A New Hybrid Algorithm for a Pair of Quasi-𝜙-Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces

Abstract

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi-𝜙-asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. We also assume that 𝐸 is a real Banach space, 𝐸^∗ is the dual space of 𝐸, 𝐶 is a nonempty closed convex subset of 𝐸, and ⟨⋅,⋅⟩ is the pairing between 𝐸 and 𝐸^∗.

Let 𝜓∶𝐶→ℝ be a real-valued function, Θ∶𝐶×𝐶→ℝ a bifunction, and 𝐴∶𝐶→𝐸^∗ a nonlinear mapping. The “so-called” generalized mixed equilibrium problem is to find 𝑢∈𝐶 such that Θ𝑢,𝑦+𝐴𝑢,𝑦−𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶. (1.1) The set of solutions for (1.1) is denoted by Ω, that is, Ω=𝑢∈𝐶∶Θ𝑢,𝑦+𝐴𝑢,𝑦−𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶. (1.2)

Special examples are as follows.(I)If 𝜓=0, the problem (1.1) is equivalent to finding 𝑢∈𝐶 such that Θ𝑢,𝑦+𝐴𝑢,𝑦−𝑢≥0,∀𝑦∈𝐶, (1.3) which is called the generalized equilibrium problem. The set of solutions for (1.3) is denoted by GEP.(II)If 𝐴=0, the problem (1.1) is equivalent to finding 𝑢∈𝐶 such that Θ𝑢,𝑦+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶, (1.4) which is called the mixed equilibrium problem (MEP) [1]. The set of solutions for (1.4) is denoted by MEP.(III)If Θ=0, the problem (1.1) is equivalent to finding 𝑢∈𝐶 such that 𝐴𝑢,𝑦−𝑢+𝜓𝑦−𝜓(𝑢)≥0,∀𝑦∈𝐶, (1.5) which is called the mixed variational inequality of Browder type (VI) [2]. The set of solutions for (1.5) is denoted by VI(𝐶,𝐴,𝜓).(IV)If 𝜓=0 and 𝐴=0, the problem (1.1) is equivalent to finding 𝑢∈𝐶 such that Θ𝑢,𝑦≥0,∀𝑦∈𝐶, (1.6) which is called the equilibrium problem. The set of solutions for (1.6) is denoted by EP(Θ).(V)If 𝜓=0 and Θ=0, the problem (1.1) is equivalent to finding 𝑢∈𝐶 such that 𝐴𝑢,𝑦−𝑢≥0,∀𝑦∈𝐶, (1.7) which is called the variational inequality of Browder type. The set of solutions for (1.7) is denoted by VI(𝐶,𝐴).

The problem (1.1) is very general in the sense that numerous problems in physics, optimiztion and economics reduce to finding a solution for (1.1). Some methods have been proposed for solving the generalized equilibrium problem and the equilibrium problem in Hilbert space (see, e.g., [3–6]).

A mapping 𝑆∶𝐶→𝐸 is called nonexpansive if 𝑆𝑥−𝑆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.8) We denote the fixed point set of 𝑆 by 𝐹(𝑆).

In 2008, S. Takahashi and W. Takahashi [6] proved some strong convergence theorems for finding an element or a common element of EP, EP(𝑓)∩𝐹(𝑆) or VI(𝐶,𝐴)∩𝐹(𝑆), respectively, in a Hilbert space.

Recently, Takahashi and Zembayashi [7, 8] proved some weak and strong convergence theorems for finding a common element of the set of solutions for equilibrium (1.6) and the set of fixed points of a relatively nonexpansive mapping in a Banach space.

In 2010, Chang et al. [9] proved a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem (1.3) and the set of common fixed points of a pair of relatively nonexpansive mappings in a Banach space.

Motivated and inspired by [4–9], we intend in this paper, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem (1.1) and the set of common fixed points of a pair of quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

2. Preliminaries

For the sake of convenience, we first recall some definitions and conclusions which will be needed in proving our main results.

The mapping 𝐽∶𝐸→2^𝐸∗ defined by 𝐽(𝑥)=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖=𝑥^∗,𝑥∈𝐸, (2.1) is called the normalized duality mapping. By the Hahn-Banach theorem, 𝐽(𝑥)≠∅ for each 𝑥∈𝐸.

In the sequel, we denote the strong convergence and weak convergence of a sequence {𝑥_𝑛} by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively.

A Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝑈={𝑧∈𝐸∶‖𝑧‖=1} with 𝑥≠𝑦. 𝐸 is said to be uniformly convex if, for each 𝜖∈(0,2], there exists 𝛿>0 such that ‖𝑥+𝑦‖/2<1−𝛿 for all 𝑥,𝑦∈𝑈 with ||𝑥−𝑦||≥𝜖. 𝐸 is said to be smooth if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2.2) exists for all 𝑥,𝑦∈𝑈. 𝐸 is said to be uniformly smooth if the above limit exists uniformly in 𝑥,𝑦∈𝑈.

Remark 2.1. The following basic properties can be found in Cioranescu [10].(i)If 𝐸 is a uniformly smooth Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.(ii)If 𝐸 is a reflexive and strictly convex Banach space, then 𝐽⁻¹ is hemicontinuous.(iii)If 𝐸 is a smooth, strictly convex, and reflexive Banach space, then 𝐽 is singlevalued, one-to-one and onto.(iv)A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex.(v)Each uniformly convex Banach space 𝐸 has the Kadec-Klee property, that is, for any sequence {𝑥_𝑛}⊂𝐸, if 𝑥_𝑛⇀𝑥∈𝐸 and ||𝑥_𝑛||→||𝑥||, then 𝑥_𝑛→𝑥.

Next we assume that 𝐸 is a smooth, strictly convex, and reflexive Banach space and 𝐶 is a nonempty closed convex subset of 𝐸. In the sequel, we always use 𝜙∶𝐸×𝐸→ℝ⁺ to denote the Lyapunov functional defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (2.3)

It is obvious from the definition of 𝜙 that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,∀𝑥,𝑦∈𝐸. (2.4)

Following Alber [11], the generalized projection Π_𝐶∶𝐸→𝐶 is defined by Π_𝐶(𝑥)=arginf_𝑦∈𝐶𝜙𝑦,𝑥,∀𝑥∈𝐸. (2.5)

Lemma 2.2 (see [11, 12]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and 𝐶 a nonempty closed convex subset of 𝐸. Then, the following conclusions hold:(a)𝜙(𝑥,Π_𝐶𝑦)+𝜙(Π_𝐶𝑦,𝑦)≤𝜙(𝑥,𝑦) for all 𝑥∈𝐶 and 𝑦∈𝐸;(b)if 𝑥∈𝐸 and 𝑧∈𝐶, then 𝑧=Π_𝐶𝑥⟺𝑧−𝑦,𝐽𝑥−𝐽𝑧≥0,∀𝑦∈𝐶; (2.6)(c)for 𝑥,𝑦∈𝐸, 𝜙(𝑥,𝑦)=0 if and only 𝑥=𝑦.

Remark 2.3. If 𝐸 is a real Hilbert space 𝐻, then 𝜙(𝑥,𝑦)=||𝑥−𝑦||² and Π_𝐶 is the metric projection 𝑃_𝐶 of 𝐻 onto 𝐶.

Let 𝐸 be a smooth, strictly, convex and reflexive Banach space, 𝐶 a nonempty closed convex subset of 𝐸, 𝑇∶𝐶→𝐶 a mapping, and 𝐹(𝑇) the set of fixed points of 𝑇. A point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛⇀𝑝 and ||𝑥_𝑛−𝑇𝑥_𝑛||→0. We denoted the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).

Definition 2.4 (see [13]). (1) A mapping 𝑇∶𝐶→𝐶 is said to be relatively nonexpansive if 𝐹(𝑇)≠∅, 𝐹(𝑇)=𝐹(𝑇), and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (2.7)
(2) A mapping 𝑇∶𝐶→𝐶 is said to be closed if, for any sequence {𝑥_𝑛}⊂𝐶 with 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, 𝑇𝑥=𝑦.

Definition 2.5 (see [14]). (1) A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (2.8)
(2) A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-asymptotically nonexpansive if 𝐹(𝑇)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that 𝜙𝑝,𝑇^𝑛𝑥≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇). (2.9)
(3) A pair of mappings 𝑇₁,𝑇₂∶𝐶→𝐶 is said to be uniformly quasi-𝜙-asymptotically nonexpansive if 𝐹(𝑇₁)⋂𝐹(𝑇₂)≠∅ and there exists a real sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that for 𝑖=1,2𝜙𝑝,𝑇^𝑛_𝑖𝑥≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹𝑇₁∩𝐹𝑇₂. (2.10)
(4) A mapping 𝑇∶𝐶→𝐶 is said to be uniformly 𝐿-Lipschitz continuous if there exists a constant 𝐿>0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.11)

Remark 2.6. (1) From the definition, it is easy to know that each relatively nonexpansive mapping is closed.
(2) The class of quasi-𝜙-asymptotically nonexpansive mappings contains properly the class of quasi-𝜙-nonexpansive mappings as a subclass, and the class of quasi-𝜙-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

Lemma 2.7 (see [15]). Let 𝐸 be a uniformly convex Banach space, 𝑟>0 a positive number, and 𝐵_𝑟(0) a closed ball of 𝐸. Then, for any given subset {𝑥₁,𝑥₂,…,𝑥_𝑁}⊂𝐵_𝑟(0) and for any positive numbers {𝜆₁,𝜆₂,…,𝜆_𝑁} with ∑^𝑁_𝑖=1𝜆_𝑖=1, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that, for any 𝑖,𝑗∈{1,2,…,𝑁} with 𝑖<𝑗, ^𝑁_𝑛=1𝜆_𝑛𝑥_𝑛²≤^𝑁_𝑛=1𝜆_𝑛𝑥_𝑛²−𝜆_𝑖𝜆_𝑗𝑔𝑥_𝑖−𝑥_𝑗. (2.12)

Lemma 2.8 (see [15]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝐶 be a closed and quasi-𝜙-asymptotically nonexpansive mapping with a sequence {𝑘_𝑛}⊂[1,∞), 𝑘_𝑛→1. Then 𝐹(𝑇) is a closed convex subset of 𝐶.

For solving the generalized mixed equilibrium problem (1.1), let us assume that the function 𝜓∶𝐶→ℝ is convex and lower semicontinuous, the nonlinear mapping 𝐴∶𝐶→𝐸^∗ is continuous and monotone, and the bifunction Θ∶𝐶×𝐶→ℝ satisfies the following conditions:(A₁) Θ(𝑥,𝑥)=0, for all 𝑥∈𝐶,(A₂) Θ is monotone, that is, Θ(𝑥,𝑦)+Θ(𝑦,𝑥)≤0, ∀𝑥,𝑦∈𝐶,(A₃)limsup_𝑡↓0Θ(𝑥+𝑡(𝑧−𝑥),𝑦)≤Θ(𝑥,𝑦)∀𝑥,𝑧,𝑦∈𝐶,(A₄)the function 𝑦↦Θ(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 2.9. Let 𝐸 be a smooth, strictly convex, and reflexive Banach space and 𝐶 a nonempty closed convex subset of 𝐸. Let Θ∶𝐶×𝐶→ℝ a bifunction satisfying the conditions (A₁)–(A₄). Let 𝑟>0 and 𝑥∈𝐸. Then, the followings hold.(i)(Blum and Oettli [3]) there exists 𝑧∈𝐶 such that Θ+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (2.13)(ii)(Takahashi and Zembayashi [8]) Define a mapping 𝑇_𝑟∶𝐸→𝐶 by 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶Θ𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶,𝑥∈𝐸. (2.14)Then, the following conclusions hold:(a) 𝑇_𝑟 is single-valued,(b) 𝑇_𝑟 is a firmly nonexpansive-type mapping, that is, ∀𝑧,𝑦∈𝐸, 𝑇_𝑟𝑧−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑧−𝐽𝑇_𝑟𝑦≤𝑇_𝑟𝑧−𝑇_𝑟𝑦,𝐽𝑧−𝐽𝑦, (2.15)(c) 𝐹(𝑇_𝑟)=EP(Θ)=𝐹(𝑇_𝑟),(d) EP(Θ) is closed and convex,(e) 𝜙(𝑞,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥),∀𝑞∈𝐹(𝑇_𝑟).

Lemma 2.10 (see [16]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ a lower semicontinuous and convex function, and Θ∶𝐶×𝐶→ℝ a bifunction satisfying conditions (A₁)–(A₄). Let 𝑟>0 be any given number and 𝑥∈𝐸 any given point. Then, the following hold.(i)There exists 𝑢∈𝐶 such that Θ𝑦(𝑢)+1𝑢,𝑦+⟨𝐴𝑢,𝑦−𝑢⟩+𝜓−𝜓𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶. (2.16)(ii)If we define a mapping 𝐾_𝑟∶𝐶→𝐶 by𝐾_𝑟(𝑥)=+𝑦(𝑢)+1𝑢∈𝐶∶Θ𝑢,𝑦𝐴𝑢,𝑦−𝑢+𝜓−𝜓𝑟𝑦−𝑢,𝐽𝑢−𝐽𝑥≥0,∀𝑦∈𝐶,∀𝑥∈𝐶. (2.17) Then, the mapping 𝐾_𝑟 has the following properties:(a) 𝐾_𝑟 is single valued,(b) 𝐾_𝑟 is a firmly nonexpansive-type mapping, that is, 𝐾_𝑟𝑧−𝐾_𝑟𝑦,𝐽𝐾_𝑟𝑧−𝐽𝐾_𝑟𝑦≤𝐾_𝑟𝑧−𝐾_𝑟𝑦,𝐽𝑧−𝐽𝑦,∀𝑧,𝑦∈𝐸, (2.18)(c) 𝐹(𝐾_𝑟)=Ω=𝐹(𝐾_𝑟),(d) Ω is closed and convex,(e)  𝜙𝑞,𝐾_𝑟𝑧+𝜙𝐾_𝑟𝑧,𝑧≤𝜙𝑞,𝑧,∀𝑞∈𝐹𝐾_𝑟,𝑧∈𝐸. (2.19)

Remark 2.11. It follows from Lemma 2.9 that the mapping 𝐾_𝑟 is a relatively nonexpansive mapping. Thus, it is quasi-𝜙-nonexpansive.

3. Main Results

In this section, we will prove a strong convergence theorem for finding a common element of the set of solutions for the generalized mixed equilibrium problem (1.1) and the set of common fixed points for a pair of quasi-𝜙-asymptotically nonexpansive mappings in Banach spaces.

Theorem 3.1. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ a lower semicontinuous and convex, function, and Θ∶𝐶×𝐶→ℝ a bifunction satisfying conditions (A₁)–(A₄). Let 𝑆,𝑇∶𝐶→𝐶 be two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly 𝐿-Lipschitz continuous and that 𝐺=𝐹(𝑇)⋂𝐹(𝑆)⋂Ω is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthat,Θ𝑢∀𝑦∈𝐶,_𝑛,𝑦+𝐴𝑢_𝑛,𝑦−𝑢_𝑛+𝜓𝑦−𝜓𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.1) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1] and {𝑟_𝑛}⊂[a,∞) for some 𝑎>0, 𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). Suppose that the following conditions are satisfied:(i) liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0,(ii) liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0.Then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩Ω}𝑥₀, where Π_{𝐹(𝑆)∩𝐹(𝑇)∩Ω} is the generalized projection of 𝐸 onto 𝐹(𝑆)∩𝐹(𝑇)∩Ω.

Proof. Firstly, we define two functions 𝐻∶𝐶×𝐶→ℝ and 𝐾_𝑟∶𝐶→𝐶 by 𝐻+𝑦(𝑥)𝐾𝑥,𝑦=Θ𝑥,𝑦𝐴𝑥,𝑦−𝑥+𝜓−𝜓,∀𝑥,𝑦∈𝐶,_𝑟(𝑥)=+1𝑢∈𝐶∶𝐻𝑢,𝑦𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶,𝑥∈𝐶. (3.2) By Lemma 2.10, we know that the function 𝐻 satisfies conditions (A₁)–(A₄) and 𝐾_𝑟 has properties (a)–(e). Therefore, (3.1) is equivalent to 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthat,𝐻𝑢∀𝑦∈𝐶,_𝑛+1,𝑦𝑟_𝑛⟨𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶⟩≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥⟩≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0. (3.3) We divide the proof of Theorem 3.1 into five steps.
(I) First we prove that 𝐶_𝑛 and 𝑄_𝑛 are both closed and convex subsets of 𝐶 for all 𝑛≥0.
In fact, it is obvious that 𝑄_𝑛 is closed and convex for all 𝑛≥0. Again we have that 𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛⟺2𝑣,𝐽𝑥_𝑛−𝐽𝑧_𝑛≤𝑥_𝑛²−𝑧_𝑛²+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛⟺2𝑣,𝐽𝑥_𝑛−𝐽𝑢_𝑛≤𝑥_𝑛²−𝑢_𝑛²+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛. (3.4) Hence 𝐶_𝑛, ∀𝑛≥0, is closed and convex, and so 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛≥0.
(II) Next we prove that 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶_𝑛∩𝑄_𝑛, ∀𝑛≥0.
Putting 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛, ∀𝑛≥0, by Lemma 2.10 and Remark 2.11, 𝐾_𝑟𝑛 is relatively nonexpansive. Again since 𝑆 and 𝑇 are quasi-𝜙-asymptotically nonexpansive, for any given 𝑢∈𝐹(𝑆)∩𝐹(𝑇)∩Ω, we have that 𝜙𝑢,𝑧_𝑛=𝜙𝑢,𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛=‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛²≤‖𝑢‖²−2𝛼_𝑛𝑢,𝐽𝑥_𝑛−21−𝛼_𝑛𝑢,𝐽𝑇^𝑛𝑥_𝑛+𝛼_𝑛𝑥_𝑛²+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛²−𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛=𝛼_𝑛𝜙𝑢,𝑥_𝑛+1−𝛼_𝑛𝜙𝑢,𝑇^𝑛𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛≤𝛼_𝑛𝜙𝑢,𝑥_𝑛+1−𝛼_𝑛𝑘_𝑛𝜙𝑢,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛≤𝑘_𝑛𝜙𝑢,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛≤𝜙𝑢,𝑥_𝑛+sup_𝑝∈𝐺𝑘_𝑛−1𝜙𝑝,𝑥_𝑛−𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛=𝜙𝑢,𝑥_𝑛+𝜉_𝑛−𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛≤𝜙𝑢,𝑥_𝑛+𝜉_𝑛. (3.5) From (3.5) we have that 𝜙𝑢,𝑢_𝑛=𝜙𝑢,𝐾_𝑟𝑛𝑦_𝑛≤𝜙𝑢,𝑦_𝑛≤𝜙𝑢,𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛=‖𝑢‖²−2𝑢,𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛+𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛²≤‖𝑢‖²−2𝛽_𝑛𝑢,𝐽𝑥_𝑛−21−𝛽_𝑛𝑢,𝐽𝑆^𝑛𝑧_𝑛+𝛽_𝑛𝑥_𝑛²+1−𝛽𝑆^𝑛𝑧_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛=𝛽_𝑛𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝜙𝑢,𝑆^𝑛𝑧_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛≤𝛽_𝑛𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝑘_𝑛𝜙𝑢,𝑧_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛≤𝛽_𝑛𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝑘_𝑛𝜙𝑢,𝑥_𝑛+𝜉_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛≤𝛽_𝑛𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝜙𝑢,𝑥_𝑛+𝜉_𝑛+1−𝛽_𝑛𝑘_𝑛𝜉_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛≤𝜙𝑢,𝑥_𝑛+1−𝛽_𝑛𝜉_𝑛+1−𝛽_𝑛𝑘_𝑛𝜉_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛≤𝜙𝑢,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛−𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛≤𝜙𝑢,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛∀𝑛≥0. (3.6) This implies that 𝑢∈𝐶_𝑛, ∀𝑛≥0, and so 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶_𝑛, ∀𝑛≥0.
Now we prove that 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶_𝑛∩𝑄_𝑛, ∀𝑛≥0.
In fact, from 𝑄₀=𝐶, we have that 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶₀∩𝑄₀. Suppose that 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶_𝑘∩𝑄_𝑘, for some 𝑘≥0. Now we prove that 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶_𝑘+1∩𝑄_𝑘+1. In fact, since 𝑥_𝑘+1=Π_{𝐶𝑘∩𝑄𝑘}𝑥₀, we have that 𝑥_𝑘+1−𝑧,𝐽𝑥₀−𝐽𝑥_𝑘+1≥0,∀𝑧∈𝐶_𝑘∩𝑄_𝑘. (3.7) Since 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝐶_𝑘∩𝑄_𝑘, for any 𝑧∈𝐹(𝑇)∩𝐹(𝑆)∩Ω, we have that 𝑥_𝑘+1−𝑧,𝐽𝑥₀−𝐽𝑥_𝑘+1≥0. (3.8) This shows that 𝑧∈𝑄_𝑘+1, and so 𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝑄_𝑘+1. The conclusion is proved.
(III) Now we prove that {𝑥_𝑛} is bounded.
From the definition of 𝑄_𝑛, we have that 𝑥_𝑛=Π_𝑄𝑛𝑥₀, ∀𝑛≥0. Hence, from Lemma 2.2(1), 𝜙𝑥_𝑛,𝑥₀=𝜙Π_𝑄𝑛𝑥₀,𝑥₀≤𝜙𝑢,𝑥₀−𝜙𝑢,Π_𝑄𝑛𝑥₀≤𝜙𝑢,𝑥₀,∀𝑢∈𝐹(𝑇)∩𝐹(𝑆)∩Ω⊂𝑄_𝑛,∀𝑛≥0. (3.9) This implies that {𝜙(𝑥_𝑛,𝑥₀)} is bounded. By virtue of (2.4), {𝑥_𝑛} is bounded. Denote 𝑀=sup_𝑛≥0𝑥_𝑛<∞. (3.10) Since 𝑥_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀∈𝐶_𝑛∩𝑄_𝑛⊂𝑄_𝑛 and 𝑥_𝑛=Π_𝑄𝑛𝑥₀, from the definition of Π_𝑄𝑛, we have that 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀≤𝑀+𝑥₀²,∀𝑛≥0. (3.11) This implies that {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing, and so the limit lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. Without loss of generality, we can assume that lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀=𝑟≥0. (3.12) By the way, from the definition of {𝜉_𝑛}, (2.4), and (3.10), it is easy to see that 𝜉_𝑛=sup_𝑢∈𝐺𝑘_𝑛−1𝜙𝑢,𝑥_𝑛≤sup_𝑢∈𝐺𝑘_𝑛−1(‖𝑢‖+𝑀)²⟶0(as𝑛⟶∞). (3.13)
(IV) Now, we prove that {𝑥_𝑛} converges strongly to some point 𝑝∈𝐺=𝐹(𝑇)∩𝐹(𝑆)∩Ω.
In fact, since {𝑥_𝑛} is bounded in 𝐶 and 𝐸 is reflexive, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝. Again since Q_𝑛 is closed and convex for each 𝑛≥0, it is weakly closed, and so 𝑝∈𝑄_𝑛 for each 𝑛≥0. Since 𝑥_𝑛=Π_𝑄𝑛𝑥₀, from the defintion of Π_𝑄𝑛, we have that 𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀,𝑛≥0. (3.14) Since liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀=liminf_{𝑛𝑖→∞}𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑥₀+𝑥₀²≥𝑝²−2𝑝,𝐽𝑥₀+𝑥₀²=𝜙𝑝,𝑥₀, (3.15) we have that 𝜙𝑝,𝑥₀≤liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤limsup_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀. (3.16) This implies that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₀)=𝜙(𝑝,𝑥₀), that is, ||𝑥_𝑛𝑖||→||𝑝||. In view of the Kadec-Klee property of 𝐸, we obtain that lim_𝑛→∞𝑥_𝑛𝑖=𝑝.
Now we first prove that 𝑥_𝑛→𝑝(𝑛→∞). In fact, if there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑞, then we have that 𝜙𝑝,𝑞=lim_{𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗≤lim_{𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₀−𝜙Π_𝑄𝑛𝑗𝑥₀,𝑥₀=lim_{𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₀−𝜙𝑥_𝑛𝑗,𝑥₀=0by(3.12). (3.17) Therefore we have that 𝑝=𝑞. This implies that lim_𝑛→∞𝑥_𝑛=𝑝. (3.18)
Now we first prove that 𝑝∈𝐹(𝑇)∩𝐹(𝑆). In fact, by the construction of 𝑄_𝑛, we have that 𝑥_𝑛=Π_𝑄𝑛𝑥₀. Therefore, by Lemma 2.2(a) we have that 𝜙𝑥_𝑛+1,𝑥_𝑛=𝜙𝑥_𝑛+1,Π_𝑄𝑛𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀−𝜙Π_𝑄𝑛𝑥₀,𝑥₀=𝜙𝑥_𝑛+1,𝑥₀−𝜙𝑥_𝑛,𝑥₀⟶0(as𝑛⟶∞). (3.19) In view of 𝑥_𝑛+1∈𝐶_𝑛∩𝑄_𝑛⊂𝐶_𝑛 and noting the construction of 𝐶_𝑛 we obtain 𝜙𝑥_𝑛+1,𝑧_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛,𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛. (3.20) From (3.13) and (3.19), we have that lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=0,lim_𝑛→∞𝜙𝑥_𝑛+1,𝑧_𝑛=0. (3.21)
From (2.4) it yields that (||𝑥_𝑛+1||−||𝑢_𝑛||)²→0 and (||𝑥_𝑛+1||−||𝑧_𝑛||)²→0. Since ||𝑥_𝑛+1||→||𝑝||, we have that 𝑢_𝑛⟶𝑝,𝑧_𝑛⟶𝑝(as𝑛⟶∞). (3.22) Hence, we have that 𝐽𝑢_𝑛⟶𝐽𝑝,𝐽𝑧_𝑛⟶𝐽𝑝(as𝑛⟶∞). (3.23)
This implies that {𝐽𝑧_𝑛} is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, there exists a subsequence {𝐽𝑧_𝑛𝑖}⊂{𝐽𝑧_𝑛} such that 𝐽𝑧_𝑛𝑖⇀𝑝₀∈𝐸^∗. In view of the reflexiveness of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence, there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑝₀. Since 𝜙𝑥_𝑛𝑖+1,𝑧_𝑛𝑖=𝑥_𝑛𝑖+1²−2𝑥_𝑛𝑖+1,𝐽𝑧_𝑛𝑖+𝑧_𝑛𝑖²=𝑥_𝑛𝑖+1²−2𝑥_𝑛𝑖+1,𝐽𝑧_𝑛𝑖+𝐽𝑧_𝑛𝑖², (3.24) taking liminf_𝑛→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ||⋅||, it yields that 0≥𝑝²−2𝑝,𝑝₀+𝑝₀²=𝑝²−2𝑝,𝐽𝑥+‖𝐽𝑥‖²=𝑝²−2𝑝,𝐽𝑥+‖𝑥‖²=𝜙𝑝,𝑥, (3.25) that is, 𝑝=𝑥. This implies that 𝑝₀=𝐽𝑝, and so 𝐽𝑧_𝑛⇀𝐽𝑝. It follows from (3.23) and the Kadec-Klee property of 𝐸^∗ that 𝐽𝑧_𝑛𝑖→𝐽𝑝 (as 𝑛→∞). Noting that 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it yields that 𝑧_𝑛𝑖⇀𝑝. It follows from (3.22) and the Kadec-Klee property of 𝐸 that lim_{𝑛𝑖→∞}𝑧_𝑛𝑖=𝑝.
By the same way as given in the proof of (3.18), we can also prove that lim_𝑛→∞𝑧_𝑛=𝑝. (3.26)
From (3.18) and (3.26), we have that 𝑥_𝑛−𝑧_𝑛⟶0(as𝑛⟶∞). (3.27) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have that 𝐽𝑥_𝑛−𝐽𝑧_𝑛⟶0(as𝑛⟶∞). (3.28) For any 𝑢∈𝐹(𝑇)⋂𝐹(𝑆)⋂Ω, it follows from (3.5) that 𝛼_𝑛1−𝛼_𝑛𝑔𝐽𝑥_𝑛−𝑇^𝑛𝑥_𝑛≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑧_𝑛+𝜉_𝑛. (3.29) Since 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑧_𝑛=𝑥_𝑛²−z_𝑛²−2⟨𝑢,𝐽𝑥_𝑛−𝐽𝑧_𝑛⟩≤𝑥_𝑛²−𝑧_𝑛²+2‖𝑢‖⋅𝐽𝑥_𝑛−𝐽𝑧_𝑛≤𝑥_𝑛−𝑧_𝑛𝑥_𝑛+𝑧_𝑛+2‖𝑢‖⋅𝐽𝑥_𝑛−𝐽𝑧_𝑛, (3.30) From (3.27) and (3.28), it follows that 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑧_𝑛⟶0(as𝑛⟶∞). (3.31)
In view of condition (i) and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, we see that 𝑔𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛⟶0(as𝑛⟶∞). (3.32) It follows from the property of 𝑔 that 𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑥_𝑛⟶0(as𝑛⟶∞). (3.33) Since 𝑥_𝑛→𝑝 and 𝐽 is uniformly continuous, it yields that 𝐽𝑥_𝑛→𝐽𝑝. Hence from (3.33) we have that 𝐽𝑇^𝑛𝑥_𝑛⟶𝐽𝑝(as𝑛⟶∞). (3.34) Since 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it follows that 𝑇^𝑛𝑥_𝑛⇀𝑝. (3.35) On the other hand, we have that 𝑇^𝑛𝑥_𝑛−𝑝=𝐽𝑇^𝑛𝑥_𝑛−𝐽𝑝≤𝐽𝑇^𝑛𝑥_𝑛−𝐽𝑝⟶0(as𝑛⟶∞). (3.36) This together with (3.35) shows that 𝑇^𝑛𝑥_𝑛⟶𝑝. (3.37)
Furthermore, by the assumption that 𝑇 is uniformly 𝐿-Lipschitz continuous, we have that 𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛𝑥_𝑛≤𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛+1𝑥_𝑛+1+𝑇^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇^𝑛𝑥_𝑛≤(𝐿+1)𝑥_𝑛+1−𝑥_𝑛+𝑇^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+𝑥_𝑛−𝑇^𝑛𝑥_𝑛. (3.38)
This together with (3.18) and (3.37), yields ||𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛𝑥_𝑛||→0 (as 𝑛→∞). Hence from (3.37) we have that 𝑇^𝑛+1𝑥_𝑛→𝑝, that is, 𝑇𝑇^𝑛𝑥_𝑛→𝑝. In view of (3.37) and the closeness of 𝑇, it yields that 𝑇𝑝=𝑝. This implies that 𝑝∈𝐹(𝑇).
By the same way as given in the proof of (3.23) to (3.31), we can also prove that lim_𝑛→∞𝑢_𝑛=𝑝,𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛⟶0(as𝑛⟶∞). (3.39)
Since 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛, from (2.19), (3.6), (3.13), and (3.39), we have that 𝜙𝑢_𝑛,𝑦_𝑛=𝜙𝐾_𝑟𝑛𝑦_𝑛,𝑦_𝑛≤𝜙𝑢,𝑦_𝑛−𝜙𝑢,𝑢_𝑛≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛⟶0(as𝑛⟶∞). (3.40) From (2.4) it yields that (||𝑢_𝑛||−||𝑦_𝑛||)²→0. Since ||𝑢_𝑛||→||𝑝||, we have that 𝑦_𝑛⟶𝑝(as𝑛⟶∞). (3.41)
Hence we have that 𝐽𝑦_𝑛⟶𝐽𝑝(as𝑛⟶∞). (3.42) By the same way as given in the proof of (3.26), we can also prove that lim_𝑛→∞𝑦_𝑛=𝑝. (3.43) From (3.39) and (3.43) we have that 𝑢_𝑛−𝑦_𝑛⟶0(as𝑛⟶∞). (3.44) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have that 𝐽𝑢_𝑛−𝐽𝑦_𝑛⟶0(as𝑛⟶∞). (3.45) For any 𝑢∈𝐹(𝑇)⋂𝐹(𝑆)⋂Ω, it follows from (3.6), (3.13), and (3.39) that 𝛽_𝑛1−𝛽_𝑛𝑔𝐽𝑥_𝑛−𝑆^𝑛𝑧_𝑛≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛⟶0. (3.46) In view of condition (ii) and liminf_𝑛→∞𝛽_𝑛(1−𝛽_𝑛)>0, we see that 𝑔𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛⟶0(as𝑛⟶∞). (3.47) It follows from the property of 𝑔 that 𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑧_𝑛⟶0(as𝑛⟶∞). (3.48) Since 𝑥_𝑛→𝑝 and 𝐽 is uniformly continuous, it yields, 𝐽𝑥_𝑛→𝐽𝑝. Hence from (3.48) we have that 𝐽𝑆^𝑛𝑧_𝑛⟶𝐽𝑝(as𝑛⟶∞). (3.49) Since 𝐽⁻¹∶𝐸^∗→𝐸 is hemicontinuous, it follows that 𝑆^𝑛𝑧_𝑛⇀𝑝. (3.50) On the other hand, we have that 𝑆^𝑛𝑧_𝑛−𝑝=𝐽𝑆^𝑛𝑧_𝑛−𝐽𝑝≤𝐽𝑆^𝑛𝑧_𝑛−𝐽𝑝⟶0(as𝑛⟶∞). (3.51) This together with (3.50) shows that 𝑆^𝑛𝑧_𝑛⟶𝑝. (3.52)
Furthermore, by the assumption that 𝑆 is uniformly 𝐿-Lipschitz continuous, we have that 𝑆^𝑛+1𝑧_𝑛−𝑆^𝑛𝑧_𝑛≤𝑆^𝑛+1𝑧_𝑛−𝑆^𝑛+1𝑧_𝑛+1+𝑆^𝑛+1𝑧_𝑛+1−𝑧_𝑛+1+𝑧_𝑛+1−𝑧_𝑛+𝑧_𝑛−𝑆^𝑛𝑧_𝑛≤(𝐿+1)𝑧_𝑛+1−𝑧_𝑛+𝑆^𝑛+1𝑧_𝑛+1−𝑧_𝑛+1+𝑧_𝑛−𝑆^𝑛𝑧_𝑛. (3.53)
This together with (3.26) and (3.52), yields that ||𝑆^𝑛+1𝑧_𝑛−𝑆^𝑛𝑧_𝑛||→0 (as 𝑛→∞). Hence from (3.52) we have that 𝑆^𝑛+1𝑧_𝑛→𝑝, that is, 𝑆𝑆^𝑛𝑧_𝑛→𝑝. In view of (3.52) and the closeness of 𝑇, it yields that 𝑆𝑝=𝑝. This implies that 𝑝∈𝐹(𝑆).
Next we prove that 𝑝∈Ω. From (3.45) and the assumption that 𝑟_𝑛≥𝑎, ∀𝑛≥0, we have that lim_𝑛→∞𝐽𝑢_𝑛−𝐽𝑦_𝑛𝑟_𝑛=0. (3.54) Since 𝑢_𝑛=𝐾_𝑟𝑛𝑦_𝑛, we have that 𝐻𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛≥0,∀𝑦∈𝐶. (3.55) Replacing 𝑛 by 𝑛_𝑘 in (3.55), from condition (A₂), we have that 1𝑟_𝑛𝑘𝑦−𝑢_𝑛𝑘,𝐽𝑢_𝑛𝑘−𝐽𝑦_𝑛𝑘≥−𝐻𝑢_𝑛𝑘,𝑦≥𝐻𝑦,𝑢_𝑛𝑘,∀𝑦∈𝐶. (3.56) By the assumption that 𝑦↦𝐻(𝑥,𝑦) is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting 𝑛_𝑘→∞ in (3.55), from (3.54) and condition (A₄), we have that 𝐻(𝑦,𝑝)≤0, ∀𝑦∈𝐶.
For 𝑡∈(0,1] and 𝑦∈𝐶, letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝, there are 𝑦_𝑡∈𝐶 and 𝐻(𝑦_𝑡,𝑝)≤0. By conditions (A₁) and (A₄), we have that 0=𝐻𝑦_𝑡,𝑦_𝑡≤𝑡𝐻𝑦_𝑡,𝑦+(1−𝑡)𝐻𝑦_𝑡,𝑝≤𝑡𝐻𝑦_𝑡,𝑦. (3.57) Dividing both sides of the above equation by 𝑡, we have that 𝐻(𝑦_𝑡,𝑦)≥0, ∀𝑦∈𝐶. Letting 𝑡↓0, from condition (A₃), we have that 𝐻(𝑝,𝑦)≥0, ∀𝑦∈𝐶, that is, Θ(𝑝,𝑦)+⟨𝐴𝑝,𝑦−𝑝⟩+𝜓(𝑦)−𝜓(𝑝)≥0, ∀𝑦∈𝐶. Therefore 𝑝∈Ω, and so 𝑝∈𝐹(𝑇)⋂𝐹(𝑆)⋂Ω.
(V) Finally, we prove that 𝑥_𝑛→Π_{⋂⋂Ω𝐹(𝑇)𝐹(𝑆)}𝑥₀.
Let 𝑤=Π_{⋂⋂Ω𝐹(𝑇)𝐹(𝑆)}𝑥₀. From 𝑤∈𝐹(𝑇)⋂𝐹(𝑆)⋂Ω⊂𝐶_𝑛∩𝑄_𝑛, and 𝑥_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀, we have that 𝜙𝑥_𝑛+1,𝑥₀≤𝜙𝑤,𝑥₀,∀𝑛≥0. (3.58) Since the norm is weakly lower semicontinuous, this implies that 𝜙𝑝,𝑥₀=𝑝²−2𝑝,𝐽𝑥₀+𝑥₀²≤lim_{𝑛𝑘→∞}inf𝑥_𝑛𝑘²−2𝑥_𝑛𝑘,𝐽𝑥₀+𝑥₀²≤lim_{𝑛𝑘→∞}inf𝜙𝑥_𝑛𝑘,𝑥₀≤lim_{𝑛𝑘→∞}sup𝜙𝑥_𝑛𝑘,𝑥₀≤𝜙𝑤,𝑥₀. (3.59) It follows from the definition of Π_{⋂⋂Ω𝐹(𝑇)𝐹(𝑆)}𝑥₀ and (3.59) that we have 𝑝=𝑤. Therefore, 𝑥_𝑛→Π_{⋂⋂Ω𝐹(𝑇)𝐹(𝑆)}𝑥₀. This completes the proof of Theorem 3.1.

Remark 3.2. Theorem 3.1 improves and extends the corresponding results in [7–9].(a)For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property(note that each uniformly convex Banach space must have the Kadec-Klee property).(b)For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or weak relatively nonexpansive mappings to a pair of quasi-𝜙-asymptotically nonexpansive mappings.(c)For the equilibrium problem, we extend the generalized equilibrium problem to the generalized mixed equilibrium problem.

The following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.3. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping and Θ∶𝐶×𝐶→ℝ a bifunction satisfying conditions (A₁)–(A₄). Let 𝑆,𝑇∶𝐶→𝐶 be two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly 𝐿-Lipschitz continuous and that 𝐺=𝐹(𝑇)⋂𝐹(𝑆)⋂GEP is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈C,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthat,Θ𝑢∀𝑦∈𝐶,_𝑛,𝑦+𝐴𝑢_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙v,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.60) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0, 𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (i)-(ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩GEP}𝑥₀, where GEP is the set for the solutions of generalized equilibrium problem (1.3).

Proof. Putting 𝜓=0 in Theorem 3.1, the conclusion of Theorem 3.3 can be obtained from Theorem 3.1.

Theorem 3.4. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝜓∶𝐶→ℝ be a lower semicontinuous and convex function and Θ∶𝐶×𝐶→ℝ a bifunction satisfying conditions (A₁)–(A₄). Let 𝑆,𝑇∶𝐶→𝐶 be two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly 𝐿-Lipschitz continuous and that 𝐺=𝐹(𝑇)⋂𝐹(𝑆)⋂MEP is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthatΘ𝑢,∀𝑦∈𝐶,_𝑛,𝑦+𝜓𝑦−𝜓𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.61) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0,𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (i)-(ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩MEP}𝑥₀, where MEP is the set of solutions for mixed equilibrium problem (1.4).

Proof. Putting 𝐴=0 in Theorem 3.1, the conclusion of Theorem 3.4 can be obtained from Theorem 3.1.

Theorem 3.5. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping and 𝜓∶𝐶→ℝ a lower semicontinuous and convex function. Let 𝑆,𝑇∶𝐶→𝐶 be two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly 𝐿-Lipschitz continuous and that 𝐺=𝐹(𝑇)⋂𝐹(𝑆)⋂VI(𝐶,𝐴,𝜓) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthat,∀𝑦∈𝐶,𝐴𝑢_𝑛,𝑦−𝑢_𝑛+𝜓𝑦−𝜓𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.62) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0,𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (i)-(ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩VI(𝐶,𝐴,𝜓)}𝑥₀, where VI(𝐶,𝐴,𝜓) is the set of solutions for the mixed variational inequality (1.5).

Proof. Putting Θ=0 in Theorem 3.1, the conclusion of Theorem 3.5 can be obtained from Theorem 3.1.

Theorem 3.6. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let Θ∶𝐶×𝐶→ℝ be a bifunction satisfying conditions (A₁)–(A₄). Let 𝑆,𝑇∶𝐶→𝐶 be two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly L -Lipschitz continuous and that 𝐺=𝐹(𝑇)⋂𝐹(𝑆)⋂EP(Θ) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthatΘ𝑢,∀𝑦∈𝐶,_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩Q𝑛}𝑥₀,∀𝑛≥0, (3.63) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0, 𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (i)-(ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩EP(Θ)}𝑥₀, where EP(Θ) is the set of solutions for the equilibrium problem (1.6).

Proof. Putting 𝜓=0 and 𝐴=0 in Theorem 3.1, the conclusion of Theorem 3.6 can be obtained from Theorem 3.1.

Theorem 3.7. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping and 𝑆,𝑇∶𝐶→𝐶 two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly 𝐿-Lipschitz continuous and that 𝐺=𝐹(𝑇)⋂𝐹(𝑆)⋂VI(𝐶,𝐴) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈C,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛∈𝐶suchthat,∀𝑦∈𝐶,𝐴𝑢_𝑛,𝑦−𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+𝑘_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.64) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0, 𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (i)-(ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)∩VI(𝐶,𝐴)}𝑥₀, where VI(𝐶,𝐴) is the set of solutions for the variational inequality (1.7)

Proof. Putting 𝜓=0 and Θ=0 in Theorem 3.1, the conclusion of Theorem 3.7 can be obtained from Theorem 3.1.

Theorem 3.8. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ a lower semicontinuous and convex function, and Θ∶𝐶×𝐶→ℝ a bifunction satisfying conditions (A₁)–(A₄). Let 𝑆∶𝐶→𝐶 be a closed and quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 is uniformly 𝐿-Lipschitz continuous and that 𝐹(𝑆)⋂Ω is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑦=𝐶,_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑥_𝑛,𝑢_𝑛∈𝐶suchthatΘ𝑢,∀𝑦∈𝐶,_𝑛,𝑦+𝐴𝑢_𝑛,𝑦−𝑢_𝑛+𝜓𝑦−𝜓𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.65) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0,𝜉_𝑛=sup_{𝑢∈𝐹(𝑆)∩Ω}(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛽_𝑛} satisfy condition (ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩Ω}𝑥₀.

Proof. Taking 𝑇=𝐼 in Theorem 3.1, we have that 𝑧_𝑛=𝑥_𝑛, ∀𝑛≥0. Hence, the conclusion of Theorem 3.8 is obtained.

Theorem 3.9. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous and monotone mapping, 𝜓∶𝐶→ℝ a lower semicontinuous and convex function, and Θ∶𝐶×𝐶→ℝ a bifunction satisfying conditions (A₁)–(A₄). Suppose that Ω is a nonempty subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑢=𝐶,_𝑛∈𝐶suchthatΘ𝑢,∀𝑦∈𝐶,_𝑛,𝑦+𝐴𝑢_𝑛,𝑦−𝑢_𝑛+𝜓𝑦−𝜓𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑥_𝑛𝐶≥0,_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.66) where {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. Then {𝑥_𝑛} converges strongly to Π_Ω𝑥₀.

Proof. Taking 𝑇=𝑆=𝐼 in Theorem 3.1, the conclusion is obtained.

Theorem 3.10. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and 𝐶 a nonempty closed convex subset of 𝐸. Let 𝑆,𝑇∶𝐶→𝐶 be two closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that 𝑆 and 𝑇 are uniformly 𝐿-Lipschitz continuous and that 𝐹(𝑇)⋂𝐹(𝑆) is a nonempty and bounded subset in 𝐶. Let {𝑥_𝑛} be the sequence generated by 𝑥₀∈𝐶,𝐶₀=𝐶,𝑄₀𝑧=𝐶,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇^𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑆^𝑛𝑧_𝑛,𝑢_𝑛=Π_𝐶𝑦_𝑛,𝐶_𝑛=𝑣∈𝐶_𝑛−1∶𝜙𝑣,𝑧_𝑛≤𝜙𝑣,𝑥_𝑛+𝜉_𝑛,𝜙𝑣,𝑢_𝑛≤𝜙𝑣,𝑥_𝑛+1+k_𝑛1−𝛽_𝑛𝜉_𝑛,𝑄_𝑛=𝑧∈𝑄_𝑛−1∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛≥0, (3.67) where 𝐽∶𝐸→𝐸^∗ is the normalized duality mapping, {𝛼_𝑛} and {𝛽_𝑛} are sequences in [0,1], and 𝜉_𝑛=sup_{𝑢∈𝐹(𝑆)∩𝐹(𝑇)}(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛). If {𝛼_𝑛} and {𝛽_𝑛} satisfy conditions (i)-(ii) in Theorem 3.1, then {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩𝐹(𝑇)}𝑥₀.

Proof. Taking 𝐴=Θ=0 and 𝑟_𝑛=1, ∀𝑛≥0 in Theorem 3.1, the conclusion of Theorem 3.10 is obtained.