Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Abstract

Let 𝐸 be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from 𝐸 to 𝐸^∗. Let 𝒮={𝑇(𝑠)∶0≤𝑠<∞} be a nonexpansive semigroup on 𝐸 such that Fix(𝒮)∶=⋂_𝑡≥0Fix(𝑇(𝑡))≠∅, and 𝑓 is a contraction on 𝐸 with coefficient 0<𝛼<1. Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1 and 𝛾 a positive real number such that 𝛾<1/𝛼(1−1−𝛿/𝜆). When the sequences of real numbers {𝛼_𝑛} and {𝑡_𝑛} satisfy some appropriate conditions, the three iterative processes given as follows: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐹)𝑇(𝑡_𝑛)𝑥_𝑛, 𝑛≥0, 𝑦_𝑛+1=𝛼_𝑛𝛾𝑓(𝑇(𝑡_𝑛)𝑦_𝑛)+(𝐼−𝛼_𝑛𝐹)𝑇(𝑡_𝑛)𝑦_𝑛, 𝑛≥0, and 𝑧_𝑛+1=𝑇(𝑡_𝑛)(𝛼_𝑛𝛾𝑓(𝑧_𝑛)+(𝐼−𝛼_𝑛𝐹)𝑧_𝑛), 𝑛≥0 converge strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality ⟨(𝐹−𝛾𝑓)𝑥,𝑗(𝑥−𝑥)⟩≥0, 𝑥∈Fix(𝒮). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.

1. Introduction

Let 𝐸 be a real Banach space. A mapping 𝑇 of 𝐸 into itself is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for each 𝑥,𝑦∈𝐸. We denote by Fix(𝑇) the set of fixed points of 𝑇. A mapping 𝑓∶𝐸→𝐸 is called 𝛼-contraction if there exists a constant 0<𝛼<1 such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸. A family 𝒮={𝑇(𝑡)∶0≤𝑡<∞} of mappings of 𝐸 into itself is called a nonexpansive semigroup 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)‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐸 and 𝑡≥0; (iv) for all 𝑥∈𝐸, the mapping 𝑡↦𝑇(𝑡)𝑥 is continuous.

We denote by Fix(𝒮) the set of all common fixed points of 𝒮, that is, Fix(𝒮)∶={𝑥∈𝐸∶𝑇(𝑡)𝑥=𝑥,0≤𝑡<∞}=_𝑡≥0Fix(𝑇(𝑡)). (1.1)

In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space𝑥_𝑛=𝛼_𝑛𝑥+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ, (1.2) where {𝛼_𝑛} is a sequence in (0,1) and {𝑡_𝑛} is a sequence of positive real numbers which diverges to ∞. Under certain restrictions on the sequence {𝛼_𝑛}, Shioji and Takahashi [1] proved strong convergence of the sequence {𝑥_𝑛} to a member of 𝐹(𝒮). In [2], Shimizu and Takahashi studied the strong convergence of the sequence {𝑥_𝑛} defined by𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ (1.3) in a real Hilbert space where {𝑇(𝑡)∶𝑡≥0} is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset 𝐶 of a Banach space 𝐸 and lim_𝑛→∞𝑡_𝑛=∞. Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup {𝑇(𝑡)∶𝑡≥0} with Fix(𝒮)≠∅ in a Banach space:𝑥_𝑛+1=𝛼_𝑛𝑓(𝑥)+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ, (1.4) where 𝑓 is a contraction. Note however that their iterate 𝑥_𝑛 at step 𝑛 is constructed through the average of the semigroup over the interval (0,𝑡). Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process:𝑥_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,∀𝑛∈ℕ, (1.5) for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if 𝒮 satisfies an asymptotic regularity condition and {𝛼_𝑛} fulfills the control conditions lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, and lim_𝑛→∞𝛼_𝑛/𝛼_𝑛+1=0, then both the implicit iteration process (1.5) and the explicit iteration process (1.6),𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,∀𝑛∈ℕ, (1.6) converge to a same point of 𝐹(𝒮). In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping. Recently, Chen and He [7] introduced the viscosity approximation process:𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑇𝑡_𝑛𝑥_𝑛,∀𝑛∈ℕ, (1.7) where 𝑓 is a contraction and {𝛼_𝑛} is a sequence in (0,1) and a nonexpansive semigroup {𝑇(𝑡)∶𝑡≥0}. The strong convergence theorem of {𝑥_𝑛} is proved in a reflexive Banach space which admits a weakly sequentially continuous duality mapping. In [8], Chen et al. introduced and studied modified Mann iteration for nonexpansive mapping in a uniformly convex Banach space.

On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [9–11] and the references therein. 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. (1.8) 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⟩, (1.9) 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 [10] 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_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑢,𝑛≥0, (1.10) converges strongly to the unique solution of the minimization problem (1.9) provided the sequence {𝛼_𝑛} satisfies certain conditions. Using the viscosity approximation method, Moudafi [12] introduced the following iterative process for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial 𝑥₀∈𝐻, define a sequence {𝑥_𝑛} recursively by𝑥_𝑛+1=1−𝛼_𝑛𝑇𝑥_𝑛+𝛼_𝑛𝑓𝑥_𝑛,𝑛≥0, (1.11) where {𝛼_𝑛} is a sequence in (0,1). It is proved [12, 13] that, under certain appropriate conditions imposed on {𝛼_𝑛}, the sequence {𝑥_𝑛} generated by (1.11) strongly converges to the unique solution 𝑥^∗ in 𝐶 of the variational inequality𝐼−𝑓𝑥^∗,𝑥−𝑥^∗≥0,𝑥∈𝐻. (1.12) Recently, Marino and Xu [14] mixed the iterative method (1.10) and the viscosity approximation method (1.11) and considered the following general iterative method:𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0, (1.13) where 𝐴 is a strongly positive bounded linear operator on 𝐻. They proved that if the sequence {𝛼_𝑛} of parameters satisfies the certain conditions, then the sequence {𝑥_𝑛} generated by (1.13) converges strongly to the unique solution 𝑥^∗ in 𝐻 of the variational inequality𝐴−𝛾𝑓𝑥^∗,𝑥−𝑥^∗≥0,𝑥∈𝐻 (1.14) 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).

Very recently, Li et al. [15] introduced the following iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup on a Hilbert space 𝐻:𝑥₀=𝑥∈𝐻,𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0, (1.15) where 𝐴 is a strongly positive bounded linear operator on 𝐻.

Let 𝛿 and 𝜆 be two positive real numbers such that 𝛿,𝜆<1. Recall that a mapping 𝐹 with domain 𝐷(𝐹) and range 𝑅(𝐹) in 𝐸 is called 𝛿-strongly accretive if, for each 𝑥,𝑦∈𝐷(𝐹), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that𝐹𝑥−𝐹𝑦,𝑗𝑥−𝑦≥𝛿𝑥−𝑦², (1.16) where 𝐽 is the normalized duality mapping from 𝐸 into the dual space 𝐸^∗. Recall also that a mapping 𝐹 is called 𝜆-strictly pseudocontractive if, for each 𝑥,𝑦∈𝐷(𝐹), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that𝐹𝑥−𝐹𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−𝐹𝑥−𝐹𝑦². (1.17) It is easy to see that (1.17) can be rewritten as(𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦,𝑗𝑥−𝑦≥𝜆(𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦², (1.18) see [16].

In this paper, motivated by the above results, we introduce and study the strong convergence theorems of the general iterative scheme {𝑥_𝑛} defined by (1.19) in the framework of a reflexive Banach space 𝐸 which admits a weakly sequentially continuous duality mapping:𝑥₀=𝑥∈𝐸,𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛,𝑛≥0, (1.19) where 𝐹 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓 is a contraction on 𝐸 with coefficient 0<𝛼<1, 𝛾 is a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆), and 𝒮={𝑇(𝑡)∶0≤𝑡<∞} is a nonexpansive semigroup on 𝐸. The strong convergence theorems are proved under some appropriate control conditions on parameters {𝛼_𝑛} and {𝑡_𝑛}. Furthermore, by using these results, we obtain strong convergence theorems of the following new general iterative schemes {𝑦_𝑛} and {𝑧_𝑛} defined by𝑦₀=𝑦∈𝐸,𝑦_𝑛+1=𝛼_𝑛𝛾𝑓𝑇𝑡_𝑛𝑦_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑦_𝑛𝑧,𝑛≥0,₀=𝑧∈𝐸,𝑧_𝑛+1=𝑇𝑡_𝑛𝛼_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐹𝑧_𝑛,𝑛≥0. (1.20)(1.21) The results presented in this paper extend and improve the main results in Li et al. [15], Chen and He [7], and many others.

2. Preliminaries

Throughout this paper, it is assumed that 𝐸 is a real Banach space with norm ‖⋅‖ and let 𝐽 denote the normalized duality mapping from 𝐸 into 𝐸^∗ given by 𝐽(𝑥)=𝑓∈𝐸^∗∶𝑥,𝑓=‖𝑥‖²=𝑓² (2.1) for each 𝑥∈𝐸, where 𝐸^∗ denotes the dual space of 𝐸,⟨⋅,⋅⟩ denotes the generalized duality pairing, and ℕ denotes the set of all positive integers. In the sequel, we will denote the single-valued duality mapping by 𝑗, and consider 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. When {𝑥_𝑛} is a sequence in 𝐸, then 𝑥_𝑛→𝑥 (resp., 𝑥_𝑛⇀𝑥, 𝑥⇀_𝑛^∗𝑥 ) will denote strong (resp., weak, weak*) convergence of the sequence {𝑥_𝑛} to 𝑥. In a Banach space 𝐸, the following result (the subdifferential inequality) is well known [17, Theorem  4.2.1]: for all 𝑥,𝑦∈𝐸, for all 𝑗(𝑥+𝑦)∈𝐽(𝑥+𝑦), for all 𝑗(𝑥)∈𝐽(𝑥),‖𝑥‖²+2𝑦,𝑗(𝑥)≤𝑥+𝑦²≤‖𝑥‖²+𝑦,𝑗𝑥+𝑦. (2.2) A real Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. It is said to be uniformly convex if, for all 𝜖∈[0,2], there exits 𝛿_𝜖>0 such that ‖𝑥‖=𝑦=1with𝑥−𝑦≥𝜖implies𝑥+𝑦2<1−𝛿_𝜖. (2.3) The following results are well known and can be founded in [17]:(i)a uniformly convex Banach space 𝐸 is reflexive and strictly convex [17, Theorems  4.2.1 and 4.1.6],(ii)if 𝐸 is a strictly convex Banach space and 𝑇∶𝐸→𝐸 is a nonexpansive mapping, then fixed point set 𝐹(𝑇) of 𝑇 is a closed convex subset of 𝐸 [17, Theorem  4.5.3].

If a Banach space 𝐸 admits a sequentially continuous duality mapping 𝐽 from weak topology to weak star topology, then from Lemma  1 of [18], it follows that the duality mapping 𝐽 is single-valued and also 𝐸 is smooth. In this case, duality mapping 𝐽 is also said to be weakly sequentially continuous, that is, for each {𝑥_𝑛}⊂𝐸 with 𝑥_𝑛⇀𝑥, then 𝐽(𝑥_𝑛)⇀^∗𝐽(𝑥) (see [18, 19]).

In the sequel, we will denote the single-valued duality mapping by 𝑗. A Banach space 𝐸 is said to satisfy Opial's condition if, for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥 as 𝑛→∞ implieslimsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦∀𝑦∈𝐸with𝑥≠𝑦. (2.4) By Theorem  1 of [18], we know that if 𝐸 admits a weakly sequentially continuous duality mapping, then 𝐸 satisfies Opial’s condition and 𝐸 is smooth; for the details, see [18].

Now, we present the concept of uniformly asymptotically regular semigroup (also see [20, 21]). Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸, 𝒮={𝑇(𝑡)∶0≤𝑡<∞} a continuous operator semigroup on 𝐶. Then, 𝒮 is said to be uniformly asymptotically regular (in short, u.a.r.) on 𝐶 if, for all ℎ≥0 and any bounded subset 𝐷 of 𝐶,lim_𝑡→∞sup_𝑥∈𝐷‖𝑇(ℎ)(𝑇(𝑡)𝑥)−𝑇(𝑡)𝑥‖=0. (2.5) The nonexpansive semigroup {𝜎_𝑡∶𝑡>0} defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [20, Examples  17 and 18].

Lemma 2.1 (see [3, Lemma  2.7]). Let 𝐶 be a nonempty closed convex subset of a uniformly convex Banach space 𝐸, 𝐷 a bounded closed convex subset of 𝐶, and 𝒮={𝑇(𝑠)∶0≤𝑠<∞} a nonexpansive semigroup on 𝐶 such that 𝐹(𝒮)≠∅. For each ℎ>0, set 𝜎_𝑡(𝑥)=(1/𝑡)∫^𝑡₀𝑇(𝑠)𝑥𝑑𝑠, then lim_𝑡→∞sup_𝑥∈𝐷𝜎_𝑡(𝑥)−𝑇(ℎ)𝜎_𝑡(𝑥)=0. (2.6)

Example 2.2. The set {𝜎_𝑡∶𝑡>0} defined by Lemma 2.1 is u.a.r. nonexpansive semigroup. In fact, it is obvious that {𝜎_𝑡∶𝑡>0} is a nonexpansive semigroup. For each ℎ>0, we have 𝜎_𝑡(𝑥)−𝜎_ℎ𝜎_𝑡(𝑥)=𝜎_𝑡(𝑥)−1ℎ^ℎ₀𝑇(𝑠)𝜎_𝑡(𝑥)=1𝑑𝑠ℎ^ℎ₀𝜎_𝑡(𝑥)−𝑇(𝑠)𝜎_𝑡(𝑥)≤1𝑑𝑠ℎ^ℎ₀𝜎_𝑡(𝑥)−𝑇(𝑠)𝜎_𝑡(𝑥)𝑑𝑠. (2.7) Applying Lemma 2.1, we have lim_𝑡→∞sup_𝑥∈𝐷𝜎_𝑡(𝑥)−𝜎_ℎ𝜎_𝑡(𝑥≤1)ℎ^ℎ₀lim_𝑡→∞sup_𝑥∈𝐷𝜎_𝑡(𝑥)−𝑇(𝑠)𝜎_𝑡(𝑥)𝑑𝑠=0. (2.8)
Let 𝐶 be a nonempty closed and convex subset of a Banach space 𝐸 and 𝐷 a nonempty subset of 𝐶. A mapping 𝑄∶𝐶→𝐷 is said to be sunny if 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥, (2.9) whenever 𝑄𝑥+𝑡(𝑥−𝑄𝑥)∈𝐶 for 𝑥∈𝐶 and 𝑡=0. A mapping 𝑄∶𝐶→𝐷 is called a retraction if 𝑄𝑥=𝑥 for all 𝑥∈𝐷. Furthermore, 𝑄 is a sunny nonexpansive retraction from 𝐶 onto 𝐷 if 𝑄 is a retraction from 𝐶 onto 𝐷 which is also sunny and nonexpansive. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retraction of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.3 (see [22, 23]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸. Let 𝐷 be a nonempty subset of 𝐶 and 𝑄∶𝐶→𝐷 be a retraction. Then, 𝑄 is sunny and nonexpansive if and only if 𝑢−𝑄𝑢,𝑗𝑦−𝑄𝑢≤0 (2.10) for all 𝑢∈𝐶 and 𝑦∈𝐷.

Lemma 2.4 (see [24, Lemma  2.3]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the property 𝑎_𝑛+1≤1−𝑡_𝑛𝑎_𝑛+𝑡_𝑛𝑐_𝑛+𝑏_𝑛,∀𝑛≥0, (2.11) where {𝑡_𝑛},{𝑏_𝑛},and{𝑐_𝑛} satisfy the restrictions (i)∑^∞_𝑛=1𝑡_𝑛=∞; (ii)∑^∞_𝑛=1𝑏_𝑛<∞; (iii)limsup_𝑛→∞𝑐_𝑛≤0.
Then, lim_𝑛→∞𝑎_𝑛=0.

The following lemma will be frequently used throughout the paper and can be found in [25].

Lemma 2.5 (see [25, Lemma  2.7]). Let 𝐸 be a real smooth Banach space and 𝐹∶𝐸→𝐸 a mapping.
(i)If 𝐹 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, then 𝐼−𝐹 is contractive with constant (1−𝛿)/𝜆.
(i)If 𝐹 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, then, for any fixed number 𝜏∈(0,1), 𝐼−𝜏𝐹 is contractive with constant 1−𝜏(1−(1−𝛿)/𝜆).

3. Main Results

Now, we are in a position to state and prove our main results.

Theorem 3.1. Let 𝐸 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0≤𝑡<∞} be a u.a.r. nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Suppose that the real sequences {𝛼_𝑛}⊂[0,1], {𝑡_𝑛}⊂(0,∞) satisfy the conditions lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,lim_𝑛→∞𝑡_𝑛=∞. (3.1) Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑥_𝑛} defined by (1.19) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.2) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Proof. Note that Fix(𝒮) is a nonempty closed convex set. We first show that {𝑥_𝑛} is bounded. Let 𝑞∈Fix(𝒮). Thus, by Lemma 2.5, we have 𝑥_𝑛+1−𝑞=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝐹𝑞−𝛼_𝑛𝐹𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐹𝑞+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛−𝑞≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑞+𝛼_𝑛𝛾𝑓𝑞−𝐹𝑞+𝐼−𝛼_𝑛𝐹𝑥_𝑛−𝑞≤𝛼_𝑛𝛼𝛾𝑥_𝑛−𝑞+𝛼_𝑛𝑞+𝛾𝑓−𝐹𝑞1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛=−𝑞1−𝛼_𝑛1−1−𝛿𝜆−𝛼𝛾𝑥_𝑛−𝑞+𝛼_𝑛1−1−𝛿𝜆𝑞−𝛼𝛾𝛾𝑓−𝐹𝑞1−()𝑥1−𝛿/𝜆−𝛼𝛾≤max_𝑛,1−𝑞1−()𝑞1−𝛿/𝜆−𝛼𝛾𝛾𝑓−𝐹𝑞,∀𝑛≥0. (3.3) By induction, we get 𝑥_𝑛−𝑞≤max𝑥₀,1−𝑞1−()𝑞1−𝛿/𝜆−𝛼𝛾𝛾𝑓−𝐹𝑞,𝑛≥0. (3.4) This implies that {𝑥_𝑛} is bounded and, hence, so are {𝑓(𝑥_𝑛)} and {𝐹𝑇(𝑡_𝑛)𝑥_𝑛}. This implies that lim_𝑛→∞𝑥_𝑛+1−𝑇𝑡_𝑛𝑥_𝑛=lim_𝑛→∞𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐹𝑇𝑡_𝑛𝑥_𝑛=0. (3.5) Since {𝑇(𝑡)} is a u.a.r. nonexpansive semigroup and lim_𝑛→∞𝑡_𝑛=∞, we have, for all ℎ>0, lim_𝑛→∞𝑇(ℎ)𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑥_𝑛≤lim_𝑛→∞sup_{𝑥𝑥∈𝑛}𝑇(ℎ)𝑇𝑡_𝑛𝑥−𝑇𝑡_𝑛𝑥=0. (3.6) Hence, for all ℎ>0, 𝑥_𝑛+1−𝑇(ℎ)𝑥_𝑛+1≤𝑥_𝑛+1−𝑇𝑡_𝑛𝑥_𝑛+𝑇𝑡_𝑛𝑥_𝑛−𝑇(ℎ)𝑇𝑡_𝑛𝑥_𝑛+𝑇(ℎ)𝑇𝑡_𝑛𝑥_𝑛−𝑇(ℎ)𝑥_𝑛+1≤2𝑥_𝑛+1−𝑇𝑡_𝑛𝑥_𝑛+𝑇𝑡_𝑛𝑥_𝑛−𝑇(ℎ)𝑇𝑡_𝑛𝑥_𝑛⟶0. (3.7) That is, for all ℎ>0, lim_𝑛→∞𝑥_𝑛−𝑇(ℎ)𝑥_𝑛=0. (3.8) Let Φ=𝑄_Fix(𝒮). Then, Φ(𝐼−𝐹−𝛾𝑓) is a contraction on 𝐸. In fact, from Lemma 2.5(i), we have Φ𝑦≤𝑦≤)()𝑦𝑓(𝑥)𝑦≤𝐼−𝐹−𝛾𝑓𝑥−Φ𝐼−𝐹−𝛾𝑓𝐼−𝐹−𝛾𝑓𝑥−𝐼−𝐹−𝛾𝑓(𝐼−𝐹𝑥−𝐼−𝐹+𝛾−𝑓1−𝛿𝜆=𝑥−𝑦+𝛼𝛾𝑥−𝑦1−𝛿𝜆+𝛼𝛾𝑥−𝑦,∀𝑥,𝑦∈𝐸. (3.9) Therefore, Φ(𝐼−𝐹−𝛾𝑓) is a contraction on 𝐸 due to ((1−𝛿)/𝜆+𝛼𝛾)∈(0,1). Thus, by Banach contraction principle, 𝑄_Fix(𝒮)(𝐼−𝐹−𝛾𝑓) has a unique fixed point 𝑥. Then, using Lemma 2.3, 𝑥 is the unique solution in Fix(S) of the variational inequality (3.2). Next, we show that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛−𝑥≤0. (3.10) Indeed, we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛−𝑥=lim_𝑘→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛𝑘−𝑥. (3.11) We may assume that 𝑥_𝑛𝑘⇀𝑝∈𝐸 as 𝑘→∞, since a Banach space 𝐸 has a weakly sequentially continuous duality mapping 𝐽 satisfying Opial's condition [13]. We will prove that 𝑝∈Fix(𝒮). Suppose the contrary, 𝑝∉Fix(𝒮), that is, 𝑇(ℎ₀)𝑝≠𝑝 for some ℎ₀>0. It follows from (3.8) and Opial’s condition that liminf_𝑘→∞𝑥_𝑛𝑘−𝑝<liminf_𝑘→∞𝑥_𝑛𝑘−𝑇ℎ₀𝑝≤liminf_𝑘→∞𝑥_𝑛𝑘−𝑇ℎ₀𝑥_𝑛𝑘+𝑇ℎ₀𝑥_𝑛𝑘−𝑇ℎ₀𝑝≤liminf_𝑘→∞𝑥_𝑛𝑘−𝑇ℎ₀𝑥_𝑛𝑘+𝑥_𝑛𝑘−𝑝=liminf_𝑘→∞𝑥_𝑛𝑘−𝑝. (3.12) This is a contradiction, which shows that 𝑝∈𝐹(𝑇(ℎ)) for all ℎ>0, that is, 𝑝∈Fix(𝒮). In view of the variational inequality (3.2) and the assumption that duality mapping 𝐽 is weakly sequentially continuous, we conclude limsup_𝑛→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛−𝑥=lim_𝑘→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛𝑘−𝑥≤𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑝−𝑥≤0. (3.13) Finally, we will show that 𝑥_𝑛→𝑥. For each 𝑛≥0, we have 𝑥_𝑛+1−𝑥²=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝐹𝑥−𝛼_𝑛𝐹𝑥²≤𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛼_𝑛𝐹𝑥+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝐹𝑥²=𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝐹𝑥²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥≤1−𝛼_𝑛1−1−𝛿𝜆²𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑥),𝑗𝑥_𝑛+1−𝑥+2𝛼_𝑛𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥. (3.14) On the other hand, 𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑥),𝑗𝑥_𝑛+1−𝑥≤𝛾𝛼𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥≤𝛾𝛼𝑥_𝑛−𝑥1−𝛼_𝑛1−1−𝛿𝜆²𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥≤𝛾𝛼1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑥²+𝛾𝛼𝑥_𝑛−𝑥2𝛾𝑓𝑥_𝑛−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥√𝛼_𝑛≤𝛾𝛼1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑥²+√𝛼_𝑛𝑀₀, (3.15) where 𝑀₀ is a constant satisfying 𝑀₀≥𝛾𝛼‖𝑥_𝑛−𝑥‖2|⟨𝛾𝑓(𝑥_𝑛)−𝐹𝑥,𝑗(𝑥_𝑛+1−𝑥)⟩|. Substituting (3.15) in (3.14), we obtain 𝑥_𝑛+1−𝑥²≤1−𝛼_𝑛1−1−𝛿𝜆²𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝛼1−𝛼_𝑛1−1−𝛿𝜆×𝑥_𝑛−𝑥²+2𝛼_𝑛√𝛼_𝑛𝑀₀+2𝛼_𝑛𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥=1−2𝛼_𝑛1−1−𝛿𝜆+𝛼²_𝑛1−1−𝛿𝜆²𝑥_𝑛−𝑥²+2𝛼_𝑛𝛾𝛼1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑥²+2𝛼_𝑛√𝛼_𝑛𝑀₀+2𝛼_𝑛𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥=1−2𝛼_𝑛1−1−𝛿𝜆−𝛼𝛾+𝛼_𝑛𝛾𝛼1−1−𝛿𝜆𝑥_𝑛−𝑥²+𝛼_𝑛𝛼_𝑛1−1−𝛿𝜆²𝑥_𝑛−𝑥²+2𝑀₀√𝛼_𝑛+2𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥=1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝑥²+𝛼_𝑛𝛾_𝑛𝛽_𝑛𝛾_𝑛, (3.16) where 𝛾_𝑛=21−1−𝛿𝜆−𝛼𝛾+𝛼_𝑛𝛾𝛼1−1−𝛿𝜆,𝛽_𝑛=𝛼_𝑛1−1−𝛿𝜆²𝑥_𝑛−𝑥²+2𝑀₀√𝛼_𝑛+2𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛+1−𝑥. (3.17) It is easily seen that ∑^∞_𝑛=1𝛼_𝑛𝛾_𝑛=∞. Since {𝑥_𝑛} is bounded and lim_𝑛→∞𝛼_𝑛=0, by (3.46), we obtain limsup_𝑛→∞𝛽_𝑛/𝛾_𝑛≤0, applying Lemma 2.4 to (3.16) to conclude 𝑥_𝑛→𝑥 as 𝑛→∞. This completes the proof.

Using Theorem 3.1, we obtain the following two strong convergence theorems of new iterative approximation methods for a nonexpansive semigroup {𝑇(𝑡)∶0≤𝑡<∞}.

Corollary 3.2. Let 𝐸 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0≤𝑡<∞} be a u.a.r. nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Suppose that the real sequences {𝛼_𝑛}⊂[0,1], {𝑡_𝑛}⊂(0,∞) satisfy the conditions lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,lim_𝑛→∞𝑡_𝑛=∞. (3.18) Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑦_𝑛} defined by (1.20) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.19) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Proof. Let {𝑥_𝑛} be the sequence given by 𝑥₀=𝑦₀ and 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛,∀𝑛≥0. (3.20) Form Theorem 3.1, 𝑥_𝑛→𝑥. We claim that 𝑦_𝑛→𝑥. Indeed, we estimate 𝑥_𝑛+1−𝑦_𝑛+1≤𝛼_𝑛𝛾𝑓𝑇𝑡_𝑛𝑦_𝑛−𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛−𝑇𝑡_𝑛𝑦_𝑛≤𝛼_𝑛𝛾𝛼𝑇𝑡_𝑛𝑦_𝑛−𝑥_𝑛+1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑦_𝑛≤𝛼_𝑛𝛾𝛼𝑇𝑡_𝑛𝑦_𝑛−𝑇𝑡_𝑛𝑥+𝛼_𝑛𝛾𝛼𝑇𝑡_𝑛𝑥−𝑥_𝑛+1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑦_𝑛≤𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑥+𝛼_𝑛𝛾𝛼𝑥−𝑥_𝑛+1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑦_𝑛≤𝛼_𝑛𝛾𝛼𝑦_𝑛−𝑥_𝑛+𝛼_𝑛𝛾𝛼𝑥_𝑛−𝑥+𝛼_𝑛𝛾𝛼𝑥−𝑥_𝑛+1−𝛼_𝑛1−1−𝛿𝜆𝑥_𝑛−𝑦_𝑛=1−𝛼_𝑛1−1−𝛿𝜆−𝛾𝛼𝑥_𝑛−𝑦_𝑛+𝛼_𝑛1−1−𝛿𝜆−𝛾𝛼2𝛼𝛾1−(1−𝛿)/𝜆−𝛾𝛼𝑥−𝑥_𝑛. (3.21) It follows from ∑^∞_𝑛=1𝛼_𝑛=∞, lim_𝑛→∞‖𝑥_𝑛−𝑥‖=0, and Lemma 2.4 that ‖𝑥_𝑛−𝑦_𝑛‖→0. Consequently, 𝑦_𝑛→𝑥 as required.

Corollary 3.3. Let 𝐸 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0≤𝑡<∞} be a u.a.r. nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Suppose that the real sequences {𝛼_𝑛}⊂[0,1], {𝑡_𝑛}⊂(0,∞) satisfy the conditions lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,lim_𝑛→∞𝑡_𝑛=∞. (3.22) Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑧_𝑛} defined by (1.21) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.23) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Proof. Define the sequences {𝑦_𝑛} and {𝛽_𝑛} by 𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐹𝑧_𝑛,𝛽_𝑛=𝛼_𝑛+1∀𝑛∈ℕ. (3.24) Taking 𝑝∈Fix(𝒮), we have 𝑧_𝑛+1−𝑝=𝑇𝑡_𝑛𝑦_𝑛−𝑇𝑡_𝑛𝑝≤𝑦_𝑛=𝛼−𝑝_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐹𝑧_𝑛−𝐼−𝛼_𝑛𝐹𝑝−𝛼_𝑛≤𝐹𝑝1−𝛼_𝑛1−1−𝛿𝜆𝑧_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑧_𝑛𝑝=−𝐹1−𝛼_𝑛1−1−𝛿𝜆𝑧_𝑛−𝑝+𝛼_𝑛1−1−𝛿𝜆𝛾𝑓𝑧_𝑛−𝐹𝑝1−().1−𝛿/𝜆 (3.25) It follows from induction that 𝑧_𝑛+1−𝑝≤max𝑧₀,𝑧−𝑝𝛾𝑓₀−𝐹𝑝1−(1−𝛿)/𝜆,𝑛≥0. (3.26) Thus, both {𝑧_𝑛} and {𝑦_𝑛} are bounded. We observe that 𝑦_𝑛+1=𝛼_𝑛+1𝛾𝑓𝑧_𝑛+1+𝐼−𝛼_𝑛+1𝐹𝑧_𝑛+1=𝛽_𝑛𝛾𝑓𝑇𝑡_𝑛𝑦_𝑛+𝐼−𝛽_𝑛𝐹𝑇𝑡_𝑛𝑦_𝑛. (3.27) Thus, Corollary 3.2 implies that {𝑦_𝑛} converges strongly to some point 𝑥. In this case, we also have 𝑧_𝑛−𝑥≤𝑧_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥=𝛼_𝑛𝛾𝑓𝑧_𝑛−𝐹𝑧_𝑛+𝑦_𝑛−𝑥⟶0. (3.28) Hence, the sequence {𝑧_𝑛} converges strongly to some point 𝑥. This complete the proof.

Using Theorem 3.1, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.4. Let 𝐸 be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Suppose that the real sequences {𝛼_𝑛}⊂[0,1], {𝑡_𝑛}⊂(0,∞) satisfy the conditions lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,lim_𝑛→∞𝑡_𝑛=∞. (3.29) Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑥_𝑛} defined by 𝑥₀𝑥=𝑥∈𝐸,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹1𝑡_𝑛^𝑡𝑛₀𝑇(𝑡)𝑥_𝑛𝑑𝑠,𝑛≥0 (3.30) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.31) or equivalently 𝑥=𝑄_Fix(𝒮)((𝐼−𝐹+𝛾𝑓)𝑥), where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Corollary 3.5. Let 𝐻 be a real Hilbert space. Let 𝒮={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐻 such that Fix(𝒮)≠∅. Suppose that the real sequences {𝛼_𝑛}⊂[0,1], {𝑡_𝑛}⊂(0,∞) satisfy the conditions lim_𝑛→∞𝛼_𝑛=0,^∞_𝑛=0𝛼_𝑛=∞,lim_𝑛→∞𝑡_𝑛=∞. (3.32) Let 𝑓∶𝐸→𝐸 be a contraction mapping with coefficient 𝛼∈(0,1) and 𝐴 a strongly positive bounded linear operator with coefficient 𝛾>1/2 and 0<𝛾<(1−2−2𝛾)/𝛼. Then, the sequence {𝑥_𝑛} defined by 𝑥₀𝑥=𝑥∈𝐸,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑡)𝑥_𝑛𝑑𝑠,𝑛≥0 (3.33) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐴−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.34) or equivalently 𝑥=𝑄_Fix(𝒮)((𝐼−𝐴+𝛾𝑓)𝑥), where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Proof. Since 𝐴 is a strongly positive bounded linear operator with coefficient 𝛾, we have 𝐴𝑥−𝐴𝑦,𝑥−𝑦≥𝛾𝑥−𝑦². (3.35) Therefore, 𝐴 is 𝛾-strongly accretive. On the other hand, (𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²=−,−=+=𝑥−𝑦𝐴𝑥−𝐴𝑦𝑥−𝑦𝐴𝑥−𝐴𝑦𝑥−𝑦,𝑥−𝑦−2𝐴𝑥−𝐴𝑦,𝑥−𝑦𝐴𝑥−𝐴𝑦,𝐴𝑥−𝐴𝑦𝑥−𝑦²−2𝐴𝑥−𝐴𝑦,𝑥−𝑦+𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²−2𝐴𝑥−𝐴𝑦,𝑥−𝑦+‖𝐴‖²𝑥−𝑦². (3.36) Since 𝐴 is strongly positive if and only if (1/‖𝐴‖)𝐴 is strongly positive, we may assume, without loss of generality, that ‖𝐴‖=1, so that 𝐴𝑥−𝐴𝑦,𝑥−𝑦≤𝑥−𝑦²−12(𝐼−𝐴)𝑥−(𝐼−𝐴)𝑦²=𝑥−𝑦²−12𝑥−𝑦−𝐴𝑥−𝐴𝑦². (3.37) Hence, 𝐴 is 1⁄2-strongly pseudocontractive. Applying Corollary 3.4, we conclude the result.

Theorem 3.6. Let 𝐸 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0<𝑡<∞} be a u.a.r. nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real number satisfying 0<𝛼_𝑛<1,^∞_𝑛=0𝛼_𝑛=∞,𝑡_𝑛>0,lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛼_𝑛𝑡_𝑛=0. (3.38) Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑥_𝑛} defined by 𝑥₀𝑥=𝑥∈𝐸,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑥_𝑛,𝑛≥0 (3.39) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.40) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Proof. By the same argument as in the proof of Theorem 3.1, we can obtain that {𝑥_𝑛}, {𝑓(𝑥_𝑛)}, and {𝐹𝑇(𝑡_𝑛)𝑥_𝑛} are bounded and 𝑄_Fix(𝒮)(𝐼−𝐹−𝛾𝑓) is a contraction on 𝐸. Thus, by Banach contraction principle, 𝑄_Fix(𝒮)(𝐼−𝐹−𝛾𝑓) has a unique fixed point 𝑥. Then, using Lemma 2.3, 𝑥 is the unique solution in Fix(𝒮) of the variational inequality (3.40). Next, we show that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛−𝑥≤0. (3.41) Indeed, we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛−𝑥=lim_𝑘→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛𝑘−𝑥. (3.42) We may assume that 𝑥_𝑛𝑘⇀𝑝∈𝐸 as 𝑘→∞. Now, we show that 𝑝∈Fix(𝒮). Put 𝑥_𝑘=𝑥_𝑛𝑘,𝛼_𝑘=𝛼_𝑛𝑘𝑠_𝑘=𝑡_𝑛𝑘∀𝑘∈ℕ. (3.43) Fix 𝑡>0, then we have 𝑥_𝑘−𝑇(𝑡)𝑝=^{[𝑡/𝑠𝑖]−1}_𝑖=0𝑇(𝑖+1)𝑠_𝑘𝑥_𝑘−𝑇𝑖𝑠_𝑘𝑥_𝑘+𝑇𝑡𝑠_𝑘𝑠_𝑘𝑥_𝑘𝑡−𝑇𝑠_𝑘𝑠_𝑘𝑝+𝑇𝑡𝑠_𝑘𝑠_𝑘(𝑡)𝑝≤𝑡𝑝−𝑇𝑠_𝑘𝑇𝑠_𝑘𝑥_𝑘−𝑥_𝑘+1+𝑥_𝑘+1+𝑇𝑡−𝑝𝑡−𝑠_𝑘𝑠_𝑘≤𝑡𝑝−𝑝𝑠_𝑘𝛼_𝑘𝐹𝑇𝑠_𝑘𝑥_𝑘−𝑓𝑥_𝑘+𝑥_𝑘+1+𝑇𝑡−𝑝𝑡−𝑠_𝑘𝑠_𝑘≤𝑝−𝑝𝑡𝛼_𝑘𝑠_𝑘𝐹𝑇𝑠_𝑘𝑥_𝑘−𝑓𝑥_𝑘+𝑥_𝑘+1−𝑝+max𝑇(𝑠)𝑝−𝑝∶0≤𝑠≤𝑠_𝑘. (3.44) Thus, for all 𝑘∈ℕ, we obtain limsup_𝑘→∞𝑥_𝑘−𝑇(𝑡)𝑝≤limsup_𝑘→∞𝑥_𝑘+1−𝑝=limsup_𝑘→∞𝑥_𝑘−𝑝. (3.45) Since Banach space 𝐸 has a weakly sequentially continuous duality mapping satisfying Opial's condition [13], we can conclude that 𝑇(𝑡)𝑝=𝑝 for all 𝑡>0, that is, 𝑝∈Fix(𝒮). In view of the variational inequality (3.2) and the assumption that duality mapping 𝐽 is weakly sequentially continuous, we conclude limsup_𝑛→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛−𝑥=lim_𝑘→∞𝛾𝑓(𝑥)−𝐹𝑥,𝑗𝑥_𝑛𝑘−𝑥≤𝛾𝑓(𝑥)−𝐹𝑥,𝐽𝑝−𝑥≤0. (3.46) By the same argument as in the proof of Theorem 3.1, we conclude that 𝑥_𝑛→𝑥 as 𝑛→∞. This completes the proof.

Using Theorem 3.6 and the method as in the proof of Corollary 3.7, we have the following result.

Corollary 3.7. Let 𝐸 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0<𝑡<∞} be a u.a.r. nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real number satisfying 0<𝛼_𝑛<1,^∞_𝑛=0𝛼_𝑛=∞,𝑡_𝑛>0,lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛼_𝑛𝑡_𝑛=0. (3.47) Let 𝐹 be a 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 is a positive real number such that 𝛾<1/𝛼(1−(1−𝛿)/𝜆). Then, the sequence {𝑦_𝑛} defined by 𝑦₀𝑦=𝑦∈𝐸,_𝑛+1=𝛼_𝑛𝛾𝑓𝑇𝑡_𝑛𝑦_𝑛+𝐼−𝛼_𝑛𝐹𝑇𝑡_𝑛𝑦_𝑛,𝑛≥0 (3.48) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.49) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Using Theorem 3.6 and the method as in the proof of Corollary 3.8, we have the following result.

Corollary 3.8. Let 𝐸 be a reflexive Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0<𝑡<∞} be a u.a.r. nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real number satisfying 0<𝛼_𝑛<1,^∞_𝑛=0𝛼_𝑛=∞,𝑡_𝑛>0,lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛼_𝑛𝑡_𝑛=0. (3.50) Let 𝐹 be a 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 is a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑧_𝑛} defined by 𝑧₀𝑧=𝑧∈𝐸,_𝑛+1=𝑇𝑡_𝑛𝛼_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐹𝑧_𝑛,𝑛≥0 (3.51) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.52) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Using Theorem 3.6, Lemma 2.1, and Example 2.2, we have the following result.

Corollary 3.9. Let 𝐸 be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping 𝐽. Let 𝒮={𝑇(𝑡)∶0<𝑡<∞} be a nonexpansive semigroup on 𝐸 such that Fix(𝒮)≠∅. Let {𝛼_𝑛} and {𝑡_𝑛} be sequences of real numbers satisfying 0<𝛼_𝑛<1,^∞_𝑛=0𝛼_𝑛=∞,𝑡_𝑛>0,lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛼_𝑛𝑡_𝑛=0. (3.53) Let 𝐹 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, 𝑓∶𝐸→𝐸 a contraction mapping with coefficient 𝛼∈(0,1), and 𝛾 a positive real number such that 𝛾<(1/𝛼)(1−(1−𝛿)/𝜆). Then, the sequence {𝑥_𝑛} defined by 𝑥₀𝑥=𝑥∈𝐸,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐹1𝑡_𝑛^𝑡𝑛₀𝑇(𝑡)𝑥_𝑛𝑑𝑠,𝑛≥0 (3.54) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐹−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.55) or equivalently 𝑥=𝑄_Fix(𝒮)(𝐼−𝐹+𝛾𝑓)𝑥, where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Corollary 3.10. Let 𝐻 be a real Hilbert space. Let 𝒮={𝑇(𝑡)∶0≤𝑡<∞} be a nonexpansive semigroup on 𝐻 such that Fix(𝒮)≠∅. Suppose that the real sequences {𝛼_𝑛}⊂[0,1], {𝑡_𝑛}⊂(0,∞) satisfy the conditions 0<𝛼_𝑛<1,^∞_𝑛=0𝛼_𝑛=∞,𝑡_𝑛>0,lim_𝑛→∞𝛼_𝑛=lim_𝑛→∞𝛼_𝑛𝑡_𝑛=0. (3.56) Let 𝑓∶𝐸→𝐸 be a contraction mapping with coefficient 𝛼∈(0,1) and 𝐴 a strongly positive bounded linear operator with coefficient 𝛾>1/2 and 0<𝛾<(1−2−2𝛾)/𝛼. Then, the sequence {𝑥_𝑛} defined by 𝑥₀𝑥=𝑥∈𝐸,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑡)𝑥_𝑛𝑑𝑠,𝑛≥0 (3.57) converges strongly to 𝑥, where 𝑥 is the unique solution in Fix(𝒮) of the variational inequality 𝐴−𝛾𝑓𝑥,𝑗(𝑥−𝑥)≥0,𝑥∈Fix(𝒮) (3.58) or equivalently 𝑥=𝑄_Fix(𝒮)((𝐼−𝐴+𝛾𝑓)𝑥), where 𝑄_Fix(𝒮) is the sunny nonexpansive retraction of 𝐸 onto Fix(𝒮).

Acknowledgment

The project was supported by the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand.