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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 804538, 16 pages
http://dx.doi.org/10.1155/2012/804538
Research Article

Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

1School of Science, University of Phayao, Phayao 56000, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 13 February 2012; Accepted 9 March 2012

Academic Editor: Rudong Chen

Copyright © 2012 Kamonrat Nammanee et al. 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 hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

1. Introduction

Let 𝐸 be a real Banach space and 𝐶 a nonempty subset of 𝐸. Let 𝐸 be the dual space of 𝐸. We denote the value of 𝑥𝐸 at 𝑥𝐸 by 𝑥,𝑥. Let 𝑇𝐶𝐶 be a nonlinear mapping. We denote by 𝐹(𝑇) the fixed points set of 𝑇, that is, 𝐹(𝑇)={𝑥𝐶𝑥=𝑇𝑥}. Let 𝐴𝐸2𝐸 be a set-valued mapping. We denote 𝐷(𝐴) by the domain of 𝐴, that is, 𝐷(𝐴)={𝑥𝐸𝐴𝑥} and also denote 𝐺(𝐴) by the graph of 𝐴, that is, 𝐺(𝐴)={(𝑥,𝑥)𝐸×𝐸𝑥𝐴𝑥}. A set-valued mapping 𝐴 is said to be monotone if 𝑥𝑦,𝑥𝑦0 whenever (𝑥,𝑥),(𝑦,𝑦)𝐺(𝐴). It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operators on 𝐸. It is known that if 𝐴 is maximal monotone, then the set 𝐴1(0)={𝑧𝐸0𝐴𝑧} is closed and convex.

The problem of finding a zero point of maximal monotone operators plays an important role in optimizations. This is because it can be reformulated to a convex minimization problem and a variational inequality problem. Many authors have studied the convergence of such problems in various spaces (see, e.g., [116]). Initiated by Martinet [17], in a real Hilbert space 𝐻, Rockafellar [18] introduced the following iterative scheme: 𝑥1𝐻 and 𝑥𝑛+1=𝐽𝜆𝑛𝑥𝑛,𝑛1,(1.1) where {𝜆𝑛}(0,), 𝐽𝜆 is the resolvent of 𝐴 defined by 𝐽𝜆=𝐽𝜆𝐴=(𝐼+𝜆𝐴)1 for all 𝜆>0, and 𝐴 is a maximal monotone operator on 𝐻. Such an algorithm is called the proximal point algorithm. It was proved that the sequence {𝑥𝑛} generated by (1.1) converges weakly to an element in 𝐴1(0) provided that liminf𝑛𝜆𝑛>0. Recently, Kamimura and Takahashi [19] introduced the following iteration in a real Hilbert space: 𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑥𝑛+1𝛼𝑛𝐽𝜆𝑛𝑥𝑛,𝑛1,(1.2) where {𝛼𝑛}[0,1] and {𝜆𝑛}(0,). The weak convergence theorems are also established in a real Hilbert space under suitable conditions imposed on {𝛼𝑛} and {𝜆𝑛}.

In 2004, Kamimura et al. [20] extended the above iteration process to a much more general setting. In fact, they proposed the following algorithm: 𝑥1𝐸 and 𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝐽𝜆𝑛𝑥𝑛,𝑛1,(1.3) where {𝛼𝑛}[0,1], {𝜆𝑛}(0,), and 𝐽𝜆=𝐽𝜆𝐴=(𝐽+𝜆𝐴)1𝐽 for all 𝜆>0. They proved, in a uniformly smooth and uniformly convex Banach space, a weak convergence theorem.

Let 𝐹𝐶×𝐶, where is the set of real numbers, be a bifunction. The equilibrium problem is to find 𝑥𝐶 such that 𝐹(𝑥,𝑦)0,𝑦𝐶.(1.4) The solutions set of (1.4) is denoted by EP(𝐹).

For solving the equilibrium problem, we assume that(A1)𝐹(𝑥,𝑥)=0 for all 𝑥𝐶,(A2)𝐹 is monotone, that is 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)0 for all 𝑥,𝑦𝐶,(A3) for all 𝑥,𝑦,𝑧𝐶, limsup𝑡0𝐹(𝑡𝑧+(1𝑡)𝑥,𝑦)𝐹(𝑥,𝑦),(A4) for all 𝑥𝐶,𝐹(𝑥,) is convex and lower semi-continuous.

Recently, Takahashi and Zembayashi [21] introduced the following iterative scheme for a relatively nonexpansive mapping 𝑇𝐶𝐶 in a uniformly smooth and uniformly convex Banach space: 𝑥1𝐶 and 𝐶1𝑦=𝐶,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑥𝑛,𝑢𝑛𝑢𝐶suchthat𝐹𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝐶0𝑦𝐶,𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑢𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥1,𝑛1,(1.5) where {𝛼𝑛}[0,1] and {𝑟𝑛}(0,). Such an algorithm is called the shrinking projection method which was introduced by Takahashi et al. [22]. They proved that the sequence {𝑥𝑛} converges strongly to an element in 𝐹(𝑇)EP(𝐹) under appropriate conditions. The equilibrium problem has been intensively studied by many authors (see, e.g., [2331]).

Motivated by the previous results, we introduce a hybrid-iterative scheme for finding a zero point of maximal monotone operators 𝐴𝑖𝐸2𝐸 (𝑖=1,2,,𝑁) which is also a common element in the solutions set of an equilibrium problem for 𝐹 and in the fixed points set of weak relatively nonexpansive mappings 𝑇𝑖𝐶𝐶  (𝑖=1,2,). Using the projection technique, we also prove that the sequence generated by a constructed algorithm converges strongly to an element in [𝑁𝑖=1𝐴𝑖1(0)][𝑖=1𝐹(𝑇𝑖)]EP(𝐹) in a uniformly smooth and uniformly convex Banach space. Finally, we apply our results to a system of convex minimization problems.

2. Preliminaries and Lemmas

In this section, we give some useful preliminaries and lemmas which will be used in the sequel.

Let 𝐸 be a real Banach space and let 𝑈={𝑥𝐸𝑥=1} be the unit sphere of 𝐸. A Banach space 𝐸 is said to be strictly convex if for any 𝑥,𝑦𝑈, 𝑥𝑦implies𝑥+𝑦<2.(2.1) A Banach space 𝐸 is said to be uniformly convex if, for each 𝜀(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦𝑈, 𝑥𝑦𝜀implies𝑥+𝑦<2(1𝛿).(2.2) It is known that a uniformly convex Banach space is reflexive and strictly convex. The function 𝛿[0,2][0,1] which is called the modulus of convexity of 𝐸 is defined as follows: 𝛿(𝜀)=inf1𝑥+𝑦2.𝑥,𝑦𝐸,𝑥=𝑦=1,𝑥𝑦𝜀(2.3) Then 𝐸 is uniformly convex if and only if 𝛿(𝜀)>0 for all 𝜀(0,2]. A Banach space 𝐸 is said to be smooth if the limit lim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.4) exists for all 𝑥,𝑦𝑈. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for 𝑥,𝑦𝑈. The duality mapping 𝐽𝐸2𝐸 is defined by 𝑥𝐽(𝑥)=𝐸𝑥,𝑥=𝑥2=𝑥2(2.5) for all 𝑥𝐸. It is also known that if 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸 (see [32] for more details).

Let 𝐸 be a smooth Banach space. The function 𝜙𝐸×𝐸 is defined by 𝜙(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2(2.6) for all 𝑥,𝑦𝐸. From the definition of 𝜙, we see that ()𝑥𝑦2)𝜙(𝑥,𝑦)(𝑥+𝑦2,𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2𝑥𝑧,𝐽𝑧𝐽𝑦(2.7) for all 𝑥,𝑦,𝑧𝐸.

Let 𝐶 be a closed and convex subset of 𝐸, and let 𝑇 be a mapping from 𝐶 into itself. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 [33] if 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛𝑥𝑛𝑇𝑥𝑛=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 is said to be relatively nonexpansive [33, 34] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑝𝐹(𝑇) and 𝑥𝐶. A point 𝑝 in 𝐶 is said to be a strong asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥𝑛} which converges strongly to 𝑝 such that lim𝑛𝑥𝑛𝑇𝑥𝑛=0. The set of strong asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 is said to be weak relatively nonexpansive [35] if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑝𝐹(𝑇) and 𝑥𝐶. It is obvious by definition that the class of weak relatively nonexpansive mappings contains the class of relatively nonexpansive mappings. Indeed, for any mapping 𝑇𝐶𝐶, we see that 𝐹(𝑇)𝐹(𝑇)𝐹(𝑇). Therefore, if 𝑇 is a relatively nonexpansive mapping, then 𝐹(𝑇)=𝐹(𝑇)=𝐹(𝑇).

Nontrivial examples of weak relatively nonexpansive mappings which are not relatively nonexpansive can be found in [36].

Let 𝐸 be a reflexive, strictly convex and smooth Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. The generalized projection mapping, introduced by Alber [37], is a mapping Π𝐶𝐸𝐶, that assigns to an arbitrary point 𝑥𝐸 the minimum point of the function 𝜙(𝑦,𝑥), that is, Π𝐶(𝑥)=𝑥, where 𝑥 is the solution to the minimization problem 𝜙𝑥,𝑥=min{𝜙(𝑦,𝑥)𝑦𝐶}.(2.8) In a Hilbert space, Π𝐶 is coincident with the metric projection denoted by 𝑃𝐶.

Lemma 2.1 (see [38]). Let 𝐸 be a uniformly convex and smooth Banach space and let {𝑥𝑛},{𝑦𝑛} be two sequences in 𝐸. If lim𝑛𝜙(𝑥𝑛,𝑦𝑛)=0 and either {𝑥𝑛} or {𝑦𝑛} is bounded, then lim𝑛𝑥𝑛𝑦𝑛=0.

Lemma 2.2 (see [37, 38]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex and reflexive Banach space 𝐸, let 𝑥𝐸 and let 𝑧𝐶. Then 𝑧=Π𝐶(𝑥) if and only if 𝑦𝑧,𝐽𝑥𝐽𝑧0 for all 𝑦𝐶.

Lemma 2.3 (see [37, 38]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Then 𝜙𝑥,Π𝐶𝑦Π+𝜙𝐶𝑦,𝑦𝜙(𝑥,𝑦)𝑥𝐶,𝑦𝐸.(2.9)

Lemma 2.4 (see [39]). Let 𝐸 be a smooth and strictly convex Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑇 be a mapping from 𝐶 into itself such that 𝐹(𝑇) is nonempty and 𝜙(𝑢,𝑇𝑥)𝜙(𝑢,𝑥) for all (𝑢,𝑥)𝐹(𝑇)×𝐶. Then 𝐹(𝑇) is closed and convex.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space. It is known that 𝐴𝐸2𝐸 is maximal monotone if and only if 𝑅(𝐽+𝜆𝐴)=𝐸 for all 𝜆>0, where 𝑅(𝐵) stands for the range of 𝐵.

Define the resolvent of 𝐴 by 𝐽𝜆𝐴=(𝐽+𝜆𝐴)1𝐽 for all 𝜆>0. It is known that 𝐽𝜆𝐴 is a single-valued mapping from 𝐸 to 𝐷(𝐴) and 𝐴1(0)=𝐹(𝐽𝜆𝐴) for all 𝜆>0. For each 𝜆>0, the Yosida approximation of 𝐴 is defined by 𝐴𝜆1(𝑥)=𝜆𝐽(𝑥)𝐽𝐽𝜆𝐴(𝑥)(2.10) for all 𝑥𝐸. We know that 𝐴𝜆(𝑥)𝐴(𝐽𝜆𝐴(𝑥)) for all 𝜆>0 and 𝑥𝐸.

Lemma 2.5 (see [5]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐴𝐸×𝐸 be a maximal monotone operator with 𝐴1(0), and let 𝐽𝜆𝐴=(𝐽+𝜆𝐴)1𝐽 for each 𝜆>0. Then 𝜙𝑝,𝐽𝜆𝐴𝐽(𝑥)+𝜙𝜆𝐴(𝑥),𝑥𝜙(𝑝,𝑥)(2.11) for all 𝜆>0, 𝑝𝐴1(0), and 𝑥𝐸.

Lemma 2.6 (see[40]). Let 𝐶 be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, let 𝐹 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4), and let 𝑟>0 and 𝑥𝐸. Then, there exists 𝑧𝐶 such that 1𝐹(𝑧,𝑦)+𝑟𝑦𝑧,𝐽𝑧𝐽𝑥0,𝑦𝐶.(2.12)

Lemma 2.7 (see [41]). Let 𝐶 be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸, and let 𝐹 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4). For all 𝑟>0 and 𝑥𝐸, define the mapping 𝑇𝑟𝐸𝐶 as follows: 𝑇𝑟1(𝑥)=𝑧𝐶𝐹(𝑧,𝑦)+𝑟.𝑦𝑧,𝐽𝑧𝐽𝑥0,𝑦𝐶(2.13) Then, the following holds:(1)𝑇𝑟 is single-valued;(2)𝑇𝑟 is a firmly nonexpansive-type mapping [42], that is, for all 𝑥,𝑦𝐸, 𝑇𝑟𝑥𝑇𝑟𝑦,𝐽𝑇𝑟𝑥𝐽𝑇𝑟𝑦𝑇𝑟𝑥𝑇𝑟𝑦,𝐽𝑥𝐽𝑦;(2.14)(3)𝐹(𝑇𝑟)=EP(𝐹);(4)EP(𝐹) is closed and convex.

Lemma 2.8 (see [41]). Let 𝐶 be a closed and convex subset of a smooth, strictly, and reflexive Banach space 𝐸, let 𝐹 be a bifunction from 𝐶×𝐶 to satisfying (A1)–(A4), let 𝑟>0. Then 𝜙𝑝,𝑇𝑟𝑥𝑇+𝜙𝑟𝑥,𝑥𝜙(𝑝,𝑥),(2.15) for all 𝑥𝐸 and 𝑝𝐹(𝑇𝑟).

3. Strong Convergence Theorems

In this section, we are now ready to prove our main theorem.

Theorem 3.1. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty, closed and convex subset of 𝐸. Let 𝐴𝑖𝐸2𝐸 (𝑖=1,2,,𝑁) be maximal monotone operators, let 𝐹𝐶×𝐶 be a bifunction, and let 𝑇𝑖𝐶𝐶  (𝑖=1,2,) be weak relatively nonexpansive mappings such that =[𝑁𝑖=1𝐴𝑖1(0)][𝑖=1𝐹(𝑇𝑖)]𝐸𝑃(𝐹). Let {𝑒𝑛}𝑛=1𝐸 be the sequence such that lim𝑛𝑒𝑛=0. Define the sequence {𝑥𝑛}𝑛=1 in 𝐶 as follows: 𝑥1𝐶1𝑦=𝐶,𝑛=𝐽𝜆𝑁𝑛𝐴𝑁𝐽𝜆𝑛𝑁1𝐴𝑁1𝐽𝜆1𝑛𝐴1𝑥𝑛+𝑒𝑛,𝑢𝑛=𝑇𝑟𝑛𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛sup𝑖1𝜙𝑧,𝑇𝑖𝑢𝑛𝜙𝑧,𝑥𝑛+𝑒𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥1,𝑛1.(3.1) If liminf𝑛𝜆𝑖𝑛>0 for each 𝑖=1,2,,𝑁 and liminf𝑛𝑟𝑛>0, then the sequence {𝑥𝑛} converges strongly to 𝑞=Π(𝑥1).

Proof. We split the proof into several steps as follows.
Step  1. 𝐶𝑛 for all 𝑛1.
From Lemma 2.4, we know that 𝑖=1𝐹(𝑇𝑖) is closed and convex. From Lemma 2.7(4), we also know that EP(𝐹) is closed and convex. On the other hand, since 𝐴𝑖 (𝑖=1,2,,𝑁) are maximal monotone, 𝐴𝑖1(0) are closed and convex for each 𝑖=1,2,,𝑁; consequently, 𝑁𝑖=1𝐴𝑖1(0) is closed and convex. Hence is a nonempty, closed, and convex subset of 𝐶.
We next show that 𝐶𝑛 is closed and convex for all 𝑛1. Obviously, 𝐶1=𝐶 is closed and convex. Now suppose that 𝐶𝑘 is closed and convex for some 𝑘. Then, for each 𝑧𝐶𝑘 and 𝑖1, we see that 𝜙(𝑧,𝑇𝑖𝑢𝑘)𝜙(𝑧,𝑥𝑘) is equivalent to 2𝑧,𝐽𝑥𝑘2𝑧,𝐽𝑇𝑖𝑢𝑘𝑥𝑘2𝑇𝑖𝑢𝑘2.(3.2) By the construction of the set 𝐶𝑘+1, we see that 𝐶𝑘+1=𝑧𝐶𝑘sup𝑖1𝜙𝑧,𝑇𝑖𝑢𝑘𝜙𝑧,𝑥𝑘=𝑖=1𝑧𝐶𝑘𝜙𝑧,𝑇𝑖𝑢𝑘𝜙𝑧,𝑥𝑘.(3.3) Hence, 𝐶𝑘+1 is closed and convex. This shows, by induction, that 𝐶𝑛 is closed and convex for all 𝑛1. It is obvious that 𝐶1=𝐶. Now, suppose that 𝐶𝑘 for some 𝑘. For any 𝑝, by Lemmas 2.5 and 2.8, we have 𝜙𝑝,𝑇𝑖𝑢𝑘𝜙𝑝,𝑢𝑘=𝜙𝑝,𝑇𝑟𝑘𝑦𝑘𝜙𝑝,𝑦𝑘=𝜙𝑝,𝐽𝜆𝑁𝑘𝐴𝑁𝐽𝜆𝑘𝑁1𝐴𝑁1𝐽𝜆1𝑘𝐴1𝑥𝑘+𝑒𝑘𝜙𝑝,𝐽𝜆𝑘𝑁1𝐴𝑁1𝐽𝜆𝑘𝑁2𝐴𝑁2𝐽𝜆1𝑘𝐴1𝑥𝑘+𝑒𝑘𝜙𝑝,𝐽𝜆2𝑘𝐴2𝐽𝜆1𝑘𝐴1𝑥𝑘+𝑒𝑘𝜙𝑝,𝐽𝜆1𝑘𝐴1𝑥𝑘+𝑒𝑘𝜙𝑝,𝑥𝑘+𝑒𝑘.(3.4) This shows that 𝐶𝑘+1. By induction, we can conclude that 𝐶𝑛 for all 𝑛1.
Step  2. lim𝑛𝜙(𝑥𝑛,𝑥1) exists.
From 𝑥𝑛=Π𝐶𝑛(𝑥1) and 𝑥𝑛+1=Π𝐶𝑛+1(𝑥1)𝐶𝑛+1𝐶𝑛, we have 𝜙𝑥𝑛,𝑥1𝑥𝜙𝑛+1,𝑥1,𝑛1.(3.5) From Lemma 2.3, for any 𝑝𝐶𝑛, we have 𝜙𝑥𝑛,𝑥1Π=𝜙𝐶𝑛𝑥1,𝑥1𝜙𝑝,𝑥1𝜙𝑝,𝑥𝑛𝜙𝑝,𝑥1.(3.6) Combining (3.5) and (3.6), we conclude that lim𝑛𝜙(𝑥𝑛,𝑥1) exists.
Step  3. lim𝑛𝐽(𝑇𝑖𝑦𝑛)𝐽(𝑥𝑛+𝑒𝑛)=0.
Since 𝑥𝑚=Π𝐶𝑚(𝑥1)𝐶𝑚𝐶𝑛 for 𝑚>𝑛1, by Lemma 2.3, it follows that 𝜙𝑥𝑚,𝑥𝑛𝑥=𝜙𝑚,Π𝐶𝑛𝑥1𝑥𝜙𝑚,𝑥1Π𝜙𝐶𝑛𝑥1,𝑥1𝑥=𝜙𝑚,𝑥1𝑥𝜙𝑛,𝑥1.(3.7) Letting 𝑚,𝑛, we have 𝜙(𝑥𝑚,𝑥𝑛)0. By Lemma 2.1, it follows that 𝑥𝑚𝑥𝑛0 as 𝑚,𝑛. Therefore, {𝑥𝑛} is a Cauchy sequence. By the completeness of the space 𝐸 and the closedness of 𝐶, we can assume that 𝑥𝑛𝑞𝐶 as 𝑛. In particular, we obtain that lim𝑛𝑥𝑛+1𝑥𝑛=0.(3.8) Since 𝑒𝑛0, we have lim𝑛𝑥𝑛+1𝑥𝑛+𝑒𝑛=0.(3.9) Since 𝑥𝑛+1=Π𝐶𝑛+1(𝑥1)𝐶𝑛+1, for each 𝑖1, 𝜙𝑥𝑛+1,𝑇𝑖𝑢𝑛𝑥𝜙𝑛+1,𝑥𝑛+𝑒𝑛=𝑥𝑛+1𝑥,𝐽𝑛+1𝑥𝐽𝑛+𝑒𝑛+𝑥𝑛+1𝑥𝑛+𝑒𝑛𝑥,𝐽𝑛+1.(3.10) Since 𝐸 is uniformly smooth, 𝐽 is uniformly norm-to-norm continuous on bounded sets. It follows from (3.9) and by the boundedness of {𝑥𝑛} that lim𝑛𝜙𝑥𝑛+1,𝑇𝑖𝑢𝑛=0(3.11) for all 𝑖=1,2,. So from Lemma 2.1, we have lim𝑛𝑥𝑛+1𝑇𝑖𝑢𝑛=0,lim𝑛𝑇𝑖𝑢𝑛𝑥𝑛=0,(3.12) and, since 𝑒𝑛0, therefore lim𝑛𝑇𝑖𝑢𝑛𝑥𝑛+𝑒𝑛=0,(3.13) for all 𝑖=1,2,. Since 𝐽 is uniformly norm-to-norm continuous on bounded subsets of 𝐸, lim𝑛𝐽𝑇𝑖𝑢𝑛𝑥𝐽𝑛+𝑒𝑛=0(3.14) for all 𝑖=1,2,.
Step  4. lim𝑛𝑇𝑖𝑢𝑛𝑢𝑛=0 for all 𝑖=1,2,.
Denote that Θ𝑖𝑛=𝐽𝜆𝑖𝑛𝐴𝑖𝐽𝜆𝑛𝑖1𝐴𝑖1𝐽𝜆1𝑛𝐴1 for each 𝑖{1,2,,𝑁} and Θ0𝑛=𝐼 for each 𝑛1. We note that 𝑦𝑛=Θ𝑁𝑛(𝑥𝑛+𝑒𝑛) for each 𝑛1.
To this end, we will show that lim𝑛𝐽Θ𝑖𝑛𝑥𝑛+𝑒𝑛Θ𝐽𝑛𝑖1𝑥𝑛+𝑒𝑛=0(3.15) for all 𝑖=1,2,,𝑁.
For any 𝑝, by (3.4), we see that 𝜙𝑝,Θ𝑛𝑁1𝑥𝑛+𝑒𝑛𝜙𝑝,Θ𝑛𝑁2𝑥𝑛+𝑒𝑛𝜙𝑝,Θ𝑛𝑁3𝑥𝑛+𝑒𝑛𝑥𝜙𝑝,𝑛+𝑒𝑛.(3.16) Since 𝑝, by Lemma 2.5 and (3.16), it follows that 𝜙𝑦𝑛,Θ𝑛𝑁1𝑥𝑛+𝑒𝑛𝜙𝑝,Θ𝑛𝑁1𝑥𝑛+𝑒𝑛𝜙𝑝,𝑦𝑛𝑥𝜙𝑝,𝑛+𝑒𝑛𝜙𝑝,𝑦𝑛𝑥𝜙𝑝,𝑛+𝑒𝑛𝜙𝑝,𝑢𝑛𝑥𝜙𝑝,𝑛+𝑒𝑛𝜙𝑝,𝑇𝑖𝑢𝑛=𝑥𝑛+𝑒𝑛2𝑇𝑖𝑢𝑛2𝑥2𝑝,𝐽𝑛+𝑒𝑛𝑇𝐽𝑖𝑢𝑛.(3.17) From (3.13) and (3.14), we get that lim𝑛𝜙(𝑦𝑛,Θ𝑛𝑁1(𝑥𝑛+𝑒𝑛))=0. So we obtain that lim𝑛𝑦𝑛Θ𝑛𝑁1𝑥𝑛+𝑒𝑛=0.(3.18) Again, since 𝑝, 𝜙Θ𝑛𝑁1𝑥𝑛+𝑒𝑛,Θ𝑛𝑁2𝑥𝑛+𝑒𝑛𝜙𝑝,Θ𝑛𝑁2𝑥𝑛+𝑒𝑛𝜙𝑝,Θ𝑛𝑁1𝑥𝑛+𝑒𝑛𝑥𝜙p,𝑛+𝑒𝑛𝜙𝑝,Θ𝑛𝑁1𝑥𝑛+𝑒𝑛𝑥𝜙𝑝,𝑛+𝑒𝑛𝜙𝑝,𝑇𝑖𝑢𝑛.(3.19) From (3.13) and (3.14), we get that lim𝑛𝜙Θ𝑛𝑁1𝑥𝑛+𝑒𝑛,Θ𝑛𝑁2𝑥𝑛+𝑒𝑛=0.(3.20) It also follows that lim𝑛Θ𝑛𝑁1𝑥𝑛+𝑒𝑛Θ𝑛𝑁2𝑥𝑛+𝑒𝑛=0.(3.21) Continuing in this process, we can show that lim𝑛Θ𝑛𝑁2𝑥𝑛+𝑒𝑛Θ𝑛𝑁3𝑥𝑛+𝑒𝑛==lim𝑛Θ1𝑛𝑥𝑛+𝑒𝑛𝑥𝑛+𝑒𝑛=0.(3.22) So, we now conclude that lim𝑛Θ𝑖𝑛𝑥𝑛+𝑒𝑛Θ𝑛𝑖1𝑥𝑛+𝑒𝑛=0(3.23) for each 𝑖=1,2,,𝑁. By the uniform norm-to-norm continuity of 𝐽, we also have lim𝑛𝐽Θ𝑖𝑛𝑥𝑛+𝑒𝑛Θ𝐽𝑛𝑖1𝑥𝑛+𝑒𝑛=0(3.24) for each 𝑖=1,2,,𝑁. Using (3.23), it is easily seen that lim𝑛𝑦𝑛𝑥𝑛+𝑒𝑛=0.(3.25) From 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛, by Lemma 2.8, it follows that 𝜙𝑢𝑛,𝑦𝑛𝑇=𝜙𝑟𝑛𝑦𝑛,𝑦𝑛𝜙𝑝,𝑦𝑛𝜙𝑝,𝑇𝑟𝑛𝑦𝑛𝜙𝑝,𝑥𝑛+𝑒𝑛𝜙𝑝,𝑢𝑛𝜙𝑝,𝑥𝑛+𝑒𝑛𝜙𝑝,𝑇𝑖𝑢𝑛.(3.26) This implies that lim𝑛𝜙(𝑢𝑛,𝑦𝑛)=0 and hence lim𝑛𝑢𝑛𝑦𝑛=0.(3.27) Combining (3.13), (3.25), and (3.27), we obtain that lim𝑛𝑇𝑖𝑢𝑛𝑢𝑛=0(3.28) for all 𝑖1.
Step  5. 𝑞𝑖=1𝐹(𝑇𝑖).
Since 𝑥𝑛𝑞 and 𝑒𝑛0, 𝑥𝑛+𝑒𝑛𝑞. So from (3.25) and (3.27), we have 𝑢𝑛𝑞. Note that 𝑇𝑖 (𝑖=1,2,) are weak relatively nonexpansive. Using (3.28), we can conclude that 𝑞𝐹(𝑇𝑖)=𝐹(𝑇𝑖) for all 𝑖1. Hence 𝑞𝑖=1𝐹(𝑇𝑖).
Step  6. 𝑞𝑁𝑖=1𝐴𝑖1(0).
Noting that Θ𝑖𝑛(𝑥𝑛+𝑒𝑛)=𝐽𝜆𝑖𝑛𝐴𝑖Θ𝑛𝑖1(𝑥𝑛+𝑒𝑛) for each 𝑖=1,2,,𝑁, we obtain that 𝐴𝜆𝑖𝑛Θ𝑛𝑖1𝑥𝑛+𝑒𝑛=1𝜆𝑖𝑛𝐽Θ𝑛𝑖1𝑥𝑛+𝑒𝑛Θ𝐽𝑖𝑛𝑥𝑛+𝑒𝑛.(3.29) From (3.24) and liminf𝑛𝜆𝑖𝑛>0, we have lim𝑛𝐴𝜆𝑖𝑛Θ𝑛𝑖1𝑥𝑛+𝑒𝑛=0.(3.30) We note that (Θ𝑖𝑛(𝑥𝑛+𝑒𝑛),𝐴𝜆𝑖𝑛Θ𝑛𝑖1(𝑥𝑛+𝑒𝑛))𝐺(𝐴𝑖) for each 𝑖=1,2,,𝑁. If (𝑤,𝑤)𝐺(𝐴𝑖) for each 𝑖=1,2,,𝑁, then it follows from the monotonicity of 𝐴𝑖 that 𝑤𝐴𝜆𝑖𝑛Θ𝑛𝑖1𝑥𝑛+𝑒𝑛,𝑤Θ𝑖𝑛𝑥𝑛+𝑒𝑛0.(3.31) We see that Θ𝑖𝑛(𝑥𝑛+𝑒𝑛)𝑞 for each 𝑖=1,2,,𝑁. Thus, from (3.30) and (3.31), we have 𝑤,𝑤𝑞0.(3.32) By the maximality of 𝐴𝑖, it follows that 𝑞𝐴𝑖1(0) for each 𝑖=1,2,,𝑁. Therefore, 𝑞𝑁𝑖=1𝐴𝑖1(0).
Step  7. 𝑞EP(𝐹).
From 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛, we have 𝐹𝑢𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛0,𝑦𝐶.(3.33) By (A2), we have 𝑦𝑢𝑛𝐽𝑢𝑛𝐽𝑦𝑛𝑟𝑛1𝑟𝑛𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑦𝑛𝑢𝐹𝑛,𝑦𝐹𝑦,𝑢𝑛,𝑦𝐶.(3.34) Note that 𝐽𝑢𝑛𝐽𝑦𝑛/𝑟𝑛0 since liminf𝑛𝑟𝑛>0. From (A4) and 𝑢𝑛𝑞, we get 𝐹(𝑦,𝑞)0 for all 𝑦𝐶. For 0<𝑡<1 and 𝑦𝐶, define that 𝑦𝑡=𝑡𝑦+(1𝑡)𝑞. Then 𝑦𝑡𝐶, which implies that 𝐹(𝑦𝑡,𝑞)0. From (A1), we obtain that 0=𝐹(𝑦𝑡,𝑦𝑡)𝑡𝐹(𝑦𝑡,𝑦)+(1𝑡)𝐹(𝑦𝑡,𝑞)𝑡𝐹(𝑦𝑡,𝑦). Thus, 𝐹(𝑦𝑡,𝑦)0. From (A3), we have 𝐹(𝑞,𝑦)0 for all 𝑦𝐶. Hence, 𝑞EP(𝐹). From Steps 5, 6, and 7, we now can conclude that 𝑞.
Step  8. 𝑞=Π(𝑥1).
From 𝑥𝑛=Π𝐶𝑛(𝑥1), we have 𝐽𝑥1𝑥𝐽𝑛,𝑥𝑛𝑧0,𝑧𝐶𝑛.(3.35) Since 𝐶𝑛, we also have 𝐽𝑥1𝑥𝐽𝑛,𝑥𝑛𝑧0,𝑧.(3.36) Letting 𝑛 in (3.36), we obtain that 𝐽𝑥1𝐽(𝑞),𝑞𝑧0,𝑧.(3.37) This shows that 𝑞=Π(𝑥1) by Lemma 2.2. We thus complete the proof.

As a direct consequence of Theorem 3.1, we can also apply to a system of convex minimization problems.

Theorem 3.2. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let 𝑓𝑖𝐸(,] (𝑖=1,2,,𝑁) be proper lower semicontinuous convex functions, let 𝐹𝐶×𝐶 be a bifunction, and let 𝑇𝑖𝐶𝐶 (𝑖=1,2,) be weak relatively nonexpansive mappings such that =[𝑁𝑖=1(𝜕𝑓𝑖1)(0)][𝑖=1𝐹(𝑇𝑖)]𝐸𝑃(𝐹). Let {𝑒𝑛}𝑛=1𝐸 be the sequence such that lim𝑛𝑒𝑛=0. Define the sequence {𝑥𝑛}𝑛=1 in 𝐶 as follows: 𝑥1𝐶1𝑧=𝐶,1𝑛=argmin𝑦𝐸𝑓11(𝑦)+2𝜆1𝑛𝑦2+1𝜆1𝑛𝑥𝑦,𝐽𝑛+𝑒𝑛,𝑧𝑛𝑁1=argmin𝑦𝐸𝑓𝑁11(𝑦)+2𝜆𝑛𝑁1𝑦2+1𝜆𝑛𝑁1𝑧𝑦,𝐽𝑛𝑁2,𝑦𝑛=argmin𝑦𝐸𝑓𝑁1(𝑦)+2𝜆𝑁𝑛𝑦2+1𝜆𝑁𝑛𝑧𝑦,𝐽𝑛𝑁1,𝑢𝑛=𝑇𝑟𝑛𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛sup𝑖1𝜙𝑧,𝑇𝑖𝑢𝑛𝜙𝑧,𝑥𝑛+𝑒𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥1,𝑛1.(3.38) If liminf𝑛𝜆𝑖𝑛>0 for each 𝑖=1,2,,𝑁 and liminf𝑛𝑟𝑛>0, then the sequence {𝑥𝑛} converges strongly to 𝑞=Π(𝑥1).

Proof. By Rockafellar's theorem [43, 44], 𝜕𝑓𝑖 are maximal monotone operators for each 𝑖=1,2,,𝑁. Let 𝜆𝑖>0 for each 𝑖=1,2,,𝑁. Then, 𝑧𝑖=𝐽𝜆𝑖𝜕𝑓𝑖(𝑥) if and only if 0𝜕𝑓𝑖𝑧𝑖+1𝜆𝑖𝐽𝑧𝑖𝑓𝐽(𝑥)=𝜕𝑖+1𝜆𝑖22𝑧𝐽(𝑥)𝑖,(3.39) which is equivalent to 𝑧𝑖=argmin𝑦𝐸𝑓𝑖1(𝑦)+𝜆𝑖𝑦22.𝑦,𝐽(𝑥)(3.40) Using Theorem 3.1, we thus complete the proof.

If 𝐸=𝐻 is a real Hilbert space, we then obtain the following results.

Corollary 3.3. Let 𝐶 be a nonempty, closed and convex subset of a real Hilbert space 𝐻. Let 𝐴𝑖𝐻2𝐻 (𝑖=1,2,,𝑁) be maximal monotone operators, let 𝐹𝐶×𝐶 be a bifunction, and let 𝑇𝑖𝐶𝐶 (𝑖=1,2,) be weak relatively nonexpansive mappings such that =[𝑁𝑖=1𝐴𝑖1(0)][𝑖=1𝐹(𝑇𝑖)]𝐸𝑃(𝐹). Let {𝑒𝑛}𝑛=1𝐻 be the sequence such that lim𝑛𝑒𝑛=0. Define the sequence {𝑥𝑛}𝑛=1 in 𝐶 as follows: 𝑥1𝐶1𝑦=𝐶,𝑛=𝐽𝜆𝑁𝑛𝐴𝑁J𝜆𝑛𝑁1𝐴𝑁1𝐽𝜆1𝑛𝐴1𝑥𝑛+𝑒𝑛,𝑢𝑛=𝑇𝑟𝑛𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛sup𝑖1𝑧𝑇𝑖𝑢𝑛𝑥𝑧𝑛+𝑒𝑛,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥1,𝑛1.(3.41) If liminf𝑛𝜆𝑖𝑛>0 for each 𝑖=1,2,,𝑁 and liminf𝑛𝑟𝑛>0, then the sequence {𝑥𝑛} converges strongly to 𝑞=𝑃(𝑥1).

Corollary 3.4. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝑓𝑖𝐻(,] (𝑖=1,2,,𝑁) be proper lower semi-continuous convex functions, let 𝐹𝐶×𝐶 be a bifunction, and let 𝑇𝑖𝐶𝐶 (𝑖=1,2,) be weak relatively nonexpansive mappings such that =[𝑁𝑖=1(𝜕𝑓𝑖1)(0)][𝑖=1𝐹(𝑇𝑖)]𝐸𝑃(𝐹). Let {𝑒𝑛}𝑛=1𝐻 be the sequence such that lim𝑛𝑒𝑛=0. Define the sequence {𝑥𝑛}𝑛=1 in 𝐶 as follows: 𝑥1𝐶1𝑧=𝐶,1𝑛=argmin𝑦𝐻𝑓11(𝑦)+2𝜆1𝑛𝑦2+1𝜆1𝑛𝑦,𝑥𝑛+𝑒𝑛,𝑧𝑛𝑁1=argmin𝑦𝐻𝑓𝑁11(𝑦)+2𝜆𝑛𝑁1𝑦2+1𝜆𝑛𝑁1𝑦,𝑧𝑛𝑁2,𝑦𝑛=argmin𝑦𝐻𝑓𝑁1(𝑦)+2𝜆𝑁𝑛𝑦2+1𝜆𝑁𝑛𝑦,𝑧𝑛𝑁1,𝑢𝑛=𝑇𝑟𝑛𝑦𝑛,𝐶𝑛+1=𝑧𝐶𝑛sup𝑖1𝑧𝑇𝑖𝑢𝑛𝑥𝑧𝑛+𝑒𝑛,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥1,𝑛1.(3.42) If liminf𝑛𝜆𝑖𝑛>0 for each 𝑖=1,2,,𝑁 and liminf𝑛𝑟𝑛>0, then the sequence {𝑥𝑛} converges strongly to 𝑞=𝑃(𝑥1).

Remark 3.5. Using the shrinking projection method, we can construct a hybrid-proximal point algorithm for solving a system of the zero-finding problems, the equilibrium problems, and the fixed point problems of weak relatively nonexpansive mappings.

Remark 3.6. Since every relatively nonexpansive mapping is weak relatively nonexpansive, our results also hold if 𝑇𝑖𝐶𝐶 (𝑖=1,2,) are relatively nonexpansive mappings.

Acknowledgments

The authors thank the editor and the referee(s) for valuable suggestions. The first author was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. The second and the third authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

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