A New Hybrid Iterative Scheme for Countable Families of Relatively Quasi-Nonexpansive Mappings and System of 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 closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized 𝑓-projection operator. Using this result, we discuss strong convergence theorem concerning variational inequality and convex minimization 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 𝐶 a 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) It is obvious from (1.3) that‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,∀𝑥,𝑦∈𝐸. (1.4)

Definition 1.1. 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., [4–9]) if the following conditions are satisfied:(R1)𝐹(𝑇)≠∅,(R2)𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),forall𝑥∈𝐶,𝑝∈𝐹(𝑇),(R3)𝐹(𝑇)=𝐹(𝑇).

If 𝑇 satisfies (R1) and (R2), then 𝑇 is said to be relatively quasi-nonexpansive. It is easy to see that the class of relatively quasi-nonexpansive mappings contains the class of relatively nonexpansive mappings. Many authors have studied the methods of approximating the fixed points of relatively quasi-nonexpansive mappings (see, e.g., [10–12] and the references cited therein). Clearly, in Hilbert space 𝐻, relatively quasi-nonexpansive mappings and quasi-nonexpansive mappings are the same, for 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖²,forall𝑥,𝑦∈𝐻, and this implies that 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥⟺𝑇𝑥−𝑝≤𝑥−𝑝,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (1.5) The examples of relatively quasi-nonexpansive mappings are given in [11].

Let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ. The equilibrium problem (see, e.g., [13–25]) is to find 𝑥^∗∈𝐶 such that𝐹𝑥^∗,𝑦≥0, (1.6) for all 𝑦∈𝐶. We will denote the solutions set of (1.6) by EP(𝐹). Numerous problems in physics, optimization, and economics reduce to find a solution of problem (1.6). The equilibrium problems include fixed point problems, optimization problems, and variational inequality problems as special cases (see, e.g., [26]).

In [7], 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≥0_𝑛+1=Π_{𝐻𝑛∩𝑊𝑛}𝑥₀,𝑛≥0. (1.7) They proved that {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_𝐹(𝑇)𝑥₀, where 𝐹(𝑇)≠∅.

In [27], 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, (1.8) 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 (1.8) converges strongly to a common fixed point of 𝑇 and 𝑆.

In [9], 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, (1.9) where 𝐽 is the duality mapping on 𝐸. Then, they proved that {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_𝐹𝑥₀, where 𝐹=EP(𝐹)∩𝐹(𝑇)≠∅.

Furthermore, in [28], Qin et al. introduced the following hybrid iterative algorithm for approximation of common fixed point of two countable families of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniform smooth real Banach space: 𝑧_𝑖,𝑛=𝐽⁻¹𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛,𝑦_𝑖,𝑛=𝐽⁻¹𝛼_𝑛,𝑖𝐽𝑥₀+1−𝛼_𝑛,𝑖𝐽𝑧_𝑖,𝑛,𝐶_𝑛,𝑖=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑖,𝑛≤𝜙𝑧,𝑥_𝑛+𝛼_𝑛,𝑖𝑥₀²+2𝑧,𝐽𝑥_𝑛−𝐽𝑥₀,𝐶_𝑛=_𝑖∈𝐼𝐶_𝑛,𝑖,𝑄₀𝑄=𝐶,_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥0. (1.10) They proved that the sequence {𝑥_𝑛} converges strongly to a common fixed point of the countable families {𝑇_𝑖} and {𝑆_𝑖} of closed relatively quasi-nonexpansive mappings in a uniformly convex and uniformly smooth Banach space under some appropriate conditions on {𝛽⁽¹⁾_𝑛,𝑖}, {𝛽⁽²⁾_𝑛,𝑖}, {𝛽⁽³⁾_𝑛,𝑖}, and {𝛼_𝑛,𝑖}.

Recently, Li et al. [29] 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, 𝐶₀=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≥0. (1.11) 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. [29] extended and improved on the results of Matsushita and Takahashi [7].

Quite recently, motivated by the results of Takahashi and Zembayashi [9], Cholamjiak and Suantai [30] proved the following strong convergence theorem by hybrid iterative scheme for approximation of common fixed point of a countable family of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in uniformly convex and uniformly smooth Banach space.

Theorem 1.2. Let 𝐸 be a uniformly convex real Banach space which is also uniformly smooth, and 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). Suppose {𝑇_𝑖}^∞_𝑖=1 is an infinitely countable family of closed and relatively quasi-nonexpansive mappings of 𝐶 into itself such that Ω∶=⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩(⋂^∞_𝑖=1𝐹(𝑇_𝑖))≠∅. Suppose {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,𝐶₀=𝐶, 𝑦_𝑖,𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇_𝑖𝑥_𝑛,𝑢_𝑖,𝑛=𝑇^𝐹𝑚_{𝑟𝑚,𝑛}𝑇^𝐹𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝐹2_𝑟2,𝑛𝑇^𝐹1_𝑟1,𝑛𝑦_𝑖,𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶sup_𝑖≥1𝜙𝑧,𝑢_𝑖,𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥0. (1.12) Assume that {𝛼_𝑛}^∞_𝑛=1 and {𝑟_𝑘,𝑛}^∞_𝑛=1(𝑘=1,2,…,𝑚) are sequences which satisfy the following conditions:(i)limsup_𝑛→∞𝛼_𝑛<1,(ii)liminf_𝑛→∞𝑟_𝑘,𝑛>0(k=1,2,…,𝑚). Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀.

Motivated by the above-mentioned results and the on-going research, it is our purpose in this paper to prove a strong convergence theorem for two countable families of closed relatively quasi-nonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized 𝑓-projection operator. Our results extend the results of Matsushita and Takahashi [7], Takahashi and Zembayashi [9], Qin et al. [28], Cholamjiak and Suantai [30], Li et al. [29], 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 dim 𝐸≥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,forall𝜆∈(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 [31], the generalized projection Π_𝐶 from 𝐸 onto 𝐶 is defined byΠ_𝐶(𝑥)∶=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, 31–34]). 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,𝜑=𝜉²−2𝜉,𝜑+𝜑²+2𝜌𝑓(𝜉), (2.5) where 𝜉∈𝐶, 𝜑∈𝐸^∗, 𝜌 is a positive number, and 𝑓∶𝐶→ℝ∪{+∞} is proper, convex, and lower semicontinuous. From the definitions of 𝐺 and 𝑓, it is easy to see the following properties:(i)𝐺(𝜉,𝜑) is convex and continuous with respect to 𝜑 when 𝜉 is fixed,(ii)𝐺(𝜉,𝜑) is convex and lower semicontinuous with respect to 𝜉 when 𝜑 is fixed.

Definition 2.1 (see Wu and Huang [35]). Let 𝐸 be a real Banach space with its dual 𝐸^∗. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. We say that Π^𝑓_𝐶∶𝐸^∗→2^𝐶 is a generalized 𝑓-projection operator if Π^𝑓_𝐶𝜑=𝑢∈𝐶∶𝐺𝑢,𝜑=inf_𝜉∈𝐶𝐺𝜉,𝜑,∀𝜑∈𝐸^∗. (2.6)

For the generalized 𝑓-projection operator, Wu and Huang [35] proved the following theorem basic properties.

Lemma 2.2 (see Wu and Huang [35]). 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 𝜑∈𝐸^∗,(ii)if 𝐸 is smooth, then, for all 𝜑∈𝐸^∗,𝑥∈Π^𝑓_𝐶 if and only if 𝑥−𝑦,𝜑−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. [36] showed that the condition 𝑓 is positive homogeneous which appeared in Lemma 2.2 can be removed.

Lemma 2.3 (see Fan et al. [36]). 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 𝜑∈𝐸^∗ such that 𝜑=𝐽𝑥 for each 𝑥∈𝐸. This substitution in (2.5) 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. We say 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∶𝐶→𝐶 is a relatively quasi-nonexpansive mapping and is equivalent to(𝑅^′1)𝐹(𝑇)≠∅, (𝑅^′2)𝐺(𝑝,𝐽𝑇𝑥)≤𝐺(𝑝,𝐽𝑥),forall𝑥∈𝐶,𝑝∈𝐹(𝑇).

Lemma 2.5 (see Li et al. [29]). Let 𝐸 be a Banach space, and let 𝑓∶𝐸→ℝ∪{+∞} be a lower semicontinuous 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 (see Li et al. [29]). 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 (see Li et al. [29]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and reflexive Banach space 𝐸. Let 𝑥∈𝐸 and 𝑥∈Π^𝑓_𝐶𝑥. Then, 𝜙y,𝑥+𝐺(𝑥,𝐽𝑥)≤𝐺𝑦,𝐽𝑥,∀𝑦∈𝐶. (2.12)

The fixed points set 𝐹(𝑇) of a relatively quasi-nonexpansive mapping is closed and convex as given in the following lemma.

Lemma 2.8 (see Chang et al. [37]). Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸 which also has Kadec-Klee property. Let 𝑇 be a closed relatively quasi-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 (see Cho et al. [38]). 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∈𝐶,lim_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 2.10 (see Blum and Oettli [26]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸, and let 𝐹 be a bifunction of 𝐶×𝐶 into ℝ satisfying (A1)–(A4). Let 𝑟>0 and 𝑥∈𝐸. Then, there exists 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐾. (2.15)

Lemma 2.11 (see Takahashi and Zembayashi [39]). Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Assume that 𝐹∶𝐶×𝐶→ℝ satisfies (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇^𝐹_𝑟∶𝐸→𝐶 as follows: 𝑇^𝐹_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶 (2.16) for all 𝑧∈𝐸. Then, the following hold: (1)𝑇^𝐹_𝑟 is singlevalued,(2)𝑇^𝐹_𝑟 is firmly nonexpansive-type mapping, that is, for any 𝑥,𝑦∈𝐸, 𝑇^𝐹_𝑟𝑥−𝑇^𝐹_𝑟𝑦,𝐽𝑇^𝐹_𝑟𝑥−𝐽𝑇^𝐹_𝑟𝑦≤𝑇^𝐹_𝑟𝑥−𝑇^𝐹_𝑟𝑦,𝐽𝑥−𝐽𝑦, (2.17)(3)𝐹(𝑇^𝐹_𝑟)=EP(𝐹),(4)EP(𝐹) is closed and convex.

Lemma 2.12 (see Takahashi and Zembayashi [39]). Let 𝐶 be a nonempty closed 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^𝐹_𝑟), 𝜙𝑞,𝑇^𝐹_𝑟𝑥+𝜙𝑇^𝐹_𝑟𝑥,𝑥≤𝜙𝑞,𝑥. (2.18)

For the rest of this paper, the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝 and will be denoted by 𝑥_𝑛→𝑝 as 𝑛→∞, {𝑥_𝑛}^∞_𝑛=0 converges weakly to 𝑝 and will be denoted by 𝑥_𝑛⇀𝑝 and we will assume that 𝛽⁽¹⁾_𝑛,𝑖,𝛽⁽²⁾_𝑛,𝑖,𝛽⁽³⁾_𝑛,𝑖∈[0,1],forall𝑖=1,2,3,… such that 𝛽⁽¹⁾_𝑛,𝑖+𝛽⁽²⁾_𝑛,𝑖+𝛽⁽³⁾_𝑛,𝑖=1,forall𝑛≥0.

We recall that a Banach space 𝐸 has Kadec-Klee property if, for any sequence {𝑥_𝑛}^∞_𝑛=0⊂𝐸 and 𝑥∈𝐸 with 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖, 𝑥_𝑛→𝑥 as 𝑛→∞. We note that every uniformly convex Banach space has the Kadec-Klee property. For more details on Kadec-Klee property, the reader is referred to [2, 33].

Lemma 2.13 (see Li et al. [29]). Let 𝐸 be a Banach space and 𝑦∈𝐸. Let 𝑓∶𝐸→ℝ∪{+∞} be a proper, convex, and lower semicontinuous mapping with convex domain 𝐷(𝑓). If {𝑥_𝑛} is a sequence in 𝐷(𝑓) such that 𝑥_𝑛⇀𝑥∈int(𝐷(𝑓)) and lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑦)=𝐺(𝑥,𝐽𝑦), then lim_𝑛→∞‖𝑥_𝑛‖=‖𝑥‖.

3. Main Results

Theorem 3.1. Let 𝐸 be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. 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). Suppose {𝑇_𝑖}^∞_𝑖=1 and {𝑆_𝑖}^∞_𝑖=1 are two countable families of closed relatively quasi-nonexpansive mappings of 𝐶 into itself such that Ω∶=⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩(⋂^∞_𝑖=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖))≠∅. Let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)), and suppose {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=∩^∞_𝑖=1𝐶_1,𝑖,𝑥₁=Π^𝑓_𝐶1𝑥₀, 𝑧_𝑛,𝑖=𝐽⁻¹𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛,𝑦_𝑛,𝑖=𝐽⁻¹𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖,𝑢_𝑛,𝑖=𝑇^𝐹𝑚_{𝑟𝑚,𝑛}𝑇^𝐹𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝐹2_𝑟2,𝑛𝑇^𝐹1_𝑟1,𝑛𝑦_𝑛,𝑖,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝐺𝑧,𝐽𝑢_𝑛,𝑖≤𝐺𝑧,𝐽𝑥_𝑛,𝐶_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1, (3.1) with the conditions(i)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖>0,(ii)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽³⁾_𝑛,𝑖>0,(iii)0≤𝛼_𝑛,𝑖≤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. We first show that 𝐶_𝑛,forall𝑛≥1 is closed and convex. It is obvious that 𝐶_1,𝑖=𝐶 is closed and convex. Suppose 𝐶_𝑘,𝑖 is closed and convex for some 𝑘>1. For each 𝑧∈𝐶_𝑘,𝑖, we see that 𝐺(𝑧,𝐽𝑢_𝑘,𝑖)≤𝐺(𝑧,𝐽𝑥_𝑘) is equivalent to 2𝑧,𝐽𝑥_𝑘−𝑧,𝐽𝑢_𝑘,𝑖≤𝑥_𝑘²−𝑢_𝑘,𝑖². (3.2) By the construction of the set 𝐶_𝑘+1,𝑖, we see that 𝐶_𝑘+1,𝑖 is closed and convex. Therefore, 𝐶_𝑘+1=⋂^∞_𝑖=1𝐶_𝑘+1,𝑖 is also closed and convex. Hence, 𝐶_𝑛,forall𝑛≥1 is closed and convex.
By taking 𝜃^𝑘_𝑛=𝑇^𝐹𝑘_{𝑟𝑘,𝑛}𝑇^𝐹𝑘−1_{𝑟𝑘−1,𝑛}⋯𝑇^𝐹2_𝑟2,𝑛𝑇^𝐹1_𝑟1,𝑛,𝑘=1,2,…,𝑚 and 𝜃⁰_𝑛=𝐼 for all 𝑛≥1, we obtain 𝑢_𝑛,𝑖=𝜃^𝑚_𝑛𝑦_𝑛,𝑖.
We next show that Ω⊂𝐶_𝑛,forall𝑛≥1. For 𝑛=1, we have Ω⊂𝐶=𝐶₁. Then, for each 𝑥^∗∈Ω, we obtain 𝐺𝑥^∗,𝐽𝑢_𝑛,𝑖=𝐺𝑥^∗,𝐽𝜃^𝑚_𝑛𝑦_𝑛,𝑖≤𝐺𝑥^∗,𝐽𝑦_𝑛,𝑖=𝐺𝑥^∗,𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖=𝑥^∗2−2𝛼_𝑛,𝑖𝑥^∗,𝐽𝑥_𝑛−21−𝛼_𝑛,𝑖𝑥^∗,𝐽𝑧_𝑛,𝑖+𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖²+2𝜌𝑓𝑥^∗≤𝑥^∗2−2𝛼_𝑛,𝑖𝑥^∗,𝐽𝑥_𝑛−21−𝛼_𝑛,𝑖𝑥^∗,𝐽𝑧_𝑛,𝑖+𝛼_𝑛,𝑖𝑥_𝑛²+1−𝛼_𝑛,𝑖𝑧_𝑛,𝑖²+2𝜌𝑓𝑥^∗=𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑧_𝑛,𝑖=𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐺𝑥^∗,𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛≤𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝑥^∗2−2𝛽⁽¹⁾_𝑛,𝑖𝑥^∗,𝐽𝑥_𝑛−2𝛽⁽²⁾_𝑛,𝑖𝑥^∗,𝐽𝑇_𝑖𝑥_𝑛−2𝛽⁽³⁾_𝑛,𝑖𝑥^∗,𝐽𝑆_𝑖𝑥_𝑛+𝛽⁽¹⁾_𝑛,𝑖𝑥_𝑛²+𝛽⁽²⁾_𝑛,𝑖𝑇_𝑖𝑥_𝑛²+𝛽⁽³⁾_𝑛,𝑖𝑆_𝑖𝑥_𝑛²+2𝜌𝑓𝑥^∗=𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽x_𝑛+1−𝛼_𝑛,𝑖𝛽⁽¹⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑆_𝑖𝑥_𝑛≤𝐺𝑥^∗,𝐽𝑥_𝑛. (3.3) So, 𝑥^∗∈𝐶_𝑛. This implies that Ω⊂𝐶_𝑛,forall𝑛≥1. Therefore, {𝑥_𝑛} is well defined.
We now show that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀) exists. Since 𝑓∶𝐸→ℝ is convex and lower semicontinuous, 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_𝑛2−2𝑥_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^∗∈𝐹. 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_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀∈𝐶_𝑛. It then follows from Lemma 2.7 that 𝜙𝑥_𝑛+1,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑥_𝑛+1,𝐽𝑥₀. (3.7) It is obvious that 𝜙𝑥_𝑛+1,𝑥_𝑛≥𝑥_𝑛+1−𝑥_𝑛²≥0, (3.8) and so {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 is nondecreasing. It follows that the limit of {𝐺(𝑥_𝑛,𝐽𝑥₀)}^∞_𝑛=0 exists.
Now since {𝑥_𝑛}^∞_𝑛=0 is bounded in 𝐶 and 𝐸 is reflexive, we may assume that 𝑥_𝑛⇀𝑝, and since 𝐶_𝑛 is closed and convex for each 𝑛≥1, it is easy to see that 𝑝∈𝐶_𝑛 for each 𝑛≥1. Again since 𝑥_𝑛=Π^𝑓_𝐶𝑛𝑥₀, from the definition of Π^𝑓_𝐶𝑛, we obtain 𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑝,𝐽𝑥₀,∀𝑛≥1. (3.9) Since liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀=liminf_𝑛→∞𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑥_𝑛≥𝑝²−2𝑝,𝐽𝑥₀+𝑥₀²+2𝜌𝑓𝑝=𝐺𝑝,𝐽𝑥₀, (3.10) then we obtain 𝐺𝑝,𝐽𝑥₀≤liminf_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤limsup_𝑛→∞𝐺𝑥_𝑛,𝐽𝑥₀≤𝐺𝑝,𝐽𝑥₀. (3.11) This implies that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥₀)=𝐺(𝑝,𝐽𝑥₀). By Lemma 2.13, we obtain lim_𝑛→∞‖𝑥_𝑛‖=‖𝑝‖. In view of Kadec-Klee property of 𝐸, we have that lim_𝑛→∞𝑥_𝑛=𝑝.
We next show that 𝑝∈⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩(⋂^∞_𝑖=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖)). We first show that 𝑝∈⋂^∞_𝑖=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖). By the fact that 𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀∈𝐶_𝑛, we obtain 𝜙𝑥_𝑛+1,𝑢_𝑛,𝑖≤𝜙𝑥_𝑛+1,𝑥_𝑛. (3.12) Now, (3.7) implies that 𝜙𝑥_𝑛+1,𝑢_𝑛,𝑖≤𝜙𝑥_𝑛+1,𝑥_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥₀−𝐺𝑥_𝑛,𝐽𝑥₀. (3.13) Taking the limit as 𝑛→∞ in (3.13), we obtain lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0. (3.14) Therefore, lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛,𝑖=0,∀𝑖≥1. (3.15) It then yields that lim_𝑛→∞(‖𝑥_𝑛+1‖−‖𝑢_𝑛,𝑖‖)=0,forall𝑖≥1. Since lim_𝑛→∞‖𝑥_𝑛+1‖=‖𝑝‖, we have lim_𝑛→∞𝑢_𝑛,𝑖=𝑝,∀𝑖≥1. (3.16) Hence, lim_𝑛→∞𝐽𝑢_𝑛,𝑖=𝐽𝑝,∀𝑖≥1. (3.17) This implies that {‖𝐽𝑢_𝑛,𝑖‖}^∞_𝑛=0,𝑖≥1 is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, we can then assume that 𝐽𝑢_𝑛,𝑖⇀𝑓₀∈𝐸^∗,forall𝑖≥1. In view of reflexivity of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence, there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑓₀. Since 𝜙𝑥_𝑛+1,𝑢_𝑛,𝑖=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑢_𝑛,𝑖+𝑢_𝑛,𝑖²=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑢_𝑛,𝑖+𝐽𝑢_𝑛,𝑖², (3.18) taking the limit inferior of both sides of (3.18) and in view of weak lower semicontinuity of ‖⋅‖, we have 0≥𝑝²−2𝑝,𝑓₀+𝑓₀²=𝑝²−2𝑝,𝐽𝑥+‖𝐽𝑥‖²=𝑝²−2𝑝,𝐽𝑥+‖𝑥‖²=𝜙𝑝,𝑥, (3.19) that is, 𝑝=𝑥. This implies that 𝑓₀=𝐽𝑝 and so 𝐽𝑢_𝑛,𝑖⇀𝐽𝑝,forall𝑖≥1. It follows from lim_𝑛→∞‖𝐽𝑢_𝑛,𝑖‖=‖𝐽𝑝‖,forall𝑖≥1 and Kadec-Klee property of 𝐸^∗ that 𝐽𝑢_𝑛,𝑖→𝐽𝑝,forall𝑖≥1. Note that 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous; it yields that 𝑢_𝑛,𝑖⇀𝑝,forall𝑖≥1. It then follows from lim_𝑛→∞‖𝑢_𝑛,𝑖‖=‖𝑝‖,forall𝑖≥1 and Kadec-Klee property of 𝐸 that lim_𝑛→∞𝑢_𝑛,𝑖=𝑝,forall𝑖≥1. Hence, lim_𝑛→∞𝑥_𝑛−𝑢_𝑛,𝑖=0,∀𝑖≥1. (3.20) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets and lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛,𝑖‖=0,forall𝑖≥1, we obtain lim_𝑛→∞𝐽𝑥_𝑛−𝐽𝑢_𝑛,𝑖=0,∀𝑖≥1. (3.21) Since {𝑥_𝑛} is bounded, so are {𝑧_𝑛,𝑖}, {𝐽𝑇_𝑖𝑥_𝑛}, and {𝐽𝑆_𝑖𝑥_𝑛}. Also, since 𝐸 is uniformly smooth, 𝐸^∗ is uniformly convex. Then, from Lemma 2.9, we have 𝐺𝑥^∗,𝐽𝑢_𝑛,𝑖=𝐺𝑥^∗,𝐽𝜃^𝑚_𝑛𝑦_𝑛,𝑖≤𝐺𝑥^∗,𝐽𝑦_𝑛,𝑖=𝐺𝑥^∗,𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖=𝑥^∗2−2𝛼_𝑛,𝑖𝑥^∗,𝐽𝑥_𝑛−21−𝛼_𝑛,𝑖𝑥^∗,𝐽𝑧_𝑛,𝑖+𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖²+2𝜌𝑓𝑥^∗≤𝑥^∗2−2𝛼_𝑛,𝑖𝑥^∗,𝐽𝑥_𝑛−21−𝛼_𝑛,𝑖𝑥^∗,𝐽𝑧_𝑛,𝑖+𝛼_𝑛,𝑖𝑥_𝑛²+1−𝛼_𝑛,𝑖𝑧_𝑛,𝑖²+2𝜌𝑓𝑥^∗=𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑧_𝑛,𝑖=𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐺𝑥^∗,𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛≤𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝑥^∗2−2𝛽⁽¹⁾_𝑛,𝑖𝑥^∗,𝐽𝑥_𝑛−2𝛽⁽²⁾_𝑛,𝑖𝑥^∗,𝐽𝑇_𝑖𝑥_𝑛−2𝛽⁽³⁾_𝑛,𝑖𝑥^∗,𝐽𝑆_𝑖𝑥_𝑛+𝛽⁽¹⁾_𝑛,𝑖𝑥_𝑛²+𝛽⁽²⁾_𝑛,𝑖𝑇_𝑖𝑥_𝑛²+𝛽⁽³⁾_𝑛,𝑖𝑆_𝑖𝑥_𝑛²−𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛+2𝜌𝑓𝑥^∗=𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝛽⁽¹⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑆_𝑖𝑥_𝑛−𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛≤𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝛽⁽¹⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛−𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛=𝛼_𝑛,𝑖𝐺𝑥^∗,𝑥_𝑛+1−𝛼_𝑛,𝑖𝐺𝑥^∗,𝐽𝑥_𝑛−𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛≤𝐺𝑥^∗,𝐽𝑥_𝑛−1−𝛼_𝑛,𝑖𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛. (3.22) It then follows that (1−𝛼)𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛≤1−𝛼_𝑛,𝑖𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛≤𝐺𝑥^∗,𝐽𝑥_𝑛−𝐺𝑥^∗,𝐽𝑢_𝑛,𝑖. (3.23) But 𝐺𝑥^∗,𝐽𝑥_𝑛−𝐺𝑥^∗,𝐽𝑢_𝑛,𝑖=𝑥_𝑛²−𝑢_𝑛,𝑖²−2𝑥^∗,𝐽𝑥_𝑛−𝐽𝑢_𝑛,𝑖≤𝑥_𝑛²−𝑢_𝑛,𝑖²+2𝑥^∗,𝐽𝑥_𝑛−𝐽𝑢_𝑛,𝑖≤𝑥_𝑛−𝑢_𝑛,𝑖𝑥_𝑛+𝑢_𝑛,𝑖+2𝑥^∗𝐽𝑥_𝑛−𝐽𝑢_𝑛,𝑖≤𝑥_𝑛−𝑢_𝑛,𝑖𝑥_𝑛+𝑢_𝑛,𝑖+2𝑥^∗𝐽𝑥_𝑛−𝐽𝑢_𝑛,𝑖. (3.24) From lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛,𝑖‖=0 and lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑢_𝑛,𝑖‖=0, we obtain 𝐺𝑥^∗,𝐽𝑥_𝑛−𝐺𝑥^∗,𝐽𝑢_𝑛,𝑖⟶0,𝑛⟶∞. (3.25) Using the condition liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖>0, we have lim_𝑛→∞𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛=0,∀𝑖≥1. (3.26) By property of 𝑔, we have lim_𝑛→∞‖𝐽𝑥_𝑛−𝐽𝑇_𝑖𝑥_𝑛‖=0,forall𝑖≥1. Since 𝐽⁻¹ is also uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞𝑥_𝑛−𝑇_𝑖𝑥_𝑛=0,∀𝑖≥1. (3.27) Similarly, we can show that lim_𝑛→∞𝑥_𝑛−𝑆_𝑖𝑥_𝑛=0,∀𝑖≥1. (3.28) Since 𝑥_𝑛→𝑝 and 𝑇_𝑖,𝑆_𝑖 are closed, we have 𝑝∈(⋂^∞_𝑖=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖)).
Next, we show that 𝑝∈⋂^𝑚_𝑘=1EP(𝐹_𝑘). Now, by Lemma 2.12, we obtain 𝜙𝑢_𝑛,𝑖,𝑦_𝑛,𝑖=𝜙𝜃^𝑚_𝑛𝑦_𝑛,𝑖,𝑦_𝑛,𝑖≤𝜙𝑥^∗,𝑦_𝑛,𝑖−𝜙𝑥^∗,𝜃^𝑚_𝑛𝑦_𝑛,𝑖≤𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥^∗,𝑢_𝑛,𝑖⟶0,𝑛⟶∞. (3.29) It then yields that lim_𝑛→∞(‖𝑢_𝑛,𝑖‖−‖𝑦_𝑛,𝑖‖)=0. Since lim_𝑛→∞‖𝑢_𝑛,𝑖‖=‖𝑝‖,𝑖≥1, we have lim_𝑛→∞𝑦_𝑛,𝑖=𝑝,𝑖≥1. (3.30) Hence, lim_𝑛→∞𝐽𝑦_𝑛,𝑖=𝐽𝑝,𝑖≥1. (3.31) This implies that {‖𝐽𝑦_𝑛,𝑖‖}^∞_𝑛=0 is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, we can then assume that 𝐽𝑦_𝑛,𝑖⇀𝑓₁∈𝐸^∗. In view of reflexivity of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence, there exists 𝑧∈𝐸 such that 𝐽𝑧=𝑓₁. Since 𝜙𝑢_𝑛,𝑖,𝑦_𝑛,𝑖=𝑢_𝑛,𝑖²−2𝑢_𝑛,𝑖,𝐽𝑦_𝑛,𝑖+𝑦_𝑛,𝑖²=𝑢_𝑛,𝑖²−2𝑢_𝑛,𝑖,𝐽𝑦_𝑛,𝑖+𝐽𝑦_𝑛,𝑖², (3.32) taking the limit inferior of both sides of (3.32) and in view of weak lower semicontinuity of ‖⋅‖, we have 0≥𝑝²−2𝑝,𝑓₁+𝑓₁²=𝑝²−2𝑝,𝐽𝑧+‖𝐽𝑧‖²=𝑝²−2𝑝,𝐽𝑧+‖𝑧‖²=𝜙𝑝,𝑧, (3.33) that is, 𝑝=𝑧. This implies that 𝑓₁=𝐽𝑝 and so 𝐽𝑦_𝑛,𝑖⇀𝐽𝑝. It follows from lim_𝑛→∞‖𝐽𝑦_𝑛,𝑖‖=‖𝐽𝑝‖ and Kadec-Klee property of 𝐸^∗ that 𝐽𝑦_𝑛,𝑖→𝐽𝑝. Note that 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous; it yields that 𝑦_𝑛,𝑖⇀𝑝. It then follows from lim_𝑛→∞‖𝑦_𝑛,𝑖‖=‖𝑝‖ and Kadec-Klee property of 𝐸 that lim_𝑛→∞𝑦_𝑛,𝑖=𝑝,𝑖≥1. 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,𝑖,𝑦_𝑛,𝑖≤𝜙𝑥^∗,𝑦_𝑛,𝑖−𝜙𝑥^∗,𝜃^𝑘_𝑛𝑦_𝑛,𝑖≤𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥^∗,𝜃^𝑘_𝑛𝑦_𝑛,𝑖. (3.34) Observe that 𝜙𝑥^∗,𝑢_𝑛,𝑖=𝜙𝑥^∗,𝜃^𝑚_𝑛𝑦_𝑛,𝑖=𝜙𝑥^∗,𝑇^𝐹𝑚_{𝑟𝑚,𝑛}𝑇^𝐹𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝐹𝑘_{𝑟𝑘,𝑛}𝑇^𝐹𝑘−1_{𝑟𝑘−1,𝑛}⋯𝑇^𝐹2_𝑟2,𝑛𝑇^𝐹1_𝑟1,𝑛𝑦_𝑛,𝑖=𝜙𝑥^∗,𝑇^𝐹𝑚_{𝑟𝑚,𝑛}𝑇^𝐹𝑚−1_{𝑟𝑚−1,𝑛}⋯𝜃^𝑘_𝑛𝑦_𝑛,𝑖≤𝜙𝑥^∗,𝜃^𝑘_𝑛𝑦_𝑛,𝑖. (3.35) Using (3.35) in (3.34), we obtain 𝜙𝜃^𝑘_𝑛𝑦_𝑛,𝑖,𝑦_𝑛,𝑖≤𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥^∗,𝑢_𝑛,𝑖⟶0,𝑛⟶∞. (3.36) It then yields that lim_𝑛→∞(‖𝜃^𝑘_𝑛𝑦_𝑛,𝑖‖−‖𝑦_𝑛,𝑖‖)=0. Since lim_𝑛→∞‖𝑦_𝑛,𝑖‖=‖𝑝‖, we have lim_𝑛→∞𝜃^𝑘_𝑛𝑦_𝑛,𝑖=𝑝,𝑘=1,2,…,𝑚. (3.37) This implies that {‖𝜃^𝑘_𝑛𝑦_𝑛,𝑖‖}^∞_𝑛=0 is bounded in 𝐸. Since 𝐸 is reflexive, we can then assume that 𝜃^𝑘_𝑛𝑦_𝑛,𝑖⇀𝑤∈𝐸. Since 𝜙𝜃^𝑘_𝑛𝑦_𝑛,𝑖,𝑦_𝑛,𝑖=𝜃^𝑘_𝑛𝑦_𝑛,𝑖²−2𝜃^𝑘_𝑛𝑦_𝑛,𝑖,𝐽𝑦_𝑛,𝑖+𝑦_𝑛,𝑖²=𝜃^𝑘_𝑛𝑦_𝑛,𝑖²−2𝜃^𝑘_𝑛𝑦_𝑛,𝑖,𝐽𝑦_𝑛,𝑖+𝐽𝑦_𝑛,𝑖², (3.38) taking the limit inferior of both sides of (3.38) and in view of weak lower semicontinuity of ‖⋅‖, we have 0≥‖𝑤‖²−2𝑤,𝐽𝑝+𝑝²=‖𝑤‖²−2𝑤,𝐽𝑝+𝐽𝑝²=𝜙𝑤,𝑝, (3.39) that is, 𝑝=𝑤. This implies that 𝜃^𝑘_𝑛𝑦_𝑛,𝑖⇀𝑝. It follows from lim_𝑛→∞‖𝜃^𝑘_𝑛𝑦_𝑛,𝑖‖=‖𝑝‖ and Kadec-Klee property of 𝐸 that 𝜃^𝑘_𝑛𝑦_𝑛,𝑖⟶𝑝,𝑛⟶∞,𝑘=1,2,…,𝑚. (3.40) Similarly, lim_𝑛→∞‖𝑝−𝜃_𝑛^𝑘−1𝑦_𝑛,𝑖‖=0,𝑘=1,2,…,𝑚. This further implies that lim_𝑛→∞𝜃^𝑘_𝑛𝑦_𝑛,𝑖−𝜃_𝑛^𝑘−1𝑦_𝑛,𝑖=0,𝑖≥1. (3.41) Also, since 𝐽 is uniformly norm-to-norm continuous on bounded sets and using (3.41), we obtain lim_𝑛→∞𝐽𝜃^𝑘_𝑛𝑦_𝑛,𝑖−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛,𝑖=0,𝑖≥1. (3.42) Since liminf_𝑛→∞𝑟_𝑘,𝑛>0(𝑘=1,2,…,𝑚), lim_𝑛→∞𝐽𝜃^𝑘_𝑛𝑦_𝑛,𝑖−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛,𝑖𝑟_𝑘,𝑛=0. (3.43) By Lemma 2.11, we have that for each 𝑘=1,2,…,𝑚𝐹_𝑘𝜃^𝑘_𝑛𝑦_𝑛,𝑖+1,𝑦𝑟_𝑘,𝑛𝑦−𝜃^𝑘_𝑛𝑦_𝑛,𝑖,𝐽𝜃^𝑘_𝑛𝑦_𝑛,𝑖−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛,𝑖≥0,∀𝑦∈𝐶. (3.44) Furthermore, using (A2), we obtain 1𝑟_𝑘,𝑛𝑦−𝜃^𝑘_𝑛𝑦_𝑛,𝑖,𝐽𝜃^𝑘_𝑛𝑦_𝑛,𝑖−𝐽𝜃_𝑛^𝑘−1𝑦_𝑛,𝑖≥𝐹_𝑘𝑦,𝜃^𝑘_𝑛𝑦_𝑛,𝑖. (3.45) By (A4), (3.43), and 𝜃^𝑘_𝑛𝑦_𝑛,𝑖→𝑝, we have for each 𝑘=1,2,…,𝑚𝐹_𝑘𝑦,𝑝≤0,∀𝑦∈𝐶. (3.46) For fixed 𝑦∈𝐶, let 𝑧_𝑡,𝑦∶=𝑡𝑦+(1−𝑡)𝑝 for all 𝑡∈(0,1]. This implies that 𝑧_𝑡,𝑦∈𝐶. This yields that 𝐹_𝑘(𝑧_𝑡,𝑦,𝑝)≤0. It follows from (A1) and (A4) that 0=𝐹_𝑘𝑧_𝑡,𝑦,𝑧_𝑡,𝑦≤𝑡𝐹_𝑘𝑧_𝑡,𝑦,𝑦+(1−𝑡)𝐹_𝑘𝑧_𝑡,𝑦,𝑝≤𝑡𝐹_𝑘𝑧_𝑡,𝑦,𝑦, (3.47) and hence 0≤𝐹_𝑘𝑧_𝑡,𝑦,𝑦. (3.48) From condition (A3), we obtain 𝐹_𝑘𝑝,𝑦≥0,∀𝑦∈𝐶. (3.49) This implies that 𝑝∈EP(𝐹_𝑘),𝑘=1,2,…,𝑚. Thus, 𝑝∈⋂^𝑚_𝑘=1EP(𝐹_𝑘). Hence, we have 𝑝∈Ω=⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩(⋂^∞_𝑛=0𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖)).
Finally, we show that 𝑝=Π^𝑓_Ω𝑥₀. Since Ω=⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩(⋂^∞_𝑛=0𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖)) is a closed and convex set, from Lemma 2.6, we know that Π^𝑓_𝐹𝑥₀ 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≥1. (3.50) We know that 𝐺(𝜉,𝐽𝜑) is convex and lower semicontinuous 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. (3.51) From the definition of Π^𝑓_Ω𝑥₀ and 𝑝∈Ω, we see that 𝑝=𝑤. This completes the proof.

Take 𝑓(𝑥)=0 for all 𝑥∈𝐸 in Theorem 3.1, then 𝐺(𝜉,𝐽𝑥)=𝜙(𝜉,𝑥) and Π^𝑓_𝐶𝑥₀=Π_𝐶𝑥₀. Then we obtain the following corollary.

Corollary 3.2. Let 𝐸 be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. 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). Suppose {𝑇_𝑖}^∞_𝑖=1 and {𝑆_𝑖}^∞_𝑖=1 are two countable families of closed relatively quasi-nonexpansive mappings of 𝐶 into itself such that Ω∶=⋂^𝑚_𝑘=1EP(𝐹_𝑘)∩(⋂^∞_𝑛=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑛=1𝐹(𝑆_𝑖))≠∅. Suppose {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=∩^∞_𝑖=1𝐶_1,𝑖,𝑥₁=Π_𝐶1𝑥₀, 𝑧_𝑛,𝑖=𝐽⁻¹𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛,𝑦_𝑛,𝑖=𝐽⁻¹𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖,𝑢_𝑛,𝑖=𝑇^𝐹𝑚_{𝑟𝑚,𝑛}𝑇^𝐹𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝐹2_𝑟2,𝑛𝑇^𝐹1_𝑟1,𝑛𝑦_𝑛,𝑖,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝜙𝑧,𝑢_𝑛,𝑖≤𝜙𝑧,𝑥_𝑛,𝐶_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥1, (3.52) with the conditions(i)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖>0,(ii)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽³⁾_𝑛,𝑖>0,(iii)0≤𝛼_𝑛,𝑖≤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.

Corollary 3.3 (see Li et al. [29]). 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 is a relatively nonexpansive mapping of 𝐶 into itself such that Ω∶=𝐹(𝑇)≠∅. Let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)), and suppose {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶, 𝐶₀=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≥0. (3.53) Suppose {𝛼_𝑛}^∞_𝑛=1 is a sequence in (0,1) such that limsup_𝑛→∞𝛼_𝑛<1. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀.

Corollary 3.4 (see Takahashi and Zembayashi [9]). 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. Let 𝐹 be a bifunction from 𝐶×𝐶 satisfying (A1)–(A4). Suppose 𝑇 is a relatively nonexpansive mapping of 𝐶 into itself such that Ω∶=EP(𝐹)∩𝐹(𝑇)≠∅. Let {𝑥_𝑛}^∞_𝑛=0 be iteratively generated by 𝑥₀∈𝐶, 𝐶₁=𝐶, 𝑥₁=Π_𝐶1𝑥₀, 𝑦_𝑛=𝐽⁻¹𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑇𝑥_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, (3.54) where 𝐽 is the duality mapping on 𝐸. Suppose {𝛼_𝑛,𝑖}^∞_𝑛=1 is a sequence in (0,1) such that liminf_𝑛→∞𝛼_𝑛,𝑖(1−𝛼_𝑛,𝑖)>0 and {𝑟_𝑛}^∞_𝑛=1⊂(0,∞) satisfying liminf_𝑛→∞𝑟_𝑛>0. Then, {𝑥_𝑛}^∞_𝑛=0 converges strongly to Π_Ω𝑥₀.

4. Applications

Let 𝐴 be a monotone operator from 𝐶 into 𝐸^∗, the classical variational inequality is to find 𝑥^∗∈𝐶 such that𝑦−𝑥,𝐴𝑥^∗≥0,∀𝑦∈𝐶. (4.1) The set of solutions of (4.1) is denoted by VI(𝐶,𝐴).

Let 𝜑∶𝐶→ℝ be a real-valued function. The convex minimization problem is to find 𝑥^∗∈𝐶 such that𝜑𝑥^∗≤𝜑𝑦,∀𝑦∈𝐶. (4.2) The set of solutions of (4.2) is denoted by CMP(𝜑).

The following lemmas are special cases of Lemmas  2.8 and Lemma  2.9 of [39].

Lemma 4.1. Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Assume that 𝐴∶𝐶→𝐸^∗ is a continuous and monotone operator. For 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇^𝐴_𝑟∶𝐸→𝐶 as follows: 𝑇^𝐴_𝑟(𝑥)=+1𝑧∈𝐶∶𝐴𝑧,𝑦−𝑧𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (4.3) Then, the following hold:(1)𝑇^𝐴_𝑟 is singlevalued,(2)𝐹(𝑇^𝐴_𝑟)=VI(𝐶,𝐴),(3)VI(𝐶,𝐴) is closed and convex,(4)𝜙(𝑞,𝑇^𝐴_𝑟𝑥)+𝜙(𝑇^𝐴_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥),forall𝑞∈𝐹(𝑇^𝐴_𝑟).

Lemma 4.2. Let 𝐶 be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Assume that 𝜑∶𝐶→ℝ is lower semicontinuous and convex. For 𝑟>0 and 𝑥∈𝐸, define a mapping 𝑇^𝜑_𝑟∶𝐸→𝐶 as follows: 𝑇^𝜑_𝑟(x)=𝑦+1𝑧∈𝐶∶𝜑𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥𝜑(𝑧),∀𝑦∈𝐶. (4.4) Then, the following hold:(1)𝑇^𝜑_𝑟 is single valued,(2)𝐹(𝑇^𝜑_𝑟)=CMP(𝜑),(3)CMP(𝜑) is closed and convex,(4)𝜙(𝑞,𝑇^𝜑_𝑟𝑥)+𝜙(𝑇^𝜑_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥),forall𝑞∈𝐹(𝑇^𝜑_𝑟).

Then we obtain the following theorems from Theorem 3.1.

Theorem 4.3. Let 𝐸 be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. For each 𝑘=1,2,…,𝑚, let 𝐴_𝑘 be a continuous and monotone operator from 𝐶 into 𝐸^∗. Suppose {𝑇_𝑖}^∞_𝑖=1 and {𝑆_𝑖}^∞_𝑖=1 are two countable families of closed relatively quasi-nonexpansive mappings of 𝐶 into itself such that Ω∶=⋂^𝑚_𝑘=1VI(𝐶,𝐴_𝑘)∩(⋂^∞_𝑖=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖))≠∅. Let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)), and suppose {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=∩^∞_𝑖=1𝐶_1,𝑖,𝑥₁=Π^𝑓_𝐶1𝑥₀, 𝑧_𝑛,𝑖=𝐽⁻¹𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛,𝑦_𝑛,𝑖=𝐽⁻¹𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖,𝑢_𝑛,𝑖=𝑇^𝐴𝑚_{𝑟𝑚,𝑛}𝑇^𝐴𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝐴2_𝑟2,𝑛𝑇^𝐴1_𝑟1,𝑛𝑦_𝑛,𝑖,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝐺𝑧,𝐽𝑢_𝑛,𝑖≤𝐺𝑧,𝐽𝑥_𝑛,𝐶_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1, (4.5) with the conditions(i)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽²⁾_𝑛,𝑖>0,(ii)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽³⁾_𝑛,𝑖>0,(iii)0≤𝛼_𝑛,𝑖≤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.

Theorem 4.4. Let 𝐸 be a uniformly smooth and strictly convex real Banach space which also has Kadec-Klee property. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸. For each 𝑘=1,2,…,𝑚, let 𝜑_𝑘∶𝐶→ℝ be lower semicontinuous and convex. Suppose {𝑇_𝑖}^∞_𝑖=1 and {𝑆_𝑖}^∞_𝑖=1 are two countable families of closed relatively quasi-nonexpansive mappings of 𝐶 into itself such that Ω∶=⋂^𝑚_𝑘=1CMP(𝜑_𝑘)∩(⋂^∞_𝑖=1𝐹(𝑇_𝑖))∩(⋂^∞_𝑖=1𝐹(𝑆_𝑖))≠∅. Let 𝑓∶𝐸→ℝ be a convex and lower semicontinuous mapping with 𝐶⊂int(𝐷(𝑓)), and suppose {𝑥_𝑛}^∞_𝑛=0 is iteratively generated by 𝑥₀∈𝐶,𝐶_1,𝑖=𝐶,𝐶₁=⋂^∞_𝑖=1𝐶_1,𝑖,𝑥₁=Π^𝑓_𝐶1𝑥₀, 𝑧_𝑛,𝑖=𝐽⁻¹𝛽⁽¹⁾_𝑛,𝑖𝐽𝑥_𝑛+𝛽⁽²⁾_𝑛,𝑖𝐽𝑇_𝑖𝑥_𝑛+𝛽⁽³⁾_𝑛,𝑖𝐽𝑆_𝑖𝑥_𝑛,𝑦_n,𝑖=𝐽⁻¹𝛼_𝑛,𝑖𝐽𝑥_𝑛+1−𝛼_𝑛,𝑖𝐽𝑧_𝑛,𝑖,𝑢_𝑛,𝑖=𝑇^𝜑𝑚_{𝑟𝑚,𝑛}𝑇^𝜑𝑚−1_{𝑟𝑚−1,𝑛}⋯𝑇^𝜑2_𝑟2,𝑛𝑇^𝜑1_𝑟1,𝑛𝑦_𝑛,𝑖,𝐶_𝑛+1,𝑖=𝑧∈𝐶_𝑛,𝑖∶𝐺𝑧,𝐽𝑢_𝑛,𝑖≤𝐺𝑧,𝐽𝑥_𝑛,𝐶_𝑛+1=^∞_𝑖=1𝐶_𝑛+1,𝑖,𝑥_𝑛+1=Π^𝑓_𝐶𝑛+1𝑥₀,𝑛≥1, (4.6) with the conditions(i)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,i𝛽⁽²⁾_𝑛,𝑖>0,(ii)liminf_𝑛→∞𝛽⁽¹⁾_𝑛,𝑖𝛽⁽³⁾_𝑛,𝑖>0,(iii)0≤𝛼_𝑛,𝑖≤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.