A New Hybrid Algorithm for a System of Generalized Mixed Equilibrium Problems and Fixed Point of Quasi-𝜙-Asymptotically Nonexpansive Mappings

Abstract

The purpose of this paper is to use a new hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family 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^∗. 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.

Let 𝜓∶𝐶→ℝ be a real-valued function, 𝐴∶𝐶→𝐸^∗ a nonlinear mapping, and {Θ_𝑖∶𝐶×𝐶→ℝ,𝑖=1,2,…,𝑁} a finite family of equilibrium bifunctions, that is, Θ_𝑖(𝑢,𝑢)=0 for each 𝑢∈𝐶. The “so-called” system of generalized mixed equilibrium problems (SGMEP) for functions (Θ₁,Θ₂,…,Θ_𝑁,𝐴,𝜓) is to find a common element 𝑥^∗∈𝐶 such that ΘΘ₁𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶,₂Θ𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+⋮𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶,_𝑁𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶.(1.1)

We denote the set of solutions of (1.1) by Ω∶=⋂^𝑁𝑖=1Ω(Θ_𝑖,𝜓), where Ω(Θ_𝑖,𝜓) is the set of solutions to the generalized mixed equilibrium problem: Θ_𝑖𝑥^∗,𝑦+⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶.(1.2)

Special examples are as follows.(i)If 𝐴=0 and 𝑁=1, the problem (1.1) is equivalent to finding 𝑥^∗∈𝐶 such that Θ𝑥^∗,𝑦+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶,(1.3)

which is called the mixed equilibrium problem (MEP) [1]. The set of solutions to (1.3) is denoted by MEP.(ii)If Θ=0 and 𝑁=1, the problem (1.1) is equivalent to finding 𝑥^∗∈𝐶 such that ⟨𝐴𝑥^∗,𝑦−𝑥^∗⟩+𝜓(𝑦)−𝜓𝑥^∗≥0,∀𝑦∈𝐶,(1.4)

which is called the mixed variational inequality of Browder type (VI) [2]. The set of solutions to (1.4) is denoted by VI(𝐶,𝐴,𝜓).

Recently, many authors studied the problems of finding a common element of the set of fixed point for a nonexpansive mapping and the set of solutions for an equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectively (see, e.g., [3–5] and the references therein).

Motivated and inspired by the researches going on in this direction, the purpose of this paper is using a hybrid algorithm for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for an infinite family 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 improve and extend the corresponding results in [6–12].

2. Preliminaries

First, we recall some definitions and conclusions.

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.

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→0‖𝑥+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 [13]. (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 norm-weak continuous.(iii)If 𝐸 is a smooth, strictly convex, and reflexive Banach space, then 𝐽 is single-valued, 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 𝜙(𝑥,𝑦)=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+‖𝑦‖²,∀𝑥,𝑦∈𝐸.(2.3)
It is obvious from the definition of 𝜙 that (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²,∀𝑥,𝑦∈𝐸.(2.4)
Following Alber [14], the generalized projection Π_𝐶∶𝐸→𝐶 is defined by Π_𝐶(𝑥)=arginf_𝑦∈𝐶𝜙(𝑦,𝑥),∀𝑥∈𝐸.(2.5)

Lemma 2.2 (see [14, 15]). 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 all𝑥∈𝐶and𝑦∈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 [16]). (1) A mapping 𝑇∶𝐶→𝐶 is said to be relatively nonexpansive if 𝐹(𝑇)≠∅,𝐹(𝑇)=𝐹(𝑇), and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥),∀𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.7)
(2) A mapping𝑇∶𝐶→𝐶is said to beclosed if, for any sequence {𝑥_𝑛}⊂𝐶with𝑥_𝑛→𝑥 and𝑇𝑥_𝑛→𝑦, then𝑇𝑥=𝑦.

Definition 2.5 (see [9]). (1) A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,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 𝜙(𝑝,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥),∀𝑛≥1,𝑥∈𝐶,𝑝∈𝐹(𝑇).(2.9)
(3) 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‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,∀𝑛≥1.(2.10)

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 [6]). 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 sequence {𝑥_𝑖}^∞𝑖=1⊂𝐵_𝑟(0) and for any given sequence {𝜆_𝑖}^∞𝑖=1 of positive numbers with ∑^∞𝑖=1𝜆_𝑖=1, then there exists a continuous, strictly increasing, and convex function 𝑔∶[0,2𝑟)→[0,∞) with 𝑔(0)=0 such that for any positive integers 𝑖,𝑗 with 𝑖<𝑗, ‖‖‖‖^∞_𝑛=1𝜆_𝑛𝑥_𝑛‖‖‖‖²≤^∞_𝑛=1𝜆_𝑛‖‖𝑥_𝑛‖‖²−𝜆_𝑖𝜆_𝑗𝑔‖‖𝑥_𝑖−𝑥_𝑗‖‖.(2.11)

Lemma 2.8 (see [6]). Let 𝐸 be a real uniformly smooth and strictly convex Banach space with 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 system of generalized mixed equilibrium problems (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_𝑖∶𝐶×𝐶→ℝ,𝑖=1,2,…,𝑁 satisfies the following conditions:(𝐴₁)Θ_𝑖(𝑥,𝑥)=0,forall𝑥∈𝐶;(𝐴₂)Θ_𝑖 is monotone, that is, Θ_𝑖(𝑥,𝑦)+Θ_𝑖(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶;(𝐴₃)limsup_𝑡↓0Θ_𝑖(𝑥+𝑡(𝑧−𝑥),𝑦)≤Θ_𝑖(𝑥,𝑦)forall𝑥,𝑧,𝑦∈𝐶;(𝐴₄)the function 𝑦↦Θ_𝑖(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 be a bifunction satisfying conditions (𝐴₁)-(𝐴₄). Let 𝑟>0 and 𝑥∈𝐸. Then, the following hold. (i)(Blum and Oettli [17]) There exists 𝑧∈𝐶 such that Θ(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.12)(ii)(Takahashi and Zembayashi [3]) Define a mapping 𝑇_𝑟∶𝐸→𝐶 by 𝑇_𝑟(𝑥)=𝑧∈𝐶∶Θ(𝑧,𝑦)+1𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶,𝑥∈𝐸.(2.13)Then, the following conclusions hold: (a)𝑇_𝑟 is single valued;(b)𝑇_𝑟 is firmly nonexpansive-type mapping, that is, for all 𝑧,𝑦∈𝐸, ⟨𝑇_𝑟𝑧−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑧−𝐽𝑇_𝑟𝑦⟩≤⟨𝑇_𝑟𝑧−𝑇_𝑟𝑦,𝐽𝑧−𝐽𝑦⟩;(2.14)(c)𝐹(𝑇_𝑟)=EP(Θ)=F̃(T_𝑟); (d)EP(Θ) is closed and convex;(e)𝜙(𝑞,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥),forall𝑞∈𝐹(T_𝑟).

Lemma 2.10 (see [7]). 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 the conditions (𝐴₁)-(𝐴₄). Let 𝑟>0 be any given number and 𝑥∈𝐸 be any given point. Then, the following hold. (i)There exists 𝑢∈𝐶 such that Θ(𝑢,𝑦)+⟨𝐴𝑢,𝑦−𝑢⟩+𝜓(𝑦)−𝜓(𝑢)+1𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶.(2.15)(ii)If we define a mapping 𝐾^Θ𝑟∶𝐶→𝐶 by 𝐾^Θ𝑟(𝑥)=+𝑢𝜓∈(𝐶𝑦)∶−Θ𝜓(𝑢(𝑢,)𝑦)++1⟨𝐴𝑢,𝑦−𝑢⟩𝑟⟨𝑦−𝑢,𝐽𝑢−𝐽𝑥⟩≥0,∀𝑦∈𝐶,∀𝑥∈𝐶.(2.16)then, the mapping 𝐾^Θ𝑟 has the following properties: (a)𝐾^Θ𝑟 is single valued;(b)𝐾^Θ𝑟 is a firmly nonexpansive-type mapping, that is, 𝐾^Θ𝑟𝑧−𝐾^Θ𝑟𝑦,𝐽𝐾^Θ𝑟𝑧−𝐽𝐾^Θ𝑟𝑦≤𝐾^Θ𝑟𝑧−𝐾^Θ𝑟𝑦,𝐽𝑧−𝐽𝑦,∀𝑧,𝑦∈𝐸;(2.17)(c)𝐹(𝐾^Θ𝑟)=Ω(Θ,𝜓)=F̃(𝐾^Θ𝑟); (d)Ω(Θ,𝜓) is closed and convex;(e)𝜙𝑞,𝐾^Θ𝑟𝑥+𝜙𝐾^Θ𝑟𝑥,𝑥≤𝜙(𝑞,𝑥),∀𝑞∈𝐹𝐾^Θ𝑟.(2.18)

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

3. Main Results

In this section, we will use the hybrid method to prove some strong convergence theorems for finding a common element of the set of solutions for a system of the generalized mixed equilibrium problems (1.1) and the set of common fixed points for an infinite family of quasi-𝜙-asymptotically nonexpansive mappings in Banach spaces.

Theorem 3.1. Let 𝐸 be a uniformly smooth and strictly convex Banach space with Kleac-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_𝑖∶𝐶×𝐶→ℝ,𝑖=1,2,…,𝑁} be a finite family of bifunction satisfying conditions (𝐴₁)-(𝐴₄). Let {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with a sequence {𝑘_𝑛}⊂[1,∞) and 𝑘_𝑛→1. Suppose that for each 𝑖≥1,𝑆_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous and that 𝐺∶=⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂Ω is a nonempty and bounded subset in 𝐶, where Ω=⋂^𝑁𝑖=1Ω(Θ_𝑖,𝜓). Let {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑢^𝑛(𝑘)},𝑘=1,2,…,𝑁, be the sequences generated by 𝑧𝑥₀∈𝐶,𝐶₀=𝐶,_𝑛=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑦_𝑛,𝑢=𝐽⁻¹_𝑛(𝑁)𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛,Θ∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡,∀𝑦∈𝐶,_𝑁𝑢+^𝑛(𝑁),1𝑦+𝐴𝑢^𝑛(𝑁),𝑦−𝑢^𝑛(𝑁)+𝜓(𝑦)−𝜓𝑢^𝑛(𝑁)𝑟_𝑛𝑦−𝑢^𝑛(𝑁)Θ,𝐽𝑢^𝑛(𝑁)−𝐽𝑢^{𝑛(𝑁−1)}≥0,_𝑁−1𝑢+^{𝑛(𝑁−1)},1𝑦+𝐴𝑢^{𝑛(𝑁−1)},𝑦−𝑢^{𝑛(𝑁−1)}+𝜓(𝑦)−𝜓𝑢^{𝑛(𝑁−1)}𝑟_𝑛𝑦−𝑢^{𝑛(𝑁−1)}Θ,𝐽𝑢⋮^{𝑛(𝑁−1)}−𝐽𝑢^{𝑛(𝑁−2)}≥0,₁𝑢^𝑛(1),𝑦𝐶+𝐴𝑢^𝑛(1),𝑦−𝑢^𝑛(1)+𝜓(𝑦)−𝜓𝑢^𝑛(1)+𝑟1_𝑛𝑦−𝑢^𝑛(1),𝐽𝑢^𝑛(1)−𝐽𝑦_𝑛≥0,_𝑛+1=𝑣∈𝐶_𝑛∶𝜙𝑣𝑥,𝑢^𝑛(𝑁)≤𝜙𝑣,𝑥^𝑛+𝜉_𝑛,∀𝑛≥0,_𝑛+1=Π_𝐶𝑛+1𝑥₀,(3.1) where 𝑢𝑢^𝑛(1)_𝑛(𝑖)=𝐾^Θ1_𝑟𝑛𝑦_𝑛,=𝐾^Θ𝑖_𝑟𝑛𝑢^{𝑛(𝑖−1)}=𝐾^Θ𝑖_𝑟𝑛𝐾^Θ𝑖−1_𝑟𝑛𝑢^{𝑛(𝑖−2)}=𝐾^Θ𝑖_𝑟𝑛⋯𝐾^Θ2_𝑟𝑛𝑢^𝑛(1)=𝐾^Θ𝑖_𝑟𝑛⋯𝐾^Θ2_𝑟𝑛𝐾^Θ1_𝑟𝑛𝑦_𝑛,𝑖=1,2,…,𝑁,(3.2) and 𝐾^Θ𝑖_𝑟𝑛∶𝐶→𝐶, 𝑖=1,2,…,𝑁 is the mapping defined by (2.16),𝑟_𝑛∈[𝑑,∞) for some 𝑑>0, 𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛), Π_𝐶𝑛+1 is the generalized projection of 𝐸 onto the set 𝐶_𝑛+1, and for each 𝑛≥0, {𝛼_𝑛,𝑖}, {𝛼_𝑛} are sequences in [0,1] satisfying the following conditions: (a)∑^∞𝑖=0𝛼_𝑛,𝑖=1 for all 𝑛≥0;(b)liminf_𝑛→∞𝛼_𝑛,0⋅𝛼_𝑛,𝑖>0 for all 𝑖≥1;(c)0<𝛼≤𝛼_𝑛<1 for some 𝛼∈(0,1).
Then {𝑥_𝑛} converges strongly to Π_𝐺𝑥₀, where Π_𝐺 is the generalized projection from 𝐸 onto G.

Proof. We divide the proof of Theorem 3.1 into five steps.(I)We first prove that 𝐺 and 𝐶_𝑛 both are closed and convex subset of 𝐶 for all 𝑛≥0.
In fact, it follows from Lemmas 2.8 and 2.10 that 𝐹(𝑆_𝑖), 𝑖≥1, and Ω both are closed and convex. Therefore 𝐺 is a closed and convex subset in 𝐶. Furthermore, it is obvious that 𝐶₀=𝐶 is closed and convex. Suppose that 𝐶_𝑛 is closed and convex for some 𝑛≥1. Since the inequality 𝜙(𝑣,𝑢^𝑛(𝑁))≤𝜙(𝑣,𝑥_𝑛)+𝜉_𝑛 is equivalent to 2𝑣,𝐽𝑥_𝑛−𝐽𝑢^𝑛(𝑁)≤‖‖𝑥^𝑛‖‖²−‖‖𝑢^𝑛(𝑁)‖‖²+𝜉_𝑛,(3.3) therefore, we have𝐶_𝑛+1=𝑣∈𝐶_𝑛∶2𝑣,𝐽𝑥_𝑛−𝐽𝑢^𝑛(𝑁)≤‖‖𝑥^𝑛‖‖²−‖‖𝑢^𝑛(𝑁)‖‖²+𝜉_𝑛.(3.4)
This implies that 𝐶_𝑛+1 is closed and convex. The desired conclusions are proved. These in turn show that Π_𝐺𝑥₀ and Π_𝐶𝑛𝑥₀ are well defined.
(II)We prove that {𝑥_𝑛} is a bounded sequence in 𝐶.By the definition of 𝐶_𝑛, we have 𝑥_𝑛=Π_𝐶𝑛𝑥₀ for all 𝑛≥0. It follows from Lemma 2.2(a) that 𝜙𝑥_𝑛,𝑥₀=≤𝜙𝜙𝑢Π,𝑥_𝐶𝑛0𝑥₀,𝑥₀≤𝜙𝑢,𝑥₀−𝜙𝑢,Π_𝐶𝑛𝑥₀,∀𝑛≥0,𝑢∈𝐺.(3.5)
This implies that {𝜙(𝑥_𝑛,𝑥₀)} is bounded. By virtue of (2.4), {𝑥_𝑛} is bounded. Denote by 𝑀=sup_𝑛≥0‖‖𝑥_𝑛‖‖<∞.(3.6)
In view of the structure of {𝐶_𝑛}, we have 𝐶_𝑛+1⊂𝐶_𝑛, 𝑥_𝑛=Π_𝐶𝑛𝑥₀, and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. This implies that 𝑥_𝑛+1∈𝐶_𝑛 and 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥1.(3.7) Therefore, {𝜙(𝑥_𝑛,𝑥₀)} is convergent. Without loss of generality, we can assume that lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀=𝑟≥0.(3.8)
(III)Next, we prove that 𝐺∶=⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂(⋂^𝑁𝑖=1Ω(Θ_𝑖,𝜓))⊂𝐶_𝑛 for all 𝑛≥0.
Indeed, it is obvious that 𝐺⊂𝐶₀=𝐶. Suppose that 𝐺⊂𝐶_𝑛 for some 𝑛≥0. Since 𝑢^𝑛(𝑁)=𝐾^Θ𝑁_𝑟𝑛𝐾^Θ𝑁−1_𝑟𝑛⋯𝐾^Θ2_𝑟𝑛𝐾^Θ1_𝑟𝑛𝑦_𝑛, by Lemma 2.10 and Remark 2.6, 𝐾^Θ𝑖_𝑟𝑛 is quasi-𝜙-asymptotically nonexpansive. Since 𝐸 is uniformly smooth, 𝐸^∗ is uniformly convex. For any given 𝑢∈𝐺⊂𝐶_𝑛 and for any positive integer 𝑗>0, from Lemma 2.7 we have 𝜙𝑢,𝑢^𝑛(𝑁)==𝜙𝜙𝑢𝑢,,𝐽𝐾^{Θ𝑁𝑟𝑛}₋₁𝐾^Θ𝑁−1_𝑟𝑛⋯𝐾^Θ2_𝑟𝑛𝐾^Θ1_𝑟𝑛𝑦_𝑛≤𝜙𝑢,𝑦^𝑛≤‖𝑢‖₂𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛−2𝑢,𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛+‖‖𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛‖‖²≤‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛+𝛼_𝑛‖‖𝐽𝑧_𝑛‖‖²+1−𝛼_𝑛‖‖𝐽𝑥_𝑛‖‖²=‖𝑢‖²−2𝑢,𝛼_𝑛𝐽𝑧_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛+𝛼_𝑛‖‖𝑧_𝑛‖‖²+1−𝛼_𝑛‖‖𝑥_𝑛‖‖²==𝛼𝛼_𝑛𝑛𝜙𝑢,𝑧_𝑛+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛𝜙𝑢,𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛+1−𝛼^𝑛=𝛼_𝑛𝜙𝑢,𝑥_𝑛⎛⎜⎜⎝‖𝑢‖₂−2𝛼_𝑛,0⟨𝑢,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+‖‖‖‖𝛼_𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛+1−𝛼_𝑛‖‖‖‖²⎞⎟⎟⎠≤𝛼_𝑛𝜙𝑢,𝑥_𝑛‖𝑢‖²−2𝛼_𝑛,0⟨𝑢,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+𝛼_𝑛,0‖‖𝐽𝑥_𝑛‖‖²+^∞_𝑖=1≤𝛼_𝑛𝛼_𝑛,𝑖‖‖𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛‖𝑢‖²−2𝛼_𝑛,0⟨𝑢,𝐽𝑥_𝑛⟩−2^∞_𝑖=1𝛼_𝑛,𝑖𝑢,𝐽𝑆^𝑛𝑖𝑥_𝑛+𝛼_𝑛,0‖‖𝑥_𝑛‖‖²+^∞_𝑖=1=𝛼_𝑛𝛼_𝑛,𝑖‖𝑆^𝑛𝑖𝑥_𝑛‖²−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛𝛼^𝑛,0𝜙𝑢,𝑥_𝑛+^∞_𝑖=1+1−𝛼_𝑛𝛼_𝑛,𝑖𝜙𝑢,𝑆^𝑛𝑖𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖≤𝛼_𝑛𝜙𝑢,𝑥_𝑛𝛼^𝑛,0𝜙𝑢,𝑥_𝑛+^∞_𝑖=1+1−𝛼_𝑛𝛼_𝑛,𝑖𝑘_𝑛𝜙𝑢,𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖≤𝛼_𝑛𝜙𝑢,𝑥_𝑛≤𝛼_𝑛𝑘_𝑛𝜙𝑢,𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛𝜙𝑢,𝑥^𝑛+sup_𝑢∈𝐺𝑘_𝑛+1−𝛼_𝑛−1𝜙𝑢,𝑥_𝑛−𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖=𝜙𝑢,𝑥_𝑛𝜙𝑢,𝑥_𝑛≤𝜙𝑢,𝑥_𝑛+𝛼_𝑛𝜉_𝑛−𝛼_𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖+𝜉_𝑛.(3.9) Hence 𝑢∈𝐶_𝑛+1, and so 𝐺⊂𝐶_𝑛 for all 𝑛≥0. By the way, from the definition of {𝜉_𝑛}, (2.4), and (3.6), it is easy to see that 𝜉_𝑛=sup_𝑢∈𝐺𝑘_𝑛−1𝜙𝑢,𝑥_𝑛≤sup_𝑢∈𝐺𝑘_𝑛−1(‖𝑢‖+𝑀)²⟶0(as𝑛⟶∞).(3.10)
(IV)Now, we prove that {𝑥_𝑛} converges strongly to some point 𝑝∈𝐺∶=⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂Ω.
First, we prove that {𝑥_𝑛} converges strongly to some point 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖).
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 𝐶_𝑛 is closed and convex for each 𝑛≥1, it is weakly closed, and so 𝑝∈𝐶_𝑛 for each 𝑛≥0. Since 𝑥_𝑛=Π_𝐶𝑛𝑥₀, from the definition of Π_𝐶𝑛, we have 𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀,𝑛≥0.(3.11) Since liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀=≥l‖i𝑝m‖i₂nf_{𝑛𝑖→∞}‖‖𝑥^𝑛_𝑖‖‖²−2𝑥_𝑛𝑖,𝐽𝑥₀+‖‖𝑥₀‖‖²−2⟨𝑝,𝐽𝑥₀⟩+‖‖𝑥₀‖‖²=𝜙𝑝,𝑥₀,(3.12) we have 𝜙𝑝,𝑥₀≤liminf_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤limsup_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑥₀≤𝜙𝑝,𝑥₀.(3.13) 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 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 𝜙(𝑝,𝑞)==lliimm_{𝑛𝑖→∞,𝑛𝑗→∞𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗≤lim_{𝑛𝑖→∞,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖𝑥₀−𝜙Π_𝐶𝑛𝑗𝑥₀,𝑥₀𝜙𝑥_𝑛𝑖,𝑥₀−𝜙𝑥_𝑛𝑗,𝑥₀=0(by(3.8)).(3.14) Therefore we have 𝑝=𝑞. This implies that lim_𝑛→∞𝑥_𝑛=𝑝.(3.15)
Now we prove that 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖). In fact, by the construction of 𝐶_𝑛, we have that 𝐶_𝑛+1⊂𝐶_𝑛 and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀. Therefore, by Lemma 2.2(a) we have 𝜙𝑥_𝑛+1,𝑥_𝑛=≤𝜙𝜙𝑥𝑥_𝑛+1𝑛+1,Π_𝐶𝑛𝑥₀=𝜙𝑥_𝑛+1,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀,𝑥₀−𝜙𝑥_𝑛,𝑥₀⟶0(as𝑛⟶∞).(3.16) In view of 𝑥_𝑛+1∈𝐶_𝑛 and noting the construction of 𝐶_𝑛+1 we obtain that 𝜙𝑥_𝑛+1,𝑢^𝑛(𝑁)≤𝜙𝑥^𝑛+1,𝑥_𝑛+𝜉_𝑛⟶0(as𝑛⟶∞).(3.17) From (2.4) it yields (‖𝑥_𝑛+1‖−‖𝑢^𝑛(𝑁)‖)²→0. Since ‖𝑥_𝑛+1‖→‖𝑝‖, we have ‖‖𝑢^𝑛(𝑁)‖‖⟶‖𝑝‖(as𝑛⟶∞).(3.18)
Hence we have ‖‖𝐽𝑢^𝑛(𝑁)‖‖⟶‖𝐽𝑝‖(as𝑛⟶∞).(3.19)
This implies that {𝐽𝑢^𝑛(𝑁)} is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, there exists a subsequence {𝐽𝑢^𝑛(𝑁)_𝑖}⊂{𝐽𝑢^𝑛(𝑁)} such that 𝐽𝑢^𝑛(𝑁)_𝑖⇀𝑝₀∈𝐸^∗. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑝₀. Since 𝜙𝑥_𝑛𝑖,𝑢^𝑛(𝑁)_𝑖=‖‖𝑥^𝑛_𝑖‖‖²−2𝑥_𝑛𝑖,𝐽𝑢^𝑛(𝑁)_𝑖+‖‖𝑢^{𝑛(𝑁)𝑖}‖‖²=‖‖𝑥_𝑛𝑖‖‖²−2𝑥_𝑛𝑖,𝐽𝑢^𝑛(𝑁)_𝑖+‖‖𝐽𝑢^{𝑛(𝑁)𝑖}‖‖²,(3.20) taking liminf_𝑛→∞ on both sides of above equality and in view of the weak lower semicontinuity of norm ‖⋅‖, it yields that 0≥‖𝑝‖²−2⟨𝑝,𝑝₀⟩+‖‖𝑝₀‖‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑝,𝑥).(3.21) That is, 𝑝=𝑥. This implies that 𝑝₀=𝐽𝑝, and so 𝐽𝑢^𝑛(𝑁)⇀𝐽𝑝. It follows from (3.19) and the Kadec-Klee property of 𝐸^∗ that 𝐽𝑢^𝑛(𝑁)_𝑖→𝐽𝑝 (as 𝑛→∞). Noting that 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak continuous, it yields that 𝑢^𝑛(𝑁)_𝑖⇀𝑝. It follows from (3.18) and the Kadec-Klee property of 𝐸 that lim_{𝑛𝑖→∞}𝑢^𝑛(𝑁)_𝑖=𝑝.
By the similar way as given in the proof of (3.15), we can also prove that lim_𝑛→∞𝑢^𝑛(𝑁)=𝑝.(3.22)
From (3.15) and (3.22) we have that ‖‖𝑥_𝑛−𝑢^𝑛(𝑁)‖‖⟶0(as𝑛⟶∞).(3.23)
Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have ‖‖𝐽𝑥_𝑛−𝐽𝑢^𝑛(𝑁)‖‖⟶0(as𝑛⟶∞).(3.24)
For any 𝑗≥1 and any 𝑢∈𝐺, it follows from (3.9), (3.15), and (3.22) that 𝛼_𝑛𝛼_𝑛,0𝛼_𝑛,𝑗𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢^𝑛(𝑁)+𝛼^𝑛𝜉_𝑛.(3.25)
Since 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢^𝑛(𝑁)≤=‖‖|||‖‖𝑥𝑥^𝑛_𝑛‖‖²−‖‖𝑢^𝑛(𝑁)‖‖²−2𝑢,𝐽𝑥_𝑛−𝐽𝑢^𝑛(𝑁)‖‖²−‖‖𝑢^𝑛(𝑁)‖‖²|||+2‖𝑢‖⋅‖‖𝐽𝑥_𝑛≤‖‖𝑥_𝑛−𝐽𝑢^𝑛(𝑁)‖‖−𝑢^𝑛(𝑁)‖‖‖‖𝑥_𝑛‖‖+‖‖𝑢^𝑛(𝑁)‖‖+2‖𝑢‖⋅‖‖𝐽𝑥_𝑛−𝐽𝑢^𝑛(𝑁)‖‖,(3.26)
From (3.23) and (3.24), it follows that 𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝑢^𝑛(𝑁)⟶0(as𝑛⟶∞).(3.27)
In view of conditions (b) liminf_𝑛→∞𝛼_𝑛,0𝛼_𝑛,𝑗>0 and (c) 𝛼_𝑛≥𝛼>0, we see that 𝑔‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(3.28)
It follows from the property of 𝑔 that ‖‖𝐽𝑥_𝑛−𝐽𝑆^𝑛𝑖𝑥_𝑛‖‖⟶0(as𝑛⟶∞).(3.29)
Since 𝑥_𝑛→𝑝 and 𝐽 is uniformly continuous, it yields 𝐽𝑥_𝑛→𝐽𝑝. Hence from (3.29) we have 𝐽𝑆^𝑛𝑗𝑥_𝑛⟶𝐽𝑝(as𝑛⟶∞).(3.30) Since 𝐽⁻¹∶𝐸^∗→𝐸 is norm-weak continuous, it follows that 𝑆^𝑛𝑗𝑥_𝑛⇀𝑝(foreach𝑗≥1).(3.31) On the other hand, for each 𝑗≥1 we have ||‖‖𝑆^𝑛𝑗𝑥_𝑛‖‖−‖𝑝‖||=||‖‖𝐽𝑆^𝑛𝑗𝑥_𝑛‖‖−‖𝐽𝑝‖||≤‖‖𝐽𝑆^𝑛𝑗𝑥_𝑛−𝐽𝑝‖‖⟶0(as𝑛⟶∞).(3.32) This together with (3.31) shows that 𝑆^𝑛𝑗𝑥_𝑛⟶𝑝(foreach𝑗≥1).(3.33)
Furthermore, by the assumption that for each 𝑗≥1, 𝑆_𝑗 is uniformly 𝐿_𝑖-Lipschitz continuous, hence we have ‖‖𝑆^𝑗𝑛+1𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖‖≤≤‖‖𝐿𝑆^𝑗𝑛+1_𝑗𝑥_𝑛−𝑆^𝑗𝑛+1𝑥_𝑛+1‖‖+‖‖𝑆^𝑗𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖‖+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑆^𝑗𝑛+1𝑥_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖‖.(3.34)
This together with (3.15) and (3.33) yields ‖𝑆^𝑗𝑛+1𝑥_𝑛−𝑆^𝑛𝑗𝑥_𝑛‖→0 (as 𝑛→∞). Hence from (3.33) we have 𝑆^𝑗𝑛+1𝑥_𝑛→𝑝, that is, 𝑆_𝑗𝑆^𝑛𝑗𝑥_𝑛→𝑝. In view of (3.33) and the closeness of 𝑆_𝑗, it yields that 𝑆_𝑗𝑝=𝑝,for all𝑗≥1. This implies that 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖).
Next, we prove that 𝑝∈⋂^𝑁𝑖=1Ω(Θ_𝑖,𝜓). Since 𝑢^𝑛(𝑁)=𝐾^Θ𝑁_𝑟𝑛𝑢^{𝑛(𝑁−1)}, from (2.18), (3.9), and (3.27), we have 𝜙𝑢^𝑛(𝑁),𝑢^{𝑛(𝑁−1)}=≤𝜙𝜙𝑢𝐾,𝑢^{Θ𝑁𝑟𝑛}𝑢^{𝑛(𝑁−1)}_{𝑛(𝑁−1)},𝑢^{𝑛(𝑁−1)}≤𝜙𝑢,𝑥_𝑛−𝜙𝑢,𝐾^{Θ𝑁𝑟𝑛}𝑢^{𝑛(𝑁−1)}=𝜙𝑢,𝑥_𝑛+𝜉_𝑛−𝜙𝑢,𝐾^Θ𝑁_𝑟𝑛𝑢^{𝑛(𝑁−1)}+𝜉_𝑛−𝜙𝑢,𝑢^𝑛(𝑁)⟶0(as𝑛⟶∞).(3.35) From (2.4) it yields (‖𝑢^𝑛(𝑁)‖−‖𝑢^{𝑛(𝑁−1)}‖)²→0. Since ‖𝑢^𝑛(𝑁)‖→‖𝑝‖, we have ‖‖𝑢^{𝑛(𝑁−1)}‖‖⟶‖𝑝‖(as𝑛⟶∞).(3.36)
Hence we have ‖‖𝐽𝑢^{𝑛(𝑁−1)}‖‖⟶‖𝐽𝑝‖(as𝑛⟶∞).(3.37)
This implies that {𝐽𝑢^{𝑛(𝑁−1)}} is bounded in 𝐸^∗. Since 𝐸 is reflexive, and so 𝐸^∗ is reflexive, there exists a subsequence {𝐽𝑢^{𝑛(𝑁−1)}_𝑖}⊂{𝐽𝑢^{𝑛(𝑁−1)}} such that 𝐽𝑢^{𝑛(𝑁−1)}_𝑖⇀𝑝₀∈𝐸^∗. In view of the reflexive of 𝐸, we see that 𝐽(𝐸)=𝐸^∗. Hence there exists 𝑥∈𝐸 such that 𝐽𝑥=𝑝₀. Since 𝜙𝑢^𝑛(𝑁)_𝑖,𝑢^{𝑛(𝑁−1)}_𝑖=‖‖𝑢^{𝑛(𝑁)𝑖}‖‖²−2𝑢^𝑛(𝑁)_𝑖,𝐽𝑢^{𝑛(𝑁−1)}_𝑖+‖‖𝑢^{𝑛(𝑁−1)𝑖}‖‖²=‖‖𝑢^𝑛(𝑁)_𝑖‖‖²−2𝑢^𝑛(𝑁)_𝑖,𝐽𝑢^{𝑛(𝑁−1)}_𝑖+‖‖𝐽𝑢^{𝑛(𝑁−1)𝑖}‖‖²,(3.38) taking liminf_𝑛→∞ on both sides of equality above and in view of the weak lower semicontinuity of norm ‖⋅‖, it yields that 0≥‖𝑝‖²−2⟨𝑝,𝑝₀⟩+‖‖𝑝₀‖‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝐽𝑥‖²=‖𝑝‖²−2⟨𝑝,𝐽𝑥⟩+‖𝑥‖²=𝜙(𝑝,𝑥).(3.39) That is, 𝑝=𝑥. This implies that 𝑝₀=𝐽𝑝, and so 𝐽𝑢^{𝑛(𝑁−1)}_𝑖⇀𝐽𝑝. It follows from (3.37) and the Kadec-Klee property of 𝐸^∗ that 𝐽𝑢^{𝑛(𝑁−1)}→𝐽𝑝 (as 𝑛→∞). Note that 𝐽⁻¹∶𝐸^∗→𝐸 is hemi-continuous, it yields that 𝑢^{𝑛(𝑁−1)}_𝑖⇀𝑝. It follows from (3.36) and the Kadec-Klee property of 𝐸 that lim_{𝑛𝑖→∞}𝑢^{𝑛(𝑁−1)}_𝑖=𝑝.
By the similar way as given in the proof of (3.15), we can also prove that lim_𝑛→∞𝑢^{𝑛(𝑁−1)}=𝑝.(3.40) From (3.22) and (3.40) we have that ‖𝑢^𝑛(𝑁)−𝑢^{𝑛(𝑁−1)}‖⟶0(as𝑛⟶∞).(3.41) Since 𝐽 is uniformly continuous on any bounded subset of 𝐸, we have ‖𝐽𝑢^𝑛(𝑁)−𝐽𝑢^{𝑛(𝑁−1)}‖⟶0(as𝑛⟶∞).(3.42)
Since 𝑢^𝑛(𝑖)=𝐾^Θ𝑖_𝑟𝑛𝑢^{𝑛(𝑖−1)},𝑖=2,3,…,𝑁,𝑢^𝑛(1)=𝐾^Θ1_𝑟𝑛𝑦_𝑛,(3.43)
by the similar way as above, we can also prove that ‖𝐽𝑢‖𝑢^𝑛(𝑖)_𝑛(𝑖)−𝑢^{𝑛(𝑖−1)}‖⟶0,‖−𝑦_𝑛𝐽𝑢^{𝑛(𝑖−1)}‖⟶0,‖𝐽𝑦_𝑛−𝑢^𝑛(1)‖⟶0,−𝐽𝑢^𝑛(1)‖⟶0(as𝑛⟶∞),𝑖=2,3,…,𝑁.(3.44) From (3.44) and the assumption that 𝑟_𝑛≥𝑑,forall𝑛≥0, we have lim_𝑛→∞‖‖𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}‖‖𝑟_𝑛=0,lim_𝑛→∞‖‖𝐽𝑦_𝑛−𝐽𝑢^𝑛(1)‖‖𝑟_𝑛=0,𝑖=2,3,…,𝑁.(3.45) Now, we define functions 𝐻_𝑖∶𝐶×𝐶→ℝ,𝑖=1,2,…,𝑁 by 𝐻_𝑖(𝑥,𝑦)=Θ_𝑖(𝑥,𝑦)+⟨𝐴𝑥,𝑦−𝑥⟩+𝜓(𝑦)−𝜓(𝑥),∀𝑥,𝑦∈𝐶.(3.46) By Lemma 2.9, we know that the function 𝐻_𝑖 satisfies conditions (𝐴₁)-(𝐴₄). Since 𝐻_𝑖𝑢^𝑛(𝑖),𝑦+1𝑟_𝑛𝑦−𝑢^𝑛(𝑖)𝐻,𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}≥0,₁𝑢^𝑛(1),𝑦+𝑟1_𝑛𝑦−𝑢^𝑛(1),𝐽𝑢^𝑛(1)−𝐽𝑦_𝑛≥0,∀𝑦∈𝐶,(3.47) by condition (𝐴₂), we have 1𝑟_𝑛𝑦−𝑢^𝑛(𝑖)1,𝐽𝑢^𝑛(𝑖)−𝐽𝑢^{𝑛(𝑖−1)}≥−𝐻^𝑖𝑢^𝑛(𝑖),𝑦≥𝐻_𝑖𝑦,𝑢^𝑛(𝑖),𝑟_𝑛𝑦−𝑢^𝑛(1),𝐽𝑢^𝑛(1)−𝐽𝑦_𝑛≥−𝐻¹𝑢^𝑛(1),𝑦≥𝐻₁𝑦,𝑢^𝑛(1),∀𝑦∈𝐶.(3.48) By the assumption that 𝑦↦𝐻_𝑖(𝑥,𝑦) is convex and lower semicontinuous, letting 𝑛→∞ in (3.48), from (3.45) and (3.48), for each 𝑖, we have 𝐻_i(𝑦,𝑝)≤0, for all 𝑦∈𝐶.
For 𝑡∈(0,1] and 𝑦∈𝐶, letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑝, there are 𝑦_𝑡∈𝐶 and 𝐻_𝑖(𝑦_𝑡,𝑝)≤0. By conditions (𝐴₁) and (𝐴₄), we have 0=𝐻_𝑖𝑦_𝑡,𝑦_𝑡≤𝑡𝐻_𝑖𝑦_𝑡,𝑦+(1−𝑡)𝐻_𝑖𝑦_𝑡,𝑝≤𝑡𝐻_𝑖𝑦_𝑡,𝑦.(3.49) Dividing both sides of the above equation by 𝑡, we have 𝐻_𝑖(𝑦_𝑡,𝑦)≥0,for all 𝑦∈𝐶. Letting 𝑡↓0, from condition (𝐴₃), we have 𝐻_𝑖(𝑝,𝑦)≥0, for all 𝑦∈𝐶, that is, 𝑝∈⋂^𝑁𝑖=1Ω(Θ_𝑖,𝜓), that is, 𝑝∈Ω, and 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂Ω.
(V)Now, we prove 𝑥_𝑛→Π_⋂^∞𝑖=1_{𝐹(𝑆𝑖)⋂Ω}𝑥₀.
Let 𝑤=Π_⋂^∞𝑖=1_{𝐹(𝑆𝑖)⋂Ω}𝑥₀. From 𝑤∈⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂Ω⊂𝐶_𝑛+1 and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀, we have 𝜙(𝑥_𝑛+1,𝑥₀)≤𝜙(𝑤,𝑥₀), for all 𝑛≥0. This implies that 𝜙𝑝,𝑥₀=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤𝜙𝑤,𝑥₀.(3.50)
By the definition of Π_⋂^∞𝑖=1_{𝐹(𝑆𝑖)⋂Ω}𝑥₀ and (3.50), we have 𝑝=𝑤. Therefore, 𝑥_𝑛→Π_⋂^∞𝑖=1_{𝐹(𝑆𝑖)⋂Ω}𝑥₀. This completes the proof of Theorem 3.1.

Theorem 3.2. Let 𝐸 be a uniformly smooth and strictly convex Banach space with Kleac-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_𝑖∶𝐶×𝐶→ℝ,𝑖=1,2,…,𝑁} a finite family of bifunction satisfying conditions (𝐴₁)-(𝐴₄). Let {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of closed and quasi-𝜙-nonexpansive mappings. Suppose that 𝐺∶=⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂Ω≠∅, where Ω=⋂^𝑁𝑖=1Ω(Θ_𝑖,𝜓). Let {𝑥_𝑛}, {𝑦_𝑛}, {𝑧_𝑛}, and {𝑢^𝑛(𝑘)}, 𝑘=1,2,…,𝑁, be the sequences generated by 𝑧𝑥₀∈𝐶,𝐶₀=𝐶,_𝑛=𝐽⁻¹𝛼^𝑛,0𝐽𝑥_𝑛+^∞_𝑖=1𝛼_𝑛,𝑖𝐽𝑆^𝑛𝑖𝑥_𝑛𝑦_𝑛,𝑢_𝑛(𝑁)=𝐽⁻¹𝛼_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,Θ∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡,∀𝑦∈𝐶,_𝑁𝑢^𝑛(𝑁),𝑦+Θ𝐴𝑢^𝑛(𝑁),𝑦−𝑢^𝑛(𝑁)+𝜓(𝑦)−𝜓𝑢^𝑛(𝑁)+𝑟1_𝑛𝑦−𝑢^𝑛(𝑁),𝐽𝑢^𝑛(𝑁)−𝐽𝑢^{𝑛(𝑁−1)}≥0,_𝑁−1𝑢+^{𝑛(𝑁−1)},𝑦1+𝐴𝑢^{𝑛(𝑁−1)},𝑦−𝑢^{𝑛(𝑁−1)}+𝜓(𝑦)−𝜓𝑢^{𝑛(𝑁−1)}𝑟_𝑛𝑦−𝑢^{𝑛(𝑁−1)}Θ,𝐽𝑢⋮^{𝑛(𝑁−1)}−𝐽𝑢^{𝑛(𝑁−2)}≥0,₁𝑢^𝑛(1),𝑦+𝐶𝐴𝑢^𝑛(1),𝑦−𝑢^𝑛(1)+𝜓(𝑦)−𝜓𝑢^𝑛(1)+𝑟1_𝑛𝑦−𝑢^𝑛(1),𝐽𝑢^𝑛(1)−𝐽𝑦_𝑛≥0,_𝑛+1=𝑣∈𝐶_𝑛∶𝜙𝑥𝑣,𝑢^𝑛(𝑁)≤𝜙𝑣,𝑥^𝑛,∀𝑛≥0,_𝑛+1=Π_𝐶𝑛+1𝑥₀,(3.51) where 𝑟_𝑛∈[𝑑,∞) for some 𝑑>0 and for 𝑖≥0,{𝛼_𝑛,𝑖},{𝛼_𝑛} are sequences in [0,1] satisfying the following conditions: (a)∑^∞𝑖=0𝛼_𝑛,𝑖=1, for all 𝑛≥0;(b)liminf_𝑛→∞𝛼_𝑛,0⋅𝛼_𝑛,𝑖>0 for all 𝑖≥1;(c)0<𝛼≤𝛼_𝑛<1 for some 𝛼∈(0,1).Then {𝑥_𝑛} converges strongly to Π_𝐺𝑥₀.

Proof. Since {𝑆_𝑖}^∞𝑖=1∶𝐶→𝐶 is an infinite family of closed quasi-𝜙-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings with sequence {𝑘_𝑛=1}. Hence 𝜉_𝑛=sup_𝑢∈𝐺(𝑘_𝑛−1)𝜙(𝑢,𝑥_𝑛)=0. Therefore, the conditions appearing in Theorem 3.1—“𝐺 is bounded subset in 𝐶” and for each 𝑖≥1,𝑆_𝑖 is uniformly 𝐿_𝑖-Lipschitz continuous— are of no use here. In fact, by the same methods as given in the proofs of (3.15), (3.22), (3.31), (3.37), and (3.42), we can prove that 𝑥_𝑛→𝑝, 𝑢^𝑛(𝑖)→𝑝, 𝑦_𝑛→𝑝, and 𝑆_𝑗𝑥_𝑛→𝑝 (as 𝑛→∞), for each 𝑗≥1, that is, 𝑝∈⋂^∞𝑖=1𝐹(𝑆_𝑖)⋂Ω. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 is obtained from Theorem 3.1 immediately.

Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results in [10–12, 18].(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 Kadec-Klee property).(b)For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or quasi-𝜙-nonexpansive mapping to an infinite family of quasi-𝜙-asymptotically nonexpansive mappings.(c)We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.