝑥).

Let 𝐵 be an inverse-strongly monotone mapping of 𝐶 into 𝐻 and let 𝑁_𝐶𝑣 be normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐶}, and define𝑀𝑣=𝐵𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (1.13) Then 𝑀 is a maximal monotone and 𝜃∈𝑀𝑣 if and only if 𝑣∈VI(𝐶,𝐵) [13].

Let 𝑀∶𝐻→2^𝐻 be a set-valued maximal monotone mapping, then the single-valued mapping 𝐽_𝑀,𝜆∶𝐻→𝐻 defined by𝐽_𝑀,𝜆(𝑥)=(𝐼+𝜆𝑀)⁻¹(𝑥),𝑥∈𝐻 (1.14) is called the resolvent operator associated with 𝑀, where 𝜆 is any positive number and 𝐼 is the identity mapping. It is worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone, and that a solution of problem (1.12) is a fixed point of the operator 𝐽_𝑀,𝜆(𝐼−𝜆𝐵) for all 𝜆>0, see [14], that is, 𝐼(𝐵,𝑀)=𝐹(𝐽_𝑀,𝜆(𝐼−𝜆𝐵)),∀𝜆>0.

In 2000, Moudafi [15] introduced the viscosity approximation method for nonexpansive mapping and proved that if 𝐻 is a real Hilbert space, the sequence {𝑥_𝑛} defined by the iterative method below, with the initial guess 𝑥₀∈𝐶 chosen arbitrarily,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,𝑛≥0, (1.15) where {𝛼_𝑛}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of 𝑆 (say 𝑥∈𝐶) which is the unique solution of the following variational inequality:𝐼−𝑓𝑥,𝑥−𝑥≥0,∀𝑥∈𝐹(𝑆). (1.16)

In 2006, Marino and Xu [7] introduced a general iterative method for nonexpansive mapping. They defined the sequence {𝑥_𝑛} generated by the algorithm 𝑥₀∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑥_𝑛,𝑛≥0, (1.17) where {𝛼_𝑛}⊂(0,1) and 𝐴 is a strongly positive linear bounded operator. They prove that if 𝐶=𝐻 and the sequence {𝛼_𝑛} satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.17) converges strongly to a fixed point of 𝑆 (say 𝑥∈𝐻) which is the unique solution of the following variational inequality:𝐴−𝛾𝑓𝑥,𝑥−𝑥≥0,∀𝑥∈𝐹(𝑆), (1.18) which is the optimality condition for the minimization problemmin_{𝑥∈𝐹(𝑆)∩EP(𝐹)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.19) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

In 2010, Yao and Liou [16] introduced the following composite iterative scheme in a real Hilbert space: 𝑥₀∈𝐶𝑥_𝑛+1=𝜇_𝑛𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛+1−𝜇_𝑛𝑇_𝑟𝑥_𝑛−𝑟𝐴𝑥_𝑛, (1.20) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝜇_𝑛}⊂[0,1). Furthermore, they proved {𝑥_𝑛} and {𝑢_𝑛} converge strongly to the same point 𝑧∈𝐹(𝑆)∩EP(𝐹), where 𝑃_𝐶 is the projection of 𝐻 onto 𝐶.

In 2011, Tan and Chang [11] introduced the following iterative process for {𝑇_𝑛∶𝐶→𝐶} be a sequence of nonexpansive mappings. Let {𝑥_𝑛} be the sequence defined by𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶1−𝑡_𝑛𝐽_𝑀,𝜆(𝐼−𝜆𝐴)𝑇_𝜇𝐼−𝜇𝐵𝑥_𝑛,∀𝑛≥0, (1.21) where {𝛼_𝑛}⊂(0,1), 𝜆∈(0,2𝛼] and 𝜇∈(0,2𝛽]. Then, the sequence {𝑥_𝑛} defined by (1.21) converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solution of the variational inequality and the generalized equilibrium problem.

In this paper, we modify the iterative methods (1.17), (1.20), and (1.21) by purposing the following new general viscosity iterative method: 𝑥₀∈𝐶,𝑥_𝑛+1=𝜉_𝑛𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛+1−𝜉_𝑛𝑇_𝑟𝑥_𝑛−𝑟Φ𝑥_𝑛, (1.22) for all 𝑛∈ℕ, where {𝛼_𝑛},{𝜉_𝑛}⊂(0,1), 𝑟∈(0,2𝜎), and 𝜆∈(0,2𝛽) satisfy some appropriate conditions. Consequently, we show that under some control conditions the sequence {𝑥_𝑛} strongly converge to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problem, and the set of solution of the variational inclusion in a real Hilbert space.

2. Preliminaries

Let 𝐻 be a real Hilbert space and 𝐶 be a nonempty closed convex subset of 𝐻. Recall that the (nearest point) projection 𝑃_𝐶 from 𝐻 onto 𝐶 assigns to each 𝑥∈𝐻, the unique point in 𝑃_𝐶𝑥∈𝐶 satisfying the property 𝑥−𝑃_𝐶𝑥=min_𝑦∈𝐶𝑥−𝑦. (2.1) The following characterizes the projection 𝑃_𝐶. We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1. The function 𝑢∈𝐶 is a solution of the variational inequality (1.4) if and only if 𝑢∈𝐶 satisfies the relation 𝑢=𝑃_𝐶(𝑢−𝜆Φ𝑢) for all 𝜆>0.

Lemma 2.2. For a given 𝑧∈𝐻, 𝑢∈𝐶, 𝑢=𝑃_𝐶𝑧⇔⟨𝑢−𝑧,𝑣−𝑢⟩≥0,∀𝑣∈𝐶.
It is well known that 𝑃_𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶 and satisfies 𝑃_𝐶𝑥−𝑃_𝐶𝑦²≤𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦,∀𝑥,𝑦∈𝐻. (2.2) Moreover, 𝑃_𝐶𝑥 is characterized by the following properties: 𝑃_𝐶𝑥∈𝐶 and for all 𝑥∈𝐻,𝑦∈𝐶, 𝑥−𝑃_𝐶𝑥,𝑦−𝑃_𝐶𝑥≤0. (2.3)

Lemma 2.3 (see [17]). Let 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping and let 𝐵∶𝐻→𝐻 be a monotone and Lipschitz continuous mapping. Then the mapping 𝐿=𝑀+𝐵∶𝐻→2^𝐻 is a maximal monotone mapping.

Lemma 2.4 (see [18]). Each Hilbert space 𝐻 satisfies Opial's condition, that is, for any sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞‖𝑥_𝑛−𝑥‖<liminf_𝑛→∞‖𝑥_𝑛−𝑦‖, hold for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.5 (see [19]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛𝛾_𝑛,∀𝑛≥0, (2.4) where {𝛾_𝑛}⊂(0,1) and {𝛿_𝑛} is a sequence in ℛ such that(i)∑^∞_𝑛=1𝛾_𝑛=∞.(ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=1|𝛿_𝑛𝛾_𝑛|<∞.Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.6 (see [20]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then 𝐼−𝑇 is demiclosed at zero, that is, 𝑥_𝑛⇀𝑥,𝑥_𝑛−𝑇𝑥_𝑛⟶0 (2.5) implies 𝑥=𝑇𝑥.

For solving the generalized mixed equilibrium problem, let us assume that the bifunction 𝐹∶𝐶×𝐶→ℛ, the nonlinear mapping Φ∶𝐶→𝐻 is continuous monotone and 𝜑∶𝐶→ℛ satisfies the following conditions: (A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶; (A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶; (A3)for each fixed 𝑦∈𝐶, 𝑥↦𝐹(𝑥,𝑦) is weakly upper semicontinuous; (A4)for each fixed 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous; (B1)for each 𝑥∈𝐶 and 𝑟>0, there exist a bounded subset 𝐷_𝑥⊆𝐶 and 𝑦_𝑥∈C such that for any 𝑧∈𝐶⧵𝐷_𝑥, 𝐹𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0, (2.6)(B2)𝐶 is a bounded set.

Lemma 2.7 (see [21]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶×𝐶→ℛ be a bifunction mapping satisfies (A1)–(A4) and let 𝜑∶𝐶→ℛ is convex and lower semicontinuous such that 𝐶∩dom𝜑≠∅. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, then there exists 𝑢∈𝐶 such that 𝐹𝑦(𝑢)+1𝑢,𝑦+𝜑−𝜑𝑟𝑦−𝑢,𝑢−𝑥. (2.7) Define a mapping 𝑇_𝑟∶𝐻→𝐶 as follows: 𝑇_𝑟(𝑥)=𝑦(𝑢)+1𝑢∈𝐶∶𝐹𝑢,𝑦+𝜑−𝜑𝑟𝑦−𝑢,𝑢−𝑥≥0,∀𝑦∈𝐶, (2.8) for all 𝑥∈𝐻. Then, the following hold: (i)𝑇_𝑟 is single-valued; (ii)𝑇_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖𝑇_𝑟𝑥−𝑇_𝑟𝑦‖²≤⟨𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝑥−𝑦⟩; (iii)𝐹(𝑇_𝑟)=MEP(𝐹,𝜑); (iv)MEP(𝐹,𝜑) is closed and convex.

Lemma 2.8 (see[7]). Assume 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹, then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of 𝐹(𝑆),GMEP(𝐹,𝜑,Φ), and 𝐼(𝐵,𝑀) of inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. Let 𝐻 be a real Hilbert space, 𝐶 be a closed convex subset of 𝐻. Let 𝐹 be a bifunction of 𝐶×𝐶 into ℛ satisfying (A1)–(A4) and 𝐵,Φ∶𝐶→𝐻 be 𝛽,𝜎-inverse-strongly monotone mappings, 𝜑∶𝐶→ℛ is convex and lower semicontinuous function, 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1), 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping and 𝐴 be a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0, assume that 0<𝛾<𝛾/𝛼. Let 𝑆 be a nonexpansive mapping of 𝐻 into itself and assume that either (B1) or (B2) holds such that Θ∶=𝐹(𝑆)∩GMEP𝐹,𝜑,Φ∩𝐼(𝐵,𝑀)≠∅. (3.1) Suppose {𝑥_𝑛} is a sequences generated by the following algorithm 𝑥₀∈𝐶 arbitrarily: 𝑥_𝑛+1=𝜉_𝑛𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛+1−𝜉_𝑛𝑇_𝑟𝑥_𝑛−𝑟Φ𝑥_𝑛, (3.2) where {𝛼_𝑛},{𝜉_𝑛}⊂(0,1),𝜆∈(0,2𝛽) such that 0<𝑎≤𝜆≤𝑏<2𝛽 and 𝑟∈(0,2𝜎) with 0<𝑐≤𝑑≤1−𝜎 satisfy the following conditions:(C1)lim_𝑛→∞𝛼_𝑛=0, Σ^∞_𝑛=0𝛼_𝑛=∞ and lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1,(C2)0<liminf_𝑛→∞𝜉_𝑛<limsup_𝑛→∞𝜉_𝑛<1 and lim_𝑛→∞((𝜉_𝑛+1−𝜉_𝑛)/𝛼_𝑛+1)=1.
Then {𝑥_𝑛} converges strongly to 𝑞∈Θ, where 𝑞=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: 𝛾𝑓−𝐴𝑞,𝑝−𝑞≤0,∀𝑝∈Θ (3.3) which is the optimality condition for the minimization problem min_𝑞∈Θ12⟨𝐴𝑞,𝑞⟩−ℎ(𝑞), (3.4) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. Because of condition (C1), we may assume without loss of generality, then 𝛼_𝑛∈(0,‖𝐴‖⁻¹) for all 𝑛. By Lemma 2.8, we have ‖𝐼−𝛼_𝑛𝐴‖≤1−𝛼_𝑛𝛾. Next, we will assume that ‖𝐼−𝐴‖≤‖1−𝛾‖.
Step 1. We will show {𝑥_𝑛},{𝑢_𝑛} are bounded.
Since 𝐵,Φ are 𝛽,𝜎-inverse-strongly monotone mappings, we have (𝐼−𝜆𝐵)𝑥−(𝐼−𝜆𝐵)𝑦²=𝑥−𝑦−𝜆𝐵𝑥−𝐵𝑦²=𝑥−𝑦²−2𝜆𝑥−𝑦,𝐵𝑥−𝐵𝑦+𝜆²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²+𝜆𝜆−2𝛽𝐵𝑥−𝐵𝑦²≤𝑥−𝑦². (3.5) In similar way, we can obtain (𝐼−𝑟Φ)𝑥−(𝐼−𝑟Φ)𝑦²≤𝑥−𝑦². (3.6) It is clear that if 0<𝜆<2𝛽,0<𝑟<2𝜎, then 𝐼−𝜆𝐵,𝐼−𝑟Φ are all nonexpansive.
Put 𝑦_𝑛=𝐽_𝑀,𝜆(𝑥_𝑛−𝜆𝐵𝑥_𝑛),𝑛≥0. It follows that 𝑦_𝑛−𝑞=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝐽_𝑀,𝜆≤𝑥𝑞−𝜆𝐵𝑞_𝑛−𝑞. (3.7) By Lemma 2.7, we have 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛−𝑟Φ𝑥_𝑛) for all 𝑛≥0. Then, we have 𝑢_𝑛−𝑞²=𝑇_𝑟𝑥_𝑛−𝑟Φ𝑥_𝑛−𝑇_𝑟𝑞−𝑟Φ𝑞²≤𝑥_𝑛−𝑟Φ𝑥_𝑛−𝑞−𝑟Φ𝑞²≤𝑥_𝑛−𝑞²+𝑟(𝑟−2𝜎)Φ𝑥_𝑛−Φ𝑞²≤𝑥_𝑛−𝑞². (3.8) Put 𝑧_𝑛=𝑃_𝐶[𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐴)𝑆𝑦_𝑛] for all 𝑛≥0. From (3.2), we deduce that ‖𝑥_𝑛+1−𝑞‖=𝜉_𝑛𝑧_𝑛−𝑞+1−𝜉_𝑛𝑢_𝑛−𝑞≤𝜉_𝑛𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑃_𝐶𝑞+1−𝜉_𝑛𝑢_𝑛−𝑞≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞+1−𝜉_𝑛𝑢_𝑛−𝑞=𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞+1−𝜉_𝑛𝑢_𝑛−𝑞≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝜉_𝑛1−𝛼_𝑛𝛾𝑦_𝑛−𝑞+1−𝜉_𝑛𝑢_𝑛−𝑞≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓𝑞+𝜉_𝑛𝛼_𝑛𝛾𝑓𝑞−𝐴𝑞+𝜉_𝑛1−𝛼_𝑛𝛾𝑥_𝑛−𝑞+1−𝜉_𝑛𝑥_𝑛−𝑞≤𝜉_𝑛𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑞+𝜉_𝑛𝛼_𝑛𝛾𝑓𝑞−𝐴𝑞+𝜉_𝑛1−𝛼_𝑛𝛾𝑥_𝑛−𝑞+1−𝜉_𝑛𝑥_𝑛=−𝑞1−𝛾−𝛾𝛼𝜉_𝑛𝛼_𝑛𝑥_𝑛−𝑞+𝜉_𝑛𝛼_𝑛𝑞=𝛾𝑓−𝐴𝑞1−𝛾−𝛾𝛼𝜉_𝑛𝛼_𝑛𝑥_𝑛−𝑞+𝛾−𝛾𝛼𝜉_𝑛𝛼_𝑛𝛾𝑓𝑞−𝐴𝑞𝑥𝛾−𝛾𝛼≤max_𝑛,𝑞−𝑞𝛾𝑓−𝐴𝑞.𝛾−𝛾𝛼 (3.9) It follows from induction that 𝑥_𝑛−𝑞≤max𝑥₀,𝑞−𝑞𝛾𝑓−𝐴𝑞𝛾−𝛾𝛼,𝑛≥0. (3.10) Therefore {𝑥_𝑛} is bounded, so are {𝑦_𝑛}, {𝑆𝑦_𝑛}, {𝐵𝑥_𝑛}, {𝑓(𝑥_𝑛)}, and {𝐴𝑆𝑦_𝑛}.Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+2−𝑥_𝑛+1‖=0. From (3.2), we have 𝑥_𝑛+2−𝑥_𝑛+1=𝜉_𝑛+1𝑧_𝑛+1+1−𝜉_𝑛+1𝑢_𝑛+1−𝜉_𝑛𝑧_𝑛−1−𝜉_𝑛𝑢_𝑛=𝜉_𝑛+1𝑧_𝑛+1−𝑧_𝑛+𝜉_𝑛+1−𝜉_𝑛𝑧_𝑛+1−𝜉_𝑛+1𝑢_𝑛+1−𝑢_𝑛+𝜉_𝑛+1−𝜉_𝑛𝑢_𝑛≤𝜉_𝑛+1𝑧_𝑛+1−𝑧_𝑛+1−𝜉_𝑛+1𝑢_𝑛+1−𝑢_𝑛+𝜉_𝑛+1−𝜉_𝑛𝑧_𝑛+𝑢_𝑛. (3.11) We estimate ‖𝑢_𝑛+1−𝑢_𝑛‖, so we have 𝑢_𝑛+1−𝑢_𝑛=𝑇_𝑟𝑥_𝑛+1−𝑟Φ𝑥_𝑛+1−𝑇_𝑟𝑥_𝑛−𝑟Φ𝑥_𝑛≤𝑥_𝑛+1−𝑟Φ𝑥_𝑛+1−𝑥_𝑛−𝑟Φ𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (3.12) Substituting (3.12) into (3.11) that 𝑥_𝑛+2−𝑥_𝑛+1≤𝜉_𝑛+1𝑧_𝑛+1−𝑧_𝑛+1−𝜉_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝜉_𝑛+1−𝜉_𝑛𝑧_𝑛+𝑢_𝑛. (3.13) We note that 𝑧_𝑛+1−𝑧_𝑛=𝑃_𝐶𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐼−𝛼_𝑛+1𝐴𝑆𝑦_𝑛+1−𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛≤𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+𝐼−𝛼_𝑛+1𝐴𝑆𝑦_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛=𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛+1𝐴𝑆𝑦_𝑛+1−𝑆𝑦_𝑛+𝛼_𝑛−𝛼_𝑛+1𝐴𝑆𝑦_𝑛≤𝛼_𝑛+1𝛾𝛼𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+1−𝛼_𝑛+1𝛾𝑦_𝑛+1−𝑦_𝑛+𝛼_𝑛+1−𝛼_𝑛𝐴𝑆𝑦_𝑛=𝛼_𝑛+1𝛾𝛼𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐴𝑆𝑦_𝑛+1−𝛼_𝑛+1𝛾𝑦_𝑛+1−𝑦_𝑛. (3.14) Next, we estimate ‖𝑦_𝑛+1−𝑦_𝑛‖, then we get 𝑦_𝑛+1−𝑦_𝑛=𝐽_𝑀,𝜆𝑥_𝑛+1−𝜆𝐵𝑥_𝑛+1−𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛≤𝑥_𝑛+1−𝜆𝐵𝑥_𝑛+1−𝑥_𝑛−𝜆𝐵𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (3.15) Substituting (3.15) into (3.14), we obtain that 𝑧_𝑛+1−𝑧_𝑛≤𝛼_𝑛+1𝛾𝛼𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐴𝑆𝑦_𝑛+1−𝛼_𝑛+1𝛾𝑥_𝑛+1−𝑥_𝑛. (3.16) And substituting (3.12), (3.16) into (3.11), we get 𝑥_𝑛+2−𝑥_𝑛+1≤𝜉_𝑛+1𝛼_𝑛+1𝛾𝛼𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐴𝑆𝑦_𝑛+1−𝛼_𝑛+1𝛾𝑥_𝑛+1−𝑥_𝑛+1−𝜉_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝜉_𝑛+1−𝜉_𝑛𝑧_𝑛+𝑢_𝑛≤1−𝛾−𝛾𝛼𝜉_𝑛+1𝛼_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛+𝜉_𝑛+1−𝜉_𝑛𝑀, (3.17) where 𝑀>0 is a constant satisfying sup_𝑛𝛾𝑓𝑥_𝑛+𝐴𝑆𝑦_𝑛,𝑧_𝑛+𝑢_𝑛≤𝑀. (3.18) This together with (C1), (C2), and Lemma 2.5, implies that lim_𝑛→∞𝑥_𝑛+2−𝑥_𝑛+1=0. (3.19) From (3.15), we also have ‖𝑦_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞.Step 3. We show the following: (i)lim_𝑛→∞‖𝐵𝑥_𝑛−𝐵𝑞‖=0; (ii)lim_𝑛→∞‖Φ𝑥_𝑛−Φ𝑞‖=0. For 𝑞∈Ω and 𝑞=𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞), then we get 𝑦_𝑛−𝑞²=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝐽_𝑀,𝜆𝑞−𝜆𝐵𝑞²≤𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝑞−𝜆𝐵𝑞²≤𝑥_𝑛−𝑞²+𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞². (3.20) It follows that 𝑧_𝑛−𝑞²=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑃_𝐶𝑞²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+1−𝛼_𝑛𝛾𝑦_𝑛−𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²+𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+𝑥−𝑞_𝑛−𝑞²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞². (3.21) By the convexity of the norm ‖⋅‖, we have 𝑥_𝑛+1−𝑞²=𝜉_𝑛𝑧_𝑛+1−𝜉_𝑛𝑢_𝑛−𝑞²=𝜉_𝑛𝑧_𝑛−𝑞+1−𝜉_𝑛𝑢_𝑛−𝑞²≤𝜉_𝑛𝑧_𝑛−𝑞²+1−𝜉_𝑛𝑢_𝑛−𝑞². (3.22) Substituting (3.8), (3.21) into (3.22), we obtain 𝑥_𝑛+1−𝑞²≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞+𝑥_𝑛−𝑞²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+1−𝜉_𝑛𝑥_𝑛−𝑞²=𝜉_𝑛𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝜉_𝑛𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞+𝜉_𝑛𝑥_𝑛−𝑞²+𝜉_𝑛1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+1−𝜉_𝑛𝑥_𝑛−𝑞². (3.23) So, we obtain 𝜉_𝑛1−𝛼_𝑛𝛾𝜆2𝛽−𝜆𝐵𝑥_𝑛−𝐵𝑞²≤𝜉_𝑛𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+𝜖_𝑛+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑞+𝑥_𝑛+1−𝑞, (3.24) where 𝜖_𝑛=2𝜉_𝑛𝛼_𝑛(1−𝛼_𝑛𝛾)‖𝛾𝑓(𝑥_𝑛)𝐴𝑞‖‖𝑦_𝑛−𝑞‖. Since condition (C1), (C2) and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 then we obtain that ‖𝐵𝑥_𝑛−𝐵𝑞‖→0 as 𝑛→∞. We consider this inequality in (3.21) that 𝑧_𝑛−𝑞²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+1−𝛼_𝑛𝛾𝑦_𝑛−𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞. (3.25) Substituting (3.20) into (3.25), we have 𝑧_𝑛−𝑞²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²+𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞=𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝑥_𝑛−𝑞²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞. (3.26) Substituting (3.8) and (3.26) into (3.22), we obtain 𝑥_𝑛+1−𝑞²≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝑥_𝑛−𝑞²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝜉_𝑛𝑥_𝑛−𝑞²+𝑟(𝑟−2𝜎)Φ𝑥_𝑛−Φ𝑞²=𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝜉_𝑛𝑥_𝑛−𝑞²+𝜉_𝑛1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞²+2𝜉_𝑛𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞+1−𝜉_𝑛𝑥_𝑛−𝑞²+1−𝜉_𝑛𝑟(𝑟−2𝜎)Φ𝑥_𝑛−Φ𝑞². (3.27) So, we also have 1−𝜉_𝑛𝑟(2𝜎−𝑟)Φ𝑥_𝑛−Φ𝑞²≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝜖_𝑛+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑞+𝑥_𝑛+1−𝑞+𝜉_𝑛1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑥_𝑛−𝐵𝑞², (3.28) where 𝜖_𝑛=2𝜉_𝑛𝛼_𝑛(1−𝛼_𝑛𝛾)‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖‖𝑦_𝑛−𝑞‖. Since condition (C1), (C2), lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝐵𝑥_𝑛−𝐵𝑞‖=0 then we obtain that ‖Φ𝑥_𝑛−Φ𝑞‖→0 as 𝑛→∞.Step 4. We show the following:(i)lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0; (ii)lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0; (iii)lim_𝑛→∞‖𝑦_𝑛−𝑆𝑦_𝑛‖=0. Since 𝑇_𝑟 is firmly nonexpansive, we observe that 𝑢_𝑛−𝑞²=𝑇_𝑟𝑥_𝑛−𝑟Φ𝑥_𝑛−𝑇_𝑟𝑞−𝑟Φ𝑞²≤𝑥_𝑛−𝑟Φ𝑥_𝑛−𝑞−𝑟Φ𝑞,𝑢_𝑛=1−𝑞2𝑥_𝑛−𝑟Φ𝑥_𝑛−𝑞−𝑟Φ𝑞²+𝑢_𝑛−𝑞²−𝑥_𝑛−𝑟Φ𝑥_𝑛−𝑞−𝑟Φ𝑞−𝑢_𝑛−𝑞²≤12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛−𝑟Φ𝑥_𝑛−Φ𝑞²=12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟Φ𝑥_𝑛−Φ𝑞,𝑥_𝑛−𝑢_𝑛−𝑟²Φ𝑥_𝑛−Φ𝑞². (3.29) Hence, we have 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟Φ𝑥_𝑛−Φ𝑞𝑥_𝑛−𝑢_𝑛. (3.30) Since 𝐽_𝑀,𝜆 is 1-inverse-strongly monotone, we have 𝑦_𝑛−𝑞²=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝐽_𝑀,𝜆𝑞−𝜆𝐵𝑞²≤𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝑞−𝜆𝐵𝑞,𝑦_𝑛=1−𝑞2𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝑞−𝜆𝐵𝑞²+𝑦_𝑛−𝑞²−𝑥_𝑛−𝜆𝐵𝑥_𝑛−𝑞−𝜆𝐵𝑞−𝑦_𝑛−𝑞²≤12𝑥_𝑛−𝑞²+𝑦_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛−𝜆𝐵𝑥_𝑛−𝐵𝑞²=12𝑥_𝑛−𝑞²+𝑦_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛²+2𝜆𝑥_𝑛−𝑦_𝑛,𝐵𝑥_𝑛−𝐵𝑞−𝜆²𝐵𝑥_𝑛−𝐵𝑞², (3.31) which implies that 𝑦_𝑛−𝑞²≤𝑥_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛²+2𝜆𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞. (3.32) Substituting (3.32) into (3.25), we have 𝑧_𝑛−𝑞²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛²+2𝜆𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²−1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛²+2𝜆1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞. (3.33) Substituting (3.30) and (3.33) into (3.22), we obtain 𝑥_𝑛+1−𝑞²≤𝜉_𝑛𝑧_𝑛−𝑞²+1−𝜉_𝑛𝑢_𝑛−𝑞²≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²−1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛²+2𝜆1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝜉_𝑛𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟Φ𝑥_𝑛−Φ𝑞𝑥_𝑛−𝑢_𝑛≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛²+2𝜆1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝜉_𝑛𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟Φ𝑥_𝑛−Φ𝑞𝑥_𝑛−𝑢_𝑛≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝜉_𝑛𝑥_𝑛−𝑞²−𝑥_𝑛−𝑦_𝑛²+2𝜆𝜉_𝑛1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝜉_𝑛𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝜉_𝑛𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟1−𝜉_𝑛Φ𝑥_𝑛−Φ𝑞𝑥_𝑛−𝑢_𝑛. (3.34) Then, we derive 𝑥_𝑛−𝑢_𝑛²+𝑥_𝑛−𝑦_𝑛²=𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝑥_𝑛−𝑞²−𝑥_𝑛+1−𝑞²+2𝜉_𝑛𝜆1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝜉_𝑛𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞‖𝑦_𝑛−𝑞‖+2𝑟1−𝜉_𝑛Φ𝑥_𝑛−Φ𝑞𝑥_𝑛−𝑢_𝑛≤𝜉_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝑥_𝑛+1−𝑥_𝑛𝑥_𝑛−𝑞+𝑥_𝑛+1−𝑞+2𝜉_𝑛𝜆1−𝛼_𝑛𝛾𝑥_𝑛−𝑦_𝑛𝐵𝑥_𝑛−𝐵𝑞+2𝜉_𝑛𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞‖𝑦_𝑛−𝑞‖+2𝑟1−𝜉_𝑛Φ𝑥_𝑛−Φ𝑞𝑥_𝑛−𝑢_𝑛. (3.35) By condition (C1), (C2), lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0,lim_𝑛→∞‖Φ𝑥_𝑛−Φ𝑞‖=0, and lim_𝑛→∞‖𝐵𝑥_𝑛−𝐵𝑞‖=0. So, we have ‖𝑥_𝑛−𝑢_𝑛‖→0,‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. It follows that 𝑢_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑢_𝑛+𝑥_𝑛−𝑦_𝑛⟶0,as𝑛⟶∞. (3.36) We note that 𝑥_𝑛+1−𝑥_𝑛=𝜉_𝑛(𝑧_𝑛−𝑥_𝑛)+(1−𝜉_𝑛)(𝑢_𝑛−𝑥_𝑛). From lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0,lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, and hence lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (3.37) Since 𝑧_𝑛−𝑦_𝑛≤𝑧_𝑛−𝑥_𝑛+𝑥_𝑛−𝑦_𝑛. (3.38) So, by (3.37) and lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0, we obtain lim_𝑛→∞𝑧_𝑛−𝑦_𝑛=0. (3.39) Therefore, we observe that 𝑆𝑦_𝑛−𝑧_𝑛≤𝑆𝑦_𝑛−𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛≤𝑆𝑦_𝑛−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑆𝑦_𝑛. (3.40) By condition (C1), we have ‖𝑆𝑦_𝑛−𝑧_𝑛‖→0 as 𝑛→∞. Next, we observe that 𝑆𝑦_𝑛−𝑦_𝑛≤𝑆𝑦_𝑛−𝑧_𝑛+𝑧_𝑛−𝑦_𝑛. (3.41) By (3.39) and (3.40), we have ‖𝑆𝑦_𝑛−𝑦_𝑛‖→0 as 𝑛→∞.Step 5. We show that 𝑞∈Θ∶=𝐹(𝑆)∩GMEP(𝐹,𝜑,Φ)∩𝐼(𝐵,𝑀) and limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑆𝑦_𝑛−𝑞⟩≤0. It is easy to see that 𝑃_Θ(𝛾𝑓+(𝐼−𝐴)) is a contraction of 𝐻 into itself. Indeed, since 0<𝛾<𝛾/𝛼 we have 𝑃_Θ𝛾𝑓+(𝐼−𝐴)𝑥−𝑃_Θ()𝑦≤()()𝑦𝑓(𝑥)𝑦+|+𝛾𝑓+𝐼−𝐴𝛾𝑓+𝐼−𝐴𝑥−𝛾𝑓+𝐼−𝐴≤𝛾−𝑓𝐼−𝐴|𝑥−𝑦≤𝛾𝛼𝑥−𝑦1−𝛾=𝑥−𝑦1−𝛾+𝛾𝛼𝑥−𝑦. (3.42) Since 𝐻 is complete, there exists a unique fixed point 𝑞∈𝐻 such that 𝑞=𝑃_Θ(𝛾𝑓+(𝐼−𝐴))(𝑞). By Lemma 2.2, we obtain that ⟨(𝛾𝑓−𝐴)𝑞,𝑤−𝑞⟩≤0 for all 𝑤∈Θ.
Next, we show that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑆𝑦_𝑛−𝑞⟩≤0, where 𝑞=𝑃_Θ(𝛾𝑓+𝐼−𝐴)(𝑞) is the unique solution of the variational inequality ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≥0,forall𝑝∈Θ. We can choose a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that limsup_𝑛→∞𝛾𝑓−𝐴𝑞,S𝑦_𝑛−𝑞=lim_𝑖→∞𝛾𝑓−𝐴𝑞,𝑆𝑦_𝑛𝑖−𝑞. (3.43) Since {𝑦_𝑛𝑖} is bounded, there exists a subsequence {𝑦_𝑛𝑖𝑗} of {𝑦_𝑛𝑖} which converges weakly to 𝑤. We may assume without loss of generality that 𝑦_𝑛𝑖⇀𝑤. We claim that 𝑤∈Θ, since lim_𝑛→∞‖𝑦_𝑛−𝑆𝑦_𝑛‖=0 and by Lemma 2.6, we have 𝑤∈𝐹(𝑆).
Next, we show that 𝑤∈GMEP(𝐹,𝜑,Φ). Since 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛−𝑟Φ𝑥_𝑛), we know that 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+Φ𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,∀𝑦∈𝐶. (3.44) It follows by (A2) that 𝜑𝑦−𝜑𝑢_𝑛+Φ𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶. (3.45) Hence, 𝜑𝑦−𝜑𝑢_𝑛𝑖+Φ𝑥_𝑛𝑖,𝑦−𝑢_𝑛𝑖+1𝑟𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖≥𝐹𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (3.46) For 𝑡∈(0,1] and 𝑦∈𝐻, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. From (3.46) we have 𝑦_𝑡−𝑢_𝑛𝑖,Φ𝑦_𝑡≥𝑦_𝑡−𝑢_𝑛𝑖,Φ𝑦_𝑡−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−Φ𝑥_𝑛𝑖,𝑦_𝑡−𝑢_𝑛𝑖−1𝑟𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖+𝐹𝑦_𝑡,𝑢_𝑛𝑖=𝑦_𝑡−𝑢_𝑛𝑖,Φ𝑦_𝑡−Φ𝑢_𝑛𝑖+𝑦_𝑡−𝑢_𝑛𝑖,Φ𝑢_𝑛𝑖−Φ𝑥_𝑛𝑖−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−1𝑟𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖+𝐹𝑦_𝑡,𝑢_𝑛𝑖. (3.47) From ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0, we have ‖Φ𝑢_𝑛𝑖−Φ𝑥_𝑛𝑖‖→0. Further, from (A4) and the weakly lower semicontinuity of 𝜑,(𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟→0 and 𝑢_𝑛𝑖⇀𝑤, we have 𝑦_𝑡−𝑤,Φ𝑦_𝑡≥−𝜑𝑦_𝑡+𝜑(𝑤)+𝐹𝑦_𝑡,𝑤. (3.48) From (A1), (A4), and (3.48), we have 0=𝐹𝑦_𝑡,𝑦_𝑡−𝜑𝑦_𝑡+𝜑𝑦_𝑡≤𝑡𝐹𝑦_𝑡