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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 506976, 14 pages
http://dx.doi.org/10.1155/2012/506976
Research Article

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

1School of Science, University of Phayao, Phayao 56000, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 23 November 2011; Accepted 27 January 2012

Academic Editor: Giuseppe Marino

Copyright © 2012 Kamonrat Nammanee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝐽𝜑, 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)𝑡0} of self-mappings of 𝐾 such that(i)𝑇(0)𝑥=𝑥 for all 𝑥𝐾,(ii)𝑇(𝑡+𝑠)𝑥=𝑇(𝑡)𝑇(𝑠)𝑥 for all 𝑡,𝑠0 and 𝑥𝐾,(iii)for each 𝑥𝐾, the mapping 𝑇()𝑥 is continuous,(iv)for each 𝑡0, 𝑇(𝑡) is nonexpansive, that is,𝑇(𝑡)𝑥𝑇(𝑡)𝑦𝑥𝑦,𝑥,𝑦𝐾.(1.1) We denote 𝐹 by the common fixed points set of 𝔉, that is, 𝐹=𝑡0𝐹(𝑇(𝑡)).

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping 𝑇𝐾𝐾 in a real Hilbert space:𝑥𝑛+1=𝛼𝑛𝑢+1𝛼𝑛𝑇𝑥𝑛,𝑛0,(1.2) where {𝛼𝑛}(0,1) and 𝑢𝐾.

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

C1: lim𝑛𝛼𝑛=0; C2: 𝑛=0𝛼𝑛=; C3: lim𝑛(𝛼𝑛𝛼𝑛1)/𝛼2𝑛=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 {𝛼𝑛} excluded the canonical choice 𝛼𝑛=1/(𝑛+1).

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

C1: lim𝑛𝛼𝑛=0; C2: 𝑛=0𝛼𝑛=; C3: 𝑛=0|𝛼𝑛+1𝛼𝑛|<.

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., [612] and the reference cited therein).

Let 𝑓𝐾𝐾 be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping 𝑇 as follows:𝑥𝑛+1=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑇𝑥𝑛,𝑛0,(1.3) where 𝛼𝑛(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 that𝐴𝑥,𝑥𝛾𝑥2,𝑥𝐻.(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:min𝑥𝐶12𝐴𝑥,𝑥𝑥,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:𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑥𝑛,𝑛1,(1.6) where {𝛼𝑛}(0,1), 𝑓 is a contraction on 𝐻, and 𝐴 is a strongly positive bounded linear operator on 𝐻. They proved that the sequence {𝑥𝑛} generated by (1.6) converges strongly to a fixed point 𝑥𝐹(𝑇) which also solves the variational inequality(𝐴𝛾𝑓)𝑥,𝑥𝑥0,𝑥𝐹(𝑇),(1.7) which is the optimality condition for the minimization problem: min𝑥𝐶(1/2)𝐴𝑥,𝑥(𝑥), where is a potential function for 𝛾𝑓 (i.e., (𝑥)=𝛾𝑓(𝑥) for 𝑥𝐻).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup {𝑇(𝑡)𝑡0} in a Hilbert space:𝑥𝑛=𝛼𝑛𝑢+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑛1,(1.8) where {𝛼𝑛}(0,1) and 𝑢𝐾. 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:𝑥𝑛=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛𝑥,𝑛1,(1.9)𝑛+1=𝛼𝑛𝑓𝑥𝑛+1𝛼𝑛𝑇𝑡𝑛𝑥𝑛,𝑛1,(1.10) where {𝛼𝑛}(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., [2438]).

Let 𝐸 be a real reflexive Banach space which admits the duality mapping 𝐽𝜑 with a gauge 𝜑. Let {𝑇(𝑡)𝑡0} 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 that𝐴𝑥,𝐽𝜑(𝑥)(𝛾𝑥𝜑𝑥),𝛼𝐼𝛽𝐴=sup𝑥1||(𝛼𝐼𝛽𝐴)𝑥,𝐽𝜑||,(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:𝑥𝑛=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛𝑥,𝑛1,(1.12)𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛,𝑛1,(1.13) where {𝛼𝑛}(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,𝑦𝐸 with 𝑥=𝑦=1 and 𝑥𝑦. A Banach space 𝐸 is called uniformly convex if for each 𝜖>0 there is a 𝛿>0 such that for 𝑥,𝑦𝐸 with 𝑥,𝑦1 and 𝑥𝑦𝜖,𝑥+𝑦2(1𝛿) holds. The modulus of convexity of 𝐸 is defined by𝛿𝐸1(𝜖)=inf12,(𝑥+𝑦)𝑥,𝑦1,𝑥𝑦𝜖(2.1) for all 𝜖[0,2]. 𝐸 is uniformly convex if 𝛿𝐸(0)=0, and 𝛿𝐸(𝜖)>0 for all 0<𝜖2. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let 𝑆(𝐸)={𝑥𝐸𝑥=1}. Then the norm of 𝐸 is said to be Gâteaux differentiable iflim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.2) exists for each 𝑥,𝑦𝑆(𝐸). In this case 𝐸 is called smooth. The norm of 𝐸 is said to be Fréchet differentiable if for each 𝑥𝑆(𝐸), the limit is attained uniformly for 𝑦𝑆(𝐸). The norm of 𝐸 is called uniformly Fréchet differentiable, if the limit is attained uniformly for 𝑥,𝑦𝑆(𝐸). It is well known that (uniformly) Fréchet differentiability of the norm of 𝐸 implies (uniformly) Gâteaux differentiability of the norm of 𝐸.

Let 𝜌𝐸[0,)[0,) be the modulus of smoothness of 𝐸 defined by𝜌𝐸1(𝑡)=sup2(.𝑥+𝑦+𝑥𝑦)1𝑥𝑆(𝐸),𝑦𝑡(2.3)

A Banach space 𝐸 is called uniformly smooth if 𝜌𝐸(𝑡)/𝑡0 as 𝑡0. See [4042] 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 lim𝑡𝜑(𝑡)=.

Definition 2.2. Let 𝐸 be a normed space and 𝜑 a gauge function. Then the mapping 𝐽𝜑𝐸2𝐸 defined by 𝐽𝜑𝑓(𝑥)=𝐸𝑥,𝑓(=𝑥𝜑𝑥),𝑓()=𝜑𝑥,𝑥𝐸,(2.4) is called the duality mapping with gauge function 𝜑.

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

In the case 𝜑(𝑡)=𝑡𝑞1,𝑞>1, the duality mapping 𝐽𝜑=𝐽𝑞 is called the generalized duality mapping. It follows from the definition that 𝐽𝜑(𝑥)=𝜑(𝑥)/𝑥𝐽(𝑥) and 𝐽𝑞(𝑥)=𝑥𝑞2𝐽(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 Φ1.

It is noted that if 0𝑘1, then 𝜑(𝑘𝑥)𝜑(𝑥). FurtherΦ(𝑘𝑡)=0𝑘𝑡𝜑(𝑠)𝑑𝑠=𝑘𝑡0𝜑(𝑘𝑥)𝑑𝑥𝑘𝑡0𝜑(𝑥)𝑑𝑥=𝑘Φ(𝑡).(2.6)

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

We also know the following facts:(i)𝐽𝜑 is a nonempty, closed, and convex set in 𝐸 for each 𝑥𝐸,(ii)𝐽𝜑 is a function when 𝐸 is strictly convex,(iii)If 𝐽𝜑 is single-valued, then𝐽𝜑(𝜆𝑥)=𝑠𝑖𝑔𝑛(𝜆)𝜑(𝜆𝑥)𝐽𝜑(𝑥)𝜑(𝑥),𝑥𝐸,𝜆,𝑥𝑦,𝐽𝜑(𝑥)𝐽𝜑(𝑦)(𝜑(𝑥)𝜑(𝑦))(𝑥𝑦),𝑥,𝑦𝐸.(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 𝐽𝜑 is single-valued and continuous from the weak topology to the weak* topology, that is, for any {𝑥𝑛} with 𝑥𝑛𝑥, the sequence {𝐽𝜑(𝑥𝑛)} converges weakly* to 𝐽𝜑(𝑥). It is known that the space 𝑝 has a weakly continuous duality mapping with a gauge function 𝜑(𝑡)=𝑡𝑝1 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 𝐽𝜑 with gauge 𝜑.(i)For all 𝑥,𝑦𝐸, the following inequality holds: Φ(𝑥+𝑦)Φ(𝑥)+𝑦,𝐽𝜑.(𝑥+𝑦)(2.8) In particular, for all 𝑥,𝑦𝐸, 𝑥+𝑦2𝑥2+2𝑦,𝐽(𝑥+𝑦).(2.9)(ii)Assume that a sequence {𝑥𝑛} in 𝐸 converges weakly to a point 𝑥𝐸. Then the following holds: limsup𝑛Φ𝑥𝑛𝑦=limsup𝑛Φ𝑥𝑛()𝑥+Φ𝑥𝑦(2.10) for all 𝑥,𝑦𝐸.

Lemma 2.6 (See [23]). Assume that a Banach space 𝐸 has a weakly continuous duality mapping 𝐽𝜑 with gauge 𝜑. Let 𝐴 be a strongly positive bounded linear operator on 𝐸 with coefficient 𝛾>0 and 0<𝜌𝜑(1)𝐴1. Then 𝐼𝜌𝐴𝜑(1)(1𝜌𝛾).

Lemma 2.7 (See [12]). Assume that {𝑎𝑛} is a sequence of nonnegative real numbers such that 𝑎𝑛+11𝛾𝑛𝑎𝑛+𝛾𝑛𝛿𝑛,𝑛1,(2.11) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in such that
(a) 𝑛=1𝛾𝑛=; (b) limsup𝑛𝛿𝑛0 or 𝑛=1|𝛾𝑛𝛿𝑛|<.
Then lim𝑛𝑎𝑛=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 𝐽𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let 𝔉={𝑇(𝑡)𝑡0} 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 {𝛼𝑛} and {𝑡𝑛} be real sequences satisfying 0<𝛼𝑛<1, 𝑡𝑛>0 and lim𝑛𝑡𝑛=lim𝑛𝛼𝑛/𝑡𝑛=0. Then {𝑥𝑛} defined by (1.12) converges strongly to 𝑞𝐹 which solves the following variational inequality: (𝐴𝛾𝑓)(𝑞),𝐽𝜑(𝑞𝑤)0,𝑤𝐹.(3.1)

Proof. First, we prove the uniqueness of the solution to the variational inequality (3.1) in 𝐹. Suppose that 𝑝,𝑞𝐹 satisfy (3.1), so we have (𝐴𝛾𝑓)(𝑝),𝐽𝜑(𝑝𝑞)0,(𝐴𝛾𝑓)(𝑞),𝐽𝜑(𝑞𝑝)0.(3.2) Adding the above inequalities, we get 𝐴(𝑝)𝐴(𝑞)𝛾(𝑓(𝑝)𝑓(𝑞)),𝐽𝜑(𝑝𝑞)0.(3.3) This shows that 𝐴(𝑝𝑞),𝐽𝜑𝑓(𝑝𝑞)𝛾(𝑝)𝑓(𝑞),𝐽𝜑,(𝑝𝑞)(3.4) which implies by the strong positivity of 𝐴(𝐴𝛾𝑝𝑞𝜑𝑝𝑞)(𝑝𝑞),𝐽𝜑((𝑝𝑞)𝛾𝛼𝑝𝑞𝜑𝑝𝑞).(3.5) Since 𝜑 is invariant on [0,1], 𝜑(1)𝛾𝑝𝑞𝜑(𝑝𝑞)𝛾𝛼𝑝𝑞𝜑(𝑝𝑞).(3.6) It follows that 𝜑(1)(𝛾𝛾𝛼𝑝𝑞𝜑𝑝𝑞)0.(3.7) Therefore 𝑝=𝑞 since 0<𝛾<(𝛾𝜑(1))/𝛼.
We next prove that {𝑥𝑛} is bounded. For each 𝑤𝐹, by Lemma 2.6, we have𝑥𝑛=𝛼𝑤𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛=𝑤𝐼𝛼𝑛𝐴𝑇𝑡𝑛x𝑛𝐼𝛼𝑛𝐴𝑤+𝛼𝑛𝑥𝛾𝑓𝑛𝐴(𝑤)𝜑(1)1𝛼𝑛𝛾𝑥𝑛𝑤+𝛼𝑛𝑥𝛾𝛼𝑛𝑥𝑤+𝛾𝑓(𝑤)𝐴(𝑤)𝑛𝑤𝛼𝑛𝜑(1)𝛾𝑥𝑛𝑤+𝛼𝑛𝑥𝛾𝛼𝑛𝑤+𝛼𝑛𝛾𝑓(𝑤)𝐴(𝑤),(3.8) which yields 𝑥𝑛1𝑤𝜑(1)𝛾𝛾𝛼𝛾𝑓(𝑤)𝐴(𝑤).(3.9) Hence {𝑥𝑛} is bounded. So are {𝑓(𝑥𝑛)} and {𝐴𝑇(𝑡𝑛)𝑥𝑛}.
We next prove that {𝑥𝑛} is relatively sequentially compact. By the reflexivity of 𝐸 and the boundedness of {𝑥𝑛}, there exists a subsequence {𝑥𝑛𝑗} of {𝑥𝑛} and a point 𝑝 in 𝐸 such that 𝑥𝑛𝑗𝑝 as 𝑗. Now we show that 𝑝𝐹. Put 𝑥𝑗=𝑥𝑛𝑗, 𝛽𝑗=𝛼𝑛𝑗 and 𝑠𝑗=𝑡𝑛𝑗 for 𝑗, fix 𝑡>0. We see that𝑥𝑗𝑇(𝑡)𝑝[𝑡/𝑠𝑗]1𝑘=0𝑇(𝑘+1)𝑠𝑗𝑥𝑗𝑇𝑘𝑠𝑗𝑥𝑗+1+𝑇𝑡𝑠𝑗𝑠𝑗𝑥𝑗𝑡𝑇𝑠𝑗𝑠𝑗𝑝+𝑇𝑡𝑠𝑗𝑠𝑗𝑡𝑝𝑇(𝑡)𝑝𝑠𝑗𝑇𝑠𝑗𝑥𝑗𝑥𝑗+𝑥𝑗+𝑇𝑡𝑝𝑡𝑠𝑗𝑠𝑗=𝑡𝑝𝑝𝑠𝑗𝛽𝑗𝑠𝐴𝑇𝑗𝑥𝑗𝑥𝛾𝑓𝑗+𝑥𝑗+𝑇𝑡𝑝𝑡𝑠𝑗𝑠𝑗𝑝𝑝𝑡𝛽𝑗𝑠𝑗𝑠𝐴𝑇𝑗𝑥𝑗𝑥𝛾𝑓𝑗+𝑥𝑗𝑝+max𝑇(𝑠)𝑝𝑝0𝑠𝑠𝑗.(3.10) So we have limsup𝑗Φ𝑥𝑗𝑇(𝑡)𝑝limsup𝑗Φ𝑥𝑗.𝑝(3.11) On the other hand, by Lemma 2.5 (ii), we have limsup𝑗Φ𝑥𝑗𝑇(𝑡)𝑝=limsup𝑗Φ𝑥𝑗(𝑝+Φ𝑇(𝑡)𝑝𝑝).(3.12) Combining (3.11) and (3.12), we have Φ(𝑇(𝑡)𝑝𝑝)0.(3.13) This implies that 𝑝𝐹. Further, we see that 𝑥𝑗𝑝𝜑𝑥𝑗=𝑥𝑝𝑗𝑝,𝐽𝜑𝑥𝑗=𝑝𝐼𝛽𝑗𝐴𝑇𝑠𝑗𝑥𝑗𝐼𝛽𝑗𝐴𝑝,𝐽𝜑𝑥𝑗𝑝+𝛽𝑗𝑥𝛾𝑓𝑗𝛾𝑓(𝑝),𝐽𝜑𝑥𝑗𝑝+𝛽𝑗𝛾𝑓(𝑝)𝐴(𝑝),𝐽𝜑𝑥𝑗𝑝𝜑(1)1𝛽𝑗𝛾𝑥𝑗𝜑𝑥𝑝𝑗𝑝+𝛽𝑗𝑥𝛾𝛼𝑗𝜑𝑥𝑝𝑗𝑝+𝛽𝑗𝛾𝑓(𝑝)𝐴(𝑝),𝐽𝜑𝑥𝑗.𝑝(3.14) So we have 𝑥𝑗𝜑𝑥𝑝𝑗1𝑝𝜑(1)𝛾𝛾𝛼𝛾𝑓(𝑝)𝐴(𝑝),𝐽𝜑𝑥𝑗.𝑝(3.15) By the definition of Φ, it is easily seen that Φ𝑥𝑗𝑥𝑝𝑗𝜑𝑥𝑝𝑗.𝑝(3.16) Hence Φ𝑥𝑗1𝑝𝜑(1)𝛾𝛾𝛼𝛾𝑓(𝑝)𝐴(𝑝),𝐽𝜑𝑥𝑗.𝑝(3.17) Therefore Φ(𝑥𝑗𝑝)0 as 𝑗 since 𝐽𝜑 is weakly continuous; consequently, 𝑥𝑗𝑝 as 𝑗 by the continuity of Φ. Hence {𝑥𝑛} is relatively sequentially compact.
Finally, we prove that 𝑝 is a solution in 𝐹 to the variational inequality (3.1). For any 𝑤𝐹, we see that𝑡𝐼𝑇𝑛𝑥𝑛𝑡𝐼𝑇𝑛𝑤,𝐽𝜑𝑥𝑛=𝑥𝑤𝑛𝑤,𝐽𝜑𝑥𝑛𝑇𝑡𝑤𝑛𝑥𝑛𝑡𝑇𝑛𝑤,𝐽𝜑𝑥𝑛𝑥𝑤𝑛𝜑𝑥𝑤𝑛𝑇𝑡𝑤𝑛𝑥𝑛𝑡𝑇𝑛𝑤𝐽𝜑𝑥𝑛𝑥𝑤𝑛𝜑𝑥𝑤𝑛𝑥𝑤𝑛𝐽𝑤𝜑𝑥𝑛𝑤=0.(3.18) On the other hand, we have 𝑥(𝐴𝛾𝑓)𝑛1=𝛼𝑛𝐼𝛼𝑛𝐴𝑡𝐼𝑇𝑛𝑥𝑛,(3.19) which implies 𝑥(𝐴𝛾𝑓)𝑛,𝐽𝜑𝑥𝑛1𝑤=𝛼𝑛𝑡𝐼𝑇𝑛𝑥𝑛𝑡𝐼𝑇𝑛𝑤,𝐽𝜑𝑥𝑛+𝐴𝑡𝑤𝐼𝑇𝑛𝑥𝑛,𝐽𝜑𝑥𝑛𝐴𝑡𝑤𝐼𝑇𝑛𝑥𝑛,𝐽𝜑𝑥𝑛.𝑤(3.20) Observe 𝑥𝑗𝑠𝑇𝑗𝑥𝑗=𝛽𝑗𝑥𝛾𝑓𝑗𝑠𝐴𝑇𝑗𝑥𝑗0,(3.21) as 𝑗. Replacing 𝑛 by 𝑛𝑗 and letting 𝑗 in (3.20), we obtain (𝐴𝛾𝑓)(𝑝),𝐽𝜑(𝑝𝑤)0,𝑤𝐹.(3.22) So 𝑝𝐹 is a solution of variational inequality (3.1); and hence 𝑝=𝑞 by the uniqueness. In a summary, we have proved that {𝑥𝑛} is relatively sequentially compact and each cluster point of {𝑥𝑛} (as 𝑛) equals 𝑞. Therefore 𝑥𝑛𝑞 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 𝐽𝜑.

Theorem 4.1. Let 𝐸 be a reflexive Banach space which admits a weakly continuous duality mapping 𝐽𝜑 with gauge 𝜑 such that 𝜑 is invariant on [0,1]. Let {𝑇(𝑡)𝑡0} 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 {𝛼𝑛} and {𝑡𝑛} be real sequences satisfying 0<𝛼𝑛<1, 𝑛=1𝛼𝑛=, 𝑡𝑛>0 and lim𝑛𝑡𝑛=lim𝑛𝛼𝑛/𝑡𝑛=0. Then {𝑥𝑛} defined by (1.13) converges strongly to 𝑞𝐹 which also solves the variational inequality (3.1).

Proof. Since 𝛼𝑛0, we may assume that 𝛼𝑛<𝜑(1)𝐴1 and 1𝛼𝑛(𝜑(1)𝛾𝛾𝛼)>0 for all 𝑛. First we prove that {𝑥𝑛} is bounded. For each 𝑤𝐹, by Lemma 2.6, we have 𝑥𝑛+1=𝛼𝑤𝑛𝑥𝛾𝑓𝑛+𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛=𝑤𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛𝐼𝛼𝑛𝐴𝑤+𝛼𝑛𝑥𝛾𝑓𝑛𝐴(𝑤)𝜑(1)1𝛼𝑛𝛾𝑥𝑛𝑤+𝛼𝑛𝑥𝛾𝛼𝑛𝑤+𝛼𝑛=𝛾𝑓(𝑤)𝐴(𝑤)𝜑(1)𝛼𝑛𝜑(1)𝑥𝛾𝛾𝛼𝑛𝑤+𝛼𝑛(𝛾𝑓𝑤)𝐴(𝑤)1𝛼𝑛𝜑(1)𝑥𝛾𝛾𝛼𝑛𝑤+𝛼𝑛𝜑(1)𝛾𝛾𝛼𝛾𝑓(𝑤)𝐴(𝑤)𝜑(1).𝛾𝛾𝛼(4.1) It follows from induction that 𝑥𝑛+1𝑥𝑤max1,(𝑤𝛾𝑓𝑤)𝐴(𝑤)𝜑(1)𝛾𝛾𝛼,𝑛1.(4.2) Thus {𝑥𝑛} is bounded, and hence so are {𝑓(𝑥𝑛)} and {𝐴𝑇(𝑡𝑛)𝑥𝑛}. From Theorem 3.1, there is a unique solution 𝑞𝐹 to the following variational inequality: (𝐴𝛾𝑓)𝑞,𝐽𝜑(𝑞𝑤)0,𝑤𝐹.(4.3) Next we prove that limsup𝑛(𝐴𝛾𝑓)𝑞,𝐽𝜑𝑞𝑥𝑛+10.(4.4) Indeed, we can choose a subsequence {𝑥𝑛𝑗} of {𝑥𝑛} such that limsup𝑛(𝐴𝛾𝑓)𝑞,𝐽𝜑𝑞𝑥𝑛=limsup𝑗(𝐴𝛾𝑓)𝑞,𝐽𝜑𝑞𝑥𝑛𝑗.(4.5) Further, we can assume that 𝑥𝑛𝑗𝑝𝐸 by the reflexivity of 𝐸 and the boundedness of {𝑥𝑛}. Now we show that 𝑝𝐹. Put 𝑥𝑗=𝑥𝑛𝑗,𝛽𝑗=𝛼𝑛𝑗 and 𝑠𝑗=𝑡𝑛𝑗 for 𝑗, fix 𝑡>0. We obtain 𝑥𝑗+1𝑇(𝑡)𝑝[𝑡/𝑠𝑗]1𝑘=0𝑇(𝑘+1)𝑠𝑗𝑥𝑗𝑇𝑘𝑠𝑗𝑥𝑗+1+𝑇𝑡𝑠𝑗𝑠𝑗𝑥𝑗𝑡𝑇𝑠𝑗𝑠𝑗𝑝+𝑇𝑡𝑠𝑗𝑠𝑗𝑡𝑝𝑇(𝑡)𝑝s𝑗𝑇𝑠𝑗𝑥𝑗𝑥𝑗+1+𝑥𝑗+𝑇𝑡𝑝𝑡𝑠𝑗𝑠𝑗=𝑡𝑝𝑝𝑠𝑗𝛽𝑗𝑠𝐴𝑇𝑗𝑥𝑗𝑥𝛾𝑓𝑗+𝑥𝑗+𝑇𝑡𝑝𝑡𝑠𝑗𝑠𝑗𝑝𝑝𝑡𝛽𝑗𝑠𝑗𝑠𝐴𝑇𝑗𝑥𝑗𝑥𝛾𝑓𝑗+𝑥𝑗𝑝+max𝑇(𝑠)𝑝𝑝0𝑠𝑠𝑗.(4.6) It follows that limsup𝑛Φ(𝑥𝑗𝑇(𝑡)𝑝)limsup𝑛Φ(𝑥𝑗𝑝). From Lemma 2.5 (ii) we have limsup𝑛Φ𝑥𝑗𝑇(𝑡)𝑝=limsup𝑛Φ𝑥𝑗(𝑝+Φ𝑇(𝑡)𝑝𝑝).(4.7) So we have Φ(𝑇(𝑡)𝑝𝑝)0 and hence 𝑝𝐹. Since the duality mapping 𝐽𝜑 is weakly sequentially continuous, limsup𝑛(𝐴𝛾𝑓)𝑞,𝐽𝜑𝑞𝑥𝑛+1=limsup𝑗(𝐴𝛾𝑓)𝑞,𝐽𝜑𝑞𝑥𝑛𝑗+1=(𝐴𝛾𝑓)𝑞,𝐽𝜑(𝑞𝑝)0.(4.8) Finally, we show that 𝑥𝑛𝑞. From Lemma 2.5 (i), we have Φ𝑥𝑛+1𝑞=Φ𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛𝐼𝛼𝑛𝐴𝑞+𝛼𝑛𝑥𝛾𝑓𝑛𝛾𝑓(𝑞)+𝛼𝑛(𝛾𝑓(𝑞)𝐴(𝑞))Φ𝐼𝛼𝑛𝐴𝑇𝑡𝑛𝑥𝑛𝑞+𝛼𝑛𝑥𝛾𝑓𝑛𝛾𝑓(𝑞)+𝛼𝑛𝛾𝑓(𝑞)𝐴(𝑞),𝐽𝜑𝑥𝑛+1𝑞Φ𝜑(1)1𝛼𝑛𝛾𝑥𝑛𝑞+𝛼𝑛𝑥𝛾𝛼𝑛𝑞+𝛼𝑛𝛾𝑓(𝑞)𝐴(𝑞),𝐽𝜑𝑥𝑛+1𝑞=Φ𝜑(1)𝛼𝑛𝜑(1)𝑥𝛾𝛾𝛼𝑛𝑞+𝛼𝑛𝛾𝑓(𝑞)𝐴(𝑞),𝐽𝜑𝑥𝑛+1𝑞1𝛼𝑛𝜑(1)Φ𝑥𝛾𝛾𝛼𝑛𝑞+𝛼𝑛𝛾𝑓(𝑞)𝐴(𝑞),𝐽𝜑𝑥𝑛+1.𝑞(4.9) Note that 𝑛=1𝛼𝑛= and limsup𝑛𝛾𝑓(𝑞)𝐴(𝑞),𝐽𝜑(𝑥𝑛+1𝑞)0. Using Lemma 2.7, we have 𝑥𝑛𝑞 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.

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