Implicit Iterative Scheme for a Countable Family of Nonexpansive Mappings in 2-Uniformly Smooth Banach Spaces

Abstract

Implicit Mann process and Halpern-type iteration have been extensively studied by many others. In this paper, in order to find a common fixed point of a countable family of nonexpansive mappings in the framework of Banach spaces, we propose a new implicit iterative algorithm related to a strongly accretive and Lipschitzian continuous operator 𝐹∶𝑥_𝑛=𝛼_𝑛𝛾𝑉(𝑥_𝑛)+𝛽_𝑛𝑥_𝑛−1+((1−𝛽_𝑛)𝐼−𝛼_𝑛𝜇𝐹)𝑇_𝑛𝑥_𝑛 and get strong convergence under some mild assumptions. Our results improve and extend the corresponding conclusions announced by many others.

1. Introduction

Let 𝑋 be a real 𝑞-uniformly smooth Banach space with induced norm ‖⋅‖, 𝑞>1. Let 𝑋^∗ be the dual space of 𝑋. Let 𝐽_𝑞 denote the generalized duality mapping from 𝑋 into 2^𝑋∗ given by 𝐽_𝑞(𝑥)={𝑓∈𝑋^∗∶⟨𝑥,𝑓⟩=‖𝑥‖^𝑞,‖𝑓‖=‖𝑥‖^𝑞−1,𝑥∈𝑋}. In our paper, we consider real 2-uniformly smooth Banach spaces, that is, 𝑞=2, so the normalized duality mapping is 𝐽(𝑥)={𝑓∈𝑋^∗∶⟨𝑥,𝑓⟩=‖𝑥‖²,‖𝑓‖=‖𝑥‖,𝑥∈𝑋}. If 𝑋 is smooth, then 𝐽 is single valued. Throughout this paper, we use Fix(𝑇) to denote the fixed points set of the mapping 𝑇.

In what follows, we write 𝑥_𝑛⇀𝑥 to indicate that the sequence converges weakly to 𝑥. 𝑥_𝑛→𝑥 implies that the sequence converges strongly to 𝑥.

Given a nonlinear operator Γ∶𝑋→𝑋, it is well-known that the generalized variational inequality problem VIP(Γ,𝑋) over 𝑋 is to find a 𝑥^∗∈𝑋, such that Γ𝑥^∗,𝑗𝑥−𝑥^∗≥0,∀𝑥∈𝑋. (1) Scholars mainly proposed iterative algorithms to solve the generalized variational inequalities, and some of them focused on the existence of the solutions of generalized variational inequalities; see [1, 2] and references therein.

Variational inequalities are developed from operator equations and have been playing an essential role in management science, mechanics, and finance. As for mathematics, variational inequality problems mainly originate from partial differential equations, optimization problems; see [3–7] and references therein.

Definition 1. A mapping 𝑇 is said to be(1)𝜂-strongly accretive if for each 𝑥,𝑦∈𝑋, there exists a 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) and 𝜂>0, such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≥𝜂𝑥−𝑦^𝑞; (2)(2)𝐿-Lipschitzian continuous if for each 𝑥,𝑦∈𝑋, there exists a constant 𝐿>0, such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦. (3)
In particular, 𝑇 is called nonexpansive if 𝐿=1; it is said to be contractive if 𝐿<1.
Yamada [3] introduced the hybrid steepest descent method: 𝑥_𝑛+1=𝐼−𝜇𝜆_𝑛𝐹𝑇𝑥_𝑛,∀𝑛≥0, (4) where 𝑇 is a nonexpansive mapping in Hilbert spaces. Under some appropriate conditions, Yamada [3] proved that the sequence {𝑥_𝑛} generated by (4) converges strongly to the unique solution 𝑥^∗∈Fix(𝑇) of the variational inequality: VIP(𝐹,𝑋)∶⟨𝐹𝑥^∗,𝑥−𝑥^∗⟩≥0, for all 𝑥∈Fix(𝑇).
Moudafi [4] introduced the classical viscosity approximation method for nonexpansive mappings and defined a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (5) where {𝛼_𝑛} is a sequence in (0,1). Xu [6] proved that under certain appropriate conditions on {𝛼_𝑛}, the sequence {𝑥_𝑛} generated by (5) converges strongly to the unique solution 𝑥^∗∈𝐶 of the variational inequality: ⟨(𝐼−𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0, for all 𝑥∈Fix(𝑇), (where 𝐶=Fix(𝑇)) in Hilbert spaces as well as in some Banach spaces.
Marino and Xu [7] considered the following general iterative method in Hilbert spaces: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,∀𝑛≥0, (6) where 𝐴 is a strongly positive bounded linear operator. It is proved that if the sequence {𝛼_𝑛} satisfies appropriate conditions, the sequence {𝑥_𝑛} generated by (6) converges strongly to the unique solution 𝑥∈𝐶 of the variational inequality: ⟨(𝛾𝑓−𝐴)𝑥,𝑥−𝑥⟩≤0, for all 𝑥∈𝐶, where 𝐶 is the fixed points set of a nonexpansive mapping 𝑇.
Tian [8] considered the following general iterative algorithm (GIA) in Hilbert spaces: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛,∀𝑛≥0. (7) It is proved that if the sequence {𝛼_𝑛} satisfies appropriate conditions, the sequence {𝑥_𝑛} generated by (7) converges strongly to the unique solution 𝑥∈Fix(𝑇) of the variational inequality: ⟨(𝛾𝑓−𝜇𝐹)𝑥,𝑥−𝑥⟩≤0, for all 𝑥∈Fix(𝑇).
In 2001, Soltuz [9] introduced the following backward Mann scheme iteration: 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥1, (8) where 𝑇 is a nonexpansive mapping and got strong convergence in Hilbert spaces.
In order to find a common fixed point of a finite family of nonexpansive mappings {𝑇_𝑖∶𝑖∈𝐽}, where 𝐽 stands for {1,2,3,…,𝑁}, in 2001, Xu and Ori [10] introduced the following implicit process: 𝑥_𝑛=𝛼_𝑛𝑥_𝑛−1+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,∀𝑛≥1, (9) where 𝑇_𝑛=𝑇_{𝑛(mod𝑁)}, and a weak convergence is obtained in real Hilbert spaces.
Ceng et al. [11] introduced an iterative algorithm to find a common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces with a weak sequentially continuous duality mapping, which satisfy the uniformly asymptotical regularity condition: 𝑥_𝑛+1=1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛+𝛼_𝑛𝑓𝑦_𝑛+𝛽_𝑛𝑇_𝑟𝑛𝑦_𝑛,𝑦_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇_𝑟𝑛𝑥_𝑛. (10) Under some appropriate conditions one the parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and the sequence {𝑥_𝑛} generated by (10) converges strongly to the approximate solution of a variational inequality problem.
In order to find a common element of the solution set of a general system of variational inequalities and the fixed-point set of the mapping 𝑆, Ceng et al. [12] constructed a new relaxed extragradient iterative method: 𝑢_𝑛=𝑃_𝐶𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝜇₁𝐵₁𝑃_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,𝑧_𝑛=𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴𝑢_𝑛,𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑃_𝐶𝑢_𝑛−𝜆_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑦_𝑛+𝛿_𝑛𝑆𝑦_𝑛,∀𝑛≥0. (11) Under mild assumptions, they obtained a strong convergence theorem.
Yao et al. [13] introduced the following Halpern-type implicit iterative method where 𝑇 is a continuous pseudocontraction: 𝑥_𝑛=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑇𝑥_𝑛,∀𝑛≥1 (12) and obtained a strong convergence theorem in Banach spaces.
Hu [14] introduced an iteration for a nonexpansive mapping in Banach spaces, which guarantee a uniformly Gêteaux differentiable norm as follows: 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (13) and several strong convergent theorems are obtained.
Very recently, Jung [15] proposed an iterative process in the frame of Hilbert spaces as follows: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑃_𝐶𝑆𝑥_𝑛,∀𝑛≥0, (14) where 𝑆 is a mapping defined by 𝑆𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥 and 𝑇 is a 𝑘-strictly pseudocontraction. Strong convergence theorems are established.
Motivated and inspired by Soltuz [9], Xu and Ori [10], Ceng et al. [11], Ceng et al. [12], Yao et al. [13], Hu [14], and Jung [15], we consider the following new implicit iteration in real 2-uniformly smooth Banach spaces: 𝑥_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1+1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛,∀𝑛≥1, (15) where {𝛼_𝑛} and {𝛽_𝑛} are real sequences in (0,1), 𝑉 is an 𝐿-Lipschitzian continuous with Lipschitzian constant 𝐿>0, 𝐹 is an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous mapping with 𝜅>0 and 𝜂>0, and {𝑇_𝑖}^∞_𝑖=1 is a countable family of nonexpansive mappings.
In this paper, we prove that the implicit iterative process (15) has strong convergence and find the unique solution 𝑥 of variational inequality: 𝜇𝐹−𝛾𝑉𝑥,𝑗(𝑥−𝑥)≥0,∀𝑥∈𝑆=^∞_𝑛=1Fix𝑇_𝑛≠∅. (16) Our results improve and extend the corresponding conclusions announced by many others.

2. Preliminaries

Let 𝑆_𝑋={𝑥∈𝑋∶‖𝑥‖=1}. Then the norm of 𝑋 is said to be Gâteaux differentiable if Δ=lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (17) exists for each 𝑥,𝑦∈𝑆_𝑋. In this case, 𝑋 is said to be smooth. The norm of 𝑋 is called uniformly Gâteaux differentiable, if for each 𝑦∈𝑆_𝑋, Δ is attained uniformly for 𝑥∈𝑆_𝑋. The norm of 𝑋 is called Fréchet differentiable, if for each 𝑥∈𝑆_𝑋, Δ is attained uniformly for 𝑦∈𝑆_𝑋. The norm of 𝑋 is called uniformly Fréchet differentiable, if Δ is attained uniformly for 𝑥,𝑦∈𝑆_𝑋. It is well known that (uniformly) Fréchet differentiability of the norm of 𝑋 implies (uniformly) Gâteaux differentiability of the norm of 𝑋. If the norm on 𝑋 is uniformly Gâteaux differentiable, the generalized duality mapping 𝐽_𝑞 is single-valued and strong-weak^∗ uniformly continuous on any bounded subsets of 𝑋.

Let 𝜌_𝑋∶[0,∞)→∞ be the modulus of smoothness of 𝑋 defined by 𝜌_𝑋(𝑡)1=sup2𝑥+𝑦+𝑥−𝑦−1∶𝑥∈𝑆_𝑋,𝑦≤𝑡. (18) A Banach space 𝑋 is said to be uniformly smooth if 𝜌_𝑋(𝑡)/𝑡→0 as 𝑡→0. A Banach space is said to be 𝑞-uniformly smooth, if there exists a fixed constant 𝑐>0, such that 𝜌_𝑋(𝑡)≤𝑐𝑡^𝑞. It is well known that the 𝑋 is uniformly smooth if and only if the norm of 𝑋 is uniformly Fréchet differentiable [16].

The so-called gauge function 𝜑 is defined as follows: let 𝜑∶[0,∞)∶=ℝ⁺→ℝ⁺ be a continuous strictly increasing function, such that 𝜑(0)=0 and 𝜑(𝑡)→∞ as 𝑡→∞. The duality mapping 𝐽_𝜑∶𝑋→2^𝑋∗ associated with a gauge function 𝜑 is defined by 𝐽_𝜑(𝑥)=𝑓∈𝑋^∗∶𝑥,𝑓=‖𝑥‖𝜑(‖𝑥‖),𝑓=𝜑(‖𝑥‖),𝑥∈𝑋. (19) It is known that real 2-uniformly smooth Banach spaces have a weakly continuous duality mapping with a guage function 𝜑(𝑡)=𝑡, which is the same as the normalized duality mapping 𝐽. Set Φ(𝑡)=∫^𝑡₀𝜑(𝜏)𝑑𝜏, for all 𝑡≥0, then 𝐽_𝜑(𝑥)=𝜕Φ(‖𝑥‖), where 𝜕 denotes the subdifferential in the sense of convex analysis. In fact, for 0≤𝑘≤1, we have 𝜑(𝑘𝑡)≤𝜑(𝑡) and Φ(𝑘𝑡)=₀^𝑘𝑡𝜑(𝜏)𝑑𝜏=𝑘^𝑡₀𝜑(𝑘𝜏)𝑑𝜏≤𝑘^𝑡₀𝜑(𝜏)𝑑𝜏=𝑘Φ(𝑡). (20)

Lemma 2 (see [17]). Let 𝑋 be a real 𝑞-uniformly smooth Banach space for some 𝑞>1, then there exists some positive constant 𝑑_𝑞, such that 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝑗_𝑞(𝑥)+𝑑_𝑞𝑦^𝑞,∀𝑥,𝑦∈𝑋,𝑗_𝑞(𝑥)∈𝐽_𝑞(𝑥), (21) in particular, if 𝑋 is a real 2-uniformly smooth Banach space, then there exists a best smooth constant 𝐾>0, such that 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗(𝑥)+2𝐾𝑦²,∀𝑥,𝑦∈𝑋,𝑗_𝑞(𝑥)∈𝐽_𝑞(𝑥). (22)

Lemma 3 (see [18]). Assume that a Banach space 𝑋 has a weakly continuous duality mapping 𝐽_𝑞:(i) for all 𝑥,𝑦∈𝑋, the following inequality holds: Φ𝑥+𝑦≤Φ(‖𝑥‖)+𝑦,𝑗_𝑞𝑥+𝑦, (23)  in particular, for all 𝑥,𝑦∈𝑋, there holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦; (24)(ii) assume that a sequence {𝑥_𝑛}⊂𝑋 converges weakly to a point 𝑥∈𝑋, then the following equation holds: limsup_𝑛→∞Φ𝑥_𝑛−𝑦=limsup_𝑛→∞Φ𝑥_𝑛−𝑥+Φ𝑦−𝑥,∀𝑥,𝑦∈𝑋. (25)

Lemma 4 (see [19]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers, such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,∀𝑛≥0, (26) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ, such that (a)∑^∞_𝑛=0𝛾_𝑛=∞; (b)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=0|𝛾_𝑛𝛿_𝑛|<∞.
Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 5. Let 𝑋 be a real 2-uniformly smooth Banach space. Let 𝑇 be a nonexpansive mapping over 𝑋, and let 𝐹∶𝑋→𝑋 be an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous mapping with 𝜅>0 and 𝜂>0. For 0<𝑡<𝜎≤1 and 𝜇∈(0,min{1,𝜂/𝐾²𝜅²}), set 𝜏=𝜇(𝜂−𝜇𝐾²𝜅²), and define a mapping 𝑇^𝑡∶𝑋→𝑋 by 𝑇^𝑡∶=𝜎𝐼−𝑡𝜇𝐹. Then 𝑇^𝑡 is a contraction on 𝑋; that is, ‖𝑇^𝑡𝑥−𝑇^𝑡𝑦‖≤(𝜎−𝑡𝜏)‖𝑥−𝑦‖.

Proof. From 0<𝜇<𝜂/𝐾²𝜅², we have 𝜂−𝜇𝐾²𝜅²>0. Setting 𝜂<1/2, we have 0<2(𝜂−𝜇𝐾²𝜅²)<1. For each 𝑥,𝑦∈𝑋, by Lemma 2, we have 𝑇^𝑡𝑥−𝑇^𝑡𝑦²=𝜎(𝑥−𝑦)−𝑡𝜇(𝐹𝑥−𝐹𝑦)²≤𝜎²𝑥−𝑦²−𝜎2𝑡𝜇𝐹𝑥−𝐹𝑦,𝑗𝑥−𝑦+2𝐾²𝑡²𝜇²𝐹𝑥−𝐹𝑦²≤𝜎²𝑥−𝑦²−2𝜎𝑡𝜇𝜂𝑥−𝑦²+2𝐾²𝑡²𝜇²𝜅²𝑥−𝑦²≤𝜎²−2𝜎𝑡𝜇𝜂−𝜇𝐾²𝜅²𝑥−𝑦²=𝜎²−2𝜎𝑡𝜏𝑥−𝑦²≤(𝜎−𝑡𝜏)²𝑥−𝑦². (27)
Hence, it implies that 𝑇^𝑡𝑥−𝑇^𝑡𝑦≤(𝜎−𝑡𝜏)𝑥−𝑦. (28) This completes the proof.

To deal with a family of mappings, we will introduce the following concept called the AKTT condition.

Definition 6 (see [20]). Let 𝑋 be a real Banach space, let 𝐶 be a nonempty subset of 𝑋, and let {𝑇_𝑛}^∞_𝑛=1 be a countable family of mappings of 𝐶 with ⋂^∞_𝑛=1Fix(𝑇_𝑛)≠∅. Then {𝑇_𝑛} is said to satisfy the AKTT condition, if for any bounded subset 𝐷 of 𝐶, the following inequality holds: sup𝑇^∞_𝑛=1𝑛+1𝑥−𝑇_𝑛𝑥∶𝑥∈𝐷<∞. (29)

Lemma 7 (see [20]). Let 𝑋 be a Banach space, let 𝐶 be a nonempty closed subset of 𝑋, and let {𝑇_𝑛} be a family of self-mappings of 𝐶 satisfying the AKTT condition. Then for each 𝑥∈𝐶, {𝑇_𝑛𝑥} converges strongly to a point in 𝐶. Moreover, let the mapping 𝑇 be defined by 𝑇𝑥=lim𝑇_𝑛𝑥,∀𝑥∈𝐶. (30) Then for any bounded subset 𝐷 of 𝐶, the following equality holds: limsup_𝑛→∞sup𝑇𝑥−𝑇_𝑛𝑥∶𝑥∈𝐷=0. (31)

Lemma 8 (see [1]). Suppose that 𝑞>1. Then the following inequality holds: 1𝑎𝑏≤𝑞𝑎^𝑞+𝑞−1𝑞𝑏^{𝑞/(𝑞−1)}, (32) for arbitrary positive real numbers 𝑎 and 𝑏.

3. Main Results

In order to obtain the main results, we divide this section into 3 parts. In Proposition 9, we give the path convergence. In Proposition 10, under the demiclosed assumption and combined with Proposition 9, we find the unique solution of a variational inequality. In Theorem 11, we prove that the sequence {𝑥_𝑛} defined by the implicit scheme (15) converges strongly to the unique solution of (16).

Throughout this paper, we assume that 𝑋 is a real 2-uniformly smooth Banach space, which guarantees a weakly continuous duality 𝐽 as proposed in Section 1.

Proposition 9 (the path convergence). Let 𝑇∶𝑋→𝑋 be a nonexpansive mapping with Fix(𝑇)≠∅, and let 𝑉∶𝑋→𝑋 be an 𝐿-Lipschitzian continuous mapping with Lipschitzian constant 𝐿>0. 𝐹∶𝑋→𝑋 is an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous mapping with 𝜅>0 and 𝜂>0. For 𝑡∈(0,1), let 𝜇∈(0,min{1,𝜂/𝐾²𝜅²}), and set 𝜏=𝜇(𝜂−𝜇𝐾²𝜅²) and 0<𝛾<𝜏/𝐿. Then assume that {𝑥_𝑡} is defined by 𝑥_𝑡=𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑥_𝑡. (33) Then {𝑥_𝑡} converges strongly as 𝑡→0⁺ to a fixed point 𝑥 of  𝑇, which is the unique solution of the variational inequality VIP: 𝜇𝐹−𝛾𝑉𝑥,𝑗(𝑥−𝑥)≥0,∀𝑥∈Fix(𝑇). (34)

Proof. Consider a mapping 𝑆_𝑡 on 𝑋 defined by 𝑆_𝑡𝑥=𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑥_𝑡,∀𝑥∈𝑋. (35) It is easy to see that 𝑆_𝑡 is a contraction. Indeed, for any 𝑥,𝑦∈𝑋, by Lemma 5, we have 𝑆_𝑡𝑥−𝑆_𝑡𝑦𝑉(𝑥)𝑦+≤.≤𝑡𝛾−𝑉𝐼−𝑡𝜇𝐹𝑇𝑥−𝐼−𝑡𝜇𝐹𝑇𝑦1−𝑡𝜏−𝛾𝐿𝑥−𝑦 (36) Hence, by the Banach contraction mapping principle, 𝑆_𝑡 has a unique fixed point, denoted by 𝑥_𝑡, which uniquely solves the fixed point equation (33).
We divided the proof into several steps.
Step 1. We show the uniqueness of the solution of the variational inequality (34). Assume that both 𝑥₁∈Fix(𝑇) and 𝑥₂∈Fix(𝑇) are solutions of the variational inequality (34), then we have 𝜇𝐹−𝛾𝑉𝑥₁,𝑗_𝑞𝑥₂−𝑥₁𝑥≥0,𝜇𝐹−𝛾𝑉₂,𝑗𝑥₁−𝑥₂≥0. (37) Adding up (37) yields 𝜇𝐹−𝛾𝑉𝑥₁−𝜇𝐹−𝛾𝑉𝑥₂,𝑗𝑥₂−𝑥₁≥0. (38) Indeed, from the given conditions 𝜇∈(0,min{1,𝜂/𝐾²𝜅²}), and 𝜏=𝜇(𝜂−𝜇𝐾²𝜅²), 0<𝛾<𝜏/𝐿, we have 𝜇𝐹−𝛾𝑉𝑥₁−𝜇𝐹−𝛾𝑉𝑥₂,𝑗𝑥₁−𝑥₂=𝜇𝐹𝑥₁−𝐹𝑥₂,𝑗𝑥₁−𝑥₂−𝛾𝑉𝑥₁−𝑉𝑥₂,𝑗𝑥₁−𝑥₂≥𝜇𝜂𝑥₁−𝑥₂²−𝛾𝐿𝑥₁−𝑥₂²=𝜇𝜂−𝛾𝐿𝑥₁−𝑥₂²≥0. (39)
Thus, we conclude that 𝑥₁=𝑥₂. So the uniqueness of the variational inequality (35) is guaranteed.
Step 2. We show that {𝑥_𝑡} is bounded. Taking 𝑝∈Fix(𝑇), it follows from Lemma 5 that 𝑥_𝑡−𝑝²=𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑥_𝑡−𝑝,𝑗𝑥_𝑡𝑥−𝑝=𝑡𝛾𝑉_𝑡−𝜇𝐹𝑝,𝑗𝑥_𝑡+−𝑝𝐼−𝑡𝜇𝐹𝑇𝑥_𝑡−𝐼−𝑡𝜇𝐹𝑝,𝑗𝑥_𝑡𝑥−𝑝≤𝑡𝛾𝑉_𝑡−𝛾𝑉𝑝+𝛾𝑉𝑝−𝜇𝐹𝑝,𝑗𝑥_𝑡+−𝑝𝐼−𝑡𝜇𝐹𝑇𝑥_𝑡−𝐼−𝑡𝜇𝐹𝑝𝑥_𝑡𝑥−𝑝≤𝑡𝛾𝐿_𝑡−𝑝²+𝑡𝛾𝑉𝑝−𝜇𝐹𝑝𝑥_𝑡+(𝑥−𝑝1−𝑡𝜏)_𝑡−𝑝²≤1−𝑡𝜏−𝛾𝐿𝑥_𝑡−𝑝²+𝑡𝛾𝑉𝑝−𝜇𝐹𝑝𝑥_𝑡−𝑝. (40) It follows that 𝑥_𝑡≤1−𝑝𝑝.𝜏−𝛾𝐿𝛾𝑉−𝜇𝐹𝑝 (41) Hence {𝑥_𝑡} is bounded, so are {𝑉(𝑥_𝑡)} and {𝐹𝑇𝑥_𝑡)}.
Step 3. Next, we will show that {𝑥_𝑡} has a subsequence converging strongly to 𝑥^∗∈Fix(𝑇).
Assume 𝑡_𝑛→0, and set 𝑥_𝑛∶=𝑥_𝑡𝑛. By the definition of {𝑥_𝑛}, we have 𝑥_𝑛−𝑇𝑥_𝑛=𝑡_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇𝑥_𝑛⟶0. (42)
Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} converging weakly to 𝑥^∗∈𝑋 as 𝑘→∞.
Set 𝑥_𝑛∶=𝑥_𝑛𝑘. Define a mapping 𝐵∶𝑋→ℝ by 𝐵(𝑥)=limsup_𝑛→∞Φ𝑥_𝑛−𝑥,∀𝑥∈𝑋. (43) Again, 𝐽 is weakly continuous, by Lemma 3, and it follows that 𝐵𝑦=𝐵(𝑥)+Φ𝑦−𝑥^∗,∀𝑦∈𝑋. (44) From (42), we have 𝐵𝑇𝑥^∗=limsup_𝑛→∞Φ𝑥_𝑛−𝑇𝑥^∗=limsup_𝑛→∞Φ𝑇𝑥_𝑛−𝑇𝑥^∗+𝑥_𝑛−𝑇𝑥_𝑛=limsup_𝑛→∞Φ𝑇𝑥_𝑛−𝑇𝑥^∗≤limsup_𝑛→∞Φ𝑥_𝑛−𝑥^∗=𝐵𝑥^∗, (45) and we also note that 𝐵𝑇𝑥^∗=limsup_𝑛→∞Φ𝑥_𝑛−𝑥^∗+Φ𝑇𝑥^∗−𝑥^∗=𝐵𝑥^∗+Φ𝑇𝑥^∗−𝑥^∗, (46) so, we obtain Φ𝑇𝑥^∗−𝑥^∗≤0. (47) This implies that 𝑇𝑥^∗=𝑥^∗; that is, 𝑥^∗∈Fix(𝑇).
By Lemma 5, we have 𝑥_𝑛−𝑥^∗2=𝑡_𝑛(𝛾𝑉(𝑥_𝑛)−𝜇𝐹𝑥^∗)+(𝐼−𝑡_𝑛𝜇𝐹)𝑇𝑥_𝑛−(𝐼−𝑡_𝑛𝜇𝐹)𝑥^∗2=𝐼−𝑡_𝑛𝜇𝐹𝑇𝑥_𝑛−𝐼−𝑡_𝑛𝜇𝐹𝑥^∗,𝑗𝑥_𝑛−𝑥^∗+𝑡_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑥^∗,𝑗𝑥_𝑛−𝑥^∗≤1−𝑡_𝑛𝜏𝑥_𝑛−𝑥^∗2+𝑡_𝑛𝛾𝑉𝑥_𝑛−𝛾𝑉𝑥^∗,𝑗𝑥_𝑛−𝑥^∗+𝑡_𝑛𝛾𝑉𝑥^∗−𝜇𝐹𝑥^∗,𝑗𝑥_𝑛−𝑥^∗≤1−𝑡_𝑛𝜏𝑥_𝑛−𝑥^∗2+𝑡_𝑛𝛾𝐿𝑥_𝑛−𝑥^∗2+𝑡_𝑛𝛾𝑉𝑥^∗−𝜇𝐹𝑥^∗,𝑗𝑥_𝑛−𝑥^∗=1−𝑡_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥^∗2+𝑡_𝑛𝛾𝑉𝑥^∗−𝜇𝐹𝑥^∗,𝑗𝑥_𝑛−𝑥^∗. (48) This implies that 𝑥_𝑛−𝑥^∗2≤1𝑥𝜏−𝛾𝐿𝛾𝑉^∗−𝜇𝐹𝑥^∗,𝑗𝑥_𝑛−𝑥^∗. (49) Since {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} satisfying 𝑥_𝑛𝑘−𝑥^∗2≤1𝑥𝜏−𝛾𝐿𝛾𝑉^∗−𝜇𝐹𝑥^∗,𝑗𝑥_𝑛𝑘−𝑥^∗. (50) Since the mapping 𝐽 is single-valued and weakly continuous, it follows from (50) that ‖𝑥_𝑛𝑘−𝑥^∗‖²→0 as 𝑘→∞. Thus, there exists a subsequence, such that 𝑥_𝑛𝑘→𝑥^∗.
Step 4. Finally, we show that 𝑥^∗ is the unique solution of variational inequality (34).
Since 𝑥_𝑡=𝑡𝛾𝑉(𝑥_𝑡)+(𝐼−𝑡𝜇𝐹)𝑇𝑥_𝑡, we can derive that 𝜇𝐹−𝛾𝑉𝑥_𝑡1=−𝑡(𝐼−𝑇)𝑥_𝑡+𝜇𝐹𝑥_𝑡−𝐹𝑇𝑥_𝑡. (51)
It follows that, for any 𝑥∈Fix(𝑇), 𝜇𝐹−𝛾𝑉𝑥_𝑡,𝑗𝑥_𝑡1−𝑥=−𝑡(𝐼−𝑇)𝑥_𝑡−(𝐼−𝑇)𝑥,𝑗𝑥_𝑡−𝑥+𝜇𝐹𝑥_𝑡−𝐹𝑇𝑥_𝑡,𝑗𝑥_𝑡−𝑥. (52)
Since 𝑇 is a nonexpansive mapping, for all 𝑥,𝑦∈𝑋, we conclude that )()≥(𝐼−𝑇𝑥−𝐼−𝑇𝑦,𝑗𝑥−𝑦=⟨𝑥−𝑦,𝑗𝑥−𝑦⟩−𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦𝑥−𝑦²−𝑥−𝑦²=0. (53)
Now replacing 𝑡 in (52) with 𝑡_𝑛 and letting 𝑛→∞, from (42), we have that 𝐹𝑥_𝑡𝑛−𝐹𝑇𝑥_𝑡𝑛→0, thus we can conclude that 𝜇𝐹−𝛾𝑉𝑥^∗,𝑗𝑥^∗−𝑥≤0,∀𝑥∈Fix(𝑇). (54) So, 𝑥^∗ is a solution of (34). Hence, 𝑥^∗=𝑥 by uniqueness. Therefore, 𝑥_𝑡→𝑥 as 𝑡→0⁺. This completes the proof.

Proposition 10 (the demiclosed result). Let 𝑇∶𝑋→𝑋 be a nonexpansive mapping with Fix(𝑇)≠∅, and let 𝑉 be an 𝐿-Lipschitzian continuous self-mapping on 𝑋 with Lipschitzian constant 𝐿>0. 𝐹∶𝑋→𝑋 is an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous mapping with 𝜅>0 and 𝜂>0. Assume that the net {𝑥_𝑡} is defined as Proposition 9 which converges strongly as 𝑡→0⁺ to 𝑥∈Fix(𝑇). Suppose that the sequence {𝑥_𝑛}⊂𝑋 is bounded and satisfies the condition lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0 (the so-called demiclosed property). Then the following inequality VIP holds: limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤0. (55)

Proof. Set 𝑎_𝑛(𝑡)=‖𝑇𝑥_𝑛−𝑥_𝑛‖‖𝑥_𝑡−𝑥_𝑛‖. From the given condition and the boundness of {𝑥_𝑡} and {𝑥_𝑛}, it is obvious that 𝑎_𝑛(𝑡)→0 when 𝑛→∞.
From (33) and the fact that 𝑇 is a nonexpansive mapping, we obtain that 𝑥_𝑡−𝑥_𝑛²=𝑡𝛾𝑉𝑥_𝑡+𝐼−𝑡𝜇𝐹𝑇𝑥_𝑡−𝑥_𝑛,𝑗𝑥_𝑡−𝑥_𝑛=𝑇𝑥_𝑡−𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑥_𝑛+𝑡𝛾𝑉𝑥_𝑡−𝜇𝐹𝑥_𝑡+𝑡𝜇𝐹𝑥_𝑡−𝜇𝐹𝑇𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛≤𝑥_𝑡−𝑥_𝑛²+𝑇𝑥_𝑛−𝑥_𝑛,𝑗𝑥_𝑡−𝑥_𝑛+𝑡𝛾𝑉𝑥_𝑡−𝜇𝐹𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛+𝑡𝜇𝐹𝑥_𝑡−𝜇𝐹𝑇𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛≤𝑥_𝑡−𝑥_𝑛²+𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑡−𝑥_𝑛+𝑡𝛾𝑉𝑥_𝑡−𝜇𝐹𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛+𝑡𝜇𝐹𝑥_𝑡−𝐹𝑇𝑥_𝑡𝑥_𝑡−𝑥_𝑛, (56) which implies that 𝜇𝐹−𝛾𝑉𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛≤1𝑡𝑇𝑥_𝑛−𝑥_𝑛𝑥_𝑡−𝑥_𝑛+𝜇𝐹𝑥_𝑡−𝐹𝑇𝑥_𝑡𝑥_𝑡−𝑥_𝑛. (57) It follows that limsup_𝑛→∞𝜇𝐹−𝛾𝑉𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛≤𝜇𝐹𝑥_𝑡−𝐹𝑇𝑥_𝑡limsup_𝑛→∞𝑥_𝑡−𝑥_𝑛. (58) Taking the limsup as 𝑡→0 and recalling (42) and the continuity of 𝐹, we conclude that limsup_𝑡→0limsup_𝑛→∞𝜇𝐹−𝛾𝑉𝑥_𝑡,𝑗𝑥_𝑡−𝑥_𝑛≤0. (59)
On the other hand, since 𝑋 is a real 2-uniformly smooth Banach space, and 𝐽 is single-valued and strong-weak* uniformly continuous on 𝑋, as 𝑡→0⁺, we have ⟨𝜇𝐹−𝛾𝑉𝑥,𝑗_𝑞𝑥−𝑥_𝑛⟩−𝜇𝐹𝑥_𝑡−𝛾𝑉𝑥_𝑡,𝑗_𝑞𝑥_𝑡−𝑥_𝑛=𝜇𝐹−𝛾𝑉𝑥,𝑗_𝑞𝑥−𝑥_𝑛−𝑗_𝑞𝑥_𝑡−𝑥_𝑛+𝜇𝐹(𝑥)−𝜇𝐹𝑥_𝑡+𝛾𝑉𝑥_𝑡−𝛾𝑉(𝑥),𝑗_𝑞𝑥_𝑡−𝑥_𝑛=𝜇𝐹−𝛾𝑉𝑥,𝑗_𝑞𝑥−𝑥_𝑛−𝑗_𝑞𝑥_𝑡−𝑥_𝑛+𝜇𝐹(𝑥)−𝜇𝐹𝑥_𝑡,𝑗_𝑞𝑥_𝑡−𝑥_𝑛+𝛾𝑉𝑥_𝑡−𝑉(𝑥),𝑗_𝑞𝑥_𝑡−𝑥_𝑛⟶0. (60) Thus, from (59) and (60), we obtain limsup_𝑛→∞𝜇𝐹−𝛾𝑉𝑥,𝑗_𝑞𝑥−𝑥_𝑛=limsup_𝑡→0limsup_𝑛→∞𝜇𝐹−𝛾𝑉𝑥,𝑗_𝑞𝑥−𝑥_𝑛≤0. (61) So (55) is valid. This completes the proof.

Finally, we study the following implicit iterative method process: the initial 𝑥₀∈𝑋 is arbitrarily selected, and the iterative algorithm is recursively defined by 𝑥_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐹𝑇_𝑛𝑥_𝑛,∀𝑛≥1, (62) where the sequences {𝛼_𝑛} and {𝛽_𝑛} are sequences in (0,1) and satisfy the following conditions: (C1)lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛽_𝑛=0, (C2)∑^∞_𝑛=0(𝛼_𝑛/(𝛼_𝑛+𝛽_𝑛))=∞.

Theorem 11. Let {𝑇_𝑖}^∞_𝑖=1 be a countable family of self-nonexpansive mappings on 𝑋, such that 𝑆∶=⋂^∞_𝑛=1Fix(𝑇_𝑛)≠∅. Let 𝑉 be an 𝐿-Lipschitzian continuous self-mapping on 𝑋 with Lipschitzian constant 𝐿>0. 𝐹∶𝑋→𝑋 is an 𝜂-strongly accretive and 𝜅-Lipschitzian continuous mapping with 𝜅>0 and 𝜂>0. Suppose that the sequences {𝛼_𝑛} and {𝛽_𝑛} satisfy the controlling conditions (C1)-(C2). Let 𝜇∈(0,min{1,𝜂/𝐾²𝜅²}), and set 𝜏=𝜇(𝜂−𝜇𝐾²𝜅²) and (𝜏−1)/𝐿<𝛾<𝜏/𝐿. Assume that ({𝑇_𝑛},𝑇) satisfies the AKTT condition. Then {𝑥_𝑛} defined by (62) converges strongly to a common fixed point 𝑥 of  {𝑇_𝑛}^∞_𝑛=1 which equivalently solves the following variational inequality: 𝛾𝑉−𝜇𝐹𝑥,𝑗𝑥−𝑝≥0,∀𝑝∈𝑆. (63)

Proof. First we show that {𝑥_𝑛} is well defined. Consider a mapping 𝑆_𝑛 on 𝑋 defined by 𝑆_𝑛𝑥=𝛼_𝑛𝛾𝑉𝑥+𝛽_𝑛𝑥_𝑛−1+1−𝛽_𝑛𝐼−𝜇𝛼_𝑛𝐹𝑇_𝑛𝑥. (64) It is easy to see that 𝑆_𝑛 is a contraction. Indeed, for any 𝑥,𝑦∈𝑋, by Lemma 5, we have 𝑆_𝑡𝑥−𝑆_𝑡𝑦≤𝛼_𝑛𝛾𝑉(𝑥)𝑦+−𝑉1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇𝑥−1−𝛽_𝑛𝐼−𝛼_𝑛≤𝜇𝐹𝑇𝑦1−𝛽_𝑛−𝛼_𝑛𝜏−𝛾𝐿𝑥−𝑦. (65)
Hence, 𝑆_𝑛 is a contraction. By the Banach contraction mapping principle, we conclude that 𝑆_𝑛 has a unique fixed point, denoted by 𝑥_𝑛. So (62) is well defined.
Then we show that {𝑥_𝑛} is bounded. Taking any 𝑝∈𝑆, we have 𝑥_𝑛−𝑝²=𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑝,𝑗𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−1−𝑝,𝑗𝑥_𝑛+−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑝,𝑗𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝑉𝑝,𝑗𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑉𝑝−𝜇𝐹𝑝,𝑗𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−1−𝑝,𝑗𝑥_𝑛+−𝑝1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑝,𝑗𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝐿𝑥_𝑛−𝑝²+𝛼_𝑛𝛾𝑉𝑝−𝜇𝐹𝑝,𝑗𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−1−𝑝𝑥_𝑛+−𝑝1−𝛽_𝑛−𝛼_𝑛𝜏𝑥_𝑛−𝑝²=1−𝛽_𝑛−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑝²+𝛼_𝑛𝛾𝑉𝑝−𝜇𝐹𝑝𝑥_𝑛−𝑝+𝛽_𝑛𝑥_𝑛−1−𝑝𝑥_𝑛−𝑝, (66) which implies that 𝑥_𝑛≤𝛽−𝑝^𝑛𝛽_𝑛+𝛼_𝑛𝑥𝜏−𝛾𝐿^𝑛−1+𝛼−𝑝_𝑛𝜏−𝛾𝐿𝛽_𝑛+𝛼_𝑛𝑝𝜏−𝛾𝐿𝛾𝑉−𝜇𝐹𝑝.𝜏−𝛾𝐿 (67) By induction, it follows that 𝑥_𝑛−𝑝≤max𝑥₀,𝑝−𝑝𝛾𝑉−𝜇𝐹𝑝𝜏−𝛾𝐿,𝑛≥0. (68) Hence {𝑥_𝑛} is bounded, so are the {𝑉(𝑥_𝑛)} and {𝐹(𝑥_𝑛)}.
Next, we show that 𝑥_𝑛−𝑇𝑥_𝑛⟶0. (69) From (C1) and the definition of {𝑥_𝑛}, we observe that 𝑥_𝑛−𝑇_𝑛𝑥_𝑛=𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1−𝑇_𝑛𝑥_𝑛≤𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑇_𝑛𝑥_𝑛+𝛽_𝑛𝑥_𝑛−1−𝑇_𝑛𝑥_𝑛⟶0. (70) By Lemma 7 and (70), we have 𝑥_𝑛−𝑇𝑥_𝑛=𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑇_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑇_𝑛𝑥_𝑛−𝑇𝑥_𝑛⟶0. (71)
Let 𝑥_𝑡 be defined by (33), from Propositions 9 and 10, and we have that {𝑥_𝑡} converges strongly to 𝑥∈𝑆∶=Fix(𝑇)=⋂^∞_𝑖=1Fix(𝑇_𝑖) and limsup_𝑛→∞𝛾𝑉−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤0. (72)
As required, finally we show that 𝑥_𝑛→𝑥. As a matter of fact, by Lemmas 5 and 8, we have 𝑥_𝑛−𝑥²=1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑥+𝛽_𝑛𝑥_𝑛−1−𝑥+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑇_𝑛𝑥_𝑛−1−𝛽_𝑛𝐼−𝛼_𝑛𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥+𝛽_𝑛𝑥_𝑛−1−𝑥𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤1−𝛽_𝑛−𝛼_𝑛𝜏𝑥_𝑛−𝑥²+𝛽_𝑛𝑥_𝑛−1−𝑥𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑉𝑥_𝑛−𝑉(𝑥),𝑗𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑉(𝑥)−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤1−𝛽_𝑛−𝛼_𝑛𝜏𝑥_𝑛−𝑥²+𝛽_𝑛𝑥_𝑛−1−𝑥𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝐿𝑥_𝑛−𝑥²+𝛼_𝑛𝛾𝑉(𝑥)−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤1−𝛽_𝑛−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥²+𝛽_𝑛𝑥_𝑛−1−𝑥𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑉(𝑥)−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤1−𝛽_𝑛−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥²+𝛽_𝑛2𝑥_𝑛−1−𝑥²+𝑥_𝑛−𝑥²+𝛼_𝑛𝛾𝑉(𝑥)−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥≤𝛽1−^𝑛2−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥²+𝛽_𝑛2𝑥_𝑛−1−𝑥²+𝛼_𝑛𝛾𝑉(𝑥)−𝜇𝐹𝑥,𝑗𝑥_𝑛−𝑥, (73) which implies that 𝑥_𝑛−𝑥²≤𝛽_𝑛𝛽_𝑛+2𝛼_𝑛𝑥𝜏−𝛾𝐿^𝑛−1−𝑥²+2𝛼_𝑛𝛽_𝑛+2𝛼_𝑛𝑥𝜏−𝛾𝐿𝛾𝑉−𝜇𝐹𝑥,𝑗_𝑛−𝑥=1−2𝛼_𝑛𝜏−𝛾𝐿𝛽_𝑛+2𝛼_𝑛𝑥𝜏−𝛾𝐿^𝑛−1−𝑥²+2𝛼_𝑛𝜏−𝛾𝐿𝛽_𝑛+2𝛼_𝑛𝑥𝜏−𝛾𝐿𝛾𝑉−𝜇𝐹𝑥,𝑗_𝑛−𝑥.𝜏−𝛾𝐿 (74) It is easily to see that 2𝛼_𝑛𝜏−𝛾𝐿𝛽_𝑛+2𝛼_𝑛>𝜏−𝛾𝐿2𝛼_𝑛𝜏−𝛾𝐿2𝛽_𝑛+2𝛼_𝑛=𝛼𝜏−𝛾𝐿^𝑛𝛼_𝑛+𝛽_𝑛. (75) Thus, (C2) yields that ∑^∞_𝑛=0(2𝛼_𝑛(𝜏−𝛾𝐿)/(𝛽_𝑛+2𝛼_𝑛(𝜏−𝛾𝐿)))=∞. Applying Lemma 4 and (72) to (74), we conclude that 𝑥_𝑛→𝑥.
This completes the proof.

Remark 12. Our result in Proposition 9 extends Theorem 3.1 of Tian [8] from real Hilbert spaces to real 2-uniformly smooth Banach spaces. If we set 𝛽_𝑛=0, our result in Theorem 11 extends Theorem 3.2 of Tian [8] from real Hilbert spaces to real 2-uniformly smooth Banach spaces as well as from a single nonexpansive mapping to a countable family of nonexpansive mappings.

Remark 13. In 2008, Hu [14] introduced a modified Halpern-type iteration for a single nonexpansive mapping in Banach spaces which have a uniformly Gêteaux differentiable norm as follows: 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛. (76) Under some appropriate assumptions, he proved that the sequence {𝑥_𝑛} defined by the iteration process (76) converges strongly to the fixed point of 𝑇.

Corollary 14. If we take 𝛾=1, 𝐹=𝐼, 𝜇=1, and 𝛽_𝑛=0 in (62), we extend the classical viscosity approximation [4] under a mild assumption: the contraction mapping 𝑓 is replaced by an 𝐿-Lipschitzian continuous mapping 𝑉. Our proving process needs no Banach limit and is different from the proving process given by Xu [6] in some aspects.

Remark 15. Ceng et al. [11] introduced the following iterative algorithm to find a common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces: 𝑥_𝑛+1=1−𝛼_𝑛−𝛽_𝑛𝑥_𝑛+𝛼_𝑛𝑓𝑦_𝑛+𝛽_𝑛𝑇_𝑟𝑛𝑦_𝑛,𝑦_𝑛=1−𝛾_𝑛𝑥_𝑛+𝛾_𝑛𝑇_𝑟𝑛𝑥_𝑛. (77) Under some appropriate conditions one the parameter sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and the sequence {𝑥_𝑛} converges strongly to the approximate solution of a variational inequality problem.
If we set 𝛼_𝑛+𝛽_𝑛=1 and 𝛾_𝑛=0, the algorithm is simplified into viscosity-form iterative schemes for a finite family of nonexpansive semigroups. Our algorithms are considered in full space and avoid the generalized projections or sunny nonexpansive retractions in Banach space. For further improving our works, in order to obtain more general results, we should take the results given by Ceng et al. in [11] into account.

Acknowledgments

Ming Tian was supported in part by the Fundamental Research Funds for the Central Universities (no. ZXH2012K001). Xin Jin was supported in part by Technology Innovation Funds of Civil Aviation University of China for Graduate (YJSCX12-18).