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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 414831, 35 pages
http://dx.doi.org/10.1155/2012/414831
Research Article

A System of Mixed Equilibrium Problems, a General System of Variational Inequality Problems for Relaxed Cocoercive, and Fixed Point Problems for Nonexpansive Semigroup and Strictly Pseudocontractive Mappings

1Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Received 17 November 2011; Accepted 23 January 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Poom Kumam and Phayap Katchang. 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 an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a general system of variational inequalities for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (2010) and many others.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product , and norm . Let 𝐶 be a nonempty closed convex subset of 𝐻. Recall that a mapping 𝑇𝐶𝐶 is nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐶.(1.1) We denote the set of fixed points of 𝑇 by 𝐹(𝑇), that is 𝐹(𝑇)={𝑥𝐶𝑥=𝑇𝑥}. A mapping 𝑓𝐶𝐶 is said to be an 𝛼-contraction if there exists a coefficient 𝛼(0,1) such that 𝑓(𝑥)𝑓(𝑦)𝛼𝑥𝑦,𝑥,𝑦𝐶.(1.2) Let 𝐵𝐶𝐻 be a mapping. Then 𝐵 is called:(1)monotone if 𝐵𝑥𝐵𝑦,𝑥𝑦0,𝑥,𝑦𝐶;(1.3)(2)𝑑-strongly monotone if there exists a positive real number 𝑑 such that 𝐵𝑥𝐵𝑦,𝑥𝑦𝑑𝑥𝑦2,𝑥,𝑦𝐶,(1.4) for constant 𝑑>0, this implies that 𝐵𝑥𝐵𝑦𝑑𝑥𝑦,(1.5) that is, 𝐵 is 𝑑-expansive and when 𝑑=1, it is expansive;(3)𝐿-Lipschitz continuous if there exists a positive real number 𝐿 such that𝐵𝑥𝐵𝑦𝐿𝑥𝑦,𝑥,𝑦𝐶;(1.6)(4)𝑐-cocoercive [1, 2] if there exists a positive real number 𝑐 such that 𝐵𝑥𝐵𝑦,𝑥𝑦𝑐𝐵𝑥𝐵𝑦2,𝑥,𝑦𝐶,(1.7) Clearly, every 𝑐-cocoercive map 𝐵 is (1/𝑐)-Lipschitz continuous;(5)relaxed 𝑐-cocoercive, if there exists a positive real number 𝑐 such that 𝐵𝑥𝐵𝑦,𝑥𝑦(𝑐)𝐵𝑥𝐵𝑦2,𝑥,𝑦𝐶;(1.8)(6)relaxed (𝑐,𝑑)-cocoercive, if there exists a positive real number 𝑐,𝑑 such that 𝐵𝑥𝐵𝑦,𝑥𝑦(𝑐)𝐵𝑥𝐵𝑦2+𝑑𝑥𝑦2,𝑥,𝑦𝐶,(1.9) for 𝑐=0, 𝐵 is 𝑑-strongly monotone. This class of mapping is more general than the class of strongly monotone mapping. It is easy to see that we have the following implication: 𝑑-strongly monotonicity implying relaxed (𝑐,𝑑)-cocoercivity,(7)𝑘-strictly pseudocontractive, if there exists a constant 𝑘[0,1) such that 𝐵𝑥𝐵𝑦2𝑥𝑦2+𝑘(𝐼𝐵)𝑥(𝐼𝐵)𝑦2,𝑥,𝑦𝐶.(1.10)

Remark 1.1 (see [3, Remark  1.1 pages 135-136]). If 𝐵𝐶𝐻 is a 𝐿𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿2𝐵 and 0<𝜏<2(𝑑𝑐𝐿2𝐵)/𝐿2𝐵, then 𝐼𝜏𝐵 satisfies the following: (𝐼𝜏𝐵)𝑥(𝐼𝜏𝐵)𝑦(1𝜏𝜉)𝑥𝑦,𝑥,𝑦𝐶,(1.11) where 𝜉=(𝐿2𝐵/2)[2(𝑑𝑐𝐿2𝐵)/𝐿2𝐵𝜏].
Similarly, if 𝐷𝐶𝐻is𝐿𝐷-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿2𝐷 and 0<𝛿<2(𝑑𝑐𝐿2𝐷)/𝐿2𝐷, then the mapping 𝐼𝛿𝐷 satisfies the following: (𝐼𝛿𝐷)𝑥(𝐼𝛿𝐷)𝑦1𝛿𝜉𝑥𝑦,(1.12) where 𝜉=(𝐿2𝐷/2)[2(𝑑𝑐𝐿2𝐷)/𝐿2𝐷𝛿].

Let 𝐴 be a strongly positive linear bounded operator on 𝐻 if there is a constant 𝛾>0 with the property 𝐴𝑥,𝑥𝛾𝑥2,𝑥𝐻.(1.13) We recall optimization problem (for short, OP) as the following min𝑥𝐹𝜇21𝐴𝑥,𝑥+2𝑥𝑢2(𝑥),(1.14) where 𝐹=𝑛=1𝐶𝑛,𝐶1,𝐶2, are infinitely closed convex subsets of 𝐻 such that 𝑛=1𝐶𝑛, 𝑢𝐻, 𝜇0 is a real number, 𝐴 is a strongly positive linear bounded operator on 𝐻, and is a potential function for 𝛾𝑓 (i.e., (𝑥)=𝛾𝑓(𝑥) for 𝑥𝐻). This kind of optimization problem has been studied extensively by many authors, see, for example, [47] when 𝐹=𝑛=1𝐶𝑛 and (𝑥)=𝑥,𝑏, where 𝑏 is a given point in 𝐻.

On the other hand, a family 𝒮={𝑆(𝑠)0𝑠<} of mappings of 𝐶 into itself is called a nonexpansive semigroup on 𝐶 if it satisfies the following conditions:(i)𝑆(0)𝑥=𝑥 for all 𝑥𝐶;(ii)𝑆(𝑠+𝑡)=𝑆(𝑠)𝑆(𝑡) for all 𝑠,𝑡0;(iii)𝑆(𝑠)𝑥𝑆(𝑠)𝑦𝑥𝑦 for all 𝑥,𝑦𝐶 and 𝑠0;(iv)for all 𝑥𝐶,𝑠𝑆(𝑠)𝑥 is continuous.

We denote by 𝐹(𝒮) the set of all common fixed points of 𝒮={𝑆(𝑠)𝑠0}, that is, 𝐹(𝒮)=𝑠0𝐹(𝑆(𝑠)). It is known that 𝐹(𝒮) is closed and convex.

Let 𝜙𝐶 be a real-valued function and let {Θ𝑘𝐶×𝐶,𝑘=1,2,,𝑁} be a finite family of equilibrium functions, that is, Θ𝑘(𝑢,𝑢)=0 for each 𝑢𝐶. The system of mixed equilibrium problems (for short, SMEP) for function (Θ1,Θ2,,Θ𝑁,𝜙) is to find 𝑧𝐶 such that Θ1Θ(𝑧,𝑦)+𝜙(𝑦)𝜙(𝑧)0,𝑦𝐶,2Θ(𝑧,𝑦)+𝜙(𝑦)𝜙(𝑧)0,𝑦𝐶,𝑁(𝑧,𝑦)+𝜙(𝑦)𝜙(𝑧)0,𝑦𝐶.(1.15) The set of solutions of (1.15) is denoted by 𝑁𝑘=1MEP(Θ𝑘,𝜙), where MEP(Θ𝑘,𝜙) is the set of solutions of the mixed equilibrium problem (for short, MEP), which is to find 𝑧𝐶 such that Θ𝑘(𝑧,𝑦)+𝜙(𝑦)𝜙(𝑧)0,𝑦𝐶.(1.16) In particular, if 𝜙0, and 𝑁=1, then the problem (1.15) reduces to the equilibrium problem (for short, EP), which is to find 𝑧𝐶 such that Θ(𝑧,𝑦)0,𝑦𝐶.(1.17) It is well known that the SMEP includes fixed point problem, optimization problem, variational inequality problem, and Nash equilibrium problem as its special cases (see [813] for more details).

For solving the solutions of a nonexpansive semigroup and the solutions of the system of mixed equilibrium problems were studied by many authors see [1423] and reference therein. In 2010, Chang et al. [24] studied the following approximation method: Θ1𝑢𝑛(1)𝑢,𝑥+𝜙(𝑥)𝜙𝑛(1)+1𝑟1𝐾𝑢𝑛(1)𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛(1)Θ0,𝑥𝐶,2𝑢𝑛(2)𝑢,𝑥+𝜙(𝑥)𝜙𝑛(2)+1𝑟2𝐾𝑢𝑛(2)𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛(2)Θ0,𝑥𝐶,𝑁𝑢𝑛(𝑁)𝑢,𝑥+𝜙(𝑥)𝜙𝑛(𝑁)+1𝑟𝑁𝐾𝑢𝑛(𝑁)𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛(𝑁)𝑥0,𝑥𝐶,𝑛+1=𝛼𝑛𝑓𝑊𝑛𝑥𝑛+𝛽𝑛𝑥𝑛+𝛾𝑛1𝑡𝑛𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑢𝑛(𝑁)𝑑𝑠,(1.18) where 𝑢𝑛(1)=𝐽Θ1𝑟1𝑥𝑛,𝑢𝑛(𝑘)=𝐽Θ𝑘𝑟𝑘𝑢𝑛(𝑘1)=𝐽Θ𝑘𝑟𝑘𝐽Θ𝑘1𝑟𝑘1𝑢𝑛(𝑘2)=𝐽Θ𝑘𝑟𝑘𝐽Θ2𝑟2𝑢𝑛(1),=𝐽Θ𝑘𝑟𝑘𝐽Θ2𝑟2𝐽Θ1𝑟1𝑥𝑛,𝑘=2,3,,𝑁,(1.19)𝐽Θ𝑘𝑟𝑘𝐶𝐶,𝑘=1,2,,𝑁 is the mapping defined by (2.22) below, 𝑊𝑛 is the mapping defined by (2.12), and 𝒮={𝑆(𝑠)0𝑠<} is a nonexpansive semigroup. They proved that {𝑥𝑛} converges strongly to a fixed point of 𝐹(𝒮)𝐹(𝑊)(𝑁𝑘=1MEP(Θ𝑘,𝜙)) under control conditions on the parameters.

Let 𝐵,𝐷𝐶𝐻 be two mappings. The general system of variational inequalities problem (see [25]) is to find (𝑥,𝑦)𝐶×𝐶 such that 𝜏𝐵𝑦+𝑥𝑦,𝑥𝑥0,𝑥𝐶,𝛿𝐷𝑥+𝑦𝑥,𝑥𝑦0,𝑥𝐶,(1.20) where 𝜏 and 𝛿 are two positive real numbers. The set of solutions of the general system of variational inequalities problem is denoted by SVI(𝐶,𝐵,𝐷). In particular, if 𝐵=𝐷, then the problem (1.20) reduces to the following equation: 𝜏𝐵𝑦+𝑥𝑦,𝑥𝑥0,𝑥𝐶,𝛿𝐵𝑥+𝑦𝑥,𝑥𝑦0,𝑥𝐶,(1.21) which is defined by Verma [26] (see also Verma [27]), and is called the new system of variational inequalities. Further, if we set 𝐷=0, then problem (1.20) reduces to the classical variational inequality is to find 𝑥𝐶 such that 𝐵𝑥,𝑥𝑥0,𝑥𝐶.(1.22) We denoted by VI(𝐶,𝐵) the set of solutions of the variational inequality problem. The variational inequality problem has been extensively studied in literature, see, for example, [2831] and references therein. In order to find the solutions of the general system of variational inequality problem (1.20), Wangkeeree and Kamraksa [32] considered the following iterative algorithm: Θ𝑢𝑛𝑢,𝑥+𝜙(𝑥)𝜙𝑛+1𝑟𝐾𝑢𝑛𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛𝑧0,𝑥𝐶,𝑛=𝑃𝐶𝑢𝑛𝛿𝐷𝑢𝑛,𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝛽𝑛𝑥𝑛+1𝛽𝑛𝐼𝛼𝑛𝐴𝑊𝑛𝑃𝐶𝑧𝑛𝜏𝐵𝑧𝑛,(1.23) where 𝐵,𝐷𝐶𝐻 is a 𝐿𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping and 𝐿𝐷-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping, respectively. They proved that {𝑥𝑛} converges strongly to a fixed point of 𝐹(𝑊𝑛)MEP(Θ,𝜙)SVI(𝐶,𝐵,𝐷) which is a solution of general system of variational inequality (1.20). Very recently, Jaiboon and Kumam [33] studied a new general iterative method for finding a common element of the set of solution of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces, which solves some optimization problems.

Inspired and motivated by Chang et al. [24], Jaiboon and Kumam [33], Kumam and Jaiboon [34] and Wangkeeree and Kamraksa [32], the purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroup, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings. Consequently, we prove the strong convergence theorem in Hilbert spaces under control conditions on the parameters. Furthermore, we can apply our results for solving some optimization problems. Our results extend and improve the corresponding results in Chang et al. [24], Kumam and Jaiboon [34], Wangkeeree and Kamraksa [32], and many others.

2. Preliminaries

Let 𝐻 a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. We denote strong convergence (weak convergence) by notation (). In a real Hilbert space 𝐻, it is well known that 𝑥𝑦2=𝑥2𝑦22𝑥𝑦,𝑦,(2.1)𝑥+𝑦2𝑥2+2𝑦,𝑥+𝑦,(2.2)𝑥+𝑦2𝑥2+2𝑦,𝑥,(2.3)𝜆𝑥+(1𝜆)𝑦2=𝜆𝑥2+(1𝜆)𝑦2𝜆(1𝜆)𝑥𝑦2(2.4) for all 𝑥,𝑦𝐻 and 𝜆.

Recall that for every point 𝑥𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃𝐶𝑥, such that 𝑥𝑃𝐶𝑥𝑥𝑦,𝑦𝐶.(2.5)𝑃𝐶 is called the metric projection of 𝐻 onto 𝐶. It is well known that 𝑃𝐶 is a nonexpansive mapping of 𝐻 onto 𝐶 and satisfies 𝑥𝑦,𝑃𝐶𝑥𝑃𝐶𝑃𝑦𝐶𝑥𝑃𝐶𝑦2(2.6) for every 𝑥,𝑦𝐻. Obviously, this immediately implies that 𝑃(𝑥𝑦)𝐶𝑥𝑃𝐶𝑦2𝑥𝑦2𝑃𝐶𝑥𝑃𝐶𝑦2,𝑥,𝑦𝐻.(2.7) Moreover, 𝑃𝐶𝑥 is characterized by the following properties: 𝑃𝐶𝑥𝐶 and 𝑥𝑃𝐶𝑥,𝑦𝑃𝐶𝑥0,𝑥𝑦2𝑥𝑃𝐶𝑥2+𝑦𝑃𝐶𝑥2(2.8) for all 𝑥𝐻,𝑦𝐶.

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [35]). Let 𝑉𝐶𝐻 be a k-strict pseudo-contraction, then(1)the fixed point set 𝐹(𝑉) of 𝑉 is closed convex so that the projection 𝑃𝐹(𝑉) is well defined;(2)define a mapping 𝑇𝐶𝐻 by 𝑇𝑥=𝑡𝑥+(1𝑡)𝑉𝑥,𝑥𝐶.(2.9) If 𝑡[𝑘,1), then 𝑇 is a nonexpansive mapping such that 𝐹(𝑉)=𝐹(𝑇).

A family of mappings {𝑉𝑖𝐶𝐻}𝑖=1 is called a family of uniformly 𝑘-strict pseudo-contractions, if there exists a constant 𝑘[0,1) such that 𝑉𝑖𝑥𝑉𝑖𝑦2𝑥𝑦2+𝑘𝐼𝑉𝑖𝑥𝐼𝑉𝑖𝑦2,𝑥,𝑦𝐶,𝑖1.(2.10) Let {𝑉𝑖𝐶𝐶}𝑖=1 be a countable family of uniformly 𝑘-strict pseudo-contractions. Let {𝑇𝑖𝐶𝐶}𝑖=1 be the sequence of nonexpansive mappings defined by (2.9), that is, 𝑇𝑖𝑥=𝑡𝑥+(1𝑡)𝑉𝑖[𝑥,𝑥𝐶,𝑖1,𝑡𝑘,1).(2.11)

Let {𝑇𝑖} be a sequence of nonexpansive mappings of 𝐶 into itself defined by (2.11) and let {𝜇𝑖} be a sequence of nonnegative numbers in [0,1]. For each 𝑛1, define a mapping 𝑊𝑛 of 𝐶 into itself as follows: 𝑈𝑛,𝑛+1𝑈=𝐼,𝑛,𝑛=𝜇𝑛𝑇𝑛𝑈𝑛,𝑛+1+1𝜇𝑛𝑈𝐼,𝑛,𝑛1=𝜇𝑛1𝑇𝑛1𝑈𝑛,𝑛+1𝜇𝑛1𝑈𝐼,𝑛,𝑘=𝜇𝑘𝑇𝑘𝑈𝑛,𝑘+1+1𝜇𝑘𝑈𝐼,𝑛,𝑘1=𝜇𝑘1𝑇𝑘1𝑈𝑛,𝑘+1𝜇𝑘1𝑈𝐼,𝑛,2=𝜇2𝑇2𝑈𝑛,3+1𝜇2𝑊𝐼,𝑛=𝑈𝑛,1=𝜇1𝑇1𝑈𝑛,2+1𝜇1𝐼.(2.12) Such a mapping 𝑊𝑛 is nonexpansive from 𝐶 to 𝐶 and it is called the 𝑊-mapping generated by 𝑇1,𝑇2,,𝑇𝑛 and 𝜇1,𝜇2,,𝜇𝑛.

For each 𝑛,𝑘, let the mapping 𝑈𝑛,𝑘 be defined by (2.12). Then we can have the following crucial conclusions concerning 𝑊𝑛. You can find them in [36]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.2 (see [36]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇1,𝑇2, be nonexpansive mappings of 𝐶 into itself such that 𝑛=1𝐹(𝑇𝑛) is nonempty, let 𝜇1,𝜇2, be real numbers such that 0𝜇𝑛𝑏<1 for every 𝑛1. Then,(1)𝑊𝑛 is nonexpansive and 𝐹(𝑊𝑛)=𝑛𝑖=1𝐹(𝑇𝑖), for all 𝑛1;(2)for every 𝑥𝐶 and 𝑘, the limit lim𝑛𝑈𝑛,𝑘𝑥 exists;(3)a mapping 𝑊𝐶𝐶 defined by 𝑊𝑥=lim𝑛𝑊𝑛𝑥=lim𝑛𝑈𝑛,1𝑥,𝑥𝐶(2.13) is a nonexpansive mapping satisfying 𝐹(𝑊)=𝑖=1𝐹(𝑇𝑖) and it is called the 𝑊-mapping generated by 𝑇1,𝑇2, and 𝜇1,𝜇2,.

Lemma 2.3 (see [37]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, {𝑇𝑖𝐶𝐶} a countable family of nonexpansive mappings with 𝑖=1𝐹(𝑇𝑖), {𝜇𝑖} a real sequence such that 0<𝜇𝑖𝑏<1,forall𝑖1. If 𝐷 is any bounded subset of 𝐶, then lim𝑛sup𝑥𝐷𝑊𝑥𝑊𝑛𝑥=0.(2.14)

Lemma 2.4 (see [38]). Each Hilbert space 𝐻 satisfies Opial’s condition, that is, for any sequence {𝑥𝑛}𝐻 with 𝑥𝑛𝑥, the inequality liminf𝑛𝑥𝑛𝑥<liminf𝑛𝑥𝑛𝑦(2.15) holds for each 𝑦𝐻 with 𝑦𝑥.

Lemma 2.5 (see [39]). Assume 𝐴 is a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and 0<𝜌𝐴1. Then, 𝐼𝜌𝐴1𝜌𝛾.

For solving the system of mixed equilibrium problems (1.15), let us assume that function Θ𝑘𝐻×𝐻,𝑘=1,2,,𝑁 satisfies the following conditions:(H1)Θ𝑘 is monotone, that is, Θ𝑘(𝑥,𝑦)+Θ𝑘(𝑦,𝑥)0, for all 𝑥,𝑦𝐻;(H2) for each fixed 𝑦𝐻, 𝑥Θ𝑘(𝑥,𝑦) is convex and upper semicontinuous;(H3) for each 𝑥𝐻,𝑦Θ𝑘(𝑥,𝑦) is convex.

Let 𝜂𝐻×𝐻𝐻 and 𝐵𝐻𝐻 be two mappings. 𝐵 is said to be(1)monotone if 𝐵𝑥𝐵𝑦,𝜂(𝑥,𝑦)0,𝑥,𝑦𝐻;(2.16)(2)𝑑-strongly monotone if there exists a positive real number 𝑑 such that 𝐵𝑥𝐵𝑦,𝜂(𝑥,𝑦)𝑑𝑥𝑦2,𝑥,𝑦𝐻;(2.17)(3)𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that 𝜂(𝑥,𝑦)𝐿𝑥𝑦,𝑥,𝑦𝐻.(2.18)

Let 𝐾𝐻 be a differentiable functional on 𝐻, which is called:(1)𝜂-convex [40] if 𝐾𝐾(𝑦)𝐾(𝑥)(𝑥),𝜂(𝑦,𝑥),𝑥,𝑦𝐻,(2.19) where 𝐾(𝑥) is the Fréchet derivative of 𝐾 at 𝑥;(2)𝜂-strongly convex [41] if there exists a constant 𝜎>0 such that 𝐾𝐾(𝑦)𝐾(𝑥)𝜎(𝑥),𝜂(𝑦,𝑥)2𝑥𝑦2,𝑥,𝑦𝐻.(2.20)

In particular, if 𝜂(𝑥,𝑦)=𝑥𝑦 for all 𝑥,𝑦𝐻, then 𝐾 is said to be strongly convex.

Lemma 2.6 (see [42]). Let 𝐻 be a real Hilbert space and let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to . Let Θ be a bifunction from 𝐻×𝐻 to satisfying (H1)–(H3). Assume that(i)𝜂𝐻×𝐻𝐻 is 𝜆-Lipschitz continuous with constant 𝜆>0 such that(a)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0,forall𝑥,𝑦𝐻, (b)𝜂(,) is affine in the first variable,(c)for each fixed 𝑥𝐻, 𝑦𝜂(𝑥,𝑦) is sequentially continuous from the weak topology to the weak topology;(ii)𝐾𝐻 is 𝜂-strongly convex with constant 𝜎>0 and its derivative 𝐾 is sequentially continuous from the weak topology to the strong topology;(iii)for each 𝑥𝐻, there exist bounded subsets 𝐸𝑥𝐻 and 𝑧𝑥𝐻 such that for any 𝑦𝐻𝐸𝑥, Θ𝑦,𝑧𝑥𝑧+𝜙𝑥1𝜙(𝑦)+𝑟𝐾(𝑦)𝐾𝑧(𝑥),𝜂𝑥,𝑦<0.(2.21) For given 𝑟>0, let 𝐽Θ𝑟𝐻𝐻 be the mapping defined by 𝐽Θ𝑟1(𝑥)=𝑦𝐻Θ(𝑦,𝑧)+𝜙(𝑧)𝜙(𝑦)+𝑟𝐾(𝑦)𝐾(𝑥),𝜂(𝑧,𝑦)0,𝑧𝐻(2.22)for all 𝑥𝐻. Then(1)𝐽Θ𝑟 is single-valued.(2)𝐹(𝐽Θ𝑟)=MEP(Θ,𝜙), where MEP(Θ,𝜙) is the set of solution of the mixed equilibrium problem, Θ(𝑥,𝑦)+𝜙(𝑦)𝜙(𝑥)0,𝑦𝐻.(2.23)(3)MEP(Θ,𝜙) is closed and convex.

Lemma 2.7 (see [43]). Let {𝑥𝑛} and {𝑣𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1. Suppose 𝑥𝑛+1=(1𝛽𝑛)𝑣𝑛+𝛽𝑛𝑥𝑛 for all integers 𝑛0 and limsup𝑛(𝑣𝑛+1𝑣𝑛𝑥𝑛+1𝑥𝑛)0. Then, lim𝑛𝑣𝑛𝑥𝑛=0.

Lemma 2.8 (see [44]). Assume {𝑥𝑛} is a sequence of nonnegative real numbers such that 𝑥𝑛+11𝑎𝑛𝑥𝑛+𝑏𝑛,𝑛0,(2.24) where {𝑎𝑛} is a sequence in (0,1) and {𝑏𝑛} is a sequence in such that(1)𝑛=1𝑎𝑛=, (2)limsup𝑛(𝑏𝑛/𝑎𝑛)0 or 𝑛=1|𝑏𝑛|<.Then, lim𝑛𝑥𝑛=0.

Lemma 2.9 (see [45]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑔𝐶{} a proper lower-semicontinuous differentiable convex function. If 𝑧 is a solution to the minimization problem 𝑔(𝑧)=inf𝑥𝐶𝑔(𝑥),(2.25) then 𝑔(𝑥),𝑥𝑧0,𝑥𝐶.(2.26) In particular, if 𝑧 solves problem 𝑂𝑃, then []𝑢+𝛾𝑓(𝐼+𝜇𝐴)𝑧,𝑥𝑧0.(2.27)

Lemma 2.10 (see [46]). Let 𝐶 be a nonempty bounded closed convex subset of a Hilbert space 𝐻 and let 𝒮={𝑆(𝑠)0𝑠<} be a nonexpansive semigroup on 𝐶, then for any 0, lim𝑡sup𝑥𝐶1𝑡𝑡01𝑇(𝑠)𝑥𝑑𝑠𝑇()𝑡𝑡0𝑇(𝑠)𝑥𝑑𝑠=0.(2.28)

Lemma 2.11 (see [47]). Let C be a nonempty bounded closed convex subset of 𝐻, {𝑥𝑛} a sequence in C, and 𝒮={𝑆(𝑠)0𝑠<} a nonexpansive semigroup on 𝐶. If the following conditions are satisfied:(i)𝑥𝑛𝑧;(ii)limsup𝑠limsup𝑛𝑆(𝑠)𝑥𝑛𝑥𝑛=0, then 𝑧𝒮.

Lemma 2.12 (see [25]). For given 𝑥,𝑦𝐶 and (𝑥,𝑦) is a solution of the problem (1.20) if and only if 𝑥 is a fixed point of the mapping 𝐺𝐶𝐶 is defined by 𝐺(𝑥)=𝑃𝐶𝑃𝐶(𝑥𝛿𝐷𝑥)𝜏𝐵𝑃𝐶(𝑥𝛿𝐷𝑥),𝑥𝐻,(2.29) where 𝑦=𝑃𝐶(𝑥𝛿𝐷𝑥), 𝛿 and 𝜏 are positive constants and 𝐵,𝐷𝐻𝐻 are two mappings.

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by SVI(𝐶,𝐵,𝐷).

Lemma 2.13 (see [32]). Let 𝐺𝐶𝐶 be defined in Lemma 2.12. If 𝐵𝐻𝐻 is a 𝐿𝐵-Lipschitzian and relaxed (𝑐,𝑑)-cocoercive mapping and 𝐷𝐻𝐻 is a 𝐿𝐷-Lipschitz and relaxed (𝑐,𝑑)-cocoercive mapping where 𝜏2(𝑑𝑐𝐿2𝐵)/𝐿2𝐵 and 𝛿2(𝑑𝑐𝐿2𝐷)/𝐿2𝐷, then G is nonexpansive.

Lemma 2.14 (demiclosedness principle [48]). Assume that 𝑆 is a nonexpansive self-mapping of a nonempty closed convex subset 𝐶 of a real Hilbert space 𝐻. If 𝑆 has a fixed point, then 𝐼𝑆 is demiclosed; that is, whenever {𝑥𝑛} is a sequence in 𝐶 converging weakly to some 𝑥𝐶 (for short, 𝑥𝑛𝑥𝐶), and the sequence {(𝐼𝑆)𝑥𝑛} converges strongly to some 𝑦 (for short, (𝐼𝑆)𝑥𝑛𝑦), it follows that (𝐼𝑆)𝑥=𝑦. Here 𝐼 is the identity operator of 𝐻.

3. Main Results

In this section, we prove a strong convergence theorem of an iterative algorithm (3.1) for finding the solutions of a common element of the set of solutions of (1.15), the set of solutions of (1.20) for Lipschitz continuous and relaxed cocoercive mappings, the set of common fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space.

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 which 𝐶+𝐶𝐶 and let 𝑓 be a contraction of 𝐶 into itself with 𝛼(0,1). Let 𝜙 be a lower semicontinuous and convex functional from 𝐻 to and let {Θ𝑘𝐻×𝐻,𝑘=1,2,,𝑁} be a finite family of equilibrium functions satisfying conditions (H1)–(H3). Let 𝒮={𝑆(𝑠)0𝑠<} be a nonexpansive semigroup on 𝐶 and let {𝑡𝑛} be a positive real divergent sequence. Let {𝑉𝑖𝐶𝐶}𝑖=1 be a countable family of uniformly 𝑘-strict pseudo-contractions, let {𝑇𝑖𝐶𝐶}𝑖=1 be the countable family of nonexpansive mappings defined by 𝑇𝑖𝑥=𝑡𝑥+(1𝑡)𝑉𝑖𝑥,forall𝑥𝐶,forall𝑖1,𝑡[𝑘,1), let 𝑊𝑛 be the 𝑊-mapping defined by (2.12), and let 𝑊 be a mapping defined by (2.13) with 𝐹(𝑊). Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾>0 and let 0<𝛾<(1+𝜇𝛾)/𝛼, 𝐵𝐻𝐻 be a 𝐿𝐵-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿2𝐵, and let 𝐷𝐻𝐻 be a 𝐿𝐷-Lipschitz continuous and relaxed (𝑐,𝑑)-cocoercive mapping with 𝑑>𝑐𝐿2𝐷. Suppose that Ω=𝐹(𝒮)𝐹(𝑊)𝔉SVI(𝐶,𝐵,𝐷), where 𝔉=(𝑁𝑘=1MEP(Θ𝑘,𝜙)). Let 𝜇>0, 𝛾>0 and 𝑟𝑘>0,𝑘=1,2,,𝑁, which are constants. For given 𝑥1𝐻 arbitrarily and fixed 𝑢𝐻, suppose {𝑥𝑛}, {𝑦𝑛}, {𝑧𝑛} and {𝑢𝑛(𝑘)},𝑘=1,2,,𝑁 are the sequences generated iteratively by Θ1𝑢𝑛(1)𝑢,𝑥+𝜙(𝑥)𝜙𝑛(1)+1𝑟1𝐾𝑢𝑛(1)𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛(1)Θ0,𝑥𝐻,2𝑢𝑛(2)𝑢,𝑥+𝜙(𝑥)𝜙𝑛(2)+1𝑟2𝐾𝑢𝑛(2)𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛(2)Θ0,𝑥𝐻,𝑁𝑢𝑛(𝑁)𝑢,𝑥+𝜙(𝑥)𝜙𝑛(𝑁)+1𝑟𝑁𝐾𝑢𝑛(𝑁)𝐾𝑥𝑛,𝜂𝑥,𝑢𝑛(𝑁)𝑧0,𝑥𝐻,𝑛=𝑃𝐶𝑢𝑛(𝑁)𝛿𝐷𝑢𝑛(𝑁),𝑦𝑛=𝑃𝐶𝑧𝑛𝜏𝐵𝑧𝑛,𝑥𝑛+1=𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+𝛽𝑛𝑥𝑛+1𝛽𝑛𝐼𝛼𝑛1(𝐼+𝜇𝐴)𝑡𝑛𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛𝑑𝑠,(3.1) where 𝑢𝑛(1)=𝐽Θ1𝑟1𝑥𝑛,𝑢𝑛(𝑘)=𝐽Θ𝑘𝑟𝑘𝑢𝑛(𝑘1)=𝐽Θ𝑘𝑟𝑘𝐽Θ𝑘1𝑟𝑘1𝑢𝑛(𝑘2)=𝐽Θ𝑘𝑟𝑘𝐽Θ2𝑟2𝑢𝑛(1),=𝐽Θ𝑘𝑟𝑘𝐽Θ2𝑟2𝐽Θ1𝑟1𝑥𝑛,𝑘=2,3,,𝑁,(3.2)𝐽Θ𝑘𝑟𝑘𝐻𝐻,𝑘=1,2,,𝑁 is the mapping defined by (2.22) and {𝛼𝑛} and {𝛽𝑛} are two sequences in (0,1) for all 𝑛. Assume the following conditions are satisfied:(C1)𝜂𝐻×𝐻𝐻 is 𝜆-Lipschitz continuous with constant 𝜆>0 such that(a)𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0,forall𝑥,𝑦𝐻, (b)𝑥𝜂(𝑥,𝑦) is affine,(c)for each fixed 𝑦𝐻, 𝑦𝜂(𝑥,𝑦) is sequentially continuous from the weak topology to the weak topology;(C2)𝐾𝐻 is 𝜂-strongly convex with constant 𝜎>0 and its derivative 𝐾 is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant 𝜈>0 such that 𝜎>𝜆𝜈;(C3) for each 𝑘{1,2,,𝑁} and for all 𝑥𝐻, there exist bounded subsets 𝐸𝑥𝐻 and 𝑧𝑥𝐻 such that for any 𝑦𝐻𝐸𝑥, Θ𝑘𝑦,𝑧𝑥𝑧+𝜙𝑥1𝜙(𝑦)+𝑟𝑘𝐾(𝑦)𝐾𝑧(𝑥),𝜂𝑥,𝑦<0;(3.3)(C4)lim𝑛𝛼𝑛=0 and 𝑛=1𝛼𝑛=;(C5)0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1; (C6)0<𝜏<2(𝑑𝑐𝐿2𝐵)/𝐿2𝐵 and 0<𝛿<2(𝑑𝑐𝐿2𝐷)/𝐿2𝐷.Then, {𝑥𝑛} converges strongly to 𝑥Ω, which solves the following optimization problem (OP): min𝑥Ω𝜇2𝐴𝑥,𝑥1+2𝑥𝑢2𝑥,(3.4) and (𝑥,𝑦) is a solution of the general system of variational inequality problem (1.20) such that 𝑦=𝑃𝐶(𝑥𝛿𝐷𝑥).

Proof. By the condition (C4) and (C5), we may assume, without loss of generality, that 𝛼𝑛(1𝛽𝑛)(1+𝜇𝐴)1 for all 𝑛. Indeed, 𝐴 is a strongly positive bounded linear operator on 𝐻, we have ||||𝐴=sup𝐴𝑥,𝑥𝑥𝐻,𝑥=1.(3.5) Observe that 1𝛽𝑛𝐼𝛼𝑛(𝐼+𝜇𝐴)𝑥,𝑥=1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝐴𝑥,𝑥1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝐴0,(3.6) so this shows that (1𝛽𝑛)𝐼𝛼𝑛(𝐼+𝜇𝐴) is positive. It follows that 1𝛽𝑛𝐼𝛼𝑛||(𝐼+𝜇𝐴)=sup1𝛽𝑛𝐼𝛼𝑛||(𝐼+𝜇𝐴)𝑥,𝑥𝑥𝐻,𝑥=1=sup1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝐴𝑥,𝑥𝑥𝐻,𝑥=11𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾.(3.7) We shall divide the proofs into several steps.Step 1. We show that {𝑥𝑛} is bounded.
Let 𝑥Ω=𝐹(𝒮)𝐹(𝑊)(𝑁𝑘=1MEP(Θ𝑘,𝜙))SVI(𝐶,𝐵,𝐷). In fact, by the assumption that for each 𝑘{1,2,,𝑁}, 𝐽Θ𝑘𝑟𝑘 is nonexpansive. Let 𝒜𝑁=𝐽Θ𝑁𝑟𝑁𝐽Θ2𝑟2𝐽Θ1𝑟1 and 𝒜0=𝐼. Then, we have 𝑥=𝒜𝑁𝑥 and 𝑢𝑛(𝑁)=𝒜𝑁𝑥𝑛. Since 𝑥SVI(𝐶,𝐵,𝐷), then 𝑥=𝑃𝐶𝑃𝐶𝑥𝛿𝐷𝑥𝜏𝐵𝑃𝐶𝑥𝛿𝐷𝑥=𝑃𝐶𝑃𝐶(𝐼𝛿𝐷)𝒜𝑁𝑥𝜏𝐵𝑃𝐶(𝐼𝛿𝐷)𝒜𝑁𝑥.(3.8) Putting 𝑦=𝑃𝐶(𝑥𝛿𝐷𝑥)=𝑃𝐶(𝐼𝛿𝐷)𝒜𝑁𝑥, we have 𝑥=𝑃𝐶(𝑦𝜏𝐵𝑦). Since 𝑥=𝑆(𝑠)𝑥,forall𝑠0 and 𝑥=𝑊𝑛𝑥,forall𝑛1, therefore, we have 𝑥=𝒜𝑁𝑥=𝑃𝐶𝑦𝜏𝐵𝑦=𝑊𝑛𝑃𝐶𝑦𝜏𝐵𝑦=𝑆(𝑠)𝑊𝑛𝑃𝐶𝑦𝜏𝐵𝑦.(3.9) Because 𝑃𝐶 and 𝒜𝑁 are nonexpansive mappings and from Remark 1.1, we have 𝑦𝑛𝑥=𝑃𝐶𝑧𝑛𝜏𝐵𝑧𝑛𝑃𝐶𝑦𝜏𝐵𝑦(𝐼𝜏𝐵)𝑧𝑛(𝐼𝜏𝐵)𝑦𝑧𝑛𝑦=𝑃𝐶𝑢𝑛(𝑁)𝛿𝐷𝑢𝑛(𝑁)𝑃𝐶𝑥𝛿𝐷𝑥(𝐼𝛿𝐷)𝑢𝑛(𝑁)(𝐼𝛿𝐷)𝑥𝑢𝑛(𝑁)𝑥=𝒜𝑁𝑥𝑛𝒜𝑁𝑥𝑥𝑛𝑥(3.10) which yields that 𝑥𝑛+1𝑥=𝛼𝑛𝑢+𝛼𝑛𝑊𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥+𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝐼𝛼𝑛(1𝐼+𝜇𝐴)𝑡𝑛𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛𝑑𝑠𝑥𝛼𝑛𝑢+𝛼𝑛𝑊𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥+𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝛼𝑛1+𝜇𝛾𝑥𝑛𝑥𝛼𝑛𝑢+𝛼𝑛𝑊𝛾𝑓𝑛𝑥𝑛𝑥𝛾𝑓+𝛼𝑛𝑥𝛾𝑓(𝐼+𝜇𝐴)𝑥+𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝛼𝑛1+𝜇𝛾𝑥𝑛𝑥𝛼𝑛𝑢+𝛼𝑛𝑥𝛾𝛼𝑛𝑥+𝛼𝑛𝑥𝛾𝑓(𝐼+𝜇𝐴)𝑥+𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝛼𝑛1+𝜇𝛾𝑥𝑛𝑥=𝛼𝑛𝑥𝑢+𝛾𝑓(𝐼+𝜇𝐴)𝑥+1𝛼𝑛1+𝜇𝛾+𝛼𝑛𝑥𝛾𝛼𝑛𝑥=1𝛼𝑛1+𝜇𝛾𝑥𝛾𝛼𝑛𝑥+𝛼𝑛1+𝜇𝛾𝑥𝛾𝛼𝑢+𝛾𝑓(𝐼+𝜇𝐴)𝑥1+𝜇𝛾.𝛾𝛼(3.11) It follows from (3.11) and induction that 𝑥𝑛𝑥𝑥max1,𝑥𝑝𝑢+𝛾𝑓(𝐼+𝜇𝐴)𝑥1+𝜇𝛾𝛾𝛼,𝑛1.(3.12) Hence, {𝑥𝑛} is bounded, so are {𝑦𝑛}, {𝑧𝑛}, {𝑊𝑛𝑥𝑛}, {𝑓(𝑊𝑛𝑥𝑛)}, {𝑢𝑛(𝑘)} for all 𝑘=1,2,,𝑁 and {𝐾𝑛𝑊𝑛𝑦𝑛}, where 𝐾𝑛=(1/𝑡𝑛)𝑡𝑛0𝑆(𝑠)𝑑𝑠.
Step 2. We prove that lim𝑛𝑥𝑛+1𝑥𝑛=0 and lim𝑛𝑢(𝑁)𝑛+1𝑢𝑛(𝑁)=0.
Again, from Remark 1.1, we have the following estimates: 𝑦𝑛+1𝑦𝑛=𝑃𝐶𝑧𝑛+1𝜏𝐵𝑧𝑛+1𝑃𝐶𝑧𝑛𝜏𝐵𝑧𝑛𝑧𝑛+1𝜏𝐵𝑧𝑛+1𝑧𝑛𝜏𝐵𝑧𝑛𝑧𝑛+1𝑧𝑛=𝑃𝐶𝑢(𝑁)𝑛+1𝛿𝐷𝑢(𝑁)𝑛+1𝑃𝐶𝑢𝑛(𝑁)𝛿𝐷𝑢𝑛(𝑁)𝑢(𝑁)𝑛+1𝛿𝐷𝑢(𝑁)𝑛+1𝑢𝑛(𝑁)𝛿𝐷𝑢𝑛(𝑁)𝑢(𝑁)𝑛+1𝑢𝑛(𝑁)=𝒜𝑁𝑥𝑛+1𝒜𝑁𝑥𝑛𝑥𝑛+1𝑥𝑛.(3.13) On the other hand, since 𝑇𝑖 and 𝑈𝑛,𝑖 are nonexpansive, we have 𝑊𝑛+1𝑦𝑛𝑊𝑛𝑦𝑛=𝜇1𝑇1𝑈𝑛+1,2𝑦𝑛𝜇1𝑇1𝑈𝑛,2𝑦𝑛𝜇1𝑈𝑛+1,2𝑦𝑛𝑈𝑛,2𝑦𝑛=𝜇1𝜇2𝑇2𝑈𝑛+1,3𝑦𝑛𝜇2𝑇2𝑈𝑛,3𝑦𝑛𝜇1𝜇2𝑈𝑛+1,3𝑦𝑛𝑈𝑛,3𝑦𝑛𝜇1𝜇2𝜇𝑛𝑈𝑛+1,𝑛+1𝑦𝑛𝑈𝑛,𝑛+1𝑦𝑛𝑀1𝑛𝑖=1𝜇𝑖,(3.14) where 𝑀10 is a constant such that 𝑈𝑛+1,𝑛+1𝑦𝑛𝑈𝑛,𝑛+1𝑦𝑛𝑀1 for all 𝑛0. It follows from (3.13) and (3.14) that we have 𝑊𝑛+1𝑦𝑛+1𝑊𝑛𝑦𝑛𝑊𝑛+1𝑦𝑛+1𝑊𝑛+1𝑦𝑛+𝑊𝑛+1𝑦𝑛𝑊𝑛𝑦𝑛𝑦𝑛+1𝑦𝑛+𝑀1𝑛𝑖=1𝜇𝑖𝑥𝑛+1𝑥𝑛+𝑀1𝑛𝑖=1𝜇𝑖.(3.15) It follows that 𝐾𝑛+1𝑊𝑛+1𝑦𝑛+1𝐾𝑛𝑊𝑛𝑦𝑛=1𝑡𝑛+1𝑡𝑛+10𝑆(𝑠)𝑊𝑛+1𝑦𝑛+11𝑑𝑠𝑡𝑛𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛1𝑑𝑠𝑡𝑛+1𝑡𝑛+10𝑆(𝑠)𝑊𝑛+1𝑦𝑛+1𝑆(𝑠)𝑊𝑛𝑦𝑛+1𝑑𝑠𝑡𝑛+1𝑡𝑛+10𝑆(𝑠)𝑊𝑛𝑦𝑛1𝑑𝑠𝑡𝑛𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛𝑊𝑑𝑠𝑛+1𝑦𝑛+1𝑊𝑛𝑦𝑛+1𝑡𝑛+1𝑡𝑛+1𝑡𝑛𝑆(𝑠)𝑊𝑛𝑦𝑛1𝑑𝑠+𝑡𝑛+1𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛1𝑑𝑠𝑡𝑛𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛𝑊𝑑𝑠𝑛+1𝑦𝑛+1𝑊𝑛𝑦𝑛+1𝑡𝑛+1𝑡𝑛+1𝑡𝑛𝑆(𝑠)𝑊𝑛𝑦𝑛+||||1𝑑𝑠𝑡𝑛+11𝑡𝑛||||𝑡𝑛0𝑆(𝑠)𝑊𝑛𝑦𝑛𝑊𝑑𝑠𝑛+1𝑦𝑛+1𝑊𝑛𝑦𝑛𝑡+21𝑛𝑡𝑛+1𝑀2𝑥𝑛+1𝑥𝑛+𝑀1𝑛𝑖=1𝜇𝑖𝑡+21𝑛𝑡𝑛+1𝑀2,(3.16) where 𝑀2=max{𝑆(𝑠)𝑊𝑛𝑦𝑛}.
Setting 𝑥𝑛+1=(1𝛽𝑛)𝑣𝑛+𝛽𝑛𝑥𝑛, for all 𝑛1, we have 𝑣𝑛=𝑥𝑛+1𝛽𝑛𝑥𝑛1𝛽𝑛=𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+1𝛽𝑛𝐼𝛼𝑛𝐾(𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛1𝛽𝑛.(3.17) Then, we obtain 𝑣𝑛+1𝑣𝑛=𝛼𝑛+1𝑊𝑢+𝛾𝑓𝑛+1𝑥𝑛+1+1𝛽𝑛+1𝐼𝛼𝑛+1𝐾(𝐼+𝜇𝐴)𝑛+1𝑊𝑛+1𝑦𝑛+11𝛽𝑛+1𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+1𝛽𝑛𝐼𝛼𝑛(𝐾𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛1𝛽𝑛=𝛼𝑛+11𝛽𝑛+1𝑊𝑢+𝛾𝑓𝑛+1𝑥𝑛+1𝛼𝑛1𝛽𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+𝐾𝑛+1𝑊𝑛+1𝑦𝑛+1𝐾𝑛𝑊𝑛𝑦𝑛+𝛼𝑛1𝛽𝑛(𝐼+𝜇𝐴)𝐾𝑛𝑊𝑛𝑦𝑛𝛼𝑛+11𝛽𝑛+1(𝐼+𝜇𝐴)𝐾𝑛+1𝑊𝑛+1𝑦𝑛+1=𝛼𝑛+11𝛽𝑛+1𝑊𝑢+𝛾𝑓𝑛+1𝑥𝑛+1(𝐼+𝜇𝐴)𝐾𝑛+1𝑊𝑛+1𝑦𝑛+1+𝛼𝑛1𝛽𝑛(𝐼+𝜇𝐴)𝐾𝑛𝑊𝑛𝑦𝑛𝑊𝑢𝛾𝑓𝑛𝑥𝑛+𝐾𝑛+1𝑊𝑛+1𝑦𝑛+1𝐾𝑛𝑊𝑛𝑦𝑛.(3.18) It follows from (3.16) and (3.18) that 𝑣𝑛+1𝑣𝑛𝑥𝑛+1𝑥𝑛𝛼𝑛+11𝛽𝑛+1𝑊𝑢+𝛾𝑓𝑛+1𝑥𝑛+1+(𝐼+𝜇𝐴)𝐾𝑛+1𝑊𝑛+1𝑦𝑛+1+𝛼𝑛1𝛽𝑛(𝐼+𝜇𝐴)𝐾𝑛𝑊𝑛𝑦𝑛𝑊+𝑢+𝛾𝑓𝑛𝑥𝑛+𝑀1𝑛𝑖=1𝜇𝑖𝑡+21𝑛𝑡𝑛+1𝑀2.(3.19) By the conditions (C4), (C5) and from 𝑡𝑛(0,), 𝑡𝑛 and 0<𝜇𝑖𝑏<1,forall𝑖1, we have limsup𝑛𝑣𝑛+1𝑣𝑛𝑥𝑛+1𝑥𝑛0.(3.20) Hence, by Lemma 2.7, we obtain lim𝑛𝑣𝑛𝑥𝑛=0.(3.21) It follows that lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛1𝛽𝑛𝑣𝑛𝑥𝑛=0.(3.22) Applying (3.22) into (3.13), we obtain that lim𝑛𝑦𝑛+1𝑦𝑛=lim𝑛𝑧𝑛+1𝑧𝑛=lim𝑛𝑢(𝑁)𝑛+1𝑢𝑛(𝑁)=0.(3.23)
Step 3. We show that lim𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑦𝑛=0, lim𝑛𝑦𝑛𝑆(𝑠)𝑦𝑛=0, and lim𝑛𝑢𝑛(𝑘+1)𝑢𝑛(𝑘)=0, where 𝐾𝑛=(1/𝑡𝑛)𝑡𝑛0𝑆(𝑠)𝑑𝑠.
Since 𝑥𝑛+1=𝛼𝑛(𝑢+𝛾𝑓(𝑊𝑛𝑥𝑛))+𝛽𝑛𝑥𝑛+((1𝛽𝑛)𝐼𝛼𝑛(𝐼+𝜇𝐴))𝐾𝑛𝑊𝑛𝑦𝑛, we have 𝑥𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝐾𝑛𝑊𝑛𝑦𝑛=𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+𝛽𝑛𝑥𝑛+1𝛽𝑛𝐼𝛼𝑛𝐾(𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛𝐾𝑛𝑊𝑛𝑦𝑛=𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝐾𝑛𝑊𝑛𝑦𝑛+𝛽𝑛𝑥𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+(𝐼+𝜇𝐴)𝐾𝑛𝑊𝑛𝑦𝑛+𝛽𝑛𝑥𝑛𝐾𝑛𝑊𝑛𝑦𝑛,(3.24) that is 𝑥𝑛𝐾𝑛𝑊𝑛𝑦𝑛11𝛽𝑛𝑥𝑛𝑥𝑛+1+𝛼𝑛1𝛽𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛+(𝐼+𝜇𝐴)𝐾𝑛𝑊𝑛𝑦𝑛.(3.25) By (C4), (C5), and (3.22) it follows that lim𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥𝑛=0.(3.26)
Since 𝐽Θ𝑁𝑟𝑁𝐶𝐶 is firmly nonexpansive, 𝑢𝑛(𝑁)=𝒜𝑁𝑥𝑛, where 𝒜𝑁=𝐽Θ𝑁𝑟𝑁𝐽Θ2𝑟2𝐽Θ1𝑟1 and 𝑥Ω, we have 𝑢𝑛(𝑁)𝑥2=𝒜𝑁𝑥𝑛𝒜𝑁𝑥2𝒜𝑁𝑥𝑛𝒜𝑁𝑥,𝑥𝑛𝑥=𝑢𝑛(𝑁)𝑥,𝑥𝑛𝑥=12𝑢𝑛(𝑁)𝑥2+𝑥𝑛𝑥2𝑥𝑛𝑢𝑛(𝑁)2,(3.27) and hence 𝑢𝑛(𝑁)𝑥2𝑥𝑛𝑥2𝑥𝑛𝑢𝑛(𝑁)2.(3.28) Observe that 𝑥𝑛+1𝑥2=1𝛽𝑛𝐼𝛼𝑛𝐾(𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛𝑥+𝛽𝑛𝑥𝑛𝑥+𝛼𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥2=1𝛽𝑛𝐼𝛼𝑛𝐾(𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛𝑥+𝛽𝑛𝑥𝑛𝑥2+𝛼2𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥2+2𝛽𝑛𝛼𝑛𝑥𝑛𝑥𝑊,𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥+2𝛼𝑛1𝛽𝑛𝐼𝛼𝑛𝐾(𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛𝑥𝑊,𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾𝐾𝑛𝑊𝑛𝑦𝑛𝑥+𝛽𝑛𝑥𝑛𝑥2+𝛼2𝑛𝑊𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥2+2𝛽𝑛𝛼𝑛𝑥𝑛𝑥𝑊,𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥+2𝛼𝑛1𝛽𝑛𝐼𝛼𝑛𝐾(𝐼+𝜇𝐴)𝑛𝑊𝑛𝑦𝑛𝑥𝑊,𝑢+𝛾𝑓𝑛𝑥𝑛(𝐼+𝜇𝐴)𝑥=1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾𝐾𝑛𝑊𝑛𝑦𝑛𝑥+𝛽𝑛𝑥𝑛𝑥2+𝑐𝑛1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾2𝐾𝑛𝑊𝑛𝑦𝑛𝑥2+𝛽2𝑛𝑥𝑛𝑥2+21𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥𝑥𝑛𝑥+𝑐𝑛1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾2𝐾𝑛𝑊𝑛𝑦𝑛𝑥2+𝛽2𝑛𝑥𝑛𝑥2+1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥2+𝑥𝑛𝑥2+𝑐𝑛=1𝛼𝑛𝛼𝑛𝜇𝛾221𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛+𝛽2𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥2+𝛽2𝑛𝑥𝑛𝑥2+1𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛𝛽2𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥2+𝑥𝑛𝑥2+𝑐𝑛=1𝛼𝑛𝛼𝑛𝜇𝛾21𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛𝐾𝑛𝑊𝑛𝑦𝑛𝑥2+1𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛𝑥𝑛𝑥2+𝑐𝑛1𝛼𝑛𝛼𝑛𝜇𝛾1𝛽𝑛𝛼𝑛𝛼𝑛𝜇𝛾𝑦𝑛𝑥2+1𝛼𝑛𝛼𝑛𝜇𝛾𝛽𝑛𝑥𝑛𝑥2+𝑐𝑛,(3.