A New Iterative Scheme for Solving the Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

Abstract

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

1. Introduction

Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻 with inner product ⟨⋅,⋅⟩. Recall that a mapping 𝑆∶𝐶→𝐶 is called nonexpansive if ‖𝑆𝑥−𝑆𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. The set of all fixed points of 𝑆 is denoted by 𝐹(𝑆)={𝑥∈𝐶∶𝑥=𝑆𝑥}. A mapping 𝑔∶𝐶→𝐶 is a contraction on 𝐶 if there is a constant 𝛼∈(0,1) such that ‖𝑔(𝑥)−𝑔(𝑦)‖≤𝛼‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. We use Π_𝐶 to denote the collection of all contractions on 𝐶. Note that each 𝑔∈Π_𝐶 has a fixed unique fixed point in 𝐶. A linear bounded operator 𝐴 is strongly positive if there is a constant 𝛾>0 with property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖² for all 𝑥∈𝐻.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [1–4] and the references therein. Convex minimization problems have a great impact and influence in 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: 𝜃(𝑥)=min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩, (1.1) where 𝐴 is a linear bounded operator, 𝐶 is the fixed point set of a nonexpansive mapping 𝑇, and 𝑏 is a given point in 𝐻. Let 𝐻 be a real Hilbert space. Recall that a linear bounded operator 𝐵 is strongly positive if there is a constant 𝛾>0 with property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²∀𝑥∈𝐻. (1.2) Recently, Marino and Xu [5] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [6]: 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝛾𝑔𝑥_𝑛,𝑛≥0, (1.3) where 𝐴 is a strongly positive bounded linear operator on 𝐻. They proved that if the sequence {𝛼_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.3) converges strongly to the unique solution of the variational inequality: 𝐴−𝛾𝑔𝑥^∗,𝑥−𝑥^∗≥0,𝑥∈𝐶, (1.4) which is the optimality condition for the minimization problem: min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.5) where ℎ is a potential function for 𝛾𝑔 (i.e., ℎ^′(𝑥)=𝛾𝑔(𝑥) for 𝑥∈𝐻).

A mapping 𝐵 of 𝐶 into 𝐻 is called monotone if ⟨𝐵𝑥−𝐵𝑦,𝑥−𝑦⟩≥0 for all 𝑥,𝑦∈𝐶. The variational inequality problem is to find 𝑥∈𝐶 such that ⟨𝐵𝑥,𝑥−𝑥⟩≥0∀𝑥∈𝐶. (1.6) The set of solutions of variational inequality is denoted by VI(𝐶,𝐵). A mapping 𝐵∶𝐶→𝐻 is called inverse-strongly monotone if there exists a positive real number 𝛽 such that 𝑥−𝑦,𝐵𝑥−𝐵𝑦≥𝛽𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶. (1.7) For such a case, 𝐵 is 𝛽-inverse-strongly monotone. If 𝐵 is a 𝛽-inverse-strongly monotone mapping of 𝐶 to 𝐻, then it is obvious that 𝐵 is (1/𝛽)-Lipschitz continuous. In 2009, Klin-eam and Suantai [7] introduced the following general iterative method: 𝑥₀∈𝐶,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐵𝑥_𝑛,𝑛≥0, (1.8) where 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶, 𝑔 is a contraction, 𝐴 is a strongly positive linear bounded operator, 𝐵 is a 𝛽-inverse strongly monotone mapping, {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 generated by (1.8) 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 solves the following variational inequality: 𝐴−𝛾𝑔𝑥,𝑥−𝑥≥0,∀𝑥∈𝐹(𝑆)∩VI(𝐶,𝐵). (1.9)

We recall the following well-known definitions. A bifunction 𝑓∶𝐶×𝐶→ℝ is called(a)monotone on 𝐶 if𝑓𝑥,𝑦+𝑓𝑦,𝑥≤0,∀𝑥,𝑦∈𝐶; (1.10)(b) pseudomonotone on 𝐶 if𝑓𝑥,𝑦≥0⟹𝑓𝑦,𝑥≤0,∀𝑥,𝑦∈𝐶; (1.11)(c) Lipschitz-type continuous on 𝐶 with two constants 𝑐₁>0 and 𝑐₂>0 if𝑓𝑥,𝑦+𝑓𝑦,𝑧≥𝑓(𝑥,𝑧)−𝑐₁𝑥−𝑦²−𝑐₂𝑦−𝑧²,∀𝑥,𝑦,𝑧∈𝐶. (1.12)

We consider the following equilibrium problems: find 𝑥∈𝐶 such that Find𝑥∈𝐶suchthat𝑓𝑥,𝑦≥0,∀𝑥∈𝐶. (1.13) The set of solution of problem (1.13) is denoted by EP(𝑓,𝐶). If 𝑓(𝑥,𝑦)∶=⟨𝐹𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶, where 𝐹 is a mapping from 𝐶 to 𝐻, then problem EP(𝑓,𝐶) reduces to the variational inequalities (1.6). It is well known that problem EP(𝑓,𝐶) covers many important problems in optimization and nonlinear analysis as well as it has found many applications in economic, transportation, and engineering.

For solving the common element of the set of fixed points of a nonexpansive mapping and the solution set of equilibrium problems, S. Takahashi and W. Takahashi [8] introduced the following viscosity approximation method: 𝑥₀∈𝐻,Find𝑦_𝑛∈𝐶suchthat𝑓𝑦_𝑛+1,𝑦𝑟_𝑛𝑦−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝑥,∀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑔𝑥_𝑛+1−𝛼_𝑛𝑆𝑦_𝑛,∀𝑛≥0, (1.14) where {𝛼_𝑛}⊂[0,1] and {𝑟_𝑛}⊂(0,∞). They showed that under certain conditions over {𝛼_𝑛} and {𝑟_𝑛}, sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑧∈𝐹(𝑆)∩EP(𝐹), where 𝑧=𝑃_{𝐹(𝑆)∩EP(𝐹)}𝑔(𝑧).

In this paper, inspired and motivated by Klin-eam and Suantai [7] and S. Takahashi and W. Takahashi [8], we introduce the new algorithm for solving the common element of the set of fixed points of a nonexpansive mapping, the solution set of equilibrium problems, and the solution set of the variational inequality problems for an inverse strongly monotone mapping. Let 𝑓 be monotone, Lipschitz-type continuous on 𝐶 with two constants 𝑐₁>0 and 𝑐₂>0, 𝐴 a strongly linear bounded operator, and 𝐵 a 𝛽-inverse strongly monotone mapping. Let 𝑔∶𝐶→𝐶 be a contraction with coefficient 𝛼 such that 0<𝛾<𝛾/𝛼 and 𝑆∶𝐶→𝐶 a nonexpansive mapping. The algorithm is now described as follows.

Step 1 (initialization). Choose positive sequences {𝛼_𝑛}⊂(0,1) and {𝜆_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1/𝐿), where 𝐿=max{2𝑐₁,2𝑐₂} and 𝛽⊂[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2𝛽.

Step 2 (solving convex problems). For a given point 𝑥₀=𝑥∈𝐶 and set 𝑛∶=0, we solve the following two strongly convex problems: 𝑦_𝑛=argmin𝜆_𝑛𝑓𝑥_𝑛+1,𝑦2𝑦−𝑥_𝑛²,𝑡∶𝑦∈𝐶_𝑛=argmin𝜆_𝑛𝑓𝑦_𝑛+1,𝑦2𝑦−𝑥_𝑛²∶𝑦∈𝐶. (1.15)

Step 3 (iteration 𝑛). Compute 𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑃_𝐶𝑡_𝑛−𝛽𝐵𝑡_𝑛, (1.16) where 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶. Increase 𝑛 by 1 and go to Step 1.

We show that under some control conditions the sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑡_𝑛} defined by (1.15) and (1.16) converge strongly to a common element of solution set of monotone, Lipschitz-type continuous equilibrium problems, and the set of fixed points of nonexpansive mappings which is a unique solution of the variational inequality problem (1.6).

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction. For solving the mixed equilibrium problem, let us assume the following conditions for a bifunction 𝑓∶𝐶×𝐶→ℝ:(A1)𝑓(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝑓 is Lipschitz-type continuous on 𝐶;(A3)𝑓 is monotone on 𝐶;(A4) for each 𝑥∈𝐶, 𝑓(𝑥,⋅) is convex and subdifferentiable on 𝐶;(A5)𝑓 is upper semicontinuous on 𝐶.

The metric (nearest point) projection 𝑃_𝐶 from a Hilbert space 𝐻 to a closed convex subset 𝐶 of 𝐻 is defined as follows: given 𝑥∈𝐻, 𝑃_𝐶𝑥 is the only point in 𝐶 such that ‖𝑥−𝑃_𝐶𝑥‖−inf{‖𝑥−𝑦‖∶𝑦∈𝐶}. In what follows lemma can be found in any standard functional analysis book.

Lemma 2.1. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Given 𝑥∈𝐻 and 𝑦∈𝐶, then(i)𝑦=𝑃_𝐶𝑥 if and only if ⟨𝑥−𝑦,𝑦−𝑧⟩≥0 for all 𝑧∈𝐶,(ii)𝑃_𝐶 is nonexpansive,(iii)⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖² for all 𝑥,𝑦∈𝐻,(iv)⟨𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦⟩ for all 𝑥∈𝐻 and 𝑦∈𝐶.

Using Lemma 2.1, one can show that the variational inequality (1.6) is equivalent to a fixed point problem.

Lemma 2.2. The point 𝑢∈𝐶 is a solution of the variational inequality (1.6) if and only if 𝑢 satisfies the relation 𝑢=𝑃_𝐶(𝑢−𝜆𝐵𝑢) for all 𝜆>0.

A set-valued mapping 𝑇∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻,𝑢∈𝑇𝑥, and 𝑣∈𝑇𝑦 imply ⟨𝑥−𝑦,𝑢−𝑣⟩≥0. A monotone mapping 𝑇∶𝐻→2^𝐻 is maximal if the graph 𝐺(𝑇) of 𝑇 is not property contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇 is maximal if and only if for (𝑥,𝑢)∈𝐻×𝐻,⟨𝑥−𝑦,𝑢−𝑣⟩≥0 for every (𝑦,𝑣)∈𝐺(𝑇) implies 𝑢∈𝑇𝑥. Let 𝐵 be an inverse-strongly monotone mapping of 𝐶 to 𝐻, let 𝑁_𝐶𝑣 be normal cone to 𝐶 at 𝑣∈𝐶, that is, 𝑁_𝐶𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,forall𝑢∈𝐶}, and define 𝑇𝑣=𝐵𝑣+𝑁_𝐶𝑣,if𝑣∈𝐶,∅,if𝑣∉𝐶. (2.1) Then 𝑇 is a maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐶,𝐵) [9].

Now we collect some useful lemmas for proving the convergence results of this paper.

Lemma 2.3 (see [10]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and ℎ∶𝐶→ℝ be convex and subdifferentiable on 𝐶. Then 𝑥^∗ is a solution to the following convex problem: min{ℎ(𝑥)∶𝑥∈𝐶} (2.2) if and only if 0∈𝜕ℎ(𝑥^∗)+𝑁_𝐶(𝑥^∗), where 𝜕ℎ(⋅) denotes the subdifferential of ℎ and 𝑁_𝐶(𝑥^∗) is the (outward) normal cone of 𝐶 at 𝑥^∗∈𝐶.

Lemma 2.4 (see [11, Lemma  3.1]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a pseudomonotone, Lipschitz-type continuous bifunction with constants 𝑐₁>0 and 𝑐₂>0. For each 𝑥∈𝐶, let 𝑓(𝑥,⋅) be convex and subdifferentiable on 𝐶. Suppose that the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} are generated by Scheme (1.15) and 𝑝∈EP(𝑓). Then 𝑡_𝑛−𝑝²≤𝑥_𝑛−𝑝²−1−2𝜆_𝑛𝑐₁𝑥_𝑛−𝑦_𝑛²−1−2𝜆_𝑛𝑐₂𝑦_𝑛−𝑡_𝑛²,∀𝑛≥0. (2.3)

Lemma 2.5 (see [12]). Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let 𝑆∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑆)≠∅. If a sequence {𝑥_𝑛} in 𝐶 such that 𝑥_𝑛⇀𝑧 and 𝑥_𝑛−𝑆𝑥_𝑛→0, then 𝑧=𝑆𝑧.

Lemma 2.6 (see [5]). 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.

In the following, we also need the following lemma that can be found in the existing literature [3, 13].

Lemma 2.7 (see [3, Lemma  2.1]). Let {𝑎_𝑛} be a sequence of non-negative real number satisfying the following property: 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛽_𝑛,𝑛≥0, (2.4) where {𝛾_𝑛}⊆(0,1) and {𝛽_𝑛}⊆ℝ such that ∑^∞_𝑛=0𝛾_𝑛=∞ and limsup_𝑛→∞𝛽_𝑛≤0. Then {𝑎_𝑛} converges to zero, as 𝑛→∞.

3. Main Theorems

In this section, we prove the strong convergence theorem for solving a common element of solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed points of nonexpansive mappings.

Theorem 3.1. Let 𝐻 be a real Hilbert space, and let 𝐶 be a closed convex subset of 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A5), let 𝐵∶𝐶→𝐻 be a 𝛽-inverse strongly monotone mapping, let 𝐴 be a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0 such that ‖𝐴‖=1 and let 𝑔∶𝐶→𝐶 be a contraction with coefficient 𝛼(𝛼∈(0,1)). Assume that 0<𝛾<𝛾/𝛼. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩EP(𝑓)∩VI(𝐶,𝐵)≠∅. Let the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} be generated by (1.15) and (1.16), where {𝛼_𝑛}⊂(0,1), {𝛽}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,2𝛽), and {𝜆_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1/𝐿), where 𝐿=max{2𝑐₁,2𝑐₂}. Suppose that the following conditions are satisfied:(B1)lim_𝑛→∞𝛼_𝑛=0;(B2)∑^∞_𝑛=1𝛼_𝑛=∞;(B3)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(B4)∑^∞_𝑛=1|𝜆_𝑛+1−𝜆_𝑛|<∞. Then the following holds.(i)𝑃_Ω(𝐼−𝐴+𝛾𝑔) is a contraction on 𝐶; hence there exists 𝑞∈𝐶 such that 𝑞=𝑃_Ω(𝐼−𝐴+𝛾𝑔)(𝑞), where 𝑃_Ω is the metric projection of 𝐻 onto 𝐶.(ii)The sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} converge strongly to the same point 𝑞.

Proof. For any 𝑥,𝑦∈𝐻, we have 𝑃_Ω𝛾𝑔+(𝐼−𝐴)𝑥−𝑃_Ω()𝑦≤()()𝑦𝑔(𝑥)𝑦+‖+𝛾𝑔+𝐼−𝐴𝛾𝑔+𝐼−𝐴𝑥−𝛾𝑔+𝐼−𝐴≤𝛾−𝑔𝐼−𝐴‖𝑥−𝑦≤𝛾𝛼𝑥−𝑦1−𝛾=𝑥−𝑦1−𝛾−𝛾𝛼𝑥−𝑦. (3.1) Banach’s contraction principle guarantees that 𝑃_Ω(𝛾𝑔+(𝐼−𝐴)) has a unique fixed point, say 𝑞∈𝐻. That is, 𝑞=𝑃_Ω(𝛾𝑔+(𝐼−𝐴))(𝑞). By Lemma 2.1(i), we obtain that 𝛾𝑔−𝐴𝑞,𝑝−𝑞≤0,∀𝑝∈Ω. (3.2)
The proof of (ii) is divided into several steps.
Step 1. 𝐼−𝛽𝐵 is nonexpansive mapping. Indeed, since 𝐵 is a 𝛽-strongly monotone mapping and 0<𝛽<2𝛽, for all 𝑥,𝑦∈𝐶, we have 𝐼−𝛽𝐵𝑥−𝐼−𝛽𝐵𝑦²=𝑥−𝑦−𝛽𝐵𝑥−𝐵𝑦²=𝑥−𝑦²−2𝛽𝑥−𝑦,𝐵𝑥−𝐵𝑦+𝛽²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²−2𝛽𝐵𝑥−𝐵𝑦²+𝛽²𝐵𝑥−𝐵𝑦²≤𝑥−𝑦²+𝛽𝛽−2𝛽𝐵𝑥−𝐵𝑦²≤𝑥−𝑦². (3.3)Step 2. We show that {𝑥_𝑛} is a bounded sequence. Put 𝑤_𝑛=𝑃_𝐶(𝑡_𝑛−𝛽𝐵𝑡_𝑛) for all 𝑛≥0. Let 𝑝∈Ω; we have 𝑤_𝑛−𝑝=𝑃_𝐶𝑡_𝑛−𝛽𝐵𝑡_𝑛−𝑃_𝐶𝑝−≤𝛽𝐵𝑝𝐼−𝛽𝐵𝑡_𝑛−𝐼−𝑝≤𝑡𝛽𝐵_𝑛−𝑝. (3.4) By Lemma 2.4, we have 𝑥_𝑛+1−𝑝=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛≤𝛼−𝑝_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑆𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+1−𝛼_𝑛𝛾𝑡_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+1−𝛼_𝑛𝛾𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑔𝑝−𝐴𝑝+1−𝛼_𝑛𝛾𝑥_𝑛≤−𝑝1−𝛼_𝑛𝛾−𝛾𝛼𝑥_𝑛−𝑝+𝛼_𝑛𝑝𝛾−𝛾𝛼𝛾𝑔−𝐴𝑝.𝛾−𝛾𝛼 (3.5) By induction, we get that 𝑥_𝑛−𝑝≤max𝑥₁,1−𝑝𝑝𝛾−𝛾𝛼𝛾𝑔−𝐴𝑝,𝑛≥0. (3.6) Hence {𝑥_𝑛} is bounded, and then {𝑤_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} are also bounded. Step 3. We show that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.7) Since 𝑓(𝑥,⋅) is convex on 𝐶 for each 𝑥∈𝐶, applying Lemma 2.3, we see that 𝑡_𝑛=argmin{(1/2)‖𝑡−𝑥_𝑛‖+𝜆_𝑛𝑓(𝑦_𝑛,𝑡)∶𝑡∈𝐶} if and only if 0∈𝜕₂𝜆_𝑛𝑓𝑦_𝑛+1,𝑡2𝑦−𝑥_𝑛²𝑡_𝑛+𝑁_𝐶𝑡_𝑛, (3.8) where 𝑁_𝐶(𝑥) is the (outward) normal cone of 𝐶 at 𝑥∈𝐶. This implies that 0=𝜆_𝑛𝑤+𝑡_𝑛−𝑥_𝑛+𝑤, where 𝑤∈𝜕₂𝑓(𝑦_𝑛,𝑡_𝑛) and 𝑤∈𝑁_𝐶𝑡_𝑛. By the definition of the normal cone 𝑁_𝐶, we have 𝑥_𝑛−𝜆_𝑛𝑤−𝑡_𝑛,𝑡−𝑡_𝑛=𝑤,𝑡−𝑡_𝑛≤0,∀𝑡∈𝐶, (3.9) and so 𝑡_𝑛−𝑥_𝑛,𝑡−𝑡_𝑛≥𝜆_𝑛𝑤,𝑡_𝑛−𝑡,∀𝑡∈𝐶. (3.10) Substituting 𝑡=𝑡_𝑛+1∈𝐶 into (3.10), we get that 𝑡_𝑛−𝑥_𝑛,𝑡_𝑛+1−𝑡_𝑛≥𝜆_𝑛𝑤,𝑡_𝑛−𝑡_𝑛+1. (3.11) Since 𝑓(𝑥,⋅) is subdifferentiable on 𝐶 and 𝑤∈𝜕₂𝑓(𝑦_𝑛,𝑡_𝑛), we have 𝑓𝑦_𝑛,𝑡−𝑓𝑦_𝑛,𝑡_𝑛≥𝑤,𝑡−𝑡_𝑛,∀𝑡∈𝐶. (3.12) From (3.11) and (3.12), we obtain that 𝑡_𝑛−𝑥_𝑛,𝑡_𝑛+1−𝑡_𝑛≥𝜆_𝑛𝑤,𝑡_𝑛−𝑡_𝑛+1≥𝜆_𝑛𝑓𝑦_𝑛,𝑡_𝑛−𝑓𝑦_𝑛,𝑡_𝑛+1. (3.13) By the similar way, we also have 𝑡_𝑛+1−𝑥_𝑛+1,𝑡_𝑛−𝑡_𝑛+1≥𝜆_𝑛+1𝑓𝑦_𝑛+1,𝑡_𝑛+1−𝑓𝑦_𝑛+1,𝑡_𝑛. (3.14) It follows from (3.13) and (3.14) and 𝑓 is Lipschitz-type continuous and monotone, we get 12𝑥_𝑛+1−𝑥_𝑛²−12𝑡_𝑛+1−𝑡_𝑛²≥𝑡_𝑛+1−𝑡_𝑛,𝑡_𝑛−𝑥_𝑛−𝑡_𝑛+1+𝑥_𝑛+1≥𝜆_𝑛𝑓𝑦_𝑛,𝑡_𝑛−𝑓𝑦_𝑛,𝑡_𝑛+1+𝜆_𝑛+1𝑓𝑦_𝑛+1,𝑡_𝑛+1−𝑓𝑦_𝑛+1,𝑡_𝑛≥𝜆_𝑛−𝑓𝑡_𝑛,𝑡_𝑛+1−𝑐₁𝑦_𝑛−𝑡_𝑛²−𝑐₂𝑡_𝑛−𝑡_𝑛+1²+𝜆_𝑛+1−𝑓𝑡_𝑛+1,𝑡_𝑛−𝑐₁𝑦_𝑛+1,𝑡_𝑛+1²−𝑐₂𝑡_𝑛−𝑡_𝑛+1²≥𝜆_𝑛+1−𝜆_𝑛𝑓𝑡_𝑛,𝑡_𝑛+1≥−𝜆_𝑛+1−𝜆_𝑛𝑓𝑡_𝑛,𝑡_𝑛+1, (3.15) and hence 𝑡_𝑛+1−𝑡_𝑛²=𝑥_𝑛+1−𝑥_𝑛²+2𝜆_𝑛+1−𝜆_𝑛𝑓𝑡_𝑛,𝑡_𝑛+1≤𝑥_𝑛+1−𝑥_𝑛2𝜆_𝑛+1−𝜆_𝑛𝑓𝑡_𝑛,𝑡_𝑛+1. (3.16) Thus, we have 𝑥_𝑛+1−𝑥_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑃_𝐶𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝐴𝑆𝑤_𝑛−1≤𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1−𝐼−𝛼_𝑛−1𝐴𝑆𝑤_𝑛−1=𝛼_𝑛𝛾𝑔𝑥_𝑛−𝛼_𝑛𝛾𝑔𝑥_𝑛−1+𝛼_𝑛𝛾𝑔𝑥_𝑛−1−𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1+𝐼−𝛼_𝑛𝐴𝑆𝑡_𝑛−𝐼−𝛼_𝑛𝐴𝑆𝑡_𝑛−1+𝐼−𝛼_𝑛𝐴𝑆𝑡_𝑛−1−𝐼−𝛼_𝑛−1𝐴𝑆𝑡_𝑛−1≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑆𝑤_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝐴𝑆𝑡_𝑛−1≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥_𝑛−1+𝐼−𝛼_𝑛𝛾𝑤_𝑛−𝑤_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1+𝐴𝑆𝑡_𝑛−1≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥_𝑛−1+𝐼−𝛼_𝑛𝛾𝑡_𝑛−𝑡_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1+𝐴𝑆𝑡_𝑛−1≤𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝛾𝑥_𝑛−𝑥_𝑛−1+2𝜆_𝑛+1−𝜆_𝑛𝑓𝑡_𝑛,𝑡_𝑛+1+𝛼_𝑛−𝛼_𝑛−1𝛾𝑔𝑥_𝑛−1+𝐴𝑆𝑡_𝑛+1≤1−𝛼_𝑛𝛾−𝛾𝛼𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝛾2𝜆_𝑛+1−𝜆_𝑛𝐾+𝛼_𝑛−𝛼_𝑛−1𝑀, (3.17) where 𝐾≥sup_𝑛≥0|𝑓(𝑡_𝑛−1,𝑡_𝑛)| and 𝑀≥sup_𝑛≥0𝛾‖𝑔(𝑥_𝑛−1)‖+‖𝐴𝑆(𝑡_𝑛−1)‖. Using (B3), (B4), and Lemma 2.7, we have lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.Step 4. We show that lim_𝑛→∞𝑥_𝑛−𝑡_𝑛=0. (3.18) Indeed, for each 𝑝∈Ω, applying Lemma 2.4, we have 𝑥_𝑛+1−𝑝²=𝑃_𝐶𝛼_𝑛𝛾𝑔(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐴)𝑆(𝑤_𝑛)−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛+(𝐼−𝛼_𝑛𝐴)𝑆(𝑤_𝑛)−𝑝²≤𝛼_𝑛𝛾𝑔(𝑥_𝑛)−𝐴𝑝+(1−𝛼_𝑛𝛾)𝑆(𝑤_𝑛)−𝑝²=𝛼²_𝑛𝛾𝑔(𝑥_𝑛)−𝐴𝑝²+1−𝛼_𝑛𝛾²𝑆(𝑤_𝑛)−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝑔𝑝𝑆𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔(𝑥_𝑛)−𝐴𝑝²+1−𝛼_𝑛𝛾𝑆(𝑤_𝑛)−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝑔𝑝𝑆𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝛾𝑔𝑝+𝛾𝑔𝑝−𝐴𝑝²+1−𝛼_𝑛𝛾𝑆(𝑤_𝑛)−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝑔𝑝𝑆𝑤_𝑛−𝑝=𝛼_𝑛𝛾𝑔(𝑥_𝑛)−𝛾𝑔(𝑝)²+2𝛼_𝑛𝛾𝑔𝑥_𝑛−𝛾𝑔𝑝𝛾𝑔𝑝−𝐴𝑝+𝛼_𝑛𝛾𝑔(𝑝)−𝐴𝑝²+1−𝛼_𝑛𝛾𝑆(𝑤_𝑛)−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝑔𝑝𝑆𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔(𝑥_𝑛)−𝛾𝑔(𝑝)²+1−𝛼_𝑛𝛾𝑡_𝑛−𝑝²+𝜃_𝑛≤𝛼_𝑛𝛾²𝛼²𝑥_𝑛−𝑝²+𝜃_𝑛+1−𝛼_𝑛𝛾𝑥_𝑛−𝑝²−1−2𝜆_𝑛𝑐₁𝑥_𝑛−𝑦_𝑛²−1−2𝜆_𝑛𝑐₂𝑦_𝑛−𝑡_𝑛²≤𝛼_𝑛𝛾𝛼²𝑥_𝑛−𝑝²+1−𝛼_𝑛𝛾𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₁𝑥_𝑛−𝑦_𝑛²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₂𝑦_𝑛−𝑡_𝑛²+𝜃_𝑛=1−𝛼_𝑛𝛾−𝛾𝛼²𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₁𝑥_𝑛−𝑦_𝑛²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₂𝑦_𝑛−𝑡_𝑛²+𝜃_𝑛≤𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₁𝑥_𝑛−𝑦_𝑛²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₂𝑦_𝑛−𝑡_𝑛²+𝜃_𝑛, (3.19) where 𝜃_𝑛=2𝛼_𝑛𝛾𝑔𝑥_𝑛−𝛾𝑔𝑝𝛾𝑔𝑝−𝐴𝑝+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛𝑝×𝑆𝑤−𝑔_𝑛−𝑝+𝛼_𝑛𝛾𝑔𝑝−𝐴𝑝². (3.20) It then follows that 1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₂𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²−1−𝛼_𝑛𝛾1−2𝜆_𝑛𝑐₁𝑦_𝑛−𝑡_𝑛²+𝜃_𝑛≤𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1+𝜃_𝑛⟶0, (3.21) as 𝑛→∞. Hence lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (3.22) By the similar way, also lim_𝑛→∞𝑦_𝑛−𝑡_𝑛=0. (3.23) From (3.22) and (3.23), we can conclude that 𝑥_𝑛−𝑡_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑡_𝑛⟶0as𝑛⟶∞. (3.24)Step 5. We show that lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (3.25) From (1.15), we get that 𝑥_𝑛+1−𝑝²=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+1−𝛼_𝑛𝛾𝑆𝑤_𝑛−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+1−𝛼_𝑛𝛾𝑤_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+1−𝛼_𝑛𝛾𝐼−𝛽𝐵𝑡_𝑛−𝐼−𝛽𝐵𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+1−𝛼_𝑛𝛾𝑡_𝑛−𝑝²+𝛽𝛽−2𝛽𝐵𝑡_𝑛−𝐵𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+𝑡_𝑛−𝑝²+1−𝛼_𝑛𝛾𝑎𝑏−2𝛽𝐵𝑡_𝑛−𝐵𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝²+1−𝛼_𝑛𝛾𝑎𝑏−2𝛽𝐵𝑡_𝑛−𝐵𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝, (3.26) and hence −1−𝛼_𝑛𝛾𝑎𝑏−2𝛽𝐵𝑡_𝑛−𝐵𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝. (3.27) Since 𝛼_𝑛→0, we get that ‖𝐵𝑡_𝑛−𝐵𝑝‖→0, as 𝑛→∞. By Lemma 2.1(iii), we have 𝑤_𝑛−𝑝²=𝑃_𝐶𝑡_𝑛−𝛽𝐵𝑡_𝑛−𝑃_𝐶𝑝−𝛽𝐵𝑝²≤𝑡_𝑛−𝛽𝐵𝑡_𝑛−𝑝−𝛽𝐵𝑝,𝑤_𝑛=1−𝑝2𝑡_𝑛−𝛽𝐵𝑡_𝑛−𝑝−𝛽𝐵𝑝²+𝑤_𝑛−𝑝²−𝑡_𝑛−𝛽𝐵𝑡_𝑛−𝑡_𝑛−𝛽𝐵𝑡_𝑛−𝑤_𝑛−𝑝²≤12𝑡_𝑛−𝑝²+𝑤_𝑛−𝑝²−𝑡_𝑛−𝑤_𝑛−𝛽𝐵𝑡_𝑛−𝐵𝑝²≤12𝑡_𝑛−𝑝²+𝑤_𝑛−𝑝²−𝑡_𝑛−𝑤_𝑛²+122𝛽𝑡_𝑛−𝑤_𝑛,𝐵𝑡_𝑛−𝐵𝑝−𝛽²_𝑛𝐵𝑡_𝑛−𝐵𝑝², (3.28) which implies that 𝑤_𝑛−𝑝²≤𝑡_𝑛−𝑝²−𝑡_𝑛−𝑤_𝑛²+2𝛽𝑡_𝑛−𝑤_𝑛,𝐵𝑡_𝑛−𝐵𝑝−𝛽²_𝑛𝐵𝑡_𝑛−𝐵𝑝². (3.29) Again from (3.29), we have 𝑥_𝑛+1−𝑝²=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝+1−𝛼_𝑛𝛾𝑆𝑤_𝑛−𝑝²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+1−𝛼_𝑛𝛾𝑤_𝑛−𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+1−𝛼_𝑛𝛾𝑡_𝑛−𝑝²−1−𝛼_𝑛𝛾𝑡_𝑛−𝑤_𝑛²+2𝛽1−𝛼_𝑛𝛾𝑡_𝑛−𝑤_𝑛,𝐵𝑡_𝑛−𝐵𝑝−𝛽²_𝑛1−𝛼_𝑛𝛾𝐵𝑡_𝑛−𝐵𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝²−1−𝛼_𝑛𝛾𝑡_𝑛−𝑤_𝑛²+2𝛽1−𝛼_𝑛𝛾𝑡_𝑛−𝑤_𝑛,𝐵𝑡_𝑛−𝐵𝑝−𝛽²_𝑛1−𝛼_𝑛𝛾𝐵𝑡_𝑛−𝐵𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝. (3.30) This implies that 1−𝛼_𝑛𝛾𝑡_𝑛−𝑤_𝑛²≤𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+2𝛽1−𝛼_𝑛𝛾𝑡_𝑛−𝑤_𝑛,𝐵𝑡_𝑛−𝐵𝑝−𝛽²_𝑛1−𝛼_𝑛𝛾𝐵𝑡_𝑛−𝐵𝑝²+2𝛼_𝑛1−𝛼_𝑛𝛾𝛾𝑔𝑥_𝑛−𝐴𝑝𝑤_𝑛−𝑝. (3.31) Since 𝛼_𝑛→0 and ‖𝐵𝑡_𝑛−𝐵𝑝‖→0, we obtain that lim_𝑛→∞𝑡_𝑛−𝑤_𝑛=0. (3.32) From (1.16), we have 𝑥_𝑛+1−𝑆𝑤_𝑛=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑃_𝐶𝑆𝑤_𝑛≤𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑆𝑤_𝑛=𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐴𝑆𝑤_𝑛⟶0,as𝑛⟶∞. (3.33) Since 𝑡_𝑛−𝑆𝑡_𝑛≤𝑡_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆𝑤_𝑛+𝑆𝑤_𝑛−𝑆𝑡_𝑛≤𝑡_𝑛−𝑥_𝑛+𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆𝑤_𝑛+𝑤_𝑛−𝑡_𝑛, (3.34) from (3.7), (3.24), (3.32), and (3.33), we obtain that ‖𝑡_𝑛−𝑆𝑡_𝑛‖→0, as 𝑛→∞. Moreover, we get that 𝑤_𝑛−𝑆𝑤_𝑛≤𝑤_𝑛−𝑡_𝑛+𝑡_𝑛−𝑆𝑡_𝑛+𝑆𝑡_𝑛−𝑆𝑤_𝑛≤2𝑤_𝑛−𝑡_𝑛+𝑡_𝑛−𝑆𝑡_𝑛⟶0,as𝑛⟶∞. (3.35) Since 𝑥_𝑛−𝑤_𝑛≤𝑥_𝑛−𝑡_𝑛+𝑡_𝑛−𝑤_𝑛, (3.36) it implies that lim_𝑛→∞𝑥_𝑛−𝑤_𝑛=0. (3.37) Since 𝑥_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆𝑤_𝑛+𝑆𝑤_𝑛−𝑆𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑆𝑤_𝑛+𝑤_𝑛−𝑥_𝑛, (3.38) we obtain that lim_𝑛→∞𝑥_𝑛−𝑆𝑥_𝑛=0. (3.39)Step 6. We show that limsup_𝑛→∞𝛾𝑔−𝐴𝑞,𝑆𝑤_𝑛−𝑞≤0. (3.40) Indeed, we choose a subsequence {𝑤_𝑛𝑘} of {𝑤_𝑛} such that limsup_𝑛→∞𝛾𝑔−𝐴𝑞,𝑆𝑤_𝑛−𝑞=lim_𝑘→∞𝛾𝑔−𝐴𝑞,𝑆𝑤_𝑛𝑘−𝑞. (3.41) Since {𝑤_𝑛} is bounded, there exists a subsequence {𝑤_𝑛𝑘𝑖} of {𝑤_𝑛𝑘} which converges weakly to 𝑝. Next we show that 𝑝∈Ω.
We prove that 𝑝∈𝐹(𝑆). We may assume without loss of generality that 𝑤_𝑛𝑘⇀𝑝. Since ‖𝑤_𝑛−𝑆𝑤_𝑛‖→0, we obtain 𝑆𝑤_𝑛𝑘⇀𝑝. Since ‖𝑥_𝑛−𝑆𝑥_𝑛‖→0,‖𝑥_𝑛−𝑤_𝑛‖→0 and by Lemma 2.5, we have 𝑝∈𝐹(𝑆).
We show that 𝑝∈EP(𝑓). From Steps 4 and 6, we have that 𝑡_𝑛𝑘⇀𝑝,𝑥_𝑛𝑘⇀𝑝,𝑦_𝑛𝑘⇀𝑝. (3.42) Since 𝑦_𝑛 is the unique solution of the convex minimization problem 1min2𝑦−𝑥_𝑛²+𝑓𝑥_𝑛,𝑦∶𝑦∈𝐶, (3.43) we have 0∈𝜕₂𝜆_𝑛𝑓𝑥_𝑛+1,𝑦2𝑦−𝑥_𝑛²𝑦_𝑛+𝑁_𝐶𝑦_𝑛. (3.44) It follows that 0=𝜆_𝑛𝑧+𝑦_𝑛−𝑥_𝑛+𝑧_𝑛, (3.45) where 𝑧∈𝜕₂𝑓(𝑥_𝑛,𝑦_𝑛) and 𝑧_𝑛∈𝑁_𝐶(𝑦_𝑛). By the definition of the normal cone 𝑁_𝐶, we get that 𝑦_𝑛−𝑥_𝑛,𝑦−𝑦_𝑛≥𝜆_𝑛𝑧,𝑦_𝑛−𝑦,∀𝑦∈𝐶. (3.46) On the other hand, since 𝑓(𝑥_𝑛,⋅) is subdifferentiable on 𝐶 and 𝑧∈𝜕₂𝑓(𝑥_𝑛,𝑦_𝑛), we have 𝑓𝑥_𝑛,𝑦−𝑓𝑥_𝑛,𝑦_𝑛≥𝑧,𝑦−𝑦_𝑛,∀𝑦∈𝐶. (3.47) Combining (3.47) with (3.46), we have 𝜆_𝑛𝑓𝑥_𝑛,𝑦−𝑓𝑥_𝑛,𝑦_𝑛≥𝑦_𝑛−𝑥_𝑛,𝑦_𝑛−𝑦,∀𝑦∈𝐶. (3.48) Hence 𝜆_𝑛𝑘𝑓𝑥_𝑛𝑘,𝑦−𝑓𝑥_𝑛𝑘,𝑦_𝑛𝑘≥𝑦_𝑛𝑘−𝑥_𝑛𝑘,𝑦_𝑛𝑘−𝑦,∀𝑦∈𝐶. (3.49) Thus, using {𝜆⊂[𝑎,𝑏]⊂(0,1/𝐿)} and the upper semicontinuity of 𝑓, we have 𝑓𝑝,𝑦≥0,∀𝑦∈𝐶. (3.50) Hence 𝑝∈EP(𝑓).
We show that 𝑝∈VI(𝐶,𝐵). Let 𝑇𝑣=𝐵𝑣+𝑁_𝐶𝑣,𝑣∈𝐶,∅,𝑣∉𝐶, (3.51) where 𝑁_𝐶𝑣 is normal cone to 𝐶 at 𝑣∈𝐶. Then 𝑇 is a maximal monotone operator. Let (𝑣,𝑢)∈𝐺(𝑇). Since 𝑢−𝐵𝑣∈𝑁_𝐶𝑣 and 𝑤_𝑛∈𝐶, we have ⟨𝑣−𝑦_𝑛,𝑢−𝐵𝑣⟩≥0. On the other hand, by Lemma 2.1(iv) and from 𝑤_𝑛=𝑃_𝐶(𝑡_𝑛−𝛽𝐵𝑡_𝑛), we have 𝑣−𝑤_𝑛,𝑤_𝑛−𝑡_𝑛−𝛽𝐵𝑡_𝑛≥0, (3.52) and hence ⟨𝑣−𝑤_𝑛,(𝑤_𝑛−𝑡_𝑛)/𝛽+𝐵𝑡_𝑛⟩≥0. Therefore, we get that 𝑣−𝑤_𝑛𝑘,𝑢≥𝑣−𝑤_𝑛𝑘≥,𝐵𝑣𝑣−𝑤_𝑛𝑘,𝐵𝑣−𝑣−𝑤_𝑛𝑘,𝑤_𝑛𝑘−𝑡_𝑛𝑘𝜆_𝑛𝑘+𝐵𝑥_𝑛𝑘=𝑣−𝑤_𝑛𝑘,𝐵𝑣−𝐵𝑡_𝑛𝑘−𝑤_𝑛𝑘−𝑡_𝑛𝑘𝜆_𝑛𝑘=𝑣−𝑤_𝑛𝑘,𝐵𝑣−𝐵𝑤_𝑛𝑘+𝑣−𝑤_𝑛𝑘,𝐵𝑤_𝑛𝑘−𝐵𝑡_𝑛𝑘−𝑣−𝑤_𝑛𝑘,𝑤_𝑛𝑘−𝑡_𝑛𝑘𝜆_𝑛𝑘≥𝑣−𝑤_𝑛𝑘,𝐵𝑤_𝑛𝑘−𝐵𝑡_𝑛𝑘−𝑣−𝑤_𝑛𝑘,𝑤_𝑛𝑘−𝑡_𝑛𝑘𝜆_𝑛𝑘. (3.53) This implies that ⟨𝑣−𝑝,𝑢⟩≥0 as 𝑘→∞. Since 𝑇 is maximal monotone, we have 𝑝∈𝑇⁻¹0 and hence 𝑝∈VI(𝐶,𝐵).
From (a), (b), and (c), we obtain that 𝑝∈Ω. This implies that limsup_𝑛→∞⟨𝛾𝑔−𝐴𝑞,𝑆𝑤_𝑛−𝑞⟩=lim_𝑘→∞⟨𝛾𝑔−𝐴𝑞,𝑆𝑤_𝑛𝑘−𝑞⟩=⟨𝛾𝑔−𝐴𝑞,𝑝−𝑞⟩≤0. (3.54)Step 7. We show that 𝑥_𝑛→𝑞. We observe that 𝑥_𝑛+1−𝑞²=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑃_𝐶(𝑞)²≤𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑞²≤𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞+𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑞²+2𝛼_𝑛𝐼−𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛−𝐴𝑞≤𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞+1−𝛼_𝑛𝛾²𝑆𝑤_𝑛−𝑞²+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛−𝐴𝑞−2𝛼²_𝑛𝐴𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²𝑤_𝑛−𝑞²+𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛−𝛾𝑔𝑞+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑞−𝐴𝑞−2𝛼²_𝑛𝐴𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²𝑡_𝑛−𝑞²+𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞+2𝛼_𝑛𝑆𝑤_𝑛−𝑞𝛾𝑔𝑥_𝑛−𝛾𝑔𝑞+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑞−𝐴𝑞−2𝛼²_𝑛𝐴𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞+2𝛾𝛼𝛼_𝑛𝑤_𝑛−𝑞𝑥_𝑛−𝑞+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑞−𝐴𝑞−2𝛼²_𝑛𝐴𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞²+2𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑞−𝐴𝑞−2𝛼²_𝑛𝐴𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑥_𝑛≤−𝐴𝑞1−𝛼_𝑛𝛾²+2𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+𝛼²_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞²+2𝛼_𝑛𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑞−𝐴𝑞+𝛼²_𝑛𝛾²𝑥_𝑛−𝑞²+2𝛼²_𝑛𝐴𝑆𝑤_𝑛−𝑞𝐴𝑞−𝛾𝑔𝑥_𝑛=1−2𝛼_𝑛𝛾−𝛾𝛼𝑥_𝑛−𝑞²+𝛼_𝑛2𝑆𝑤_𝑛−𝑞,𝛾𝑔𝑞−𝐴𝑞+𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞²+2𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑞𝐴𝑞−𝛾𝑔𝑥_𝑛+𝛼_𝑛𝛾²𝑥_𝑛−𝑞². (3.55) Since {𝑥_𝑛}, {𝑔(𝑥_𝑛)}, and ‖𝑆𝑤_𝑛‖ are bounded, we can take a constant 𝑀>0 such that 𝑀≥sup_𝑛≥0𝛼_𝑛𝛾𝑔𝑥_𝑛−𝐴𝑞²+2𝛼_𝑛𝐴𝑆𝑤_𝑛−𝑞𝐴𝑞−𝛾𝑔𝑥_𝑛+𝛼_𝑛𝛾²𝑥_𝑛−𝑞². (3.56) This implies that 𝑥_𝑛+1−𝑞²≤1−2𝛾−𝛾𝛼𝛼_𝑛𝑥_𝑛−𝑞²+𝛼_𝑛𝜎_𝑛, (3.57) where 𝜎_𝑛=2⟨𝑆𝑤_𝑛−𝑞,𝛾𝑔(𝑞)−𝐴𝑞⟩+𝑀𝛼_𝑛. From (3.40), we have limsup_𝑛→∞𝜎_𝑛≤0. Applying Lemma 2.7 to (3.57), we obtain that 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

If we put 𝛾=1 and 𝐴=𝐼 in Theorem 3.1, we immediately obtain the following corollary.

Corollary 3.2. Let 𝐻 be a real Hilbert space, and let 𝐶 be a closed convex subset of 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A5), let 𝐵∶𝐶→𝐻 be a 𝛽-inverse strongly monotone mapping, and let 𝑔∶𝐶→𝐶 be a contraction with coefficient 𝛼(𝛼∈(0,1)). Assume that 0<𝛾<𝛾/𝛼. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩EP(𝑓)∩VI(𝐶,𝐵)≠∅. Let the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} be generated by 𝑥₀𝑦=𝑥∈𝐶,_𝑛=argmin𝜆_𝑛𝑓𝑥_𝑛+1,𝑦2𝑦−𝑥_𝑛²,𝑡∶𝑦∈𝐶_𝑛=argmin𝜆_𝑛𝑓𝑦_𝑛+1,𝑦2𝑦−𝑥_𝑛²,𝑥∶𝑦∈𝐶_𝑛+1=𝛼_𝑛𝑔𝑥_𝑛+1−𝛼_𝑛SP_C𝑡_𝑛−𝛽𝐵𝑡_𝑛,𝑛≥0, (3.58) where {𝛼_𝑛}⊂(0,1), {𝛽}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,2𝛽), and {𝜆_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1/𝐿), where 𝐿=max{2𝑐₁,2𝑐₂}. Suppose that the following conditions are satisfied:(B1)lim_𝑛→∞𝛼_𝑛=0;(B2)∑^∞_𝑛=1𝛼_𝑛=∞;(B3)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(B4)∑^∞_𝑛=1|𝜆_𝑛+1−𝜆_𝑛|<∞. Then the following holds.(i)𝑃_Ω𝑔 is a contraction on 𝐶; and hence there exists 𝑞∈𝐶 such that 𝑞=𝑃_Ω𝑔(𝑞), where 𝑃_Ω is the metric projection of 𝐻 onto 𝐶.(ii)The sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} converge strongly to the same point 𝑞 which is the unique solution in the Ω to the following variational inequality:𝐼−𝑓𝑞,𝑥−𝑞≥0,∀𝑥∈Ω. (3.59)

If we put 𝑔≡𝑢 in the previous corollary, we get the following corollary.

Corollary 3.3. Let 𝐻 be a real Hilbert space, and let 𝐶 be a closed convex subset of 𝐻. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction satisfying (A1)–(A5), and let 𝐵∶𝐶→𝐻 be a 𝛽-inverse strongly monotone mapping. Assume that 0<𝛾<𝛾/𝛼. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑆)∩EP(𝑓)∩VI(𝐶,𝐵)≠∅. Let the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} be generated by 𝑥₀𝑦=𝑥∈𝐶,_𝑛=argmin𝜆_𝑛𝑓𝑥_𝑛+1,𝑦2𝑦−𝑥_𝑛²,𝑡∶𝑦∈𝐶_𝑛=argmin𝜆_𝑛𝑓𝑦_𝑛+1,𝑦2𝑦−𝑥_𝑛²,𝑥∶𝑦∈𝐶_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛SP_C𝑡_𝑛−𝛽𝐵𝑡_𝑛,𝑛≥0, (3.60) where {𝛼_𝑛}⊂(0,1), {𝛽}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,2𝛽), and {𝜆_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1/𝐿), where 𝐿=max{2𝑐₁,2𝑐₂}. Suppose that the following conditions are satisfied:(B1)lim_𝑛→∞𝛼_𝑛=0;(B2)∑^∞_𝑛=1𝛼_𝑛=∞;(B3)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(B4)∑^∞_𝑛=1|𝜆_𝑛+1−𝜆_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} converge strongly to the same point 𝑞, where 𝑞=𝑃_Ω𝑢, which is the unique solution in the Ω to the following variational inequality: 𝑞−𝑢,𝑥−𝑞≥0,∀𝑥∈Ω. (3.61)

4. Deduced Theorems

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 with inner product ⟨⋅,⋅⟩. Let 𝐹 be a nonlinear mapping from 𝐶 into 𝐻. Recall that the function 𝐹 is called(a) strongly monotone on 𝐶 if there exists 𝛽>0 such that𝐹(𝑥)−𝐹𝑦,𝑥−𝑦≥𝛽𝑥−𝑦²,∀𝑥,𝑦∈𝐶; (4.1)(b) monotone on 𝐶 if𝐹(𝑥)−𝐹𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝐶; (4.2)(c) pseudomonotone on 𝐶 if𝐹𝑦,𝑥−𝑦≥0⟹𝐹(𝑥),𝑥−𝑦≤0,∀𝑥,𝑦∈𝐶. (4.3)

Remark 4.1. Notice that if 𝐹 is 𝐿-Lipschitz on 𝐶, then for each 𝑥,𝑦∈𝐶,𝑓(𝑥,𝑦)=⟨𝐹(𝑥),𝑦−𝑥⟩ is Lipschitz-type continuous with constants 𝑐₁=𝑐₂=𝐿/2 on 𝐶. Indeed, 𝑓()(𝑥)𝐹𝑦−⟨𝐹(𝑥)⟩𝐹𝑦(𝑥)𝐹𝑦(𝑥𝐿𝑥,𝑦+𝑓𝑦,𝑧−𝑓𝑥,𝑧=⟨𝐹,𝑦−𝑥⟩+,𝑧−𝑦,𝑧−𝑥=−−𝐹,𝑦−𝑧≥−−𝐹)𝑦−𝑧≥−𝐿𝑥−𝑦𝑦−𝑧≥−2𝑥−𝑦²−𝐿2𝑦−𝑧²=−𝑐₁𝑥−𝑦²−𝑐₂𝑦−𝑧². (4.4) Thus 𝑓 is Lipschitz-type continuous on 𝐶.
Let 𝑓∶𝐶×𝐶→ℝ be defined by 𝑓(𝑥,𝑦)=⟨𝐹(𝑥),𝑦−𝑥⟩, where 𝐹∶𝐶→𝐻. Thus, by Algorithm  (1.15), we get the following: 𝑦_𝑛=argmin𝜆_𝑛𝑓𝑥_𝑛+1,𝑦2𝑦−𝑥_𝑛²𝜆𝐹𝑥∶𝑦∈𝐶=argmin_𝑛,𝑦−𝑥_𝑛+12𝑦−𝑥_𝑛²∶𝑦∈𝐶=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐹𝑥_𝑛. (4.5) Similarly, we also obtain that 𝑡_𝑛=𝑃_𝐶(𝑥_𝑛−𝜆_𝑛𝐹(𝑦_𝑛)). Applying Theorem 3.1, we obtain the convergence theorem for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set VI(𝐶,𝐵).

Corollary 4.2. Let 𝐻 be a real Hilbert space, and let 𝐶 be a closed convex subset of 𝐻. Let 𝐹∶𝐶→𝐻 be a monotone, 𝐿-Lipschitz continuous mapping, let 𝐵∶𝐶→𝐻 be a 𝛽-inverse strongly monotone mapping, also let 𝐴 be a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0 such that ‖𝐴‖=1, and let 𝑔∶𝐶→𝐶 be a contraction with coefficient 𝛼(𝛼∈(0,1)). Assume that 0<𝛾<𝛾/𝛼. Let 𝑆 be a nonexpansive mapping of 𝐶 into itself such that Ω=𝐹(𝑆)∩EP(𝑓)∩VI(𝐶,𝐵)≠∅. Let the sequence {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} be generated by 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐹𝑥_𝑛,𝑡_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐹𝑦_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝛾𝑔𝑥_𝑛+𝐼−𝛼_𝑛𝐴SP_C𝑡_𝑛−𝛽𝐵𝑡_𝑛,𝑛≥0, (4.6) where {𝛼_𝑛}⊂(0,1), {𝛽}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,2𝛽), and {𝜆_𝑛}⊂[𝑐,𝑑] for some 𝑐,𝑑∈(0,1/𝐿). Suppose that the following conditions are satisfied:(B1)lim_𝑛→∞𝛼_𝑛=0;(B2)∑^∞_𝑛=1𝛼_𝑛=∞;(B3)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(B4)∑^∞_𝑛=1|𝜆_𝑛+1−𝜆_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑡_𝑛} converge strongly to the same point 𝑞, where 𝑞=𝑃_Ω(𝐼−𝐴+𝛾𝑔)(𝑞).

Coflict of Interests

The authors declare that they have no conflict interests.

Authors’ Contribution

All authors read and approved the final paper.

Acknowledgments

The authors would like to thank 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. Finally, the first author is supported by the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.