A General Iterative Algorithm for Generalized Mixed Equilibrium Problems and Variational Inclusions Approach to Variational Inequalities

Abstract

We introduce a new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions 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), Su et al. (2008), Klin-eam and Suantai (2009), Tan and Chang (2011), and some other authors.

1. Introduction

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, Ψ∶𝐶→𝐻 a mapping, and 𝜑∶𝐶→ℛ a real-valued function. The generalized mixed equilibrium problem is for finding 𝑥∈𝐶 such that𝐹𝑥,𝑦+Ψ𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.1) The set of solutions of (1.1) is denoted by GMEP(𝐹,𝜑,Ψ), that is, GMEP𝐹,𝜑,Ψ=𝑥∈𝐶∶𝐹𝑥,𝑦+Ψ𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.2) If 𝐹≡0, the problem (1.1) is reduced into the mixed variational inequality of Browder type [1] for finding 𝑥∈𝐶 such thatΨ𝑥,𝑦−𝑥+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.3) The set of solutions of (1.3) is denoted by MVI(𝐶,𝜑,Ψ).

If Ψ≡0 and 𝜑≡0, the problem (1.1) is reduced into the equilibrium problem [2] for finding 𝑥∈𝐶 such that𝐹𝑥,𝑦≥0,∀𝑦∈𝐶. (1.4) The set of solutions of (1.4) is denoted by EP(𝐹). This problem contains fixed point problems and includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the equilibrium problem; see [3–5].

If 𝐹≡0 and 𝜑≡0, the problem (1.1) is reduced into the Hartmann-Stampacchia variational inequality [6] for finding 𝑥∈𝐶 such thatΨ𝑥,𝑦−𝑥≥0,∀𝑦∈𝐶. (1.5) The set of solutions of (1.5) is denoted by VI(𝐶,Ψ). The variational inequality has been extensively studied in the literature [7].

If 𝐹≡0 and Ψ≡0, the problem (1.1) is reduced into the minimize problem for finding 𝑥∈𝐶 such that𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.6) The set of solutions of (1.6) is denoted by Argmin(𝜑).

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻:𝜃(𝑥)=12⟨𝐴𝑥,𝑥⟩−𝑥,𝑦,∀𝑥∈𝐹(𝑆), (1.7) 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 [8].

Recall that a mapping 𝑆∶𝐶→𝐶 is said to be nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦, (1.8) 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 [9]. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. A mapping 𝐴 of 𝐶 into 𝐻 is called monotone if 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥0, (1.9) 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 that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦², (1.10) for all 𝑥,𝑦∈𝐶. It is obvious that any 𝛼-inverse-strongly monotone mapping 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, (1.11) for all 𝑥∈𝐻. A self mapping 𝑓∶𝐶→𝐶 is a contractions on 𝐶 if there exists a constant 𝛼∈(0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦, (1.12) for all 𝑥,𝑦∈𝐶. We use ∏_𝐶 to denote the collection of all contraction on C. Note that each 𝑓∈∏_𝐶 has a unique fixed point in 𝐶.

Let 𝐵∶𝐻→𝐻 be a single-valued nonlinear mapping and 𝑀∶𝐻→2^𝐻 a set-valued mapping. The variational inclusion problem is to find 𝑥∈𝐻 such that𝜃∈𝐵(𝑥)+𝑀(𝑥), (1.13) where 𝜃 is the zero vector in 𝐻. The set of solutions of problem (1.13) is denoted by 𝐼(𝐵,𝑀). The variational inclusion has been extensively studied in the literature, see, for example, [10–13] 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 define𝑇𝑣=𝐵𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (1.14) Then, 𝑇 is a maximal monotone and 𝜃∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐵) [14].

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 (1.15) 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 (1.13) is a fixed point of the operator 𝐽_𝑀,𝜆(𝐼−𝜆𝐵) for all 𝜆>0 [15].

In 2000, Moudafi [16] 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, is chosen arbitrarily,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑥_𝑛,𝑛≥0, (1.16) where {𝛼_𝑛}⊂(0,1) satisfies certain conditions and 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). (1.17)

In 2006, Marino and Xu [8] 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, (1.18) where {𝛼_𝑛}⊂(0,1) and 𝐴 is a strongly positive linear bounded operator. They proved that, if 𝐶=𝐻 and the sequence {𝛼_𝑛} satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.18) 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), (1.19) which is the optimality condition for the minimization problemmin_{𝑥∈𝐹(𝑆)∩EP(𝐹)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.20) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities, let 𝑃_𝐶 be the projection of 𝐻 onto 𝐶. In 2005, Iiduka and Takahashi [17] introduced following iterative process for 𝑥₀∈𝐶:𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥0, (1.21) where 𝑢∈𝐶, {𝛼_𝑛}⊂(0,1), and {𝜆_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2𝛽. They proved that under certain appropriate conditions imposed on {𝛼_𝑛} and {𝜆_𝑛}, the sequence {𝑥_𝑛} generated by (1.21) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say 𝑥∈𝐶) which solve some variational inequality⟨𝑥−𝑢,𝑥−𝑥⟩≥0,∀𝑥∈𝐹(𝑆)∩VI(𝐶,𝐴). (1.22)

In 2008, Su et al. [18] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: 𝑥₁,𝑢_𝑛∈𝐻𝐹𝑢_𝑛+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴𝑢_𝑛, (1.23) for all 𝑛∈ℕ, where {𝛼_𝑛}⊂[0,1) and {𝑟_𝑛}⊂(0,∞) satisfy some appropriate conditions. Furthermore, they proved that {𝑥_𝑛} and {𝑢_𝑛} converge strongly to the same point 𝑧∈𝐹(𝑆)∩VI(𝐶,𝐴)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩VI(𝐶,𝐴)∩EP(𝐹)}𝑓(𝑧).

In 2011, Tan and Chang [12] introduced following iterative process for {𝑇_𝑛∶𝐶→𝐶} which is 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, (1.24) where {𝛼_𝑛}⊂(0,1), 𝜆∈(0,2𝛼], and 𝜇∈(0,2𝛽]. The sequence {𝑥_𝑛} defined by (1.24) converges strongly to a common element of the set of fixed points of nonexpansive mapping, the set of solutions of the variational inequality, and the generalized equilibrium problem.

In this paper, we modify the iterative methods (1.18), (1.23), and (1.24) by proposing the following new general viscosity iterative method: 𝑥₀,𝑢_𝑛∈𝐶,𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+Ψ𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛, (1.25) for all 𝑛∈ℕ, where {𝛼_𝑛}⊂(0,1), {𝑟_𝑛}⊂(0,2𝜎), and 𝜆∈(0,2𝛽) satisfy some appropriate conditions. The purpose of this paper is to show that under some control conditions the sequence {𝑥_𝑛} strongly converges to a common element of the set of fixed points of nonexpansive mapping, the solution of the generalized mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space.

2. Preliminaries

Let 𝐻 be a real Hilbert space and 𝐶 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 (1.5) if and only if 𝑢∈𝐶 satisfies the relation 𝑢=𝑃_𝐶(𝑢−𝜆Ψ𝑢) for all 𝜆>0.

Lemma 2.2. For a given 𝑧∈𝐻, 𝑢∈𝐶, 𝑢=𝑃_𝐶𝑧⇔⟨𝑢−𝑧,𝑣−𝑢⟩≥0, for all 𝑣∈𝐶.
It is well known that 𝑃_𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶 and satisfies ‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖²≤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 [19]). Let 𝑀∶𝐻→2^𝐻 be a maximal monotone mapping, and let 𝐵∶𝐻→𝐻 be a monotone and Lipshitz continuous mapping. Then the mapping 𝐿=M+𝐵∶𝐻→2^𝐻 is a maximal monotone mapping.

Lemma 2.4 (see [20]). 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‖ holds for each 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 2.5 (see [21]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_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/𝛾_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞.Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.6 (see [22]). 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 Ψ∶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 [23]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐹∶𝐶×𝐶→ℛ be a bifunction mapping satisfying (A1)–(A4), and let 𝜑∶𝐶→ℛ be convex and lower semicontinuous such that 𝐶∩dom𝜑≠∅. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥∈𝐻, there exists 𝑢∈𝐶 such that 𝐹𝑦(𝑢)+1𝑢,𝑦+𝜑−𝜑𝑟𝑦−𝑢,𝑢−𝑥. (2.7) Define a mapping 𝐾_𝑟∶𝐻→𝐶 as follows: 𝐾_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)𝐾_𝑟 is single valued; (ii)𝐾_𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦∈𝐻,‖𝐾_𝑟𝑥−𝐾_𝑟𝑦‖²≤⟨𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝑥−𝑦⟩;(iii)𝐹(𝐾_𝑟)=MEP(𝐹,𝜑); (iv)MEP(𝐹,𝜑) is closed and convex.

Lemma 2.8 (see [8]). Assume that 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,Ψ), and 𝐼(𝐵,𝑀) of an inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. Let 𝐻 be a real Hilbert space, 𝐶 a closed convex subset of 𝐻, 𝐵, Ψ∶𝐶→𝐻 be 𝛽, 𝜎-inverse-strongly monotone mappings, respectively. Let 𝜑∶𝐶→ℛ be a convex and lower semicontinuous function, 𝑓∶𝐶→𝐶 a contraction with coefficient 𝛼(0<𝛼<1), 𝑀∶𝐻→2^𝐻 a maximal monotone mapping, and 𝐴 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 𝐶 into itself such that Ω∶=𝐹(𝑆)∩GMEP𝐹,𝜑,Ψ∩𝐼(𝐵,𝑀)≠∅. (3.1) Suppose that {𝑥_𝑛} is a sequences generated by the following algorithm for 𝑥₀∈𝐶 arbitrarily: 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+Ψ𝑥_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+I−𝛼_𝑛𝐴𝑆𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛 (3.2) for all 𝑛=0,1,2,…, where(C1){𝛼_𝑛}⊂(0,1), lim_𝑛→0𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞,(C2){𝑟_𝑛}⊂[𝑐,𝑑] with 𝑐,𝑑∈(0,2𝜎) and ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞,(C3)𝜆∈(0,2𝛽).
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω, where 𝑞=𝑃_Ω(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: 𝛾𝑓−𝐴𝑞,𝑝−𝑞≤0,∀𝑝∈Ω (3.3) which is the optimality condition for the minimization problem min_𝑞∈Ω12⟨𝐴𝑞,𝑞⟩−ℎ𝑞, (3.4) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. Due to condition (C1), we may assume without loss of generality, then, that 𝛼_𝑛∈(0,‖𝐴‖⁻¹) for all 𝑛. By Lemma 2.8, we have that ‖𝐼−𝛼_𝑛𝐴‖≤1−𝛼_𝑛𝛾. Next, we will assume that ‖𝐼−𝐴‖≤‖1−𝛾‖.
Next, we will divide the proof into six steps.
Step 1. We will show that {𝑥_𝑛},{𝑢_𝑛} are bounded.
Since 𝐵, Ψ are 𝛽, 𝜎-inverse-strongly monotone mappings, we have that (𝐼−𝜆𝐵)𝑥−(𝐼−𝜆𝐵)𝑦²=(𝑥−𝑦)−𝜆(𝐵𝑥−𝐵𝑦)²=𝑥−𝑦²−2𝜆⟨𝑥−𝑦,𝐵𝑥−𝐵𝑦⟩+𝜆²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²+𝜆𝜆−2𝛽𝐵𝑥−𝐵𝑦²≤𝑥−𝑦². (3.5) In a similar way, we can obtain (𝐼−𝑟_𝑛Ψ)𝑥−(𝐼−𝑟_𝑛Ψ)𝑦²≤𝑥−𝑦². (3.6) It is clear that if 0<𝜆<2𝛽, 0<𝑟_𝑛<2𝜎, then 𝐼−𝜆𝐵, 𝐼−𝑟_𝑛Ψ are all nonexpansive.
Put 𝑦_𝑛=𝐽_𝑀,𝜆(𝑢_𝑛−𝜆𝐵𝑢_𝑛), 𝑛≥0. It follows that 𝑦_𝑛−𝑞=𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝐽_𝑀,𝜆≤𝑢𝑞−𝜆𝐵𝑞_𝑛−𝑞. (3.7) By Lemma 2.7, we have that 𝑢_𝑛=𝐾_𝑟𝑛(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛) for all 𝑛≥0. Then, we have that 𝑢_𝑛−𝑞²=𝐾_𝑟𝑛(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛)−𝐾_𝑟𝑛(𝑞−𝑟_𝑛Ψ𝑞)²≤(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛)−(𝑞−𝑟_𝑛Ψ𝑞)²≤𝑥_𝑛−𝑞²+𝑟_𝑛𝑟_𝑛−2𝜎Ψ𝑥_𝑛−Ψ𝑞²=𝑥_𝑛−𝑞². (3.8) Hence, we have that 𝑦_𝑛−𝑞≤𝑥_𝑛−𝑞. (3.9) From (3.2), we deduce that 𝑥_𝑛+1−𝑞=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑃_𝐶𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+1−𝛼_𝑛𝛾𝑦_𝑛−𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓𝑞+𝛼_𝑛𝛾𝑓𝑞−𝐴𝑞+1−𝛼_𝑛𝛾𝑦_𝑛−𝑞≤𝛼𝛾𝛼_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝛾𝑓𝑞−𝐴𝑞+1−𝛼_𝑛𝛾𝑥_𝑛=−𝑞1−𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝑞=𝛾𝑓−𝐴𝑞1−𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞+𝛾−𝛾𝛼𝛼_𝑛𝛾𝑓𝑞−𝐴𝑞𝑥𝛾−𝛾𝛼≤max_𝑛,𝑞−𝑞𝛾𝑓−𝐴𝑞.𝛾−𝛾𝛼 (3.10) It follows by induction that 𝑥_𝑛−𝑞≤max𝑥₀,𝑞−𝑞𝛾𝑓−𝐴𝑞𝛾−𝛾𝛼,𝑛≥0. (3.11) Therefore {𝑥_𝑛} is bounded, so are {𝑦_𝑛}, {𝑆𝑦_𝑛}, {𝐵𝑢_𝑛}, {𝑓(𝑥_𝑛)}, and {𝐴𝑆𝑦_𝑛}.
Step 2. We claim that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. From (3.2), we have that 𝑥_𝑛+1−𝑥_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑃_𝐶𝛼_𝑛−1𝛾𝑓𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1≤𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑆𝑦_𝑛−1−𝛼_𝑛−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1+𝛾𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1≤1−𝛼_𝑛𝛾𝑦_𝑛−𝑦_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1+𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1. (3.12)
Since 𝐼−𝜆𝐵 are nonexpansive, we also have that ‖𝑦_𝑛−𝑦_𝑛−1‖=𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝐽_𝑀,𝜆𝑢_𝑛−1−𝜆𝐵𝑢_𝑛−1≤𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝑢_𝑛−1−𝜆𝐵𝑢_𝑛−1≤𝑢_𝑛−𝑢_𝑛−1. (3.13) On the other hand, from 𝑢_𝑛−1=𝐾_𝑟𝑛−1(𝑥_𝑛−1−𝑟_𝑛−1Ψ𝑥_𝑛−1) and 𝑢_𝑛=𝐾_𝑟𝑛(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛), it follows that 𝐹𝑢_𝑛−1,𝑦+Ψ𝑥_𝑛−1,𝑦−𝑢_𝑛−1+𝜑𝑦−𝜑𝑢_𝑛−1+1𝑟_𝑛−1𝑦−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1𝐹𝑢≥0,∀𝑦∈𝐶,_𝑛,𝑦+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛⟩+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶. (3.14)(3.15) Substituting 𝑦=𝑢_𝑛 into (3.14) and y=𝑢_𝑛−1 into (3.15), we get 𝐹𝑢_𝑛−1,𝑢_𝑛+⟨Ψ𝑥_𝑛−1,𝑢_𝑛−𝑢_𝑛−1⟩+𝜑𝑢_𝑛−𝜑𝑢_𝑛−1+1𝑟_𝑛−1⟨𝑢_𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑥_𝑛−1𝐹𝑢⟩≥0,_𝑛,𝑢_𝑛−1+Ψ𝑥_𝑛,𝑢_𝑛−1−𝑢_𝑛+𝜑𝑢_𝑛−1−𝜑𝑢_𝑛+1𝑟_𝑛𝑢_𝑛−1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0. (3.16) From (A2), we obtain 𝑢_𝑛−𝑢_𝑛−1,Ψ𝑥_𝑛−1−Ψ𝑥_𝑛+𝑢_𝑛−1−𝑥_𝑛−1𝑟_𝑛−1−𝑢_𝑛−𝑥_𝑛𝑟_𝑛≥0, (3.17) and then 𝑢_𝑛−𝑢_𝑛−1,𝑟_𝑛−1Ψ𝑥_𝑛−1−Ψ𝑥_𝑛+𝑢_𝑛−1−𝑥_𝑛−1−𝑟_𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛≥0. (3.18) So ⟨𝑢_𝑛−𝑢_𝑛−1,𝑟_𝑛−1Ψ𝑥_𝑛−1−𝑟_𝑛−1Ψ𝑥_𝑛+𝑢_𝑛−1−𝑢_𝑛+𝑢_𝑛−𝑥_𝑛−1−𝑟_𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛⟩≥0. (3.19) It follows that 𝑢_𝑛−𝑢_𝑛−1,𝐼−𝑟_𝑛−1Ψ𝑥_𝑛−𝐼−𝑟_𝑛−1Ψ𝑥_𝑛−1+𝑢_𝑛−1−𝑢_𝑛+𝑢_𝑛−𝑥_𝑛−𝑟_𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛≥0,⟨𝑢_𝑛−𝑢_𝑛−1,𝑢_𝑛−1−𝑢_𝑛⟩+⟨𝑢_𝑛−𝑢_𝑛−1,𝑥_𝑛−𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛⟩≥0. (3.20) Without loss of generality, let us assume that there exists a real number 𝑐 such that 𝑟_𝑛−1>𝑐>0, for all 𝑛∈ℕ. Then, we have that 𝑢_𝑛−𝑢_𝑛−1²≤𝑢_𝑛−𝑢_𝑛−1,𝑥_𝑛−𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛≤𝑢_𝑛−𝑢_𝑛−1𝑥_𝑛−𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝑢_𝑛−𝑥_𝑛, (3.21) and hence 𝑢_𝑛−𝑢_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1𝑟_𝑛𝑟_𝑛−𝑟_𝑛−1𝑢_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝑐𝑟_𝑛−𝑟_𝑛−1, (3.22) where 𝑀₁=sup{‖𝑢_𝑛−𝑥_𝑛‖∶𝑛∈ℕ}. Substituting (3.22) into (3.13), we have that 𝑦_𝑛−𝑦_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝑐𝑟_𝑛−𝑟_𝑛−1. (3.23) Substituting (3.23) into (3.12), we get 𝑥_𝑛+1−𝑥_𝑛≤1−𝛼_𝑛𝛾𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝑐𝑟_𝑛−𝑟_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1+𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1=1−𝛼_𝑛𝛾𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝛾𝑀₁𝑐𝑟_𝑛−𝑟_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1+𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1≤1−𝛾−𝛾𝛼𝛼_𝑛‖𝑥_𝑛−𝑥_𝑛−1𝑀‖+¹𝑐𝑟_𝑛−𝑟_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1+𝛾𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1≤1−𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝑐𝑟_𝑛−𝑟_𝑛−1+𝑀₂𝛼_𝑛−𝛼_𝑛−1, (3.24) where 𝑀₂=sup{max{‖𝐴𝑆𝑦_𝑛−1‖,‖𝑓(𝑥_𝑛−1)‖∶𝑛∈ℕ}}. By conditions (C1)-(C2) and Lemma 2.5, we have that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. From (3.23), we also have that ‖𝑦_𝑛+1−𝑦_𝑛‖→0 as 𝑛→∞.
Step 3. We show the following: (i)lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑞‖=0; (ii)lim_𝑛→∞‖Ψ𝑥_𝑛−Ψ𝑞‖=0. For 𝑞∈Ω and 𝑞=𝐽_𝑀,𝜆(q−𝜆𝐵𝑞), by (3.5) and (3.8), we get 𝑦_𝑛−𝑞²=𝐽_𝑀,𝜆(𝑢_𝑛−𝜆𝐵𝑢_𝑛)−𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞)²≤(𝑢_𝑛−𝜆𝐵𝑢_𝑛)−(𝑞−𝜆𝐵𝑞)²≤𝑢_𝑛−𝑞²+𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞²≤𝑥_𝑛−𝑞²+𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞². (3.25) It follows that 𝑥_𝑛+1−𝑞²=𝑃_𝐶(𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐴)𝑆𝑦_𝑛)−𝑃_𝐶(𝑞)²≤𝛼_𝑛(𝛾𝑓(𝑥_𝑛)−𝐴𝑞)+(𝐼−𝛼_𝑛𝐴)(𝑆𝑦_𝑛−𝑞)²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞+1−𝛼_𝑛𝛾𝑦_𝑛−𝑞²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+1−𝛼_𝑛𝛾𝑦_𝑛−𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝛼_𝑛𝛾‖𝑥_𝑛−𝑞‖²+𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞+‖𝑥_𝑛−𝑞‖²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽‖𝐵𝑢_𝑛−𝐵𝑞‖². (3.26) So, we obtain 1−𝛼_𝑛𝛾𝜆2𝛽−𝜆𝐵𝑢_𝑛−𝐵𝑞²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑞²+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑞+𝑥_𝑛+1−𝑞+𝜖_𝑛, (3.27) where 𝜖_𝑛=2𝛼_𝑛(1−𝛼_𝑛𝛾)‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖‖𝑦_𝑛−𝑞‖. By conditions (C1) and (C3) and lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, we obtain that ‖𝐵𝑢_𝑛−𝐵𝑞‖→0 as 𝑛→∞.
Substituting (3.8) into (3.25), we get 𝑦_𝑛−𝑞²≤𝑢_𝑛−𝑞²+𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞²≤𝑥_𝑛−𝑞²+𝑟_𝑛𝑟_𝑛−2𝜎Ψ𝑥_𝑛−Ψ𝑞²+𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞². (3.28) From (3.26), we have that 𝑥_𝑛+1−𝑞²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛+−𝑞1−𝛼_𝑛𝛾𝑥_𝑛−𝑞²+𝑟_𝑛𝑟_𝑛−2𝜎Ψ𝑥_𝑛−Ψ𝑞²+𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞+𝑥_𝑛−𝑞²+1−𝛼_𝑛𝛾𝑟_𝑛𝑟_𝑛−2𝜎Ψ𝑥_𝑛−Ψ𝑞²+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞². (3.29) So, we also have that 1−𝛼_𝑛𝛾𝑟_𝑛2𝜎−𝑟_𝑛‖Ψ𝑥_𝑛−Ψ𝑞‖²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑞+𝑥_𝑛+1−𝑞+𝜖_𝑛+1−𝛼_𝑛𝛾𝜆𝜆−2𝛽𝐵𝑢_𝑛−𝐵𝑞², (3.30) where 𝜖_𝑛=2𝛼_𝑛(1−𝛼_𝑛𝛾)‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖‖𝑦_𝑛−𝑞‖. By conditions (C1)–(C3), lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑞‖=0, we obtain that ‖Ψ𝑥_𝑛−Ψ𝑞‖→0 as 𝑛→∞.
Step 4. We show the following: (i)lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0; (ii)lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0; (iii)lim_𝑛→∞‖𝑦_𝑛−𝑆𝑦_𝑛‖=0. Since 𝐾_𝑟𝑛 is firmly nonexpansive and by (2.2), we observe that 𝑢_𝑛−𝑞²=𝐾_𝑟𝑛(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛)−𝐾_𝑟𝑛(𝑞−𝑟_𝑛Ψ𝑞)²≤𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛−𝑞−𝑟_𝑛Ψ𝑞,𝑢_𝑛=1−𝑞2(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛)−(𝑞−𝑟_𝑛Ψ𝑞)²+𝑢_𝑛−𝑞²−(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛)−(𝑞−𝑟_𝑛Ψ𝑞)−(𝑢_𝑛−𝑞)²≤12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑞²−(𝑥_𝑛−𝑢_𝑛)−𝑟_𝑛(Ψ𝑥_𝑛−Ψ𝑞)²=12𝑥_𝑛−𝑞²+𝑢_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛Ψ𝑥_𝑛−Ψ𝑞,𝑥_𝑛−𝑢_𝑛−𝑟²_𝑛Ψ𝑥_𝑛−Ψ𝑞². (3.31) Hence, we have that 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛Ψ𝑥_𝑛−Ψ𝑞𝑥_𝑛−𝑢_𝑛. (3.32) Since 𝐽_𝑀,𝜆 is 1-inverse-strongly monotone and by (2.2), we compute 𝑦_𝑛−𝑞²=𝐽_𝑀,𝜆(𝑢_𝑛−𝜆𝐵𝑢_𝑛)−𝐽_𝑀,𝜆(𝑞−𝜆𝐵𝑞)²≤𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝑞−𝜆𝐵𝑞,𝑦_𝑛=1−𝑞2𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝑞−𝜆𝐵𝑞²+𝑦_𝑛−𝑞²−𝑢_𝑛−𝜆𝐵𝑢_𝑛−𝑞−𝜆𝐵𝑞−𝑦_𝑛−𝑞²≤12𝑢_𝑛−𝑞²+𝑦_𝑛−𝑞²−𝑢_𝑛−𝑦_𝑛−𝜆𝐵𝑢_𝑛−𝐵𝑞²=12𝑢_𝑛−𝑞²+𝑦_𝑛−𝑞²−𝑢_𝑛−𝑦_𝑛²+2𝜆𝑢_𝑛−𝑦_𝑛,𝐵𝑢_𝑛−𝐵𝑞−𝜆²𝐵𝑢_𝑛−𝐵𝑞², (3.33) which implies that 𝑦_𝑛−𝑞²≤𝑢_𝑛−𝑞²−𝑢_𝑛−𝑦_𝑛²+2𝜆𝑢_𝑛−𝑦_𝑛𝐵𝑢_𝑛−𝐵𝑞. (3.34) Substituting (3.32) into (3.34), we have that 𝑦_𝑛−𝑞²≤𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛Ψ𝑥_𝑛−Ψ𝑞𝑥_𝑛−𝑢_𝑛−𝑢_𝑛−𝑦_𝑛²+2𝜆𝑢_𝑛−𝑦_𝑛𝐵𝑢_𝑛−𝐵𝑞. (3.35) Substituting (3.35) into (3.26), we get 𝑥_𝑛+1−𝑞²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+𝑦_𝑛−𝑞²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+𝑥_𝑛−𝑞²−𝑥_𝑛−𝑢_𝑛²+2𝑟_𝑛Ψ𝑥_𝑛−Ψ𝑞𝑥_𝑛−𝑢_𝑛−𝑢_𝑛−𝑦_𝑛²+2𝜆_𝑛𝑢_𝑛−𝑦_𝑛𝐵𝑢_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞. (3.36) Then, we derive 𝑥_𝑛−𝑢_𝑛²+𝑢_𝑛−𝑦_𝑛²≤𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+𝑥_𝑛−𝑞²−𝑥_𝑛+1−𝑞²+2𝑟_𝑛Ψ𝑥_𝑛−Ψ𝑞𝑥_𝑛−𝑢_𝑛+2𝜆𝑢_𝑛−𝑦_𝑛𝐵𝑢_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑓𝑥_𝑛−𝐴𝑞𝑦_𝑛−𝑞=𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+𝑥_𝑛−𝑥_𝑛+1𝑥_𝑛−𝑞+𝑥_𝑛+1−𝑞+2𝑟_𝑛Ψ𝑥_𝑛−Ψ𝑞𝑥_𝑛−𝑢_𝑛+2𝜆𝑢_𝑛−𝑦_𝑛𝐵𝑢_𝑛−𝐵𝑞+2𝛼_𝑛1−𝛼_𝑛𝛾‖𝛾𝑓𝑥_𝑛−𝐴𝑞‖‖𝑦_𝑛−𝑞‖. (3.37) By condition (C1), lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0, lim_𝑛→∞‖Ψ𝑥_𝑛−Ψ𝑞‖=0, and lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑞‖=0. So, we have that ‖𝑥_𝑛−𝑢_𝑛‖→0, ‖𝑢_𝑛−𝑦_𝑛‖→0 as 𝑛→∞. It follows that 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑦_𝑛⟶0,as𝑛⟶∞. (3.38) From (3.2), we have that 𝑥_𝑛−𝑆𝑦_𝑛≤𝑥_𝑛−𝑆𝑦_𝑛−1+𝑆𝑦_𝑛−1−𝑆𝑦_𝑛≤𝑃_𝐶𝛼_𝑛−1𝛾𝑓𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝐴𝑆𝑦_𝑛−1−𝑃_𝐶𝑆𝑦_𝑛−1+𝑦_𝑛−1−𝑦_𝑛≤𝛼_𝑛−1𝛾𝑓𝑥_𝑛−1−𝐴𝑆𝑦_𝑛−1+𝑦_𝑛−1−𝑦_𝑛. (3.39) By condition (C1) and lim_𝑛→∞‖𝑦_𝑛−1−𝑦_𝑛‖=0, we obtain that ‖𝑥_𝑛−𝑆𝑦_𝑛‖→0 as 𝑛→∞. Next, we observe that 𝑥_𝑛+1−𝑆𝑦_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑃_𝐶𝑆𝑦_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑆𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑆𝑦_𝑛. (3.40) Since {𝐴𝑆𝑦_𝑛} is bounded and by condition (C1), we have that ‖𝑥_𝑛+1−𝑆𝑦_𝑛‖→0 as 𝑛→∞, and 𝑥_𝑛−𝑆𝑦_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆𝑦_𝑛. (3.41) Since lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞‖𝑥_𝑛+1−𝑆𝑦_𝑛‖=0, it implies that ‖𝑥_𝑛−𝑆𝑦_𝑛‖→0 as 𝑛→∞. Hence, we have that 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆𝑦_𝑛+𝑆𝑦_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑆𝑦_𝑛+𝑦_𝑛−𝑥_𝑛. (3.42) By (3.38) and lim_𝑛→∞‖𝑥_𝑛−𝑆𝑦_𝑛‖=0, we obtain ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0 as 𝑛→∞. Moreover, we also have that 𝑦_𝑛−𝑆𝑦_𝑛≤𝑦_𝑛−𝑥_𝑛+𝑥_𝑛−𝑆𝑦_𝑛. (3.43) By (3.38) and lim_𝑛→∞‖𝑥_𝑛−𝑆𝑦_𝑛‖=0, we obtain ‖𝑦_𝑛−𝑆𝑦_𝑛‖→0 as 𝑛→∞.
Step 5. We show that 𝑞∈Ω∶=𝐹(𝑆)∩GMEP(𝐹,𝜑,Ψ)∩𝐼(𝐵,𝑀) and limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑆𝑦_𝑛−𝑞⟩≤0. It is easy to see that 𝑃_Ω(𝛾𝑓+(𝐼−𝐴)) is a contraction of 𝐻 into itself. Indeed, since 0<𝛾<𝛾/𝛼, we have that 𝑃_Ω𝛾𝑓+(𝐼−𝐴)𝑥−𝑃_Ω()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. (3.44) Hence 𝐻 is complete, and there exists a unique fixed point 𝑞∈𝐻 such that 𝑞=𝑃_Ω(𝛾𝑓+(𝐼−𝐴))(𝑞). By Lemma 2.2, we obtain that ⟨(𝛾𝑓−𝐴)𝑞,𝑤−𝑞⟩≤0 for all 𝑤∈Ω.
Next, we show that limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑆𝑦_𝑛−𝑞⟩≤0, where 𝑞=𝑃_Ω(𝛾𝑓+𝐼−𝐴)(𝑞) is the unique solution of the variational inequality ⟨(𝛾𝑓−𝐴)𝑞,𝑝−𝑞⟩≥0, for all 𝑝∈Ω. 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. (3.45) As {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∈Ω. Since ‖𝑦_𝑛−𝑆𝑦_𝑛‖→0, ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0, and ‖𝑥_𝑛−𝑦_𝑛‖→0 and by Lemma 2.6, we have that 𝑤∈𝐹(𝑆).
Next, we show that 𝑤∈GMEP(𝐹,𝜑,Ψ). Since 𝑢_𝑛=𝐾_𝑟𝑛(𝑥_𝑛−𝑟_𝑛Ψ𝑥_𝑛), we know that 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶. (3.46) It follows by (A2) that 𝜑𝑦−𝜑𝑢_𝑛+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶. (3.47) Hence, 𝜑𝑦−𝜑𝑢_𝑛𝑖+Ψ𝑥_𝑛𝑖,𝑦−𝑢_𝑛𝑖+1𝑟_𝑛𝑖𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖≥𝐹𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (3.48) For 𝑡∈(0,1] and 𝑦∈𝐻, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. From (3.48), we have that ⟨𝑦_𝑡−𝑢_𝑛𝑖,Ψ𝑦_𝑡⟩≥⟨𝑦_𝑡−𝑢_𝑛𝑖,Ψ𝑦_𝑡⟩−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−⟨Ψ𝑥_𝑛𝑖,𝑦_𝑡−𝑢_𝑛𝑖⟩−1𝑟_𝑛𝑖⟨𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖⟩+𝐹𝑦_𝑡,𝑢_𝑛𝑖=𝑦_𝑡−𝑢_𝑛𝑖,Ψ𝑦_𝑡−Ψ𝑢_𝑛𝑖+𝑦_𝑡−𝑢_𝑛𝑖,Ψ𝑢_𝑛𝑖−Ψ𝑥_𝑛𝑖−𝜑𝑦_𝑡+𝜑𝑢_𝑛𝑖−1r_𝑛𝑖𝑦_𝑡−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖+𝐹𝑦_𝑡,𝑢_𝑛𝑖. (3.49) From ‖𝑢_𝑛𝑖−𝑥_𝑛𝑖‖→0, we have that ‖Ψ𝑢_𝑛𝑖−Ψ𝑥_𝑛𝑖‖→0. Further, from (A4) and the weakly lower semicontinuity of 𝜑,(𝑢_𝑛𝑖−𝑥_𝑛𝑖)/𝑟_𝑛𝑖→0 and 𝑢_𝑛𝑖⇀𝑤, we have that 𝑦_𝑡−𝑤,Ψ𝑦_𝑡≥−𝜑𝑦_𝑡+𝜑(𝑤)+𝐹𝑦_𝑡,𝑤. (3.50) From (A1), (A4), and (3.50), we have that 0=𝐹𝑦_𝑡,𝑦_𝑡−𝜑𝑦_𝑡+𝜑𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+(1−𝑡)𝐹𝑦_𝑡,𝑤+𝑡𝜑𝑦+(1−𝑡)𝜑(𝑤)−𝜑𝑦_𝑡=𝑡𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝐹𝑦_𝑡,𝑤+𝜑(𝑤)−𝜑𝑦_𝑡≤𝑡𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)⟨𝑦_𝑡−𝑤,Ψ𝑦_𝑡⟩=𝑡𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑡⟨𝑦−𝑤,Ψ𝑦_𝑡⟩, (3.51) and hence 0≤𝐹𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)𝑦−𝑤,Ψ𝑦_𝑡. (3.52) Letting 𝑡→0, we have, for each 𝑦∈𝐶, that 𝐹𝑤,𝑦+𝜑𝑦−𝜑(𝑤)+⟨𝑦−𝑤,Ψ𝑤⟩≥0. (3.53) This implies that 𝑤∈GMEP(𝐹,𝜑,Ψ).
Lastly, we show that 𝑤∈𝐼(𝐵,𝑀). In fact, since 𝐵 is a 𝛽-inverse-strongly monotone, 𝐵 is monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that 𝑀+𝐵 is a maximal monotone. Let (𝑣,𝑔)∈𝐺(𝑀+𝐵), since 𝑔−𝐵𝑣∈𝑀(𝑣). Again since 𝑦_𝑛𝑖=𝐽_𝑀,𝜆(𝑢_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖), we have that 𝑢_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖∈(𝐼+𝜆𝑀)(𝑦_𝑛𝑖), that is, (1/𝜆)(𝑢_𝑛𝑖−𝑦_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖)∈𝑀(𝑦_𝑛𝑖). By virtue of the maximal monotonicity of 𝑀+𝐵, we have that 𝑣−𝑦_𝑛𝑖1,𝑔−𝐵𝑣−𝜆𝑢_𝑛𝑖−𝑦_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖≥0, (3.54) and hence 𝑣−𝑦_𝑛𝑖,𝑔≥𝑣−𝑦_𝑛𝑖1,𝐵𝑣+𝜆𝑢_𝑛𝑖−𝑦_𝑛𝑖−𝜆𝐵𝑢_𝑛𝑖=𝑣−𝑦_𝑛𝑖,𝐵𝑣−𝐵𝑦_𝑛𝑖+𝑣−𝑦_𝑛𝑖,𝐵𝑦_𝑛𝑖−𝐵𝑢_𝑛𝑖+𝑣−𝑦_𝑛𝑖,1𝜆𝑢_𝑛𝑖−𝑦_𝑛𝑖. (3.55) It follows from lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0, lim_𝑛→∞‖𝐵𝑢_𝑛−𝐵𝑦_𝑛‖=0, and 𝑦_𝑛𝑖⇀𝑤 that limsup_𝑛→∞𝑣−𝑦_𝑛,𝑔=𝑣−𝑤,𝑔≥0. (3.56) It follows from the maximal monotonicity of 𝐵+𝑀 that 𝜃∈(𝑀+𝐵)(𝑤), that is, 𝑤∈𝐼(𝐵,𝑀). Therefore, 𝑤∈Ω. It follows that limsup_𝑛→∞𝛾𝑓−𝐴𝑞,𝑆𝑦_𝑛−𝑞=lim_𝑖→∞𝛾𝑓−𝐴𝑞,𝑆𝑦_𝑛𝑖−𝑞=𝛾𝑓−𝐴𝑞,𝑤−𝑞≤0. (3.57)
Step 6. We prove that 𝑥_𝑛→𝑞. By using (3.2) and together with Schwarz inequality, we have that 𝑥_𝑛+1−𝑞²=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑃_𝐶(𝑞)²≤𝛼_𝑛(𝛾𝑓(𝑥_𝑛)−𝐴𝑞)+𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞²≤𝐼−𝛼_𝑛𝐴²𝑆𝑦_𝑛−𝑞²+𝛼²_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛𝐼−𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²𝑦_𝑛−𝑞²+𝛼²_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝐴𝑞−2𝛼²_𝑛𝐴𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛⟨𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛−𝛾𝑓𝑞⟩+2𝛼_𝑛⟨𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑞−𝐴𝑞⟩−2𝛼²_𝑛⟨𝐴𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛≤−𝐴𝑞⟩1−𝛼_𝑛𝛾²𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛼_𝑛𝑆𝑦_𝑛−𝑞𝛾𝑓𝑥_𝑛−𝛾𝑓𝑞+2𝛼_𝑛𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑞−𝐴𝑞−2𝛼²_𝑛⟨𝐴𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛≤−𝐴𝑞⟩1−𝛼_𝑛𝛾²𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛾𝛼𝛼_𝑛𝑦_𝑛−𝑞𝑥_𝑛−𝑞+2𝛼_𝑛𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑞−𝐴𝑞−2𝛼²_𝑛⟨𝐴𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛≤−𝐴𝑞⟩1−𝛼_𝑛𝛾²𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛⟨𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑞−𝐴𝑞⟩−2𝛼²_𝑛𝐴𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²+2𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑞−𝐴𝑞−2𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞𝛾𝑓𝑥_𝑛=−𝐴𝑞1−2𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛𝛼_𝑛𝛾𝑓(𝑥_𝑛)−𝐴𝑞²+2𝑆𝑦_𝑛−𝑞,𝛾𝑓𝑞−𝐴𝑞−2𝛼_𝑛𝐴𝑆𝑦_𝑛−𝑞𝛾𝑓𝑥_𝑛−𝐴𝑞+𝛼_𝑛𝛾²𝑥_𝑛−𝑞². (3.58)
Since {𝑥_𝑛} is bounded, where 𝜂≥‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖²−2‖𝐴(𝑆𝑦_𝑛−𝑞)‖‖𝛾𝑓(𝑥_𝑛)−𝐴𝑞‖+𝛾²‖𝑥_𝑛−𝑞‖² for all 𝑛≥0, it follows that 𝑥_𝑛+1−𝑞²≤1−2𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛𝜍_𝑛, (3.59) where 𝜍_𝑛=2⟨𝑆𝑦_𝑛−𝑞,𝛾𝑓(𝑞)−𝐴𝑞⟩+𝜂𝛼_𝑛. By limsup_𝑛→∞⟨(𝛾𝑓−𝐴)𝑞,𝑆𝑦_𝑛−𝑞⟩≤0, we get limsup_𝑛→∞𝜍_𝑛≤0. Applying Lemma 2.5, we can conclude that 𝑥_𝑛→𝑞. This completes the proof.

Corollary 3.2. Let H be a real Hilbert space and 𝐶 a closed convex subset of 𝐻. Let 𝐵,Ψ∶𝐶→𝐻 be 𝛽,𝜎-inverse-strongly monotone mappings and 𝜑∶𝐶→ℛ a convex and lower semicontinuous function. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1), 𝑀∶𝐻→2^𝐻 a maximal monotone mapping, and 𝑆 a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩GMEP𝐹,𝜑,Ψ∩𝐼(𝐵,𝑀)≠∅. (3.60) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀,𝑢_𝑛∈𝐶 arbitrarily: 𝐹𝑢_𝑛,𝑦+𝜑y−𝜑𝑢_𝑛+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑆𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛 (3.61) for all 𝑛=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω, where 𝑞=𝑃_Ω(𝑓+𝐼)(𝑞) which solves the following variational inequality: 𝑓−𝐼𝑞,𝑝−𝑞≤0,∀𝑝∈Ω. (3.62)

Proof. Putting 𝐴≡𝐼 and 𝛾≡1 in Theorem 3.1, we can obtain the desired conclusion immediately.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a closed convex subset of 𝐻. Let 𝐵, Ψ∶𝐶→𝐻 be 𝛽, 𝜎-inverse-strongly monotone mappings, 𝜑∶𝐶→ℛ a convex and lower semicontinuous function, and 𝑀∶𝐻→2^𝐻 a maximal monotone mapping. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩GMEP𝐹,𝜑,Ψ∩𝐼(𝐵,𝑀)≠∅. (3.63) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀,𝑢∈𝐶 and 𝑢_𝑛∈𝐶: 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝑢+1−𝛼_𝑛𝑆𝐽_𝑀,𝜆𝑢_𝑛−𝜆𝐵𝑢_𝑛 (3.64) for all 𝑛=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω, where 𝑞=𝑃_Ω(𝑞) which solves the following variational inequality: ⟨𝑢−𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω. (3.65)

Proof. Putting 𝑓(𝑥)≡𝑢, for all 𝑥∈𝐶, in Corollary 3.2, we can obtain the desired conclusion immediately.

Corollary 3.4. Let 𝐻 be a real Hilbert space, 𝐶 a closed convex subset of 𝐻, 𝐵∶𝐶→𝐻 be 𝛽-inverse-strongly monotone mappings, and 𝐴 a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and 𝑆 a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩VI(𝐶,𝐵)≠∅. (3.66) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀∈𝐶 arbitrarily: 𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑃_𝐶𝑥_𝑛−𝜆𝐵𝑥_𝑛 (3.67) for all 𝑛=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω, where 𝑞=𝑃_Ω(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: 𝛾𝑓−𝐴𝑞,𝑝−𝑞≤0,∀𝑝∈Ω. (3.68)

Proof. Taking 𝐹≡0,Ψ≡0,𝜑≡0,𝑢_𝑛=𝑥_𝑛, and 𝐽_𝑀,𝜆=𝑃_𝐶 in Theorem 3.1, we can obtain the desired conclusion immediately.

Remark 3.5. In Corollary 3.4 we generalize and improve the result of Klin-eam and Suantai [24].

4. Applications

In this section, we apply the iterative scheme (1.25) for finding a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping and also apply Theorem 3.1 for finding a common fixed point of nonexpansive mappings and inverse-strongly monotone mappings.

Definition 4.1. A mapping 𝑇∶𝐶→𝐶 is called strictly pseudocontraction if there exists a constant 0≤𝜅<1 such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝜅(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (4.1) If 𝜅=0, then 𝑆 is nonexpansive. In this case, we say that 𝑇∶𝐶→𝐶 is a 𝜅-strictly pseudocontraction. Putting 𝐵=𝐼−𝑇. Then, we have that (𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦²≤𝑥−𝑦²+𝜅𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶. (4.2) Observe that (𝐼−𝐵)𝑥−(𝐼−𝐵)𝑦²=𝑥−𝑦²+𝐵𝑥−𝐵𝑦²−2𝑥−𝑦,𝐵𝑥−𝐵𝑦,∀𝑥,𝑦∈𝐶. (4.3) Hence, we obtain 𝑥−𝑦,𝐵𝑥−𝐵𝑦≥1−𝜅2𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶. (4.4) Then, 𝐵 is ((1−𝜅)/2)-inverse-strongly monotone mapping.

Using Theorem 3.1, we first prove a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strict pseudocontraction.

Theorem 4.2. Let 𝐻 be a real Hilbert space, 𝐶 a closed convex subset of 𝐻, 𝐵, Ψ∶𝐶→𝐻 be 𝛽, 𝜎-inverse-strongly monotone mappings, 𝜑∶𝐶→ℛ a convex and lower semicontinuous function, 𝑓∶𝐶→𝐶 a contraction with coefficient 𝛼(0<𝛼<1), and 𝐴 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 𝐶 into itself, and let 𝑇 be a 𝜅-strictly pseudocontraction of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩𝐹(𝑇)∩GMEP𝐹,𝜑,Ψ≠∅. (4.5) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀,𝑢_𝑛∈𝐶 arbitrarily: 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆(1−𝜆)𝑢_𝑛+𝜆𝑇𝑢_𝑛 (4.6) for all 𝑛=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω, where 𝑞=𝑃_Ω(𝛾𝑓+𝐼−𝐴)(𝑞) which solves the following variational inequality: 𝛾𝑓−𝐴𝑞,𝑝−𝑞≤0,∀𝑝∈Ω (4.7) which is the optimality condition for the minimization problem min_𝑞∈Ω12𝐴𝑞,𝑞−ℎ𝑞, (4.8) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. Put 𝐵≡𝐼−𝑇, then 𝐵 is ((1−𝜅)/2)-inverse-strongly monotone, 𝐹(𝑇)=𝐼(𝐵,𝑀), and 𝐽_𝑀,𝜆(𝑥_𝑛−𝜆𝐵𝑥_𝑛)=(1−𝜆)𝑥_𝑛+𝜆𝑇𝑥_𝑛. So by Theorem 3.1, we obtain the desired result.

Corollary 4.3. Let 𝐻 be a real Hilbert space, 𝐶 a closed convex subset of 𝐻, 𝐵, Ψ∶𝐶→𝐻 be 𝛽, 𝜎-inverse-strongly monotone mappings, and 𝜑∶𝐶→ℛ a convex and lower semicontinuous function. Let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝛼(0<𝛼<1) and 𝑆 a nonexpansive mapping of 𝐶 into itself, and let 𝑇 be a 𝜅-strictly pseudocontraction of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩𝐹(𝑇)∩GMEP𝐹,𝜑,Ψ≠∅. (4.9) Suppose that {𝑥_𝑛} is a sequence generated by the following algorithm for 𝑥₀∈𝐶 arbitrarily: 𝐹𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+⟨Ψ𝑥_𝑛,𝑦−𝑢_𝑛1⟩+𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑥⟩≥0,∀𝑦∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝑆(1−𝜆)𝑢_𝑛+𝜆𝑇𝑢_𝑛 (4.10) for all 𝑛=0,1,2,…, by (C1)–(C3) in Theorem 3.1.
Then {𝑥_𝑛} converges strongly to 𝑞∈Ω, where 𝑞=𝑃_Ω(𝑓+𝐼)(𝑞) which solves the following variational inequality: ⟨𝑓−𝐼𝑞,𝑝−𝑞⟩≤0,∀𝑝∈Ω (4.11) which is the optimality condition for the minimization problem min_𝑞∈Ω12⟨𝐴𝑞,𝑞⟩−ℎ𝑞, (4.12) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑞)=𝛾𝑓(𝑞) for 𝑞∈𝐻).

Proof. Putting 𝐴≡𝐼 and 𝛾≡1 in Theorem 4.2, we obtain the desired result.

Acknowledgments

The authors would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support under the project NRU-CSEC no. 54000267. Furthermore, they also would like to thank the Faculty of Science (KMUTT) and the National Research Council of Thailand. Finally, the authors would like to thank Professor Vittorio Colao and the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.