A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

Abstract

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

1. Introduction

Throughout this paper we denoted by ℕ and ℝ⁺ the set of all positive integers and all positive real numbers, respectively. 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 by 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 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. As in [3], we use the notation ∏_𝐶 to denote the collection of all contractions on 𝐶, that is, ∏_𝐶={𝑓∶𝐶→𝐶acontraction}. Note that each 𝑓∈∏_𝐶 has a unique fixed point in 𝐶.

In the last ten years, 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. (1.1) 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⟩, (1.2) 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 [3] proved that the sequence {𝑥_𝑛} generated by𝑥₀𝑥∈𝐶chosenarbitrarily,_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑢,∀𝑛≥0, (1.3) converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence {𝛼_𝑛} satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if 𝐻 is a real Hilbert space, the sequence {𝑥_𝑛} generated by the following algorithm:𝑥₀𝑥∈𝐶chosenarbitrarily,_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (1.4) where 𝑓∶𝐶→𝐶 is a contraction mapping with constant 𝛼∈(0,1) and {𝛼_𝑛}⊂(0,1) satisfies certain conditions, converges strongly to a fixed point of 𝑇 in 𝐶 which is unique solution 𝑥^∗ of the variational inequality:𝑓−𝐼𝑥^∗,𝑦−𝑥^∗≤0,∀𝑦∈𝐹(𝑇). (1.5)

In 2006, Marino and Xu [8] combined the iterative method (1.3) with the viscosity approximation method (1.4) considering the following general iterative process:𝑥₀𝑥∈𝐶chosenarbitrarily,_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,∀𝑛≥0, (1.6) where 0<𝛾<𝛾/𝛼. They proved that the sequence {𝑥_𝑛} generated by (1.6) converges strongly to a unique solution 𝑥^∗ of the variational inequality:𝛾𝑓−𝐴𝑥^∗,𝑦−𝑥^∗≤0,∀𝑦∈𝐹(𝑇), (1.7) which is the optimality condition for the minimization problem:min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (1.8) 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 𝑥∈𝐻). Kim and Xu [9] studied the sequence generated by the following algorithm:𝑥₁𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑢+1−𝛽_𝑛𝑦_𝑛,∀𝑛≥0, (1.9) and proved strong convergence of scheme (1.9) in the framework of uniformly smooth Banach spaces. Later, yao, et al. [10] introduced a new iteration process by combining the modified Mann iteration [9] and the viscosity approximation method introduced by Moudafi [6]. Let 𝐶 be a closed convex subset of a Banach space, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that 𝐹(𝑇)≠∅ and 𝑓∈∏_𝐶. Define {𝑥_𝑛} in the following way:𝑥₁𝑦∈𝐶chosenarbitrarily;_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛;𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,∀𝑛≥0, (1.10) where {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1). They proved under certain different control conditions on the sequences {𝛼_𝑛} and {𝛽_𝑛} that {𝑥_𝑛} converges strongly to a fixed point of 𝑇. Recently, Chen and Song [11] studied the sequence generated by the algorithm in a uniformly convex Banach space, as follows:𝑥₁𝑥∈𝐶chosenarbitrarily;_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛∈ℕ, (1.11) and they proved that the sequence {𝑥_𝑛} defined by (1.11) converges strongly to the unique solution of the variational inequality:𝑓−𝐼𝑥^∗,𝐽𝑦−𝑥^∗≤0,∀𝑦∈𝐹(𝑇). (1.12) In 2010, Sunthrayuth and Kumam [12] introduced the a general iterative scheme generated by𝑥₀𝑥∈𝐶chosenarbitrarily;_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠,∀𝑛≥0, (1.13) for the approximation of common fixed point of a one-parameter nonexpansive semigroup in a Banach space under some appropriate control conditions. They proved strong convergence theorems of the iterative scheme which solve some variational inequality. Very recently, Kumam and Wattanawitoon [13] studied and introduced a new composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. In the same year, Sunthrayuth et al. [14] introduced a general composite iterative scheme for nonexpansive semigroups in Banach spaces. They established some strong convergence theorems of the general iteration scheme under different control conditions.

In this paper, motivated by Yao et al. [10], Sunthrayuth, and Kumam [12] and Kumam and Wattanawitoon [13] we introduce a new general iterative algorithm (3.23) for finding a common point of the set of solution of some variational inequality for nonexpansive semigroups in Banach spaces which admit a weakly continuous duality mapping and then proved the strong convergence theorem generated by the proposed iterative scheme. The results presented in this paper improve and extend some others from Hilbert spaces to Banach spaces and some others as special cases.

2. Preliminaries

Throughout this paper, 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; also, a mapping 𝐼 denote the identity mapping. Let 𝑋 be a real Banach space, and let 𝑋^∗ be its dual space. Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1}. A Banach space 𝑋 is said to be uniformly convex if, for each 𝜖∈(0,2], there exists a 𝛿>0 such that for each 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 [15]). A Banach space is said to be smooth if the limit lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖/𝑡 exists for each 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥,𝑦∈𝑈.

Let 𝜑∶[0,∞)∶=ℝ⁺→ℝ⁺ be a continuous strictly increasing function such that 𝜑(0)=0 and 𝜑(𝑡)→∞ as 𝑡→∞. This function 𝜑 is called a gauge function . The duality mapping 𝐽_𝜑∶𝑋→2^𝑋∗ associated with a gauge function 𝜑 is defined by 𝐽_𝜑(𝑥)=𝑓^∗∈𝑋^∗∶𝑥,𝑓^∗=‖𝑥‖𝜑(‖𝑥‖),𝑓^∗=𝜑(‖𝑥‖),∀𝑥∈𝑋, (2.1) where ⟨⋅,⋅⟩ denotes the generalized duality paring. 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 each 𝑥≠0 (see [16]).

Browder [16] initiated the study of certain classes of nonlinear operators by means of the duality mapping 𝐽_𝜑. Following Browder [16], we say that Banach space 𝑋 has a weakly continuous duality mapping if there exists a gauge function 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 each {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the sequence {𝐽(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, forall𝑡≥0; then 𝐽_𝜑(𝑥)=𝜕Φ(‖𝑥‖), where 𝜕 denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function 𝜙∶𝑋→ℝ at 𝑥∈𝑋 is the set 𝜕𝜙(𝑥)={𝑥^∗∈𝑋;𝜙(𝑦)≥𝜙(𝑥)+⟨𝑥^∗,𝑦−𝑥⟩,forall𝑦∈𝑋}).

In a Banach space having a weakly continuous duality mapping 𝐽_𝜑 with a gauge function 𝜑, we defined an operator 𝐴 is to be strongly positive (see [17]) if there exists a constant 𝛾>0 with the property𝐴𝑥,𝐽_𝜑(𝑥)≥𝛾‖𝑥‖𝜑(‖𝑥‖), (2.2)‖𝑎𝐼−𝑏𝐴‖=sup_{‖𝑥‖≤1}(𝑎𝐼−𝑏𝐴)𝑥,𝐽_𝜑(𝑥),𝑎∈[0,1],𝑏∈[−1,1]. (2.3) If 𝑋∶=𝐻 is a real Hilbert space, then the inequality (2.2) reduces to (1.1).

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 [18].

Lemma 2.1 (see [18]). Assume that a Banach space 𝑋 has a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑. (i) For all 𝑥,𝑦∈𝑋, the following inequality holds: Φ𝑥+𝑦≤Φ(‖𝑥‖)+𝑦,𝐽_𝜑𝑥+𝑦. (2.4) In particular, for all 𝑥,𝑦∈𝑋, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦. (2.5)(ii) Assume that a sequence {𝑥_𝑛} in 𝑋 converges weakly to a point 𝑥∈𝑋. Then the following identity holds: limsup_𝑛→∞Φ𝑥_𝑛−𝑦=limsup_𝑛→∞Φ𝑥_𝑛−𝑥+Φ𝑦−𝑥,∀𝑥,𝑦∈𝑋. (2.6)

Lemma 2.2 (see [17]). Assume that a Banach space 𝑋 has a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑. Let 𝐴 be a strongly positive linear bounded operator on 𝑋 with a coefficient 𝛾>0 and 0<𝜌≤𝜑(1)‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤𝜑(1)(1−𝜌𝛾).

Lemma 2.3 (see [11]). Let 𝐶 be a closed convex subset of a uniformly convex Banach space 𝑋 and let 𝒮={𝑇(𝑡)∶𝑡∈ℝ⁺} be 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)(ℎ)1𝑥𝑑𝑠−𝑇𝑡^𝑡₀𝑇(𝑠)𝑥𝑑𝑠=0. (2.7)

Lemma 2.4 (see [19]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜇_𝑛𝑎_𝑛+𝛿_𝑛, (2.8) where {𝜇_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (i)∑^∞_𝑛=0𝜇_𝑛=∞; (ii)limsup_𝑛→∞(𝛿_𝑛/𝜇_𝑛)≤0 or ∑^∞_𝑛=0|𝛿_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

Let 𝑋 be a Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1], and let 𝐶 be 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, let 𝑓 be a contraction mapping with a coefficient 𝛼∈(0,1), let 𝐴 be a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾𝜑(1)/𝛼, and let 𝑡∈(0,1) such that 𝑡≤𝜑(1)‖𝐴‖⁻¹ which satisfies 𝑡→0. Define the mapping 𝑇^𝑓_𝑡∶𝐶→𝐶 by 𝑇^𝑓_𝑡()1∶=𝑡𝛾𝑓+𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑑𝑠 (3.1) to be a contraction mapping. Indeed, for each 𝑥,𝑦∈𝐶,𝑇^𝑓_𝑡𝑥−𝑇^𝑓_𝑡𝑦=𝑓(𝑥)𝑦+()1𝑡𝛾−𝑓𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)(𝑠)𝑦𝑓(𝑥)𝑦+‖‖1𝑥−𝑇𝑑𝑠≤𝑡𝛾−𝑓𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)(𝑠)𝑦(1)𝑥−𝑇𝑑𝑠≤𝑡𝛾𝛼𝑥−𝑦+𝜑1−𝑡𝛾≤𝜑(1)𝑥−𝑦1−𝑡𝛾−𝛾𝛼𝑥−𝑦. (3.2) Thus, by Banach contraction mapping principle, there exists a unique fixed point 𝑥_𝑡∈𝐶, that is,𝑥_𝑡=𝑡𝛾𝑓𝑥_𝑡+()1𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠. (3.3)

Remark 3.1. We note that space 𝑙_𝑝 has a weakly continuous duality mapping with a gauge function 𝜑(𝑡)=𝑡^𝑝−1 for all 1<𝑝<∞. This shows that 𝜑 is invariant on [0,1].

Lemma 3.2. Let 𝑋 be a uniformly convex Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1], and let 𝐶 be 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 𝑓 be a contraction mapping with a coefficient 𝛼∈(0,1), let 𝐴 be a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾𝜑(1)/𝛼, and let 𝑡∈(0,1) such that 𝑡≤𝜑(1)‖𝐴‖⁻¹ which satisfies 𝑡→0. Then the net {𝑥_𝑡} defined by (3.3) with {𝜆_𝑡}_0<𝑡<1 is a positive real divergent sequence, converges strongly as 𝑡→0 to a common fixed point 𝑥^∗, in which 𝑥^∗∈𝐹(𝒮), and is the unique solution of the variational inequality: 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥−𝑥^∗≤0,∀𝑥∈𝐹(𝒮). (3.4)

Proof. Firstly, we show the uniqueness of a solution of the variational inequality (3.4). Suppose that 𝑥,𝑥^∗∈𝐹(𝒮) are solutions of (3.4); then 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥−𝑥^∗(𝑥)≤0,𝛾𝑓−𝐴𝑥,𝐽_𝜑𝑥^∗−𝑥≤0. (3.5) Adding up (3.5), we obtain 0≥𝛾𝑓𝑥^∗−𝐴𝑥^∗−𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥−𝑥^∗=𝐴𝑥−𝑥^∗,𝐽_𝜑𝑥−𝑥^∗−𝛾𝑓(𝑥)−𝑓𝑥^∗,𝐽_𝜑𝑥−𝑥^∗≥𝛾𝑥−𝑥^∗𝜑𝑥−𝑥^∗−𝛾𝑓(𝑥)−𝑓𝑥^∗𝐽_𝜑𝑥−𝑥^∗≥𝛾Φ𝑥−𝑥^∗−𝛾𝛼Φ𝑥−𝑥^∗=𝛾−𝛾𝛼Φ𝑥−𝑥^∗≥𝜑(1)𝛾−𝛾𝛼Φ𝑥−𝑥^∗, (3.6) which is a contradiction, we must have 𝑥=𝑥^∗, and the uniqueness is proved. Here in after, we use 𝑥 to denote the unique solution of the variational inequality (3.4).
Next, we show that {𝑥_𝑡} is bounded. Indeed, for each 𝑝∈𝐹(𝒮), we have 𝑥_𝑡−𝑝=𝑡𝛾𝑓𝑥_𝑡+()1−𝐴𝑝𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑥−𝑝𝑑𝑠≤𝑡𝛾𝑓_𝑡+‖‖1−𝐴𝑝𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑓𝑥−𝑝𝑑𝑠≤𝑡𝛾_𝑡−𝑓𝑝+𝑡𝛾𝑓𝑝−𝐴𝑝+𝜑(1)1−𝑡𝛾𝑥_𝑡≤𝜑(1)−𝑝1−𝑡𝛾−𝛾𝛼𝑥_𝑡−𝑝+𝑡𝛾𝑓𝑝−𝐴𝑝. (3.7) It follows that 𝑥_𝑡≤1−𝑝𝜑(1)𝑝𝛾−𝛾𝛼𝛾𝑓−𝐴𝑝. (3.8) Hence, {𝑥_𝑡} is bounded, so are {𝑓(𝑥_𝑡)} and {𝐴((1/𝜆_𝑡)∫^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠)}.
Next, we show that ‖𝑥_𝑡−𝑇(ℎ)𝑥_𝑡‖→0 as 𝑡→0. We note that 𝑥_𝑡−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠=𝑡𝛾𝑓𝑥_𝑡1−𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠. (3.9) Moreover, we note that 𝑥_𝑡−𝑇(ℎ)𝑥_𝑡≤𝑥_𝑡−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡+1𝑑𝑠𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡(ℎ)1𝑑𝑠−𝑇𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡+𝑇(ℎ)1𝑑𝑠𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑇(ℎ)𝑥_𝑡≤2𝑥_𝑡−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡+1𝑑𝑠𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡(ℎ)1𝑑𝑠−𝑇𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠, (3.10) for all ℎ≥0. Define the set 𝐾={𝑧∈𝐶∶‖𝑧−𝑝‖≤‖𝛾𝑓(𝑝)−𝐴𝑝‖/(𝜑(1)𝛾−𝛾𝛼)}; then 𝐾 is a nonempty bounded closed convex subset of 𝐶 which is 𝑇(𝑠)-invariant for each ℎ≥0. Since {𝑥_𝑡}⊂𝐾 and 𝐾 is bounded, there exists 𝑟>0 such that 𝐾⊂𝐵_𝑟, and it follows by Lemma 2.3 that lim_{𝜆𝑡→∞}1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡(ℎ)1𝑑𝑠−𝑇𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠=0, (3.11) for each ℎ≥0. From (3.9)-(3.10), letting 𝑡→0 and noting (3.11) then, for each ℎ≥0, we obtain 𝑥_𝑡−𝑇(ℎ)𝑥_𝑡⟶0. (3.12) Assume that {𝑡_𝑛}^∞_𝑛=1⊂(0,1) is such that 𝑡_𝑛→0 as 𝑛→∞. Put 𝑥_𝑛∶=𝑥_𝑡𝑛 and 𝜆_𝑛∶=𝜆_𝑡𝑛. We will show that {𝑥_𝑛} contains a subsequence converging strongly to 𝑥∈𝐹(𝒮). Since {𝑥_𝑛} is bounded sequence and Banach space 𝑋 is a uniformly convex, hence it is reflexive, and there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} which converges weakly to some 𝑥∈𝐶 as 𝑗→∞. Again, since 𝐽_𝜑 is weakly sequentially continuous, we have by Lemma 2.1 that limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑧=limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑥+Φ(‖𝑧−𝑥‖). (3.13) Let 𝐻(𝑧)=limsup_𝑗→∞Φ(‖𝑥_𝑛𝑗−𝑧‖), for all 𝑧∈𝐶.
It follows that 𝐻(𝑧)=𝐻(𝑥)+Φ(‖𝑧−𝑥‖), forall𝑧∈𝐶. From (3.12), we have 𝐻(𝑇(ℎ)𝑥)=limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑇(ℎ)𝑥=limsup_𝑗→∞Φ𝑇(ℎ)𝑥_𝑛𝑗−𝑇(ℎ)𝑥≤limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑥=𝐻(𝑥). (3.14) On the other hand, we note that 𝐻(𝑇(ℎ)𝑥)=limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑥𝑇(ℎ)𝑥(𝑥)𝑇(ℎ)𝑥.+Φ(‖𝑥−‖)=𝐻+Φ(‖𝑥−‖) (3.15) Combining (3.14) with (3.15), we obtain Φ(‖𝑇(ℎ)𝑥−𝑥‖)≤0. This implies that 𝑇(ℎ)𝑥=𝑥, that is, 𝑥∈𝐹(𝒮). In fact, since Φ(𝑡)=∫^𝑡₀𝜑(𝜏)𝑑𝜏, forall𝑡≥0 and 𝜑∶ℝ⁺→ℝ⁺ is the gauge function, then for 1≥𝑘≥0, 𝜑(𝑘𝑦)≤𝜑(𝑦) and Φ(𝑘𝑡)=₀^𝑘𝑡𝜑(𝜏)𝑑𝜏=𝑘^𝑡₀𝜑𝑘𝑦𝑑𝑦≤𝑘^𝑡₀𝜑𝑦𝑑𝑦=𝑘Φ(𝑡). (3.16) By Lemma 2.1, we have Φ𝑥_𝑛−𝑥=Φ𝑡_𝑛𝛾𝑓𝑥_𝑛𝑥+()1−𝐴𝐼−𝑡𝐴𝜆_𝑛^𝜆𝑡₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑥≤Φ𝐼−𝑡_𝑛𝐴1𝜆_𝑛^𝜆𝑛₀𝑇(𝑠)𝑥_𝑛−𝑥𝑑𝑠+𝑡_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥≤Φ𝜑(1)1−𝑡_𝑛𝛾𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑥),𝐽_𝜑𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥≤𝜑(1)1−𝑡_𝑛𝛾Φ𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝛼𝑥_𝑛−𝑥𝐽_𝜑𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥=𝜑(1)1−𝑡_𝑛𝛾Φ𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝛼Φ𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥≤1−𝑡_𝑛𝜑(1)𝛾−𝛾𝛼Φ𝑥_𝑛−𝑥+𝑡_𝑛𝛾𝑓(𝑥)−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥. (3.17) This implies that Φ𝑥_𝑛−𝑥≤1𝜑(1)(𝑥)𝛾−𝛾𝛼𝛾𝑓−𝐴𝑥,𝐽_𝜑𝑥_𝑛−𝑥. (3.18) In particular, we have Φ𝑥_𝑛𝑗−𝑥≤1𝜑(1)(𝑥)𝛾−𝛾𝛼𝛾𝑓−𝐴𝑥,𝐽_𝜑𝑥_𝑛𝑗−𝑥. (3.19) Since the mapping 𝐽_𝜑 is single-valued and weakly continuous, it follows from (3.19) that Φ(‖𝑥_𝑛𝑗−𝑥‖)→0 as 𝑗→∞. This implies that 𝑥_𝑛𝑗→𝑥 as 𝑗→∞.
Next, we show that 𝑥 solves the variational inequality (3.4), for each 𝑥∈𝐹(𝒮). From (3.3), we derive that 𝛾𝑓−𝐴𝑥_𝑡1=−𝑡()1𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑥_𝑡. (3.20) Now, we observe that 1𝜆_𝑡^𝜆𝑡₀((𝑠1𝐼−𝑇))𝑥𝑑𝑠−𝜆_𝑡^𝜆𝑡₀(𝐼−𝑇(𝑠))𝑥_𝑡𝑑𝑠,𝐽_𝜑𝑥−𝑥_𝑡=𝑥−𝑥_𝑡,𝐽_𝜑𝑥−𝑥_𝑡−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥−𝑇(𝑠)𝑥_𝑡𝑑𝑠,𝐽_𝜑𝑥−𝑥_𝑡≥𝑥−𝑥_𝑡𝐽_𝜑𝑥−𝑥_𝑡−1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥−𝑇(𝑠)𝑥_𝑡𝑑𝑠𝐽_𝜑𝑥−𝑥_𝑡≥Φ𝑥−𝑥_𝑡−𝑥−𝑥_𝑡𝐽_𝜑𝑥−𝑥_𝑡=Φ𝑥−𝑥_𝑡−Φ𝑥−𝑥_𝑡=0. (3.21) It follows from (3.20) that 𝛾𝑓−𝐴𝑥_𝑡,𝐽_𝜑𝑥−𝑥_𝑡1=−𝑡)1(𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡𝑑𝑠−𝑥_𝑡,𝐽_𝜑𝑥−𝑥_𝑡1=−𝑡)1(𝐼−𝑡𝐴𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)𝑥_𝑡1𝑑𝑠−𝜆_𝑡^𝜆𝑡₀𝑥_𝑡𝑑𝑠,𝐽_𝜑𝑥−𝑥_𝑡1=−𝑡1𝜆_𝑡^𝜆𝑡₀((𝑠1𝐼−𝑇))𝑥𝑑𝑠−𝜆_𝑡^𝜆𝑡₀(𝐼−𝑇(𝑠))𝑥_𝑡𝑑𝑠,𝐽_𝜑𝑥−𝑥_𝑡+𝐴1𝜆_𝑡^𝜆𝑡₀(𝑇(𝑠)−𝐼)𝑥_𝑡𝑑𝑠,𝐽_𝜑𝑥−𝑥_𝑡≤𝐴1𝜆_𝑡^𝜆𝑡₀𝑇(𝑠)−𝐼))𝑥_𝑡𝑑𝑠,𝐽_𝜑𝑥−𝑥_𝑡. (3.22) Now, replacing 𝑡 and 𝜆_𝑡 with 𝑡_𝑛𝑗 and 𝜆_𝑛𝑗, respectively, in (3.22), and letting 𝑗→∞, and we notice that (𝑇(𝑠)−𝐼)𝑥_𝑛𝑗→(𝑇(𝑠)−𝐼)𝑥=0 for 𝑥∈𝐹(𝒮), we obtain that ⟨(𝛾𝑓−𝐴)𝑥,𝐽_𝜑(𝑥−𝑥)⟩≤0. That is, 𝑥 is a solution of the variational inequality (3.4). By uniqueness, as 𝑥=𝑥^∗, we have shown that each cluster point of the net {𝑥_𝑡} is equal to 𝑥^∗. Then, we conclude that 𝑥_𝑡→𝑥^∗ as 𝑡→0. This proof is complete.

Theorem 3.3. Let 𝑋 be a uniformly convex Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with the gauge function 𝜑 such that 𝜑 is invariant in [0,1], and let 𝐶 be 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 𝑓 be a contraction mapping with a coefficient 𝛼∈(0,1), and let 𝐴 be a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾𝜑(1)/𝛼. Let {𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0,{𝛾_𝑛}^∞_𝑛=0 be the sequences in (0,1) and let {𝑡_𝑛}^∞_𝑛=0 be a positive real divergent sequence. Assume that the following conditions hold: (C1)lim_𝑛→∞𝛼_{_𝑛}=0 and ∑^∞_𝑛=0𝛼_𝑛=∞, (C2)lim_𝑛→∞𝛾_𝑛=0, (C3)𝛽_𝑛=𝑜(𝛼_𝑛), Then the sequence {𝑥_𝑛} defined by 𝑥₀𝑧∈𝐶chosenarbitrarily;_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑦𝑑𝑠;_𝑛=𝛼_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐴𝑧_𝑛;𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,∀𝑛≥0, (3.23) converges strongly to the common fixed point 𝑥^∗ that is obtained in Lemma 3.2.

Proof. From the condition (C1), we may assume, with no loss of generality, that 𝛼_𝑛≤𝜑(1)‖𝐴‖⁻¹ for each 𝑛≥0. From Lemma 2.2, we have ‖𝐼−𝛼_𝑛𝐴‖≤𝜑(1)(1−𝛼_𝑛𝛾).
Firstly, we show that {𝑥_𝑛} is bounded. Let 𝑝∈𝐹(𝒮); we get 𝑧_𝑛−𝑝=𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛−𝑝𝑑𝑠≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛−𝑝𝑑𝑠≤𝛾_𝑛𝑥_𝑛−𝑝+1−𝛾_𝑛𝑥_𝑛=𝑥−𝑝_𝑛,𝑦−𝑝_𝑛−𝑝=𝛼_𝑛𝛾𝑓𝑧_𝑛−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑧_𝑛−𝑝≤𝛼_𝑛𝛾𝑓𝑧_𝑛−𝑓𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝+𝐼−𝛼_𝑛𝐴𝑧_𝑛≤−𝑝1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼𝑥_𝑛−𝑝+𝛼_𝑛𝛾𝑓𝑝−𝐴𝑝. (3.24) It follows that 𝑥_𝑛+1−𝑝=𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑦_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼𝑥_𝑛−𝑝+𝛼_𝑛𝑝=𝛾𝑓−𝐴𝑝1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼1−𝛽_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝜑(1)𝛾−𝛾𝛼1−𝛽_𝑛𝛾𝑓𝑝−𝐴𝑝𝜑(1)𝑥𝛾−𝛾𝛼≤max_𝑛,𝑝−𝑝𝛾𝑓−𝐴𝑝𝜑(1).𝛾−𝛾𝛼 (3.25) By induction on 𝑛, we have 𝑥_𝑛−𝑝≤max𝑥₀,𝑝−𝑝𝛾𝑓−𝐴𝑝𝜑(1)𝛾−𝛾𝛼,∀𝑛≥0. (3.26) Thus, {𝑥_𝑛} is bounded. Since {𝑥_𝑛} is bounded, then ‖(1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑝‖≤‖𝑥_𝑛−𝑝‖ and {𝐴((1/𝑡_𝑛)∫^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠)} and {𝑓(𝑧_𝑛)} are also bounded.
Next, we show that lim_𝑛→∞‖𝑥_𝑛−𝑇(ℎ)𝑥_𝑛‖=0, forallℎ≥0. From (3.23), we note that 𝑥_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠≤𝑥_𝑛+1−𝑦_𝑛+𝑦_𝑛−𝑧_𝑛+𝑧_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠≤𝛽_𝑛𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝛾𝑓𝑧_𝑛−𝐴𝑧_𝑛+𝛾_𝑛𝑥_𝑛−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠. (3.27) By the conditions (C1)–(C3), then (3.27), we obtain lim_𝑛→∞𝑥_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠=0. (3.28) Moreover, we note that 𝑥_𝑛+1−𝑇(ℎ)𝑥_𝑛+1≤𝑥_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛+1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛(ℎ)1𝑑𝑠−𝑇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛+𝑇(ℎ1𝑑𝑠)𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠−𝑇(ℎ)𝑥_𝑛+1≤2𝑥_𝑛+1−1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛+1𝑑𝑠𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛(ℎ)1𝑑𝑠−𝑇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠. (3.29) Define the set 𝐾={𝑧∈𝐶∶‖𝑧−𝑝‖≤‖𝑥₀−𝑝‖+‖𝛾𝑓(𝑝)−𝐴𝑝‖/(𝜑(1)𝛾−𝛾𝛼)}. Then 𝐾 is a nonempty bounded closed convex subset of 𝐶, which is 𝑇(𝑠)—invariant for each 𝑠≥0 and contains {𝑥_𝑛}; it follows from Lemma 2.3 that lim_𝑛→∞1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛(ℎ)1𝑑𝑠−𝑇𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠=0,∀ℎ≥0. (3.30) Then, for all ℎ≥0, from (3.28) and (3.30), into (3.29), we obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑇(ℎ)𝑥_𝑛+1‖=0, and hence lim_𝑛→∞𝑥_𝑛−𝑇(ℎ)𝑥_𝑛=0,∀ℎ≥0. (3.31) Next, we show that limsup_𝑛→∞⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝐽_𝜑(𝑥_𝑛−𝑥^∗)⟩≤0. We can take subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that lim_𝑗→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥_𝑛𝑗−𝑥^∗=limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥_𝑛−𝑥^∗. (3.32) By the assumption that 𝑋 is uniformly convex, hence it is reflexive and {𝑥_𝑛} is bounded; then there exists a subsequence {𝑥_𝑛𝑗} which converges weakly to some 𝑥∈𝐶 as 𝑗→∞. Since 𝐽_𝜑 is weakly continuous, from Lemma 2.1, we have limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑧=limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑥+Φ(‖𝑧−𝑥‖),∀𝑧∈𝐶. (3.33) Let 𝐻(𝑧)=limsup_𝑗→∞Φ(‖𝑥_𝑛𝑗−𝑧‖), forall𝑧∈𝐶.
It follows that 𝐻(𝑧)=𝐻(𝑥)+Φ(‖𝑧−𝑥‖), forall𝑧∈𝐶.
From (3.31), we have 𝐻(𝑇(ℎ)𝑥)=limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑇(ℎ)𝑥=limsup_𝑗→∞Φ𝑇(ℎ)𝑥_𝑛𝑗−𝑇(ℎ)𝑥≤limsup_𝑗→∞Φ𝑥_𝑛𝑗−𝑥=𝐻(𝑥). (3.34) On the other hand, we note that 𝐻(𝑇(ℎ)𝑥)=limsup_𝑗→∞Φ𝑥_𝑛𝑗𝑇(ℎ)(𝑥)𝑇(ℎ).−𝑥+Φ(‖𝑥−𝑥‖)=𝐻+Φ(‖𝑥−𝑥‖) (3.35) Combining (3.34) with (3.35), we obtain Φ(‖𝑇(ℎ)𝑥−𝑥‖)≤0.
This implies that 𝑇(ℎ)𝑥=𝑥; that is, 𝑥∈𝐹(𝒮).
Since the duality map 𝐽_𝜑 is single-valued and weakly continuous, we get that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥_𝑛−𝑥^∗=lim_𝑗→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥_𝑛𝑗−𝑥^∗=𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥−𝑥^∗≤0, (3.36) as required. Hence, limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥_𝑛+1−𝑥^∗≤0. (3.37) Since ‖𝑥_𝑛+1−𝑦_𝑛‖=𝛽_𝑛‖𝑥_𝑛−𝑦_𝑛‖, by condition (C3), we obtain that lim_𝑛→∞‖𝑥_𝑛+1−𝑦_𝑛‖=0. It follows from (3.37), that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑦_𝑛−𝑥^∗≤0. (3.38) Finally, we show that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. Now, from Lemma 2.1, we have Φ𝑦_𝑛−𝑥^∗=Φ𝛼_𝑛𝛾𝑓𝑧_𝑛−𝐴𝑥^∗+𝐼−𝛼_𝑛𝐴𝑧_𝑛−𝑥^∗=Φ𝛼_𝑛𝛾𝑓𝑧_𝑛−𝛾𝑓𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗+𝐼−𝛼_𝑛𝐴𝑧_𝑛−𝑥^∗≤Φ𝛼_𝑛𝛾𝑓𝑧_𝑛−𝛾𝑓𝑥^∗+𝐼−𝛼_𝑛𝐴𝑧_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑦_𝑛−𝑥^∗≤1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼Φ𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑦_𝑛−𝑥^∗. (3.39) On the other hand, we note that Φ𝑥_𝑛+1−𝑥^∗=Φ𝛽_𝑛𝑥_𝑛−𝑥^∗+1−𝛽_𝑛𝑦_𝑛−𝑥^∗≤1−𝛽_𝑛Φ𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗,𝐽_𝜑𝑥_𝑛+1−𝑥^∗. (3.40) It follows from (3.40) that Φ𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼Φ𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑦_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗,𝐽_𝜑𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼Φ𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑦_𝑛−𝑥^∗+𝛽_𝑛𝛼_𝑛𝑀, (3.41) where 𝑀=sup_𝑛≥0{‖𝑥_𝑛−𝑥^∗‖𝜑(‖𝑥_𝑛+1−𝑥^∗‖)}.
Put 𝜇_𝑛∶=𝛼_𝑛(𝜑(1)𝛾−𝛾𝛼) and 𝛿_𝑛∶=𝛼_𝑛[⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝐽_𝜑(𝑦_𝑛−𝑥^∗)⟩+(𝛽_𝑛/𝛼_𝑛)𝑀]. Then (3.41) reduces to formula Φ(‖𝑥_𝑛+1−𝑥^∗‖)≤(1−𝜇_𝑛)Φ(‖𝑥_𝑛−𝑥^∗‖)+𝛿_𝑛. By conditions (C1) and (C3) and noting (3.38), it is easy to see that ∑^∞_𝑛=0𝜇_𝑛=∞ and limsup_𝑛→∞(𝛿_𝑛/𝜇_𝑛)=limsup_𝑛→∞(1/𝜑(1)𝛾−𝛾𝛼)[⟨𝛾𝑓(𝑥^∗)−𝐴𝑥^∗,𝐽_𝜑(𝑦_𝑛−𝑥^∗)⟩+(𝛽_𝑛/𝛼_𝑛)𝑀]≤0. Applying Lemma 2.4, we obtain Φ(‖𝑥_𝑛−𝑥^∗‖)→0 as 𝑛→∞ this implies that 𝑥_𝑛→𝑥^∗ as 𝑛→∞. This completes the proof.

Taking 𝛾_𝑛=0 in (3.23), we can get the following corollary easily.

Corollary 3.4. Let 𝑋 be a uniformly convex Banach space which admits a weakly continuous duality mapping 𝐽_𝜑 with the gauge function 𝜑 such that 𝜑 invariant in [0,1], 𝐶 be 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)≠∅, 𝑓 be a contraction mapping with a coefficient 𝛼∈(0,1) and 𝐴 be a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾𝜑(1)/𝛼. Let {𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0 be the sequences in (0,1) and {𝑡_𝑛}^∞_𝑛=0 be a positive real divergent sequence. Assume the following conditions are hold: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞; (C2)𝛽_𝑛=𝑜(𝛼_𝑛). Then the sequence {𝑥_𝑛} defined by 𝑥₀𝑦∈𝐶chosenarbitrarily,_𝑛=𝛼_𝑛1𝛾𝑓𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑑𝑠+𝐼−𝛼_𝑛𝐴1𝑡_𝑛^𝑡𝑛₀𝑇(𝑠)𝑥_𝑛𝑥𝑑𝑠,_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,∀𝑛≥0, (3.42) converges strongly to the common fixed point 𝑥^∗, in which 𝑥^∗∈𝐹(𝒮) is the unique solution of the variational inequality: 𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝐽_𝜑𝑥−𝑥^∗≤0,∀𝑥∈𝐹(𝒮). (3.43)

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon [20] and it was generalized to that for nonlinear semigroups by Reich et al. [21–23]. It is clear that Theorem 3.3 are valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.5. Let 𝐻 be a real Hilbert space, and let𝐶 be 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)≠∅, 𝑓 be a contraction mapping with a coefficient 𝛼∈(0,1), and let 𝐴 be a strongly positive linear bounded operator with a coefficient 𝛾>0 such that 0<𝛾<𝛾𝜑(1)/𝛼. Let {𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0,and{𝛾_𝑛}^∞_𝑛=0 be the sequences in (0,1) and let {𝑡_𝑛}^∞_𝑛=0 be a positive real divergent sequence. Assume that the following conditions are hold: (C1)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞; (C2)lim_𝑛→∞𝛾_𝑛=0; (C3)𝛽_𝑛=𝑜(𝛼_𝑛). Then the sequence {𝑥_𝑛} defined by 𝑥₀𝑧∈𝐶chosenarbitrarily,_𝑛=𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛1𝑛+1^𝑛_𝑗=0𝑇^𝑗𝑥_𝑛,𝑦_𝑛=𝛼_𝑛𝛾𝑓𝑧_𝑛+𝐼−𝛼_𝑛𝐴𝑧_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛,∀𝑛≥0, (3.44) converges strongly to the common fixed point 𝑥^∗, in which 𝑥^∗∈𝐹(𝑇) is the unique solution of the variational inequality: ⟨𝛾𝑓𝑥^∗−𝐴𝑥^∗,𝑥−𝑥^∗⟩≤0,∀𝑥∈𝐹(𝑇). (3.45)

Acknowledgments

The authors are grateful for the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work. They would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support under NRU-CSEC project no. 54000267.