A Viscosity Approximation Method with a Weakly Contractive Mapping of General Iterative Processes for Nonexpansive Semigroups in Banach Spaces

Abstract

We introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for finding common fixed point of nonexpansive semigroups {𝑇(𝑡)∶𝑡∈ℝ⁺} in the framework of Banach spaces. We proved that under some mild conditions these iterative processes converge strongly to the common fixed point of {𝑇(𝑡)∶𝑡∈ℝ⁺}, which is the unique solution of some variational inequality. The results obtained in this paper extend and improve on the recent results of Li et al. (2009), Chen and He (2007), and many others as special cases.

1. Introduction

Let 𝑋 be a real Banach space, and let 𝐶 be a nonempty closed convex subset of 𝑋. A mapping 𝑇 of 𝐶 into itself is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for each 𝑥,𝑦∈𝐶. We denote 𝐹(𝑇) as the set of fixed points of 𝑇. We know that 𝐹(𝑇) is nonempty if 𝐶 is bounded; for more detail see [1]. Throughout this paper we denote by ℕ and ℝ⁺ the set of all positive integers and all positive real numbers, respectively. A one-parameter family 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} from 𝐶 of 𝑋 into itself is said to be a nonexpansive semigroup on 𝐶 if it satisfies the following conditions: (i)𝑇(0)𝑥=𝑥forall𝑥∈𝐶,(ii)𝑇(𝑠+𝑡)=𝑇(𝑠)∘𝑇(𝑡) for all 𝑠,𝑡∈ℝ⁺,(iii)for each 𝑥∈𝐶 the mapping 𝑡↦𝑇(𝑡)𝑥 is continuous,(iv)‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶 and 𝑡∈ℝ⁺.

We denote by 𝐹(𝒮) the set of all common fixed points of 𝒮, that is 𝐹(𝒮)∶=⋂_𝑡∈ℝ+𝐹(𝑇(𝑡))={𝑥∈𝐶∶𝑇(𝑡)𝑥=𝑥}. We know that 𝐹(𝒮) is nonempty if 𝐶 is bounded; see [2]. Recall that a self-mapping 𝑓∶𝐶→𝐶 is a contraction if there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖ for each 𝑥,𝑦∈𝐶.

Definition 1.1. A mapping 𝜙∶𝐶→𝐶 is said to be weakly contractive if there exists a continuous and strictly increasing function 𝜓∶ℝ⁺→ℝ⁺,𝜓(0)=0 such that ‖𝜙(𝑥)−𝜙(𝑦)‖≤‖𝑥−𝑦‖−𝜓(‖𝑥−𝑦‖),∀𝑥,𝑦∈𝐶.(1.1)

Remark 1.2. As a special case, if we consider 𝜓(𝑡)∶=(1−𝛼)𝑡, for each 𝑡∈[0,∞) and 𝛼∈(0,1), then we get the contraction mapping with the coefficient 𝛼.

In the last ten years or so, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let 𝐻 be a real Hilbert space, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. Let 𝐴 be a strongly positive bounded linear operator on 𝐻: that is, there is a constant 𝛾>0 with property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²∀𝑥∈𝐻.(1.2) A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻: min_𝑥∈𝐹12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.3) where 𝐶 is the fixed point set of a nonexpansive mapping 𝑇 on 𝐻 and 𝑏 is a given point in 𝐻.

In 2003, Xu [5] proved that the sequence {𝑥_𝑛} defined by the iterative method below, with the initial guess 𝑥₀∈𝐻 chosen arbitrarily: 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑢,∀𝑛≥0,(1.4) converges strongly to the unique solution of the minimization problem (1.3) provided the sequence {𝛼_𝑛} satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the following iterative process for nonexpansive mappings (see [7, 8] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial 𝑥₀∈𝐻, we defined the sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝜎_𝑛𝑓𝑥_𝑛+1−𝜎_𝑛𝑇𝑥_𝑛,∀𝑛≥0,(1.5) where {𝜎_𝑛} is a sequence in (0,1). It is proved in [6, 8] that under certain appropriate conditions imposed on {𝜎_𝑛}, the sequence {𝑥_𝑛} generated by (1.5) strongly converges to a unique solution 𝑥^∗ of the variational inequality ⟨(𝑓−𝐼)𝑥^∗,𝜔−𝑥^∗⟩≤0,∀𝜔∈𝐹(𝑇).(1.6) Recently, Marino and Xu [9] combined the iterative method (1.4) with the viscosity approximation method (1.5) considering the following general iterative process: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,∀𝑛≥0,(1.7) where 0<𝛾<(𝛾/𝛼). They proved that the sequence {𝑥_𝑛} generated by (1.7) converges strongly to a unique solution 𝑥^∗ of the variational inequality ⟨(𝛾𝑓−𝐴)𝑥^∗,𝜔−𝑥^∗⟩≤0,∀𝜔∈𝐹(𝑇),(1.8) which is the optimality condition for the minimization problem: min_𝑥∈𝐶(1/2)⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

On the other hand, Shioji and Takahashi [10] introduced in a Hilbert space the implicit iteration as follows: 𝑥_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ,(1.9) where {𝛼_𝑛} is a sequence in (0,1)and{𝑡_𝑛} is a positive real divergent sequence and for 𝑢∈𝐶. They proved, under certain restrictions on the sequence {𝛼_𝑛}, that the sequence {𝑥_𝑛} defined by (1.9) converges strongly to a member of 𝐹(𝒮) (see also [11]).

Chen and Song [12] studied the strong convergence of the following sequence (1.10) for a nonexpansive semigroup 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} with 𝐹(𝒮)≠∅ in a uniformly convex Banach space: 𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ.(1.10) Suzuki [13] was the first to introduced again in a Hilbert space the following implicit iteration process: 𝑥_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,∀𝑛∈ℕ,(1.11) for a nonexpansive semigroup case. Xu [14] established a Banach space version of the sequence (1.11) of Suzuki [13].

In 2007, Chen and He [15] extended the result of Suzuki [13] and Xu [14] and studied the strong convergence theorem of viscosity implicit iteration process for a nonexpansive semigroup 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} with 𝐹(𝒮)≠∅, in Banach spaces as follows: 𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,∀𝑛∈ℕ.(1.12) Very recently, S. Li et al. [16] considered a general iterative process for a nonexpansive semigroup 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} in a Hilbert space as follows: 𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ,(1.13) where {𝛼_𝑛}⊂(0,1] and {𝑡_𝑛} are two sequences satisfying certain conditions. They proved that the sequence {𝑥_𝑛} defined by (1.13) converges strongly to 𝑥^∗∈𝐹(𝒮), which solves the following variational inequality (1.8).

Question 1. Can Theorem of S. Li et al. [16] be extended from Hilbert space to a general Banach space?

Question 2. Can we extend the iterative method of algorithms (1.10) and (1.12) to general algorithms?

Question 3. We know that the weakly contractive mapping is more general than the contractive mapping. What happens if the contractive mapping is replaced by the weakly contractive mapping?

The purpose of this paper is to give affirmative answer to these questions mentioned above. Motivated by the iterative process (1.10), (1.12), and (1.13), we introduce a new viscosity approximation method with a weakly contractive mapping of general iterative processes for nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of these iterative processes in a Banach space which admits a weakly sequentially continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Chen and He [15] and S. Li et al. [16] and many others as special cases.

2. Preliminaries

Throughout this paper, let 𝑋 be a real Banach space and 𝐶 a closed convex subset of 𝑋. Let 𝐽∶𝑋→2^𝑋∗ be a normalized duality mapping by 𝐽(𝑥)={𝑓^∗∈𝑋^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²=‖𝑓^∗‖²}, where 𝑋^∗ denotes the dual space of 𝑋 and ⟨⋅,⋅⟩ denotes the generalized duality paring. In the following, the notations ⇀ and → denote the weak and strong convergence, respectively. Also, a mapping 𝐼∶𝐶→𝐶 denotes the identity mapping.

The norm of a Banach space 𝑋 is said to be Gâteaux differentiable if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.1) exists for each 𝑥,𝑦∈𝐶 on the unit sphere 𝑆(𝑋) of 𝑋. In this case 𝑋 is smooth. Recall that the Banach space 𝑋 is said to be smooth if duality mapping 𝐽 is single valued. In a smooth Banach space, we always assume that 𝐴 is strongly positive (see [17]), that is, a constant 𝛾>0 with the property ⟨𝐴𝑥,𝐽(𝑥)⟩≥𝛾‖𝑥‖²,‖𝑎𝐼−𝑏𝐴‖=sup_{‖𝑥‖≤1}||⟨(𝑎𝐼−𝑏𝐴)𝑥,𝐽(𝑥)⟩||𝑎∈[0,1],𝑏∈[−1,1].(2.2)

Moreover, if for each 𝑦 in 𝑆(𝑋) the limit (2.1) is uniformly attained for 𝑥∈𝑆(𝑋), we say that the norm 𝑋 is uniformly Gâteaux differentiable. The norm of 𝑋 is said to be Frêchet differentiable if, for each 𝑥∈𝑆(𝑋), the limit (2.1) is attained uniformly for 𝑦∈𝑆(𝑋). The norm of 𝑋 is said to be uniformly Frêchet differentiable (or 𝑋 is said to be uniformly smooth) if the limit (2.1) is attained uniformly for (𝑥,𝑦)∈𝑆(𝑋)×𝑆(𝑋). A Banach space 𝑋 is said to be strictly convex if ‖𝑥‖=‖𝑦‖=1,𝑥≠𝑦, implies ‖𝑥+𝑦‖/2<1 and uniformly convex if 𝛿_𝑋(𝜖)>0 for all 𝜖>0, where 𝛿_𝑋(𝜖) is modulus of convexity of 𝑋 defined by 𝛿_𝑋(𝜖)=inf{1−(‖𝑥+𝑦‖/2)∶‖𝑥‖≤1,‖𝑦‖≤1,‖𝑥+𝑦‖≥𝜖}, for all 𝜖∈[0,2]. A uniformly convex Banach space 𝑋 is reflexive and strictly convex (see Theorems 4.1.6, and 4.1.2 of [18]) and every uniformly smooth Banach space 𝑋 is a reflexive Banach with uniformly Gâteaux differentiable norm (see Theorems 4.3.7, and 4.1.6 of [18] and also [19]).

In the sequel we will use the following lemmas, which will be used in the proofs for the main results in the next section.

Lemma 2.1 (see [17]). Assume that A is a strongly positive linear bounded operator on a smooth Banach space 𝑋 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 2.2 (see [20]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 a weakly contractive mapping. Then, 𝑇 has a unique fixed point in 𝑋.

If a Banach space 𝑋 admits a sequentially continuous duality mapping 𝐽 from weak topology to weak star topology, then, by Lemma 1 of [21], we have that duality mapping 𝐽 is a single value. In this case, the duality mapping 𝐽 is said to be a weakly sequentially continuous duality mapping, that is, for each {𝑥_𝑛}⊂𝑋 with 𝑥_𝑛⇀𝑥, we have 𝐽(𝑥_𝑛)⇀^∗𝐽(𝑥) (see [21–23] for more details).

A Banach space 𝑋 is said to be satisfying Opial’s condition if for any sequence 𝑥_𝑛⇀𝑥 for all 𝑥∈𝑋 implies limsup_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<limsup_𝑛→∞‖‖𝑥_𝑛−𝑦‖‖∀𝑦∈𝑋,with𝑥≠𝑦.(2.3)

By Theorem 1 in [21], it is well known that, if 𝑋 admits a weakly sequentially continuous duality mapping, then 𝑋 satisfies Opial’s condition and 𝑋 is smooth.

Lemma 2.3 ([22] (Demiclosed Principle)). Let C be a nonempty closed convex subset of a reflexive Banach space 𝑋 which satisfies Opial's condition, and that suppose 𝑇∶𝐶→𝑋 is nonexpansive. Then the mapping 𝐼−𝑇 is demiclosed at zero, that is, 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0 imply that 𝑥=𝑇𝑥.

Lemma 2.4 (see [15]). Let 𝐶 be a closed convex subset of a uniformly convex Banach space 𝑋 and 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} a nonexpansive semigroup on 𝐶 such that 𝐹(𝒮)≠∅. Then, for each 𝑟>0 and ℎ≥0, lim_t→∞sup_{𝑥∈𝐶∩𝐵𝑟}‖‖‖1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠−𝑇(ℎ)1𝑡^𝑡0𝑇(𝑠)𝑥𝑑𝑠‖‖‖=0.(2.4)

3. Main Results

In this section, we prove our main results.

Theorem 3.1. Let 𝑋 be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 into 𝑋^∗ and 𝐶 a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐶 into itself such that 𝐹(𝒮)≠∅. Let 𝜙 be a weakly contractive mapping and 𝐴 a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾. Let {𝑥_𝑛} be a sequence defined by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛1𝑡_𝑛^𝑡𝑛₀𝑥𝑇(𝑠)𝑥_𝑛𝑑𝑠,_𝑛=𝛼_𝑛𝛾𝜙𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0,(3.1) where {𝛼_𝑛},{𝛽_𝑛} are two sequences in (0,1) and {𝑡_𝑛} is a positive real divergent sequence satisfying the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to the common fixed point 𝑥^∗∈𝐹(𝒮), where 𝑥^∗ is the unique solution of the variational inequality 𝛾𝜙𝑥^∗−𝐴𝑥^∗,𝐽𝜔−𝑥^∗≤0,∀𝜔∈𝐹(𝒮).(3.2)

Proof. Firstly, we show that {𝑥_𝑛} defined by (3.1) is well define. Since lim_𝑛→∞𝛼_𝑛=0, we may assume, with no loss of generality, that 𝛼_𝑛<‖𝐴‖⁻¹ for each 𝑛≥0. Define the mapping 𝑇^𝜙𝑛∶𝐶→𝐶 by 𝑇^𝜙𝑛∶=𝛼_𝑛𝛾𝜙+𝐼−𝛼_𝑛𝐴𝛽_𝑛𝐼+1−𝛽_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑑𝑠.(3.3) Indeed, from Lemma 2.1, we have, for all 𝑥,𝑦∈𝐶, ‖‖𝑇^𝜙𝑛𝑥−𝑇^𝜙𝑛𝑦‖‖=+𝛼_𝑛𝛾1(−𝜙(𝛽𝑥_𝑛)−𝜙(𝑦))+𝛽_𝑛𝐼−𝛼_𝑛𝐴(𝑥−𝑦)≤𝛼_𝑛𝐼−𝛼_𝑛𝐴𝑡1_𝑛^𝑡𝑛₀(𝑇(𝑠)𝑥−𝑇(𝑠)𝑦)𝑑𝑠+𝛾1‖−𝜙𝛽(𝑥_𝑛)−𝜙(𝑦)‖+𝛽_𝑛‖‖𝐼−𝛼_𝑛𝐴‖‖‖𝑥−𝑦‖≤𝛼_𝑛‖‖𝐼−𝛼_𝑛𝐴‖‖𝑡1_𝑛^𝑡𝑛₀‖𝑇(𝑠)𝑥−𝑇(𝑠)𝑦‖𝑑𝑠+𝛾1[−‖𝑥𝛽−_𝑛𝑦‖−𝜓(‖𝑥−𝑦‖)]+𝛽_𝑛1−𝛼_𝑛𝛾‖𝑥−𝑦‖=𝛼_𝑛1−𝛼_𝑛𝛾‖𝑥−𝑦‖=1𝛾−[‖𝛼𝑥_𝑛−𝑦‖−𝜓(‖𝑥−𝑦‖)]+1−𝛼_𝑛𝛾‖𝑥−𝑦‖≤‖𝑥−𝑦‖𝛾−−𝛼𝛾_𝑛‖𝑥−𝑦‖−𝛼_𝑛𝛾𝜓(‖𝑥−𝑦‖)𝛾𝜓(‖𝑥−𝑦‖).(3.4) This shows that 𝑇^𝜙𝑛 is weakly contractive. It follows from Lemma 2.2 that 𝑇^𝜙𝑛 has a unique fixed point 𝑥_𝑛∈𝐶, that is, {𝑥_𝑛} defined by (3.1) is well defined.
Next, we show that {𝑥_𝑛} is bounded. Letting 𝑝∈𝐹(𝒮), we get‖‖𝑦_𝑛−𝑝‖‖=‖‖‖𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛≤𝛽_𝑛−𝑝𝑑𝑠‖‖‖‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛1𝑡_𝑛^𝑡𝑛₀≤𝛽_𝑛‖‖𝑇(𝑠)𝑥_𝑛−𝑝‖‖𝑑𝑠=‖‖𝑥‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑝_𝑛‖‖‖‖𝑥−𝑝_𝑛‖‖,−𝑝‖‖²==𝛼_𝑛𝛼_𝑛𝛾𝜙𝑥_𝑛−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑝,𝐽𝑥_𝑛−𝑝+𝐼𝛾−𝜙𝛼_𝑛𝑥_𝑛−𝛾𝜙(𝑝)+(𝛾𝜙(𝑝)−𝐴𝑝),𝐽𝑥_𝑛−𝑝≤𝛼_𝑛𝐴𝑦_𝑛−𝑝,𝐽𝑥_𝑛−𝑝+‖‖𝛾𝐼‖‖−𝜙𝛼_𝑛𝑥_𝑛−𝜙(𝑝)‖‖‖‖𝐽𝑥_𝑛−𝑝‖‖+𝛼_𝑛𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝≤𝛼_𝑛𝐴‖‖‖‖𝑦_𝑛−𝑝‖‖‖‖𝐽𝑥_𝑛−𝑝‖‖+1𝛾−‖‖𝑥𝛼_𝑛𝑛−𝑝‖‖−𝛾𝜓‖‖𝑥_𝑛−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝𝛾‖‖𝑥_𝑛−𝑝‖‖²=+1𝛼−_𝑛𝛼_𝑛𝛾−𝛾‖‖𝑥_𝑛−𝑝‖‖²−𝛼_𝑛𝛾𝜓‖‖𝑥_𝑛−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝,(3.5) and so 𝛾𝜓(‖𝑥−𝑝‖)‖‖𝑥_𝑛−𝑝‖‖+𝛾−𝛾‖‖𝑥_𝑛−𝑝‖‖≤𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝.(3.6) Thus, 𝛾𝜓(‖𝑥−𝑝‖)‖‖𝑥_𝑛−𝑝‖‖≤≤‖𝛾𝛾𝜙𝜙((𝑝𝑝))−−𝐴𝐴𝑝𝑝‖,𝐽‖‖𝐽𝑥_𝑛𝑥−_𝑛𝑝=‖𝛾𝜙(𝑝)−𝐴𝑝‖‖‖𝑥_𝑛−𝑝‖‖−𝑝‖‖.(3.7) It follows that 𝛾𝜓(‖𝑥_𝑛−𝑝‖)≤‖𝛾𝜙(𝑝)−𝐴𝑝‖. Hence, ‖‖𝑥_𝑛−𝑝‖‖≤𝜓⁻¹‖𝛾𝜙(𝑝)𝛾−𝐴𝑝‖,(3.8) which implies that {𝑥_𝑛} is bounded. Since 𝜙 is weakly contractive, we have ‖‖𝜙𝑥_𝑛−𝜙(𝑝)‖‖≤‖‖𝑥_𝑛−𝑝‖‖−𝜓‖‖𝑥_𝑛−𝑝‖‖≤‖‖𝑥_𝑛−𝑝‖‖.(3.9) Then, {𝜙(𝑥_𝑛)} is bounded. From (3.1), we have ‖‖𝑦_𝑛𝐼−𝛼_𝑛𝐴𝑦_𝑛=𝑥_𝑛−𝛼_𝑛𝛾𝜙𝑥_𝑛‖‖≤11−𝛼_𝑛𝛾‖‖𝑥^𝑛‖‖+𝛼^𝑛𝛾1−𝛼_𝑛𝛾‖‖𝜙𝑥^𝑛‖‖.(3.10) Thus, {𝑦_𝑛} is also bounded. We denoted 𝑧_𝑛∶=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠. And since 𝑧_𝑛=11−𝛽_𝑛𝑦_𝑛−𝛽^𝑛1−𝛽_𝑛‖‖𝑧_𝑛𝑥_𝑛,‖‖≤11−𝛽_𝑛‖‖𝑦_𝑛‖‖+𝛽^𝑛1−𝛽_𝑛‖‖𝑥_𝑛‖‖,(3.11){𝑧_𝑛} is also bounded by the boundedness of {𝑥_𝑛} and {𝑦_𝑛}.
Next, we show that ‖𝑥_𝑛−𝑇(ℎ)𝑥_𝑛‖→0 as 𝑛→∞, for all ℎ≥0. We note that‖‖𝑥_𝑛−𝑧_𝑛‖‖≤=𝛼‖‖_𝑛𝑥_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑧_𝑛‖‖‖‖𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑧_𝑛‖‖.(3.12) It follows that 1−𝛽_𝑛‖‖𝑥_𝑛−𝑧_𝑛‖‖≤𝛼_𝑛‖‖𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛‖‖.(3.13) By conditions (𝐶1) and (𝐶2), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑧_𝑛‖‖=0.(3.14) Moreover, we note that, for all ℎ≥0, ‖‖𝑥_𝑛−𝑇(ℎ)𝑥_𝑛‖‖≤≤2‖‖‖‖𝑥𝑥_𝑛𝑛−𝑧_𝑛‖‖+‖‖𝑧_𝑛−𝑇(ℎ)𝑧_𝑛‖‖+‖‖𝑇(ℎ)𝑧_𝑛−𝑇(ℎ)𝑥_𝑛‖‖−𝑧_𝑛‖‖+‖‖𝑧_𝑛−𝑇(ℎ)𝑧_𝑛‖‖.(3.15) Define the set 𝐾={𝑧∈𝐶∶‖𝑧−𝑝‖≤𝜓⁻¹(‖𝛾𝜙(𝑝)−𝐴𝑝‖/𝛾)}; then 𝐾 is a nonempty bounded closed convex subset of 𝐶, which is𝑇(ℎ)-invariant for each ℎ≥0 (i.e., 𝑇(ℎ)𝐶⊂𝐶). Since {𝑥_𝑛}⊂𝐶 and 𝐶 is bounded, there exists 𝑟>0 such that 𝐾⊂𝐵_𝑟, and it follows from Lemma 2.4 that lim_𝑛→∞‖‖𝑧_𝑛−𝑇(ℎ)𝑧_𝑛‖‖=0,∀ℎ≥0.(3.16) Noting (3.14) and (3.16), then, from (3.15), we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑇(ℎ)𝑥_𝑛‖‖=0,∀ℎ≥0.(3.17)
Next, we show that {𝑥_𝑛} contains a subsequence converging strongly to 𝑥̃∈𝐹(𝒮). Since {𝑥_𝑛} is bounded and Banach space 𝑋 is a uniformly convex, it is reflexive and there exists a subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛}, which converges weakly to some 𝑥̃∈𝐶 as 𝑗→∞. Again since Banach space 𝑋 has a weakly sequentially continuous duality mapping satisfying Opial's condition, noting (3.17) and by Lemma 2.3, we have 𝑥̃∈𝐹(𝒮). From (3.7), replace 𝑝 by 𝑥̃ to obtain𝛾𝜓(‖𝑥−𝑥̃‖)‖‖𝑥_𝑛𝑗−𝑥̃‖‖≤𝛾𝜙(𝑥̃)−𝐴𝑥̃,𝐽𝑥_𝑛𝑗−𝑥̃.(3.18) Since 𝐽 is single valued and weakly sequentially continuous from 𝑋 to 𝑋^∗, we get that lim_𝑗→∞𝛾𝜓‖‖𝑥_𝑛𝑗−𝑥̃‖‖‖‖𝑥_𝑛𝑗−𝑥̃‖‖≤lim_𝑗→∞𝛾𝜙(𝑥̃)−𝐴𝑥̃,𝐽𝑥^𝑛_𝑗−𝑥̃=0.(3.19) Thus, 𝑥_𝑛𝑗→𝑥̃ as 𝑗→∞.
Next, we show that 𝑥̃ is a solution of the variational inequality (3.2). Firstly, since‖‖𝑥_𝑛−𝑦_𝑛‖‖=𝛼_𝑛‖‖𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛‖‖,(3.20) by condition (𝐶1), we obtain lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. Since 𝑥_𝑛𝑗→𝑥̃∈𝐹(𝒮), then 𝑦_𝑛𝑗→𝑥̃∈𝐹(𝒮).
From (3.1), we derive that 𝛾𝜙(𝑥_𝑛)−𝐴𝑦_𝑛=(1/𝛼_𝑛)(𝑥_𝑛−𝑦_𝑛). Then, for each 𝜔∈𝐹(𝒮),𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛,𝐽𝜔−𝑥_𝑛==𝛼11_𝑛𝑥_𝑛−𝑦_𝑛,𝐽𝜔−𝑥_𝑛𝛼_𝑛1−𝛽_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀=1𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝐽𝜔−𝑥_𝑛𝛼_𝑛1−𝛽_𝑛𝜔−𝑡1_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝐽𝜔−𝑥_𝑛−‖‖𝜔−𝑥_𝑛‖‖²≤1𝛼_𝑛1−𝛽_𝑛‖‖𝜔−𝑥_𝑛=0.‖‖²−‖‖𝜔−𝑥_𝑛‖‖²(3.21) Therefore, ⟨𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛,𝐽𝜔−𝑥_𝑛≤0.(3.22) Since the duality mapping 𝐽 is single-valued and weakly sequentially continuous duality mapping from 𝑋 to 𝑋^∗, for each 𝜔∈𝐹(𝒮),𝑥_𝑛𝑗→𝑥̃ and 𝑦_𝑛𝑗→𝑥̃, then, from (3.22), we obtain lim_𝑗→∞𝛾𝜙𝑥^𝑛_𝑗−𝐴𝑦^𝑛_𝑗,𝐽𝜔−𝑥_𝑛𝑗=⟨𝛾𝜙(𝑥̃)−𝐴𝑥̃,𝐽(𝜔−𝑥̃)⟩≤0.(3.23) That is, 𝑥̃∈𝐹(𝒮) is a solution of the variational inequality (3.2).
Next, we show the uniqueness of the solution of the variational inequality (3.2). Suppose that 𝑥̃,𝑥^∗∈𝐹(𝒮) satisfy (3.2). Then,𝛾𝜙𝛾𝜙𝑥(𝑥_∗̃)−𝐴𝑥̃,𝐽𝑥^∗−𝑥̃≤0,−𝐴𝑥^∗,𝐽𝑥̃−𝑥^∗≤0.(3.24) Adding up (3.24), we get 0≥=𝐴(𝛾𝜙𝑥−𝐴)𝑥̃−(𝛾𝜙−𝐴)𝑥^∗,𝐽𝑥^∗−𝑥̃_∗≥−𝑥̃,𝐽𝑥^∗−𝑥̃−𝛾𝜙𝑥^∗−𝜙(𝑥̃),𝐽𝑥^∗−𝑥̃≥𝛾‖𝑥^∗−𝑥̃‖²−𝛾‖‖𝜙𝑥^∗−𝜙(𝑥̃)‖‖‖‖𝐽𝑥^∗−𝑥̃‖‖𝛾‖𝑥^∗−𝑥̃‖²−𝛾‖𝑥^∗−𝑥̃‖²+𝛾𝜓‖𝑥^∗−𝑥̃‖‖𝑥^∗−𝑥̃‖.(3.25) Thus 𝛾𝜓(‖𝑥^∗−𝑥̃‖)≤(𝛾−𝛾)‖𝑥^∗−𝑥̃‖. By the property of 𝜓, we must have 𝑥̃=𝑥^∗ and the uniqueness is proved.
Finally, we show that {𝑥_𝑛} converges strongly to 𝑥̃∈𝐹(𝒮). Suppose that there exists another subsequence 𝑥_𝑛𝑖→𝑥̂ as 𝑖→∞. We note that 𝑥̂∈𝐹(𝒮) is the solution of the variational inequality (3.2). Hence, 𝑥̃=𝑥̂=𝑥^∗ by uniqueness. In summary, we have shown that {𝑥_𝑛} is sequentially compact and each cluster point of the sequence {𝑥_𝑛} is equal to 𝑥^∗. Therefore, we conclude that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. This proof is complete.

Remark 3.2. (1) Theorem 3.1 improves and generalizes Theorem 3.1 of S. Li et al. [16] from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.
(2) Theorem 3.1 also improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping and from Hilbert spaces to Banach spaces.

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [24], and it was generalized to that for nonlinear semigroups by Reich [25–27]. It is clear that Theorem 3.1 is valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻 such that 𝐶±𝐶⊂𝐶. Let T be a nonexpansive mapping from C into itself such that 𝐹(𝑇)≠∅. Let 𝜙 be a weakly contractive mapping and 𝐴 a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾. Let {𝑥_𝑛} be a sequence defined by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛1𝑛+1^𝑛_𝑗=0𝑥_𝑛𝑇^𝑗𝑥_𝑛,=𝛼_𝑛𝛾𝜙𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0,(3.26) where {𝛼_𝑛},{𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then the sequence {𝑥_𝑛} defined by (3.26) converges strongly to the common fixed point 𝑥^∗∈𝐹(𝒮), where 𝑥^∗ is the unique solution of the variational inequality 𝛾𝜙𝑥^∗−𝐴𝑥^∗,𝐽𝜔−𝑥^∗≤0,∀𝜔∈𝐹(𝒮).(3.27)

Taking 𝐴=𝐼 and 𝛾=1 in Theorem 3.1, we get the following corollary.

Corollary 3.4. Let 𝑋 be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 into 𝑋^∗, and 𝐶 a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐶 into itself such that 𝐹(𝒮)≠∅. Let 𝜙 be a weakly contractive mapping. Let {𝑥_𝑛} be a sequence defined by 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛1𝑡_𝑛^𝑡𝑛₀𝑥𝑇(𝑠)𝑥_𝑛𝑑𝑠,_𝑛=𝛼_𝑛𝜙𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛≥0,(3.28) where {𝛼_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝑡_𝑛} is a positive real divergent sequence satisfying the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Then the sequence {𝑥_𝑛} defined by (3.28) converges strongly to the common fixed point 𝑥^∗∈𝐹(𝒮), where 𝑥^∗ is the unique solution of the variational inequality 𝜙𝑥^∗−𝑥^∗,𝐽𝜔−𝑥^∗≤0,∀𝜔∈𝐹(𝒮).(3.29)

Next, we prove a strong convergence theorem under different conditions.

Theorem 3.5. Let 𝑋 be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 into 𝑋^∗ and 𝐶 a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐶 into itself such that 𝐹(𝒮)≠∅. Let 𝜙 be a weakly contractive mapping and 𝐴 a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾. Let {𝑥_𝑛} be a sequence defined by 𝑥_𝑛𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑡_𝑛𝑥_𝑛,=𝛼_𝑛𝛾𝜙𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑦_𝑛,∀𝑛≥0,(3.30) where {𝛼_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝑡_𝑛} is a positive real sequence satisfying the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0,(𝐶2)lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝛼_𝑛/𝑡_𝑛)=0. Then, the sequence {𝑥_𝑛} defined by (3.30) converges strongly to the common fixed point 𝑥^∗∈𝐹(𝒮), where 𝑥^∗ is the unique solution of the variational inequality 𝛾𝜙𝑥^∗−𝐴𝑥^∗,𝐽𝜔−𝑥^∗≤0,∀𝜔∈𝐹(𝒮).(3.31)

Proof. Firstly, we show that {𝑥_𝑛} defined by (3.30) is well defined. Since lim_𝑛→∞𝛼_𝑛=0, we may assume, with no loss of generality, that 𝛼_𝑛<‖𝐴‖⁻¹ for each 𝑛≥0. Define the mapping 𝑇^𝜙𝑛∶𝐶→𝐶 by 𝑇^𝜙𝑛∶=𝛼_𝑛𝛾𝜙+𝐼−𝛼_𝑛𝐴𝛽_𝑛𝐼+1−𝛽_𝑛𝑇𝑡_𝑛.(3.32) From Lemma 2.1, we have for all 𝑥,𝑦∈𝐶, ‖‖𝑇^𝜙𝑛𝑥−𝑇^𝜙𝑛𝑦‖‖=+𝛼_𝑛𝛾1(−𝜙(𝛽𝑥_𝑛)−𝜙(𝑦))+𝛽_𝑛𝐼−𝛼_𝑛𝐴(𝑥−𝑦)≤𝛼_𝑛𝐼−𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥−𝑇𝑡_𝑛𝑦+𝛾1‖−𝜙𝛽(𝑥_𝑛)−𝜙(𝑦)‖+𝛽_𝑛‖‖𝐼−𝛼_𝑛𝐴‖‖‖𝑥−𝑦‖≤𝛼_𝑛‖‖𝐼−𝛼_𝑛𝐴‖‖‖‖𝑇𝑡_𝑛𝑥−𝑇𝑡_𝑛𝑦‖‖+𝛾1[−‖𝑥𝛽−_𝑛𝑦‖−𝜓(‖𝑥−𝑦‖)]+𝛽_𝑛1−𝛼_𝑛𝛾‖𝑥−𝑦‖=𝛼_𝑛1−𝛼_𝑛𝛾‖𝑥−𝑦‖=1𝛾−[‖𝛼𝑥_𝑛−𝑦‖−𝜓(‖𝑥−𝑦‖)]+1−𝛼_𝑛𝛾‖𝑥−𝑦‖≤‖𝑥−𝑦‖𝛾−−𝛼𝛾_𝑛‖𝑥−𝑦‖−𝛼_𝑛𝛾𝜓(‖𝑥−𝑦‖)𝛾𝜓(‖𝑥−𝑦‖).(3.33) This show that 𝑇^𝜙𝑛 is weakly contractive. It follows from Lemma 2.2 that 𝑇^𝜙𝑛 has a unique fixed point 𝑥_𝑛∈𝐶, that is, {𝑥_𝑛} defined by (3.30) is well defined.
Next, we show that {𝑥_𝑛} is bounded. Letting 𝑝∈𝐹(𝒮), we get‖‖𝑦_𝑛−𝑝‖‖=≤‖‖𝛽𝛽_𝑛𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝑝‖‖≤𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛‖‖𝑇𝑡_𝑛𝑥_𝑛−𝑝‖‖=‖‖𝑥‖‖𝑥_𝑛−𝑝‖‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑝_𝑛‖‖‖‖𝑥−𝑝_𝑛‖‖,−𝑝‖‖²==𝛼_𝑛𝛼_𝑛𝛾𝜙𝑥_𝑛−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑦_𝑛−𝑝,𝐽𝑥_𝑛−𝑝+𝐼𝛾−𝜙𝛼_𝑛𝑥_𝑛−𝛾𝜙(𝑝)+(𝛾𝜙(𝑝)−𝐴𝑝),𝐽𝑥_𝑛−𝑝≤𝛼_𝑛𝐴𝑦_𝑛−𝑝,𝐽𝑥_𝑛−𝑝+‖‖𝛾𝐼‖‖−𝜙𝛼_𝑛𝑥_𝑛−𝜙(𝑝)‖‖‖‖𝐽𝑥_𝑛−𝑝‖‖+𝛼_𝑛𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝≤𝛼_𝑛𝐴‖‖‖‖𝑦_𝑛−𝑝‖‖‖‖𝐽𝑥_𝑛−𝑝‖‖+1𝛾−‖‖𝑥𝛼_𝑛𝑛−𝑝‖‖−𝛾𝜓‖‖𝑥_𝑛−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝𝛾‖‖𝑥_𝑛−𝑝‖‖²=+1𝛼−_𝑛𝛼_𝑛𝛾−𝛾‖‖𝑥_𝑛−𝑝‖‖²−𝛼_𝑛𝛾𝜓‖‖𝑥_𝑛−𝑝‖‖‖‖𝑥_𝑛−𝑝‖‖𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝,(3.34) and so 𝛾𝜓(‖𝑥−𝑝‖)‖‖𝑥_𝑛−𝑝‖‖+𝛾−𝛾‖‖𝑥_𝑛−𝑝‖‖≤𝛾𝜙(𝑝)−𝐴𝑝,𝐽𝑥_𝑛−𝑝.(3.35) Thus, 𝛾𝜓(‖𝑥−𝑝‖)‖‖𝑥_𝑛−𝑝‖‖≤≤‖𝛾𝛾𝜙𝜙((𝑝𝑝))−−𝐴𝐴𝑝𝑝‖,𝐽‖‖𝐽𝑥_𝑛𝑥−_𝑛𝑝=‖𝛾𝜙(𝑝)−𝐴𝑝‖‖‖𝑥_𝑛−𝑝‖‖−𝑝‖‖.(3.36) It follows that 𝛾𝜓(‖𝑥_𝑛−𝑝‖)≤‖𝛾𝜙(𝑝)−𝐴𝑝‖. Hence, ‖‖𝑥_𝑛−𝑝‖‖≤𝜓⁻¹‖𝛾𝜙(𝑝)𝛾−𝐴𝑝‖.(3.37) This implies that {𝑥_𝑛} is bounded, so are {𝜙(𝑥_𝑛)}, {𝐴𝑦_𝑛} and {𝑇(𝑡_𝑛)𝑥_𝑛}.
Next, we show that {𝑥_𝑛} contains a subsequence converging strongly to 𝑥̃∈𝐹(𝒮). By reflexivity of 𝑋 and boundedness of the sequence {𝑥_𝑛}, there exists subsequence {𝑥_𝑛𝑗}⊆{𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥̃∈𝐹(𝒮) as 𝑗→∞. Now, we show that 𝑥̃∈𝐹(𝒮). Put 𝑥_𝑗∶=𝑥_𝑛𝑗, 𝑦_𝑗∶=𝑦_𝑛𝑗, 𝛼_𝑗∶=𝛼_𝑛𝑗, 𝛽_𝑗∶=𝛽_𝑛𝑗, and 𝑡_𝑗∶=𝑡_𝑛𝑗 for 𝑗∈ℕ, and fix 𝑡>0. We note that‖‖𝑥_𝑗−𝑇(𝑡)𝑥̃‖‖≤^{[𝑡/𝑡𝑗]−1}_𝑘=0‖‖𝑇(𝑘+1)𝑡_𝑗𝑥_𝑗−𝑇𝑘𝑡_𝑗𝑥_𝑗‖‖+‖‖‖𝑇𝑡𝑡_𝑗𝑡^𝑗𝑥^𝑗−𝑇𝑡+‖‖‖𝑇𝑡𝑡_𝑗𝑡^𝑗𝑥̃‖‖‖≤𝑡𝑡_𝑗𝑡^𝑗𝑥̃−𝑇(𝑡)𝑥̃‖‖‖𝑡_𝑗‖‖𝑇𝑡^𝑗𝑥_𝑗−𝑥_𝑗‖‖+‖‖𝑥_𝑗−𝑥̃‖‖+‖‖‖𝑇𝑡−𝑡=𝑡𝑡_𝑗𝑡^𝑗𝑥̃−𝑥̃‖‖‖𝑡_𝑗1−𝛼^𝑗𝛽_𝑗‖‖𝛾𝜙𝑥_𝑗−𝐴𝑦_𝑗‖‖+‖‖𝑥_𝑗−𝑥̃‖‖+‖‖‖𝑇𝑡−𝑡≤𝑡𝑡_𝑗𝑡^𝑗𝑥̃−𝑥̃‖‖‖1−𝛽_𝑗𝛼_𝑗𝑡_𝑗‖‖𝛾𝜙𝑥_𝑗−𝐴𝑦_𝑗‖‖+‖‖𝑥_𝑗−𝑥̃‖‖+max‖𝑇(𝑠)𝑥̃−𝑥̃‖∶0≤𝑠≤𝑡_𝑗.(3.38) For all 𝑗∈ℕ, we have limsup_𝑗→∞‖‖𝑥_𝑗−𝑇(𝑡)𝑥̃‖‖≤limsup_𝑗→∞‖‖𝑥_𝑗−𝑥̃‖‖.(3.39) By the assumption that Banach space 𝑋 has a weakly sequentially continuous duality mapping satisfying Opial’s condition, (3.39) implies that 𝑇(𝑡)𝑥̃=𝑥̃, and we get that 𝑥̃∈𝐹(𝒮). From (3.36), replace 𝑝 by 𝑥̃ to obtain 𝛾𝜓‖‖𝑥_𝑗−𝑥̃‖‖‖‖𝑥_𝑗−𝑥̃‖‖≤𝛾𝜙(𝑥̃)−𝐴𝑥̃,𝐽𝑥_𝑗−𝑥̃.(3.40) Since 𝐽 is singlevalued and weakly sequentially continuous from 𝑋 to 𝑋^∗, we get that lim_𝑗→∞𝛾𝜓‖𝑥_𝑗−𝑥̃‖‖𝑥_𝑗−𝑥̃‖≤lim_𝑗→∞⟨𝛾𝜙(𝑥̃)−𝐴𝑥̃,𝐽𝑥_𝑗−𝑥̃⟩=0.(3.41) Thus, 𝑥_𝑗→𝑥̃ as 𝑗→∞, namely, there is a subsequence {𝑥_𝑛𝑗}⊆{𝑥_𝑛} such that 𝑥_𝑛𝑗→𝑥̃ as 𝑗→∞.
Next, we show that 𝑥̃ is a solution of the variational inequality (3.31). Firstly, since‖‖𝑥_𝑛−𝑦_𝑛‖‖=𝛼_𝑛‖‖𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛‖‖.(3.42) by condition (𝐶1), we obtain lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0. Since 𝑥_𝑛𝑗→𝑥̃∈𝐹(𝒮), then 𝑦_𝑛𝑗→𝑥̃∈𝐹(𝒮).
From (3.30), we derive that 𝛾𝜙(𝑥_𝑛)−𝐴𝑦_𝑛=(1/𝛼_𝑛)(𝑥_𝑛−𝑦_𝑛). Then, for each 𝜔∈𝐹(𝒮),𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛,𝐽𝜔−𝑥_𝑛==𝛼11_𝑛𝑥_𝑛−𝑦_𝑛,𝐽𝜔−𝑥_𝑛𝛼_𝑛1−𝛽_𝑛𝑥_𝑛=1−𝑇𝑡_𝑛𝑥_𝑛,𝐽𝜔−𝑥_𝑛𝛼_𝑛1−𝛽_𝑛𝜔−𝑇𝑡_𝑛𝑥_𝑛≤1,𝐽𝜔−𝑥_𝑛−‖‖𝜔−𝑥_𝑛‖‖²𝛼_𝑛1−𝛽_𝑛‖‖𝜔−𝑥_𝑛=0.‖‖²−‖‖𝜔−𝑥_𝑛‖‖²(3.43) Therefore, 𝛾𝜙𝑥_𝑛−𝐴𝑦_𝑛,𝐽𝜔−𝑥_𝑛≤0.(3.44) By using the same argument and techniques as those of Theorem 3.1, we note that the variational inequality (3.31) has a unique solution. We denoted by 𝑥^∗∈𝐹(𝒮) the unique solution of (3.31). Therefore, 𝑥_𝑛→𝑥^∗ as 𝑛→∞. The proof is completed.

Remark 3.6. (1) Theorem 3.5 improves and generalizes Theorem 3.2 of Marino and Xu [9] from a nonexpansive mapping to a nonexpansive semigroup, from a contractive mapping to a weakly contractive mapping, and from Hilbert spaces to Banach spaces.
(2) Theorem 3.5 also improves and generalizes Theorem 3.1 of Chen and He [15] from a contractive mapping to a weakly contractive mapping.

Taking 𝐴=𝐼 and 𝛾=1 in Theorem 3.5, we get the following corollary.

Corollary 3.7. Let 𝑋 be a uniformly convex, smooth Banach space which admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 into 𝑋^∗ and 𝐶 a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be a nonexpansive semigroup from 𝐶 into itself such that 𝐹(𝒮)≠∅. Let 𝜙 be a weakly contractive mapping. Let {𝑥_𝑛} be a sequence defined by 𝑥_𝑛𝑦_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑡_𝑛𝑥_𝑛,=𝛼_𝑛𝜙𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛≥0,(3.45) where {𝛼_𝑛}, {𝛽_𝑛} are two sequences in (0,1) and {𝑡_𝑛} is a positive real sequence satisfying the following conditions: (𝐶1)lim_𝑛→∞𝛼_𝑛=0, (𝐶2)lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞(𝛼_𝑛/𝑡_𝑛)=0. Then the sequence {𝑥_𝑛} defined by (3.45) converges strongly to the common fixed point 𝑥^∗∈𝐹(𝒮), where 𝑥^∗ is the unique solution of the variational inequality 𝜙𝑥^∗−𝑥^∗,𝐽𝜔−𝑥^∗≤0,∀𝜔∈𝐹(𝒮).(3.46)

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under CSEC project no. 54000267). The first author would like to give thanks to the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand for their financial support.