A General Iterative Method for a Nonexpansive Semigroup in Banach Spaces with Gauge Functions

Abstract

We study strong convergence of the sequence generated by implicit and explicit general iterative methods for a one-parameter nonexpansive semigroup in a reflexive Banach space which admits the duality mapping 𝐽_𝜑, where 𝜑 is a gauge function on [0,∞). Our results improve and extend those announced by G. Marino and H.-K. Xu (2006) and many authors.

1. Introduction

Let 𝐸 be a real Banach space and 𝐸^∗ the dual space of 𝐸. Let 𝐾 be a nonempty, closed, and convex subset of 𝐸. A (one-parameter) nonexpansive semigroup is a family 𝔉={𝑇(𝑡)∶𝑡≥0} of self-mappings of 𝐾 such that(i)𝑇(0)𝑥=𝑥 for all 𝑥∈𝐾,(ii)𝑇(𝑡+𝑠)𝑥=𝑇(𝑡)𝑇(𝑠)𝑥 for all 𝑡,𝑠≥0 and 𝑥∈𝐾,(iii)for each 𝑥∈𝐾, the mapping 𝑇(⋅)𝑥 is continuous,(iv)for each 𝑡≥0, 𝑇(𝑡) is nonexpansive, that is,‖𝑇(𝑡)𝑥−𝑇(𝑡)𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐾.(1.1) We denote 𝐹 by the common fixed points set of 𝔉, that is, 𝐹∶=⋂_𝑡≥0𝐹(𝑇(𝑡)).

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping 𝑇∶𝐾→𝐾 in a real Hilbert space:𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0,(1.2) where {𝛼_𝑛}⊂(0,1) and 𝑢∈𝐾.

In 1977, Lions [2] obtained a strong convergence provide the real sequence {𝛼_𝑛} satisfies the following conditions:

C1: lim_𝑛→∞𝛼_𝑛=0; C2: ∑^∞𝑛=0𝛼_𝑛=∞; C3: lim_𝑛→∞(𝛼_𝑛−𝛼_𝑛−1)/𝛼^2𝑛=0.

Reich [3] also extended the result of Halpern from Hilbert spaces to uniformly smooth Banach spaces. However, both Halpern’s and Lion’s conditions imposed on the real sequence {𝛼_𝑛} excluded the canonical choice 𝛼_𝑛=1/(𝑛+1).

In 1992, Wittmann [4] proved that the sequence {𝑥_𝑛} converges strongly to a fixed point of 𝑇 if {𝛼_𝑛} satisfies the following conditions:

C1: lim_𝑛→∞𝛼_𝑛=0; C2: ∑^∞𝑛=0𝛼_𝑛=∞; C3: ∑^∞𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞.

Shioji and Takahashi [5] extended Wittmann’s result to real Banach spaces with uniformly Gâteaux differentiable norms and in which each nonempty closed convex and bounded subset has the fixed point property for nonexpansive mappings. The concept of the Halpern iterative scheme has been widely used to approximate the fixed points for nonexpansive mappings (see, e.g., [6–12] and the reference cited therein).

Let 𝑓∶𝐾→𝐾 be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping 𝑇 as follows:𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0,(1.3) where 𝛼_𝑛∈(0,1). Xu [14] also studied the iteration process (1.3) in uniformly smooth Banach spaces.

Let 𝐴 be a strongly positive bounded linear operator on a real Hilbert space 𝐻, that is, there is a constant 𝛾>0 such that⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻.(1.4)

A typical problem is to minimize a quadratic function over the fixed points set of a nonexpansive mapping on a Hilbert space 𝐻:min_𝑥∈𝐶12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.5) where 𝐶 is the fixed points set of a nonexpansive mapping 𝑇 on 𝐻 and 𝑏 is a given point in 𝐻.

In 2006, Marino and Xu [15] introduced the following general iterative method for a nonexpansive mapping 𝑇 in a Hilbert space 𝐻:𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,𝑛≥1,(1.6) where {𝛼_𝑛}⊂(0,1), 𝑓 is a contraction on 𝐻, and 𝐴 is a strongly positive bounded linear operator on 𝐻. They proved that the sequence {𝑥_𝑛} generated by (1.6) converges strongly to a fixed point 𝑥^∗∈𝐹(𝑇) which also solves the variational inequality⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐹(𝑇),(1.7) 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).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup {𝑇(𝑡)∶𝑡≥0} in a Hilbert space:𝑥_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,𝑛≥1,(1.8) where {𝛼_𝑛}⊂(0,1) and 𝑢∈𝐾. He proved strong convergence of iteration (1.8) under suitable conditions. Subsequently, Xu [17] extended Suzuki’s [16] result from a Hilbert space to a uniformly convex Banach space which admits a weakly sequentially continuous normalized duality mapping.

Motivated by Chen and Song [18], in 2007, Chen and He [19] investigated the implicit and explicit viscosity methods for a nonexpansive semigroup without integral in a reflexive Banach space which admits a weakly sequentially continuous normalized duality mapping:𝑥𝑥_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,𝑛≥1,(1.9)_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛,𝑛≥1,(1.10) where {𝛼_𝑛}⊂(0,1).

In 2008, Song and Xu [20] also studied the iterations (1.9) and (1.10) in a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. Subsequently, Cholamjiak and Suantai [21] extended Song and Xu’s results to a Banach space which admits duality mapping with a gauge function. Wangkeeree and Kamraksa [22] and Wangkeeree et al. [23] obtained the convergence results concerning the duality mapping with a gauge function in Banach spaces. The convergence of iterations for a nonexpansive semigroup and nonlinear mappings has been studied by many authors (see, e.g., [24–38]).

Let 𝐸 be a real reflexive Banach space which admits the duality mapping 𝐽_𝜑 with a gauge 𝜑. Let {𝑇(𝑡)∶𝑡≥0} be a nonexpansive semigroup on 𝐸. Recall that an operator 𝐴 is said to be strongly positive if there exists a constant 𝛾>0 such that‖𝛼𝐼−𝛽𝐴𝐴𝑥‖,𝐽=_𝜑(s𝑥u)p_{‖𝑥‖≤1}≥𝛾‖𝑥‖𝜑(‖𝑥‖),||(𝛼𝐼−𝛽𝐴)𝑥,𝐽_𝜑(𝑥)||,(1.11) where 𝛼∈[0,1] and 𝛽∈[−1,1].

Motivated by Chen and Song [18], Chen and He [19], Marino and Xu [15], Colao et al. [39], and Wangkeeree et al. [23], we study strong convergence of the following general iterative methods:𝑥𝑥_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥_𝑛,𝑛≥1,(1.12)_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥_𝑛,𝑛≥1,(1.13) where {𝛼_𝑛}⊂(0,1), 𝑓 is a contraction on 𝐸 and 𝐴 is a positive bounded linear operator on 𝐸.

2. Preliminaries

A Banach space 𝐸 is called strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. A Banach space 𝐸 is called uniformly convex if for each 𝜖>0 there is a 𝛿>0 such that for 𝑥,𝑦∈𝐸 with ‖𝑥‖,‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜖,‖𝑥+𝑦‖≤2(1−𝛿) holds. The modulus of convexity of 𝐸 is defined by𝛿_𝐸(𝜖)=inf1−‖‖‖12(𝑥+𝑦)‖‖‖∶‖𝑥‖,‖𝑦‖≤1,‖𝑥−𝑦‖≥𝜖,(2.1) for all 𝜖∈[0,2]. 𝐸 is uniformly convex if 𝛿_𝐸(0)=0, and 𝛿_𝐸(𝜖)>0 for all 0<𝜖≤2. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1}. Then the norm of 𝐸 is said to be Gâteaux differentiable iflim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.2) exists for each 𝑥,𝑦∈𝑆(𝐸). In this case 𝐸 is called smooth. The norm of 𝐸 is said to be Fréchet differentiable if for each 𝑥∈𝑆(𝐸), the limit is attained uniformly for 𝑦∈𝑆(𝐸). The norm of 𝐸 is called uniformly Fréchet differentiable, if the limit is attained uniformly for 𝑥,𝑦∈𝑆(𝐸). It is well known that (uniformly) Fréchet differentiability of the norm of 𝐸 implies (uniformly) Gâteaux differentiability of the norm of 𝐸.

Let 𝜌_𝐸∶[0,∞)→[0,∞) be the modulus of smoothness of 𝐸 defined by𝜌_𝐸(𝑡)=sup12(‖𝑥+𝑦‖+‖𝑥−𝑦‖)−1∶𝑥∈𝑆(𝐸),‖𝑦‖≤𝑡.(2.3)

A Banach space 𝐸 is called uniformly smooth if 𝜌_𝐸(𝑡)/𝑡→0 as 𝑡→0. See [40–42] for more details.

We need the following definitions and results which can be found in [40, 41, 43].

Definition 2.1. A continuous strictly increasing function 𝜑∶[0,∞)→[0,∞) is said to be gauge function if 𝜑(0)=0 and lim_𝑡→∞𝜑(𝑡)=∞.

Definition 2.2. Let 𝐸 be a normed space and 𝜑 a gauge function. Then the mapping 𝐽_𝜑∶𝐸→2^𝐸∗ defined by 𝐽_𝜑(𝑥)=𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖𝜑(‖𝑥‖),‖𝑓^∗‖=𝜑(‖𝑥‖),𝑥∈𝐸,(2.4) is called the duality mapping with gauge function 𝜑.

In the particular case 𝜑(𝑡)=𝑡, the duality mapping 𝐽_𝜑=𝐽 is called the normalized duality mapping.

In the case 𝜑(𝑡)=𝑡^𝑞−1,𝑞>1, the duality mapping 𝐽_𝜑=𝐽_𝑞 is called the generalized duality mapping. It follows from the definition that 𝐽_𝜑(𝑥)=𝜑(‖𝑥‖)/‖𝑥‖𝐽(𝑥) and 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−2𝐽(𝑥),𝑞>1.

Remark 2.3. For the gauge function 𝜑, the function Φ∶[0,∞)→[0,∞) defined by Φ(𝑡)=^𝑡0𝜑(𝑠)𝑑𝑠(2.5) is a continuous convex and strictly increasing function on [0,∞). Therefore, Φ has a continuous inverse function Φ⁻¹.

It is noted that if 0≤𝑘≤1, then 𝜑(𝑘𝑥)≤𝜑(𝑥). FurtherΦ(𝑘𝑡)=^0𝑘𝑡𝜑(𝑠)𝑑𝑠=𝑘^𝑡0𝜑(𝑘𝑥)𝑑𝑥≤𝑘^𝑡0𝜑(𝑥)𝑑𝑥=𝑘Φ(𝑡).(2.6)

Remark 2.4. For each 𝑥 in a Banach space 𝐸, 𝐽_𝜑(𝑥)=𝜕Φ(‖𝑥‖), where 𝜕 denotes the sub-differential.

We also know the following facts:(i)𝐽_𝜑 is a nonempty, closed, and convex set in 𝐸^∗ for each 𝑥∈𝐸,(ii)𝐽_𝜑 is a function when 𝐸^∗ is strictly convex,(iii)If 𝐽_𝜑 is single-valued, then𝐽_𝜑(𝜆𝑥)=𝑠𝑖𝑔𝑛(𝜆)𝜑(‖𝜆𝑥‖)𝜑(‖𝑥‖)𝐽^𝜑𝑥−𝑦,𝐽_𝜑(𝑥),∀𝑥∈𝐸,𝜆∈ℝ,(𝑥)−𝐽_𝜑(𝑦)≥(𝜑(‖𝑥‖)−𝜑(‖𝑦‖))(‖𝑥‖−‖𝑦‖),∀𝑥,𝑦∈𝐸.(2.7) Following Browder [43], 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 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 the space ℓ^𝑝 has a weakly continuous duality mapping with a gauge function 𝜑(𝑡)=𝑡^𝑝−1 for all 1<𝑝<∞. Moreover, 𝜑 is invariant on [0,1].

Lemma 2.5 (See [44]). 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: Φ(‖𝑥+𝑦‖)≤Φ(‖𝑥‖)+𝑦,𝐽_𝜑(𝑥+𝑦).(2.8) In particular, for all 𝑥,𝑦∈𝐸, ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝐽(𝑥+𝑦)⟩.(2.9)(ii)Assume that a sequence {𝑥_𝑛} in 𝐸 converges weakly to a point 𝑥∈𝐸. Then the following holds: limsup_𝑛→∞Φ‖‖𝑥_𝑛−𝑦‖‖=limsup_𝑛→∞Φ‖‖𝑥_𝑛−𝑥‖‖+Φ(‖𝑥−𝑦‖)(2.10) for all 𝑥,𝑦∈𝐸.

Lemma 2.6 (See [23]). Assume that a Banach space 𝐸 has a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑. Let 𝐴 be a strongly positive bounded linear operator on 𝐸 with coefficient 𝛾>0 and 0<𝜌≤𝜑(1)‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤𝜑(1)(1−𝜌𝛾).

Lemma 2.7 (See [12]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛,𝑛≥1,(2.11) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that
(a) ∑^∞𝑛=1𝛾_𝑛=∞; (b) limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞𝑛=1|𝛾_𝑛𝛿_𝑛|<∞.
Then lim_𝑛→∞𝑎_𝑛=0.

3. Implicit Iteration Scheme

In this section, we prove a strong convergence theorem of an implicit iterative method (1.12).

Theorem 3.1. Let 𝐸 be a reflexive which admits a weakly continuous duality mapping 𝐽_𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let 𝔉={𝑇(𝑡)∶𝑡≥0} be a nonexpansive semigroup on 𝐸 such that 𝐹≠∅. Let 𝑓 be a contraction on 𝐸 with the coefficient 𝛼∈(0,1) and 𝐴 a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<𝛾𝜑(1)/𝛼. Let {𝛼_𝑛} and {𝑡_𝑛} be real sequences satisfying 0<𝛼_𝑛<1, 𝑡_𝑛>0 and lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝛼_𝑛/𝑡_𝑛=0. Then {𝑥_𝑛} defined by (1.12) converges strongly to 𝑞∈𝐹 which solves the following variational inequality: (𝐴−𝛾𝑓)(𝑞),𝐽_𝜑(𝑞−𝑤)≤0,∀𝑤∈𝐹.(3.1)

Proof. First, we prove the uniqueness of the solution to the variational inequality (3.1) in 𝐹. Suppose that 𝑝,𝑞∈𝐹 satisfy (3.1), so we have ((𝐴𝐴−−𝛾𝛾𝑓𝑓))((𝑝𝑞)),,𝐽𝐽_𝜑𝜑(𝑝−𝑞)≤0,(𝑞−𝑝)≤0.(3.2) Adding the above inequalities, we get 𝐴(𝑝)−𝐴(𝑞)−𝛾(𝑓(𝑝)−𝑓(𝑞)),𝐽_𝜑(𝑝−𝑞)≤0.(3.3) This shows that 𝐴(𝑝−𝑞),𝐽_𝜑(𝑝−𝑞)≤𝛾𝑓(𝑝)−𝑓(𝑞),𝐽_𝜑(𝑝−𝑞),(3.4) which implies by the strong positivity of 𝐴𝛾‖𝑝−𝑞‖𝜑(‖𝑝−𝑞‖)≤𝐴(𝑝−𝑞),𝐽_𝜑(𝑝−𝑞)≤𝛾𝛼‖𝑝−𝑞‖𝜑(‖𝑝−𝑞‖).(3.5) Since 𝜑 is invariant on [0,1], 𝜑(1)𝛾‖𝑝−𝑞‖𝜑(‖𝑝−𝑞‖)≤𝛾𝛼‖𝑝−𝑞‖𝜑(‖𝑝−𝑞‖).(3.6) It follows that 𝜑(1)𝛾−𝛾𝛼‖𝑝−𝑞‖𝜑(‖𝑝−𝑞‖)≤0.(3.7) Therefore 𝑝=𝑞 since 0<𝛾<(𝛾𝜑(1))/𝛼.
We next prove that {𝑥_𝑛} is bounded. For each 𝑤∈𝐹, by Lemma 2.6, we have‖‖𝑥_𝑛−𝑤‖‖==‖‖‖‖𝛼_𝑛𝐼𝛾−𝑓𝛼𝑥_𝑛𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥_𝑛−𝑤‖‖≤𝜑(1)1−𝐴𝛼_𝑛𝑇𝑡_𝑛x_𝑛−𝐼−𝛼_𝑛𝐴𝑤+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴(𝑤)‖‖≤‖‖𝑥𝛾‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑤‖‖+‖𝛾𝑓(𝑤)−𝐴(𝑤)_𝑛‖−𝑤‖‖−𝛼_𝑛𝜑(1)𝛾‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛‖𝛾𝑓(𝑤)−𝐴(𝑤)‖,(3.8) which yields ‖‖𝑥_𝑛−𝑤‖‖≤1𝜑(1)𝛾−𝛾𝛼‖𝛾𝑓(𝑤)−𝐴(𝑤)‖.(3.9) Hence {𝑥_𝑛} is bounded. So are {𝑓(𝑥_𝑛)} and {𝐴𝑇(𝑡_𝑛)𝑥_𝑛}.
We next prove that {𝑥_𝑛} is relatively sequentially compact. By the reflexivity of 𝐸 and the boundedness of {𝑥_𝑛}, there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} and a point 𝑝 in 𝐸 such that 𝑥_𝑛𝑗⇀𝑝 as 𝑗→∞. Now we show that 𝑝∈𝐹. Put 𝑥_𝑗=𝑥_𝑛𝑗, 𝛽_𝑗=𝛼_𝑛𝑗 and 𝑠_𝑗=𝑡_𝑛𝑗 for 𝑗∈ℕ, fix 𝑡>0. We see that‖‖𝑥_𝑗−𝑇(𝑡)𝑝‖‖≤^{[𝑡/𝑠𝑗]−1}_𝑘=0+‖‖‖𝑇‖‖𝑇𝑡(𝑘+1)𝑠_𝑗𝑥_𝑗−𝑇𝑘𝑠_𝑗𝑥_𝑗+1‖‖𝑠_𝑗𝑠^𝑗𝑥^𝑗−𝑇𝑡≤𝑡𝑠_𝑗𝑠^𝑗𝑝‖‖‖+‖‖‖𝑇𝑠𝑡_𝑗𝑠^𝑗𝑝−𝑇(𝑡)𝑝‖‖‖𝑠_𝑗‖‖𝑇𝑠^𝑗𝑥_𝑗−𝑥_𝑗‖‖+‖‖𝑥_𝑗−𝑝‖‖+‖‖‖𝑇𝑡−𝑡=𝑡𝑠_𝑗𝑠^𝑗𝑝−𝑝‖‖‖𝑠_𝑗𝛽^𝑗‖‖𝐴𝑇𝑠_𝑗𝑥_𝑗−𝛾𝑓𝑥_𝑗‖‖+‖‖𝑥_𝑗−𝑝‖‖+‖‖‖𝑇𝑡−𝑡≤𝑡𝛽_𝑗𝑠_𝑗𝑠^𝑗𝑝−𝑝‖‖‖𝑠_𝑗+ma‖‖x𝐴𝑇‖𝑇𝑠_𝑗(𝑠)𝑥𝑝_𝑗−−𝛾𝑝𝑓‖∶𝑥0_𝑗≤‖‖𝑠+≤‖‖𝑥𝑠_𝑗𝑗−𝑝‖‖.(3.10) So we have limsup_𝑗→∞Φ‖‖𝑥_𝑗−𝑇(𝑡)𝑝‖‖≤limsup_𝑗→∞Φ‖‖𝑥_𝑗−𝑝‖‖.(3.11) On the other hand, by Lemma 2.5 (ii), we have limsup_𝑗→∞Φ‖‖𝑥_𝑗−𝑇(𝑡)𝑝‖‖=limsup_𝑗→∞Φ‖‖𝑥_𝑗−𝑝‖‖+Φ(‖𝑇(𝑡)𝑝−𝑝‖).(3.12) Combining (3.11) and (3.12), we have Φ(‖𝑇(𝑡)𝑝−𝑝‖)≤0.(3.13) This implies that 𝑝∈𝐹. Further, we see that ‖𝑥_𝑗−𝑝‖𝜑‖𝑥_𝑗−𝑝‖==𝑥_𝑗𝐼−−𝑝𝛽,_𝑗𝐽_𝜑𝑥_𝑗−𝑝+𝛽_𝑗𝐴𝑇𝑠_𝑗𝑥_𝑗−𝐼−𝛽_𝑗𝐴𝑝,𝐽_𝜑𝑥_𝑗−𝑝≤𝜑(1)𝛾1𝑓−𝑥𝛽_𝑗𝑗−𝛾𝑓(𝑝),𝐽_𝜑𝑥_𝑗−𝑝+𝛽_𝑗𝛾𝑓(𝑝)−𝐴(𝑝),𝐽_𝜑𝑥_𝑗−𝑝+𝛽_𝑗𝛾‖‖𝑥_𝑗−𝑝‖‖𝜑‖‖𝑥_𝑗−𝑝‖‖𝛾𝛼‖‖𝑥_𝑗−𝑝‖‖𝜑‖‖𝑥_𝑗−𝑝‖‖+𝛽_𝑗𝛾𝑓(𝑝)−𝐴(𝑝),𝐽_𝜑𝑥_𝑗−𝑝.(3.14) So we have ‖‖𝑥_𝑗−𝑝‖‖𝜑‖‖𝑥_𝑗−𝑝‖‖≤1𝜑(1)𝛾−𝛾𝛼𝛾𝑓(𝑝)−𝐴(𝑝),𝐽^𝜑𝑥_𝑗−𝑝.(3.15) By the definition of Φ, it is easily seen that Φ‖‖𝑥_𝑗−𝑝‖‖≤‖‖𝑥_𝑗−𝑝‖‖𝜑‖‖𝑥_𝑗−𝑝‖‖.(3.16) Hence Φ‖‖𝑥_𝑗−𝑝‖‖≤1𝜑(1)𝛾−𝛾𝛼𝛾𝑓(𝑝)−𝐴(𝑝),𝐽^𝜑𝑥_𝑗−𝑝.(3.17) Therefore Φ(‖𝑥_𝑗−𝑝‖)→0 as 𝑗→∞ since 𝐽_𝜑 is weakly continuous; consequently, 𝑥_𝑗→𝑝 as 𝑗→∞ by the continuity of Φ. Hence {𝑥_𝑛} is relatively sequentially compact.
Finally, we prove that 𝑝 is a solution in 𝐹 to the variational inequality (3.1). For any 𝑤∈𝐹, we see that𝐼−𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝑇𝑡_𝑛𝑤,𝐽_𝜑𝑥_𝑛−𝑤=−𝑥_𝑛𝑇−𝑡𝑤,𝐽_𝜑𝑥_𝑛−𝑤_𝑛≥‖‖𝑥𝑥_𝑛−𝑇𝑡_𝑛𝑤,𝐽_𝜑𝑥_𝑛−𝑤_𝑛−‖‖𝑇−𝑡𝑤‖‖𝜑‖‖𝑥_𝑛−𝑤_𝑛‖‖≥‖‖𝑥𝑥_𝑛−𝑇𝑡_𝑛𝑤‖‖‖‖𝐽_𝜑𝑥_𝑛−𝑤_𝑛‖‖−‖‖𝑥−𝑤‖‖𝜑‖‖𝑥_𝑛−𝑤_𝑛‖‖=0.−𝑤‖‖‖‖𝐽_𝜑𝑥_𝑛−𝑤‖‖(3.18) On the other hand, we have (𝐴−𝛾𝑓)𝑥_𝑛=−1𝛼_𝑛𝐼−𝛼_𝑛𝐴𝐼−𝑇𝑡_𝑛𝑥_𝑛,(3.19) which implies (𝐴−𝛾𝑓)𝑥_𝑛,𝐽_𝜑𝑥_𝑛−𝑤=+−𝛼1𝐴_𝑛𝐼𝐼−−𝑇𝑇𝑡_𝑛𝑡_𝑛𝑥_𝑛−𝐼−𝑇𝑡_𝑛𝑤,𝐽_𝜑𝑥_𝑛−𝑤≤𝐴𝐼−𝑇𝑡_𝑛𝑥_𝑛,𝐽_𝜑𝑥_𝑛−𝑤𝑥_𝑛,𝐽_𝜑𝑥_𝑛−𝑤.(3.20) Observe ‖‖𝑥_𝑗−𝑇𝑠_𝑗𝑥_𝑗‖‖=𝛽_𝑗‖‖𝛾𝑓𝑥_𝑗−𝐴𝑇𝑠_𝑗𝑥_𝑗‖‖→0,(3.21) as 𝑗→∞. Replacing 𝑛 by 𝑛_𝑗 and letting 𝑗→∞ in (3.20), we obtain (𝐴−𝛾𝑓)(𝑝),𝐽_𝜑(𝑝−𝑤)≤0,∀𝑤∈𝐹.(3.22) So 𝑝∈𝐹 is a solution of variational inequality (3.1); and hence 𝑝=𝑞 by the uniqueness. In a summary, we have proved that {𝑥_𝑛} is relatively sequentially compact and each cluster point of {𝑥_𝑛} (as 𝑛→∞) equals 𝑞. Therefore 𝑥_𝑛→𝑞 as 𝑛→∞. This completes the proof.

4. Explicit Iteration Scheme

In this section, utilizing the implicit version in Theorem 3.1, we consider the explicit one in a reflexive Banach space which admits the duality mapping 𝐽_𝜑.

Theorem 4.1. 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 {𝑇(𝑡)∶𝑡≥0} be a nonexpansive semigroup on 𝐸 such that 𝐹≠∅. Let 𝑓 be a contraction on 𝐸 with the coefficient 𝛼∈(0,1) and 𝐴 a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<𝛾𝜑(1)/𝛼. Let {𝛼_𝑛} and {𝑡_𝑛} be real sequences satisfying 0<𝛼_𝑛<1, ∑^∞𝑛=1𝛼_𝑛=∞, 𝑡_𝑛>0 and lim_𝑛→∞𝑡_𝑛=lim_𝑛→∞𝛼_𝑛/𝑡_𝑛=0. Then {𝑥_𝑛} defined by (1.13) converges strongly to 𝑞∈𝐹 which also solves the variational inequality (3.1).

Proof. Since 𝛼_𝑛→0, we may assume that 𝛼_𝑛<𝜑(1)‖𝐴‖⁻¹ and 1−𝛼_𝑛(𝜑(1)𝛾−𝛾𝛼)>0 for all 𝑛. First we prove that {𝑥_𝑛} is bounded. For each 𝑤∈𝐹, by Lemma 2.6, we have ‖‖𝑥_𝑛+1−𝑤‖‖==‖‖‖‖𝛼_𝑛𝐼𝛾−𝑓𝛼𝑥_𝑛𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥_𝑛−𝑤‖‖≤𝜑(1)1−𝐴𝛼_𝑛𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝐴𝑤+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴(𝑤)‖‖=𝜑(1)−𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛‖𝛾𝑓(𝑤)−𝐴(𝑤)‖≤1−𝛼_𝑛𝜑(1)𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛‖𝛾𝑓(𝑤)−𝐴(𝑤)‖𝜑(1)𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑤‖‖+𝛼_𝑛𝜑(1)𝛾−𝛾𝛼‖𝛾𝑓(𝑤)−𝐴(𝑤)‖𝜑(1)𝛾−𝛾𝛼.(4.1) It follows from induction that ‖‖𝑥_𝑛+1−𝑤‖‖≤max‖‖𝑥₁−𝑤‖‖,‖𝛾𝑓(𝑤)−𝐴(𝑤)‖𝜑(1)𝛾−𝛾𝛼,𝑛≥1.(4.2) Thus {𝑥_𝑛} is bounded, and hence so are {𝑓(𝑥_𝑛)} and {𝐴𝑇(𝑡_𝑛)𝑥_𝑛}. From Theorem 3.1, there is a unique solution 𝑞∈𝐹 to the following variational inequality: (𝐴−𝛾𝑓)𝑞,𝐽_𝜑(𝑞−𝑤)≤0,∀𝑤∈𝐹.(4.3) Next we prove that limsup_𝑛→∞(𝐴−𝛾𝑓)𝑞,𝐽_𝜑𝑞−𝑥_𝑛+1≤0.(4.4) Indeed, we can choose a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that limsup_𝑛→∞(𝐴−𝛾𝑓)𝑞,𝐽_𝜑𝑞−𝑥_𝑛=limsup_𝑗→∞(𝐴−𝛾𝑓)𝑞,𝐽^𝜑𝑞−𝑥^𝑛_𝑗.(4.5) Further, we can assume that 𝑥_𝑛𝑗⇀𝑝∈𝐸 by the reflexivity of 𝐸 and the boundedness of {𝑥_𝑛}. Now we show that 𝑝∈𝐹. Put 𝑥_𝑗=𝑥_𝑛𝑗,𝛽_𝑗=𝛼_𝑛𝑗 and 𝑠_𝑗=𝑡_𝑛𝑗 for 𝑗∈ℕ, fix 𝑡>0. We obtain ‖‖𝑥_𝑗+1−𝑇(𝑡)𝑝‖‖≤^{[𝑡/𝑠𝑗]−1}_𝑘=0+‖‖‖𝑇‖‖𝑇𝑡(𝑘+1)𝑠_𝑗𝑥_𝑗−𝑇𝑘𝑠_𝑗𝑥_𝑗+1‖‖𝑠_𝑗𝑠^𝑗𝑥^𝑗−𝑇𝑡≤𝑡𝑠_𝑗𝑠^𝑗𝑝‖‖‖+‖‖‖𝑇𝑠𝑡_𝑗𝑠^𝑗𝑝−𝑇(𝑡)𝑝‖‖‖s_𝑗‖‖𝑇𝑠^𝑗𝑥_𝑗−𝑥_𝑗+1‖‖+‖‖𝑥_𝑗−𝑝‖‖+‖‖‖𝑇𝑡−𝑡=𝑡𝑠_𝑗𝑠^𝑗𝑝−𝑝‖‖‖𝑠_𝑗𝛽^𝑗‖‖𝐴𝑇𝑠_𝑗𝑥_𝑗−𝛾𝑓𝑥_𝑗‖‖+‖‖𝑥_𝑗−𝑝‖‖+‖‖‖𝑇𝑡−𝑡≤𝑡𝛽_𝑗𝑠_𝑗𝑠^𝑗𝑝−𝑝‖‖‖𝑠_𝑗+ma‖‖x𝐴𝑇‖𝑇𝑠_𝑗(𝑠)𝑥𝑝_𝑗−−𝛾𝑝𝑓‖∶𝑥0_𝑗≤‖‖𝑠+≤‖‖𝑥𝑠_𝑗𝑗−𝑝‖‖.(4.6) It follows that limsup_𝑛→∞Φ(‖𝑥_𝑗−𝑇(𝑡)𝑝‖)≤limsup_𝑛→∞Φ(‖𝑥_𝑗−𝑝‖). From Lemma 2.5 (ii) we have limsup_𝑛→∞Φ‖‖𝑥_𝑗−𝑇(𝑡)𝑝‖‖=limsup_𝑛→∞Φ‖‖𝑥_𝑗−𝑝‖‖+Φ(‖𝑇(𝑡)𝑝−𝑝‖).(4.7) So we have Φ(‖𝑇(𝑡)𝑝−𝑝‖)≤0 and hence 𝑝∈𝐹. Since the duality mapping 𝐽_𝜑 is weakly sequentially continuous, limsup_𝑛→∞(𝐴−𝛾𝑓)𝑞,𝐽_𝜑𝑞−𝑥_𝑛+1==lim(𝐴su−p_𝑗→∞𝛾𝑓()𝐴𝑞,−𝐽_𝜑𝛾𝑓)𝑞,𝐽^𝜑𝑞−𝑥^𝑛_𝑗⁺¹(𝑞−𝑝)≤0.(4.8) Finally, we show that 𝑥_𝑛→𝑞. From Lemma 2.5 (i), we have Φ‖‖𝑥_𝑛+1−𝑞‖‖=Φ+‖‖𝛼_𝑛𝐼−𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥_𝑛−𝐼−𝛼_𝑛𝐴𝑞+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑞)≤Φ‖‖𝐼(−𝛾𝑓𝛼(_𝑛𝑞)−𝐴(𝑞))‖‖+𝛼_𝑛𝐴𝑇𝑡_𝑛𝑥_𝑛−𝑞+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑞)‖‖≤Φ𝜑(𝛾1𝑓)(𝑞1)−−𝛼𝐴_𝑛(𝑞),𝐽_𝜑𝑥_𝑛+1−𝑞+𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑞‖‖+𝛼_𝑛𝛾𝛼‖‖𝑥_𝑛−𝑞‖‖=Φ𝜑𝛾(𝑓1)(𝑞−)𝛼−_𝑛𝐴(𝑞),𝐽_𝜑𝑥_𝑛+1−𝑞+𝛼_𝑛𝜑(1)𝛾−𝛾𝛼‖‖𝑥_𝑛−𝑞‖‖≤1−𝛼𝛾_𝑛𝑓(𝑞)−𝐴(𝑞),𝐽_𝜑𝑥_𝑛+1−𝑞+𝛼_𝑛𝜑(1)𝛾−𝛾𝛼Φ‖‖𝑥_𝑛−𝑞‖‖𝛾𝑓(𝑞)−𝐴(𝑞),𝐽_𝜑𝑥_𝑛+1−𝑞.(4.9) Note that ∑^∞𝑛=1𝛼_𝑛=∞ and limsup_𝑛→∞⟨𝛾𝑓(𝑞)−𝐴(𝑞),𝐽_𝜑(𝑥_𝑛+1−𝑞)⟩≤0. Using Lemma 2.7, we have 𝑥_𝑛→𝑞 as 𝑛→∞ by the continuity of Φ. This completes the proof.

Remark 4.2. Theorems 3.1 and 4.1 improve and extend the main results proved in [15] in the following senses:(i)from a nonexpansive mapping to a nonexpansive semigroup,(ii)from a real Hilbert space to a reflexive Banach space which admits a weakly continuous duality mapping with gauge functions.

Acknowledgments

The authors wish to thank the editor and the referee for valuable suggestions. K. Nammanee was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. S. Suantai and P. Cholamjiak wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.