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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 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: GiuseppeΒ Marino

Copyright Β© 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 π½πœ‘, where πœ‘ 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 πΈβˆ— 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(𝑑)βˆΆπ‘‘β‰₯0} of self-mappings of 𝐾 such that(i)𝑇(0)π‘₯=π‘₯ for all π‘₯∈𝐾,(ii)𝑇(𝑑+𝑠)π‘₯=𝑇(𝑑)𝑇(𝑠)π‘₯ for all 𝑑,𝑠β‰₯0 and π‘₯∈𝐾,(iii)for each π‘₯∈𝐾, the mapping 𝑇(β‹…)π‘₯ is continuous,(iv)for each 𝑑β‰₯0, 𝑇(𝑑) is nonexpansive, that is,‖𝑇(𝑑)π‘₯βˆ’π‘‡(𝑑)𝑦‖≀‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΎ.(1.1) We denote 𝐹 by the common fixed points set of 𝔉, that is, β‹‚πΉβˆΆ=𝑑β‰₯0𝐹(𝑇(𝑑)).

In 1967, Halpern [1] introduced the following classical iteration for a nonexpansive mapping π‘‡βˆΆπΎβ†’πΎ in a real Hilbert space:π‘₯𝑛+1=𝛼𝑛𝑒+1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0,(1.2) where {𝛼𝑛}βŠ‚(0,1) and π‘’βˆˆπΎ.

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

C1: limπ‘›β†’βˆžπ›Όπ‘›=0; C2: βˆ‘βˆžπ‘›=0𝛼𝑛=∞; C3: limπ‘›β†’βˆž(π›Όπ‘›βˆ’π›Όπ‘›βˆ’1)/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 {π‘₯𝑛} converges strongly to a fixed point of 𝑇 if {𝛼𝑛} satisfies the following conditions:

C1: limπ‘›β†’βˆžπ›Όπ‘›=0; C2: βˆ‘βˆžπ‘›=0𝛼𝑛=∞; C3: βˆ‘βˆžπ‘›=0|𝛼𝑛+1βˆ’π›Όπ‘›|<∞.

Shioji and Takahashi [5] extended Wittmann’s result to real Banach spaces with uniformly GΓ’teaux 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 π‘“βˆΆπΎβ†’πΎ be a contraction. In 2000, Moudafi [13] introduced the explicit viscosity approximation method for a nonexpansive mapping 𝑇 as follows:π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0,(1.3) where π›Όπ‘›βˆˆ(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⟨𝐴π‘₯,π‘₯⟩β‰₯𝛾‖π‘₯β€–2,βˆ€π‘₯∈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:minπ‘₯∈𝐢12⟨𝐴π‘₯,π‘₯βŸ©βˆ’βŸ¨π‘₯,π‘βŸ©,(1.5) where 𝐢 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:π‘₯𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯1,(1.6) where {𝛼𝑛}βŠ‚(0,1), 𝑓 is a contraction on 𝐻, and 𝐴 is a strongly positive bounded linear operator on 𝐻. They proved that the sequence {π‘₯𝑛} generated by (1.6) converges strongly to a fixed point π‘₯βˆ—βˆˆπΉ(𝑇) which also solves the variational inequality⟨(π΄βˆ’π›Ύπ‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐹(𝑇),(1.7) which is the optimality condition for the minimization problem: minπ‘₯∈𝐢(1/2)⟨𝐴π‘₯,π‘₯βŸ©βˆ’β„Ž(π‘₯), where β„Ž is a potential function for 𝛾𝑓 (i.e., β„Žβ€²(π‘₯)=𝛾𝑓(π‘₯) for π‘₯∈𝐻).

Suzuki [16] first introduced the following implicit viscosity method for a nonexpansive semigroup {𝑇(𝑑)βˆΆπ‘‘β‰₯0} in a Hilbert space:π‘₯𝑛=𝛼𝑛𝑒+1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛,𝑛β‰₯1,(1.8) where {𝛼𝑛}βŠ‚(0,1) and π‘’βˆˆπΎ. 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:π‘₯𝑛=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛π‘₯,𝑛β‰₯1,(1.9)𝑛+1=𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛,𝑛β‰₯1,(1.10) where {𝛼𝑛}βŠ‚(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 GΓ’teaux 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 π½πœ‘ with a gauge πœ‘. Let {𝑇(𝑑)βˆΆπ‘‘β‰₯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𝐴π‘₯,π½πœ‘ξ¬β‰₯(π‘₯)(𝛾‖π‘₯β€–πœ‘β€–π‘₯β€–),β€–π›ΌπΌβˆ’π›½π΄β€–=supβ€–π‘₯‖≀1||(π›ΌπΌβˆ’π›½π΄)π‘₯,π½πœ‘ξ¬||,(π‘₯)(1.11) where π›Όβˆˆ[0,1] and π›½βˆˆ[βˆ’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:π‘₯𝑛=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛π‘₯,𝑛β‰₯1,(1.12)𝑛+1=𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛,𝑛β‰₯1,(1.13) where {𝛼𝑛}βŠ‚(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 β€–π‘₯+𝑦‖/2<1 for all π‘₯,π‘¦βˆˆπΈ with β€–π‘₯β€–=‖𝑦‖=1 and π‘₯≠𝑦. A Banach space 𝐸 is called uniformly convex if for each πœ–>0 there is a 𝛿>0 such that for π‘₯,π‘¦βˆˆπΈ with β€–π‘₯β€–,‖𝑦‖≀1 and β€–π‘₯βˆ’π‘¦β€–β‰₯πœ–,β€–π‘₯+𝑦‖≀2(1βˆ’π›Ώ) holds. The modulus of convexity of 𝐸 is defined by𝛿𝐸‖‖‖1(πœ–)=inf1βˆ’2β€–β€–β€–βˆΆξ‚‡,(π‘₯+𝑦)β€–π‘₯β€–,‖𝑦‖≀1,β€–π‘₯βˆ’π‘¦β€–β‰₯πœ–(2.1) for all πœ–βˆˆ[0,2]. 𝐸 is uniformly convex if 𝛿𝐸(0)=0, and 𝛿𝐸(πœ–)>0 for all 0<πœ–β‰€2. It is known that every uniformly convex Banach space is strictly convex and reflexive. Let 𝑆(𝐸)={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–=1}. Then the norm of 𝐸 is said to be GΓ’teaux differentiable iflim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(2.2) exists for each π‘₯,π‘¦βˆˆπ‘†(𝐸). In this case 𝐸 is called smooth. The norm of 𝐸 is said to be FrΓ©chet differentiable if for each π‘₯βˆˆπ‘†(𝐸), the limit is attained uniformly for π‘¦βˆˆπ‘†(𝐸). The norm of 𝐸 is called uniformly FrΓ©chet differentiable, if the limit is attained uniformly for π‘₯,π‘¦βˆˆπ‘†(𝐸). It is well known that (uniformly) FrΓ©chet differentiability of the norm of 𝐸 implies (uniformly) GΓ’teaux differentiability of the norm of 𝐸.

Let 𝜌𝐸∢[0,∞)β†’[0,∞) be the modulus of smoothness of 𝐸 defined byπœŒπΈξ‚†1(𝑑)=sup2(.β€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–)βˆ’1∢π‘₯βˆˆπ‘†(𝐸),‖𝑦‖≀𝑑(2.3)

A Banach space 𝐸 is called uniformly smooth if 𝜌𝐸(𝑑)/𝑑→0 as 𝑑→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 πœ‘βˆΆ[0,∞)β†’[0,∞) is said to be gauge function if πœ‘(0)=0 and limπ‘‘β†’βˆžπœ‘(𝑑)=∞.

Definition 2.2. Let 𝐸 be a normed space and πœ‘ a gauge function. Then the mapping π½πœ‘βˆΆπΈβ†’2πΈβˆ— defined by π½πœ‘ξ€½π‘“(π‘₯)=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βˆ—(⟩=β€–π‘₯β€–πœ‘β€–π‘₯β€–),β€–π‘“βˆ—()ξ€Ύβ€–=πœ‘β€–π‘₯β€–,π‘₯∈𝐸,(2.4) is called the duality mapping with gauge function πœ‘.

In the particular case πœ‘(𝑑)=𝑑, the duality mapping π½πœ‘=𝐽 is called the normalized duality mapping.

In the case πœ‘(𝑑)=π‘‘π‘žβˆ’1,π‘ž>1, the duality mapping π½πœ‘=π½π‘ž is called the generalized duality mapping. It follows from the definition that π½πœ‘(π‘₯)=πœ‘(β€–π‘₯β€–)/β€–π‘₯‖𝐽(π‘₯) and π½π‘ž(π‘₯)=β€–π‘₯β€–π‘žβˆ’2𝐽(π‘₯),π‘ž>1.

Remark 2.3. For the gauge function πœ‘, the function Φ∢[0,∞)β†’[0,∞) defined by ξ€œΞ¦(𝑑)=𝑑0πœ‘(𝑠)𝑑𝑠(2.5) is a continuous convex and strictly increasing function on [0,∞). Therefore, Ξ¦ has a continuous inverse function Ξ¦βˆ’1.

It is noted that if 0β‰€π‘˜β‰€1, then πœ‘(π‘˜π‘₯)β‰€πœ‘(π‘₯). Furtherξ€œΞ¦(π‘˜π‘‘)=0π‘˜π‘‘ξ€œπœ‘(𝑠)𝑑𝑠=π‘˜π‘‘0ξ€œπœ‘(π‘˜π‘₯)𝑑π‘₯β‰€π‘˜π‘‘0πœ‘(π‘₯)𝑑π‘₯=π‘˜Ξ¦(𝑑).(2.6)

Remark 2.4. For each π‘₯ in a Banach space 𝐸, π½πœ‘(π‘₯)=πœ•Ξ¦(β€–π‘₯β€–), where πœ• denotes the sub-differential.

We also know the following facts:(i)π½πœ‘ is a nonempty, closed, and convex set in πΈβˆ— for each π‘₯∈𝐸,(ii)π½πœ‘ is a function when πΈβˆ— is strictly convex,(iii)If π½πœ‘ is single-valued, thenπ½πœ‘(πœ†π‘₯)=𝑠𝑖𝑔𝑛(πœ†)πœ‘(β€–πœ†π‘₯β€–)π½πœ‘(β€–π‘₯β€–)πœ‘ξ«(π‘₯),βˆ€π‘₯∈𝐸,πœ†βˆˆβ„,π‘₯βˆ’π‘¦,π½πœ‘(π‘₯)βˆ’π½πœ‘ξ¬(𝑦)β‰₯(πœ‘(β€–π‘₯β€–)βˆ’πœ‘(‖𝑦‖))(β€–π‘₯β€–βˆ’β€–π‘¦β€–),βˆ€π‘₯,π‘¦βˆˆπΈ.(2.7) Following Browder [43], we say that a Banach space 𝐸 has a weakly continuous duality mapping if there exists a gauge πœ‘ for which the duality mapping π½πœ‘ is single-valued and continuous from the weak topology to the weak* topology, that is, for any {π‘₯𝑛} with π‘₯𝑛⇀π‘₯, the sequence {π½πœ‘(π‘₯𝑛)} converges weakly* to π½πœ‘(π‘₯). It is known that the space ℓ𝑝 has a weakly continuous duality mapping with a gauge function πœ‘(𝑑)=π‘‘π‘βˆ’1 for all 1<𝑝<∞. Moreover, πœ‘ is invariant on [0,1].

Lemma 2.5 (See [44]). Assume that a Banach space 𝐸 has a weakly continuous duality mapping π½πœ‘ with gauge πœ‘.(i)For all π‘₯,π‘¦βˆˆπΈ, the following inequality holds: Ξ¦(‖π‘₯+𝑦‖)≀Φ(β€–π‘₯β€–)+𝑦,π½πœ‘ξ¬.(π‘₯+𝑦)(2.8) In particular, for all π‘₯,π‘¦βˆˆπΈ, β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,𝐽(π‘₯+𝑦)⟩.(2.9)(ii)Assume that a sequence {π‘₯𝑛} in 𝐸 converges weakly to a point π‘₯∈𝐸. Then the following holds: limsupπ‘›β†’βˆžΞ¦ξ€·β€–β€–π‘₯π‘›β€–β€–ξ€Έβˆ’π‘¦=limsupπ‘›β†’βˆžΞ¦ξ€·β€–β€–π‘₯𝑛‖‖()βˆ’π‘₯+Ξ¦β€–π‘₯βˆ’π‘¦β€–(2.10) for all π‘₯,π‘¦βˆˆπΈ.

Lemma 2.6 (See [23]). Assume that a Banach space 𝐸 has a weakly continuous duality mapping π½πœ‘ with gauge πœ‘. Let 𝐴 be a strongly positive bounded linear operator on 𝐸 with coefficient 𝛾>0 and 0<πœŒβ‰€πœ‘(1)β€–π΄β€–βˆ’1. Then β€–πΌβˆ’πœŒπ΄β€–β‰€πœ‘(1)(1βˆ’πœŒπ›Ύ).

Lemma 2.7 (See [12]). Assume that {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛾𝑛𝛿𝑛,𝑛β‰₯1,(2.11) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in ℝ such that
(a) βˆ‘βˆžπ‘›=1𝛾𝑛=∞; (b) limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0 or βˆ‘βˆžπ‘›=1|𝛾𝑛𝛿𝑛|<∞.
Then limπ‘›β†’βˆžπ‘Žπ‘›=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 π½πœ‘ with gauge πœ‘ such that πœ‘ is invariant on [0,1]. Let 𝔉={𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a nonexpansive semigroup on 𝐸 such that πΉβ‰ βˆ…. Let 𝑓 be a contraction on 𝐸 with the coefficient π›Όβˆˆ(0,1) and 𝐴 a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<π›Ύπœ‘(1)/𝛼. Let {𝛼𝑛} and {𝑑𝑛} be real sequences satisfying 0<𝛼𝑛<1, 𝑑𝑛>0 and limπ‘›β†’βˆžπ‘‘π‘›=limπ‘›β†’βˆžπ›Όπ‘›/𝑑𝑛=0. Then {π‘₯𝑛} defined by (1.12) converges strongly to π‘žβˆˆπΉ which solves the following variational inequality: (π΄βˆ’π›Ύπ‘“)(π‘ž),π½πœ‘ξ¬(π‘žβˆ’π‘€)≀0,βˆ€π‘€βˆˆπΉ.(3.1)

Proof. First, we prove the uniqueness of the solution to the variational inequality (3.1) in 𝐹. Suppose that 𝑝,π‘žβˆˆπΉ satisfy (3.1), so we have (π΄βˆ’π›Ύπ‘“)(𝑝),π½πœ‘ξ¬ξ«(π‘βˆ’π‘ž)≀0,(π΄βˆ’π›Ύπ‘“)(π‘ž),π½πœ‘ξ¬(π‘žβˆ’π‘)≀0.(3.2) Adding the above inequalities, we get 𝐴(𝑝)βˆ’π΄(π‘ž)βˆ’π›Ύ(𝑓(𝑝)βˆ’π‘“(π‘ž)),π½πœ‘ξ¬(π‘βˆ’π‘ž)≀0.(3.3) This shows that 𝐴(π‘βˆ’π‘ž),π½πœ‘ξ¬ξ«π‘“(π‘βˆ’π‘ž)≀𝛾(𝑝)βˆ’π‘“(π‘ž),π½πœ‘ξ¬,(π‘βˆ’π‘ž)(3.4) which implies by the strong positivity of 𝐴(ξ«π΄π›Ύβ€–π‘βˆ’π‘žβ€–πœ‘β€–π‘βˆ’π‘žβ€–)≀(π‘βˆ’π‘ž),π½πœ‘ξ¬((π‘βˆ’π‘ž)β‰€π›Ύπ›Όβ€–π‘βˆ’π‘žβ€–πœ‘β€–π‘βˆ’π‘žβ€–).(3.5) Since πœ‘ is invariant on [0,1], πœ‘(1)π›Ύβ€–π‘βˆ’π‘žβ€–πœ‘(β€–π‘βˆ’π‘žβ€–)β‰€π›Ύπ›Όβ€–π‘βˆ’π‘žβ€–πœ‘(β€–π‘βˆ’π‘žβ€–).(3.6) It follows that ξ€·πœ‘(1)ξ€Έ(π›Ύβˆ’π›Ύπ›Όβ€–π‘βˆ’π‘žβ€–πœ‘β€–π‘βˆ’π‘žβ€–)≀0.(3.7) Therefore 𝑝=π‘ž since 0<𝛾<(π›Ύπœ‘(1))/𝛼.
We next prove that {π‘₯𝑛} is bounded. For each π‘€βˆˆπΉ, by Lemma 2.6, we haveβ€–β€–π‘₯𝑛‖‖=β€–β€–π›Όβˆ’π‘€π‘›ξ€·π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛‖‖=β€–β€–ξ€·βˆ’π‘€πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έxπ‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘€+𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έξ€Έβ€–β€–ξ€·βˆ’π΄(𝑀)β‰€πœ‘(1)1βˆ’π›Όπ‘›π›Ύξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘€+𝛼𝑛‖‖π‘₯𝛾𝛼𝑛‖‖‖≀‖‖π‘₯βˆ’π‘€+‖𝛾𝑓(𝑀)βˆ’π΄(𝑀)π‘›β€–β€–βˆ’π‘€βˆ’π›Όπ‘›πœ‘(1)𝛾‖‖π‘₯π‘›β€–β€–βˆ’π‘€+𝛼𝑛‖‖π‘₯π›Ύπ›Όπ‘›β€–β€–βˆ’π‘€+𝛼𝑛‖𝛾𝑓(𝑀)βˆ’π΄(𝑀)β€–,(3.8) which yields β€–β€–π‘₯𝑛‖‖≀1βˆ’π‘€πœ‘(1)π›Ύβˆ’π›Ύπ›Όβ€–π›Ύπ‘“(𝑀)βˆ’π΄(𝑀)β€–.(3.9) Hence {π‘₯𝑛} is bounded. So are {𝑓(π‘₯𝑛)} and {𝐴𝑇(𝑑𝑛)π‘₯𝑛}.
We next prove that {π‘₯𝑛} is relatively sequentially compact. By the reflexivity of 𝐸 and the boundedness of {π‘₯𝑛}, there exists a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} and a point 𝑝 in 𝐸 such that π‘₯𝑛𝑗⇀𝑝 as π‘—β†’βˆž. Now we show that π‘βˆˆπΉ. Put π‘₯𝑗=π‘₯𝑛𝑗, 𝛽𝑗=𝛼𝑛𝑗 and 𝑠𝑗=𝑑𝑛𝑗 for π‘—βˆˆβ„•, fix 𝑑>0. We see thatβ€–β€–π‘₯π‘—β€–β€–β‰€βˆ’π‘‡(𝑑)𝑝[𝑑/𝑠𝑗]βˆ’1ξ“π‘˜=0‖‖𝑇(π‘˜+1)𝑠𝑗π‘₯π‘—ξ€·βˆ’π‘‡π‘˜π‘ π‘—ξ€Έπ‘₯𝑗+1β€–β€–+‖‖‖𝑇𝑑𝑠𝑗𝑠𝑗π‘₯π‘—π‘‘βˆ’π‘‡ξ‚΅ξ‚Έπ‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άπ‘β€–β€–β€–+β€–β€–β€–π‘‡π‘‘ξ‚΅ξ‚Έπ‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άβ€–β€–β€–β‰€ξ‚Έπ‘‘π‘βˆ’π‘‡(𝑑)𝑝𝑠𝑗‖‖𝑇𝑠𝑗π‘₯π‘—βˆ’π‘₯𝑗‖‖+β€–β€–π‘₯𝑗‖‖+β€–β€–β€–π‘‡ξ‚΅ξ‚Έπ‘‘βˆ’π‘π‘‘βˆ’π‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άβ€–β€–β€–=ξ‚Έπ‘‘π‘βˆ’π‘π‘ π‘—ξ‚Ήπ›½π‘—β€–β€–ξ€·π‘ π΄π‘‡π‘—ξ€Έπ‘₯𝑗π‘₯βˆ’π›Ύπ‘“π‘—ξ€Έβ€–β€–+β€–β€–π‘₯𝑗‖‖+β€–β€–β€–π‘‡ξ‚΅ξ‚Έπ‘‘βˆ’π‘π‘‘βˆ’π‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άβ€–β€–β€–β‰€π‘βˆ’π‘π‘‘π›½π‘—π‘ π‘—β€–β€–ξ€·π‘ π΄π‘‡π‘—ξ€Έπ‘₯𝑗π‘₯βˆ’π›Ύπ‘“π‘—ξ€Έβ€–β€–+β€–β€–π‘₯π‘—β€–β€–ξ€½βˆ’π‘+max‖𝑇(𝑠)π‘βˆ’π‘β€–βˆΆ0≀𝑠≀𝑠𝑗.(3.10) So we have limsupπ‘—β†’βˆžΞ¦ξ€·β€–β€–π‘₯π‘—β€–β€–ξ€Έβˆ’π‘‡(𝑑)𝑝≀limsupπ‘—β†’βˆžΞ¦ξ€·β€–β€–π‘₯𝑗‖‖.βˆ’π‘(3.11) On the other hand, by Lemma 2.5 (ii), we have limsupπ‘—β†’βˆžΞ¦ξ€·β€–β€–π‘₯π‘—β€–β€–ξ€Έβˆ’π‘‡(𝑑)𝑝=limsupπ‘—β†’βˆžΞ¦ξ€·β€–β€–π‘₯𝑗‖‖(β€–βˆ’π‘+Φ‖𝑇(𝑑)π‘βˆ’π‘).(3.12) Combining (3.11) and (3.12), we have Ξ¦(‖𝑇(𝑑)π‘βˆ’π‘β€–)≀0.(3.13) This implies that π‘βˆˆπΉ. Further, we see that β€–π‘₯π‘—ξ€·βˆ’π‘β€–πœ‘β€–π‘₯𝑗=π‘₯βˆ’π‘β€–π‘—βˆ’π‘,π½πœ‘ξ€·π‘₯𝑗=βˆ’π‘ξ€Έξ¬ξ«ξ€·πΌβˆ’π›½π‘—π΄ξ€Έπ‘‡ξ€·π‘ π‘—ξ€Έπ‘₯π‘—βˆ’ξ€·πΌβˆ’π›½π‘—π΄ξ€Έπ‘,π½πœ‘ξ€·π‘₯π‘—βˆ’π‘ξ€Έξ¬+𝛽𝑗π‘₯π›Ύπ‘“π‘—ξ€Έβˆ’π›Ύπ‘“(𝑝),π½πœ‘ξ€·π‘₯π‘—βˆ’π‘ξ€Έξ¬+𝛽𝑗𝛾𝑓(𝑝)βˆ’π΄(𝑝),π½πœ‘ξ€·π‘₯π‘—ξ€·βˆ’π‘ξ€Έξ¬β‰€πœ‘(1)1βˆ’π›½π‘—π›Ύξ€Έβ€–β€–π‘₯π‘—β€–β€–πœ‘ξ€·β€–β€–π‘₯βˆ’π‘π‘—β€–β€–ξ€Έβˆ’π‘+𝛽𝑗‖‖π‘₯π›Ύπ›Όπ‘—β€–β€–πœ‘ξ€·β€–β€–π‘₯βˆ’π‘π‘—β€–β€–ξ€Έβˆ’π‘+𝛽𝑗𝛾𝑓(𝑝)βˆ’π΄(𝑝),π½πœ‘ξ€·π‘₯𝑗.βˆ’π‘ξ€Έξ¬(3.14) So we have β€–β€–π‘₯π‘—β€–β€–πœ‘ξ€·β€–β€–π‘₯βˆ’π‘π‘—β€–β€–ξ€Έβ‰€1βˆ’π‘πœ‘(1)ξ«π›Ύβˆ’π›Ύπ›Όπ›Ύπ‘“(𝑝)βˆ’π΄(𝑝),π½πœ‘ξ€·π‘₯𝑗.βˆ’π‘ξ€Έξ¬(3.15) By the definition of Ξ¦, it is easily seen that Ξ¦ξ€·β€–β€–π‘₯𝑗‖‖≀‖‖π‘₯βˆ’π‘π‘—β€–β€–πœ‘ξ€·β€–β€–π‘₯βˆ’π‘π‘—β€–β€–ξ€Έ.βˆ’π‘(3.16) Hence Ξ¦ξ€·β€–β€–π‘₯𝑗‖‖≀1βˆ’π‘πœ‘(1)ξ«π›Ύβˆ’π›Ύπ›Όπ›Ύπ‘“(𝑝)βˆ’π΄(𝑝),π½πœ‘ξ€·π‘₯𝑗.βˆ’π‘ξ€Έξ¬(3.17) Therefore Ξ¦(β€–π‘₯π‘—βˆ’π‘β€–)β†’0 as π‘—β†’βˆž since π½πœ‘ is weakly continuous; consequently, π‘₯𝑗→𝑝 as π‘—β†’βˆž by the continuity of Ξ¦. Hence {π‘₯𝑛} is relatively sequentially compact.
Finally, we prove that 𝑝 is a solution in 𝐹 to the variational inequality (3.1). For any π‘€βˆˆπΉ, we see thatξ€·π‘‘ξ«ξ€·πΌβˆ’π‘‡π‘›π‘₯ξ€Έξ€Έπ‘›βˆ’ξ€·ξ€·π‘‘πΌβˆ’π‘‡π‘›ξ€Έξ€Έπ‘€,π½πœ‘ξ€·π‘₯𝑛=π‘₯βˆ’π‘€ξ€Έξ¬π‘›βˆ’π‘€,π½πœ‘ξ€·π‘₯π‘›βˆ’ξ«π‘‡ξ€·π‘‘βˆ’π‘€ξ€Έξ¬π‘›ξ€Έπ‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘€,π½πœ‘ξ€·π‘₯𝑛β‰₯β€–β€–π‘₯βˆ’π‘€ξ€Έξ¬π‘›β€–β€–πœ‘β€–β€–π‘₯βˆ’π‘€π‘›β€–β€–βˆ’β€–β€–π‘‡ξ€·π‘‘βˆ’π‘€π‘›ξ€Έπ‘₯π‘›ξ€·π‘‘βˆ’π‘‡π‘›ξ€Έπ‘€β€–β€–β€–β€–π½πœ‘ξ€·π‘₯𝑛‖‖β‰₯β€–β€–π‘₯βˆ’π‘€π‘›β€–β€–πœ‘β€–β€–π‘₯βˆ’π‘€π‘›β€–β€–βˆ’β€–β€–π‘₯βˆ’π‘€π‘›β€–β€–β€–β€–π½βˆ’π‘€πœ‘ξ€·π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘€=0.(3.18) On the other hand, we have ξ€·π‘₯(π΄βˆ’π›Ύπ‘“)𝑛1=βˆ’π›Όπ‘›ξ€·πΌβˆ’π›Όπ‘›π΄ξ€·π‘‘ξ€Έξ€·πΌβˆ’π‘‡π‘›π‘₯𝑛,(3.19) which implies π‘₯(π΄βˆ’π›Ύπ‘“)𝑛,π½πœ‘ξ€·π‘₯𝑛1βˆ’π‘€ξ€Έξ¬=βˆ’π›Όπ‘›ξ€·π‘‘ξ«ξ€·πΌβˆ’π‘‡π‘›π‘₯ξ€Έξ€Έπ‘›βˆ’ξ€·ξ€·π‘‘πΌβˆ’π‘‡π‘›ξ€Έξ€Έπ‘€,π½πœ‘ξ€·π‘₯𝑛+ξ«π΄ξ€·ξ€·π‘‘βˆ’π‘€ξ€Έξ¬πΌβˆ’π‘‡π‘›π‘₯𝑛,π½πœ‘ξ€·π‘₯π‘›β‰€ξ«π΄ξ€·ξ€·π‘‘βˆ’π‘€ξ€Έξ¬πΌβˆ’π‘‡π‘›π‘₯𝑛,π½πœ‘ξ€·π‘₯𝑛.βˆ’π‘€ξ€Έξ¬(3.20) Observe β€–β€–π‘₯π‘—ξ€·π‘ βˆ’π‘‡π‘—ξ€Έπ‘₯𝑗‖‖=𝛽𝑗‖‖π‘₯π›Ύπ‘“π‘—ξ€Έξ€·π‘ βˆ’π΄π‘‡π‘—ξ€Έπ‘₯𝑗‖‖→0,(3.21) as π‘—β†’βˆž. Replacing 𝑛 by 𝑛𝑗 and letting π‘—β†’βˆž in (3.20), we obtain (π΄βˆ’π›Ύπ‘“)(𝑝),π½πœ‘ξ¬(π‘βˆ’π‘€)≀0,βˆ€π‘€βˆˆπΉ.(3.22) So π‘βˆˆπΉ is a solution of variational inequality (3.1); and hence 𝑝=π‘ž by the uniqueness. In a summary, we have proved that {π‘₯𝑛} is relatively sequentially compact and each cluster point of {π‘₯𝑛} (as π‘›β†’βˆž) equals π‘ž. Therefore π‘₯π‘›β†’π‘ž as π‘›β†’βˆž. 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 π½πœ‘.

Theorem 4.1. Let 𝐸 be a reflexive Banach space which admits a weakly continuous duality mapping π½πœ‘ with gauge πœ‘ such that πœ‘ is invariant on [0,1]. Let {𝑇(𝑑)βˆΆπ‘‘β‰₯0} be a nonexpansive semigroup on 𝐸 such that πΉβ‰ βˆ…. Let 𝑓 be a contraction on 𝐸 with the coefficient π›Όβˆˆ(0,1) and 𝐴 a strongly positive bounded linear operator with coefficient 𝛾>0 and 0<𝛾<π›Ύπœ‘(1)/𝛼. Let {𝛼𝑛} and {𝑑𝑛} be real sequences satisfying 0<𝛼𝑛<1, βˆ‘βˆžπ‘›=1𝛼𝑛=∞, 𝑑𝑛>0 and limπ‘›β†’βˆžπ‘‘π‘›=limπ‘›β†’βˆžπ›Όπ‘›/𝑑𝑛=0. Then {π‘₯𝑛} defined by (1.13) converges strongly to π‘žβˆˆπΉ which also solves the variational inequality (3.1).

Proof. Since 𝛼𝑛→0, we may assume that 𝛼𝑛<πœ‘(1)β€–π΄β€–βˆ’1 and 1βˆ’π›Όπ‘›(πœ‘(1)π›Ύβˆ’π›Ύπ›Ό)>0 for all 𝑛. First we prove that {π‘₯𝑛} is bounded. For each π‘€βˆˆπΉ, by Lemma 2.6, we have β€–β€–π‘₯𝑛+1β€–β€–=β€–β€–π›Όβˆ’π‘€π‘›ξ€·π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯𝑛‖‖=β€–β€–ξ€·βˆ’π‘€πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘€+𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έξ€Έβ€–β€–ξ€·βˆ’π΄(𝑀)β‰€πœ‘(1)1βˆ’π›Όπ‘›π›Ύξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘€+𝛼𝑛‖‖π‘₯π›Ύπ›Όπ‘›β€–β€–βˆ’π‘€+𝛼𝑛‖=‖𝛾𝑓(𝑀)βˆ’π΄(𝑀)πœ‘(1)βˆ’π›Όπ‘›ξ€·πœ‘(1)β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€Έπ‘›β€–β€–βˆ’π‘€+𝛼𝑛(≀‖𝛾𝑓𝑀)βˆ’π΄(𝑀)β€–1βˆ’π›Όπ‘›ξ€·πœ‘(1)β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€Έπ‘›β€–β€–βˆ’π‘€+π›Όπ‘›ξ€·πœ‘(1)π›Ύβˆ’π›Ύπ›Όξ€Έξ€Έβ€–π›Ύπ‘“(𝑀)βˆ’π΄(𝑀)β€–πœ‘(1).π›Ύβˆ’π›Ύπ›Ό(4.1) It follows from induction that β€–β€–π‘₯𝑛+1β€–β€–ξ‚»β€–β€–π‘₯βˆ’π‘€β‰€max1β€–β€–,(βˆ’π‘€β€–π›Ύπ‘“π‘€)βˆ’π΄(𝑀)β€–πœ‘(1)ξ‚Όπ›Ύβˆ’π›Ύπ›Ό,𝑛β‰₯1.(4.2) Thus {π‘₯𝑛} is bounded, and hence so are {𝑓(π‘₯𝑛)} and {𝐴𝑇(𝑑𝑛)π‘₯𝑛}. From Theorem 3.1, there is a unique solution π‘žβˆˆπΉ to the following variational inequality: (π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ¬(π‘žβˆ’π‘€)≀0,βˆ€π‘€βˆˆπΉ.(4.3) Next we prove that limsupπ‘›β†’βˆžξ«(π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ€·π‘žβˆ’π‘₯𝑛+1≀0.(4.4) Indeed, we can choose a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} such that limsupπ‘›β†’βˆžξ«(π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ€·π‘žβˆ’π‘₯𝑛=limsupπ‘—β†’βˆžξ‚¬(π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ‚€π‘žβˆ’π‘₯𝑛𝑗.(4.5) Further, we can assume that π‘₯π‘›π‘—β‡€π‘βˆˆπΈ by the reflexivity of 𝐸 and the boundedness of {π‘₯𝑛}. Now we show that π‘βˆˆπΉ. Put π‘₯𝑗=π‘₯𝑛𝑗,𝛽𝑗=𝛼𝑛𝑗 and 𝑠𝑗=𝑑𝑛𝑗 for π‘—βˆˆβ„•, fix 𝑑>0. We obtain β€–β€–π‘₯𝑗+1β€–β€–β‰€βˆ’π‘‡(𝑑)𝑝[𝑑/𝑠𝑗]βˆ’1ξ“π‘˜=0‖‖𝑇(π‘˜+1)𝑠𝑗π‘₯π‘—ξ€·βˆ’π‘‡π‘˜π‘ π‘—ξ€Έπ‘₯𝑗+1β€–β€–+‖‖‖𝑇𝑑𝑠𝑗𝑠𝑗π‘₯π‘—π‘‘βˆ’π‘‡ξ‚΅ξ‚Έπ‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άπ‘β€–β€–β€–+β€–β€–β€–π‘‡π‘‘ξ‚΅ξ‚Έπ‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άβ€–β€–β€–β‰€ξ‚Έπ‘‘π‘βˆ’π‘‡(𝑑)𝑝s𝑗‖‖𝑇𝑠𝑗π‘₯π‘—βˆ’π‘₯𝑗+1β€–β€–+β€–β€–π‘₯𝑗‖‖+β€–β€–β€–π‘‡ξ‚΅ξ‚Έπ‘‘βˆ’π‘π‘‘βˆ’π‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άβ€–β€–β€–=ξ‚Έπ‘‘π‘βˆ’π‘π‘ π‘—ξ‚Ήπ›½π‘—β€–β€–ξ€·π‘ π΄π‘‡π‘—ξ€Έπ‘₯𝑗π‘₯βˆ’π›Ύπ‘“π‘—ξ€Έβ€–β€–+β€–β€–π‘₯𝑗‖‖+β€–β€–β€–π‘‡ξ‚΅ξ‚Έπ‘‘βˆ’π‘π‘‘βˆ’π‘ π‘—ξ‚Ήπ‘ π‘—ξ‚Άβ€–β€–β€–β‰€π‘βˆ’π‘π‘‘π›½π‘—π‘ π‘—β€–β€–ξ€·π‘ π΄π‘‡π‘—ξ€Έπ‘₯𝑗π‘₯βˆ’π›Ύπ‘“π‘—ξ€Έβ€–β€–+β€–β€–π‘₯π‘—β€–β€–ξ€½βˆ’π‘+max‖𝑇(𝑠)π‘βˆ’π‘β€–βˆΆ0≀𝑠≀𝑠𝑗.(4.6) It follows that limsupπ‘›β†’βˆžΞ¦(β€–π‘₯π‘—βˆ’π‘‡(𝑑)𝑝‖)≀limsupπ‘›β†’βˆžΞ¦(β€–π‘₯π‘—βˆ’π‘β€–). From Lemma 2.5 (ii) we have limsupπ‘›β†’βˆžΞ¦ξ€·β€–β€–π‘₯π‘—β€–β€–ξ€Έβˆ’π‘‡(𝑑)𝑝=limsupπ‘›β†’βˆžΞ¦ξ€·β€–β€–π‘₯𝑗‖‖(β€–βˆ’π‘+Φ‖𝑇(𝑑)π‘βˆ’π‘).(4.7) So we have Ξ¦(‖𝑇(𝑑)π‘βˆ’π‘β€–)≀0 and hence π‘βˆˆπΉ. Since the duality mapping π½πœ‘ is weakly sequentially continuous, limsupπ‘›β†’βˆžξ«(π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ€·π‘žβˆ’π‘₯𝑛+1=limsupπ‘—β†’βˆžξ‚¬(π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ‚€π‘žβˆ’π‘₯𝑛𝑗+1=(π΄βˆ’π›Ύπ‘“)π‘ž,π½πœ‘ξ¬(π‘žβˆ’π‘)≀0.(4.8) Finally, we show that π‘₯π‘›β†’π‘ž. From Lemma 2.5 (i), we have Ξ¦ξ€·β€–β€–π‘₯𝑛+1β€–β€–ξ€Έξ€·β€–β€–ξ€·βˆ’π‘ž=Ξ¦πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡ξ€·π‘‘π‘›ξ€Έπ‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘ž+𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έξ€Έβˆ’π›Ύπ‘“(π‘ž)+𝛼𝑛(‖‖‖‖𝛾𝑓(π‘ž)βˆ’π΄(π‘ž))β‰€Ξ¦πΌβˆ’π›Όπ‘›π΄π‘‡ξ€·π‘‘ξ€Έξ€·π‘›ξ€Έπ‘₯π‘›ξ€Έβˆ’π‘ž+𝛼𝑛π‘₯π›Ύπ‘“π‘›ξ€Έξ€Έβ€–β€–ξ€Έβˆ’π›Ύπ‘“(π‘ž)+𝛼𝑛𝛾𝑓(π‘ž)βˆ’π΄(π‘ž),π½πœ‘ξ€·π‘₯𝑛+1ξ€·ξ€·βˆ’π‘žξ€Έξ¬β‰€Ξ¦πœ‘(1)1βˆ’π›Όπ‘›π›Ύξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘ž+𝛼𝑛‖‖π‘₯π›Ύπ›Όπ‘›β€–β€–ξ€Έβˆ’π‘ž+𝛼𝑛𝛾𝑓(π‘ž)βˆ’π΄(π‘ž),π½πœ‘ξ€·π‘₯𝑛+1βˆ’π‘žξ€Έξ¬=Ξ¦ξ€·ξ€·πœ‘(1)βˆ’π›Όπ‘›ξ€·πœ‘(1)β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€Έπ‘›β€–β€–ξ€Έβˆ’π‘ž+𝛼𝑛𝛾𝑓(π‘ž)βˆ’π΄(π‘ž),π½πœ‘ξ€·π‘₯𝑛+1β‰€ξ€·βˆ’π‘žξ€Έξ¬1βˆ’π›Όπ‘›ξ€·πœ‘(1)Ξ¦ξ€·β€–β€–π‘₯π›Ύβˆ’π›Ύπ›Όξ€Έξ€Έπ‘›β€–β€–ξ€Έβˆ’π‘ž+𝛼𝑛𝛾𝑓(π‘ž)βˆ’π΄(π‘ž),π½πœ‘ξ€·π‘₯𝑛+1.βˆ’π‘žξ€Έξ¬(4.9) Note that βˆ‘βˆžπ‘›=1𝛼𝑛=∞ and limsupπ‘›β†’βˆžβŸ¨π›Ύπ‘“(π‘ž)βˆ’π΄(π‘ž),π½πœ‘(π‘₯𝑛+1βˆ’π‘ž)βŸ©β‰€0. Using Lemma 2.7, we have π‘₯π‘›β†’π‘ž as π‘›β†’βˆž by the continuity of Ξ¦. 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.

References

  1. B. Halpern, β€œFixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. P.-L. Lions, β€œApproximation de points fixes de contractions,” Comptes Rendus de l'AcadΓ©mie des Sciences, vol. 284, no. 21, pp. A1357–A1359, 1977. View at Zentralblatt MATH
  3. S. Reich, β€œApproximating fixed points of nonexpansive mappings,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 23–28, 1994. View at Zentralblatt MATH
  4. R. Wittmann, β€œApproximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486–491, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. N. Shioji and W. Takahashi, β€œStrong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, β€œApproximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. C. E. Chidume and C. O. Chidume, β€œIterative approximation of fixed points of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 288–295, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. Y. J. Cho, S. M. Kang, and H. Zhou, β€œSome control conditions on iterative methods,” Communications on Applied Nonlinear Analysis, vol. 12, no. 2, pp. 27–34, 2005. View at Zentralblatt MATH
  9. T.-H. Kim and H.-K. Xu, β€œStrong convergence of modified Mann iterations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. S. Reich, β€œStrong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. H.-K. Xu, β€œAnother control condition in an iterative method for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109–113, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. H.-K. Xu, β€œIterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. A. Moudafi, β€œViscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. H.-K. Xu, β€œViscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. G. Marino and H.-K. Xu, β€œA general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. T. Suzuki, β€œOn strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. H.-K. Xu, β€œA strong convergence theorem for contraction semigroups in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. R. Chen and Y. Song, β€œConvergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. R. Chen and H. He, β€œViscosity approximation of common fixed points of nonexpansive semigroups in Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. Y. Song and S. Xu, β€œStrong convergence theorems for nonexpansive semigroup in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 152–161, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  21. P. Cholamjiak and S. Suantai, β€œViscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions,” Journal of Global Optimization. In press. View at Publisher Β· View at Google Scholar
  22. R. Wangkeeree and U. Kamraksa, β€œStrong convergence theorems of viscosity iterative methods for a countable family of strict pseudo-contractions in Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 579725, 21 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  23. R. Wangkeeree, N. Petrot, and R. Wangkeeree, β€œThe general iterative methods for nonexpansive mappings in Banach spaces,” Journal of Global Optimization, vol. 51, no. 1, pp. 27–46, 2011. View at Publisher Β· View at Google Scholar
  24. I. K. Argyros, Y. J. Cho, and X. Qin, β€œOn the implicit iterative process for strictly pseudo-contractive mappings in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 208–216, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  25. S.-S. Chang, Y. J. Cho, H. W. J. Lee, and C. K. Chan, β€œStrong convergence theorems for Lipschitzian demicontraction semigroups in Banach spaces,” Fixed Point Theory and Applications, vol. 11, Article ID 583423, 10 pages, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. Y. J. Cho, L. Δ†iriΔ‡, and S.-H. Wang, β€œConvergence theorems for nonexpansive semigroups in CAT(0) spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6050–6059, 2011. View at Publisher Β· View at Google Scholar
  27. Y. J. Cho, S. M. Kang, and X. Qin, β€œSome results on k-strictly pseudo-contractive mappings in Hilbert spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 1956–1964, 2009. View at Publisher Β· View at Google Scholar
  28. Y. J. Cho, S. M. Kang, and X. Qin, β€œStrong convergence of an implicit iterative process for an infinite family of strict pseudocontractions,” Bulletin of the Korean Mathematical Society, vol. 47, no. 6, pp. 1259–1268, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  29. Y. J. Cho, X. Qin, and S. M. Kang, β€œStrong convergence of the modified Halpern-type iterative algorithms in Banach spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 17, no. 1, pp. 51–67, 2009.
  30. W. Guo and Y. J. Cho, β€œOn the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings,” Applied Mathematics Letters, vol. 21, no. 10, pp. 1046–1052, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  31. H. He, S. Liu, and Y. J. Cho, β€œAn explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4128–4139, 2011. View at Publisher Β· View at Google Scholar
  32. J. Kang, Y. Su, and X. Zhang, β€œGeneral iterative algorithm for nonexpansive semigroups and variational inequalities in Hilbert spaces,” journal of Inequalities and Applications, vol. 2010, Article ID 264052, 10 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  33. X. N. Li and J. S. Gu, β€œStrong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 4, pp. 1085–1092, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  34. S. Li, L. Li, and Y. Su, β€œGeneral iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  35. Q. Lin, β€œViscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5451–5457, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  36. S. Plubtieng and R. Punpaeng, β€œFixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279–286, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  37. X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, β€œConvergence of a modified Halpern-type iteration algorithm for quasi-Ο•-nonexpansive mappings,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1051–1055, 2009. View at Publisher Β· View at Google Scholar
  38. T. Suzuki, β€œStrong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  39. V. Colao, G. Marino, and H.-K. Xu, β€œAn iterative method for finding common solutions of equilibrium and fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340–352, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  40. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed Point Theory for Lipschitzian-Type Mappings with Applications, vol. 6, Springer, New York, NY, USA, 2009. View at Zentralblatt MATH
  41. C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.
  42. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  43. F. E. Browder, β€œConvergence theorems for sequences of nonlinear operators in Banach spaces,” Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  44. T.-C. Lim and H. K. Xu, β€œFixed point theorems for asymptotically nonexpansive mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 22, no. 11, pp. 1345–1355, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH