About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 538912, 29 pages
http://dx.doi.org/10.1155/2012/538912
Research Article

Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 7 October 2011; Accepted 1 November 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Tanom Chamnarnpan and Poom Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of 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. Furthermore, we give a numerical example which supports our main theorem in the last part.

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 , where is the set of real numbers, 𝜑𝐶 be a real-valued function. Let Λ be arbitrary index set. The system of mixed equilibrium problem is for finding 𝑥𝐶 such that𝐹𝑘(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)0,𝑘Λ,𝑦𝐶.(1.1) The set of solutions of (1.1) is denoted by SMEP(𝐹𝑘), that is, 𝐹SMEP𝑘=𝑥𝐶=𝐹𝑘(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)0,𝑘Λ,𝑦𝐶.(1.2) If Λ is a singleton, then problem (1.1) becomes the following mixed equilibrium problem: finding 𝑥𝐶 such that𝐹(𝑥,𝑦)+𝜑(𝑦)𝜑(𝑥)0,𝑦𝐶.(1.3) The set of solutions of (1.3) is denoted by MEP(𝐹).

If 𝜑0, the problem (1.3) is reduced into the equilibrium problem [1] for finding 𝑥𝐶 such that𝐹(𝑥,𝑦)0,𝑦𝐶.(1.4) The set of solutions of (1.4) 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 system of mixed equilibrium problem and the equilibrium problem, please consult [219].

Recall that, a mapping 𝑆𝐶𝐶 is said to be nonexpansive if 𝑆𝑥𝑆𝑦𝑥𝑦,(1.5) for all 𝑥,𝑦𝐶. If 𝐶 is a bounded closed convex and 𝑆 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑆) is nonempty [20]. Let 𝐴𝐶𝐻 be a mapping, the Hartmann-Stampacchia variational inequality for finding 𝑥𝐶 such that𝐴𝑥,𝑦𝑥0,𝑦𝐶.(1.6) The set of solutions of (1.6) is denoted by VI(𝐶,𝐴). The variational inequality has been extensively studied in the literature [2128].

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:1𝜃(𝑥)=2𝐴𝑥,𝑥𝑥,𝑦,𝑥𝐹(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 [29].

We denote weak convergence and strong convergence by notations and , respectively. A mapping 𝐴 of 𝐶 into 𝐻 is called monotone if 𝐴𝑥𝐴𝑦,𝑥𝑦0,(1.8) for all 𝑥,𝑦𝐶. A mapping 𝐴 of 𝐶 into 𝐻 is called 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼 such that 𝐴𝑥𝐴𝑦,𝑥𝑦𝛼𝐴𝑥𝐴𝑦2,(1.9) for all 𝑥,𝑦𝐶. It is obvious that any 𝛼-inverse-strongly monotone mappings 𝐴 are monotone and Lipschitz continuous mapping. A linear bounded operator 𝐴 is strongly positive if there exists a constant 𝛾>0 with the property 𝐴𝑥,𝑥𝛾𝑥2,(1.10) for all 𝑥𝐻. A self-mapping 𝑓𝐶𝐶 is a contraction on 𝐶 if there exists a constant 𝛼(0,1) such that 𝑓(𝑥)𝑓(𝑦)𝛼𝑥𝑦,(1.11) for all 𝑥,𝑦𝐶. We use Π𝐶 to denote the collection of all contraction on 𝐶. Note that each 𝑓Π𝐶 has a unique fixed point in 𝐶.

Let 𝐵𝐻𝐻 be a single-valued nonlinear mapping and 𝑀𝐻2𝐻 be a set-valued mapping. The variational inclusion problem is to find 𝑥𝐻 such that𝜃𝐵(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, [3032] and the reference therein.

A set-valued mapping 𝑀𝐻2𝐻 is called monotone if for all 𝑥,𝑦𝐻, 𝑓𝑀(𝑥), and 𝑔𝑀(𝑦) impling 𝑥𝑦,𝑓𝑔0. A monotone mapping 𝑀 is maximal if its graph 𝐺(𝑀)={(𝑓,𝑥)𝐻×𝐻𝑓𝑀(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,𝑥𝑦,𝑓𝑔0 for all (𝑦,𝑔)𝐺(𝑀) impling 𝑓𝑀(𝑥).

Let 𝐵 be an inverse-strongly monotone mapping of 𝐶 into 𝐻, and let 𝑁𝐶𝑣 be normal cone to 𝐶 at 𝑣𝐶, that is, 𝑁𝐶𝑣={𝑤𝐻𝑣𝑢,𝑤0,forall𝑢𝐶}, and define𝑇𝑣=𝐵𝑣+𝑁𝐶𝑣,if𝑣𝐶,,if𝑣𝐶.(1.13) Then, 𝑇 is a maximal monotone and 𝜃𝑇𝑣 if and only if 𝑣VI(𝐶,𝐵) (see [33]).

Let 𝑀𝐻2𝐻 be a set-valued maximal monotone mapping, then the single-valued mapping 𝐽𝑀,𝜆𝐻𝐻 defined by𝐽𝑀,𝜆(𝑥)=(𝐼+𝜆𝑀)1(𝑥),𝑥𝐻,(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 𝐽𝑀,𝜆(𝐼𝜆𝐵) for all 𝜆>0, (for more details see [34]).

In 2000, Moudafi [35] introduced the viscosity approximation method for nonexpansive mappings and proved that if 𝐻 is a real Hilbert space, the sequence {𝑥𝑛} defined by the iterative method below, with the initial guess 𝑥0𝐶 is chosen arbitrarily,𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑆𝑥𝑛,𝑛0,(1.15) where {𝛼𝑛}(0,1) satisfies certain conditions and converges strongly to a fixed point of 𝑆 (say 𝑥𝐶), which is then a unique solution of the following variational inequality:(𝐼𝑓)𝑥,𝑥𝑥0,𝑥𝐹(𝑆).(1.16)

In 2006, Marino and Xu [29] introduced a general iterative method for nonexpansive mapping. They defined the sequence {𝑥𝑛} generated by the algorithm 𝑥0𝐶,𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑆𝑥𝑛,𝑛0,(1.17) where {𝛼𝑛}(0,1), and 𝐴 is a strongly positive linear bounded operator. They proved that if 𝐶=𝐻, and the sequence {𝛼𝑛} satisfies appropriate conditions, then the sequence {𝑥𝑛} generated by (1.17) converges strongly to a fixed point of 𝑆 (say 𝑥𝐻) which is the unique solution of the following variational inequality:(𝐴𝛾𝑓)𝑥,𝑥𝑥0,𝑥𝐹(𝑆),(1.18) which is the optimality condition for the minimization problemmin𝑥𝐹(𝑆)EP(𝐹)12𝐴𝑥,𝑥(𝑥),(1.19) where is a potential function for 𝛾𝑓 (i.e., (𝑥)=𝛾𝑓(𝑥) for 𝑥𝐻).

For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities. Let 𝑃𝐶 be the projection of 𝐻 onto 𝐶. In 2005, Iiduka and Takahashi [36] introduced the following iterative process for 𝑥0𝐶,𝑥𝑛+1=𝛼𝑛𝑢+1𝛼𝑛𝑆𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑛0,(1.20) where 𝑢𝐶, {𝛼𝑛}(0,1), and {𝜆𝑛}[𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<2𝛽. They proved that under certain appropriate conditions imposed on {𝛼𝑛} and {𝜆𝑛}, the sequence {𝑥𝑛} generated by (1.20) converges strongly to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say 𝑥𝐶) which solve some variational inequality𝑥𝑢,𝑥𝑥0,𝑥𝐹(𝑆)VI(𝐶,A).(1.21)

In 2008, Su et al. [37] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: 𝑥1,𝑢𝑛𝐻𝐹𝑢𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝑥0,𝑦𝐶,𝑛+1=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑆𝑃𝐶𝑢𝑛𝜆𝑛𝐴𝑢𝑛,(1.22) for all 𝑛, where {𝛼𝑛}[0,1) and {𝑟𝑛}(0,) satisfing some appropriate conditions. Furthermore, they proved that {𝑥𝑛} and {𝑢𝑛} converge strongly to the same point 𝑧𝐹(𝑆)VI(𝐶,𝐴)EP(𝐹), where 𝑧=𝑃𝐹(𝑆)VI(𝐶,𝐴)EP(𝐹)𝑓(𝑧).

Let {𝑇𝑖} be an infinite family of nonexpansive mappings of 𝐻 into itself, and let {𝜆𝑖} be a real sequence such that 0𝜆𝑖1 for every 𝑖𝑁. For 𝑛1, we defined a mapping 𝑊𝑛 of 𝐻 into itself as follows:𝑈𝑛,𝑛+1𝑈=𝐼,𝑛,𝑛=𝜆𝑛𝑇𝑛𝑈𝑛,𝑛+1+1𝜆𝑛𝑈𝐼,𝑛,𝑘=𝜆𝑘𝑇𝑘𝑈𝑛,𝑘+1+1𝜆𝑘𝑈𝐼,𝑛,2=𝜆2𝑇2𝑈𝑛,3+1𝜆2𝑊𝐼,𝑛=𝑈𝑛,1=𝜆1𝑇1𝑈𝑛,2+1𝜆1𝐼.(1.23)

In 2011, He et al. [38] introduced the following iterative process for {𝑇𝑛𝐶𝐶} which is a sequence of nonexpansive mappings. Let {𝑧𝑛} be the sequence defined by𝑧𝑛+1=𝜖𝑛𝑧𝛾𝑓𝑛+𝐼𝜖𝑛𝑊𝑛𝐾1𝑟1,𝑛𝐾2𝑟2,𝑛𝐾𝐾𝑟𝐾,𝑛𝑧𝑛,𝑛𝑁.(1.24) The sequence {𝑧𝑛} defined by (1.24) converges strongly to a common element of the set of fixed points of nonexpansive mappings, the set of solutions of the variational inequality, and the generalized equilibrium problem. Recently, Jitpeera and Kumam [39] introduced the following 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.

In this paper, we modify the iterative methods (1.17), (1.22), and (1.24) by purposing the following new general viscosity iterative method: 𝑥0,𝑢𝑛𝐶,𝑢𝑛=𝐾𝐹𝑁𝑟𝑛,𝑛𝐾𝐹𝑁1𝑟𝑛1,𝑛𝐾𝐹𝑁2𝑟𝑛2,𝑛𝐾𝐹2𝑟2,𝑛𝐾𝐹1𝑟1,𝑛𝑥𝑛𝑥,𝑛𝑁𝑛+1=𝑃𝐶𝜖𝑛𝑥𝛾𝑓𝑛+𝐼𝜖𝑛𝐴𝑊𝑛𝐽𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛,(1.25) for all 𝑛, where {𝛼𝑛}(0,1), {𝑟𝑛}(0,2𝜎), and 𝜆(0,2𝛽) satisfy some appropriate conditions. The purpose of this paper shows that under some control conditions the sequence {𝑥𝑛} converges strongly to a common element of the set of common fixed points of nonexpansive mappings, the solution of the system of mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space. Moreover, we apply our results to the class of strictly pseudocontractive mappings. Finally, we give a numerical example which supports our main theorem in the last part. Our results improve and extend the corresponding results of Marino and Xu [29], Su et al. [37], He et al. [38], and some authors.

2. Preliminaries

Let 𝐻 be a real Hilbert space and 𝐶 be a nonempty closed and convex subset of 𝐻. Recall that the (nearest point) projection 𝑃𝐶 from 𝐻 onto 𝐶 assigns to each 𝑥𝐻 and the unique point in 𝑃𝐶𝑥𝐶 satisfies the property 𝑥𝑃𝐶𝑥=min𝑦𝐶𝑥𝑦,(2.1) which is equivalent to the following inequality𝑥𝑃𝐶𝑥,𝑃𝐶𝑥𝑦0,𝑦𝐶.(2.2) The following characterizes the projection 𝑃𝐶. We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1. The function 𝑢𝐶 is a solution of the variational inequality if and only if 𝑢𝐶 satisfies the relation 𝑢=𝑃𝐶(𝑢𝜆𝐵𝑢) for all 𝜆>0.

Lemma 2.2. For a given 𝑧𝐻, 𝑢𝐶, 𝑢=𝑃𝐶𝑧𝑢𝑧,𝑣𝑢0,𝑣𝐶.
It is well known that 𝑃𝐶 is a firmly nonexpansive mapping of 𝐻 onto 𝐶 and satisfies 𝑃𝐶𝑥𝑃𝐶𝑦2𝑃𝐶𝑥𝑃𝐶𝑦,𝑥𝑦,𝑥,𝑦𝐻.(2.3) Moreover, 𝑃𝐶𝑥 is characterized by the following properties: 𝑃𝐶𝑥𝐶 and for all 𝑥𝐻,𝑦𝐶, 𝑥𝑃𝐶𝑥,𝑦𝑃𝐶𝑥0.(2.4)

Lemma 2.3 (see [40]). Let 𝑀𝐻2𝐻 be a maximal monotone mapping, and let 𝐵𝐻𝐻 be a monotone and Lipshitz continuous mapping. Then the mapping 𝐿=𝑀+𝐵𝐻2𝐻 is a maximal monotone mapping.

Lemma 2.4 (see [41]). Each Hilbert space 𝐻 satisfies Opial's condition, that is, for any sequence {𝑥𝑛}𝐻 with 𝑥𝑛𝑥, the inequality liminf𝑛𝑥𝑛𝑥<liminf𝑛𝑥𝑛𝑦, hold for each 𝑦𝐻 with 𝑦𝑥.

Lemma 2.5 (see [42]). Assume {𝑎𝑛} is a sequence of nonnegative real numbers such that 𝑎𝑛+11𝛾𝑛𝑎𝑛+𝛿𝑛,𝑛0,(2.5) where {𝛾𝑛}(0,1) and {𝛿𝑛} is a sequence in such that(i)𝑛=1𝛾𝑛=, (ii)limsup𝑛𝛿𝑛/𝛾𝑛0 or 𝑛=1|𝛿𝑛|<.Then lim𝑛𝑎𝑛=0.

Lemma 2.6 (see [43]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻, and let 𝑇𝐶𝐶 be a nonexpansive mapping. Then 𝐼𝑇 is demiclosed at zero, that is, 𝑥𝑛𝑥,𝑥𝑛𝑇𝑥𝑛0,(2.6) implying 𝑥=𝑇𝑥.

For solving the mixed equilibrium problem, let us assume that the bifunction 𝐹𝐶×𝐶 and the nonlinear mapping 𝜑𝐶 satisfy the following conditions: (A1)𝐹(𝑥,𝑥)=0 for all 𝑥𝐶; (A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)0 for any 𝑥,𝑦𝐶; (A3)for each fixed 𝑦𝐶, 𝑥𝐹(𝑥,𝑦) is weakly upper semicontinuous; (A4)for each fixed 𝑥𝐶, 𝑦𝐹(𝑥,𝑦) is convex and lower semicontinuous; (B1)for each 𝑥𝐶 and 𝑟>0, there exist a bounded subset 𝐷𝑥𝐶 and 𝑦𝑥𝐶 such that for any 𝑧𝐶𝐷𝑥,𝐹𝑧,𝑦𝑥𝑦+𝜑𝑥1𝜑(𝑧)+𝑟𝑦𝑥𝑧,𝑧𝑥<0,(2.7)(B2)𝐶 is a bounded set.

Lemma 2.7 (see [44]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐹𝐶×𝐶 be a bifunction mapping satisfying (A1)–(A4), and let 𝜑𝐶 be a convex and lower semicontinuous function such that 𝐶dom𝜑. Assume that either (B1) or (B2) holds. For 𝑟>0 and 𝑥𝐻, then there exists 𝑢𝐶 such that 1𝐹(𝑢,𝑦)+𝜑(𝑦)𝜑(𝑢)+𝑟𝑦𝑢,𝑢𝑥0.(2.8) Define a mapping 𝐾𝑟𝐻𝐶 as follows: 𝐾𝑟1(𝑥)=𝑢𝐶𝐹(𝑢,𝑦)+𝜑(𝑦)𝜑(𝑢)+𝑟,𝑦𝑢,𝑢𝑥0,𝑦𝐶(2.9) for all 𝑥𝐻. Then, the following hold: (i)𝐾𝑟 is single-valued; (ii)𝐾𝑟 is firmly nonexpansive, that is, for any 𝑥,𝑦𝐻, 𝐾𝑟𝑥𝐾𝑟𝑦2𝐾𝑟𝑥𝐾𝑟𝑦,𝑥𝑦;(iii)𝐹(𝐾𝑟)=MEP(𝐹); (iv)MEP(𝐹) is closed and convex.

Lemma 2.8 (see [29]). Assume 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌𝐴1, then 𝐼𝜌𝐴1𝜌𝛾.

Lemma 2.9 (see [38]). Let C be a nonempty closed and convex subset of a strictly convex Banach space. Let {𝑇𝑖}𝑖𝑁 be an infinite family of nonexpansive mappings of C into itself such that 𝑖𝑁𝐹(𝑇𝑖), and let {𝜆𝑖} be a real sequence such that 0𝜆𝑖b<1 for every 𝑖𝑁. Then F(W)=𝑖𝑁𝐹(𝑇𝑖).

Lemma 2.10 (see [38]). Let C be a nonempty closed and convex subset of a strictly convex Banach space. Let {𝑇𝑖} be an infinite family of nonexpansive mappings of C into itself, and let {𝜆𝑖} be a real sequence such that 0𝜆𝑖b<1 for every 𝑖𝑁. Then, for every 𝑥𝐶 and 𝑘𝑁, the limit lim𝑛𝑈𝑛,𝑘 exist.
In view of the previous lemma, we define 𝑊𝑥=lim𝑛𝑈𝑛,1𝑥=lim𝑛𝑊𝑛𝑥.(2.10)

3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the common fixed points, the common solution of a system of mixed equilibrium problems and variational inclusion of inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty close and convex subset of 𝐻, and let 𝐵 be a 𝛽-inverse-strongly monotone mapping. Let 𝜑𝐶𝑅 be a convex and lower semicontinuous function, 𝑓𝐶𝐶 a contraction mapping with coefficient 𝛼(0<𝛼<1), and 𝑀𝐻2𝐻 a maximal monotone mapping. Let 𝐴 be a strongly positive linear bounded operator of 𝐻 into itself with coefficient 𝛾>0. Assume that 0<𝛾<𝛾/𝛼 and 𝜆(0,2𝛽). Let {𝑇𝑛} be a family of nonexpansive mappings of 𝐻 into itself such that 𝜃=𝑛=1𝐹𝑇𝑛𝑁𝑘=1𝐹SMEP𝑘𝐼(𝐵,𝑀).(3.1) Suppose that {𝑥𝑛} is a sequence generated by the following algorithm for 𝑥0𝐶 arbitrarily and 𝑢𝑛=𝐾𝐹𝑁𝑟𝑛,𝑛𝐾𝐹𝑁1𝑟𝑛1,𝑛𝐾𝐹𝑁2𝑟𝑛2,𝑛𝐾𝐹2𝑟2,𝑛𝐾𝐹1𝑟1,𝑛𝑥𝑛𝑥,𝑛𝑁𝑛+1=𝑃𝐶𝜖𝑛𝑥𝛾𝑓𝑛+𝐼𝜖𝑛𝐴𝑊𝑛𝐽𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛,(3.2) for all 𝑛=1,2,3,, where K𝐹𝑖𝑟𝑖,𝑛𝑢(𝑥)=𝑛𝐶𝐹𝑖𝑢𝑛𝑢,𝑦+𝜑(𝑦)𝜑𝑛+1𝑟𝑖,𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛,0,𝑦𝐶𝑖=1,2,3,,𝑁,(3.3) and the following conditions are satisfied(C1):{𝜖𝑛}(0,1),lim𝑛0𝜖𝑛=0,𝑛=1𝜖𝑛=,𝑛=1|𝜖𝑛+1𝜖𝑛|<; (C2):{𝑟𝑛}[𝑐,𝑑] with 𝑐,𝑑(0,2𝜎) and 𝑛=1|𝑟𝑛+1𝑟𝑛|<.
Then, the sequence {𝑥𝑛} converges strongly to 𝑞𝜃, where 𝑞=𝑃𝜃(𝛾𝑓+𝐼𝐴)(𝑞) which solves the following variational inequality: (𝛾𝑓𝐴)𝑞,𝑝𝑞0,𝑝𝜃,(3.4) which is the optimality condition for the minimization problem min𝑞𝜃12𝐴𝑞,𝑞(𝑞),(3.5) where h is a potential function for 𝛾𝑓 (i.e., (𝑞)=𝛾𝑓(𝑞) for 𝑞𝐻).

Proof. For condition (C1), we may assume without loss of generality, and 𝜖𝑛(0,𝐴1) for all 𝑛. By Lemma 2.8, we have 𝐼𝜖𝑛𝐴1𝜖𝑛𝛾. Next, we will assume that 𝐼𝐴1𝛾.
Next, we will divide the proof into six steps.
Step 1. First, we will show that {𝑥𝑛} and {𝑢𝑛} are bounded. Since 𝐵 is 𝛽-inverse-strongly monotone mappings, we have (𝐼𝜆𝐵)𝑥(𝐼𝜆𝐵)𝑦2=𝐼𝑥𝜆𝐵𝑥𝐼𝑦+𝜆𝐵𝑦2=𝑥𝑦𝜆𝐵𝑥+𝜆𝐵𝑦2=(𝑥𝑦)𝜆(𝐵𝑥+𝐵𝑦)2𝑥𝑦22𝜆𝑥𝑦𝐵𝑥+𝐵𝑦+𝜆2𝐵𝑥𝐵𝑦2𝑥𝑦22𝜆𝛽𝐵𝑥+𝐵𝑦2+𝜆2𝐵𝑥𝐵𝑦2𝑥𝑦2+𝜆(𝜆2𝛽)𝐵𝑥+𝐵𝑦2,(3.6) if 0<𝜆<2𝛽, then 𝐼𝜆𝐵 is nonexpansive.
Put 𝑦𝑛=𝐽𝑀,𝜆(𝑢𝑛𝜆𝐵𝑢𝑛),𝑛0. Since 𝐽𝑀,𝜆 and 𝐼𝜆𝐵 are nonexpansive mapping, it follows that 𝑦𝑛=𝐽𝑞𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛𝐽𝑀,𝜆𝑢(𝑞𝜆𝐵𝑞)𝑛𝜆𝐵𝑢𝑛𝑢(𝑞𝜆𝐵𝑞)𝑛.𝑞(3.7) By Lemma 2.7, we have 𝑢𝑛=𝐾𝐹𝑁𝑟𝑛,𝑛𝐾𝐹𝑁1𝑟𝑛1,𝑛𝐾𝐹𝑁2𝑟𝑛2,𝑛𝐾𝐹2𝑟2,𝑛𝐾𝐹1𝑟1,𝑛𝑥𝑛𝜏,for𝑛0𝑘𝑛=𝐾𝐹𝑘𝑟𝑘,𝑛𝐾𝐹𝑘1𝑟𝑘1,𝑛𝐾𝐹2𝑟2,𝑛𝐾𝐹1𝑟1,𝑛,for𝑘{0,1,2,,𝑁},(3.8) and 𝜏0𝑛=𝐼forall𝑛𝑁,𝑞=𝜏𝐹𝑘𝑟𝑘,𝑛𝑞,𝑢𝑛=𝜏𝑁𝑟𝑘,𝑁𝑥𝑛 Then, we have 𝑢𝑛𝑞2=𝜏𝑁𝑟𝑘,𝑛𝑥𝑛𝜏𝐹𝑘𝑟𝑘,𝑛𝑞2=𝑥𝑛𝑞2.(3.9) Hence, we get 𝑦𝑛𝑥𝑞𝑛.𝑞(3.10) From (3.2), we deduce that 𝑥𝑛+1=𝑃𝑞𝐶𝜖𝑛𝑥𝛾𝑓𝑛+𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑃𝐶𝑞𝜖𝑛𝑥𝛾𝑓𝑛+𝐴𝑞𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑞𝜖𝑛𝑥𝛾𝑓𝑛+𝐴𝑞1𝜖𝑛𝛾𝑦𝑛𝑞𝜖𝛾𝜖𝑛𝑥𝑛𝑞+𝜖𝑛(+𝛾𝑓𝑔)𝐴𝑞1𝜖𝑛𝛾𝑥𝑛=𝑞1𝜖𝛾𝛾𝜖𝑛𝑥𝑛𝑞𝜖𝑛=𝛾𝑓(𝑞)𝐴𝑞1𝜖𝛾𝛾𝜖𝑛𝑥𝑛+𝑞𝜖𝛾𝛾𝜖𝑛𝛾𝑓(𝑞)𝐴𝑞𝑥𝛾𝛾𝜖max𝑛,𝑞𝛾𝑓(𝑞)𝐴𝑞.𝛾𝛾𝜖(3.11) It follows by induction that 𝑥𝑛𝑥𝑞max0,(𝑞𝛾𝑓𝑞)𝐴𝑞𝛾𝛾𝜖,𝑛0.(3.12) Therefore {𝑥𝑛} is bounded, so are {𝑦𝑛},{𝐵𝑢𝑛},{𝑓(𝑥𝑛)}, and {𝐴𝑊𝑛𝑦𝑛}.
Step 2. We claim that lim𝑛𝑥𝑛+1𝑥𝑛=0 and lim𝑛𝑦𝑛+1𝑦𝑛=0. From (3.2), we have 𝑥𝑛+1𝑥𝑛=𝑃𝐶𝜖𝑛𝑥𝛾𝑓𝑛+𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑃𝐶𝜖𝑛1𝑥𝛾𝑓𝑛1+𝐼𝜖𝑛1𝐴𝑊𝑛𝑦𝑛1𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑊𝑛𝑦𝑛1𝜖𝑛𝜖𝑛1𝐴𝑊𝑛𝑦𝑛1+𝛾𝜖𝑛𝑓𝑥𝑛𝑥𝑓𝑛1𝜖+𝛾𝑛𝜖𝑛1𝑓𝑥𝑛11𝜖𝑛𝛾𝑦𝑛𝑦𝑛1+||𝜖𝑛𝜖𝑛1||𝐴𝑊𝑛𝑦𝑛+𝛾𝜖𝜖𝑛𝑥𝑛𝑥𝑛1||𝜖+𝛾𝑛𝜖𝑛1||𝑓𝑥𝑛1.(3.13) Since 𝐽𝑀,𝜆 and 𝐼𝜆𝐵 are nonexpansive, we also have 𝑦𝑛𝑦𝑛1=𝐽𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛𝐽𝑀,𝜆𝑢𝑛1𝜆𝐵𝑢𝑛1𝑢𝑛𝜆𝐵𝑢𝑛𝑢𝑛1𝜆𝐵𝑢𝑛1𝑢𝑛𝑢𝑛1.(3.14) On the other hand, from 𝑢𝑛1=𝜏𝑁𝑟𝑘,𝑛1𝑥𝑛1 and 𝑢𝑛=𝜏𝑁𝑟𝑘,𝑛𝑥𝑛, it follows that 𝐹𝑢𝑛1𝑢,𝑦+𝜑(𝑦)𝜑𝑛1+1𝑟𝑛1𝑦𝑢𝑛1,𝑢𝑛1𝑥𝑛1𝐹𝑢0,𝑦𝐶,(3.15)𝑛𝑢,𝑦+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛0,𝑦𝐶.(3.16) Substituting 𝑦=𝑢𝑛 into (3.15) and 𝑦=𝑢𝑛1 into (3.16), we get 𝐹𝑢𝑛1,𝑢𝑛𝑢+𝜑𝑛𝑢𝜑𝑛1+1𝑟𝑛1𝑢𝑛𝑢𝑛1,𝑢n1𝑥𝑛1𝐹𝑢0,𝑛,𝑢𝑛+1𝑢+𝜑𝑛+1𝑢𝜑𝑛+1𝑟𝑛𝑢𝑛+1𝑢𝑛,𝑢𝑛𝑥𝑛0.(3.17) From (A2), we obtain 𝑢𝑛𝑢𝑛1,𝑢𝑛1𝑥𝑛1𝑟𝑛1𝑢𝑛𝑥𝑛𝑟𝑛𝑢0,𝑛𝑢𝑛1,𝑢𝑛1𝑥𝑛1𝑟𝑛1𝑟𝑛𝑢𝑛𝑥𝑛0,(3.18) so, 𝑢𝑛𝑢𝑛1,𝑢𝑛1𝑢𝑛+𝑢𝑛𝑥𝑛1𝑟𝑛1𝑟𝑛𝑢𝑛𝑥𝑛0.(3.19) It follows that 𝑢𝑛𝑢𝑛1,𝑢𝑛1𝑢𝑛+𝑢𝑛𝑥𝑛𝑟𝑛1𝑟𝑛𝑢𝑛𝑥𝑛0,𝑢𝑛𝑢𝑛1,𝑢𝑛1𝑢𝑛𝑢+𝑛𝑢𝑛1,𝑟1𝑛1𝑟𝑛𝑢𝑛𝑥𝑛0.(3.20) Without loss of generality, let us assume that there exists a real number 𝑐 such that 𝑟𝑛1>𝑐>0, for all 𝑛. Then, we have 𝑢𝑛𝑢𝑛12𝑢𝑛𝑢𝑛1,𝑟1𝑛1𝑟𝑛𝑢𝑛𝑥𝑛𝑢𝑛𝑢𝑛1||||𝑟1𝑛1𝑟𝑛||||𝑢𝑛𝑥𝑛,(3.21) and hence 𝑢𝑛𝑢𝑛1𝑥𝑛𝑥𝑛1+1𝑟𝑛||𝑟𝑛𝑟𝑛1||𝑢𝑛𝑥𝑛𝑥𝑛𝑥𝑛1+𝑀1𝑐||𝑟𝑛𝑟𝑛1||,(3.22) where 𝑀1=sup{𝑢𝑛𝑥𝑛𝑛}. Substituting (3.22) into (3.14), we have 𝑦𝑛𝑦𝑛1𝑥𝑛𝑥𝑛1+𝑀1𝑐||𝑟𝑛𝑟𝑛1||.(3.23) Substituting (3.23) into (3.13), we get 𝑥𝑛+1𝑥𝑛1𝜖𝑛𝛾𝑥𝑛𝑥𝑛1+𝑀1𝑐||𝑟𝑛𝑟𝑛1||+||𝜖𝑛𝜖𝑛1||𝐴𝑊𝑛𝑦𝑛1+𝛾𝜖𝜖𝑛𝑥𝑛𝑥𝑛1||𝜖+𝛾𝑛𝜖𝑛1||𝑓𝑥𝑛1=1𝜖𝑛𝛾𝑥𝑛𝑥𝑛1+1𝜖𝑛𝛾𝑀1𝑐||𝑟𝑛𝑟𝑛1||+||𝜖𝑛𝜖𝑛1||𝐴𝑊𝑛𝑦𝑛1+𝛾𝜖𝜖𝑛𝑥𝑛𝑥𝑛1||𝜖+𝛾𝑛𝜖𝑛1||𝑓𝑥n11𝜖𝛾𝛾𝜖𝑛𝑥𝑛𝑥𝑛1+𝑀1𝑐||𝑟𝑛𝑟𝑛1||+||𝜖𝑛𝜖𝑛1||𝐴𝑊𝑛𝑦𝑛1||𝜖+𝛾𝑛𝜖𝑛1||𝑓𝑥𝑛11𝜖𝛾𝛾𝜖𝑛𝑥𝑛𝑥𝑛1+𝑀1𝑐||𝑟𝑛𝑟𝑛1||+𝑀2||𝜖𝑛𝜖𝑛1||,(3.24) where 𝑀2=sup{max{𝐴𝑊𝑛𝑦𝑛1,𝑓(𝑥𝑛1)𝑛}}. Since conditions (C1)-(C2) and by Lemma 2.5, we have 𝑥𝑛+1𝑥𝑛0 as 𝑛. From (3.23), we also have 𝑦𝑛+1𝑦𝑛0 as 𝑛.Step 3. Next, we show that lim𝑛𝐵𝑢𝑛𝐵𝑞=0.
For 𝑞𝜃 hence 𝑞=𝐽𝑀,𝜆(𝑞𝜆𝐵𝑞). By (3.6) and (3.9), we get 𝑦𝑛𝑞2=𝐽𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛𝐽𝑀,𝜆(𝑞𝜆𝐵𝑞)2𝑢𝑛𝜆𝐵𝑢𝑛(𝑞𝜆𝐵𝑞)2𝑢𝑛𝑞2+𝜆(𝜆2𝛽)𝐵𝑢𝑛𝐵𝑞2𝑥𝑛𝑞2+𝜆(𝜆2𝛽)𝐵𝑢𝑛𝐵𝑞2.(3.25) It follows that 𝑥𝑛+1𝑞2=𝑃𝐶𝜖𝑛𝑥𝛾𝑓𝑛+𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑃𝐶(𝑞)2𝜖𝑛𝑥𝛾𝑓𝑛+𝐴𝑞𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑞2𝜖𝑛𝑥𝛾𝑓𝑛+𝐴𝑞1𝜖𝑛𝛾𝑦𝑛𝑞2𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+1𝜖𝑛𝛾𝑦𝑛𝑞2+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛𝑞𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛+𝑞1𝜖𝑛𝛾𝑥𝑛𝑞2+𝜆(𝜆2𝛽)𝐵𝑢𝑛𝐵𝑞2𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛+𝑥𝑞𝑛𝑞2+1𝜖𝑛𝛾𝜆(𝜆2𝛽)𝐵𝑢𝑛𝐵𝑞2.(3.26) So, we obtain 1𝜖𝑛𝛾𝜆(2𝛽𝜆)𝐵𝑢𝑛𝐵𝑞2𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+𝑥𝑛𝑥𝑛+1𝑥𝑛+𝑥𝑞𝑛+1𝑞+𝜉𝑛,(3.27) where 𝜉𝑛=2𝜖𝑛(1𝜖𝑛𝛾)𝛾𝑓(𝑥𝑛)𝐴𝑞𝑦𝑛𝑞. By conditions (C1), (C3) and lim𝑛𝑥𝑛+1𝑥𝑛=0, then, we obtain that 𝐵𝑢𝑛𝐵𝑞0 as 𝑛.
Step 4. We show the following: (i)lim𝑛𝑥𝑛𝑢𝑛=0; (ii)lim𝑛𝑢𝑛𝑦𝑛=0; (iii)lim𝑛𝑦𝑛𝑊𝑛𝑦𝑛=0. Since 𝐾𝑟𝑛(𝑥) is firmly nonexpansive and (2.3), we observe that 𝑢𝑛𝑞2=𝜏𝑁𝑟𝑛,𝑛𝑥𝑛𝜏𝑁𝑟𝑛,𝑛𝑞2𝑥𝑛𝑞,𝑢𝑛=1𝑞2𝑥𝑛𝑞2+𝑢𝑛𝑞2𝑥𝑛𝑞𝑢𝑛𝑞212𝑥𝑛𝑞2+𝑢𝑛𝑞2𝑥𝑛𝑢𝑛2,(3.28) it follows that 𝑢𝑛𝑞2𝑥𝑛𝑞2𝑥𝑛𝑢𝑛2.(3.29)
Since 𝐽𝑀,𝜆 is 1-inverse-strongly monotone and by (2.3), we compute 𝑦𝑛𝑞2=𝐽𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛𝐽𝑀,𝜆(𝑞𝜆𝐵𝑞)2𝑢𝑛𝜆𝐵𝑢𝑛(𝑞𝜆𝐵𝑞),𝑦𝑛=1𝑞2𝑢𝑛𝜆𝐵𝑢𝑛(𝑞𝜆𝐵𝑞)2+𝑦𝑛𝑞2𝑢𝑛𝜆𝐵𝑢𝑛𝑦(𝑞𝜆𝐵𝑞)𝑛𝑞212𝑢𝑛𝑞2+𝑦𝑛𝑞2𝑢𝑛𝑦𝑛𝜆𝐵𝑢𝑛𝐵𝑞2=12𝑢𝑛𝑞2+𝑦𝑛𝑞2𝑢𝑛𝑦𝑛2+2𝜆𝑢𝑛𝑦𝑛,𝐵𝑢𝑛𝐵𝑞𝜆2𝐵𝑢𝑛𝐵𝑞2,(3.30) which implies that 𝑦𝑛𝑞2𝑢𝑛𝑞2𝑢𝑛𝑦𝑛2𝑢+2𝜆𝑛𝑦𝑛𝐵𝑢𝑛.𝐵𝑞(3.31) Substituting (3.31) into (3.26), we have 𝑥𝑛+1𝑞2𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+𝑦𝑛𝑞2+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛𝑞𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+𝑢𝑛𝑞2𝑢𝑛𝑦𝑛2+2𝜆𝑛𝑢𝑛𝑦𝑛𝐵𝑢𝑛𝐵𝑞+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛.𝑞(3.32) Then, we derive 𝑥𝑛𝑢𝑛2+𝑢𝑛𝑦𝑛2𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+𝑥𝑛𝑞2𝑥𝑛+1𝑞2𝑢+2𝜆𝑛𝑦𝑛𝐵𝑢𝑛𝐵𝑞+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛.𝑞=𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+𝑥𝑛𝑥𝑛+1𝑥𝑛+𝑥𝑞𝑛+1𝑢𝑞+2𝜆𝑛𝑦𝑛𝐵𝑢𝑛𝐵𝑞+2𝜖𝑛1𝜖𝑛𝛾𝑥𝛾𝑓𝑛𝑦𝐴𝑞𝑛.𝑞(3.33) By condition (C1), lim𝑛𝑥𝑛𝑥𝑛+1=0 and lim𝑛𝐵𝑢𝑛𝐵𝑞=0.
So, we have 𝑥𝑛𝑢𝑛0,𝑢𝑛𝑦𝑛0 as 𝑛. It follows that 𝑥𝑛𝑦𝑛𝑥𝑛𝑢𝑛+𝑢𝑛𝑦𝑛0,as𝑛.(3.34) From (3.2), we have 𝑥𝑛𝑊𝑛𝑦𝑛𝑥𝑛𝑊𝑛𝑦𝑛1+𝑊𝑛𝑦𝑛1𝑊𝑛𝑦𝑛𝑃𝐶𝜖𝑛1𝑥𝛾𝑓𝑛1+𝐼𝛼𝑛1𝐴𝑊𝑛𝑦𝑛1𝑃𝐶𝑊𝑛𝑦𝑛1+𝑦𝑛1𝑦𝑛𝜖𝑛1𝛾𝑓𝑥𝑛1𝐴𝑊𝑛𝑦𝑛1+𝑦𝑛1𝑦𝑛.(3.35) By condition (C1) and lim𝑛𝑦𝑛1𝑦𝑛=0, we obtain that 𝑥𝑛𝑊𝑛𝑦𝑛0 as 𝑛.
Hence, we have 𝑥𝑛𝑊𝑛𝑥𝑛𝑥𝑛𝑊𝑛𝑦𝑛+𝑊𝑛𝑦𝑛𝑊𝑛𝑥𝑛𝑥𝑛𝑊𝑛𝑦𝑛+𝑦𝑛𝑥𝑛.(3.36) By (3.34) and lim𝑛𝑥𝑛𝑊𝑛𝑦𝑛=0, we obtain 𝑥𝑛𝑊𝑛𝑥𝑛0 as 𝑛.
Moreover, we also have 𝑦𝑛𝑊𝑛𝑦𝑛𝑦𝑛𝑥𝑛+𝑥𝑛𝑊𝑛𝑦𝑛.(3.37) By (3.34) and lim𝑛𝑥𝑛𝑊𝑛𝑦𝑛=0, we obtain 𝑦𝑛𝑊𝑛𝑦𝑛0 as 𝑛.
Step 5. We show that 𝑞𝜃=𝑛=1𝐹(𝑇𝑛)(𝑁𝑘=1SMEP(𝐹𝑘))𝐼(𝐵,𝑀) and limsup𝑛(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑞0. It is easy to see that 𝑃𝜃(𝛾𝑓+(𝐼𝐴)) is a contraction of 𝐻 into itself.
Indeed, since 0<𝛾<𝛾/𝜖, we have 𝑃𝜃(𝛾𝑓+(𝐼𝐴))𝑥𝑃𝜃(𝛾𝑓+(𝐼𝐴))𝑦𝛾𝑓(𝑥)𝑓(𝑦)+𝐼𝐴𝑥y𝛾𝜖𝑥𝑦+1𝛾𝑥𝑦1𝛾+𝛾𝜖𝑥𝑦.(3.38) Since 𝐻 is complete, then there exists a unique fixed point 𝑞𝐻 such that 𝑞=𝑃𝜃(𝛾𝑓+(𝐼𝐴))(𝑞). By Lemma 2.2, we obtain that (𝛾𝑓𝐴)𝑞,𝑤𝑞0 for all 𝑤𝜃.
Next, we show that limsup𝑛(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑞0, where 𝑞=𝑃𝜃(𝛾𝑓+𝐼𝐴)(𝑞) is the unique solution of the variational inequality (𝛾𝑓𝐴)𝑞,𝑤𝑞0 for all 𝑤𝜃. We can choose a subsequence {𝑦𝑛𝑖} of {𝑦𝑛} such that limsup𝑛(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑞=lim𝑖(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑖.𝑞(3.39) As {𝑦𝑛𝑖} is bounded, there exists a subsequence {𝑦𝑛𝑖𝑗} of {𝑦𝑛𝑖} which converges weakly to 𝑤. We may assume without loss of generality that 𝑦𝑛𝑖𝑤.
Next we claim that 𝑤𝜃. Since 𝑦𝑛𝑊𝑛𝑦𝑛0,𝑥𝑛𝑊𝑛𝑥𝑛0, and 𝑥𝑛𝑦𝑛0, and by Lemma 2.6, we have 𝑤𝑛=1𝐹(𝑇𝑛).
Next, we show that 𝑤𝑘=1SMEP(𝐹𝑘). Since 𝑢𝑛=𝜏𝑁𝑟𝑘,𝑛𝑥𝑛, for 𝑘=1,2,3,,𝑁, we know that 𝐹𝑘𝑢𝑛𝑢,𝑦+𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛0,𝑦𝐶.(3.40) It follows by (A2) that 𝑢𝜑(𝑦)𝜑𝑛+1𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝐹𝑘𝑦,𝑢𝑛,𝑦𝐶.(3.41) Hence, for 𝑘=1,2,3,,𝑁, we get 𝑢𝜑(𝑦)𝜑𝑛𝑖+1𝑟𝑛𝑖𝑦𝑢𝑛𝑖,𝑢𝑛𝑖𝑥𝑛𝑖𝐹𝑘𝑦,𝑢𝑛𝑖,𝑦𝐶.(3.42) For 𝑡(0,1] and 𝑦𝐻, let 𝑦𝑡=𝑡𝑦+(1𝑡)𝑤. From (3.42), we have 𝑦0𝜑𝑡𝑢+𝜑𝑛𝑖1𝑟𝑛𝑖𝑦𝑡𝑢𝑛𝑖,𝑢𝑛𝑖𝑥𝑛𝑖+𝐹𝑘𝑦𝑡,𝑢𝑛𝑖.(3.43)
Since 𝑢𝑛𝑖𝑥𝑛𝑖0, from (A4) and the weakly lower semicontinuity of 𝜑, (𝑢𝑛𝑖𝑥𝑛𝑖)/𝑟𝑛𝑖0 and 𝑢𝑛𝑖𝑤. From (A1) and (A4), we have 0=𝐹𝑘𝑦𝑡,𝑦𝑡𝑦𝜑𝑡𝑦+𝜑𝑡𝑡𝐹𝑘𝑦𝑡,𝑦+(1𝑡)𝐹𝑘𝑦𝑡𝑦,𝑤+𝑡𝜑(𝑦)+(1𝑡)𝜑(𝑤)𝜑𝑡𝐹𝑡𝑘𝑦𝑡𝑦,𝑦+𝜑(𝑦)𝜑𝑡.(3.44) Dividing by 𝑡, we get 𝐹𝑘𝑦𝑡𝑦,𝑦+𝜑(𝑦)𝜑𝑡0.(3.45) The weakly lower semicontinuity of 𝜑 for 𝑘=1,2,3,,𝑁, we get 𝐹𝑘(𝑤,𝑦)+𝜑(𝑦)𝜑(𝑤).(3.46) So, we have 𝐹𝑘(𝑤,𝑦)+𝜑(𝑦)𝜑(𝑤)0,𝑘=1,2,3,,𝑁.(3.47) This implies that 𝑤𝑁𝑘=1SMEP(𝐹𝑘).
Lastly, we show that 𝑤𝐼(𝐵,𝑀). In fact, since B is 𝛽-inverse strongly monotone, hence 𝐵 is a 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 𝑔𝐵𝑣𝑀(𝑣). Again since 𝑦𝑛𝑖=𝐽𝑀,𝜆(𝑢𝑛𝑖𝜆𝐵𝑢𝑛𝑖), we have 𝑢𝑛𝑖𝜆𝐵𝑢𝑛𝑖(𝐼+𝜆𝑀)(𝑦𝑛𝑖), that is, (1/𝜆)(𝑢𝑛𝑖𝑦𝑛𝑖𝜆𝐵𝑢𝑛𝑖)𝑀(𝑦𝑛𝑖). By virtue of the maximal monotonicity of 𝑀+𝐵, we have 𝑣𝑦𝑛𝑖1,𝑔𝐵𝑣𝜆𝑢𝑛𝑖𝑦𝑛𝑖𝜆𝐵𝑢𝑛𝑖0,(3.48) and hence 𝑣𝑦𝑛𝑖,𝑔𝑣𝑦𝑛𝑖1,𝐵𝑣+𝜆𝑢𝑛𝑖𝑦𝑛𝑖𝜆𝐵𝑢𝑛𝑖=𝑣𝑦𝑛𝑖,𝐵𝑣𝐵𝑦𝑛𝑖+𝑣𝑦𝑛𝑖,𝐵𝑦𝑛𝑖𝐵𝑢𝑛𝑖+𝑣𝑦𝑛𝑖,1𝜆𝑢𝑛𝑖𝑦𝑛𝑖.(3.49) It follows from lim𝑛𝑢𝑛𝑦𝑛=0, we have lim𝑛𝐵𝑢𝑛𝐵𝑦𝑛=0 and 𝑦𝑛𝑖𝑤, it follows that limsup𝑛𝑣𝑦𝑛,𝑔=𝑣𝑤,𝑔0.(3.50) It follows from the maximal monotonicity of 𝐵+𝑀 that 𝜃(𝑀+𝐵)(𝑤), that is, 𝑤𝐼(𝐵,𝑀). Therefore, 𝑤𝜃. We observe that limsup𝑛(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑞=lim𝑖(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑖𝑞=(𝛾𝑓𝐴)𝑞,𝑤𝑞0.(3.51)
Step 6. Finally, we prove 𝑥𝑛𝑞. By using (3.2) and together with Schwarz inequality, we have 𝑥𝑛+1𝑞2=𝑃𝐶𝜖𝑛𝑥𝛾𝑓𝑛+𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑃𝐶(𝑞)2𝜖𝑛𝑥𝛾𝑓𝑛+𝐴𝑞𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑞2𝐼𝜖𝑛𝐴2𝑊𝑛𝑦𝑛𝑞2+𝜖2𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝜖𝑛𝐼𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞1𝜖𝑛𝛾2𝑦𝑛𝑞2+𝜖2𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝜖𝑛𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞2𝜖2𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞1𝜖𝑛𝛾2𝑥𝑛𝑞2+𝜖2𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝜖𝑛𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝛾𝑓(𝑞)+2𝜖𝑛𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞2𝜖2𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞1𝜖𝑛𝛾2𝑥𝑛𝑞2+𝜖2𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝜖𝑛𝑊𝑛𝑦𝑛𝑥𝑞𝛾𝑓𝑛𝛾𝑓(𝑞)+2𝜖𝑛𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞2𝜖2𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞1𝜖𝑛𝛾2𝑥𝑛𝑞2+𝜖2𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝛾𝜖𝜖𝑛𝑦𝑛𝑥𝑞𝑛𝑞+2𝜖𝑛𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞2𝜖2𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞1𝜖𝑛𝛾2𝑥𝑛𝑞2+𝜖2𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝛾𝜖𝜖𝑛𝑥𝑛𝑞2+2𝜖𝑛𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞2𝜖2𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞,𝛾𝑓𝑛𝐴𝑞1𝜖𝑛𝛾2+2𝛾𝜖𝜖𝑛𝑥𝑛𝑞2+𝜖𝑛𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞2𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞𝛾𝑓𝑛=𝐴𝑞12𝜖𝛾𝛾𝜖𝑛𝑥𝑛𝑞2+𝜖𝑛𝜖𝑛𝑥𝛾𝑓𝑛𝐴𝑞2+2𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞2𝜖𝑛𝐴𝑊𝑛𝑦𝑛𝑥𝑞𝛾𝑓𝑛𝐴𝑞+𝜖𝑛𝛾2𝑥𝑛𝑞2.(3.52)
Since {𝑥𝑛} is bounded, where 𝜂𝛾𝑓(𝑥𝑛)𝐴𝑞22𝐴(𝑊𝑛𝑦𝑛𝑞)𝛾𝑓(𝑥𝑛)𝐴𝑞+𝛾2𝑥𝑛𝑞2 for all 𝑛0. It follows that 𝑥𝑛+1𝑞212𝜖𝛾𝛾𝜖𝑛𝑥𝑛𝑞2+𝜖𝑛𝛿𝑛,(3.53) where 𝛿𝑛=2𝑊𝑛𝑦𝑛𝑞,𝛾𝑓(𝑞)𝐴𝑞+𝜂𝛼𝑛. Since limsup𝑛(𝛾𝑓𝐴)𝑞,𝑊𝑛𝑦𝑛𝑞0, we get limsup𝑛𝛿𝑛0. Applying Lemma 2.5, we can conclude that 𝑥𝑛𝑞. This completes the proof.

Corollary 3.2. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻. Let 𝐵 be 𝛽-inverse-strongly monotone and 𝜑𝐶 a convex and lower semicontinuous function. Let 𝑓𝐶𝐶 be a contraction with coefficient 𝛼(0<𝛼<1), 𝑀𝐻2𝐻 a maximal monotone mapping, and {𝑇𝑛} a family of nonexpansive mappings of 𝐻 into itself such that 𝜃=𝑛=1𝐹𝑇𝑛𝑁𝑘=1𝐹SMEP𝑘𝐼(𝐵,𝑀)0.(3.54) Suppose that {𝑥𝑛} is a sequence generated by the following algorithm for 𝑥0,𝑢𝑛𝐶 arbitrarily: 𝑢𝑛=𝐾𝐹𝑁𝑟𝑛,𝑛𝐾𝐹𝑁1𝑟𝑛1,𝑛𝐾𝐹𝑁2𝑟𝑛2,𝑛𝐾𝐹2𝑟2,𝑛𝐾𝐹1𝑟1,𝑛𝑥𝑛𝑥,𝑛𝑁𝑛+1=𝑃𝐶𝜖𝑛𝑓𝑥𝑛+𝐼𝜖𝑛𝑊𝑛𝐽𝑀,𝜆𝑢𝑛𝜆𝐵𝑢𝑛,(3.55) for all 𝑛=0,1,2,, and the conditions (C1)–(C3) in Theorem 3.1 are satisfied.
Then, the sequence {𝑥𝑛} converges strongly to 𝑞𝜃, where 𝑞=𝑃𝜃(𝑓+𝐼)(𝑞) which solves the following variational inequality: (𝑓𝐼)𝑞,𝑝𝑞0,𝑝𝜃.(3.56)

Proof. Putting 𝐴𝐼 and 𝛾1 in Theorem 3.1, we can obtain the desired conclusion immediately.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻. Let 𝐵 be 𝛽-inverse-strongly monotone, 𝜑𝐶 a convex and lower semicontinuous function, and 𝑀𝐻2𝐻 a maximal monotone mapping. Let {𝑇𝑛} be a family of nonexpansive mappings of 𝐻 into itself such that 𝜃=𝑛=1𝐹𝑇𝑛𝑁𝑘=1𝐹SMEP𝑘𝐼(𝐵,𝑀)0.(3.57) Suppose that {𝑥𝑛} is a sequence generated by the following algorithm for 𝑥0,𝑢𝐶 and