Abstract

We introduce a general composite iterative scheme for nonexpansive semigroups in Banach spaces. We establish some strong convergence theorems of the general iteration scheme under different control conditions. The results presented in this paper improve and extend the corresponding results of Marino and Xu (2006), and others, from Hilbert spaces to Banach spaces.

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) the set of fixed points of 𝑇. We know that 𝐹(𝑇) is nonempty if 𝐶 is bounded, for more detail see [1]. A one-parameter family 𝒯={𝑇(𝑡)∶0≤𝑡<∞} 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=𝑥 for all 𝑥∈𝐶;(ii)𝑇(𝑠+𝑡)=𝑇(𝑠)∘𝑇(𝑡) for all 𝑠,𝑡≥0;(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≥0.

We denote by 𝐹(𝒯) the set of all common fixed points of 𝒯, that is ⋂𝐹(𝒯)=𝑡≥0𝐹(𝑇(𝑡)). We know that 𝐹(𝒯) 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. As in [3], we use the notation Π𝐶 to denote the collection of all contraction on 𝐶, that is, Π𝐶={𝑓∶𝐶→𝐶acontraction}. Note that each 𝑓∈Π𝐶 has a unique fixed point in 𝐶. Iterative methods for nonexpansive mappings have recently been applied to solve minimization problems; see, for example, [4–10] and references therein.

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:min𝑥∈𝐹12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑢⟩,(1.1) where 𝐹 is the fixed point set of a nonexpansive mapping 𝑇 on 𝐻 and 𝑢 is a given point in 𝐻. Assume 𝐴 is strongly positive; that is there is a constant 𝛾 such that ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖2 for all 𝑥∈𝐻.

In 2003, Xu [7] proved that the sequence {𝑥𝑛} generated by𝑥0𝑥=𝑥∈𝐻chosenarbitrarily;𝑛+1=𝛼𝑛𝑢+1−𝛼𝑛𝑇𝑥𝑛,∀𝑛≥0,(1.2) converges strongly to the unique solution of the minimization problem (1.1) provided the sequence {𝛼𝑛} satisfies certain conditions. On the other hand, Moudafi [11] introduced the viscosity approximation method and proved that if 𝐻 is a real Hilbert space, the sequence {𝑥𝑛} generated by the following algorithm:𝑥0𝑥=𝑥∈𝐻chosenarbitrarily;𝑛+1=𝛼𝑛𝑓𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛,∀𝑛≥0,(1.3) where 𝑓∶𝐶→𝐶 is a contraction mapping with constant 𝛼∈(0,1) and {𝛼𝑛}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of 𝑇 in 𝐶 which is the unique solution of the following variational inequality:⟨(𝑓−𝐼)𝑥∗,𝑦−𝑥∗⟩≤0,∀𝑦∈𝐹(𝑇).(1.4)

Recently, Marino and Xu [12] combined the iterative method (1.2) with the viscosity approximation method (1.3) considering the following general iterative process:𝑥0𝑥=𝑥∈𝐻chosenarbitrarily;𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝛼𝑛𝐴𝑇𝑥𝑛,∀𝑛≥0.(1.5) They proved the sequence {𝑥𝑛}, generated by (1.5), converges strongly to the unique solution of the variational inequality⟨(𝛾𝑓−𝐴)𝑥∗,𝑦−𝑥∗⟩≤0,∀𝑦∈𝐹(𝑇),(1.6) which is the optimality condition for the minimization problemmin𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.7) where 𝐶 is the fixed point set of a nonexpansive mapping 𝑇 and ℎ is a potential function for 𝛾𝑓 (i.e., ℎ(𝑥)=𝛾𝑓(𝑥) for all 𝑥∈𝐻). Xu [3] studied further the viscosity approximation method for nonexpansive semigroup in uniformly smooth Banach spaces. This result extends Theorem 2.2 of Moudafi [11] to a Banach space. Kim and Xu [13] studied the sequence generated by the algorithm𝑥1𝑦=𝑥∈𝐶chosenarbitrarily;𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛;𝑥𝑛+1=𝛽𝑛𝑢+1−𝛽𝑛𝑦𝑛,∀𝑛≥1,(1.8) and proved strong convergence of scheme (1.8) in the framework of a uniformly smooth Banach space. Later, Yao et al. [14] introduced a new iteration process by combining the modified Mann iteration [13] and the viscosity method introduced by Moudafi [11]. Let 𝐶 be a closed convex subset of a Banach space, 𝑇∶𝐶→𝐶 a nonexpansive mapping such that 𝐹(𝑇)≠∅; and 𝑓∈Π𝐶. Define {𝑥𝑛} in the following way:𝑥1𝑦=𝑥∈𝐶chosenarbitrarily;𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑥𝑛;𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑦𝑛,∀𝑛≥1,(1.9) where {𝛼𝑛} and {𝛽𝑛} are two sequences in (0,1). They proved, under different control conditions on the sequences {𝛼𝑛} and {𝛽𝑛}, that {𝑥𝑛} converge strongly to a fixed point of 𝑇.

In 2008, Sahu and O’Regan [15] studied several strong convergence theorems for a family of nonexpansive or pseudocontractive nonself-mappings in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm. Recently, Li and Gu [16] studied the sequence generated by the algorithm in Banach spaces, as follows:𝑥1𝑦=𝑥∈𝐶chosenarbitrarily;𝑛=𝛼𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑡𝑛𝑥𝑛;𝑥𝑛+1=𝛽𝑛𝑓𝑥𝑛+1−𝛽𝑛𝑦𝑛,∀𝑛≥1,(1.10) and they proved the sequence {𝑥𝑛} defined by (1.10) that, converges strongly to the unique solution of the variational inequality:(𝑓−𝐼)𝑥∗,𝐽𝑦−𝑥∗≤0,∀𝑦∈𝐹(𝒯).(1.11) Very recently, Kumam and Wattanawitoon [5] introduced the following new composite explicit iterative schemes defined by given 𝑥0=𝑥∈𝐶 and𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝛽𝑛𝑥𝑛+1−𝛽𝑛𝐼−𝛼𝑛𝐴1𝑡𝑛𝑡𝑛0𝑇(𝑠)𝑦𝑛𝑦𝑑𝑠,𝑛=𝛿𝑛𝑥𝛾𝑓𝑛+𝛾𝑛𝑥𝑛+1−𝛾𝑛𝐼−𝛿𝑛𝐴𝑥𝑛,∀𝑛≥0,(1.12) for the approximation of common fixed point of a one parameter nonexpansive semigroup in a real Hilbert space under some appropriate control conditions. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions.

Question 1. Can the theorem of Marino and Xu [12] be extend from a Hilbert space to a general Banach space, such as uniformly smooth Banach space?

Question 2. Can we extend the iterative method of algorithm (1.10) to a general iterative process?

The purpose of this paper is to give affirmative answer to these questions mentioned above. In this paper, inspired and motivated by the iterative sequence (1.5) given by Marino and Xu [12] and (1.10) given by Li and Gu [16], we introduce a composite iterative algorithm {𝑥𝑛} in a Banach space as follows:𝑥1𝑦=𝑥∈𝐶chosenarbitrarily;𝑛=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛;𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝛿𝑛𝑥𝑛+1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛,∀𝑛≥1,(1.13) where 𝑓∶𝐶→𝐶 is a contraction mapping, 𝑇(𝑡𝑛) is a nonexpansive semigroup and 𝐴 is a strongly positive linear bounded operator, and prove, under certain different control conditions on the sequences {𝛼𝑛}, {𝛽𝑛}, and {𝛿𝑛}, that {𝑥𝑛} defined by (1.13) converges strongly to a common fixed point, which solves some variational inequality in Banach spaces. The results presented in this paper extend the corresponding results announced by Marino and Xu [12] and some others from Hilbert spaces to Banach spaces.

2. Preliminaries

Throughout this paper, let 𝑋 be a real Banach space, 𝐶 be a closed convex subset of 𝑋. Let 𝐽∶𝑋→2𝑋∗ be a normalized duality mapping by 𝑓𝐽(𝑥)=∗∈𝑋∗∶⟨𝑥,𝑓∗⟩=‖𝑥‖2=‖𝑓∗‖2,(2.1) where 𝑋∗ denotes the dual space of 𝑋 and ⟨⋅,⋅⟩ denotes the generalized duality paring. In the following, the notation ⇀ 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 limitlim𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.2) 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(2.3)

Moreover, if for each 𝑦 in 𝑆(𝑋) the limit (2.2) 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 (2.2) 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), the limit (2.2) 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;  uniformly convex if 𝛿𝑋(𝜖)>0 for all 𝜖>0, where 𝛿𝑋(𝜖) is modulus of convexity of 𝑋 defined by 𝛿𝑋(𝜖)=inf1−‖𝑥+𝑦‖2∶[]‖𝑥‖≤1,‖𝑦‖≤1,‖𝑥+𝑦‖≥𝜀,∀𝜖∈0,2.(2.4) A uniformly convex Banach space 𝑋 is reflexive and strictly convex (see [18, Theorems 4.1.6 and 4.1.2]) and every uniformly smooth Banach space 𝑋 is a reflexive Banach with uniformly Gâteaux differentiable norm (see [18, Theorems 4.3.7 and 4.1.6]) (also see [19]).

Now, we present the concept of a uniformly asymptotically regular semigroup (see [20–22]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋, 𝒯={𝑇(𝑡)∶𝑡>0} is a continuous operator semigroup on 𝐶. Then 𝒯 is said to uniformly asymptotically regular (in short, u.a.r.) on 𝐶 if for all ℎ≥0 and any bounded subset 𝐵 of 𝐶,lim𝑡→∞sup𝑥∈𝐵‖𝑇(ℎ)𝑇(𝑡)𝑥−𝑇(𝑡)𝑥‖=0.(2.5)

Lemma 2.1 (Chen and Song [23]). Let 𝐶 be a nonempty closed convex subset of a uniformly Banach space 𝑋, 𝐵 a bounded closed convex subset of 𝐶. If we denote 𝒯={𝑇(𝑡)∶𝑡>0} a nonexpansive semigroup on 𝐶 such that ⋂𝐹∶=𝑡>0𝐹(𝑇(𝑡))≠∅. For all ℎ≥0, the set 𝜎𝑡∫(𝑥)=(1/𝑡)𝑡0𝑇(𝑠)𝑥𝑑𝑠, then lim𝑡→∞sup𝑥∈𝐵‖‖𝜎𝑡(𝑥)−𝑇(ℎ)𝜎𝑡‖‖(𝑥)=0.(2.6)

It is easy to check that the set {𝜎𝑡∶𝑡>0} defined by Lemma 2.1 is a u.a.r. nonexpansive semigroup on 𝐶 (see [24] for more detail).

Lemma 2.2 (Cai and Hu [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 2.3 (Suzuki [25]). Let {𝑥𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝑛→∞𝛽𝑛≤limsup𝑛→∞𝛽𝑛<1. Suppose 𝑥𝑛+1=(1−𝛽𝑛)𝑦𝑛+𝛽𝑛𝑥𝑛 for all integers 𝑛≥0 and limsup𝑛→∞(‖𝑦𝑛+1−𝑦𝑛‖−‖𝑥𝑛+1−𝑥𝑛‖)≤0. Then, lim𝑛→∞‖𝑦𝑛−𝑥𝑛‖=0.

If a Banach space 𝑋 admits a sequentially continuous duality mapping 𝐽 from weak topology to weak star topology, then by Lemma 1 of [26], 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 {𝑥𝑛}⊂𝑋 with 𝑥𝑛⇀𝑥, we have 𝐽(𝑥𝑛)⇀∗𝐽(𝑥) (see [26–28] for more details).

A Banach space 𝑋 is said to be satisfying Opial's condition if for any sequence 𝑥𝑛⇀𝑥 for all 𝑥∈𝑋 implies limsup𝑛→∞‖‖𝑥𝑛‖‖−𝑥<limsup𝑛→∞‖‖𝑥𝑛‖‖−𝑦∀𝑦∈𝑋,with𝑥≠𝑦.(2.7)

By Theorem 1 in [26], 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 2.4 (Demiclosed principle (Jung [27])). Let 𝐶 be a nonempty closed convex subset of a reflexive Banach space 𝑋 which satisfies Opial's condition, and suppose 𝑇∶𝐶→𝑋 is nonexpansive. Then the mapping 𝐼−𝑇 is demiclosed at zero, that is, 𝑥𝑛⇀𝑥 and 𝑥𝑛−𝑇𝑥𝑛→0 implies 𝑥=𝑇𝑥.

Lemma 2.5 (Liu [29]). Let 𝑋 be a real Banach space and 𝐽∶𝑋→2𝑋∗ be the normalized duality mapping. Then, for any 𝑥,𝑦∈𝑋, we have ‖𝑥+𝑦‖2≤‖𝑥‖2+2⟨𝑦,𝑗(𝑥+𝑦)⟩,(2.8) for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦) with 𝑥≠𝑦.

Lemma 2.6 (Aoyama et al. [30]). Let {𝑠𝑛} be a sequence of nonnegative real numbers, {𝑎𝑛} a sequence of [0,1] with ∑∞𝑛=1𝑎𝑛=∞, {𝑐𝑛} a sequence of nonnegative real number with ∑∞𝑛=1𝑐𝑛<∞, and {𝑏𝑛} a sequence of real numbers with limsup𝑛→∞𝑏𝑛≤0. Suppose that 𝑠𝑛+1=1−𝑎𝑛𝑠𝑛+𝑎𝑛𝑏𝑛+𝑐𝑛,(2.9) for all 𝑛∈ℕ. Then lim𝑛→∞𝑠𝑛=0.

3. Main Results

We need following the lemma to prove our theorem.

Lemma 3.1. Let 𝐶 be a nonempty bounded closed convex subset of a reflexive, smooth Banach space 𝑋 with admits a weakly sequentially continuous duality mapping 𝐽 from 𝑋 to 𝑋∗ with 𝐶±𝐶⊂𝐶 and let 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1), 𝐴 be a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼 and {𝜆𝑡}0<𝑡<1 be a net of positive real sequence such that lim𝑡→0𝜆𝑡=∞. Suppose that {𝑇(𝜆𝑡)} is a u.a.r. nonexpansive semigroup. Then the sequence {𝑥𝑡} define by 𝑥𝑡𝑥=𝑡𝛾𝑓𝑡+𝜆(𝐼−𝑡𝐴)𝑇𝑡𝑥𝑡,(3.1) converges strongly to the common fixed point 𝑥∗, where 𝑥∗ solves the variational inequality 𝑥𝛾𝑓∗−𝐴𝑥∗,𝐽𝑦−𝑥∗≤0,∀𝑦∈𝐹(𝒯).(3.2)

Proof. Observe that for 𝑡∈(0,‖𝐴‖−1). By Lemma 2.2, we have ‖𝐼−𝑡𝐴‖≤1−𝑡𝛾.
Firstly, we show that {𝑥𝑡} defined by (3.1) is well define. Define the mapping 𝑆𝑓∶𝐶→𝐶 provided by 𝑆𝑓∶=𝑡𝛾𝑓+(𝐼−𝑡𝐴)𝑇(𝜆𝑡), for each 𝑡∈(0,1). Then, for each 𝑥,𝑦∈𝐶 that ‖‖𝑆𝑓𝑥−𝑆𝑓𝑦‖‖=‖‖𝑇𝜆𝑡𝛾(𝑓(𝑥)−𝑓(𝑦))+(𝐼−𝑡𝐴)𝑡𝜆𝑥−𝑇𝑡𝑦‖‖≤𝑡𝛾𝛼‖𝑥−𝑦‖+1−𝑡𝛾=‖𝑥−𝑦‖1−𝑡𝛾−𝛾𝛼‖𝑥−𝑦‖<‖𝑥−𝑦‖.(3.3) This show that 𝑆𝑓 is a contraction mapping. Thus, by Banach's contraction principle guarantees that 𝑆𝑓 has a unique fixed point 𝑥𝑡∈𝐶, that is, 𝑥𝑡 defined by (3.1) is well define.
Next, we show the uniqueness of a solution of the variational inequality (3.2). Supposing 𝑥∗,̃𝑥∈𝐹(𝒯) satisfy the inequality (3.2), we have𝑥(𝛾𝑓−𝐴)̃𝑥,𝐽∗(−̃𝑥≤0,𝛾𝑓−𝐴)𝑥∗,𝐽̃𝑥−𝑥∗≤0.(3.4) Adding up both equations of (3.4), we get that 0≥(𝛾𝑓−𝐴)̃𝑥−(𝛾𝑓−𝐴)𝑥∗𝑥,𝐽∗=𝐴𝑥−̃𝑥∗𝑥−̃𝑥,𝐽∗𝑓𝑥−̃𝑥−𝛾∗𝑥−𝑓(̃𝑥),𝐽∗≥−̃𝑥𝛾‖𝑥∗−̃𝑥‖2‖‖𝑓𝑥−𝛾∗‖‖‖‖𝐽𝑥−𝑓(̃𝑥)∗‖‖≥−̃𝑥𝛾‖𝑥∗−̃𝑥‖2−𝛾𝛼‖𝑥∗−̃𝑥‖2=𝛾−𝛾𝛼‖𝑥∗−̃𝑥‖2.(3.5) Since 0<𝛾<𝛾/𝛼 this implies that 𝛾−𝛾𝛼>0, which is a contradiction. Hence ̃𝑥=𝑥∗ and the uniqueness is proved.
Next, we show that {𝑥𝑡} is bounded. Indeed, for any 𝑝∈𝐹(𝒯), we have‖‖𝑥𝑡‖‖=‖‖𝑥−𝑝𝑡𝛾𝑓𝑡+𝜆(𝐼−𝑡𝐴)𝑇𝑡𝑥𝑡‖‖=‖‖𝑡𝑥−𝑝𝛾𝑓𝑡𝑇𝜆−𝐴𝑝+(𝐼−𝑡𝐴)𝑡𝑥𝑡‖‖‖‖𝑥−𝑝≤𝑡𝛾𝑓𝑡‖‖‖‖𝑇𝜆−𝐴𝑝+‖𝐼−𝑡𝐴‖𝑡𝑥𝑡‖‖‖‖𝑥−𝑝≤𝑡𝛾𝑓𝑡‖‖+−𝐴𝑝1−𝑡𝛾‖‖𝑥𝑡‖‖‖‖𝛾𝑓𝑥−𝑝≤𝑡𝑡‖‖+−𝑓(𝑝)+𝛾𝑓(𝑝)−𝐴𝑝1−𝑡𝛾‖‖𝑥𝑡‖‖‖‖𝑥−𝑝≤𝑡𝛾𝛼𝑡‖‖+−𝑝+‖𝛾𝑓(𝑝)−𝐴𝑝‖1−𝑡𝛾‖‖𝑥𝑡‖‖=−𝑝1−𝑡‖‖𝑥𝛾−𝛾𝛼𝑡‖‖−𝑝+𝑡‖𝛾𝑓(𝑝)−𝐴𝑝‖.(3.6) It follows that ‖𝑥𝑡−𝑝‖≤‖𝛾𝑓(𝑝)−𝐴𝑝‖/(𝛾−𝛾𝛼). Hence {𝑥𝑡} is bounded.
Next, we show that ‖𝑥𝑡−𝑇(ℎ)𝑥𝑡‖→0 as 𝑡→0. We observe that‖‖𝑥𝑡−𝑇(ℎ)𝑥𝑡‖‖≤‖‖𝑥𝑡𝜆−𝑇𝑡𝑥𝑡‖‖+‖‖𝑇𝜆𝑡𝑥𝑡𝜆−𝑇(ℎ)𝑇𝑡𝑥𝑡‖‖+‖‖𝑇𝜆(ℎ)𝑇𝑡𝑥𝑡−𝑇(ℎ)𝑥𝑡‖‖‖‖𝑥≤2𝑡𝜆−𝑇𝑡𝑥𝑡‖‖+‖‖𝑇𝜆𝑡𝑥𝑡𝜆−𝑇(ℎ)𝑇𝑡𝑥𝑡‖‖(3.7) for all ℎ≥0. On the other hand, we note that 𝑥𝑡𝜆−𝑇𝑡𝑥𝑡𝑥=𝑡𝛾𝑓𝑡𝜆−𝐴𝑇𝑡𝑥𝑡,(3.8) for all 𝑡>0. By assuming that lim𝑡→0𝜆𝑡=∞ and {𝑇(𝜆𝑡)} be a u.a.r. nonexpansive semigroup, then for all ℎ≥0, we get lim𝜆𝑡→∞‖‖𝑇𝜆𝑡𝑥𝑡𝜆−𝑇(ℎ)𝑇𝑡𝑥𝑡‖‖≤lim𝜆𝑡→∞sup𝑧∈𝐶‖‖𝑇𝜆𝑡𝜆𝑧−𝑇(ℎ)𝑇𝑡𝑧‖‖=0.(3.9) From (3.7)–(3.9), letting 𝑡→0, we get lim𝑡→0‖‖𝑥𝑡−𝑇(ℎ)𝑥𝑡‖‖=0(3.10) for all ℎ≥0. Assume {𝑡𝑛}∞𝑛=1⊂(0,1) is such that 𝑡𝑛→0 as 𝑛→∞. Put 𝑥𝑛∶=𝑥𝑡𝑛 and 𝜆𝑛∶=𝜆𝑡𝑛. We will show that {𝑥𝑛} contains a subsequence converges strongly to 𝑥∗, where 𝑥∗∈𝐹(𝒮). Since {𝑥𝑛} is bounded sequence and Banach space 𝑋 is reflexive, there exists a subsequence {𝑥𝑛𝑗} of {𝑥𝑛} which converges weakly to 𝑥∗∈𝐶 as 𝑗→∞. Again since Banach space 𝑋 has a weakly sequentially continuous duality mapping satisfying Opial's condition. It follows by Lemma 2.4 and noting (3.10), we have 𝑥∗∈𝐹(𝒯). For each 𝑛≥1, we note that 𝑥𝑛−𝑥∗=𝑡𝑛𝑥𝛾𝑓𝑛+𝐼−𝑡𝑛𝐴𝑇𝜆𝑛𝑥𝑛−𝑥∗=𝑡𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗+𝐼−𝑡𝑛𝐴𝑇𝜆𝑛𝑥𝑛−𝑥∗.(3.11) Thus, we have ‖‖𝑥𝑛−𝑥∗‖‖2=𝑡𝑛𝑥⟨𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗⟩+𝐼−𝑡𝑛𝐴𝑇𝜆𝑛𝑥𝑛−𝑥∗𝑥,𝐽𝑛−𝑥∗≤𝑡𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗+‖‖𝐼−𝑡𝑛𝐴‖‖‖‖𝑇𝜆𝑛𝑥𝑛−𝑥∗‖‖‖‖𝐽𝑥𝑛−𝑥∗‖‖≤𝑡𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗+1−𝑡𝑛𝛾‖‖𝑥𝑛−𝑥∗‖‖2.(3.12) It follows that ‖‖𝑥𝑛−𝑥∗‖‖2≤1𝛾𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗=1𝛾𝑥𝛾𝑓𝑛𝑥−𝛾𝑓∗𝑥,𝐽𝑛−𝑥∗+𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗≤1𝛾‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗.(3.13) Hence, ‖‖𝑥𝑛−𝑥∗‖‖2≤1𝑥𝛾−𝛾𝛼𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗.(3.14) In particular, we have ‖‖𝑥𝑛𝑗−𝑥∗‖‖2≤1𝑥𝛾−𝛾𝛼𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛𝑗−𝑥∗.(3.15) Since {𝑥𝑛} is bounded and the duality mapping 𝐽 is single-valued and weakly sequentially continuous from 𝑋 into 𝑋∗, it follows (3.15), we have 𝑥𝑛𝑗→𝑥∗ as 𝑗→∞.
Next, we show that 𝑥∗ solves the variational inequality (3.2). Since𝑥𝑡𝑥=𝑡𝛾𝑓𝑡+𝜆(𝐼−𝑡𝐴)𝑇𝑡𝑥𝑡,(3.16) we derive that (𝛾𝑓−𝐴)𝑥𝑡1=−𝑡𝑇𝜆(𝐼−𝑡𝐴)𝑡𝑥𝑡−𝑥𝑡.(3.17) Notice that 𝑇𝜆𝑡𝑥𝑡−𝑥𝑡−𝑇𝜆𝑡𝑦−𝑦,𝐽𝑦−𝑥𝑡≥‖‖𝑥𝑡‖‖−𝑦2−‖‖𝑇𝜆𝑡𝑥𝑡𝜆−𝑇𝑡𝑦‖‖‖‖𝑥𝑡‖‖≥‖‖𝑥−𝑦𝑡‖‖−𝑦2−‖‖𝑥𝑡‖‖−𝑦2=0.(3.18) It follows that, for any 𝑦∈𝐹(𝒯), (𝛾𝑓−𝐴)𝑥𝑡,𝐽𝑦−𝑥𝑡1=−𝑡𝑇𝜆(𝐼−𝑡𝐴)𝑡𝑥𝑡−𝑥𝑡,𝐽𝑦−𝑥𝑡1=−𝑡𝑇𝜆𝑡𝑥𝑡−𝑥𝑡−𝑇𝜆𝑡𝑦−𝑦,𝐽𝑦−𝑥𝑡+𝐴𝑇𝜆𝑡𝑥𝑡−𝑥𝑡,𝐽𝑦−𝑥𝑡≤𝐴𝑇𝜆𝑡𝑥𝑡−𝑥𝑡,𝐽𝑦−𝑥𝑡.(3.19) Now, replacing 𝑡 and 𝜆𝑡 with 𝑡𝑛𝑗 and 𝜆𝑛𝑗, respectively in (3.19), and letting 𝑗→∞, noting (3.8), we obtain ⟨(𝛾𝑓−𝐴)𝑥∗,𝐽(𝑦−𝑥∗)⟩≤0. That is, 𝑥∗ is a solution of variational inequality (3.2). By uniqueness, as 𝑥∗=̃𝑥, we have shown that each cluster point of the sequence {𝑥𝑡} is equal to 𝑥∗. Then, we conclude that 𝑥𝑡→𝑥∗ as 𝑡→0. This completes the proof.

Now, we prove the following theorem which is the main result of this paper.

Theorem 3.2. Let 𝐶 be a nonempty bounded closed convex subset of a uniformly smooth Banach space 𝑋 which admit a weakly sequentially continuous duality mapping from 𝑋 into 𝑋∗ with 𝐶±𝐶⊂𝐶 and 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼. Let {𝛼𝑛}∞𝑛=1, {𝛽𝑛}∞𝑛=1, {𝛿𝑛}∞𝑛=1 be the sequences in (0,1) and {𝑡𝑛}∞𝑛=1 be a positive real divergent sequence such that 𝑡𝑛≤𝑡𝑛+1 for all 𝑛∈ℕ. Assume the following control conditions are hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)∑∞𝑛=1|𝛼𝑛+1−𝛼𝑛|<∞, ∑∞𝑛=1|𝛽𝑛+1−𝛽𝑛|<∞ and ∑∞𝑛=1|𝛿𝑛+1−𝛿𝑛|<∞; (𝐶3)∑∞𝑛=1sup𝑧∈𝐶{‖𝑇(ℎ)𝑇(𝑡𝑛)𝑧−𝑇(𝑡𝑛)𝑧‖}<∞ for all ℎ≥0.Then the sequence {𝑥𝑛} defined by (1.13) converges strongly to the common fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝒯) of the variational inequality (3.2).

Proof. First, we show that {𝑥𝑛} is bounded. By the control condition (𝐶1), we may assume, without loss of generality, that 𝛼𝑛≤(1−𝛿𝑛)‖𝐴‖−1 for all 𝑛≥0. Since 𝐴 is a linear bounded operator on 𝑋, by (2.3), we have ‖𝐴‖=sup{|⟨𝐴𝑢,𝐽(𝑢)⟩|∶𝑢∈𝑋,‖𝑢‖=1}. Observe that 1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑢,𝐽(𝑢)=1−𝛿𝑛−𝛼𝑛⟨𝐴𝑢,𝐽(𝑢)⟩≥1−𝛿𝑛−𝛼𝑛‖𝐴‖≥0,(3.20) that is to say (1−𝛿𝑛)𝐼−𝛼𝑛𝐴 is positive. It follows that ‖‖1−𝛿𝑛𝐼−𝛼𝑛𝐴‖‖=sup1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑢,𝐽(𝑢)∶𝑢∈𝑋,‖𝑢‖=1=sup1−𝛿𝑛−𝛼𝑛⟨𝐴𝑢,𝐽(𝑢)⟩∶𝑢∈𝑋,‖𝑢‖=1≤1−𝛿𝑛−𝛼𝑛𝛾.(3.21) For any 𝑝∈𝐹(𝒯), we compute ‖‖𝑦𝑛‖‖=‖‖𝛽−𝑝𝑛𝑥𝑛+1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛‖‖=‖‖𝛽−𝑝𝑛𝑥𝑛+−𝑝1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛‖‖−𝑝≤𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛‖‖𝑇𝑡𝑛𝑥𝑛‖‖−𝑝≤𝛽𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛽𝑛‖‖𝑥𝑛‖‖=‖‖𝑥−𝑝𝑛‖‖,‖‖𝑥−𝑝𝑛+1‖‖=‖‖𝛼−𝑝𝑛𝑥𝛾𝑓𝑛+𝛿𝑛𝑥𝑛+1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛‖‖=‖‖𝛼−𝑝𝑛𝑥𝛾𝑓𝑛−𝐴𝑝+𝛿𝑛𝑥𝑛+−𝑝1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛‖‖−𝑝≤𝛼𝑛𝛾‖‖𝑓𝑥𝑛‖‖−𝑓(𝑝)+‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛿𝑛‖‖𝑥𝑛‖‖+‖‖−𝑝1−𝛿𝑛𝐼−𝛼𝑛𝐴‖‖‖‖𝑦𝑛‖‖−𝑝≤𝛼𝑛‖‖𝑥𝛾𝛼𝑛‖‖−𝑝+‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛿𝑛‖‖𝑥𝑛‖‖+−𝑝1−𝛿𝑛−𝛼𝑛𝛾‖‖𝑥𝑛‖‖=−𝑝1−𝛼𝛾−𝛾𝛼𝑛‖‖𝑥𝑛‖‖+−𝑝𝛼𝛾−𝛾𝛼𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖.𝛾−𝛾𝛼(3.22) By induction, we get ‖‖𝑥𝑛+1‖‖‖‖𝑥−𝑝≤max1‖‖,(−𝑝‖𝛾𝑓𝑝)−𝐴𝑝‖𝛾−𝛾𝛼,(3.23) for 𝑛≥1. Hence {𝑥𝑛} is bounded, so are {𝑦𝑛}, {𝑓(𝑥𝑛)} and {𝑇(𝑡𝑛)𝑥𝑛}.
Next, we show that lim𝑛→∞‖𝑥𝑛+1−𝑥𝑛‖=0. From definition of {𝑦𝑛}, observing that 𝑦𝑛=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛,𝑦𝑛+1=𝛽𝑛+1𝑥𝑛+1+1−𝛽𝑛+1𝑇𝑡𝑛+1𝑥𝑛+1.(3.24) We note that 𝑦𝑛+1−𝑦𝑛=𝛽𝑛+1𝑥𝑛+1+1−𝛽𝑛+1𝑇𝑡𝑛+1𝑥𝑛+1−𝛽𝑛𝑥𝑛−1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛=1−𝛽𝑛+1𝑇𝑡𝑛+1𝑥𝑛+1𝑡−𝑇𝑛𝑥𝑛+𝛽𝑛+1𝑥𝑛+1−𝑥𝑛+𝛽𝑛+1−𝛽𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛.(3.25) It follows that ‖‖𝑦𝑛+1−𝑦𝑛‖‖≤1−𝛽𝑛+1‖‖𝑇𝑡𝑛+1𝑥𝑛+1𝑡−𝑇𝑛𝑥𝑛‖‖+𝛽𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.26) Now, we consider the first term on the right side of (3.2), we have ‖‖𝑇𝑡𝑛+1𝑥𝑛+1𝑡−𝑇𝑛𝑥𝑛‖‖≤‖‖𝑇𝑡𝑛+1𝑥𝑛+1𝑡−𝑇𝑛+1𝑥𝑛‖‖+‖‖𝑇𝑡𝑛+1𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑇𝑡𝑛+1𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.27)
Substituting (3.27) into (3.26), we get‖‖𝑦𝑛+1−𝑦𝑛‖‖≤1−𝛽𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+1−𝛽𝑛+1‖‖𝑇𝑡𝑛+1𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+𝛽𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑇𝑡𝑛+1𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖=‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+sup𝑧∈𝐶‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑡𝑧−𝑇𝑛𝑧‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.28) Similarly, from definition of {𝑥𝑛}, observing that 𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝛿𝑛𝑥𝑛+1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛,𝑥𝑛+2=𝛼𝑛+1𝑥𝛾𝑓𝑛+1+𝛿𝑛+1𝑥𝑛+1+1−𝛿𝑛+1𝐼−𝛼𝑛+1𝐴𝑦𝑛+1.(3.29) We note that 𝑥𝑛+2−𝑥𝑛+1=𝛼𝑛+1𝑥𝛾𝑓𝑛+1+𝛿𝑛+1𝑥𝑛+1+1−𝛿𝑛+1𝐼−𝛼𝑛+1𝐴𝑦𝑛+1−𝛼𝑛𝑥𝛾𝑓𝑛−𝛿𝑛𝑥𝑛−1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛=1−𝛿𝑛+1𝐼−𝛼𝑛+1𝐴𝑦𝑛+1−𝑦𝑛−𝛼𝑛+1−𝛼𝑛𝐴𝑦𝑛−𝛿𝑛+1−𝛿𝑛𝑦𝑛+𝛿𝑛+1𝑥𝑛+1−𝛿𝑛𝑥𝑛𝛼+𝛾𝑛+1𝑓𝑥𝑛+1−𝛼𝑛𝑓𝑥𝑛=1−𝛿𝑛+1𝐼−𝛼𝑛+1𝐴𝑦𝑛+1−𝑦𝑛+𝛿𝑛+1𝑥𝑛+1−𝛿𝑛+1𝑥𝑛+𝛾𝛼𝑛+1𝑓𝑥𝑛+1𝑥−𝑓𝑛+𝛼𝑛+1−𝛼𝑛𝑥𝛾𝑓𝑛−𝐴𝑦𝑛+𝛿𝑛+1−𝛿𝑛𝑥𝑛−𝑦𝑛.(3.30) It follows that ‖‖𝑥𝑛+2−𝑥𝑛+1‖‖≤‖‖1−𝛿𝑛+1𝐼−𝛼𝑛+1𝐴‖‖‖‖𝑦𝑛+1−𝑦𝑛‖‖+𝛿𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+𝛾𝛼𝑛+1‖‖𝑓𝑥𝑛+1𝑥−𝑓𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||‖‖𝑥𝛾𝑓𝑛−𝐴𝑦𝑛‖‖+||𝛿𝑛+1−𝛿𝑛||‖‖𝑥𝑛−𝑦𝑛‖‖≤1−𝛿𝑛+1−𝛼𝑛+1𝛾‖‖𝑦𝑛+1−𝑦𝑛‖‖+𝛿𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+𝛾𝛼𝛼𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||𝛾‖‖𝑓𝑥𝑛‖‖+‖‖𝐴𝑦𝑛‖‖+||𝛿𝑛+1−𝛿𝑛||‖‖𝑥𝑛−𝑦𝑛‖‖.(3.31) Substituting (3.28) into (3.31), we get ‖‖𝑥𝑛+2−𝑥𝑛+1‖‖≤1−𝛼𝛾−𝛾𝛼𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+1−𝛿𝑛+1−𝛼𝑛+1𝛾sup𝑧∈𝐶‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑡𝑧−𝑇𝑛𝑧‖‖+1−𝛿𝑛+1−𝛼𝑛+1𝛾||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||𝛾‖‖𝑓𝑥𝑛‖‖+‖‖𝐴𝑦𝑛‖‖+||𝛿𝑛+1−𝛿𝑛||1−𝛽𝑛‖‖𝑇𝑡𝑛𝑥𝑛−𝑥𝑛‖‖≤1−𝛼𝛾−𝛾𝛼𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+sup𝑧∈𝐶‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑡𝑧−𝑇𝑛𝑧‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛼𝑛+1−𝛼𝑛||𝛾‖‖𝑓𝑥𝑛‖‖+‖‖𝐴𝑦𝑛‖‖+||𝛿𝑛+1−𝛿𝑛||‖‖𝑇𝑡𝑛𝑥𝑛−𝑥𝑛‖‖≤1−𝛼𝛾−𝛾𝛼𝑛+1‖‖𝑥𝑛+1−𝑥𝑛‖‖+sup𝑧∈𝐶‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑡𝑧−𝑇𝑛𝑧‖‖+𝑀1||𝛼𝑛+1−𝛼𝑛||+||𝛽𝑛+1−𝛽𝑛||+||𝛿𝑛+1−𝛿𝑛||,(3.32) where 𝑀1 is an appropriate constant such that 𝑀1≥‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖‖‖𝑓𝑥+𝛾𝑛‖‖+‖‖𝐴𝑦𝑛‖‖∀𝑛∈ℕ.(3.33) Putting 𝑐𝑛∶=𝑀1(|𝛼𝑛+1−𝛼𝑛|+|𝛽𝑛+1−𝛽𝑛|+|𝛿𝑛+1−𝛿𝑛|)+sup𝑧∈𝐶‖𝑇(𝑡𝑛+1−𝑡𝑛)𝑇(𝑡𝑛)𝑧−𝑇(𝑡𝑛)𝑧‖, observing control conditions (𝐶2) and (𝐶3), we have ∞𝑛=1𝑐𝑛=∞𝑛=1𝑀1||𝛼𝑛+1−𝛼𝑛||+||𝛽𝑛+1−𝛽𝑛||+||𝛿𝑛+1−𝛿𝑛||+sup𝑧∈𝐶‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑡𝑧−𝑇𝑛𝑧‖‖≤𝑀1∞𝑛=1||𝛼𝑛+1−𝛼𝑛||+||𝛽𝑛+1−𝛽𝑛||+||𝛿𝑛+1−𝛿𝑛||+∞𝑛=1sup𝑧∈𝐶‖‖𝑇𝑡𝑛+1−𝑡𝑛𝑇𝑡𝑛𝑡𝑧−𝑇𝑛𝑧‖‖<∞.(3.34) Hence, by Lemma 2.6 to (3.32), we get that lim𝑛→∞‖‖𝑥𝑛+1−𝑥𝑛‖‖=0.(3.35) Next, we show that lim𝑛→∞‖‖𝑇(ℎ)𝑥𝑛−𝑥𝑛‖‖=0.(3.36) Observe that ‖‖𝑦𝑛−𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑥𝑛+1‖‖+‖‖𝑥𝑛+1−𝑦𝑛‖‖=‖‖𝑥𝑛−𝑥𝑛+1‖‖+‖‖𝛼𝑛𝑥𝛾𝑓𝑛+𝛿𝑛𝑥𝑛−𝑦𝑛−𝛼𝑛𝐴𝑦𝑛‖‖≤‖‖𝑥𝑛−𝑥𝑛+1‖‖+𝛿𝑛‖‖𝑥𝑛−𝑦𝑛‖‖+𝛼𝑛𝛾‖‖𝑓𝑥𝑛‖‖+‖‖𝐴𝑦𝑛‖‖.(3.37) It follows that ‖‖𝑦𝑛−𝑥𝑛‖‖≤11−𝛿𝑛‖‖𝑥𝑛−𝑥𝑛+1‖‖+𝛼𝑛𝛾‖‖𝑓𝑥𝑛‖‖+‖‖𝐴𝑦𝑛‖‖.(3.38) Observing control condition (𝐶1) and noting (3.10), we have lim𝑛→∞‖‖𝑦𝑛−𝑥𝑛‖‖=0.(3.39) Moreover, we note that ‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑦𝑛‖‖+‖‖𝑦𝑛𝑡−𝑇𝑛𝑥𝑛‖‖=‖‖𝑥𝑛−𝑦𝑛‖‖+𝛽𝑛‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.40) It follows that ‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖≤11−𝛽𝑛‖‖𝑥𝑛−𝑦𝑛‖‖.(3.41) Noting (3.19), hence lim𝑛→∞‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖=0.(3.42) On the other hand, we note that ‖‖𝑇(ℎ)𝑥𝑛−𝑥𝑛‖‖≤‖‖𝑇(ℎ)𝑥𝑛𝑡−𝑇(ℎ)𝑇𝑛𝑥𝑛‖‖+‖‖𝑇𝑡(ℎ)𝑇𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+‖‖𝑇𝑡𝑛𝑥𝑛−𝑥𝑛‖‖‖‖𝑥≤2𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+‖‖𝑡𝑇(ℎ)𝑇𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖‖‖𝑥≤2𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+sup𝑧∈𝐶‖‖𝑇𝑡(ℎ)𝑇𝑛𝑡𝑧−𝑇𝑛𝑧‖‖.(3.43) Therefor, by (3.26) and control condition (𝐶3) imply that lim𝑛→∞sup𝑧∈𝐶‖𝑇(ℎ)𝑇(𝑡𝑛)𝑧−𝑇(𝑡𝑛)𝑧‖=0, we have lim𝑛→∞‖‖𝑇(ℎ)𝑥𝑛−𝑥𝑛‖‖=0.(3.44)
Next, we show thatlimsup𝑛→∞𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗≤0.(3.45)
For each 𝑡∈(0,1), let 𝑥𝑡 be a unique point of 𝐶 such that 𝑥𝑡=𝑡𝛾𝑓(𝑥𝑡)+(𝐼−𝑡𝐴)𝑇(𝜆𝑡)𝑥𝑡. By Lemma 3.1, we have‖‖𝑥𝑡−𝑥𝑛‖‖2≤‖‖(𝐼−𝑡𝐴)(𝑇(𝜆𝑡)𝑥𝑡−𝑥𝑛)‖‖2𝑥+2𝑡𝛾𝑓𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛≤‖𝐼−𝑡𝐴‖2‖‖𝑇(𝜆𝑡)𝑥𝑡−𝑥𝑛‖‖2𝑥+2𝑡𝛾𝑓𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛≤1−𝑡𝛾2‖‖𝑇𝜆𝑡𝑥𝑡𝜆−𝑇𝑡𝑥𝑛‖‖+‖‖𝑇𝜆𝑡𝑥𝑛−𝑥𝑛‖‖2𝑥+2𝑡𝛾𝑓𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛≤1−2𝛾𝑡+𝛾2𝑡2‖‖𝑥𝑡−𝑥𝑛‖‖2+𝑓𝑛𝑥(𝑡)+2𝑡𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑡−𝑥𝑛+2𝑡𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛,(3.46) where 𝑓𝑛(𝑡)∶=(2‖𝑥𝑡−𝑥𝑛‖+‖𝑥𝑛−𝑇(𝜆𝑡)𝑥𝑛‖)‖𝑥𝑛−𝑇(𝜆𝑡)𝑥𝑛‖.
On the other hand, observe that 𝐴 is a strongly positive linear bounded operator, it follows from (2.3), we have𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛=𝐴𝑥𝑡−𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛≥𝛾‖‖𝑥𝑡−𝑥𝑛‖‖2.(3.47) Combining (3.47) with (3.46), we have 2𝑡𝐴𝑥𝑡𝑥−𝛾𝑓𝑡𝑥,𝐽𝑡−𝑥𝑛≤𝛾2𝑡2−2‖‖𝑥𝛾𝑡𝑡−𝑥𝑛‖‖2+𝑎𝑛(𝑡)+2𝑡𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛≤𝛾𝑡2−2𝑡⟨𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛⟩+2𝑡⟨𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛⟩+𝑓𝑛=(𝑡)𝛾𝑡2𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛+𝑓𝑛(𝑡).(3.48) It follows that 𝐴𝑥𝑡𝑥−𝛾𝑓𝑡𝑥,𝐽𝑡−𝑥𝑛≤𝛾𝑡2𝐴𝑥𝑡−𝐴𝑥𝑛𝑥,𝐽𝑡−𝑥𝑛+1𝑓2𝑡𝑛(𝑡).(3.49) Now, taking limit superior as 𝑛→∞ firstly, and then as 𝑡→0 in (3.49) (using (3.44)), we have limsup𝑛→∞𝐴𝑥𝑡𝑥−𝛾𝑓𝑡𝑥,𝐽𝑡−𝑥𝑛≤𝑡2𝑀2,(3.50) where 𝑀2>0 is a constant such that 𝑀2≥𝛾⟨𝐴𝑥𝑡−𝐴𝑥𝑛,𝐽(𝑥𝑡−𝑥𝑛)⟩ for all 𝑡∈(0,1) and 𝑛≥1. Now, taking limit superior as 𝑡→0 in (3.50). Hence, we get limsup𝑡→0limsup𝑛→∞𝐴𝑥𝑡𝑥−𝛾𝑓𝑡𝑥,𝐽𝑡−𝑥𝑛≤0.(3.51) Moreover, we note that 𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗𝑥=⟨𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗𝑥⟩−𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥𝑡𝑥+⟨𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥𝑡𝑥⟩−𝛾𝑓∗−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡𝑥+⟨𝛾𝑓∗−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡𝑥⟩−𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡+𝑥𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡𝑥=⟨𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗𝑥−𝐽𝑛−𝑥𝑡⟩+𝐴𝑥𝑡−𝐴𝑥∗𝑥,𝐽𝑛−𝑥𝑡+𝑥𝛾𝑓∗𝑥−𝛾𝑓𝑡𝑥,𝐽𝑛−𝑥𝑡+𝑥𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡.(3.52) Taking limit superior as 𝑛→∞ in (3.52), we have limsup𝑛→∞𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗≤limsup𝑛→∞𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗𝑥−𝐽𝑛−𝑥𝑡‖‖𝑥+‖𝐴‖𝑡−𝑥∗‖‖limsup𝑛→∞‖‖𝑥𝑛−𝑥𝑡‖‖+‖‖𝑥𝛾𝑓∗𝑥−𝛾𝑓𝑡‖‖limsup𝑛→∞‖‖𝑥𝑛−𝑥𝑡‖‖+limsup𝑛→∞𝑥𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡≤limsup𝑛→∞𝑥⟨𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗𝑥−𝐽𝑛−𝑥𝑡⟩‖‖𝑥+(‖𝐴‖+𝛾𝛼)𝑡−𝑥∗‖‖limsup𝑛→∞‖‖𝑥𝑛−𝑥𝑡‖‖+limsup𝑛→∞𝑥𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡.(3.53) By Lemma 3.1, 𝑥𝑡→𝑥∗∈𝐹(𝒯) as 𝑡→0. Since 𝑋 is a uniformly smooth Banach space, imply that 𝐽 is norm-to-norm uniformly continuous on bounded subset of 𝐶 (see, e.g., [18, Lemma 1]), we obtain limsup𝑡→0limsup𝑛→∞𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗𝑥−𝐽𝑛−𝑥𝑡=0.(3.54) Therefore, from (3.53), we have limsup𝑛→∞𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗=limsup𝑡→0limsup𝑛→∞𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛−𝑥∗≤limsup𝑡→0limsup𝑛→∞𝑥𝛾𝑓𝑡−𝐴𝑥𝑡𝑥,𝐽𝑛−𝑥𝑡≤0.(3.55)
Finally, we show that lim𝑛→∞‖𝑥𝑛−𝑥∗‖=0. By Lemma 2.5, we have‖‖𝑥𝑛+1−𝑥∗‖‖2=‖‖𝛼𝑛𝛾𝑓(𝑥𝑛)+𝛿𝑛𝑥𝑛+1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛−𝑥∗‖‖2=‖‖1−𝛿𝑛𝐼−𝛼𝑛𝐴(𝑦𝑛−𝑥∗)+𝛿𝑛(𝑥𝑛−𝑥∗)+𝛼𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗‖‖2≤‖‖1−𝛿𝑛𝐼−𝛼𝑛𝐴(𝑦𝑛−𝑥∗)+𝛿𝑛(𝑥𝑛−𝑥∗)‖‖2+2𝛼𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗≤‖‖1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛−𝑥∗‖‖+𝛿𝑛‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗≤‖‖1−𝛿𝑛𝐼−𝛼𝑛𝐴‖‖‖‖𝑦𝑛−𝑥∗‖‖+𝛿𝑛‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗≤‖‖1−𝛿𝑛−𝛼𝑛𝛾‖‖‖‖𝑥𝑛−𝑥∗‖‖+𝛿𝑛‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛𝑥𝛾𝑓𝑛−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗≤1−𝛼𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛𝑥⟨𝛾𝑓𝑛𝑥−𝛾𝑓∗𝑥,𝐽𝑛+1−𝑥∗⟩+2𝛼𝑛𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗≤1−𝛼𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2+2𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖‖‖𝑥𝑛+1−𝑥∗‖‖+2𝛼𝑛𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗≤1−𝛼𝑛𝛾2‖‖𝑥𝑛−𝑥∗‖‖2+𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+‖‖𝑥𝑛+1−𝑥∗‖‖2+2𝛼𝑛𝑥𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗.(3.56) It follows that ‖‖𝑥𝑛+1−𝑥∗‖‖2≤1−𝛼𝑛𝛾2+𝛼𝑛𝛾𝛼1−𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+2𝛼𝑛1−𝛼𝑛𝑥𝛾𝛼𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗=1−2𝛼𝑛𝛾+𝛼𝑛𝛾𝛼1−𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+𝛼2𝑛𝛾21−𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+2𝛼𝑛1−𝛼𝑛𝑥𝛾𝛼𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗=1−2𝛼𝑛𝛾−𝛾𝛼1−𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+𝛼2𝑛𝛾21−𝛼𝑛‖‖𝑥𝛾𝛼𝑛−𝑥∗‖‖2+2𝛼𝑛1−𝛼𝑛𝑥𝛾𝛼𝛾𝑓∗−𝐴𝑥∗𝑥,𝐽𝑛+1−𝑥∗.(3.57) Put 𝑎𝑛∶=2𝛼𝑛(𝛾−𝛾𝛼)/(1−𝛼𝑛𝛾𝛼) and 𝑏𝑛∶=(𝛼𝑛𝛾2/2(𝛾−𝛾𝛼))‖𝑥𝑛−𝑥∗‖2+(1/(𝛾−𝛾𝛼))⟨𝛾𝑓(𝑥∗)−𝐴𝑥∗,𝐽(𝑥𝑛+1−𝑥∗)⟩. The above reduces to formula ‖𝑥𝑛+1−𝑥∗‖2≤(1−𝑎𝑛)‖𝑥𝑛−𝑥∗‖2+𝑎𝑛𝑏𝑛. Observing control condition (𝐶1) and noting (3.55), it is easily seen that ∑∞𝑛=1𝑎𝑛=∞ and limsup𝑛→∞𝑏𝑛≤0. By Lemma 2.6, we conclude that lim𝑛→∞‖𝑥𝑛−𝑥∗‖=0. This completes the proof.

Corollary 3.3. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻 with 𝐶±𝐶⊂𝐶 and 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼. Let {𝛼𝑛}∞𝑛=1, {𝛽𝑛}∞𝑛=1, {𝛿𝑛}∞𝑛=1 be the sequences in (0,1) and {𝑡𝑛}∞𝑛=1 be a positive real divergent sequence such that 𝑡𝑛≤𝑡𝑛+1 for all 𝑛∈ℕ. Assume that the following control conditions that hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)∑∞𝑛=1|𝛼𝑛+1−𝛼𝑛|<∞, ∑∞𝑛=1|𝛽𝑛+1−𝛽𝑛|<∞, and ∑∞𝑛=1|𝛿𝑛+1−𝛿𝑛|<∞; (𝐶3)∑∞𝑛=1sup𝑧∈𝐶{‖𝑇(ℎ)𝑇(𝑡𝑛)𝑧−𝑇(𝑡𝑛)𝑧‖}<∞ for all ℎ≥0.Then the sequence {𝑥𝑛} defined by (1.13) converges strongly to the common fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝒯) of the variational inequality (1.6).

Proof. From Theorem 3.2, if 𝑋∶=𝐻 is a real Hilbert space, we get the desired conclusion easily.

Corollary 3.4. Let 𝐶 be a nonempty bounded closed convex subset of a uniformly smooth Banach space 𝑋 which admit a weakly sequentially continuous duality mapping from 𝑋 into 𝑋∗ and 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup mapping on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1). Let {𝛼𝑛}∞𝑛=1, {𝛽𝑛}∞𝑛=1, {𝛿𝑛}∞𝑛=1 be the sequences in (0,1) and {𝑡𝑛}∞𝑛=1 be a positive real divergent sequence such that 𝑡𝑛≤𝑡𝑛+1 for all 𝑛∈ℕ. Assume that the following control conditions hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)∑∞𝑛=1|𝛼𝑛+1−𝛼𝑛|<∞, ∑∞𝑛=1|𝛽𝑛+1−𝛽𝑛|<∞, and ∑∞𝑛=1|𝛿𝑛+1−𝛿𝑛|<∞; (𝐶3)∑∞𝑛=1sup𝑧∈𝐶{‖𝑇(ℎ)𝑇(𝑡𝑛)𝑧−𝑇(𝑡𝑛)𝑧‖}<∞ for all ℎ≥0.Then the sequence {𝑥𝑛} defined by 𝑥1𝑦=𝑥∈𝐶chosenarbitrarily,𝑛=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛;𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛿𝑛𝑥𝑛+1−𝛿𝑛−𝛼𝑛𝑦𝑛,∀𝑛≥1,(3.58) converges strongly to the fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝒯) of the variational inequality (1.11).

Proof. Taking 𝐴=𝐼 and 𝛾=1, then (1.13) is reduced to (3.58).

Corollary 3.5 (Marino and Xu [12]). Let 𝐻 be a real Hilbert space. Let 𝑇 be a nonexpansive mapping on 𝐻 such that 𝐹(𝑇)≠∅, 𝑓 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼. Assume that the following control conditions hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)∑∞𝑛=1|𝛼𝑛+1−𝛼𝑛|<∞. Then the sequence {𝑥𝑛} defined by 𝑥0𝑥=𝑥∈𝐻chosenarbitrarily;𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝛼𝑛𝐴𝑇𝑥𝑛,∀𝑛≥0,(3.59) converges strongly to the fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝑇) of the variational inequality (1.6).

Proof. If 𝑋∶=𝐻 is a real Hilbert space, we get the desired conclusion easily. Taking 𝛿𝑛=0, 𝛽𝑛=0, and 𝑇(𝑡𝑛)=𝑇, then (1.13) is reduced to (3.59).

Now, we prove the following theorem, under certain different control conditions on the sequence {𝛼𝑛}, {𝛽𝑛} and {𝛿𝑛}.

Theorem 3.6. Let 𝐶 be a nonempty bounded closed convex subset of a uniformly smooth Banach space 𝑋 which admit a weakly sequentially continuous duality mapping from 𝑋 into 𝑋∗ with 𝐶±𝐶⊂𝐶 and 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼. Let {𝛼𝑛}∞𝑛=1, {𝛽𝑛}∞𝑛=1, {𝛿𝑛}∞𝑛=1 be the sequences in (0,1) and {𝑡𝑛}∞𝑛=1 be a positive real divergent sequence such that 𝑡𝑛+1=ℎ+𝑡𝑛 for all 𝑛∈ℕ and for all ℎ≥0. Assume that the following control conditions are hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)lim𝑛→∞|𝛽𝑛+1−𝛽𝑛|=0; (𝐶3)0<liminf𝑛→∞𝛿𝑛≤limsup𝑛→∞𝛿𝑛<1. Suppose that {𝑇(𝑡𝑛)} is a u.a.r. nonexpansive semigroup on 𝐶. Then the sequence {𝑥𝑛} defined by (1.13) converges strongly to the common fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝒯) of the variational inequality (3.2).

Proof. By using the same argument and techniques as those of Theorem 3.2, we note that {𝑥𝑛} is bounded, and so are the set {𝑦𝑛}, {𝑓(𝑥𝑛)} and {𝑇(𝑡𝑛)𝑥𝑛}. We will show that lim𝑛→∞‖𝑥𝑛+1−𝑥𝑛‖=0.
From Theorem 3.2, we note that‖‖𝑦𝑛+1−𝑦𝑛‖‖≤‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑇𝑡𝑛+1𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.60)
Define 𝑥𝑛+1=(1−𝛿𝑛)𝑙𝑛+𝛿𝑛𝑥𝑛. That is 𝑙𝑛=(𝑥𝑛+1−𝛿𝑛𝑥𝑛)/(1−𝛿𝑛). Now, we compute 𝑙𝑛+1−𝑙𝑛.
Then, we note that𝑙𝑛+1−𝑙𝑛=𝑥𝑛+2−𝛿𝑛+1𝑥𝑛+11−𝛿𝑛+1−𝑥𝑛+1−𝛿𝑛𝑥𝑛1−𝛿𝑛=𝛼𝑛+1𝑥𝛾𝑓𝑛+1+1−𝛿𝑛+1𝐼−𝛼𝑛+1𝐴𝑦𝑛+11−𝛿𝑛+1−𝛼𝑛𝑥𝛾𝑓𝑛+1−𝛿𝑛𝐼−𝛼𝑛𝐴𝑦𝑛1−𝛿𝑛=𝛼𝑛+11−𝛿𝑛+1𝑥𝛾𝑓𝑛+1−𝐴𝑦𝑛+1+𝛼𝑛1−𝛿𝑛𝐴𝑦𝑛𝑥−𝛾𝑓𝑛+𝑦𝑛+1−𝑦𝑛,(3.61) one has ‖‖𝑙𝑛+1−𝑙𝑛‖‖≤𝛼𝑛+11−𝛿𝑛+1‖‖𝑥𝛾𝑓𝑛+1−𝐴𝑦𝑛+1‖‖+𝛼𝑛1−𝛿𝑛‖‖𝐴𝑦𝑛𝑥−𝛾𝑓𝑛‖‖+‖‖𝑦𝑛+1−𝑦𝑛‖‖.(3.62) Substituting (3.60) into (3.62), we have ‖‖𝑙𝑛+1−𝑙𝑛‖‖≤𝛼𝑛+11−𝛿𝑛+1‖‖𝑥𝛾𝑓𝑛+1−𝐴𝑦𝑛+1‖‖+𝛼𝑛1−𝛿𝑛‖‖𝐴𝑦𝑛𝑥−𝛾𝑓𝑛‖‖+‖‖𝑥𝑛+1−𝑥𝑛‖‖+‖‖𝑇𝑡𝑛+1𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.63) By assuming that 𝑡𝑛+1=ℎ+𝑡𝑛, it follows that ‖‖𝑙𝑛+1−𝑙𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖≤𝛼𝑛+11−𝛿𝑛+1‖‖𝑥𝛾𝑓𝑛+1−𝐴𝑦𝑛+1‖‖+𝛼𝑛1−𝛿𝑛‖‖𝐴𝑦𝑛𝑥−𝛾𝑓𝑛‖‖+‖‖𝑡𝑇(ℎ)𝑇𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+||𝛽𝑛+1−𝛽𝑛||‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.64) Since {𝑇(𝑡𝑛)} is a u.a.r. nonexpansive semigroup and lim𝑛→∞𝑡𝑛=∞, then for all ℎ≥0, we have lim𝑛→∞‖‖𝑇𝑡(ℎ)𝑇𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖≤lim𝑛→∞sup𝑧∈𝐶‖‖𝑇𝑡(ℎ)𝑇𝑛𝑡𝑧−𝑇𝑛𝑧‖‖=0.(3.65) Moreover, since {𝑥𝑛}, {𝑇(𝑡𝑛)𝑥𝑛} are bounded, observing control conditions (𝐶1), (𝐶2), (𝐶3), and noting (3.65) then, inequality (3.64) implies that limsup𝑛→∞‖‖𝑙𝑛+1−𝑙𝑛‖‖−‖‖𝑥𝑛+1−𝑥𝑛‖‖≤0.(3.66) It follows by Lemma 2.5 that lim𝑛→∞‖‖𝑙𝑛−𝑥𝑛‖‖=0.(3.67) Observing that 𝑥𝑛+1−𝑥𝑛=1−𝛿𝑛𝑙𝑛+𝛿𝑛𝑥𝑛−𝑥𝑛=1−𝛿𝑛𝑙𝑛−𝑥𝑛,(3.68) and hence, lim𝑛→∞‖‖𝑥𝑛+1−𝑥𝑛‖‖=0.(3.69) By the same proof in Theorem 3.2, we note that ‖‖𝑦𝑛−𝑥𝑛‖‖≤11−𝛿𝑛‖‖𝑥𝑛−𝑥𝑛+1‖‖+𝛼𝑛𝛾‖‖𝑓𝑥𝑛‖‖+‖‖𝐴𝑦𝑛‖‖.(3.70) Hence lim𝑛→∞‖𝑦𝑛−𝑥𝑛‖=0. Moreover, by Theorem 3.2, we note that lim𝑛→∞‖‖𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖=0.(3.71) Next, we show that lim𝑛→∞‖𝑇(ℎ)𝑥𝑛−𝑥𝑛‖=0. We note that ‖‖𝑇(ℎ)𝑥𝑛−𝑥𝑛‖‖≤‖‖𝑇(ℎ)𝑥𝑛𝑡−𝑇(ℎ)𝑇𝑛𝑥𝑛‖‖+‖‖𝑇𝑡(ℎ)𝑇𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+‖‖𝑇𝑡𝑛𝑥𝑛−𝑥𝑛‖‖‖‖𝑥≤2𝑛𝑡−𝑇𝑛𝑥𝑛‖‖+‖‖𝑡𝑇(ℎ)𝑇𝑛𝑥𝑛𝑡−𝑇𝑛𝑥𝑛‖‖.(3.72) Since {𝑇(𝑡𝑛)} is a u.a.r. nonexpansive semigroup, and noting (3.65), we have lim𝑛→∞‖‖𝑇(ℎ)𝑥𝑛−𝑥𝑛‖‖=0(3.73) for all ℎ≥0. By using the same argument and techniques as those of Theorem 3.2, we have {𝑥𝑛} converges strongly to a common fixed point 𝑥∗∈𝐹(𝒯). This completes the proof.

Corollary 3.7. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻 with 𝐶±𝐶⊂𝐶 and let 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 such that 0<𝛾<𝛾/𝛼. Let {𝛼𝑛}∞𝑛=1, {𝛽𝑛}∞𝑛=1, {𝛿𝑛}∞𝑛=1 be the sequences in (0,1) and {𝑡𝑛}∞𝑛=1 be a positive real divergent sequence such that 𝑡𝑛+1=ℎ+𝑡𝑛 for all 𝑛∈ℕ and for all ℎ≥0. Assume that the following control conditions are hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)lim𝑛→∞|𝛽𝑛+1−𝛽𝑛|=0; (𝐶3)0<liminf𝑛→∞𝛿𝑛≤limsup𝑛→∞𝛿𝑛<1. Suppose that {𝑇(𝑡𝑛)} is a u.a.r. nonexpansive semigroup on 𝐶. Then the sequence {𝑥𝑛} defined by (1.13) converges strongly to the fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝒯) of the variational inequality (1.6).

Proof. From Theorem 3.6, if 𝑋∶=𝐻 is a real Hilbert space, we get the desired conclusion easily.

If we taking 𝐴=𝐼 and 𝛾=1, then (1.13) is reduced to (3.74). So, we obtain the following corollary.

Corollary 3.8. Let 𝐶 be a nonempty bounded closed convex subset of a uniformly smooth Banach space 𝑋 which admit a weakly sequentially continuous duality mapping from 𝑋 into 𝑋∗ and let 𝒯={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐶 such that 𝐹(𝒯)≠∅. Let 𝑓∶𝐶→𝐶 be a contraction mapping with a coefficient 𝛼∈(0,1) and {𝛼𝑛}∞𝑛=1, {𝛽𝑛}∞𝑛=1, {𝛿𝑛}∞𝑛=1 be the sequences in (0,1) and {𝑡𝑛}∞𝑛=1 be a positive real divergent sequence such that 𝑡𝑛+1=ℎ+𝑡𝑛 for all 𝑛∈ℕ and for all ℎ≥0. Assume that the following control conditions hold: (𝐶1)lim𝑛→∞𝛼𝑛=0 and ∑∞𝑛=1𝛼𝑛=∞; (𝐶2)lim𝑛→∞|𝛽𝑛+1−𝛽𝑛|=0; (𝐶3)0<liminf𝑛→∞𝛿𝑛≤limsup𝑛→∞𝛿𝑛<1. Suppose that {𝑇(𝑡𝑛)} is a u.a.r. nonexpansive semigroup on 𝐶. Then the sequence {𝑥𝑛} defined by 𝑥1𝑦=𝑥∈𝐶chosenarbitrarily;𝑛=𝛽𝑛𝑥𝑛+1−𝛽𝑛𝑇𝑡𝑛𝑥𝑛;𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+𝛿𝑛𝑥𝑛+1−𝛿𝑛−𝛼𝑛𝑦𝑛,∀𝑛≥1,(3.74) converges strongly to the fixed point 𝑥∗, where 𝑥∗ is the unique solution in 𝐹(𝒯) of the variational inequality (1.11).

Acknowledgment

The authors would like to thank The National Research Council of Thailand (NRCT) and the Faculty of Science KMUTT for financial support.