Abstract

A new modified Halpern-Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.

1. Introduction

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let 𝐸 be a Banach space, 𝐸 the dual space of 𝐸, and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐹𝐶×𝐶 be a bifunction. The equilibrium problem is to find 𝑥𝐶 such that𝐹(𝑥,𝑦)0𝑦𝐶.(1.1) The set of solutions of (1.1) is denoted by EP(𝐹). The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases. Some methods have been proposed to solve the equilibrium problems (see, e.g., [1, 2]). In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when EP(𝐹) is nonempty, and they also proved a strong convergence theorem.

Let 𝐸 be a smooth Banach space and 𝐽 the normalized duality mapping from 𝐸 to 𝐸. Alber [4] considered the following functional 𝜑𝐸×𝐸[0,) defined by 𝜑(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2(𝑥,𝑦𝐸).(1.2) Using this functional, Matsushita and Takahashi [5, 6] studied and investigated the following mappings in Banach spaces. A mapping 𝑆𝐶𝐸 is relatively nonexpansive if the following properties are satisfied: (R1)𝐹(𝑆), (R2)𝜑(𝑝,𝑆𝑥)𝜑(𝑝,𝑥) for all 𝑝𝐹(𝑆) and 𝑥𝐶,(R3)𝐹(𝑆)=𝐹(𝑆),

where 𝐹(𝑆) and 𝐹(𝑆) denote the set of fixed points of 𝑆 and the set of asymptotic fixed points of 𝑆, respectively. It is known that 𝑆 satisfies condition (R3) if and only if 𝐼𝑆 is demiclosed at zero, where 𝐼 is the identity mapping; that is, whenever a sequence {𝑥𝑛} in 𝐶 converges weakly to 𝑝 and {𝑥𝑛𝑆𝑥𝑛} converges strongly to 0, it follows that 𝑝𝐹(𝑆). In a Hilbert space 𝐻, the duality mapping 𝐽 is an identity mapping and 𝜑(𝑥,𝑦)=𝑥𝑦2 for all 𝑥,𝑦𝐻. Hence, if 𝑆𝐶𝐻 is nonexpansive (i.e., 𝑆𝑥𝑆𝑦𝑥𝑦 for all 𝑥,𝑦𝐶), then it is relatively nonexpansive. Several articles have appeared providing methods for approximating fixed points of relatively nonexpansive mappings (see, e.g., [519] and the references therein). Matsushita and Takahashi [5] introduced the following iteration: a sequence {𝑥𝑛} defined by𝑥𝑛+1=Π𝐶𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑆𝑥𝑛𝑛=1,2,,(1.3) where 𝑥1𝐶 is arbitrary, {𝛼𝑛} is an appropriate sequence in [0,1], 𝑆 is a relatively nonexpansive mapping, and Π𝐶 denotes the generalized projection from 𝐸 onto a closed convex subset 𝐶 of 𝐸. They proved that the sequence {𝑥𝑛} converges weakly to a fixed point of 𝑇. Moreover, Matsushita and Takahashi [6] proposed the following modification of iteration (1.3):𝑥1𝑦𝐶isarbitrary,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑆𝑥𝑛,𝐶𝑛=𝑧𝐶𝜑𝑧,𝑦𝑛𝜑𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶𝑥𝑛𝑧,𝐽𝑥1𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐶𝑛𝑄𝑛𝑥1,𝑛=1,2,,(1.4) and proved that the sequence {𝑥𝑛} converges strongly to Π𝐹(𝑆)𝑥1. The iteration (1.4) is called the hybrid method. To generate the iterative sequence, we use the generalized metric projection onto 𝐶𝑛𝑄𝑛 for 𝑛. It always exists, because each 𝐶𝑛𝑄𝑛 is nonempty, closed, and convex. However, in a practical case, it is not easy to be calculated. In particular, as 𝑛 becomes larger, the shape of 𝐶𝑛𝑄𝑛 becomes more complicate, and the projection will take much more time to be calculated.

In order to overcome this difficulty, Nilsrakoo and Saejung [15] modified Halpern and Mann's iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space as follows: 𝑥𝐸,𝑥1𝐶 and𝑥𝑛+1=Π𝐶𝐽1𝛼𝑛𝐽𝑥+𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑆𝑥𝑛,𝑛=1,2,,(1.5) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are appropriate sequences in [0,1] with 𝛼𝑛+𝛽𝑛+𝛾𝑛1, and they proved that {𝑥𝑛} converges strongly to Π𝐹(𝑆)𝑥.

Many authors studied the problems of finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [2033] and the references therein). In a Hilbert space 𝐻, S. Takahashi and W. Takahashi [34] introduced the iteration as follows: sequence {𝑥𝑛} generated by 𝑥,𝑥1𝐶, 𝑢𝑛𝑢𝐶suchthat𝐹𝑛+1,𝑦𝑟𝑛𝑦𝑢𝑛,𝑢𝑛𝑥𝑛𝑥0,𝑦𝐶,𝑛+1=𝛼𝑛𝑥+1𝛼𝑛𝑆𝑢𝑛,𝑛=1,2,,(1.6) where {𝛼𝑛} is an appropriate sequence in [0,1], 𝑆 is nonexpansive, and {𝑟𝑛} is an appropriate positive real sequence. They proved that {𝑥𝑛} converges strongly to an element in 𝐹(𝑆)EP(𝐹). In 2009, Takahashi and Zembayashi [30] proposed the iteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence {𝑥𝑛} generated by 𝑢1𝐸, 𝑥𝑛𝑥𝐶suchthat𝐹𝑛+1,𝑦𝑟𝑛𝑦𝑥𝑛,𝐽𝑥𝑛𝐽𝑢𝑛𝑢0,𝑦𝐶,𝑛+1=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑆𝑥𝑛,𝑛=1,2,,(1.7) where 𝑆 is relatively nonexpansive, {𝛼𝑛} is an appropriate sequence in [0,1], and {𝑟𝑛} is an appropriate positive real sequence. They proved that if 𝐽 is weakly sequentially continuous, then {𝑥𝑛} converges weakly to an element in 𝐹(𝑆)EP(𝐹). Consequently, there are many results presented strong convergence theorems for finding a common element of the set of fixed points for a mapping and the set of common solutions to a system of equilibrium problems by using the hybrid method. However, Nilsrakoo [35] introduced the Halpern-Mann iteration guaranteeing the strong convergence as follows: 𝑥𝐶,𝑢1𝐸 and𝑥𝑛𝑥𝐶suchthat𝐹𝑛+1,𝑦𝑟𝑛𝑦𝑥𝑛,𝐽𝑥𝑛𝐽𝑢𝑛𝑦0,𝑦𝐶,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑥+1𝛼𝑛𝐽𝑥𝑛,𝑢𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑆𝑦𝑛,𝑛=1,2,,(1.8) and proved that {𝑢𝑛} and {𝑥𝑛} converge strongly to Π𝐹(𝑆)EP(𝐹)𝑥.

Motivated by Nilsrakoo and Saejung [15] and Nilsrakoo [35], we present a strong convergence theorem of a new modified Halpern-Mann iterative scheme to find a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth. The results in this work improve on the corresponding ones announced by many others.

2. Preliminaries

We collect together some definitions and preliminaries which are needed in this paper. We say that a Banach space 𝐸 is strictly convex if the following implication holds for 𝑥,𝑦𝐸: 𝑥=𝑦=1,𝑥𝑦imply𝑥+𝑦2<1.(2.1) It is also said to be uniformly convex if for any 𝜀>0, there exists 𝛿>0 such that 𝑥=𝑦=1,𝑥𝑦𝜀imply𝑥+𝑦21𝛿.(2.2) It is known that if 𝐸 is a uniformly convex Banach space, then 𝐸 is reflexive and strictly convex. We say that 𝐸 is uniformly smooth if the dual space 𝐸 of 𝐸 is uniformly convex. A Banach space 𝐸 is smooth if the limit lim𝑡0((𝑥+𝑡𝑦𝑥)/𝑡) exists for all norm one elements 𝑥 and 𝑦 in 𝐸. It is not hard to show that if 𝐸 is reflexive, then 𝐸 is smooth if and only if 𝐸 is strictly convex.

Let 𝐸 be a smooth Banach space. The function 𝜑𝐸×𝐸 (see [4]) is defined by 𝜑(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2(𝑥,𝑦𝐸),(2.3) where the duality mapping 𝐽𝐸𝐸 is given by 𝑥,𝐽𝑥=𝑥2=𝐽𝑥2(𝑥𝐸).(2.4) It is obvious from the definition of the function 𝜑 that ()𝑥𝑦2)𝜑(𝑥,𝑦)(𝑥+𝑦2,(2.5)𝜑(𝑥,𝑦)=𝜑(𝑥,𝑧)+𝜑(𝑧,𝑦)+2𝑥𝑧,𝐽𝑧𝐽𝑦,(2.6) for all 𝑥,𝑦,𝑧𝐸. Moreover,𝜑𝑥,𝐽1𝑛𝑖=1𝜆𝑖𝐽𝑦𝑖𝑛𝑖=1𝜆𝑖𝜑𝑥,𝑦𝑖,(2.7) for all 𝜆𝑖[0,1] with 𝑛𝑖=1𝜆𝑖=1 and 𝑥,𝑦𝑖𝐸.

The following lemma is an analogue of Xu's inequality [36, Theorem  2] with respect to 𝜑.

Lemma 2.1 (see [15, Lemma  2.2]). Let 𝐸 be a uniformly smooth Banach space and 𝑟>0. Then, there exists a continuous, strictly increasing, and convex function 𝑔[0,2𝑟][0,) such that 𝑔(0)=0 and 𝜑𝑥,𝐽1(𝜆𝐽𝑦+(1𝜆)𝐽𝑧)𝜆𝜑(𝑥,𝑦)+(1𝜆)𝜑(𝑥,𝑧)𝜆(1𝜆)𝑔(𝐽𝑦𝐽𝑧),(2.8) for all 𝜆[0,1], 𝑥𝐸 and 𝑦,𝑧𝐵𝑟={𝑧𝐸𝑧𝑟}.

It is also easy to see that if {𝑥𝑛} and {𝑦𝑛} are bounded sequences of a smooth Banach space 𝐸, then 𝑥𝑛𝑦𝑛0 implies that 𝜑(𝑥𝑛,𝑦𝑛)0.

Lemma 2.2 (see [37, Proposition  2]). Let 𝐸 be a uniformly convex and smooth Banach space, and let {𝑥𝑛} and {𝑦𝑛} be two sequences of 𝐸 such that {𝑥𝑛} or {𝑦𝑛} is bounded. If 𝜑(𝑥𝑛,𝑦𝑛)0, then 𝑥𝑛𝑦𝑛0.

Remark 2.3. For any bounded sequences {𝑥𝑛} and {𝑦𝑛} in a uniformly convex and uniformly smooth Banach space 𝐸, we have 𝜑𝑥𝑛,𝑦𝑛0𝑥𝑛𝑦𝑛0𝐽𝑥𝑛𝐽𝑦𝑛0.(2.9)

Let 𝐶 be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸. It is known that [4, 37] for any 𝑥𝐸, there exists a unique point ̂𝑥𝐶 such that 𝜑(̂𝑥,𝑥)=min𝑦𝐶𝜑(𝑦,𝑥).(2.10) Following Alber [4], we denote such an element ̂𝑥 by Π𝐶𝑥. The mapping Π𝐶 is called the generalized projection from 𝐸 onto 𝐶. It is easy to see that in a Hilbert space, the mapping Π𝐶 coincides with the metric projection 𝑃𝐶. Concerning the generalized projection, the followings are well known.

Lemma 2.4 (see [37, Propositions  4 and  5]). Let 𝐶 be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space 𝐸, 𝑥𝐸 and ̂𝑥𝐶. Then, (a)̂𝑥=Π𝐶𝑥 if and only if 𝑦̂𝑥,𝐽𝑥𝐽̂𝑥0 for all 𝑦𝐶,(b)𝜑(𝑦,Π𝐶𝑥)+𝜑(Π𝐶𝑥,𝑥)𝜑(𝑦,𝑥) for all 𝑦𝐶.

Remark 2.5. The generalized projection mapping Π𝐶 above is relatively nonexpansive and 𝐹(Π𝐶)=𝐶.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space. The duality mapping 𝐽 from 𝐸 onto 𝐸=𝐸 coincides with the inverse of the duality mapping 𝐽 from 𝐸 onto 𝐸; that is, 𝐽=𝐽1. We make use of the following mapping 𝑉𝐸×𝐸 studied in Alber [4]:𝑉𝑥,𝑥=𝑥22𝑥,𝑥+𝑥2,(2.11) for all 𝑥𝐸 and 𝑥𝐸. Obviously, 𝑉(𝑥,𝑥)=𝜑(𝑥,𝐽1(𝑥)) for all 𝑥𝐸 and 𝑥𝐸. We know the following lemma (see [4] and [38, Lemma  3.2]).

Lemma 2.6. Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, and let 𝑉 be as in (2.11). Then 𝑉𝑥,𝑥𝐽+21𝑥𝑥,𝑦𝑉𝑥,𝑥+𝑦,(2.12) for all 𝑥𝐸 and 𝑥,𝑦𝐸.

Lemma 2.7 (see [39, Lemma  2.1]). Let {𝑎𝑛} be a sequence of nonnegative real numbers. Suppose that 𝑎𝑛+11𝛾𝑛𝑎𝑛+𝛾𝑛𝛿𝑛(2.13) for all 𝑛, where the sequences {𝛾𝑛} in (0,1) and {𝛿𝑛} in satisfy conditions: lim𝑛𝛾𝑛=0, 𝑛=1𝛾𝑛=, and limsup𝑛𝛿𝑛0. Then lim𝑛𝑎𝑛=0.

Lemma 2.8 (see [40, Lemma  3.1]). Let {𝑎𝑛} be a sequence of real numbers such that there exists a subsequence {𝑛𝑖} of {𝑛} such that 𝑎𝑛𝑖<𝑎𝑛𝑖+1 for all 𝑖. Then, there exists a nondecreasing sequence {𝑚𝑘} such that 𝑚𝑘 and the following properties are satisfied by all (sufficiently large) numbers 𝑘: 𝑎𝑚𝑘𝑎𝑚𝑘+1,𝑎𝑘𝑎𝑚𝑘+1.(2.14) In fact, 𝑚𝑘=max{𝑗𝑘𝑎𝑗<𝑎𝑗+1}.

For solving the equilibrium problem, we usually assume that a bifunction 𝐹𝐶×𝐶 satisfies the following conditions (see, e.g., [1, 3, 30]): (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 semicontinuous.

The following lemma is a result which appeared in Blum and Oettli [1, Corollary  1].

Lemma 2.9 (see [1, Corollary  1]). Let 𝐶 be a closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Let 𝐹𝐶×𝐶 be a bifunction satisfying conditions (A1)–(A4), and let 𝑟>0 and 𝑥𝐸. Then, there exists 𝑧𝐶 such that 1𝐹(𝑧,𝑦)+𝑟𝑦𝑧,𝐽𝑧𝐽𝑥0𝑦𝐶.(2.15)

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.10 (see [30, Lemma  2.8]). Let 𝐶 be a nonempty closed convex subset of a reflexive, strictly convex, and uniformly smooth Banach space 𝐸. Let 𝐹𝐶×𝐶 be a bifunction satisfying conditions (A1)–(A4). For 𝑟>0, define a mapping 𝑇𝐹𝑟𝐸𝐶 so-called the resolvent of 𝐹 as follows: 𝑇𝐹𝑟1(𝑥)=𝑧𝐶𝐹(𝑧,𝑦)+𝑟,𝑦𝑧,𝐽𝑧𝐽𝑥0𝑦𝐶(2.16) for all 𝑥𝐸. Then, the followings hold: (i)𝑇𝑟 is single-valued, (ii)𝑇𝑟 is a firmly nonexpansive-type mapping [11], that is, for all 𝑥,𝑦𝐸𝑇𝐹𝑟𝑥𝑇𝐹𝑟𝑦,𝐽𝑇𝐹𝑟𝑥𝐽𝑇𝐹𝑟𝑦𝑇𝐹𝑟𝑥𝑇𝐹𝑟𝑦,𝐽𝑥𝐽𝑦,(2.17)(iii)for all 𝑥𝐸 and 𝑝EP(𝐹), 𝜑𝑝,𝑇𝐹𝑟𝑥𝜑𝑧,𝑇𝐹𝑟𝑥𝑇+𝜑𝐹𝑟𝑥,𝑥𝜑(𝑝,𝑥),(2.18)(iv)𝐹(𝑇𝐹𝑟)=EP(𝐹), (v)EP(𝐹) is closed and convex.

Remark 2.11. Some well-known examples of resolvents of bifunctions satisfying conditions (A1)–(A4) are presented in [3, Lemma  2.15].

Lemma 2.12  2.12 (see [8, Lemma  2.3]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸, 𝐹 a bifunction from 𝐶×𝐶 satisfying conditions (A1)–(A4), and 𝑧𝐶. Then, 𝑧EP(𝐹) if and only if 𝐹(𝑦,𝑧)0 for all 𝑦𝐶.

Lemma 2.13  2.13 (see [6], Proposition  2.4). Let 𝐶 be a nonempty closed convex subset of a strictly convex and smooth Banach space 𝐸 and 𝑆𝐶𝐸 a relatively nonexpansive mapping. Then 𝐹(𝑆) is closed and convex.

3. Main Results

In this section, we introduce a modified Halpern-Mann type iteration without using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.

Theorem 3.1. Let 𝐸 a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝐸, {𝐹𝑖}𝑚𝑖=1 be a finite family of a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑆𝐶𝐸 a relatively nonexpansive mapping such that Ω=𝐹(𝑆)(𝑚𝑖=1EP(𝐹𝑖)). Let {𝑇𝐹𝑖𝑟𝑖,𝑛}𝑚𝑖=1 be a finite family of the resolvents of 𝐹𝑖 with positive real sequences {𝑟𝑖,𝑛} such that liminf𝑛𝑟𝑖,𝑛>0 for all 𝑖=1,2,,𝑚. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐸 and 𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥+𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑆𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,𝑇𝐹1𝑟1,𝑛𝑥𝑛(𝑛1),(3.1) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=,(iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then, {𝑥𝑛} converges strongly to ΠΩ𝑥.

Proof. For each 𝑛1, setting 𝑧𝑘𝑛=𝑇𝐹𝑘𝑟𝑘,𝑛𝑇𝐹𝑘1𝑟𝑘1,𝑛𝑇𝐹1𝑟1,𝑛𝑥𝑛𝑦,(𝑘=1,2,,𝑚),𝑛=𝐽1𝛽𝑛1𝛼𝑛𝐽𝑥𝑛+𝛾𝑛1𝛼𝑛𝐽𝑆𝑧𝑚𝑛.(3.2) We can see that 𝑧𝑘𝑛=𝑇𝐹𝑘𝑘,𝑛𝑧𝑛𝑘1. Since Ω is nonempty, closed, and convex, we put ̂𝑥=ΠΩ𝑥. By Lemma 2.10(iii), we get 𝜑̂𝑥,𝑧𝑚𝑛𝜑̂𝑥,𝑧𝑛𝑚1𝑧𝜑𝑚𝑛,𝑧𝑛𝑚1𝜑̂𝑥,𝑧𝑛𝑚2𝑧𝜑𝑛𝑚1,𝑧𝑛𝑚2𝑧𝜑𝑚𝑛,𝑧𝑛𝑚1𝜑̂𝑥,𝑥𝑛𝑚𝑘=1𝜑𝑧𝑘𝑛,𝑧𝑛𝑘1,(3.3) where 𝑧0𝑛=𝑥𝑛. This together with (2.7) gives 𝜑̂𝑥,𝑦𝑛𝛽𝑛1𝛼𝑛𝜑̂𝑥,𝑥𝑛+𝛾𝑛1𝛼𝑛𝜑̂𝑥,𝑆𝑧𝑚𝑛𝛽𝑛1𝛼𝑛𝜑̂𝑥,𝑥𝑛+𝛾𝑛1𝛼𝑛𝜑̂𝑥,𝑧𝑚𝑛𝜑̂𝑥,𝑥𝑛.(3.4) By Lemma 2.6, we obtain 𝜑̂𝑥,𝑥𝑛+1=𝑉̂𝑥,𝐽𝑥𝑛+1𝑉̂𝑥,𝐽𝑥𝑛+1𝛼𝑛(𝑥𝐽𝑥𝐽̂𝑥)2𝑛+1̂𝑥,𝛼𝑛(𝐽𝑥𝐽̂𝑥)=𝜑̂𝑥,𝐽1𝛼𝑛𝐽̂𝑥+1𝛼𝑛𝐽𝑦𝑛+2𝛼𝑛𝑥𝑛+1̂𝑥,𝐽𝑥𝐽̂𝑥𝛼𝑛𝜑(̂𝑥,̂𝑥)+1𝛼𝑛𝜑̂𝑥,𝑦𝑛+2𝛼𝑛𝑥𝑛+1̂𝑥,𝐽𝑥𝐽̂𝑥1𝛼𝑛𝜑̂𝑥,𝑥𝑛+2𝛼𝑛𝑥𝑛+1.̂𝑥,𝐽𝑥𝐽̂𝑥(3.5) Next, we show that {𝑥𝑛} is bounded. We consider 𝜑̂𝑥,𝑥𝑛+1𝜑̂𝑥,𝐽1𝛼𝑛𝐽𝑥+𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑆𝑧𝑚𝑛=𝜑̂𝑥,𝐽1𝛼𝑛𝐽𝑥+1𝛼𝑛𝐽𝑦𝑛𝛼𝑛𝜑(̂𝑥,𝑥)+1𝛼𝑛𝜑̂𝑥,𝑦𝑛𝛼𝑛𝜑(̂𝑥,𝑥)+1𝛼𝑛𝜑̂𝑥,𝑥𝑛max𝜑(̂𝑥,𝑥),𝜑̂𝑥,𝑥𝑛.(3.6) By induction, we have 𝜑̂𝑥,𝑥𝑛+1𝜑max(̂𝑥,𝑥),𝜑̂𝑥,𝑥1,(3.7) for all 𝑛1. This implies that {𝑥𝑛} is bounded, and so are {𝑥𝑛}, {𝑢𝑛}, {𝑦𝑛}, {𝑧𝑚𝑛}, and {𝑆𝑧𝑚𝑛}. Let 𝑔[0,2𝑟][0,) be a function satisfying the properties of Lemma 2.1, where 𝑟=sup{𝑥𝑛,𝑆𝑧𝑚𝑛𝑛1}. It follows from (3.3) that 𝜑̂𝑥,𝑦𝑛𝛽𝑛1𝛼𝑛𝜑̂𝑥,𝑥𝑛+𝛾𝑛1𝛼𝑛𝜑̂𝑥,𝑆𝑧𝑚𝑛𝛽𝑛𝛾𝑛1𝛼𝑛2𝑔𝐽𝑥𝑛𝐽𝑆𝑧𝑚𝑛𝛽𝑛1𝛼𝑛𝜑̂𝑥,𝑥𝑛+𝛾𝑛1𝛼𝑛𝜑̂𝑥,𝑧𝑚𝑛𝛽𝑛𝛾𝑛1𝛼𝑛2𝑔𝐽𝑥𝑛𝐽𝑆𝑧𝑚𝑛𝜑̂𝑥,𝑥𝑛𝛾𝑛1𝛼𝑛𝑚𝑘=1𝜑𝑧𝑘𝑛,𝑧𝑛𝑘1𝛽𝑛𝛾𝑛1𝛼𝑛2𝑔𝐽𝑥𝑛𝐽𝑆𝑧𝑚𝑛.(3.8)
The rest of the proof will be divided into two cases.Case 1. Suppose that there exists 𝑛0 such that {𝜑(̂𝑥,𝑥𝑛)}𝑛=𝑛0 is nonincreasing. In this situation, {𝜑(̂𝑥,𝑥𝑛)} is then convergent. Then, 𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑥𝑛+10.(3.9) Notice that 𝜑̂𝑥,𝑥𝑛+1𝛼𝑛𝜑(̂𝑥,𝑥)+1𝛼𝑛𝜑̂𝑥,𝑦𝑛.(3.10) From condition (ii), 𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑦𝑛=𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑥𝑛+1+𝜑̂𝑥,𝑥𝑛+1𝜑̂𝑥,𝑦𝑛𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑥𝑛+1+𝛼𝑛𝜑(̂𝑥,𝑥)𝜑̂𝑥,𝑦𝑛0.(3.11) It follows from (3.8) that 𝛾𝑛1𝛼𝑛𝑚𝑘=1𝜑𝑧𝑘𝑛,𝑧𝑛𝑘1+𝛽𝑛𝛾𝑛1𝛼𝑛2𝑔𝐽𝑥𝑛𝐽𝑆𝑧𝑚𝑛0.(3.12) By the assumptions (i), (ii), and (iv), 𝜑𝑧𝑘𝑛,𝑧𝑛𝑘10(𝑘=1,2,,𝑚),𝑔𝐽𝑥𝑛𝐽𝑆𝑧𝑚𝑛0.(3.13) By Remark 2.3, we get 𝑧𝑘𝑛𝑧𝑛𝑘10(𝑘=1,2,,𝑚).(3.14) From 𝑔 is continuous strictly increasing with 𝑔(0)=0, we have 𝑧𝑚𝑛𝑆𝑧𝑚𝑛𝑥0,𝜑𝑛,𝑆𝑧𝑚𝑛0.(3.15) Consequently, 𝜑𝑥𝑛,𝑦𝑛𝛽𝑛1𝛼𝑛𝜑𝑥𝑛,𝑥𝑛+𝛾𝑛1𝛼𝑛𝜑𝑥𝑛,𝑆𝑧𝑚𝑛=𝛾𝑛1𝛼𝑛𝜑𝑥𝑛,𝑆𝑧𝑚𝑛𝜑𝑦0,𝑛,𝑥𝑛+1𝛼𝑛𝜑𝑦𝑛+,𝑥1𝛼𝑛𝜑𝑦𝑛,𝑦𝑛=𝛼𝑛𝜑𝑦𝑛,𝑥0.(3.16) This implies that 𝑥𝑛+1𝑥𝑛0.(3.17) Since {𝑥𝑛} is bounded and 𝐸 is reflexive, we choose a subsequence {𝑥𝑛𝑗} of {𝑥𝑛} such that 𝑥𝑛𝑗𝑤 and limsup𝑛𝑥𝑛̂𝑥,𝐽𝑥𝐽̂𝑥=lim𝑗𝑥𝑛𝑗̂𝑥,𝐽𝑥𝐽̂𝑥=𝑤̂𝑥,𝐽𝑥𝐽̂𝑥.(3.18) Let 𝑘=1,2,,𝑚 be fixed. Then, 𝑧𝑘𝑛𝑗𝑤 as 𝑗. From liminf𝑛𝑟𝑘,𝑛>0 and (3.14), we have lim𝑛1𝑟𝑘,𝑛𝐽𝑧𝑘𝑛𝐽𝑧𝑛𝑘1=0.(3.19) Then, 𝐹𝑘𝑧𝑘𝑛+1,𝑦𝑟𝑘,𝑛𝑦𝑧𝑘𝑛,𝐽𝑧𝑘𝑛𝐽𝑧𝑛𝑘10,𝑦𝐶.(3.20) Replacing 𝑛 by 𝑛𝑗, we have from (A2) that 1𝑟𝑘,𝑛𝑗𝑦𝑧𝑘𝑛𝑗,𝐽𝑧𝑘𝑛𝑗𝐽𝑧𝑛𝑘1𝑗𝐹𝑘𝑧𝑘𝑛𝑗,𝑦𝐹𝑘𝑦,𝑧𝑘𝑛𝑗,𝑦𝐶.(3.21) Letting 𝑗, we have from (3.19) and (A4) that 𝐹𝑘(𝑦,𝑤)0,𝑦𝐶.(3.22)From Lemma 2.12, we have 𝑤EP(𝐹𝑘). Since 𝑆 satisfies condition (R3) and 𝑧𝑚𝑛𝑆𝑧𝑚𝑛0, we have 𝑤𝐹(𝑆). It follows that 𝑤Ω. By Lemma 2.4(a), we immediately obtain that limsup𝑛𝑥𝑛+1̂𝑥,𝐽𝑥𝐽̂𝑥=limsup𝑛𝑥𝑛̂𝑥,𝐽𝑥𝐽̂𝑥=𝑤̂𝑥,𝐽𝑥𝐽̂𝑥0.(3.23) It follows from Lemma 2.7 and (3.5) that 𝜑(̂𝑥,𝑥𝑛)0. Then, 𝑥𝑛̂𝑥.Case 2. Suppose that there exists a subsequence {𝑛𝑖} of {𝑛} such that 𝜑̂𝑥,𝑥𝑛𝑖<𝜑̂𝑥,𝑥𝑛𝑖+1,(3.24) for all 𝑖. Then, by Lemma 2.8, there exists a nondecreasing sequence of positive integer numbers {𝑗} such that 𝑗, 𝜑̂𝑥,𝑥𝑗𝜑̂𝑥,𝑥𝑗+1,𝜑̂𝑥,𝑥𝑗𝜑̂𝑥,𝑥𝑗+1,(3.25) for all sufficiently large numbers 𝑗. We may assume without loss of generality that 𝛼𝑗>0 for all sufficiently large numbers 𝑗. Since 𝜑̂𝑥,𝑥𝑗+1𝛼𝑗𝜑(̂𝑥,𝑥)+1𝛼𝑗𝜑̂𝑥,𝑦𝑗,(3.26) we obtain 𝜑̂𝑥,𝑥𝑗𝜑̂𝑥,𝑦𝑗=𝜑̂𝑥,𝑥𝑗𝜑̂𝑥,𝑥𝑗+1+𝜑̂𝑥,𝑥𝑗+1𝜑̂𝑥,𝑦𝑗𝛼𝑗𝜑(̂𝑥,𝑥)𝜑̂𝑥,𝑦𝑗0.(3.27) It follows from (3.8) that 𝛾𝑗1𝛼𝑗𝑚𝑘=1𝜑𝑧𝑘𝑗,𝑧𝑘1𝑗+𝛽𝑗𝛾𝑗1𝛼𝑗2𝑔𝐽𝑥𝑗𝐽𝑆𝑧𝑚𝑗0.(3.28) Using the same proof of Case 1, we also obtain limsup𝑗𝑥𝑗+1̂𝑥,𝐽𝑥𝐽̂𝑥0.(3.29) From (3.5), we have 𝜑̂𝑥,𝑥𝑗+11𝛼𝑗𝜑̂𝑥,𝑥𝑗+2𝛼𝑗𝑥𝑗+1̂𝑥,𝐽𝑥𝐽̂𝑥.(3.30) Since 𝜑(̂𝑥,𝑥𝑗)𝜑(̂𝑥,𝑥𝑗+1), we have 𝛼𝑗𝜑̂𝑥,𝑥𝛼𝜑̂𝑥,𝑥𝑗𝜑̂𝑥,𝑥𝑗+1+2𝛼𝑗𝑥𝑗+1̂𝑥,𝐽𝑥𝐽̂𝑥2𝛼𝑗𝑥𝑗+1.̂𝑥,𝐽𝑥𝐽̂𝑥(3.31) In particular, since 𝛼𝑗>0, we get 𝜑̂𝑥,𝑥𝑚𝑘𝑥2𝑗+1̂𝑥,𝐽𝑥𝐽̂𝑥.(3.32) It follows from (3.29) that 𝜑(̂𝑥,𝑥𝑗)0. This together with (3.30) gives 𝜑̂𝑥,𝑥𝑗+10.(3.33) But 𝜑(̂𝑥,𝑥𝑗)𝜑(̂𝑥,𝑥𝑗+1) for all sufficiently large numbers 𝑗, we conclude that 𝑥𝑗̂𝑥.
From the two cases, we can conclude that {𝑥𝑛} converges strongly to ̂𝑥 and the proof is finished.

Setting 𝑚=1, 𝐹1=𝐹0, and 𝑟1,𝑛𝑟𝑛 in Theorem 3.1, we have the following.

Corollary 3.2. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝐸, 𝐹 a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑆𝐶𝐸 be a relatively nonexpansive mapping such that 𝐹(𝑆)EP(𝐹). Let 𝑇𝐹𝑟𝑛 be the resolvent of 𝐹 with a positive real sequence {𝑟𝑛} such that liminf𝑛𝑟𝑛>0. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐸 and 𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥+𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑆𝑇𝐹𝑟𝑛𝑥𝑛(𝑛1),(3.34) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then, {𝑥𝑛} converges strongly to Π𝐹(𝑆)EP(𝐹)𝑥.

Setting 𝐹10 and 𝑟1,𝑛1 in Corollary 3.2, we have the following result.

Corollary 3.3. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑆𝐶𝐸 a relatively nonexpansive mapping such that 𝐹(𝑆). Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐸 and 𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥+𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑆Π𝐶𝑥𝑛(𝑛1),(3.35) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then {𝑥𝑛} converges strongly to Π𝐹(𝑆)𝑥.

Next, we prove a strong convergence theorem for finding an element of the set of solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth Banach space.

Theorem 3.4. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty closed convex subset of 𝐸, {𝐹𝑖}𝑚𝑖=1 a finite family of a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑚𝑖=1EP(𝐹𝑖). Let {𝑇𝐹𝑖𝑟𝑖,𝑛}𝑚𝑖=1 be a finite family of the resolvents of 𝐹𝑖 with positive real sequences {𝑟𝑖,𝑛} such that liminf𝑛𝑟𝑖,n>0 for all 𝑖=1,2,,𝑚. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐸 and 𝑥𝑛+1=𝐽1𝛼𝑛𝐽𝑥+𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑥𝑛(𝑛1),(3.36) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0 or liminf𝑛𝛽𝑛=0. Then, {𝑥𝑛} converges strongly to Π𝑚𝑖=1EP(𝐹𝑖)𝑥.

Proof. For each 𝑛1, setting 𝑧𝑘𝑛=𝑇𝐹𝑘𝑟𝑘,𝑛𝑇𝐹𝑘1𝑟𝑘1,𝑛𝑇𝐹1𝑟1,𝑛𝑥𝑛𝑦,(𝑘=1,2,,𝑚),𝑛=𝐽1𝛽𝑛1𝛼𝑛𝐽𝑥𝑛+𝛾𝑛1𝛼𝑛𝐽𝑧𝑚𝑛.(3.37) Since 𝑚𝑖=1EP(𝐹𝑖) is nonempty, closed, and convex, we put ̂𝑥=Π𝑚𝑖=1EP(𝐹𝑖)𝑥. Using the same proof of Theorem 3.1 when 𝑆 is the identity operator, we can see that 𝜑̂𝑥,𝑦𝑛𝜑̂𝑥,𝑥𝑛𝛾𝑛1𝛼𝑛𝑚𝑘=1𝜑𝑧𝑘𝑛,𝑧𝑛𝑘1,𝜑(3.38)̂𝑥,𝑥𝑛+11𝛼𝑛𝜑̂𝑥,𝑥𝑛+2𝛼𝑛𝑥𝑛+1̂𝑥,𝐽𝑥𝐽̂𝑥.(3.39)
The rest of the proof will be divided into two cases.Case 1. Suppose that there exists 𝑛0 such that {𝜑(̂𝑥,𝑥𝑛)}𝑛=𝑛0 is non-increasing. In this situation, {𝜑(̂𝑥,𝑥𝑛)} is then convergent. Then, 𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑥𝑛+10.(3.40) Notice that 𝜑̂𝑥,𝑥𝑛+1𝛼𝑛𝜑(̂𝑥,𝑥)+1𝛼𝑛𝜑̂𝑥,𝑦𝑛.(3.41) From condition (ii), 𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑦𝑛=𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑥𝑛+1+𝜑̂𝑥,𝑥𝑛+1𝜑̂𝑥,𝑦𝑛𝜑̂𝑥,𝑥𝑛𝜑̂𝑥,𝑥𝑛+1+𝛼𝑛𝜑(̂𝑥,𝑥)𝜑̂𝑥,𝑦𝑛0.(3.42) It follows from (3.38) that 𝛾𝑛1𝛼𝑛𝑚𝑘=1𝜑𝑧𝑘𝑛,𝑧𝑛𝑘10.(3.43) By the assumptions (i), (ii), and (iv), 𝜑𝑧𝑘𝑛,𝑧𝑛𝑘10(𝑘=1,2,,𝑚).(3.44) By Remark 2.3, we get 𝑧𝑘𝑛𝑧𝑛𝑘10(𝑘=1,2,,𝑚).(3.45) Consequently, 𝜑𝑥𝑛,𝑦𝑛𝛽𝑛1𝛼𝑛𝜑𝑥𝑛,𝑥𝑛+𝛾𝑛1𝛼𝑛𝜑𝑥𝑛,𝑧𝑚𝑛=𝛾𝑛1𝛼𝑛𝜑𝑧0𝑛,𝑧𝑚𝑛𝜑𝑦0,𝑛,𝑥𝑛+1𝛼𝑛𝜑𝑦𝑛+,𝑥1𝛼𝑛𝜑𝑦𝑛,𝑦𝑛=𝛼𝑛𝜑𝑦𝑛,𝑥0.(3.46) This implies that 𝑥𝑛+1𝑥𝑛0.(3.47) Since {𝑥𝑛} is bounded and 𝐸 is reflexive, we choose a subsequence {𝑥𝑛𝑗} of {𝑥𝑛} such that 𝑥𝑛𝑗𝑤 and limsup𝑛𝑥𝑛̂𝑥,𝐽𝑥𝐽̂𝑥=lim𝑗𝑥𝑛𝑗̂𝑥,𝐽𝑥𝐽̂𝑥=𝑤̂𝑥,𝐽𝑥𝐽̂𝑥.(3.48) Let 𝑘=1,2,,𝑚 be fixed. Then, 𝑧𝑘𝑛𝑗𝑤 as 𝑗. From liminf𝑛𝑟𝑘,𝑛>0 and (3.14), we have lim𝑛1𝑟𝑘,𝑛𝐽z𝑘𝑛𝐽𝑧𝑛𝑘1=0.(3.49) Then, 𝐹𝑘𝑧𝑘𝑛+1,𝑦𝑟𝑘,𝑛𝑦𝑧𝑘𝑛,𝐽𝑧𝑘𝑛𝐽𝑧𝑛𝑘10,𝑦𝐶.(3.50) Replacing 𝑛 by 𝑛𝑗, we have from (A2) that 1𝑟𝑘,𝑛𝑗𝑦𝑧𝑘𝑛𝑗,𝐽𝑧𝑘𝑛𝑗𝐽𝑧𝑛𝑘1𝑗𝐹𝑘𝑧𝑘𝑛𝑗,𝑦𝐹𝑘𝑦,𝑧𝑘𝑛𝑗,𝑦𝐶.(3.51) Letting 𝑗, we have from (3.49) and (A4) that 𝐹𝑘(𝑦,𝑤)0,𝑦𝐶.(3.52) From Lemma 2.12, we have 𝑤EP(𝐹𝑘). By Lemma 2.4(a), we immediately obtain that limsup𝑛𝑥𝑛+1̂𝑥,𝐽𝑥𝐽̂𝑥=limsup𝑛𝑥𝑛̂𝑥,𝐽𝑥𝐽̂𝑥=𝑤̂𝑥,𝐽𝑥𝐽̂𝑥0.(3.53) It follows from Lemma 2.7 and (3.39) that 𝜑(̂𝑥,𝑥𝑛)0. Then, 𝑥𝑛̂𝑥.Case 2. Suppose that there exists a subsequence {𝑛𝑖} of {𝑛} such that 𝜑̂𝑥,𝑥𝑛𝑖<𝜑̂𝑥,𝑥𝑛𝑖+1,(3.54) for all 𝑖. Using the same proof of Case 2 in Theorem 3.1, we also conclude that 𝑥𝑗̂𝑥.
From the two cases, we can conclude that {𝑥𝑛} converges strongly to ̂𝑥.

Finally, we give two explicit examples validating the assumptions in Theorem 3.1 as follows.

Example 3.5 (Optimization). Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty bounded closed convex subset of 𝐸, and 𝑓𝐶 a lower semicontinuous and convex functional. For instance, let 𝐸=, 𝐶=[0,1] and 𝑓[0,1] be defined dy 𝑓(𝑥)=0,if𝑥=0,1;𝑥log𝑥+(1𝑥)log(1𝑥),if𝑥(0,1).(3.55) Then 𝑓 is lower semicontinuous and convex. For each 𝑖=1,2,,𝑚, let 𝐹𝑖𝐶×𝐶 be defined by 𝐹𝑖(𝑥,𝑦)=𝑓(𝑦)𝑓(𝑥) for all 𝑥,𝑦𝐶. It is known [1, 11] that 𝐹𝑖 satisfies conditions (A1)–(A4), and EP(𝐹𝑖). Let 𝑆=Π𝐶. Then, 𝑆 is relatively nonexpansive of 𝐸 into 𝐶 (see [5, 6]) and 𝐹(𝑆)=𝐶. Then, Ω=𝐹(𝑆)(𝑚𝑖=1EP(𝐹𝑖))=EP(𝐹𝑖). Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to ΠΩ𝑥.

Example 3.6 (The convex feasibility problem). Let 𝐸 be a real Hilbert space, let 𝐶1,𝐶2,,𝐶𝑚 be nonempty closed convex subsets of 𝐸 satisfying 𝐶=𝑚𝑖=1𝐶𝑖 (e.g., 𝐶1=𝐶2==𝐶𝑚=𝐶). Let {𝐹𝑖}𝑚𝑖=1 be a finite family of bifunctions of 𝐸×𝐸 into defined by 𝐹𝑖1(𝑥,𝑦)=2𝑦𝑥,𝑥𝑃𝐶𝑖𝑥𝑥,𝑦𝐸,(3.56) where 𝑃𝐶𝑖 is a metric projection from 𝐸 onto 𝐶𝑖. It is known [3, Lemma  2.15(iv)] that 𝐹𝑖 satisfies conditions (A1)–(A4) and EP(𝐹𝑖)=𝐶𝑖. Let 𝑆=𝑃𝐶. Then, 𝑆 is relatively nonexpansive of 𝐸 into 𝐶 (see [5, 6]) and then Ω=𝐹(𝑆)(𝑚𝑖=1EP(𝐹𝑖))=𝐶. Applying Theorem 3.1, we conclude that the sequence defined by (3.1) converges strongly to ΠΩ𝑥.

4. Deduced Theorems in Hilbert Spaces

In Hilbert spaces, if 𝑆 is quasi-nonexpansive such that 𝐼𝑆 is demiclosed at zero, then 𝑆 is relatively nonexpansive. We obtain the following result.

Theorem 4.1. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, {𝐹𝑖}𝑚𝑖=1 a finite family of a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑆𝐶𝐸 a quasi-nonexpansive mapping such that 𝐼𝑆 is demiclosed at zero and Ω=𝐹(𝑆)(𝑚𝑖=1EP(𝐹𝑖)). Let {𝑇𝐹𝑖𝑟𝑖,𝑛}𝑚𝑖=1 be a finite family of the resolvents of 𝐹𝑖 with real sequences {𝑟𝑖,𝑛} such that liminf𝑛𝑟𝑖,𝑛>0 for all 𝑖=1,2,,𝑚. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑥+𝛽𝑛𝑥𝑛+𝛾𝑛𝑆𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑥𝑛(𝑛1),(4.1) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then {𝑥𝑛} converges strongly to 𝑃Ω𝑥.

Applying Theorem 4.1 and using the technique in [41], we have the following result.

Theorem 4.2. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑓 a contraction of 𝐻 into itself (i.e., there is 𝑎(0,1) such that 𝑓(𝑥)𝑓(𝑦)𝑎𝑥𝑦 for all 𝑥,𝑦𝐻), {𝐹𝑖}𝑚𝑖=1 a finite family of a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑆𝐶𝐸 be a nonexpansive mapping such that Ω=𝐹(𝑆)(𝑚𝑖=1EP(𝐹𝑖)). Let {𝑇𝐹𝑖𝑟𝑖,𝑛}𝑚𝑖=1 be a finite family of the resolvents of 𝐹𝑖 with real sequences {𝑟𝑖,𝑛} such that liminf𝑛𝑟𝑖,𝑛>0 for all 𝑖=1,2,,𝑚. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛽𝑛𝑥𝑛+𝛾𝑛𝑆𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑥𝑛,(𝑛1),(4.2) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then, {𝑥𝑛} converges strongly to 𝑧 such that 𝑧=𝑃Ω𝑓(𝑧).

Proof. We note that 𝑃Ω𝑓 is contraction. By Banach contraction principle, let 𝑧 be the fixed point of 𝑃Ω𝑓 and {𝑦𝑛} a sequence generated by 𝑦1=𝑥1𝐻 and 𝑦𝑛+1=𝛼𝑛𝑓(𝑧)+𝛽𝑛𝑦𝑛+𝛾𝑛𝑆𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑦𝑛,(𝑛1).(4.3) Using Theorem 4.1, we have 𝑦𝑛𝑧=𝑃Ω𝑓(𝑧). Since 𝑆 and 𝑇𝐹𝑘𝑟𝑘,𝑛(𝑘=1,2,,𝑚) are nonexpansive, 𝑦𝑛+1𝑥𝑛+1𝛼𝑛𝑓𝑥𝑛𝑓(𝑧)+𝛽𝑛𝑦𝑛𝑥𝑛+𝛾𝑛𝑆𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑦𝑛𝑆𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑥𝑛𝛼𝑛𝑎𝑥𝑛+𝛽𝑧𝑛+𝛾𝑛𝑦𝑛𝑛𝑛𝛼𝑛𝑎𝑥𝑛𝑦𝑛+𝑦𝑛+𝛽𝑧𝑛+𝛾𝑛𝑥𝑛𝑦𝑛=1𝛼𝑛𝑦(1𝑎)𝑛𝑥𝑛+𝛼𝑛𝑎(1𝑎)𝑦1𝑎𝑛.𝑧(4.4) Applying Lemma 2.7, 𝑦𝑛𝑥𝑛0 and so 𝑥𝑛𝑧=𝑃Ω𝑓(𝑧).

Setting 𝑚=1, 𝐹1=𝐹0, and 𝑟1,𝑛𝑟𝑛 in Theorem 4.1, we have the following.

Corollary 4.3. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝐹 a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑆𝐶𝐸 a quasi-nonexpansive mapping such that 𝐼𝑆 is demiclosed at zero and 𝐹(𝑆)EP(𝐹). Let 𝑇𝐹𝑟𝑛 be the resolvent of 𝐹 with a positive real sequence {𝑟𝑛} such that liminf𝑛𝑟𝑛>0. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑥+𝛽𝑛𝑥𝑛+𝛾𝑛𝑆𝑇𝐹𝑟𝑛𝑥𝑛(𝑛1),(4.5) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then, {𝑥𝑛} converges strongly to 𝑃𝐹(𝑆)EP(𝐹)𝑥.

Corollary 4.4. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝑓 a contraction of 𝐻 into itself, 𝐹 a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑆𝐶𝐸 a nonexpansive mapping such that 𝐹(𝑆)EP(𝐹). Let 𝑇𝐹𝑟𝑛 be the resolvent of 𝐹 with a positive real sequence {𝑟𝑛} such that liminf𝑛𝑟𝑛>0. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛽𝑛𝑥𝑛+𝛾𝑛𝑆𝑇𝐹𝑟𝑛𝑥𝑛(𝑛1),(4.6) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then, {𝑥𝑛} converges strongly to 𝑧 such that 𝑧=𝑃𝐹(𝑆)EP(𝐹)𝑓(𝑧).

Remark 4.5. Corollary 4.4 improves and extends [42, Theorem  5]. More precisely, the conditions lim𝑛(𝑟𝑛+1𝑟𝑛)= are removed.

Setting 𝐹0 and 𝑟𝑛1 in Corollary 4.3, we have the following.

Corollary 4.6. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, and 𝑆𝐶𝐸 a quasi-nonexpansive mapping such that 𝐼𝑆 is demiclosed at zero and 𝐹(𝑆). Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑥+𝛽𝑛𝑥𝑛+𝛾𝑛𝑆𝑃𝐶𝑥𝑛(𝑛1),(4.7) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=, (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0. Then, {𝑥𝑛} converges strongly to 𝑃𝐹(𝑆)𝑥.

Applying Theorem 3.4, we have the following result.

Theorem 4.7. Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, {𝐹𝑖}𝑚𝑖=1 a finite family of a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and 𝑚𝑖=1EP(𝐹𝑖). Let {𝑇𝐹𝑖𝑟𝑖,𝑛}𝑚𝑖=1 be a finite family of the resolvents of 𝐹𝑖 with positive real sequences {𝑟𝑖,𝑛} such that liminf𝑛𝑟𝑖,𝑛>0 for all 𝑖=1,2,,𝑚. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑥+𝛽𝑛𝑥𝑛+𝛾𝑛𝑇𝐹𝑚𝑟𝑚,𝑛𝑇𝐹𝑚1𝑟𝑚1,𝑛,,𝑇𝐹1𝑟1,𝑛𝑥𝑛(𝑛1),(4.8) where {𝛼𝑛}, {𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions: (i)𝛼𝑛+𝛽𝑛+𝛾𝑛1, (ii)lim𝑛𝛼𝑛=0, (iii)𝑛=1𝛼𝑛=,s (iv)liminf𝑛𝛽𝑛(1𝛽𝑛)>0 or liminf𝑛𝛽𝑛=0. Then {𝑥𝑛} converges strongly to 𝑃𝑚𝑖=1EP(𝐹𝑖)𝑥.

Setting 𝑚=1, 𝐹1=𝐹0, 𝑟1,𝑛𝑟𝑛, and 𝛽𝑛0 in Theorem  4.4, we have the following result.

Corollary 4.8 (see [35, Corollary  4.4]). Let 𝐻 be a Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻, 𝐹 a bifunction of 𝐶×𝐶 into satisfying conditions (A1)–(A4), and EP(𝐹). Let 𝑇𝐹𝑟𝑛 the resolvent of 𝐹 with a positive real sequence {𝑟𝑛} such that liminf𝑛𝑟𝑛>0. Let {𝑥𝑛} be a sequence generated by 𝑥,𝑥1𝐻 and 𝑥𝑛+1=𝛼𝑛𝑥+1𝛼𝑛𝑇𝐹𝑟𝑛𝑥𝑛(𝑛1),(4.9) where {𝛼𝑛} is a sequence in [0,1] satisfying the following conditions: (i)lim𝑛𝛼𝑛=0, (ii)𝑛=1𝛼𝑛=, Then, {𝑥𝑛} converges strongly to 𝑃EP(𝐹)𝑥.

Acknowledgments

The authors would like to thank the referees for their comments and helpful suggestions. The corresponding author was supported by the Thailand Research Fund and the Commission on Higher Education (MRG5480166).