The General Iterative Methods for Asymptotically Nonexpansive Semigroups in Banach Spaces

Abstract

We introduce the general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups which is a unique solution of some variational inequalities. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature.

1. Introduction

Let 𝐸 be a normed linear space. Let 𝑇 be a self-mapping on 𝐸. Then 𝑇 is said to be asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1 such that for each 𝑥,𝑦∈𝐸, 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑘_𝑛𝑥−𝑦,∀𝑛≥1. (1.1) The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [1] as an important generalization of the class of nonexpansive maps (i.e., mappings 𝑇∶𝐸→𝐸 such that ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐸). We use 𝐹(𝑇) to denote the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐸∶𝑇𝑥=𝑥}. A self-mapping 𝑓∶𝐸→𝐸 is a contraction on 𝐸 if there exists a constant 𝛼∈(0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝛼𝑥−𝑦,∀𝑥,𝑦∈𝐸. (1.2) We use Π_𝐸 to denote the collection of all contractions on 𝐸. That is, Π_𝐸={𝑓∶𝑓  is a contraction on  𝐸}.

A family 𝒮={𝑇(𝑠)∶0≤𝑠<∞} of mappings of 𝐸 into itself is called a strongly continuous semigroup of Lipschitzian mappings on 𝐸 if it satisfies the following conditions: (i)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐸; (ii)𝑇(𝑠+𝑡)=𝑇(𝑠)𝑇(𝑡) for all 𝑠,𝑡≥0; (iii)for each 𝑡>0, there exists a bounded measurable function 𝐿_𝑡∶(0,∞)→[0,∞) such that ‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤𝐿_𝑡‖𝑥−𝑦‖, for all  𝑥,𝑦∈𝐸; (iv)for all 𝑥∈𝐸, the mapping 𝑡↦𝑇(𝑡)𝑥 is continuous.

A strongly continuous semigroup of Lipchitszian mappings 𝒮 is called strongly continuous semigroup of nonexpansive mappings if 𝐿_𝑡=1 for all 𝑡>0 and strongly continuous semigroup of asymptotically nonexpansive if limsup_𝑡→∞𝐿_𝑡≤1. Note that for asymptotically nonexpansive semigroup 𝒮, we can always assume that the Lipchitszian constant {𝐿_𝑡}_𝑡>0 is such that 𝐿_𝑡≥1 for each 𝑡>0,  𝐿_𝑡 is nonincreasing in 𝑡, and lim_𝑛→∞𝐿_𝑡=1; otherwise we replace 𝐿_𝑡, for each 𝑡>0, with 𝐿_𝑡∶=max{sup_𝑠≥𝑡𝐿_𝑠,1}. We denote by 𝐹(𝒮) the set of all common fixed points of 𝒮, that is, 𝐹(𝒮)∶={𝑥∈𝐸∶𝑇(𝑡)𝑥=𝑥,0≤𝑡<∞}=_𝑡≥0𝐹(𝑇(𝑡)). (1.3)𝒮 is called uniformly asymptotically regular on 𝐶 [2, 3] if for all ℎ≥0 and any bounded subset 𝐵 of 𝐶, lim_𝑡→∞sup_𝑥∈𝐵‖𝑇(ℎ)𝑇(𝑡)𝑥−𝑇(𝑡)𝑥‖=0, (1.4) and almost uniformly asymtotically regular on 𝐶 [4] if lim_𝑡→∞sup_𝑥∈𝐵𝑇(ℎ)1𝑡^𝑡₀𝑇(𝑠)1𝑥𝑑𝑠−𝑡^𝑡₀𝑇(𝑠)𝑥𝑑𝑠=0. (1.5)

Let 𝑢∈𝐶. Then, for each 𝑡∈(0,1) and for a nonexpansive map 𝑇, there exists a unique point 𝑥_𝑡∈𝐶 satisfying the following condition: 𝑥_𝑡=(1−𝑡)𝑇𝑥_𝑡+𝑡𝑢, (1.6) since the mapping 𝐺_𝑡(𝑥)=(1−𝑡)𝑇𝑥+𝑡𝑢 is a contraction. When 𝐻 is a Hilbert space and 𝑇 is a self-map, Browder [5] showed that {𝑥_𝑡} converges strongly to an element of 𝐹(𝑇) which is nearest to 𝑢 as 𝑡→0⁺. This result was extended to more various general Banach space by Morales and Jung [6], Takahashi and Ueda [7], Reich [8], and a host of other authors.

Many authors (see, e.g., [9, 10]) have also shown the convergence of the path 𝑥_𝑛=(1−𝛼_𝑛)𝑇^𝑛𝑥_𝑛+𝛼_𝑛𝑢, in Banach spaces for asymptotically nonexpansive mapping self-map 𝑇 under some conditions on 𝛼_𝑛.

It is an interesting problem to extend the above results to a strongly continuous semigroup of nonexpansive mappings and a strongly continuous semigroup of asymptotically nonexpansive mappings.

Let 𝒮 be a strongly continuous semigroup of nonexpansive self-mappings. In 1998 Shioji and Takahashi [11] introduced, in Hilbert space, the implicit iteration 𝑢_𝑛=1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑢_𝑛𝑑𝑠+𝛼_𝑛𝑢,𝑢∈𝐶,𝑛≥0, (1.7) where {𝛼_𝑛} is a sequence in (0,1), {𝑡_𝑛} is a sequence of positive real numbers divergent to ∞. Under certain restrictions to the sequence {𝛼_𝑛}, Shioji and Takahashi proved strong convergence of (1.7) to a member of 𝐹(𝒮). Recently, Zegeye et al. [4] introduced the implicit (1.7) and the following explicit iteration process for a semigroup of asymptotically nonexpansive mappings: 𝑢_𝑛+1=1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑢_𝑛𝑑𝑠+𝛼_𝑛𝑢,𝑢∈𝐶,𝑛≥0, (1.8) where 𝑡_𝑛∈ℝ⁺ and 𝛼_𝑛∈(0,1) in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm. Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular. Then the implicit sequence (1.7) and explicit sequence (1.8) converge strongly to a point of 𝐹(𝒮).

On the other hand, by a gauge function 𝜑 we mean a continuous strictly increasing function 𝜑∶[0,∞)→[0,∞) such that 𝜑(0)=0 and 𝜑(𝑡)→∞ as 𝑡→∞. Let 𝐸^∗ be the dual space of 𝐸. The duality mapping 𝐽_𝜑∶𝐸→2^𝐸∗ associated to a gauge function 𝜑 is defined by 𝐽_𝜑(𝑥)=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖𝜑(‖𝑥‖),𝑓^∗=𝜑(‖𝑥‖),∀𝑥∈𝐸. (1.9) In particular, the duality mapping with the gauge function 𝜑(𝑡)=𝑡, denoted by 𝐽, is referred to as the normalized duality mapping. Clearly, there holds the relation 𝐽_𝜑(𝑥)=(𝜑(‖𝑥‖)/‖𝑥‖)𝐽(𝑥) for all 𝑥≠0 (see [12]). Browder [12] initiated the study of certain classes of nonlinear operators by means of the duality mapping 𝐽_𝜑. Following Browder [12], we say that a Banach space 𝐸 has a weakly continuous duality mapping if there exists a gauge 𝜑 for which the duality mapping 𝐽_𝜑(𝑥) is single valued and continuous from the weak topology to the weak* topology; that is, for any {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the sequence {𝐽_𝜑(𝑥_𝑛)} converges weakly* to 𝐽_𝜑(𝑥). It is known that 𝑙^𝑝 has a weakly continuous duality mapping with a gauge function 𝜑(𝑡)=𝑡^𝑝−1 for all 1<𝑝<∞. Set Φ(𝑡)=^𝑡₀𝜑(𝜏)𝑑𝜏,∀𝑡≥0, (1.10) then 𝐽_𝜑(𝑥)=𝜕Φ(‖𝑥‖),∀𝑥∈𝐸, (1.11) where 𝜕 denotes the subdifferential in the sense of convex analysis.

In a Banach space 𝐸 having a weakly continuous duality mapping 𝐽_𝜑 with a gauge function 𝜑, an operator 𝐴 is said to be strongly positive [13] if there exists a constant 𝛾>0 with the property 𝐴𝑥,𝐽_𝜑(𝑥)≥𝛾‖𝑥‖𝜑(‖𝑥‖),𝛼𝐼−𝛽𝐴=sup_{‖𝑥‖≤1}𝛼𝐼−𝛽𝐴𝑥,𝐽_𝜑(𝑥),𝛼∈[0,1],𝛽∈[−1,1], (1.12)(1.13) where 𝐼 is the identity mapping. If 𝐸∶=𝐻 is a real Hilbert space, then the inequality (1.12) reduces to ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²∀𝑥∈𝐻. (1.14) 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∈𝐶12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩, (1.15) where 𝐶 is the fixed point set of a nonexpansive mapping 𝑇 on 𝐻 and 𝑏 is a given point in 𝐻. In 2009, motivated and inspired by Marino and Xu [14], Li et al. [15] introduced the following general iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup {𝑇(𝑠)∶𝑠≥0} on a nonempty closed convex subset 𝐶 in a Hilbert space: 𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑦_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑥𝑑𝑠,𝑛≥0,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0, (1.16) where {𝛼_𝑛} and {𝑡_𝑛} are sequences in [0,1] and (0,∞), respectively, 𝐴 is a strongly positive bounded linear operator on 𝐻, and 𝑓 is a contraction on 𝐻. And their convergence theorems can be proved under some appropriate control conditions on parameter {𝛼_𝑛} and {𝑡_𝑛}. Furthermore, by using these results, they obtained two mean ergodic theorems for nonexpansive mappings in a Hilbert space.

All of the above brings us to the following conjectures.

Question 1. Could we obtain strong convergence theorems for the general class of strongly continuous semigroup of asymptotically nonexpansive mappings in more general Banach spaces? such as a reflexive Banach space which admits a weakly continuous duality mapping 𝐽_𝜑, where 𝜑∶[0,∞)→[0,∞) is a gauge function.

In this paper, inspired and motivated by Shioji and Takahashi [11], Zegeye et al. [4], Marino and Xu [14], Li et al. [15], and Wangkeeree et al. [13], we prove the strong convergence theorems of the iterative approximation methods (1.16) for the general class of the strongly continuous semigroup of asymptotically nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping 𝐽_𝜑, where 𝜑∶[0,∞)→[0,∞) is a gauge function and 𝐴 is a strongly positive bounded linear operator on a Banach space 𝐸. The results in this paper generalize and improve some well-known results in Shioji and Takahashi [11], Li et al. [15], and many others.

2. Preliminaries

Throughout this paper, let 𝐸 be a real Banach space and 𝐸^∗ be its dual space. We write 𝑥_𝑛⇀𝑥 (resp., 𝑥_𝑛⇀^∗𝑥) to indicate that the sequence {𝑥_𝑛} weakly (resp., weak*) converges to 𝑥; as usual 𝑥_𝑛→𝑥 will symbolize strong convergence. Let 𝑈={𝑥∈𝐸∶‖𝑥‖=1}. A Banach space 𝐸 is said to uniformly convex if, for any 𝜖∈(0,2], there exists 𝛿>0 such that, for any 𝑥,𝑦∈𝑈, ‖𝑥−𝑦‖≥𝜖 implies ‖(𝑥+𝑦)/2‖≤1−𝛿. It is known that a uniformly convex Banach space is reflexive and strictly convex (see also [16]). A Banach space 𝐸 is said to be smooth if the limit lim_𝑡→0((‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡) exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝑈.

Now we collect some useful lemmas for proving the convergence result of this paper.

The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [10].

Lemma 2.1 (see [10]). Assume that a Banach space 𝐸 has a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑.(i)For all 𝑥,𝑦∈𝐸, the following inequality holds: Φ𝑥+𝑦≤Φ(‖𝑥‖)+𝑦,𝐽_𝜑𝑥+𝑦. (2.1) In particular, for all 𝑥,𝑦∈𝐸, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦. (2.2)(ii)Assume that a sequence {𝑥_𝑛} in 𝐸 converges weakly to a point 𝑥∈𝐸.
Then the following identity holds: limsup_𝑛→∞Φ𝑥_𝑛−𝑦=limsup_𝑛→∞Φ𝑥_𝑛−𝑥+Φ𝑦−𝑥,∀𝑥,𝑦∈𝐸. (2.3)

The next valuable lemma is proved for applying our main results.

Lemma 2.2 (see [13, Lemma 3.1]). Assume that a Banach space 𝐸 has a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑. Let 𝐴 be a strong positive linear bounded operator on 𝐸 with coefficient 𝛾>0 and 0<𝜌≤𝜑(1)‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤𝜑(1)(1−𝜌𝛾).

In the following, we also need the following lemma that can be found in the existing literature [17, 18].

Lemma 2.3 (see [18, Lemma 2.1]). Let {𝑎_𝑛} be a sequence of a nonnegative real number satisfying the property 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛽_𝑛,𝑛≥0, (2.4) where {𝛾_𝑛}⊆(0,1) and {𝛽_𝑛}⊆ℝ such that ∑^∞_𝑛=0𝛾_𝑛=∞ and limsup_𝑛→∞𝛽_𝑛≤0. Then {𝑎_𝑛} converges to zero, as 𝑛→∞.

3. Main Theorem

Theorem 3.1. Let 𝐸 be a reflexive Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let 𝒮={𝑇(𝑠)∶𝑠≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings on 𝐸 with a sequence {𝐿_𝑡}⊂[1,∞) and 𝐹(𝒮)≠∅. Let 𝑓∈Π_𝐸 with coefficient 𝛼∈(0,1), let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<𝜑(1)𝛾/𝛼, and let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers such that 0<𝛼_𝑛<1,  𝑡_𝑛>0. Then the following holds.(i)If ((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛<𝜑(1)𝛾−𝛾𝛼, for all  𝑛≥0, then there exists a sequence {𝑦_𝑛}⊂𝐸 defined by 𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑦_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝑛≥0. (3.1)(ii)Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular and the real sequences {𝛼_𝑛} and {𝑡_𝑛} satisfy the following conditions: (B1)lim_𝑛→∞𝑡_𝑛=∞; (B2)lim_𝑛→∞𝛼_𝑛=0; (B3)lim_𝑛→∞((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛=0.
Then {𝑦_𝑛} converges strongly as 𝑛→∞ to a common fixed point 𝑥 in 𝐹(𝒮) which solves the variational inequality: 𝐴−𝛾𝑓𝑥,𝐽_𝜑(𝑥−𝑧)≤0,𝑧∈𝐹(𝒮). (3.2)

Proof. We first show the uniqueness of a solution of the variational inequality (3.2). Suppose both 𝑥∈𝐹(𝒮) and 𝑥^∗∈𝐹(𝒮) are solutions to (3.2), then 𝐴−𝛾𝑓𝑥,𝐽_𝜑𝑥−𝑥^∗𝑥≤0,𝐴−𝛾𝑓^∗,𝐽_𝜑𝑥^∗−𝑥≤0. (3.3) Adding (3.3), we obtain 𝐴−𝛾𝑓𝑥−𝐴−𝛾𝑓𝑥^∗,𝐽_𝜑𝑥−𝑥^∗≤0. (3.4) Noticing that for any 𝑥,𝑦∈𝐸, 𝐴−𝛾𝑓𝑥−𝐴−𝛾𝑓𝑦,𝐽_𝜑=𝐴𝑥−𝑦𝑥−𝑦,𝐽_𝜑𝑥−𝑦−𝛾𝑓(𝑥)−𝑓𝑦,𝐽_𝜑≥𝑥−𝑦𝛾𝑥−𝑦𝜑𝑥−𝑦−𝛾𝑓(𝑥)−𝑓𝑦𝐽_𝜑≥𝑥−𝑦=𝛾Φ𝑥−𝑦−𝛾𝛼Φ𝑥−𝑦Φ≥𝛾−𝛾𝛼𝑥−𝑦𝛾𝜑(1)−𝛾𝛼Φ𝑥−𝑦≥0. (3.5) Therefore 𝑥=𝑥^∗ and the uniqueness is proved. Below, we use 𝑥 to denote the unique solution of (3.2).
Since lim_𝑛→∞𝛼_𝑛=0, we may assume, without the loss of generality, that 𝛼_𝑛<𝜑(1)‖𝐴‖⁻¹.
For each integer 𝑛≥0, define a mapping 𝐺_𝑛∶𝐸→𝐸 by 𝐺_𝑛𝑦=𝛼_𝑛𝛾𝑓𝑦+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦𝑑𝑠,∀𝑦∈𝐸. (3.6) We show that 𝐺_𝑛 is a contraction mapping. For any 𝑥,𝑦∈𝐸, 𝐺_𝑛(𝑥)−𝐺_𝑛𝑦=𝛼_𝑛𝛾𝑓(𝑥)+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥𝑑𝑠−𝛼_𝑛𝛾𝑓𝑦−𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)≤𝛼𝑦𝑑𝑠_𝑛𝛾𝑓(𝑥)−𝑓𝑦+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)1𝑥𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦𝑑𝑠≤𝛼_𝑛𝛾𝛼𝑥−𝑦+𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠=𝛼𝑑𝑠𝑥−𝑦_𝑛(1)1𝛾𝛼+𝜑𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝜑(1)𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠≤1𝑑𝑠𝑥−𝑦𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛼_𝑛𝜑(1)𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛾𝛼𝑥−𝑦. (3.7) Since 0<((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛<𝜑(1)𝛾−𝛾𝛼, we have 0<1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1𝛼_𝑛<𝜑(1)𝛾−𝛾𝛼≤𝜑(1)𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛾𝛼. (3.8) It then follows that 0<((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛼_𝑛(𝜑(1)𝛾(1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛾𝛼))<1. We have 𝐺_𝑛 is a contraction map with coefficient ((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛼_𝑛(𝜑(1)𝛾(1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝛾𝛼)). Then, for each 𝑛≥0, there exists a unique 𝑦_𝑛∈𝐸 such that 𝐺_𝑛𝑦_𝑛=𝑦_𝑛, that is, 𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑦_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝑛≥0. (3.9) Hence (i) is proved.
(ii) We first show that {𝑦_𝑛} is bounded. Letting 𝑝∈𝐹(𝒮) and using Lemma 2.2, we can calculate the following: 𝑦_𝑛−𝑝=𝛼_𝑛𝛾𝑓𝑦_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛=𝛼𝑑𝑠−𝑝_𝑛𝛾𝑓𝑦_𝑛−𝛼_𝑛𝛾𝑓𝑝+𝛼_𝑛𝑝+𝛾𝑓𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠−𝐼−𝛼_𝑛𝐴𝑝−𝛼_𝑛𝐴𝑝≤𝛼_𝑛𝛾𝑓𝑦_𝑛−𝑓𝑝+𝛼_𝑛𝑝𝑝(1)𝛾𝑓−𝐴+𝜑1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑝𝑑𝑠≤𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛−𝑝≤𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑝+𝛼_𝑛𝑝𝑝(1)1𝛾𝑓−𝐴+𝜑𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛(1)𝛼−𝑝−𝜑_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛−𝑝≤𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑝+𝛼_𝑛𝑝𝑝+1𝛾𝑓−𝐴𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛(1)𝛼−𝑝−𝜑_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛−𝑝. (3.10) Thus, we get that 𝑦_𝑛≤𝛼−𝑝^𝑛𝛾𝑓𝑝−𝐴𝑝1−𝛼_𝑛𝛾𝛼−1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠+𝜑(1)𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠.𝑑𝑠 (3.11) Calculating the right-hand side of the above inequality, we have 𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝1−𝛼_𝑛𝛾𝛼−1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠+𝜑(1)𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝜑(1)𝛼_𝑛𝛾=𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝1−𝛼_𝑛𝛾𝛼−1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠+𝜑(1)𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝜑(1)𝛼_𝑛𝛾−𝜑(1)𝛼_𝑛𝛾=𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝𝜑(1)𝛼_𝑛𝛾−𝛼_𝑛𝛼𝛾+1−1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝜑(1)𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝜑(1)𝛼_𝑛𝛾=𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝𝛼_𝑛𝜑(1)𝛾−𝛼𝛾−1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1+𝜑(1)𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠=𝛼𝑑𝑠−1_𝑛𝛾𝑓𝑝−𝐴𝑝𝛼_𝑛𝜑(1)𝛾−𝛼𝛾−1−𝜑(1)𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠.𝑑𝑠−1 (3.12) Thus, we get that 𝑦_𝑛≤𝑝𝑝−𝑝𝛾𝑓−𝐴𝜑(1)𝛾−𝛼𝛾−1−𝛼_𝑛𝛾𝑑_𝑛, (3.13) where 𝑑_𝑛=((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛. Thus, there exists 𝑁>0 such that ‖𝑦_𝑛−𝑝‖≤‖𝛾𝑓(𝑝)−𝐴(𝑝)‖/(𝜑(1)𝛾−𝛾𝛼), for all 𝑛≥𝑁. Therefore, {𝑦_𝑛} is bounded and hence {𝑓(𝑦_𝑛)} and {(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠} are also bounded.
Let 𝛿_𝑡𝑛(𝑦_𝑛)=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠. Then, from (3.1), we get 𝑦_𝑛−𝛿_𝑡𝑛𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑦_𝑛−𝐴𝛿_𝑡𝑛𝑦_𝑛→0as𝑛→∞. (3.14) Since 𝒮 is almost uniformly asymptotically regular and (3.14), we have 𝑦_𝑛−𝑇(ℎ)𝑦_𝑛≤𝑦_𝑛−𝛿_𝑡𝑛𝑦_𝑛+𝛿_𝑡𝑛𝑦_𝑛−𝑇(ℎ)𝛿_𝑡𝑛𝑦_𝑛+𝑇(ℎ)𝛿_𝑡𝑛𝑦_𝑛−𝑇(ℎ)𝑦_𝑛≤𝑦_𝑛−𝛿_𝑡𝑛𝑦_𝑛+𝛿_𝑡𝑛𝑦_𝑛−𝑇(ℎ)𝛿_𝑡𝑛𝑦_𝑛+𝐿_𝑠𝛿_𝑡𝑛𝑦_𝑛−𝑦_𝑛→0. (3.15) It follows from the reflexivity of 𝐸 and the boundedness of sequence {𝑦_𝑛} that there exists {𝑦_𝑛𝑗} which is a subsequence of {𝑦_𝑛} converging weakly to 𝑤∈𝐸 as 𝑗→∞. Since 𝐽_𝜑 is weakly sequentially continuous, we have by Lemma 2.1 that limsup_𝑗→∞Φ𝑦_𝑛𝑗−𝑦=limsup_𝑗→∞Φ𝑦_𝑛𝑗−𝑤+Φ𝑦−𝑤,∀𝑥∈𝐸. (3.16) Let 𝐻(𝑥)=limsup_𝑗→∞Φ𝑦_𝑛𝑗−𝑦,∀𝑦∈𝐸. (3.17) It follows that 𝐻𝑦=𝐻(𝑤)+Φ𝑦−𝑤,∀𝑦∈𝐸. (3.18) For ℎ≥0, from (3.15) we obtain 𝐻(𝑇(ℎ)𝑤)=limsup_𝑗→∞Φ𝑦_𝑛𝑗−𝑇(ℎ)𝑤=limsup_𝑗→∞Φ𝑇(ℎ)𝑦_𝑛𝑗−𝑇(ℎ)𝑤≤limsup_𝑗→∞Φ𝑦_𝑛𝑗−𝑤=𝐻(𝑤). (3.19) On the other hand, however, 𝐻(𝑇(ℎ)𝑤)=𝐻(𝑤)+Φ(‖𝑇(ℎ)𝑤−𝑤‖). (3.20) It follows from (3.19) and (3.20) that Φ(‖𝑇(ℎ)𝑤−𝑤‖)=𝐻(𝑇(ℎ)𝑤)−𝐻(𝑤)≤0. (3.21) This implies that 𝑇(ℎ)𝑤=𝑤 for all ℎ≥0, and so 𝑤∈𝐹(𝒮). Next, we show that 𝑦_𝑛𝑗→𝑤 as 𝑗→∞. In fact, since Φ(𝑡)=∫^𝑡₀𝜑(𝜏)𝑑𝜏, for all 𝑡≥0, and 𝜑∶[0,∞)→[0,∞) is a gauge function, then for 1≥𝑘≥0,  𝜑(𝑘𝑥)≤𝜑(𝑥) and Φ(𝑘𝑡)=₀^𝑘𝑡𝜑(𝜏)𝑑𝜏=𝑘^𝑡₀𝜑(𝑘𝑥)𝑑𝑥≤𝑘^𝑡₀𝜑(𝑥)𝑑𝑥=𝑘Φ(𝑡). (3.22) Following Lemma 2.1, we have Φ𝑦_𝑛−𝑤=Φ𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠−𝐼−𝛼_𝑛𝐴𝑤+𝛼_𝑛𝛾𝑓𝑦_𝑛−𝛾𝑓(𝑤)+𝛾𝑓(𝑤)−𝐴(𝑤)≤Φ𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠−𝐼−𝛼_𝑛𝐴𝑤+𝛼_𝑛𝛾𝑓𝑦_𝑛−𝑓(𝑤)+𝛼_𝑛𝛾𝑓(𝑤)−𝐴(𝑤),𝐽_𝜑𝑦_𝑛𝜑(1−𝑤≤Φ)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑤𝑑𝑠+𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑤+𝛼_𝑛𝛾𝑓(𝑤)−𝑓(𝑤),𝐽_𝜑𝑦_𝑛𝜑(1)−𝑤≤Φ1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛−𝑤+𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑤+𝛼_𝑛𝛾𝑓(𝑤)−𝐴(𝑤),𝐽_𝜑𝑦_𝑛𝜑(1)−𝑤≤Φ1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑤+𝛼_𝑛𝛾𝑓(𝑤)−𝐴(𝑤),𝐽_𝜑𝑦_𝑛≤𝜑(1)−𝑤1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝛼_𝑛𝛾𝛼Φ𝑦_𝑛−𝑤+𝛼_𝑛𝛾𝑓(𝑤)−𝐴(𝑤),𝐽_𝜑𝑦_𝑛−𝑤. (3.23) This implies that Φ𝑦_𝑛≤1−𝑤1−𝜑(1)1−𝛼_𝑛𝛾1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝛼_𝑛𝛼𝛾𝛼^𝑛𝛾𝑓(𝑤)−𝐴(𝑤),𝐽_𝜑𝑦_𝑛−𝑤, (3.24) also Φ𝑦_𝑛≤1−𝑤𝜑(1)𝛾−𝛼𝛾−1−𝛼_𝑛𝛾𝑑_𝑛𝛾𝑓(𝑤)−𝐴(𝑤),𝐽_𝜑𝑦_𝑛−𝑤, (3.25) where 𝑑_𝑛=((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛. Now observing that 𝑦_𝑛𝑗⇀𝑤 implies 𝐽_𝜑(𝑦_𝑛𝑗−𝑤)⇀^∗0, we conclude from the above inequality that Φ𝑦_𝑛𝑗−𝑤→0as𝑗→∞. (3.26) Hence 𝑦_𝑛𝑗→𝑤 as 𝑗→∞. Next, we prove that 𝑤 solves the variational inequality (3.2). For any 𝑧∈𝐹(𝒮), we observe that 𝑦_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛−1𝑑𝑠𝑧−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑧𝑑𝑠,𝐽_𝜑𝑦_𝑛=𝑦−𝑧_𝑛−𝑧,𝐽_𝜑𝑦_𝑛−1−𝑧𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑧𝑑𝑠,𝐽_𝜑𝑦_𝑛𝑦−𝑧≥Φ_𝑛−1−𝑧𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑧𝑑𝑠𝐽_𝜑𝑦_𝑛𝑦−𝑧≥Φ_𝑛−1−𝑧𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑦_𝑛−𝑧𝐽_𝜑𝑦_𝑛𝑦−𝑧=Φ_𝑛−1−𝑧𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠Φ𝑦_𝑛=1−𝑧1−𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠Φ𝑦_𝑛−𝑧. (3.27) Since 𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑦_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠, (3.28) we can derive that 𝐴−𝛾𝑓𝑦_𝑛1=−𝛼_𝑛𝑦_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠+𝐴𝑦_𝑛1−𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠. (3.29) Since Φ is strictly increasing and ‖𝑦_𝑛−𝑝‖≤𝑀 for some 𝑀>0, we have Φ(‖𝑦_𝑛−𝑝‖)≤Φ(𝑀). Thus 𝐴−𝛾𝑓𝑦_𝑛,𝐽_𝜑𝑦_𝑛1−𝑧=−𝛼_𝑛𝑦_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛−1𝑑𝑠𝑧−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑧𝑑𝑠,𝐽_𝜑𝑦_𝑛+𝐴𝑦−𝑧_𝑛1−𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝐽_𝜑𝑦_𝑛≤−𝑧1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1𝛼_𝑛Φ𝑦_𝑛−𝑧+𝐴𝑦_𝑛1−𝐴𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝐽_𝜑𝑦_𝑛≤−𝑧1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1𝛼_𝑛Φ(𝑀)+𝐴𝑦_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝐽_𝜑𝑦_𝑛−𝑧. (3.30) Notice that 𝑦_𝑛𝑗−1𝑡_𝑛𝑗^𝑡𝑛𝑗₀𝑇(𝑠)𝑦_𝑛𝑗1𝑑𝑠→𝑤−𝑡_𝑛𝑗^𝑡𝑛𝑗₀𝑇(𝑠)𝑤𝑑𝑠=𝑤−𝑤=0. (3.31) Now using (B3) and replacing 𝑛 with 𝑛_𝑗 in (3.30) and letting 𝑗→∞, we have 𝐴−𝛾𝑓𝑤,𝐽_𝜑(𝑤−𝑧)≤0. (3.32) So, 𝑤∈𝐹(𝒮) is a solution of the variational inequality (3.2), and hence 𝑤=𝑥 by the uniqueness. In a summary, we have shown that each cluster point of {𝑦_𝑛} (at 𝑛→∞) equals 𝑥. Therefore, 𝑦_𝑛→𝑥 as 𝑛→∞. This completes the proof.

If 𝐴≡𝐼, the identity mapping on 𝐸, and 𝛾=1, then Theorem 3.1 reduces to the following corollary.

Corollary 3.2. Let 𝐸 be a reflexive Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let 𝒮={𝑇(𝑠)∶𝑠≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings on 𝐸 with a sequence {𝐿_𝑡}⊂[1,∞) and 𝐹(𝒮)≠∅. Let 𝑓∈Π_𝐸 with coefficient 𝛼∈(0,1) and let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers such that 0<𝛼_𝑛<1 and 𝑡_𝑛>0. Then the following holds.(i)If (1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1<𝛼_𝑛(1−𝛼),  for all  𝑛∈ℕ, then there exists a sequence {𝑦_𝑛} defined by 𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝑛≥0. (3.33)(ii)Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular and the real sequences {𝛼_𝑛} and {𝑡_𝑛} satisfy the following: (B1)lim_𝑛→∞𝑡_𝑛=∞; (B2)lim_𝑛→∞𝛼_𝑛=0; (B3)lim_𝑛→∞((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛=0.
Then {𝑦_𝑛} converges strongly as 𝑛→∞ to a common fixed point 𝑥 in 𝐹(𝒮) which solves the variational inequality: 𝐼−𝑓𝑥,𝐽_𝜑(𝑥−𝑧)≤0,𝑧∈𝐹(𝒮). (3.34)

If 𝐸∶=𝐻 is a Hilbert space and 𝒮={𝑇(𝑠)∶𝑠≥0} is a strongly continuous semigroup of nonexpansive mappings on 𝐻, then we have 𝐿_𝑡≡1 and Theorem 3.1 reduces to the following corollary.

Corollary 3.3 (see [15, Theorem 3.1]). Let 𝐻 be a real Hilbert space. Suppose that 𝑓∶𝐻→𝐻 is a contraction with coefficient 𝛼∈(0,1) and 𝒮={𝑇(𝑠)∶𝑠≥0} a strongly continuous semigroup of nonexpansive mappings on 𝐻 such that 𝐹(𝒮)≠∅. Let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 and let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers such that 0<𝛼_𝑛<1,  𝑡_𝑛>0 such that lim_𝑛→∞𝑡_𝑛=∞ and lim_𝑛→∞𝛼_𝑛=0, then for any 0<𝛾<𝛾/𝛼, there is a unique {𝑦_𝑛} in 𝐻 such that 𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑦_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑦_𝑛𝑑𝑠,𝑛≥0 (3.35) and the iterative sequence {𝑦_𝑛} converges strongly as 𝑛→∞ to a common fixed point 𝑥 in 𝐹(𝒮) which solves the variational inequality: 𝐴−𝛾𝑓𝑥,𝑥−𝑧≤0,𝑧∈𝐹(𝒮). (3.36)

Theorem 3.4. Let 𝐸 be a reflexive strictly convex Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let 𝒮={𝑇(𝑠)∶𝑠≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings on 𝐸 with a sequence {𝐿_𝑡}⊂[1,∞) and 𝐹(𝒮)≠∅. Let 𝑓∈Π_𝐸 with coefficient 𝛼∈(0,1); let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<𝜑(1)𝛾/𝛼. For any 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be defined by 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0. (3.37) Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular. Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers such that 0<𝛼_𝑛<1,  𝑡_𝑛>0, (C1)lim_𝑛→∞𝑡_𝑛=∞; (C2)lim_𝑛→∞𝛼_𝑛=0,  ∑^∞_𝑛=0𝛼_𝑛=∞; (C3)lim_𝑛→∞((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛=0.
Then {𝑥_𝑛} converges strongly as 𝑛→∞ to a common fixed point 𝑥 in 𝐹(𝒮) which solves the variational inequality (3.2).

Proof. First we show that {𝑥_𝑛} is bounded. By condition (C3) and given 0<𝜀<𝜑(1)𝛾−𝛼𝛾 there exists 𝑁>0 such that ((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛<𝜀 for all 𝑛≥𝑁. Thus 1−𝜑(1)𝛾𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠≤1𝑑𝑠−1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1<𝜀𝛼_𝑛, (3.38) for all 𝑛≥𝑁. Since lim_𝑛→∞𝛼_𝑛=0, we may assume, without the loss of generality, that 𝛼_𝑛<𝜑(1)‖𝐴‖⁻¹.
Claim that ‖𝑥_𝑛−𝑝‖≤𝑀,  𝑛≥0, where 𝑀∶=max{‖𝑥₀−𝑝‖,…,‖𝑥_𝑁−𝑝‖,‖𝑓(𝑝)−𝑝‖/(𝜑(1)𝛾−𝛼𝛾−𝜀)}.
Let 𝑝∈𝐹(𝒮). Then from (3.56) we get that 𝑥_𝑛+1−𝑝=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛≤𝛼𝑑𝑠−𝑝_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝛼_𝑛𝐴𝑝−𝐼−𝛼_𝑛𝐴𝑝≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛼_𝑛𝛾𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝛼_𝑛𝐴𝑝+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)≤𝛼𝑝𝑑𝑠_𝑛𝛾𝑓𝑥_𝑛−𝛼_𝑛𝛾𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝛼_𝑛𝐴𝑝+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑝𝑑𝑠≤𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+𝛼_𝑛𝛼𝛾𝑥_𝑛−𝑝+𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑥_𝑛−𝑝=𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+𝛼_𝑛𝛼𝛾+𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+𝛼_𝑛1𝛼𝛾+𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−𝜑(1)𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑥_𝑛−𝑝≤𝛼_𝑛𝑝+(1)𝛼𝛾𝑓−𝐴𝑝1+1−𝜑_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1−𝛼_𝑛𝜑(1)𝛾−𝛼𝛾𝑥_𝑛−𝑝≤𝛼_𝑛𝑓𝑝−𝑝+1−𝛼_𝑛𝜑(1)𝛾−𝛼𝛾+𝜀𝛼_𝑛𝑥_𝑛−𝑝=𝛼_𝑛𝑓𝑝−𝑝+1−𝛼_𝑛𝜑(1)𝛾−𝛼𝛾−𝜀𝑥_𝑛𝑓𝑝−𝑝≤max−𝑝𝜑(1),𝑥𝛾−𝛼𝛾−𝜀^𝑛−𝑝. (3.39) By induction, 𝑥_𝑛𝑓𝑝−𝑝≤max−𝑝𝜑(1),𝑥𝛾−𝛼𝛾−𝜀^𝑁−𝑝,∀𝑛≥𝑁, (3.40) and hence {𝑥_𝑛} is bounded, so are {𝑓(𝑥_𝑛)} and {(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠}. Let 𝛿_𝑡𝑛(𝑥_𝑛)∶=(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠. Then, since 𝛼_𝑛→0 as 𝑛→∞, we obtain that 𝑥_𝑛+1−𝛿_𝑡𝑛𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝛿_𝑡𝑛𝑥_𝑛→0as𝑛→∞. (3.41) For any ℎ>0, we have 𝑇(ℎ)𝑥_𝑛+1−𝑥_𝑛+1≤𝑇(ℎ)𝑥_𝑛+1−𝑇(ℎ)𝛿_𝑡𝑛𝑥_𝑛+𝑇(ℎ)𝛿_𝑡𝑛𝑥_𝑛−𝛿_𝑡𝑛𝑥_𝑛+𝛿_𝑡𝑛𝑥_𝑛−𝑥_𝑛+1≤𝐿_ℎ𝑥_𝑛+1−𝛿_𝑡𝑛𝑥_𝑛+𝑇(ℎ)𝛿_𝑡𝑛𝑥_𝑛−𝛿_𝑡𝑛𝑥_𝑛+𝛿_𝑡𝑛𝑥_𝑛−𝑥_𝑛+1, (3.42) it follows from (3.41) and 𝒮 is almost uniformly asymptotically regular that 𝑇(ℎ)𝑥_𝑛+1−𝑥_𝑛+1→0as𝑛→∞. (3.43) Next, we prove that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥≤0. (3.44) Let {𝑥_𝑛𝑘} be a subsequence of {𝑥_𝑛} such that lim_𝑘→∞𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛𝑘−𝑥=limsup_𝑛→∞𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥. (3.45) It follows from the reflexivity of 𝐸 and the boundedness of sequence {𝑥_𝑛𝑘} that there exists {𝑥_𝑛𝑘𝑖} which is a subsequence of {𝑥_𝑛𝑘} converging weakly to 𝑤∈𝐸 as 𝑖→∞. Since 𝐽_𝜑 is weakly continuous, we have by Lemma 2.1 that limsup_𝑖→∞Φ𝑥_𝑛𝑘𝑖−𝑥=limsup_𝑖→∞Φ𝑥_𝑛𝑘𝑖−𝑤+Φ(‖𝑥−𝑤‖),∀𝑥∈𝐸. (3.46) Let 𝐻(𝑥)=limsup_𝑖→∞Φ𝑥_𝑛𝑘𝑖−𝑥,∀𝑥∈𝐸. (3.47) It follows that 𝐻(𝑥)=𝐻(𝑤)+Φ(‖𝑥−𝑤‖),∀𝑥∈𝐸. (3.48) From (3.43), for each ℎ>0, we obtain 𝐻(𝑇(ℎ)𝑤)=limsup_𝑖→∞Φ𝑥_𝑛𝑘𝑖−𝑇(ℎ)𝑤=limsup_𝑖→∞Φ𝑇(ℎ)𝑥_𝑛𝑘𝑖−𝑇(ℎ)𝑤≤limsup_𝑖→∞Φ𝑥_𝑛𝑘𝑖−𝑤=𝐻(𝑤). (3.49) On the other hand, however, 𝐻(𝑇(ℎ)𝑤)=𝐻(𝑤)+Φ(‖𝑇(ℎ)𝑤−𝑤‖). (3.50) It follows from (3.49) and (3.50) that Φ(‖𝑇(ℎ)𝑤−𝑤‖)=𝐻(𝑇(ℎ)𝑤)−𝐻(𝑤)≤0. (3.51) This implies that 𝑇(ℎ)𝑤=𝑤 for all ℎ>0, and so 𝑤∈𝐹(𝒮). Since the duality map 𝐽_𝜑 is single valued and weakly continuous, we get that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥=lim_𝑘→∞𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛𝑘−𝑥=lim_𝑖→∞𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛𝑘𝑖−𝑥=𝐴−𝛾𝑓𝑥,𝐽_𝜑(𝑥−𝑤)≤0 (3.52) as required. Finally, we show that 𝑥_𝑛→𝑥 as 𝑛→∞. It follows from Lemma 2.1(i) that Φ𝑥_𝑛+1−𝑥=Φ𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝐼−𝛼_𝑛𝐴𝑥+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑥)+𝛾𝑓(𝑥)−𝐴(𝑥)≤Φ𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝐼−𝛼_𝑛𝐴𝑥+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓(𝑥)+𝛼_𝑛𝛾𝑓(𝑥)−𝐴(𝑥),𝐽_𝜑𝑦_𝑛−𝑥≤Φ𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛1𝑑𝑠−𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥𝑑𝑠+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝑓(𝑥),𝐽_𝜑𝑦_𝑛−𝑥≤Φ𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝐴(𝑥),𝐽_𝜑𝑥_𝑛−𝑥≤Φ𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝐴(𝑥),𝐽_𝜑𝑥_𝑛−𝑥≤𝜑(1)1−𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠+𝛼_𝑛𝛾𝛼Φ𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝐴(𝑥),𝐽_𝜑𝑥_𝑛−𝑥≤1−𝜑(1)𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1+1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼Φ𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝑓(𝑥)−𝐴(𝑥),𝐽_𝜑𝑥_𝑛−𝑥≤1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼Φ𝑥_𝑛𝑋−𝑥+1−𝜑(1)𝛼_𝑛𝛾1𝑡_𝑛^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1𝑀+𝛼_𝑛𝛾𝑓(𝑥)−𝐴(𝑥),𝐽_𝜑𝑥_𝑛−𝑥, (3.53) where 𝑀>0 such that Φ(‖𝑥_𝑛−𝑥‖)≤𝑀. Put 𝑠_𝑛=𝛼_𝑛𝜑(1),𝑡𝛾−𝛾𝛼_𝑛=1−𝜑(1)𝛼_𝑛𝛾𝜑(1)𝛾−𝛾𝛼1/𝑡_𝑛∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1𝛼_𝑛1𝑀+𝜑(1)(𝑥)(𝑥)𝛾−𝛾𝛼𝛾𝑓−𝐴,𝐽_𝜑𝑥_𝑛−𝑥. (3.54) Then (3.53) is reduced to Φ𝑥_𝑛+1−𝑥≤1−𝑠_𝑛Φ𝑥_𝑛−𝑥+𝑠_𝑛𝑡_𝑛. (3.55) Applying Lemma 2.3 to (3.55), we conclude that Φ(‖𝑥_𝑛+1−𝑥‖)→0 as 𝑛→∞; that is, 𝑥_𝑛→𝑥 as 𝑛→∞. This completes the proof.

If 𝐴≡𝐼, the identity mapping on 𝐸, and 𝛾=1, then Theorem 3.4 reduces to the following corollary.

Corollary 3.5. Let 𝐸 be a reflexive strictly convex Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let 𝒮={𝑇(𝑠)∶𝑠≥0} be a strongly continuous semigroup of asymptotically nonexpansive mappings from 𝐶 into 𝐶 with a sequence {𝐿_𝑡}⊂[1,∞), 𝐹(𝒮)≠∅. Let 𝑓∈Π_𝐸 with coefficient 𝛼∈(0,1) and the sequence {𝑥_𝑛} be defined by 𝑥₀∈𝐶, 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0. (3.56) Suppose, in addition, that 𝒮 is almost uniformly asymptotically regular. Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers such that 0<𝛼_𝑛<1,  𝑡_𝑛>0, (C1)lim_𝑛→∞𝑡_𝑛=∞; (C2)lim_𝑛→∞𝛼_𝑛=0,  ∑^∞_𝑛=0𝛼_𝑛=∞; (C3)lim_𝑛→∞((1/𝑡_𝑛)∫^𝑡𝑛₀𝐿_𝑠𝑑𝑠−1)/𝛼_𝑛=0.
Then {𝑥_𝑛} converges strongly as 𝑛→∞ to a common fixed point 𝑥 in 𝐹(𝒮) which solves the variational inequality: 𝐼−𝑓𝑥,𝐽_𝜑(𝑥−𝑧)≤0,𝑧∈𝐹(𝒮). (3.57)

If 𝐸∶=𝐻 is a Hilbert space and 𝒮={𝑇(𝑠)∶𝑠≥0} is a strongly continuous semigroup of nonexpansive mappings on 𝐻, then we have 𝐿_𝑡≡1 and Theorem 3.4 reduces to the following corollary.

Corollary 3.6 (see [15, Theorem 3.2]). Let 𝐻 be a real Hilbert space. Suppose that 𝑓∶𝐻→𝐻 is a contraction with coefficient 𝛼∈(0,1) and 𝒮={𝑇(𝑠)∶𝑠≥0} a strongly continuous semigroup of nonexpansive mappings on 𝐻 such that 𝐹(𝒮)≠∅. Let 𝐴 be a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<𝛾/𝛼 and let the sequence {𝑥_𝑛} be defined by 𝑥₀∈𝐶, 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,𝑛≥0. (3.58) Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers such that 0<𝛼_𝑛<1,  𝑡_𝑛>0, (C1)lim_𝑛→∞𝑡_𝑛=∞; (C2)lim_𝑛→∞𝛼_𝑛=0,  ∑^∞_𝑛=0𝛼_𝑛=∞.
Then {𝑥_𝑛} converges strongly as 𝑛→∞ to a common fixed point 𝑥 in 𝐹(𝒮) which solves the variational inequality: 𝐴−𝛾𝑓𝑥,𝑥−𝑧≤0,𝑧∈𝐹(𝒮). (3.59)

Acknowledgment

R. Wangkeeree is supported by Naresuan University.