A New Iterative Scheme for Countable Families of Weak Relatively Nonexpansive Mappings and System of Generalized Mixed Equilibrium Problems

Abstract

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth using the properties of generalized 𝑓-projection operator. Using this result, we discuss strong convergence theorem concerning general 𝐻-monotone mappings and system of generalized mixed equilibrium problems in Banach spaces. Our results extend many known recent results in the literature.

1. Introduction

Let 𝐸 be a real Banach space with dual 𝐸^∗, and let 𝐶 be nonempty, closed and convex subset of 𝐸. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.1) A point 𝑥∈𝐶 is called a fixed point of 𝑇 if 𝑇𝑥=𝑥. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}.

We denote by 𝐽 the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓². (1.2) The following properties of 𝐽 are well known (the reader can consult [1–3] for more details). (1)If 𝐸 is uniformly smooth, then 𝐽 is norm-to-norm uniformly continuous on each bounded subset of 𝐸. (2)𝐽(𝑥)≠∅,  𝑥∈𝐸. (3)If 𝐸 is reflexive, then 𝐽 is a mapping from 𝐸 onto 𝐸^∗.(4)If 𝐸 is smooth, then 𝐽 is single valued.

Throughout this paper, we denote by 𝜙, the functional on 𝐸×𝐸 defined by𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (1.3) From [4], in uniformly convex and uniformly smooth Banach spaces, we have(‖𝑥‖−𝑦)²≤𝜙𝑥,𝑦≤(‖𝑥‖+𝑦)²,∀𝑥,𝑦∈𝐸. (1.4)

Definition 1.1. Let 𝐶 be a nonempty subset of 𝐸 and let {𝑇_𝑛}^∞_𝑛=0 be a countable family of mappings from 𝐶 into 𝐸. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 if 𝐶 contains a sequence {𝑥_𝑛}^∞_𝑛=0 which converges weakly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 is denoted by 𝐹({𝑇_𝑛}^∞_𝑛=0). One says that {𝑇_𝑛}^∞_𝑛=0 is countable family of relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1)𝐹({𝑇_𝑛}^∞_𝑛=0)≠∅;(R2)𝜙(𝑝,𝑇_𝑛𝑥)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐶,  𝑝∈𝐹(𝑇_𝑛),  𝑛≥0;(R3)∩^∞_𝑛=0𝐹(𝑇_𝑛)=𝐹({𝑇_𝑛}^∞_𝑛=0).

Definition 1.2. A point 𝑝∈𝐶 is said to be a strong asymptotic fixed point of {𝑇_𝑛}^∞_𝑛=0 if 𝐶 contains a sequence {𝑥_𝑛}^∞_𝑛=0 which converges strongly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0. The set of strong asymptotic fixed points of 𝑇 is denoted by 𝐹({𝑇_𝑛}^∞_𝑛=0). One says that a mapping {𝑇_𝑛}^∞_𝑛=0 is countable family of weak relatively nonexpansive mappings (see, e.g., [5]) if the following conditions are satisfied:(R1)𝐹({𝑇_𝑛}^∞_𝑛=0)≠∅;(R2)𝜙(𝑝,𝑇_𝑛𝑥)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐶,  𝑝∈𝐹(𝑇_𝑛),  𝑛≥0;(R3)∩^∞_𝑛=0𝐹(𝑇_𝑛)=𝐹({𝑇_𝑛}^∞_𝑛=0).

Definition 1.3. Let 𝐶 be a nonempty subset of 𝐸 and let 𝑇 be a mapping from 𝐶 into 𝐸. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛}^∞_𝑛=0 which converges weakly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of asymptotic fixed points of 𝑇 is denoted by 𝐹(𝑇). We say that a mapping 𝑇 is relatively nonexpansive (see, e.g., [6–11]) if the following conditions are satisfied:(R1)𝐹(𝑇)≠∅;(R2)𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐶,  𝑝∈𝐹(𝑇);(R3)𝐹(𝑇)=𝐹(𝑇).

Definition 1.4. A point 𝑝∈C is said to be an strong asymptotic fixed point of 𝑇 if 𝐶 contains a sequence {𝑥_𝑛}^∞_𝑛=0 which converges strongly to 𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. The set of strong asymptotic fixed points of 𝑇 is denoted by 𝐹(𝑇). We say that a mapping 𝑇 is weak relatively nonexpansive (see, e.g., [12, 13]) if the following conditions are satisfied:(R1)𝐹(𝑇)≠∅;(R2)𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐶,  𝑝∈𝐹(𝑇);(R3)𝐹(𝑇)=𝐹(𝑇).

Definition 1.3 (Definition 1.4) is a special form of Definition 1.1 (Definition 1.2) as 𝑇_𝑛≡𝑇, for all 𝑛≥0. Furthermore, Su et al. [5] gave an example which is a countable family of weak relatively nonexpansive mappings but not a countable family of relatively nonexpansive mappings. It is obvious that relatively nonexpansive mapping is weak relatively nonexpansive mapping. In fact, for any mapping 𝑇∶𝐶→𝐶, we have 𝐹(𝑇)⊂𝐹(𝑇)⊂𝐹(𝑇). Therefore, if 𝑇 is relatively nonexpansive mapping, then 𝐹(𝑇)=𝐹(𝑇)=𝐹(𝑇). Kang et al. [12] gave an example of a weak relatively nonexpansive mapping which is not relatively nonexpansive.

Let 𝐹∶𝐶×𝐶→ℝ be a bifunction, 𝐴∶𝐶→𝐸^∗ a mapping and 𝜑∶𝐶→ℝ a real-valued function. The generalized mixed equilibrium problem is to find 𝑥∈𝐶 (see, e.g., [14–19]) such that𝐹𝑥,𝑦+⟨𝐴𝑥,𝑦−𝑥⟩+𝜑𝑦−𝜑(𝑥)≥0, (1.5) for all 𝑦∈𝐶. We will denote the solutions set of (1.5) by GMEP(𝐹,𝜑). Thus GMEP𝐹,𝐴,𝜑∶=𝑥^∗∈𝐶∶𝐹𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜑𝑦−𝜑𝑥^∗≥0,∀𝑦∈𝐶. (1.6) If 𝜑=0 and 𝐴=0, then problem (1.5) reduces to an equilibrium problem studied by many authors (see, e.g., [20–28]), which is to find 𝑥^∗∈𝐶 such that𝐹𝑥^∗,𝑦≥0 (1.7) for all 𝑦∈𝐶. We shall denote the solutions set of (1.7) by EP(𝐹).

If 𝜑=0 and 𝐸=𝐻 (a real Hilbert space), then problem (1.5) reduces to a generalized equilibrium problem studied by many authors (see, e.g., [29–31]), which is to find 𝑥^∗∈𝐶 such that𝐹𝑥^∗,𝑦+𝐴𝑥^∗,𝑦−𝑥^∗≥0 (1.8) for all 𝑦∈𝐶.

If 𝐴=0 and 𝐸=𝐻, then problem (1.5) reduces to mixed equilibrium problem considered by many authors (see, e.g., [32–34]), which is to find 𝑥^∗∈𝐶 such that𝐹𝑥^∗,𝑦+𝜑𝑦−𝜑𝑥^∗≥0 (1.9) for all 𝑦∈𝐶.

The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases (see, e.g., [35]). Some methods have been proposed to solve the mixed equilibrium problem (see, e.g., [33, 34, 36]). Numerous problems in physics, optimization and economics reduce to find a solution of problem (1.8).

In [9], Matsushita and Takahashi introduced a hybrid iterative scheme for approximation of fixed points of relatively nonexpansive mapping in a uniformly convex real Banach space which is also uniformly smooth: 𝑥₀∈𝐶,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐻_𝑛=𝑤∈𝐶∶𝜙𝑤,𝑦_𝑛≤𝜙𝑤,𝑥_𝑛,𝑊_𝑛=𝑤∈𝐶∶𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛≥0 (1.10) They proved that {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_𝐹(𝑇)𝑥₀, where 𝐹(𝑇)≠∅.

In [37], Plubtieng and Ungchittrakool introduced the following hybrid projection algorithm for a pair of relatively nonexpansive mappings: 𝑥₀∈𝐶,𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₀+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛+𝛼_𝑛𝑥₀²+2𝑤,𝐽𝑥_𝑛−𝐽𝑥₀,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥_𝑛−𝐽𝑥₀,𝑥≤0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.11) where {𝛼_𝑛}, {𝛽_𝑛⁽¹⁾}, {𝛽_𝑛⁽²⁾}, and {𝛽_𝑛⁽³⁾} are sequences in (0,1) satisfying 𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽³⁾=1 and 𝑇 and 𝑆 are relatively nonexpansive mappings and 𝐽 is the single-valued duality mapping on 𝐸. They proved under the appropriate conditions on the parameters that the sequence {𝑥_𝑛} generated by (1.11) converges strongly to a common fixed point of 𝑇 and 𝑆.

In [10], Takahashi and Zembayashi introduced the following hybrid iterative scheme for approximation of fixed point of relatively nonexpansive mapping which is also a solution to an equilibrium problem in a uniformly convex real Banach space which is also uniformly smooth: 𝑥₀∈𝐶, 𝐶₁=𝐶, 𝑥₁=Π_𝐶1𝑥₀,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐹𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,∀𝑦∈𝐶,_𝑛+1=𝑤∈𝐶_𝑛∶𝜙𝑤,𝑢_𝑛≤𝜙𝑤,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥1, (1.12) where 𝐽 is the duality mapping on 𝐸. Then, they proved that {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀, where Ω=EP(𝐹)∩𝐹(𝑇)≠∅.

Recently, Li et al. [38] introduced the following hybrid iterative scheme for approximation of fixed points of a relatively nonexpansive mapping using the properties of generalized 𝑓-projection operator in a uniformly smooth real Banach space which is also uniformly convex: 𝑥₀∈𝐶,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1. (1.13) They proved a strong convergence theorem for finding an element in the fixed points set of 𝑇. We remark here that the results of Li et al. [38] extended and improved on the results of Matsushita and Takahashi [9].

Quite recently, motivated by the results of Matsushita and Takahashi [9] and Plubtieng and Ungchittrakool [37], Su et al. [5] proved the following strong convergence theorem by hybrid iterative scheme for approximation of common fixed point of two countable families of weak relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach space.

Theorem 1.5. Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth. Let 𝐶 be a nonempty, closed and convex subset of 𝐸. Suppose {𝑇_𝑛}^∞_𝑛=1 and {𝑆_𝑛}^∞_𝑛=1 are two countable families of weak relatively nonexpansive mappings of 𝐶 into itself such that Ω∶=(∩^∞_𝑛=1𝐹(𝑇_𝑛))∩(∩^∞_𝑛=1𝐹(𝑆_𝑛))≠∅. Suppose that {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶, 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝐶_𝑛=𝑤∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝜙𝑤,𝑦_𝑛≤𝜙𝑤,𝑥_𝑛,𝐶₀=𝑤∈𝐶∶𝜙𝑤,𝑦₀≤𝜙𝑤,𝑥₀,𝑄_𝑛=𝑤∈𝐶_𝑛−1∩𝑄_𝑛−1∶𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛,𝑄≥0₀𝑥=𝐶,_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥1, (1.14) with the conditions(i)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iii)0≤𝛼_𝑛≤𝛼<1 for some 𝛼∈(0,1).
Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀.

Motivated by the above-mentioned results and the ongoing research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of weak relatively nonexpansive mappings in a uniformly convex real Banach space which is also uniformly smooth using the properties of generalized 𝑓-projection operator. Our results extend the results of Li et al. [38], Su et al. [5] and many other recent known results in the literature.

2. Preliminaries

Let 𝐸 be a real Banach space. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝑡)1∶=sup2𝑥+𝑦+𝑥−𝑦−1∶‖𝑥‖≤1,𝑦≤𝑡. (2.1)𝐸 is uniformly smooth if and only if lim_𝑡→0𝜌_𝐸(𝑡)𝑡=0. (2.2) Let dim𝐸≥2. The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)∶=inf1−𝑥+𝑦2∶‖𝑥‖=𝑦=1;𝜖=𝑥−𝑦. (2.3)𝐸 is uniformly convex if for any 𝜖∈(0,2], there exists a 𝛿=𝛿(𝜖)>0 such that if 𝑥,𝑦∈𝐸 with ‖𝑥‖≤1, ‖𝑦‖≤1, and ‖𝑥−𝑦‖≥𝜖, then ‖(1/2)(𝑥+𝑦)‖≤1−𝛿. Equivalently, 𝐸 is uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2]. A normed space 𝐸 is called strictly convex if for all 𝑥,𝑦∈𝐸,  𝑥≠𝑦,  ‖𝑥‖=‖𝑦‖=1, we have ‖𝜆𝑥+(1−𝜆)𝑦‖<1, for all 𝜆∈(0,1).

Let 𝐸 be a smooth, strictly convex and reflexive real Banach space and let 𝐶 be a nonempty, closed and convex subset of 𝐸. Following Alber [39], the generalized projection Π_𝐶 from 𝐸 onto 𝐶 is defined by Π_𝐶𝑥∶=argmin_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝐸. (2.4) The existence and uniqueness of Π_𝐶 follows from the property of the functional 𝜙(𝑥,𝑦) and strict monotonicity of the mapping 𝐽 (see, e.g., [3, 4, 39–41]). If 𝐸 is a Hilbert space, then Π_𝐶 is the metric projection of 𝐻 onto 𝐶.

Next, we recall the concept of generalized 𝑓-projector operator, together with its properties. Let 𝐺∶𝐶×𝐸^∗→ℝ∪{+∞} be a functional defined as follows:𝐺𝜉,𝜓=𝜉²−2𝜉,𝜓+𝜓²+2𝜌𝑓(𝜉), (2.5) where 𝜉∈𝐶,  𝜓∈𝐸^∗,  𝜌 is a positive number and 𝑓∶𝐶→ℝ∪{+∞} is proper, convex and lower semi-continuous. From the definitions of 𝐺 and 𝑓, it is easy to see the following properties:(i)𝐺(𝜉,𝜓) is convex and continuous with respect to 𝜓 when 𝜉 is fixed;(ii)𝐺(𝜉,𝜓) is convex and lower semi-continuous with respect to 𝜉 when 𝜓 is fixed.

Definition 2.1 (Wu and Huang [42]). Let 𝐸 be a real Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty, closed and convex subset of 𝐸. One says that Π^𝑓_𝐶∶𝐸^∗→2^𝐶 is a generalized 𝑓-projection operator if Π^𝑓_𝐶𝜓=𝑢∈𝐶∶𝐺𝑢,𝜓=inf_𝜉∈𝐶G𝜉,𝜓,∀𝜓∈𝐸^∗. (2.6)

For the generalized 𝑓-projection operator, Wu and Huang [42] proved the following theorem basic properties.

Lemma 2.2 (Wu and Huang [42]). Let 𝐸 be a real reflexive Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Then the following statements hold:(i)Π^𝑓_𝐶 is a nonempty closed convex subset of 𝐶 for all 𝜓∈𝐸^∗;(ii)if 𝐸 is smooth, then for all 𝜓∈𝐸^∗,  𝑥∈Π^𝑓_𝐶 if and only if 𝑥−𝑦,𝜓−𝐽𝑥+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶; (2.7)(iii)if 𝐸 is strictly convex and 𝑓∶𝐶→ℝ∪{+∞} is positive homogeneous (i.e., 𝑓(𝑡𝑥)=𝑡𝑓(𝑥) for all 𝑡>0 such that 𝑡𝑥∈𝐶 where 𝑥∈𝐶), then Π^𝑓_𝐶 is a single valued mapping.

Fan et al. [43] showed that the condition 𝑓 is positive homogeneous which appeared in Lemma 2.2 and can be removed.

Lemma 2.3 (Fan et al. [43]). Let 𝐸 be a real reflexive Banach space with its dual 𝐸^∗ and 𝐶 a nonempty, closed and convex subset of 𝐸. Then if E is strictly convex, then Π^𝑓_𝐶 is a single-valued mapping.

Recall that 𝐽 is a single valued mapping when 𝐸 is a smooth Banach space. There exists a unique element 𝜓∈𝐸^∗ such that 𝜓=𝐽𝑥 for each 𝑥∈𝐸. This substitution in (4.3) gives𝐺(𝜉,𝐽𝑥)=𝜉²−2𝜉,𝐽𝑥+‖𝑥‖²+2𝜌𝑓(𝜉). (2.8) Now, we consider the second generalized 𝑓-projection operator in a Banach space.

Definition 2.4. Let 𝐸 be a real Banach space and 𝐶 a nonempty, closed, and convex subset of 𝐸. One says that Π^𝑓_𝐶∶𝐸→2^𝐶 is a generalized 𝑓-projection operator if Π^𝑓_𝐶𝑥=𝑢∈𝐶∶𝐺(𝑢,𝐽𝑥)=inf_𝜉∈𝐶𝐺(𝜉,𝐽𝑥),∀𝑥∈𝐸. (2.9)

Obviously, the definition of {𝑇_𝑛}^∞_𝑛=0 is a countably family of weak relatively nonexpansive mappings is equivalent to(R^′1)𝐹({𝑇_𝑛}^∞_𝑛=0)≠∅;(R^′2)𝐺(𝑝,𝐽𝑇_𝑛𝑥)≤𝐺(𝑝,𝐽𝑥), for all 𝑥∈𝐶,  𝑝∈𝐹(𝑇_𝑛),  𝑛≥0;(R^′3)∩^∞_𝑛=0𝐹(𝑇_𝑛)=𝐹({𝑇_𝑛}^∞_𝑛=0).

Lemma 2.5 (Li et al. [38]). Let 𝐸 be a Banach space and 𝑓∶𝐸→ℝ∪{+∞} a lower semi-continuous convex functional. Then there exists 𝑥^∗∈𝐸^∗ and 𝛼∈ℝ such that 𝑓(𝑥)≥⟨𝑥,𝑥^∗⟩+𝛼,∀𝑥∈𝐸. (2.10)

We know that the following lemmas hold for operator Π^𝑓_𝐶.

Lemma 2.6 (Li et al. [38]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and reflexive Banach space 𝐸. Then the following statements hold:(i)Π^𝑓_𝐶𝑥 is a nonempty closed and convex subset of 𝐶 for all 𝑥∈𝐸;(ii)for all 𝑥∈𝐸, 𝑥∈Π^𝑓_𝐶𝑥 if and only if 𝑥−𝑦,𝐽𝑥−𝐽𝑥+𝜌𝑓𝑦−𝜌𝑓(𝑥)≥0,∀𝑦∈𝐶; (2.11)(iii)if 𝐸 is strictly convex, then Π^𝑓_𝐶𝑥 is a single-valued mapping.

Lemma 2.7 (Li et al. [38]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and reflexive Banach space 𝐸. Let 𝑥∈𝐸 and 𝑥∈Π^𝑓_𝐶. Then 𝜙𝑦,𝑥+𝐺(𝑥,𝐽𝑥)≤𝐺𝑦,𝐽𝑥,∀𝑦∈𝐶. (2.12)

Lemma 2.8 (Su et al. [5]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex Banach space 𝐸. Let 𝑇 be a weak relatively nonexpansive mapping of 𝐶 into itself. Then 𝐹(𝑇) is closed and convex.

Also, this following lemma will be used in the sequel.

Lemma 2.9 (Kamimura and Takahashi [4]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, uniformly convex Banach space 𝐸. Let {𝑥_𝑛}^∞_𝑛=1 and {𝑦_𝑛}^∞_𝑛=1 be sequences in 𝐸 such that either {𝑥_𝑛}^∞_𝑛=1 or {𝑦_𝑛}^∞_𝑛=1 is bounded. If lim_𝑛→∞𝜙(𝑥_𝑛,𝑦_𝑛)=0, then lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0.

Lemma 2.10 (Cho et al. [44]). Let 𝐸 be a uniformly convex real Banach space. For arbitrary 𝑟>0, let 𝐵_𝑟(0)∶={𝑥∈𝐸∶‖𝑥‖≤𝑟} and 𝜆,𝜇,𝛾∈[0,1] such that 𝜆+𝜇+𝛾=1. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,2𝑟]⟶ℝ,𝑔(0)=0 (2.13) such that for every 𝑥,𝑦,𝑧∈𝐵_𝑟(0), the following inequality holds: 𝜆𝑥+𝜇𝑦+𝛾𝑧²≤𝜆‖𝑥‖²+𝜇𝑦²−𝜆𝜇𝑔𝑥−𝑦. (2.14)

For solving the equilibrium problem for a bifunction 𝐹∶𝐶×𝐶→ℝ, let us assume that 𝐹 satisfies the following conditions:(A1)𝐹(𝑥,𝑥)=0 for all 𝑥∈𝐶;(A2)𝐹 is monotone, that is, 𝐹(𝑥,𝑦)+𝐹(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶;(A3)for each 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡↓0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦);(A4)for each 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.11 (Liu et al. [14] and Zhang [19]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex and reflexive Banach space 𝐸. Assume that 𝐹∶𝐶×𝐶→ℝ satisfies (A1)–(A4), 𝐴∶𝐶→𝐸^∗ a continuous and monotone mapping, and 𝜑∶𝐶→ℝ a lower semicontinuous and convex functional. For 𝑟>0 and 𝑥∈𝐸, there exists 𝑧∈𝐶 such that 𝑄+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶, (2.15) where 𝑄(𝑧,𝑦)=𝐹(𝑧,𝑦)+⟨𝐴𝑧,𝑦−𝑧⟩+𝜑(𝑦)−𝜑(𝑧),𝑧,𝑦∈𝐶. Furthermore, define a mapping 𝑇_𝑟∶𝐸→𝐶 as follows: 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝑄𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (2.16) Then, the following hold:(i)𝑇_𝑟 is single-valued;(ii)for any 𝑥,𝑦∈𝐸, 𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦; (2.17)(iii)𝐹(𝑇_𝑟)=GMEP(𝐹,𝐴,𝜑);(iv)GMEP(𝐹,𝐴,𝜑) is closed and convex.

Lemma 2.12 (Zhang [19]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Assume that 𝐹∶𝐶×𝐶→ℝ satisfies (A1)–(A4), and let 𝑟>0. Then for each 𝑥∈𝐸 and 𝑞∈𝐹(𝑇_𝑟), 𝜙𝑞,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙𝑞,𝑥. (2.18)

For the rest of this paper, the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝 shall be denoted by 𝑥_𝑛→𝑝 as 𝑛→∞, and we shall assume that 𝛽_𝑛⁽¹⁾,𝛽_𝑛⁽²⁾,𝛽_𝑛⁽¹⁾∈[0,1] such that 𝛽_𝑛⁽¹⁾+𝛽_𝑛⁽²⁾+𝛽_𝑛⁽¹⁾=1, for all 𝑛≥0.

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. For each 𝑘=1,2,…,𝑚, let 𝐹_𝑘 be a bifunction from 𝐶×𝐶 satisfying (A1)–(A4), 𝐴_𝑘∶𝐶→𝐸^∗ a continuous and monotone mapping and 𝜑_𝑘∶𝐶→ℝ a lower semicontinuous and convex functional. Suppose that {𝑇_𝑛}^∞_𝑛=0 and {𝑆_𝑛}^∞_𝑛=0 are two countable families of weak relatively nonexpansive mappings of 𝐶 into itself such that Ω∶=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩(∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛))≠∅. Let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)), and suppose that {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,  𝐶₀=𝐶, 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑢_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥0, (3.1) with the conditions(i)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iii)0≤𝛼_𝑛≤𝛼<1 for some 𝛼∈(0,1);(iv){𝑟_𝑘,𝑛}^∞_𝑛=1⊂(0,∞),  (𝑘=1,2,…,𝑚) satisfying liminf_𝑛→∞𝑟_𝑘,𝑛>0,  (𝑘=1,2,…,𝑚). Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π^𝑓_Ω𝑥₀.

Proof. By Lemma 2.8, we know that (∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛) is closed and convex. We also know from Lemma 2.11(iv) that ∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘) is closed and convex. Hence, Ω=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩(∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛)) is a nonempty, closed and convex subset of 𝐶. Consequently, Π^𝑓_Ω𝑥₀ is well defined. We first show that 𝐶_𝑛, for all 𝑛≥0 is closed and convex. It is obvious that 𝐶₀=𝐶 is closed and convex. Thus, we only need to show that 𝐶_𝑛 is closed and convex for each 𝑛≥1. Since 𝐺(𝑧,𝐽𝑢_𝑛)≤𝐺(𝑧,𝐽𝑥_𝑛) is equivalent to 2𝑧,𝐽𝑥_𝑛−𝑧,𝐽𝑢_𝑛≤𝑥_𝑛²−𝑢_𝑛², (3.2) this implies that 𝐶_𝑛+1 is closed and convex for all 𝑛≥0. This shows that Π^𝑓_𝐶𝑛+1𝑥₀ is well defined for all 𝑛≥0. By taking 𝜃^𝑘_𝑛=𝑇^𝑄𝑘_{𝑟𝑘,𝑛}𝑇^𝑄𝑘−1_{𝑟𝑘−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛, 𝑘=1,2,…,𝑚 and 𝜃⁰_𝑛=𝐼 for all 𝑛≥1, we obtain 𝑢_𝑛=𝜃^𝑚_𝑛𝑦_𝑛. We next show that Ω⊂𝐶_𝑛, for all 𝑛≥0. For 𝑛=0, we have Ω⊂𝐶=𝐶₀. Then for each 𝑥^∗∈Ω, we obtain 𝐺𝑥^∗,𝐽𝑢_𝑛≤𝐺𝑥^∗,𝐽𝜃^𝑚_𝑛𝑦_𝑛≤𝐺𝑥^∗,𝐽𝑦_𝑛=𝐺𝑥^∗,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛=𝑥^∗2−2𝛼_𝑛𝑥^∗,𝐽𝑥_𝑛−21−𝛼_𝑛𝑥^∗,𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛+(1−𝛼_𝑛)𝐽𝑧_𝑛²+2𝜌𝑓𝑥^∗≤𝑥^∗2−2𝛼_𝑛⟨𝑥^∗,𝐽𝑥_𝑛⟩−21−𝛼_𝑛𝑥^∗,𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛²+1−𝛼_𝑛𝐽𝑧_𝑛²+2𝜌𝑓𝑥^∗=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑥^∗,𝐽𝑧_𝑛=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑥^∗,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛≤𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝑥^∗2−2𝛽_𝑛⁽¹⁾⟨𝑥^∗,𝐽𝑥_𝑛⟩−2𝛽_𝑛⁽²⁾𝑥^∗,𝐽𝑇_𝑛𝑥_𝑛−2𝛽_𝑛⁽³⁾⟨𝑥^∗,𝐽𝑆_𝑛𝑥_𝑛⟩+𝛽_𝑛⁽¹⁾𝑥_𝑛²+𝛽_𝑛⁽²⁾𝑇_𝑛𝑥_𝑛²+𝛽_𝑛⁽³⁾𝑆_𝑛𝑥_𝑛²+2𝜌𝑓𝑥^∗=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐺𝑥^∗,𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐺𝑥^∗,𝐽𝑆_𝑛𝑥_𝑛≤𝐺𝑥^∗,𝐽𝑥_𝑛. (3.3) So, 𝑥^∗∈𝐶_𝑛. This implies that ∅≠Ω⊂𝐶_𝑛, for all 𝑛≥0. It follows that {𝑥_𝑛}^∞_𝑛=0 is well defined for all 𝑛≥0.
We now show that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀) exists. Since 𝑓∶𝐸→ℝ is a convex and lower semi-continuous, applying Lemma 2.5, we see that there exists 𝑢^∗∈𝐸^∗ and 𝛼∈ℝ such that 𝑓𝑦≥⟨𝑦,𝑢^∗⟩+𝛼,∀𝑦∈𝐸. (3.4) It follows that 𝐺𝑥_𝑛,𝐽𝑥₀=𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑥_𝑛≥𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²+2𝜌𝑥_𝑛,𝑢^∗=𝑥+2𝜌𝛼_𝑛2−2⟨𝑥_𝑛,𝐽𝑥₀−𝜌𝑢^∗⟩+𝑥₀²≥𝑥+2𝜌𝛼_𝑛2−2𝑥_𝑛𝐽𝑥₀−𝜌𝑢^∗+𝑥₀²=𝑥+2𝜌𝛼_𝑛−𝐽𝑥₀−𝜌𝑢^∗2+𝑥₀²−𝐽𝑥₀−𝜌𝑢^∗2+2𝜌𝛼. (3.5) Since 𝑥_𝑛=Π^𝑓_𝐶𝑛𝑥₀, it follows from (3.5) that 𝐺𝑥^∗,𝐽𝑥₀≥𝐺𝑥_𝑛,𝐽𝑥₀≥𝑥_𝑛−𝐽𝑥₀−𝜌𝑢^∗2+𝑥₀²−𝐽𝑥₀−𝜌𝑢^∗2+2𝜌𝛼 (3.6) for each 𝑥^∗∈Ω. This implies that {𝑥_𝑛}^∞_𝑛=0 is bounded and so is {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0. By the construction of 𝐶_𝑛, we have that 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=Π^𝑓_𝐶𝑚𝑥₀∈𝐶_𝑛 for any positive integer 𝑚≥𝑛. It then follows from Lemma 2.7 that 𝜙𝑥_𝑚,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑚,𝐽𝑥₀. (3.7) It is obvious that 𝜙𝑥_𝑚,𝑥_𝑛≥𝑥_𝑚−𝑥_𝑛²≥0. (3.8) In particular, 𝜙𝑥_𝑛+1,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+1,𝐽𝑥₀,𝜙𝑥_𝑛+1,𝑥_𝑛≥𝑥_𝑛+1−𝑥_𝑛²≥0, (3.9) and so {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 is nondecreasing. It follows that the limit of {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 exists. By the fact that 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=Π^𝑓_𝐶𝑚𝑥₀∈𝐶_𝑛 for any positive integer 𝑚≥𝑛, we obtain 𝜙𝑥_𝑚,𝑢_𝑛≤𝜙𝑥_𝑚,𝑥_𝑛. (3.10) Now, (3.7) implies that 𝜙𝑥_𝑚,𝑢_𝑛≤𝜙𝑥_𝑚,𝑥_𝑛≤𝐺𝑥_𝑚,𝐽𝑥₀−𝐺𝑥_𝑛,𝐽𝑥₀. (3.11) Taking the limit as 𝑚,𝑛→∞ in (3.11), we obtain lim_𝑛→∞𝜙𝑥_𝑚,𝑥_𝑛=0. (3.12) It then follows from Lemma 2.9 that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. Hence, {𝑥_𝑛}^∞_𝑛=0 is Cauchy. Since 𝐸 is a Banach space and 𝐶 is closed and convex, then there exists 𝑝∈𝐶 such that 𝑥_𝑛→𝑝 as 𝑛→∞.
Now since 𝜙(𝑥_𝑚,𝑥_𝑛)→0 as 𝑚,𝑛→∞ we have in particular that 𝜙(𝑥_𝑛+1,𝑥_𝑛)→0 as 𝑛→∞ and this further implies that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Since 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1, we have 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛,∀𝑛≥0. (3.13) Then, we obtain lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=0. (3.14) Since 𝐸 is uniformly convex and smooth, we have from Lemma 2.9 that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0=lim_𝑛→∞𝑥_𝑛+1−𝑢_𝑛. (3.15) So, 𝑥_𝑛−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑢_𝑛. (3.16) Hence, lim_𝑛→∞𝑥_𝑛−𝑢_𝑛=0. (3.17) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets and lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0, we obtain lim_𝑛→∞𝐽𝑥_𝑛−𝑢_𝑛=0. (3.18) Since {𝑥_𝑛} is bounded, so are {𝑧_𝑛}, {𝐽𝑇_𝑛𝑥_𝑛}, and {𝐽𝑆_𝑛𝑥_𝑛}. Let 𝑟∶=sup_𝑛≥0{‖𝑥_𝑛‖,‖𝑇_𝑛𝑥_𝑛‖,‖𝑆_𝑛𝑥_𝑛‖}. Then from Lemma 2.10, we have 𝐺𝑥^∗,𝐽𝑢_𝑛≤𝐺𝑥^∗,𝐽𝜃^𝑚_𝑛𝑦_𝑛≤𝐺𝑥^∗,𝐽𝑦_𝑛=𝐺𝑥^∗,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛=𝑥^∗2−2𝛼_𝑛𝑥^∗,𝐽𝑥_𝑛−21−𝛼_𝑛𝑥^∗,𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛²+2𝜌𝑓𝑥^∗≤𝑥^∗2−2𝛼_𝑛⟨𝑥^∗,𝐽𝑥_𝑛⟩−21−𝛼_𝑛𝑥^∗,𝐽𝑧_𝑛+𝛼_𝑛𝐽𝑥_𝑛²+1−𝛼_𝑛𝐽𝑧_𝑛²+2𝜌𝑓𝑥^∗=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑥^∗,𝐽𝑧_𝑛=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑥^∗,𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛≤𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝑥^∗2−2𝛽_𝑛⁽¹⁾⟨𝑥^∗,𝐽𝑥_𝑛⟩−2𝛽_𝑛⁽²⁾⟨𝑥^∗,𝐽𝑇_𝑛𝑥_𝑛⟩−2𝛽_𝑛⁽³⁾𝑥^∗,𝐽𝑆_𝑛𝑥_𝑛+𝛽_𝑛⁽¹⁾𝑥_𝑛²+𝛽_𝑛⁽²⁾𝑇_𝑛𝑥_𝑛²+𝛽_𝑛⁽³⁾𝑆_𝑛𝑥_𝑛²−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛+2𝜌𝑓𝑥^∗=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐺𝑥^∗,𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐺𝑥^∗,𝐽𝑆_𝑛𝑥_𝑛−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽_𝑛⁽³⁾𝐺𝑥^∗,𝐽𝑥_𝑛−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛=𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛𝐺𝑥^∗,𝐽𝑥_𝑛−𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝐺𝑥^∗,𝐽𝑥_𝑛−1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛. (3.19) It then follows that (1−𝛼)𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤1−𝛼_𝑛𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛≤𝐺𝑥^∗,𝐽𝑥_𝑛−𝐺𝑥^∗,𝐽𝑢_𝑛. (3.20) But 𝐺𝑥^∗,𝐽𝑥_𝑛−𝐺𝑥^∗,𝐽𝑢_𝑛=𝑥_𝑛²−𝑢_𝑛²−2⟨𝑥^∗,𝐽𝑥_𝑛−𝐽𝑢_𝑛⟩≤𝑥_𝑛²−𝑢_𝑛²+2𝑥^∗,𝐽𝑥_𝑛−𝐽𝑢_𝑛≤𝑥_𝑛−𝑢_𝑛𝑥_𝑛−𝑢_𝑛+2𝑥^∗𝐽𝑥_𝑛−𝐽𝑢_𝑛≤𝑥_𝑛−𝑢_𝑛𝑥_𝑛+𝑢_𝑛+2𝑥^∗𝐽𝑥_𝑛−𝐽𝑢_𝑛. (3.21) From lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0 and lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑢_𝑛‖=0, we obtain G𝑥^∗,𝐽𝑥_𝑛−𝐺𝑥^∗,𝐽𝑢_𝑛⟶0,𝑛⟶∞. (3.22) Using the condition liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0, we have lim_𝑛→∞𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛=0. (3.23) By property of 𝑔, we have lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑥_𝑛‖=0. Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞𝑥_𝑛−𝑇_𝑛𝑥_𝑛=0. (3.24) Similarly, we can show that lim_𝑛→∞𝑥_𝑛−𝑆_𝑛𝑥_𝑛=0. (3.25) Since 𝑥_𝑛→𝑝 and {𝑇_𝑛}, {𝑆_𝑛} are uniformly closed, we have 𝑝∈(∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛)).
Next, we show that 𝑝∈∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘). Now, by Lemma 2.12, we obtain 𝜙𝑢_𝑛,𝑦_𝑛=𝜙𝜃^𝑚_𝑛𝑦_𝑛,𝑦_𝑛≤𝜙𝑥^∗,𝑦_𝑛−𝜙𝑥^∗,𝜃^𝑚_𝑛𝑦_𝑛≤𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥^∗,𝑢_𝑛⟶0,𝑛⟶∞. (3.26) Using Lemma 2.9, we have lim_𝑛→∞‖𝑢_𝑛−𝑦_𝑛‖=0. Furthermore, 𝑥_𝑛−𝑦_𝑛≤𝑥_𝑛−𝑢_𝑛+𝑢_𝑛−𝑦_𝑛⟶0,𝑛⟶∞. (3.27) Since 𝑥_𝑛→𝑝 as 𝑛→∞ and ‖𝑥_𝑛−𝑦_𝑛‖→0 as 𝑛→∞, then 𝑦_𝑛→𝑝 as 𝑛→∞. By the fact that 𝜃^𝑘_𝑛, 𝑘=1,2,…,𝑚 is relatively nonexpansive and using Lemma 2.12 again, we have that 𝜙𝜃^𝑘_𝑛𝑦_𝑛,𝑦_𝑛≤𝜙𝑥^∗,𝑦_𝑛−𝜙𝑥^∗,𝜃^𝑘_𝑛𝑦_𝑛≤𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥^∗,𝜃^𝑘_𝑛𝑦_𝑛. (3.28) Observe that 𝜙𝑥^∗,𝑢_𝑛=𝜙𝑥^∗,𝜃^𝑚_𝑛𝑦_𝑛=𝜙𝑥^∗,𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄𝑘_{𝑟𝑘,𝑛}𝑇^𝑄𝑘−1_{𝑟𝑘−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛=𝜙𝑥^∗,𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝜃^𝑘_𝑛𝑦_𝑛≤𝜙𝑥^∗,𝜃^𝑘_𝑛𝑦_𝑛. (3.29) Using (3.29) in (3.28), we obtain 𝜙𝜃^𝑘_𝑛𝑦_𝑛,𝑦_𝑛≤𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥^∗,𝑢_𝑛⟶0,𝑛⟶∞. (3.30) Then Lemma 2.9 implies that lim_𝑛→∞‖𝑦_𝑛−𝜃^𝑘_𝑛𝑦_𝑛‖=0,  𝑘=1,2,…,𝑚. Now 𝑝−𝜃^𝑘_𝑛𝑦_𝑛≤𝑦_𝑛−𝜃^𝑘_𝑛𝑦_𝑛+𝑦_𝑛−𝑝⟶0,𝑛⟶∞,𝑘=1,2,…,𝑚. (3.31) Similarly, lim_𝑛→∞‖𝑝−𝜃_𝑛^𝑘−1𝑦_𝑛‖=0,  𝑘=1,2,…,𝑚. This further implies that lim_𝑛→∞𝜃^𝑘_𝑛𝑦_𝑛−𝜃_𝑛^𝑘−1𝑦_𝑛=0. (3.32) Also, since 𝐽 is uniformly norm-to-norm continuous on bounded sets and using (3.32), we obtain lim_𝑛→∞𝐽𝜃^𝑘_𝑛𝑦_𝑛−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛=0. (3.33) Since liminf_𝑛→∞𝑟_𝑘,𝑛>0,  (𝑘=1,2,…,𝑚), lim_𝑛→∞𝐽𝜃^𝑘_𝑛𝑦_𝑛−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛𝑟_𝑘,𝑛=0. (3.34) By Lemma 2.11, we have that for each 𝑘=1,2,…,𝑚𝑄_𝑘𝜃^𝑘_𝑛𝑦_𝑛+1,𝑦𝑟_𝑘,𝑛𝑦−𝜃^𝑘_𝑛𝑦_𝑛,𝐽𝜃^𝑘_𝑛𝑦_𝑛−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛≥0,∀𝑦∈𝐶. (3.35) Furthermore, using (A2) we obtain 1𝑟_𝑘,𝑛𝑦−𝜃^𝑘_𝑛𝑦_𝑛,𝐽𝜃^𝑘_𝑛𝑦_𝑛−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛≥𝑄_𝑘𝑦,𝜃^𝑘_𝑛𝑦_𝑛. (3.36) By (A4), (3.34), and 𝜃^𝑘_𝑛𝑦_𝑛→𝑝, we have for each 𝑘=1,2,…,𝑚𝑄_𝑘𝑦,𝑝≤0,∀𝑦∈𝐶. (3.37) For fixed 𝑦∈𝐶, let 𝑧_𝑡,𝑦∶=𝑡𝑦+(1−𝑡)𝑝 for all 𝑡∈(0,1]. This implies that 𝑧_𝑡∈𝐶. This yields that 𝑄_𝑘(𝑧_𝑡,𝑝)≤0. It follows from (A1) and (A4) that 0=𝑄_𝑘𝑧_𝑡,𝑧_𝑡≤𝑡𝑄_𝑘𝑧_𝑡,𝑦+(1−𝑡)𝑄_𝑘𝑧_𝑡,𝑝≤𝑡𝑄_𝑘𝑧_𝑡,𝑦 (3.38) and hence 0≤𝑄_𝑘𝑧_𝑡,𝑦. (3.39) From condition (A3), we obtain 𝑄_𝑘𝑝,𝑦≥0,∀𝑦∈𝐶. (3.40) This implies that 𝑝∈GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘),  𝑘=1,2,…,𝑚. Thus, 𝑝∈∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘). Hence, we have 𝑝∈Ω=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩(∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛)).
Finally, we show that 𝑝=Π^𝑓_Ω𝑥₀. Since Ω=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩(∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛)) is a closed and convex set, from Lemma 2.6, we know that Π^𝑓_Ω𝑥₀ is single valued and denote 𝑤=Π^𝑓_Ω𝑥₀. Since 𝑥_𝑛=Π^𝑓_𝐶𝑛𝑥₀ and 𝑤∈Ω⊂𝐶_𝑛, we have 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑤,𝐽𝑥₀,∀𝑛≥0. (3.41) We know that 𝐺(𝜉,𝐽𝜑) is convex and lower semi-continuous with respect to 𝜉 when 𝜑 is fixed. This implies that 𝐺𝑝,𝐽𝑥₀≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑤,𝐽𝑥₀. (3.42) From the definition of Π^𝑓_Ω𝑥₀ and 𝑝∈Ω, we see that 𝑝=𝑤. This completes the proof.

Corollary 3.2 (Li et al. [38]). Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Suppose that 𝑇∶𝐶→𝐶 is a relatively nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑇)≠∅ and 𝑓∶𝐸→ℝ is a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)). Suppose that {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,  𝐶₀=𝐶𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥0, (3.43) where 𝐽 is the duality mapping on 𝐸. Suppose that {𝛼_𝑛}^∞_𝑛=0 is a sequence in (0,1) such that limsup_𝑛→∞𝛼_𝑛<1. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π^𝑓_𝐹(𝑇)𝑥₀.

Corollary 3.3. Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth. Let 𝐶 be a nonempty, closed and convex subset of 𝐸. For each 𝑘=1,2,…,𝑚, let 𝐹_𝑘 be a bifunction from 𝐶×𝐶 satisfying (A1)–(A4), 𝐴_𝑘∶𝐶→𝐸^∗ a continuous and monotone mapping and 𝜑_𝑘∶𝐶→ℝ a lower semicontinuous and convex functional. Suppose {𝑇_𝑛}^∞_𝑛=0 and {𝑆_𝑛}^∞_𝑛=0 are two countable families of weak relatively nonexpansive mappings of 𝐶 into itself such that Ω∶=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩(∩^∞_𝑛=0𝐹(𝑇_𝑛))∩(∩^∞_𝑛=0𝐹(𝑆_𝑛))≠∅. Suppose that {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,  𝐶₀=𝐶𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇_𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑆_𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄m_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝜙𝑤,𝑢_𝑛≤𝜙𝑤,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥0, (3.44) with the conditions(i)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iii)0≤𝛼_𝑛≤𝛼<1 for some 𝛼∈(0,1); (iv){𝑟_𝑘,𝑛}^∞_𝑛=1⊂(0,∞),  (𝑘=1,2,…,𝑚) satisfying liminf_𝑛→∞𝑟_𝑘,𝑛>0, (𝑘=1,2,…,𝑚). Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀.

Proof. Take 𝑓(𝑥)=0 for all 𝑥∈𝐸 in Theorem 3.1, 𝐺(𝜉,𝐽𝑥)=𝜙(𝜉,𝑥) and Π^𝑓_𝐶𝑥₀=Π_𝐶𝑥₀. Then, the desired conclusion follows.

Remark 3.4. Corollary 3.3 extends and improves on Theorem 1.5. In fact, the iterative procedure (3.44) is simpler than (1.14) in the following two aspects: (a) the process of computing Q_𝑛={𝑤∈𝐶_𝑛−1∩𝑄_𝑛−1∶⟨𝑥_𝑛−𝑤,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0} is removed; (b) the process of computing Π_{𝐶𝑛∩𝑄𝑛} is replaced by computing Π_𝐶𝑛.

4. Applications

A mapping 𝐻 from 𝐸 to 𝐸^∗ is said to be (i)monotone if ⟨𝐻𝑥−𝐻𝑦,𝑥−𝑦⟩≥0, for all 𝑥,𝑦∈𝐸;(ii)strictly monotone if 𝐻 is monotone and ⟨𝐻𝑥−𝐻𝑦,𝑥−𝑦⟩=0 if and only if 𝑥=𝑦;(iii)𝛽-Lipschitz continuous if there exists a constant 𝛽≥0 such that ‖𝐻𝑥−𝐻𝑦‖≤𝛽‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐸.

Let 𝑀 be a set-valued mapping from 𝐸 to 𝐸^∗ with domain 𝐷(𝑀)={𝑧∈𝐸∶𝑀𝑧≠∅} and range 𝑅(𝑀)=∪{𝑀𝑧∶𝑧∈𝐷(𝑀)}. A set-valued mapping 𝑀 is said to be (i)monotone if ⟨𝑥₁−𝑥₂,𝑦₁−𝑦₂⟩≥0 for each 𝑥_𝑖∈𝐷(𝑀) and 𝑦_𝑖∈𝑀𝑥_𝑖,  𝑖=1,2;(ii)𝑟-strongly monotone if ⟨𝑥₁−𝑥₂,𝑦₁−𝑦₂⟩≥𝑟‖𝑥₁−𝑥₂‖² for each 𝑥_𝑖∈𝐷(𝑀) and 𝑦_𝑖∈𝑀𝑥_𝑖,  𝑖=1,2;(iii)maximal monotone if 𝑀 is monotone and its graph 𝐺(𝑀)∶={(𝑥,𝑦)∶𝑦∈𝑀𝑥} is not properly contained in the graph of any other monotone operator;(iv)a general 𝐻-monotone if 𝑀 is monotone and (𝐻+𝜆𝑀)𝐸=𝐸^∗ holds for every 𝜆>0, where 𝐻 is a mapping from 𝐸 to 𝐸^∗.

We denote the set {𝑥∈𝐸∶0∈𝑀𝑥} by 𝑀⁻¹0. From Li et al. [38], we know that if 𝐻∶𝐸→𝐸^∗ is strictly monotone and 𝑀∶𝐸→2^𝐸∗ is general 𝐻-monotone mapping, then 𝑀⁻¹0 is closed and convex. Furthermore, for every 𝜆>0 and 𝑥^∗∈𝐸^∗, there exists a unique 𝑥∈𝐷(𝑀) such that 𝑥=(𝐻+𝜆𝑀)⁻¹𝑥^∗. Thus, we can define a single-value mapping 𝑇_𝜆∶𝐸→𝐷(𝑀) by 𝑇_𝜆𝑥=(𝐻+𝜆𝑀)⁻¹𝐻𝑥. It is obvious that 𝑀⁻¹0=𝐹(𝑇_𝜆) for all 𝜆>0.

Lemma 4.1 (Alber, [39]). If 𝐸 is a uniformly convex and uniformly smooth Banach space, 𝛿_𝐸(𝜖) is the modulus of convexity of 𝐸, and 𝜌_𝐸(𝑡) is the modulus of smoothness of 𝐸, then the inequalities 8𝑑²𝛿_𝐸𝑥−𝜉()4𝑑≤𝜙𝑥,𝜉≤4𝑑²𝜌_𝐸4𝑥−𝜉𝑑 (4.1) hold for all 𝑥 and 𝜉 in 𝐸, where 𝑑=(‖𝑥‖²+‖𝜉‖²)/2.

Lemma 4.2 (Xia and Huang [45]). Let 𝐸 be a Banach space with dual space 𝐸^∗,𝐻∶𝐸→𝐸^∗ a strictly monotone mapping, and 𝑀∶𝐸→2^𝐸∗ a general 𝐻-monotone mapping. Then(i)(𝐻+𝜆𝑀)⁻¹ is a single-valued mapping;(ii)if 𝐸 is reflexive and 𝑀∶𝐸→2^𝐸∗ is 𝑟-strongly monotone, (𝐻+𝜆𝑀)⁻¹ is Lipschitz continuous with constant 1/𝜆𝑟, where 𝑟>0.

Theorem 4.3. Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth with 𝛿_𝐸(𝜖)≥𝑘𝜖² and 𝜌_𝐸(𝑡)≤𝑐𝑡² for some 𝑘,𝑐>0. For each 𝑘=1,2,…,𝑚, let 𝐹_𝑘 be a bifunction from 𝐸×𝐸 satisfying (A1)–(A4), 𝐴_𝑘∶𝐸→𝐸^∗ a continuous and monotone mapping, and 𝜑_𝑘∶𝐸→ℝ a lower semicontinuous and convex functional. Suppose that 𝐻∶𝐸→𝐸^∗ is a strictly monotone and 𝛽-Lipschitz continuous mapping and 𝑀_𝑖∶𝐸→2^𝐸∗ is a general 𝐻-monotone mapping and 𝑟_𝑖-strongly monotone mapping with 𝑟_𝑖>0,  𝑖=1,2 such that Ω∶=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩𝑀₁⁻¹0∩𝑀₂⁻¹0≠∅. Let 𝑇^𝑀𝑖_𝜆=(𝐻+𝜆𝑀_𝑖)⁻¹𝐻,  𝑖=1,2, let 𝑓∶𝐸→ℝ a convex and lower semicontinuous mapping with 𝐷(𝑓)=𝐸 and suppose for each 𝑛≥0 that there exists a 𝜆_𝑛>0 such that 64𝑐𝛽²≤min{(1/2)𝑘𝜆²_𝑛𝑟²₁,(1/2)𝑘𝜆²_𝑛𝑟²₂}. Let {𝑥_𝑛}^∞_𝑛=0 be iteratively generated by 𝑥₀∈𝐸,  𝐶₀=𝐸, 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝑇^𝑀1_𝜆𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝑇^𝑀2_𝜆𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝐺𝑤,𝐽𝑦_𝑛≤𝐺𝑤,𝐽𝑥_𝑛,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥0, (4.2) with the conditions(i)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iii)0≤𝛼_𝑛≤𝛼<1; (iv)liminf_𝑛→∞𝜆_𝑛>0; (v){𝑟_𝑘,𝑛}^∞_𝑛=1⊂(0,∞),  (𝑘=1,2,…,𝑚) satisfying liminf_𝑛→∞𝑟_𝑘,𝑛>0, (𝑘=1,2,…,𝑚). Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π^𝑓_Ω𝑥₀.

Proof. We only need to prove that {𝑇^𝑀1_𝜆𝑛} and {𝑇^𝑀2_𝜆𝑛} are countable families of weak relatively nonexpansive mappings with common fixed points sets ∩^∞_𝑛=0𝐹(𝑇^𝑀1_𝜆𝑛)=𝑀₁⁻¹0 and ∩^∞_𝑛=0𝐹(𝑇^𝑀2_𝜆𝑛)=𝑀₂⁻¹0, respectively. Firstly, we have ∩^∞_𝑛=0𝐹(𝑇^𝑀1_𝜆𝑛)=𝑀₁⁻¹0≠∅. Secondly, we show that 𝜙(𝑝,𝑇^𝑀1_𝜆𝑛𝑤)≤𝜙(𝑝,𝑤), for all 𝑤∈𝐸,  𝑝∈𝐹(𝑇^𝑀1_𝜆𝑛),  𝑛≥0. Now, by Lemma 4.2 and the Lipschitz continuity of 𝐻, we have 𝑇^𝑀1_𝜆𝑛𝑝−𝑇^𝑀1_𝜆𝑛𝑤=𝐻+𝜆_𝑛𝑀₁⁻¹𝐻𝑝−𝐻+𝜆_𝑛𝑀₁⁻¹≤1𝐻𝑤𝜆_𝑛𝑟₁≤𝛽𝐻𝑝−𝐻𝑤𝜆_𝑛𝑟₁𝑝−𝑤. (4.3) By (4.3) and Lemma 4.1, 𝜙𝑝,𝑇^𝑀1_𝜆𝑛𝑤=𝜙𝑇^𝑀1_𝜆𝑛𝑝,𝑇^𝑀1_𝜆𝑛𝑤≤4𝑑²𝜌_𝐸4𝑇^𝑀1_𝜆𝑛𝑝−𝑇^𝑀1_𝜆𝑛𝑤𝑑≤64𝑐𝑇^𝑀1_𝜆𝑛𝑝−𝑇^𝑀1_𝜆𝑛𝑤²≤64𝑐𝛽²𝜆²_𝑛𝑟²₁𝑝−𝑤²,𝜙𝑝,𝑤≥8𝑑²𝛿_𝐸𝑝−𝑤≥14𝑑2𝑘𝑝−𝑤². (4.4) Since 64𝑐𝛽²≤(1/2)𝑘𝜆²_𝑛𝑟²₁, it follows from (4.4) that 𝜙(𝑝,𝑇^𝑀1_𝜆𝑛𝑤)≤𝜙(𝑝,𝑤), for all 𝑤∈𝐸, 𝑝∈𝐹(𝑇^𝑀1_𝜆𝑛), 𝑛≥0. Thirdly, we show that 𝐹({𝑇^𝑀1_𝜆𝑛})=∩^∞_𝑛=0𝐹(𝑇^𝑀1_𝜆𝑛)=𝑀₁⁻¹0. We first show that 𝐹({𝑇^𝑀1_𝜆𝑛})⊂𝑀₁⁻¹0. Let 𝑝∈𝐹({𝑇^𝑀1_𝜆𝑛}), then there exists {𝑥_𝑛}⊂𝐸 such that 𝑥_𝑛→𝑝 and lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑀1_𝜆𝑛𝑥_𝑛‖=0. Since 𝐻 is 𝛽-Lipschitz continuous, 𝐻𝑥_𝑛−𝐻𝑇^𝑀1_𝜆𝑛𝑥_𝑛≤𝛽𝑥_𝑛−𝑇^𝑀1_𝜆𝑛𝑥_𝑛. (4.5) Letting 𝑛→∞, we obtain 1𝜆_𝑛𝐻𝑥_𝑛−𝐻𝑇^𝑀1_𝜆𝑛𝑥_𝑛⟶0. (4.6) It follows from (1/𝜆_𝑛)(𝐻𝑥_𝑛−𝐻𝑇^𝑀1_𝜆𝑛𝑥_𝑛)∈𝑀₁𝑇^𝑀1_𝜆𝑛 and the monotonicity of 𝑀₁ that 𝑥−𝑇^𝑀1_𝜆𝑛,𝑥^∗−1𝜆_𝑛𝐻𝑥_𝑛−𝐻𝑇^𝑀1_𝜆𝑛𝑥_𝑛≥0 (4.7) for all ∈𝐷(𝑀₁) and 𝑥^∗∈𝑀₁𝑥.Taking the limit as 𝑛→∞, we obtain 𝑥−𝑝,𝑥^∗≥0 (4.8) for all ∈𝐷(𝑀₁) and 𝑥^∗∈𝑀₁𝑥. By the maximality of 𝑀₁, we know that 𝑝∈𝑀₁⁻¹0. On the other hand, we know that 𝐹(𝑇^𝑀1_𝜆𝑛)=𝑀₁⁻¹0, 𝐹(𝑇^𝑀1_𝜆𝑛)⊂𝐹(𝑇^𝑀1_𝜆𝑛) for all 𝑛≥0, therefore, 𝑀₁⁻¹0=∩^∞_𝑛=0𝐹(𝑇^𝑀1_𝜆𝑛)=𝐹(∩^∞_𝑛=0𝑇^𝑀1_𝜆𝑛). Thus, we have proved that {𝑇^𝑀1_𝜆𝑛} is a countable family of weak relatively nonexpansive mappings with common fixed points sets ∩^∞_𝑛=0𝐹(𝑇^𝑀1_𝜆𝑛)=𝑀₁⁻¹0. By following the same arguments, we can show that {𝑇^𝑀2_𝜆𝑛} is a countable family of weak relatively nonexpansive mappings with common fixed points sets ∩^∞_𝑛=0𝐹(𝑇^𝑀2_𝜆𝑛)=𝑀₂⁻¹0.

Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐻=𝐽 and 𝑀 a maximal monotone mapping. Then, we can define 𝐽_𝜆=(𝐽+𝜆𝑀)⁻¹𝐽 for all 𝜆>0. We know that 𝐽_𝜆 is relatively nonexpansive and therefore weak relatively nonexpansive and 𝑀⁻¹0=𝐹(𝐽_𝜆) for all 𝜆>0 (see, e.g., [2]), where 𝐹(𝐽_𝜆) denotes the fixed points set of 𝐽_𝜆. By Corollary 3.3, we obtain the following theorem.

Theorem 4.4. Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth. For each 𝑘=1,2,…,𝑚, let 𝐹_𝑘 be a bifunction from 𝐸×𝐸 satisfying (A1)–(A4), 𝐴_𝑘∶𝐸→𝐸^∗ a continuous and monotone mapping, and 𝜑_𝑘∶𝐸→ℝ a lower semicontinuous and convex functional. For each 𝑖=1,2, let 𝑀_𝑖⊂𝐸×𝐸^∗ a maximal monotone operator, and let 𝐽^𝑀𝑖_𝜆=(𝐽+𝜆𝑀_𝑖)⁻¹J for all 𝜆>0, and suppose 𝐶 is a nonempty closed and convex subset of 𝐸 such that 𝐷(𝑀_𝑖)⊂𝐶⊂𝐽⁻¹(∩_𝜆>0𝑅(𝐽+𝜆𝑀_𝑖)),  𝑖=1,2. Assume that Ω∶=∩^𝑚_𝑘=1GMEP(𝐹_𝑘,𝐴_𝑘,𝜑_𝑘)∩𝑀₁⁻¹0∩𝑀₂⁻¹0≠∅ and let {𝑥_𝑛}^∞_𝑛=0 be iteratively generated by 𝑥₀∈𝐸,  𝐶₀=𝐸, 𝑧_𝑛=𝐽⁻¹𝛽_𝑛⁽¹⁾𝐽𝑥_𝑛+𝛽_𝑛⁽²⁾𝐽𝐽𝑇^𝑀1_𝜆𝑛𝑥_𝑛+𝛽_𝑛⁽³⁾𝐽𝐽𝑇^𝑀2_𝜆𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑢_𝑛=𝑇^𝑄𝑚_{𝑟𝑚,𝑛}𝑇^𝑄𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝑄2_𝑟2,𝑛𝑇^𝑄1_𝑟1,𝑛𝑦_𝑛,𝐶_𝑛+1=𝑤∈𝐶_𝑛∶𝜙𝑤,𝑦_𝑛≤𝜙𝑤,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥0, (4.9) with the conditions(i)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽²⁾>0;(ii)liminf_𝑛→∞𝛽_𝑛⁽¹⁾𝛽_𝑛⁽³⁾>0;(iii)0≤𝛼_𝑛≤𝛼<1; (iv)liminf_𝑛→∞𝜆_𝑛>0; (v){𝑟_𝑘,𝑛}^∞_𝑛=1⊂(0,∞), (𝑘=1,2,…,𝑚) satisfying liminf_𝑛→∞𝑟_𝑘,𝑛>0, (𝑘=1,2,…,𝑚). Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀.

Acknowledgments

The author would like to express his thanks to the referees for their valuable suggestions. This research work is dedicated to Professor Isaac U. Asuzu of the University of Nigeria, Nsukka.