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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 453452, 14 pages
http://dx.doi.org/10.1155/2012/453452
Research Article

Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 27 June 2012; Accepted 21 August 2012

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Xionghua Wu 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

Let {𝑑𝑛}βŠ‚(0,1) be such that 𝑑𝑛→1 as π‘›β†’βˆž, let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction. If 𝑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {𝑑𝑛}, we show the existence of a sequence {π‘₯𝑛}𝑛 satisfying the relation π‘₯𝑛=(1βˆ’π‘‘π‘›/π‘˜π‘›)𝑓(π‘₯𝑛)+(𝑑𝑛/π‘˜π‘›)𝑇𝑛π‘₯𝑛 and prove that {π‘₯𝑛} converges strongly to the fixed point of 𝑇, which solves some variational inequality provided 𝑇 is uniformly asymptotically regular. As an application, if 𝑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset 𝐾 of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by 𝑧0∈𝐾,𝑧𝑛+1=(1βˆ’π‘‘π‘›/π‘˜π‘›)𝑓(𝑧𝑛)+(𝛼𝑑𝑛/π‘˜π‘›)𝑇𝑛𝑧𝑛+(𝛽𝑑𝑛/π‘˜π‘›)𝑧𝑛 converges strongly to the fixed point of 𝑇.

1. Introduction

Let 𝐸 be a real Banach space with dual πΈβˆ— and 𝐾 a nonempty closed convex subset of 𝐸. Recall that a mapping π‘‡βˆΆπΎβ†’πΎ is said to be asymptotically pseudocontractive if, for each π‘›βˆˆπ‘ and π‘₯,π‘¦βˆˆπΎ, there exist π‘—βˆˆπ½(π‘₯βˆ’π‘¦) and a constant π‘˜π‘›β‰₯1 with limπ‘›β†’βˆžπ‘˜π‘›=1 such that βŸ¨π‘‡π‘›π‘₯βˆ’π‘‡π‘›π‘¦βŸ©β‰€π‘˜π‘›β€–π‘₯βˆ’π‘¦β€–2,(1.1) where π½βˆΆπΈβ†’2πΈβˆ— denote the normalized duality mapping defined by 𝐽π‘₯(π‘₯)=βˆ—βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘₯βˆ—βŸ©=β€–π‘₯β€–2,β€–π‘₯βˆ—ξ€Ύ.β€–=β€–π‘₯β€–,π‘₯∈𝐸(1.2) The class of asymptotically pseudocontractive mappings is essentially wider than the class of asymptotically nonexpansive mappings. A mapping 𝑇 is called asymptotically nonexpansive if there exists a sequence {π‘˜π‘›}βŠ‚[1,∞) with limπ‘›β†’βˆžπ‘˜π‘›=1 such that ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€π‘˜π‘›β€–π‘₯βˆ’π‘¦β€–(1.3) for all integers 𝑛β‰₯0 and all π‘₯,π‘¦βˆˆπΎ. A mapping π‘“βˆΆπΎβ†’πΎ is called a contraction if there exists a constant π›Ύβˆˆ[0,1) such that ‖𝑓(π‘₯)βˆ’π‘“(𝑦)‖≀𝛾‖π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΎ.(1.4) It is clear that every contraction is nonexpansive, every nonexpansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is asymptotically pseudocontractive. The converses do not hold. The asymptotically nonexpansive mappings are important generalizations of nonexpansive mappings. For details, you may see [1].

The mapping 𝑇 is called uniformly asymptotically regular (in short u.a.r.) if for each πœ–>0 there exists 𝑛0βˆˆπ‘ such that ‖‖𝑇𝑛+1π‘₯βˆ’π‘‡π‘›π‘₯β€–β€–β‰€πœ–,(1.5) for all 𝑛β‰₯𝑛0 and π‘₯∈𝐾 and it is called uniformly asymptotically regular with sequence {πœ–π‘›} (in short u.a.r.s.) if ‖‖𝑇𝑛+1π‘₯βˆ’π‘‡π‘›π‘₯β€–β€–β‰€πœ–π‘›,(1.6) for all integers 𝑛β‰₯1 and all π‘₯∈𝐾, where πœ–π‘›β†’0 as π‘›β†’βˆž.

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping was proposed by Moudafi [2] who proved the strong convergence of both the implicit and explicit methods in Hilbert spaces, see [2, Theorems 2.1 and 2.2]. Xu [3] studied the viscosity approximation methods proposed by Moudafi [2] for a nonexpansive mapping in a uniformly smooth Banach space.

Very recently, Shahzad and Udomene [4] obtained fixed point solutions of variational inequalities for an asymptotically nonexpansive mapping defined on a real Banach space with uniformly Gateaux differentiable norm possessing uniform normal structure. They proved the following theorem.

Theorem 1.1. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, let 𝐾 be a nonempty closed convex and bounded subset of 𝐸, let π‘‡βˆΆπΎβ†’πΎ be an asymptotically nonexpansive mapping with sequence {π‘˜π‘›}π‘›βŠ‚[1,∞), and let π‘“βˆΆπΎβ†’πΎ be a contraction with constant π›Όβˆˆ[0,1). Let {𝑑𝑛}π‘›βŠ‚(0,πœ‰π‘›) be such that limπ‘›β†’βˆžπ‘‘π‘›=1,  βˆ‘βˆžπ‘›=1𝑑𝑛(1βˆ’π‘‘π‘›)=∞, and limπ‘›β†’βˆž((π‘˜π‘›βˆ’1)/(π‘˜π‘›βˆ’π‘‘π‘›))=0, where πœ‰π‘›=min{(1βˆ’π›Ό)π‘˜π‘›/(π‘˜π‘›βˆ’π›Ό),1/π‘˜π‘›}. For an arbitrary 𝑧0∈𝐾 let the sequence {𝑧𝑛} be iteratively defined by 𝑧𝑛+1=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘§π‘›ξ€Έ+π‘‘π‘›π‘˜π‘›π‘‡π‘›π‘§π‘›,π‘›βˆˆπ‘.(1.7) Then(i) for each integer 𝑛β‰₯0, there is a unique π‘₯π‘›βˆˆπΎ such that π‘₯𝑛=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+π‘‘π‘›π‘˜π‘›π‘‡π‘›π‘₯𝑛;(1.8) if in addition limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖=0,limπ‘›β†’βˆžβ€–β€–π‘§π‘›βˆ’π‘‡π‘§π‘›β€–β€–=0,(1.9) then(ii) the sequence {𝑧𝑛}𝑛 converges strongly to the unique solution of the variational inequality: ξ«ξ€·π‘βˆˆπΉ(𝑇)suchthat(πΌβˆ’π‘“)𝑝,π‘—π‘βˆ’π‘₯βˆ—ξ€Έξ¬β‰€0,βˆ€π‘₯βˆ—βˆˆπΉ(𝑇).(1.10)

Remark 1.2. We note that ‖𝑇𝑛+1π‘₯βˆ’π‘‡π‘›π‘₯β€–β‰€π‘˜π‘›β€–π‘‡π‘₯βˆ’π‘₯β€–, then the condition (1.9) limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖=0 and limπ‘›β†’βˆžβ€–π‘§π‘›βˆ’π‘‡π‘§π‘›β€–=0 imply that limπ‘›β†’βˆžβ€–β€–π‘‡π‘›+1π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖=0,limπ‘›β†’βˆžβ€–β€–π‘‡π‘›+1π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–=0,(1.11) respectively. In other words, if an asymptotically nonexpansive mapping 𝑇 satisfies the condition (1.9) then 𝑇 must be u.a.r.s.

Inspired by the works in [4–8], in this paper, we suggest and analyze a modification of the iterative algorithm.

Let {𝑑𝑛}βŠ‚(0,1), let 𝛼 and 𝛽 be two positive numbers such that 𝛼+𝛽=1, and let 𝑓 be a contraction on 𝐾, a sequence {𝑧𝑛} iteratively defined by: 𝑧0∈𝐾, 𝑧𝑛+1=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘§π‘›ξ€Έ+π›Όπ‘‘π‘›π‘˜π‘›π‘‡π‘›π‘§π‘›+π›½π‘‘π‘›π‘˜π‘›π‘§π‘›.(1.12)

Remark 1.3. The algorithm (1.12) includes the algorithm (1.7) of Chidume et al. [5] and Shahzad and Udomene [4] as a special case.

We show the convergence of the proposed algorithm (1.12) to the unique solution of some variational inequality (some related works on VI, please see [9–12]). In this respect, our results can be considered as a refinement and improvement of the known results of Chidume et al. [5], Shahzad and Udomene [4], and Lim and Xu [13].

2. Preliminaries

Let 𝑆={π‘₯βˆˆπΈβˆΆβ€–π‘₯β€–=1} denote the unit sphere of the Banach space 𝐸. The space 𝐸 is said to have a Gateaux differentiable norm if the limit limπ‘›β†’βˆžβ€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑(2.1) exists for each π‘₯,π‘¦βˆˆπ‘†, and we call 𝐸 smooth; 𝐸 is said to have a uniformly Gateaux differentiable norm if for each π‘¦βˆˆπ‘† the limit (2.1) is attained uniformly for π‘₯βˆˆπ‘†. Further, 𝐸 is said to be uniformly smooth if the limit (2.1) exists uniformly for (π‘₯,𝑦)βˆˆπ‘†Γ—π‘†. It is well known [14] that if 𝐸 is smooth then any duality mapping on 𝐸 is single-valued, and if 𝐸 has a uniformly Gateaux differentiable norm then the duality mapping is norm-to-weak* uniformly continuous on bounded sets.

Let 𝐾 be a nonempty closed convex and bounded subset of the Banach space 𝐸 and let the diameter of 𝐾 be defined by 𝑑(𝐾)=sup{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘₯,π‘¦βˆˆπΎ}. For each π‘₯∈𝐾, let π‘Ÿ(π‘₯,𝐾)=sup{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘¦βˆˆπΎ} and let π‘Ÿ(𝐾)=inf{π‘Ÿ(π‘₯,𝐾)∢π‘₯∈𝐾} denote the Chebyshev radius of 𝐾 relative to itself. The normal structure coefficient 𝑁(𝐸) of 𝐸 is defined by 𝑁(𝐸)=inf𝑑(𝐾)ξ‚Όπ‘Ÿ(𝐾)∢𝐾isaclosedconvexandboundedsubsetof𝐸with𝑑(𝐾)>0.(2.2) A space 𝐸 such that 𝑁(𝐸)>1 is said to have uniform normal structure. It is known that every space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see [13]).

We will let LIM be a Banach limit. Recall that LIM∈(π‘™βˆž)βˆ— such that β€–LIMβ€–=1, liminfπ‘›β†’βˆžπ‘Žπ‘›β‰€LIMπ‘›π‘Žπ‘›β‰€limsupπ‘›β†’βˆžπ‘Žπ‘›, and LIMπ‘›π‘Žπ‘›=LIMπ‘›π‘Žπ‘›+1 for all {π‘Žπ‘›}π‘›βˆˆπ‘™βˆž. Let {π‘₯𝑛} be a bounded sequence of 𝐸. Then we can define the real-valued continuous convex function 𝑔 on 𝐸 by 𝑔(𝑧)=LIM𝑛‖π‘₯π‘›βˆ’π‘§β€–2 for all π‘§βˆˆπΎ.

Let π‘‡βˆΆπΎβ†’πΎ be a nonlinear mapping and 𝑀={π‘₯βˆˆπΎβˆΆπ‘”(π‘₯)=minπ‘§βˆˆπΎπ‘”(𝑧)}. 𝑇 is said to satisfy the property (S) if for any bounded sequence {π‘₯𝑛} in 𝐾, limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖=0 implies π‘€βˆ©πΉ(𝑇)β‰ βˆ….

Lemma 2.1 (see [15]). Let 𝐸 be a Banach space with the uniformly Gateaux differentiable norm and π‘’βˆˆπΈ. Then 𝑔(𝑒)=infπ‘§βˆˆπΈπ‘”(𝑧)(2.3) if and only if π‘₯LIM𝑧,π½π‘›βˆ’π‘’ξ€Έξ¬β‰€0(2.4) for all π‘§βˆˆπΈ.

Lemma 2.2 (see [16]). Assume {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛿𝑛𝛾𝑛,(2.5) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that (1)βˆ‘βˆžπ‘›=1𝛾𝑛=∞; (2)limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0 or βˆ‘βˆžπ‘›=1|𝛿𝑛𝛾𝑛|<∞. Then limπ‘›β†’βˆžπ‘Žπ‘›=0.

Lemma 2.3 (see [17]). Let {π‘₯𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1. Suppose that π‘₯𝑛+1=ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›+𝛽𝑛π‘₯𝑛(2.6) for all 𝑛β‰₯0 and limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(2.7) Then limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.4. Let 𝐸 be an arbitrary real Banach space. Then β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,𝑗(π‘₯+𝑦)⟩,(2.8) for all π‘₯,π‘¦βˆˆπΈ and for all 𝑗(π‘₯+𝑦)∈𝐽(π‘₯+𝑦).

Lemma 2.5 (see [5]). Let 𝐸 be a Banach space with uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, and π‘‡βˆΆπΎβ†’πΎ an asymptotically nonexpansive mapping. Then 𝑇 has a fixed point.

3. Main Results

Theorem 3.1. Let 𝐸 be a real reflexive Banach space with a uniformly Gateaux differentiable norm, 𝐾 a nonempty closed convex and bounded subset of 𝐸, π‘‡βˆΆπΎβ†’πΎ a continuous asymptotically pseudocontractive mapping with sequence {π‘˜π‘›}π‘›βŠ‚[1,∞), and π‘“βˆΆπΎβ†’πΎ a contraction with constant π›Ύβˆˆ[0,1). Let {𝑑𝑛}βŠ‚(0,(1βˆ’π›Ύ)π‘˜π‘›/(π‘˜π‘›βˆ’π›Ύ)) be such that limπ‘›β†’βˆžπ‘‘π‘›=1 and limπ‘›β†’βˆž((π‘˜π‘›βˆ’1)/(π‘˜π‘›βˆ’π‘‘π‘›))=0. Suppose 𝑇 satisfies the property (S). Then (i) for each integer 𝑛β‰₯0, there is a unique π‘₯π‘›βˆˆπΎ such that π‘₯𝑛=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+π‘‘π‘›π‘˜π‘›π‘‡π‘›π‘₯𝑛;(3.1) if 𝑇 is u.a.r.s., then (ii) the sequence {π‘₯𝑛}𝑛 converges strongly to the unique solution of the variational inequality: ξ«ξ€·π‘βˆˆπΉ(𝑇)suchthat(πΌβˆ’π‘“)𝑝,π‘—π‘βˆ’π‘₯βˆ—ξ€Έξ¬β‰€0,βˆ€π‘₯βˆ—βˆˆπΉ(𝑇).(3.2)

Proof. By the conditions on {𝑑𝑛}, 𝑑𝑛<(1βˆ’π›Ύ)π‘˜π‘›/(π‘˜π‘›βˆ’π›Ύ) implies (1βˆ’π‘‘π‘›/π‘˜π‘›)𝛾+𝑑𝑛<1 for each integer 𝑛β‰₯0, then the mapping π‘†π‘›βˆΆπΎβ†’πΎ defined for each π‘₯∈𝐾 by 𝑆𝑛π‘₯=(1βˆ’π‘‘π‘›/π‘˜π‘›)𝑓(π‘₯)+(𝑑𝑛/π‘˜π‘›)𝑇𝑛π‘₯ is a strictly pseudocontractive mapping.
Indeed, for all π‘₯,π‘¦βˆˆπΎ, we have βŸ¨π‘†π‘›π‘₯βˆ’π‘†π‘›ξ‚΅π‘‘π‘¦,𝑗(π‘₯βˆ’π‘¦)⟩=1βˆ’π‘›π‘˜π‘›ξ‚Ά+π‘‘βŸ¨π‘“(π‘₯)βˆ’π‘“(𝑦),𝑗(π‘₯βˆ’π‘¦)βŸ©π‘›π‘˜π‘›βŸ¨π‘‡π‘›π‘₯βˆ’π‘‡π‘›β‰€ξ‚΅π‘‘π‘¦,𝑗(π‘₯βˆ’π‘¦)⟩1βˆ’π‘›π‘˜π‘›ξ‚Άβ€–π‘“(π‘₯)βˆ’π‘“(𝑦)β€–β€–π‘₯βˆ’π‘¦β€–+𝑑𝑛‖π‘₯βˆ’π‘¦β€–2≀𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ›Ύ+𝑑𝑛‖π‘₯βˆ’π‘¦β€–2.(3.3) It follows [18, Corollary 1] that 𝑆𝑛 possesses exactly one fixed point π‘₯𝑛 in 𝐾 such that 𝑆𝑛π‘₯𝑛=π‘₯𝑛.
From (3.1), we have β€–β€–π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖=‖‖‖𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+ξ‚΅π‘‘π‘›π‘˜π‘›ξ‚Άπ‘‡βˆ’1𝑛π‘₯𝑛‖‖‖=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άβ€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘‡π‘›π‘₯𝑛‖‖→0asπ‘›β†’βˆž.(3.4) Notice that β€–β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖=‖‖‖𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άξ€·π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘‡π‘₯𝑛+π‘‘π‘›π‘˜π‘›ξ€·π‘‡π‘›π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖‖≀𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άβ€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘‡π‘₯𝑛‖‖+π‘‘π‘›π‘˜π‘›β€–β€–π‘‡π‘›π‘₯π‘›βˆ’π‘‡π‘›+1π‘₯𝑛‖‖+π‘‘π‘›π‘˜π‘›β€–β€–π‘‡π‘›+1π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖≀𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άβ€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘‡π‘₯𝑛‖‖+π‘‘π‘›π‘˜π‘›β€–β€–π‘‡π‘›π‘₯π‘›βˆ’π‘‡π‘›+1π‘₯𝑛‖‖+π‘‘π‘›π‘˜π‘›π‘˜1β€–β€–π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖.(3.5) Therefore, from (3.4), (3.5), and 𝑇 which is u.a.r.s., we obtain β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖→0 as π‘›β†’βˆž.
Define a function π‘”βˆΆπΎβ†’π‘…+ by 𝑔(𝑧)=LIM𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘§2(3.6) for all π‘§βˆˆπΎ. Since 𝑔 is continuous and convex, 𝑔(𝑧)β†’βˆž as β€–π‘§β€–β†’βˆž, and 𝐸 is reflexive, 𝑔 attains it infimum over 𝐾. Let 𝑧0∈𝐾 such that 𝑔(𝑧0)=minπ‘§βˆˆπΎπ‘”(𝑧) and let 𝑀={π‘₯βˆˆπΎβˆΆπ‘”(π‘₯)=minπ‘§βˆˆπΎπ‘”(𝑧)}. Then 𝑀 is nonempty because 𝑧0βˆˆπ‘€. Since 𝑇 satisfies the property (S), it follows that π‘€βˆ©πΉ(𝑇)β‰ βˆ…. Suppose that π‘βˆˆπ‘€βˆ©πΉ(𝑇). Then, by Lemma 2.1, we have LIM𝑛π‘₯π‘₯βˆ’π‘,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0(3.7) for all π‘₯∈𝐾. In particular, we have LIM𝑛𝑓π‘₯(𝑝)βˆ’π‘,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0.(3.8) On the other hand, from (3.1), we have π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άξ€·π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘‡π‘›π‘₯𝑛=1βˆ’π‘‘π‘›/π‘˜π‘›π‘‘π‘›/π‘˜π‘›ξ€·π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯𝑛.(3.9) Now, for any π‘βˆˆπΉ(𝑇), we have π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛π‘₯,𝐽𝑛=π‘₯βˆ’π‘ξ€Έξ¬π‘›βˆ’π‘+π‘‡π‘›π‘βˆ’π‘‡π‘›π‘₯𝑛π‘₯,π½π‘›ξ€·π‘˜βˆ’π‘ξ€Έξ¬β‰₯βˆ’π‘›ξ€Έβ€–β€–π‘₯βˆ’1π‘›β€–β€–βˆ’π‘2ξ€·π‘˜β‰₯βˆ’π‘›ξ€Έπ΅βˆ’12(3.10) for some 𝐡>0 and it follows from (3.9) that π‘₯𝑛π‘₯βˆ’π‘“π‘›ξ€Έξ€·π‘₯,π‘—π‘›β‰€π‘‘βˆ’π‘ξ€Έξ¬π‘›ξ€·π‘˜π‘›ξ€Έβˆ’1π‘˜π‘›βˆ’π‘‘π‘›π΅2,(3.11) which implies that limsupπ‘›β†’βˆžξ«π‘₯𝑛π‘₯βˆ’π‘“π‘›ξ€Έξ€·π‘₯,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0.(3.12) Consequently, similar to the lines of the proof of [4, Theorem 3.1], Theorem 3.1 is easily proved. This completes the proof.

Corollary 3.2. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, π‘‡βˆΆπΎβ†’πΎ be an asymptotically nonexpansive mapping with sequence {π‘˜π‘›}π‘›βŠ‚[1,∞), and π‘“βˆΆπΎβ†’πΎ a contraction with constant π›Ύβˆˆ[0,1). Let {𝑑𝑛}βŠ‚(0,(1βˆ’π›Ύ)π‘˜π‘›/(π‘˜π‘›βˆ’π›Ύ)) be such that limπ‘›β†’βˆžπ‘‘π‘›=1 and limπ‘›β†’βˆž((π‘˜π‘›βˆ’1)/(π‘˜π‘›βˆ’π‘‘π‘›))=0. Then (i) for each integer 𝑛β‰₯0, there is a unique π‘₯π‘›βˆˆπΎ such that π‘₯𝑛=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+π‘‘π‘›π‘˜π‘›π‘‡π‘›π‘₯𝑛;(3.13) if 𝑇 is u.a.r.s., then (ii) the sequence {π‘₯𝑛}𝑛 converges strongly to the unique solution of the variational inequality: ξ«ξ€·π‘βˆˆπΉ(𝑇)suchthat(πΌβˆ’π‘“)𝑝,π‘—π‘βˆ’π‘₯βˆ—ξ€Έξ¬β‰€0,βˆ€π‘₯βˆ—βˆˆπΉ(𝑇).(3.14)

Theorem 3.3. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, π‘‡βˆΆπΎβ†’πΎ an asymptotically nonexpansive mapping with sequence {π‘˜π‘›}π‘›βŠ‚[1,∞), and π‘“βˆΆπΎβ†’πΎ a contraction with constant π›Ύβˆˆ[0,1). Let {𝑑𝑛}βŠ‚(0,πœ‰π‘›) be such that limπ‘›β†’βˆžπ‘‘π‘›βˆ‘=1,βˆžπ‘›=1𝑑𝑛(1βˆ’π‘‘π‘›)=∞, and limπ‘›β†’βˆž((π‘˜π‘›βˆ’1)/(π‘˜π‘›βˆ’π‘‘π‘›))=0, where πœ‰π‘›=min{(1βˆ’π›Ύ)π‘˜π‘›/(π‘˜π‘›βˆ’π›Ύ),1/π‘˜π‘›}. For an arbitrary 𝑧0∈𝐾, let the sequence {𝑧𝑛}𝑛 be iteratively defined by (1.12). Then (i) for each integer 𝑛β‰₯0, there is a unique π‘₯π‘›βˆˆπΎ such that π‘₯𝑛=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+π‘‘π‘›π‘˜π‘›π‘‡π‘›π‘₯𝑛;(3.15) if 𝑇 is u.a.r.s., then (ii) the sequence {𝑧𝑛}𝑛 converges strongly to the unique solution of the variational inequality: ξ«ξ€·π‘βˆˆπΉ(𝑇)suchthat(πΌβˆ’π‘“)𝑝,π‘—π‘βˆ’π‘₯βˆ—ξ€Έξ¬β‰€0,βˆ€π‘₯βˆ—βˆˆπΉ(𝑇).(3.16)

Proof. Set 𝛼𝑛=𝑑𝑛/π‘˜π‘›, then 𝛼𝑛→1 as π‘›β†’βˆž. Define 𝑧𝑛+1=𝛽𝛼𝑛𝑧𝑛+ξ€·1βˆ’π›½π›Όπ‘›ξ€Έπ‘¦π‘›.(3.17) Observe that 𝑦𝑛+1βˆ’π‘¦π‘›=𝑧𝑛+2βˆ’π›½π›Όπ‘›+1𝑧𝑛+11βˆ’π›½π›Όπ‘›+1βˆ’π‘§π‘›+1βˆ’π›½π›Όπ‘›π‘§π‘›1βˆ’π›½π›Όπ‘›=ξ€·1βˆ’π›Όπ‘›+1𝑓𝑧𝑛+1ξ€Έ+𝛼𝛼𝑛+1𝑇𝑛+1𝑧𝑛+11βˆ’π›½π›Όπ‘›+1βˆ’ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘“ξ€·π‘§π‘›ξ€Έ+𝛼𝛼𝑛𝑇𝑛𝑧𝑛1βˆ’π›½π›Όπ‘›=1βˆ’π›Όπ‘›+11βˆ’π›½π›Όπ‘›+1𝑓𝑧𝑛+1ξ€Έξ€·π‘§βˆ’π‘“π‘›+ξ‚΅ξ€Έξ€»1βˆ’π›Όπ‘›+11βˆ’π›½π›Όπ‘›+1βˆ’1βˆ’π›Όπ‘›1βˆ’π›½π›Όπ‘›ξ‚Άπ‘“ξ€·π‘§π‘›ξ€Έ+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1𝑇𝑛+1𝑧𝑛+1βˆ’π‘‡π‘›+1𝑧𝑛+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1𝑇𝑛+1π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›ξ€Έ+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1βˆ’π›Όπ›Όπ‘›1βˆ’π›½π›Όπ‘›ξ‚Άπ‘‡π‘›π‘§π‘›.(3.18) It follows that ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–β‰€1βˆ’π›Όπ‘›+11βˆ’π›½π›Όπ‘›+1𝛾‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–+||||1βˆ’π›Όπ‘›+11βˆ’π›½π›Όπ‘›+1βˆ’1βˆ’π›Όπ‘›1βˆ’π›½π›Όπ‘›||||‖‖𝑓𝑧𝑛‖‖+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1‖‖𝑇𝑛+1𝑧𝑛+1βˆ’π‘‡π‘›+1𝑧𝑛‖‖+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1‖‖𝑇𝑛+1π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–+||||𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1βˆ’π›Όπ›Όπ‘›1βˆ’π›½π›Όπ‘›||||β€–β€–π‘‡π‘›π‘§π‘›β€–β€–βˆ’β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–β‰€||||1βˆ’π›Όπ‘›+11βˆ’π›½π›Όπ‘›+1βˆ’1βˆ’π›Όπ‘›1βˆ’π›½π›Όπ‘›||||‖‖𝑓𝑧𝑛‖‖+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1‖‖𝑇𝑛+1π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–+||||𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1βˆ’π›Όπ›Όπ‘›1βˆ’π›½π›Όπ‘›||||‖‖𝑇𝑛𝑧𝑛‖‖+ξ‚΅1βˆ’π›Όπ‘›+11βˆ’π›½π›Όπ‘›+1𝛾+𝛼𝛼𝑛+11βˆ’π›½π›Όπ‘›+1π‘˜π‘›+1ξ‚Άβ€–β€–π‘§βˆ’1𝑛+1βˆ’π‘§π‘›β€–β€–.(3.19) We note that π‘˜π‘›+1ξ€·βˆ’π›Ύβˆ’π›Όπ‘˜π‘›+1ξ€Έ=+π›½βˆ’π›Ύ(1βˆ’π›Ό)π‘˜π‘›+1βˆ’π›½β‰₯1βˆ’π›Όβˆ’π›½=0.(3.20) It follows that 𝑑𝑛+1≀(1βˆ’π›Ύ)π‘˜π‘›+1π‘˜π‘›+1β‰€βˆ’π›Ύ(1βˆ’π›Ύ)π‘˜π‘›+1π›Όπ‘˜π‘›+1,+π›½βˆ’π›Ύ(3.21) which implies that π‘˜π‘›+1𝑑𝑛+1𝛼+𝑑𝑛+1π›½βˆ’π‘‘π‘›+1𝛾≀(1βˆ’π›Ύ)π‘˜π‘›+1β‡’π›Όπ‘˜π‘›+1𝛼𝑛+1+𝛼𝑛+1π›½βˆ’π›Όπ‘›+1𝛾≀1βˆ’π›Ύβ‡’π›Όπ‘˜π‘›+1𝛼𝑛+1+ξ€·1βˆ’π›Όπ‘›+1𝛾≀1βˆ’π›Όπ‘›+1π›½β‡’π›Όπ‘˜π‘›+1𝛼𝑛+1+ξ€·1βˆ’π›Όπ‘›+1𝛾1βˆ’π›Όπ‘›+1𝛽≀1.(3.22) From (3.19) and (3.22), we obtain limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–ξ€Έβ‰€0.(3.23) Hence, by Lemma 2.3 we know limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘§π‘›β€–β€–=0,(3.24) consequently limπ‘›β†’βˆžβ€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–=0.(3.25)
On the other hand, β€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–β‰€β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–+‖‖𝑧𝑛+1βˆ’π‘‡π‘›π‘§π‘›β€–β€–β‰€β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘“ξ€·π‘§π‘›ξ€Έβˆ’π‘‡π‘›π‘§π‘›β€–β€–+π›½π›Όπ‘›β€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–,(3.26) which implies that limπ‘›β†’βˆžβ€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–=0.(3.27) Hence, we have β€–β€–π‘§π‘›βˆ’π‘‡π‘§π‘›β€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–+β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›+1𝑧𝑛‖‖+‖‖𝑇𝑛+1π‘§π‘›βˆ’π‘‡π‘§π‘›β€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–+β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›+1𝑧𝑛‖‖+π‘˜1β€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–=ξ€·1+π‘˜1ξ€Έβ€–β€–π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–β€–+β€–β€–π‘‡π‘›π‘§π‘›βˆ’π‘‡π‘›+1𝑧𝑛‖‖→0(π‘›β†’βˆž).(3.28) From (3.15), π‘₯π‘šβˆ’π‘§π‘›=(1βˆ’π›Όπ‘š)(𝑓(π‘₯π‘š)βˆ’π‘§π‘›)+π›Όπ‘š(π‘‡π‘šπ‘₯π‘šβˆ’π‘§π‘›). Applying Lemma 2.4, we estimate as follows: β€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2≀𝛼2π‘šβ€–β€–π‘‡π‘šπ‘₯π‘šβˆ’π‘§π‘›β€–β€–2ξ€·+21βˆ’π›Όπ‘šπ‘“ξ€·π‘₯ξ€Έξ«π‘šξ€Έβˆ’π‘§π‘›ξ€·π‘₯,π‘—π‘šβˆ’π‘§π‘›ξ€Έξ¬β‰€π›Ό2π‘šξ€·β€–β€–π‘‡π‘šπ‘₯π‘šβˆ’π‘‡π‘šπ‘§π‘›β€–β€–+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–ξ€Έ2ξ€·+21βˆ’π›Όπ‘šξ€Έξ‚ƒξ«π‘“ξ€·π‘₯π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘₯,π‘—π‘šβˆ’π‘§π‘›+β€–β€–π‘₯ξ€Έξ¬π‘šβˆ’π‘§π‘›β€–β€–2≀𝛼2π‘šξ€·π‘˜π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–ξ€Έ2ξ€·+21βˆ’π›Όπ‘šξ€Έξ‚€ξ«π‘“ξ€·π‘₯π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘₯,π‘—π‘šβˆ’π‘§π‘›ξ€Έξ¬+π‘˜2π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2=𝛼2π‘šξ‚€π‘˜2π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2+2π‘˜π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–2+21βˆ’π›Όπ‘šξ€Έπ‘“ξ€·π‘₯ξ‚€ξ‚¬π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘₯,π‘—π‘šβˆ’π‘§π‘›ξ€Έ+π‘˜2π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2=1βˆ’(1βˆ’π›Όπ‘š)ξ€Έ2π‘˜2π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–ξ€·2π‘˜π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–ξ€Έξ€·+21βˆ’π›Όπ‘šξ€Έξ‚€ξ«π‘“ξ€·π‘₯π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘₯,π‘—π‘šβˆ’π‘§π‘›ξ€Έξ¬+π‘˜2π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2≀1+1βˆ’π›Όπ‘šξ€Έ2ξ‚π‘˜2π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–2+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–ξ€·2π‘˜π‘šβ€–β€–π‘₯π‘šβˆ’π‘§π‘›β€–β€–+β€–β€–π‘‡π‘šπ‘§π‘›βˆ’π‘§π‘›β€–β€–ξ€Έξ€·+21βˆ’π›Όπ‘šπ‘“ξ€·π‘₯ξ€Έξ«π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘₯,π‘—π‘šβˆ’π‘§π‘›.(3.29) Since 𝐾 is bounded, for some constant 𝑀>0, it follows that limsupπ‘›β†’βˆžξ«π‘“ξ€·π‘₯π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘§,π‘—π‘›βˆ’π‘₯π‘šβ‰€ξ‚ƒπ‘˜ξ€Έξ¬2π‘šβˆ’1+π‘˜2π‘šξ€·1βˆ’π›Όπ‘šξ€Έ2ξ‚„1βˆ’π›Όπ‘šπ‘€+limsupπ‘›β†’βˆžπ‘€β€–β€–π‘§π‘›βˆ’π‘‡π‘šz𝑛‖‖1βˆ’π›Όπ‘š,(3.30) so that limsupπ‘›β†’βˆžξ«π‘“ξ€·π‘₯π‘šξ€Έβˆ’π‘₯π‘šξ€·π‘§,π‘—π‘›βˆ’π‘₯π‘šβ‰€ξ‚ƒπ‘˜ξ€Έξ¬2π‘šβˆ’1+π‘˜2π‘šξ€·1βˆ’π›Όπ‘šξ€Έ2ξ‚„1βˆ’π›Όπ‘šπ‘€.(3.31) By Corollary 3.2, π‘₯π‘šβ†’π‘βˆˆπΉ(𝑇), which solve the variational inequality (3.16). Since 𝑗 is norm to weak* continuous on bounded sets, in the limit as π‘šβ†’βˆž, we obtain that limsupπ‘›β†’βˆžξ«π‘“ξ€·π‘§(𝑝)βˆ’π‘,π‘—π‘›βˆ’π‘ξ€Έξ¬β‰€0.(3.32) From Lemma 2.4, we estimate as follows: ‖‖𝑧𝑛+1β€–β€–βˆ’π‘2=β€–β€–(1βˆ’π›Όπ‘›)(𝑓(𝑧𝑛)βˆ’π‘)+𝛼𝛼𝑛(π‘‡π‘›π‘§π‘›βˆ’π‘)+𝛽𝛼𝑛(π‘§π‘›β€–β€–βˆ’π‘)2≀‖‖𝛼𝛼𝑛(π‘‡π‘›π‘§π‘›βˆ’π‘)+𝛽𝛼𝑛(π‘§π‘›β€–β€–βˆ’π‘)2ξ€·+21βˆ’π›Όπ‘›π‘“ξ€·π‘§ξ€Έξ«π‘›ξ€Έξ€·π‘§βˆ’π‘,𝑗𝑛+1βˆ’π‘ξ€Έξ¬β‰€π›Ό2𝛼2π‘›β€–β€–π‘‡π‘›π‘§π‘›β€–β€–βˆ’π‘2+2𝛼𝛽𝛼2π‘›β€–β€–π‘‡π‘›π‘§π‘›β€–β€–β€–β€–π‘§βˆ’π‘π‘›β€–β€–βˆ’π‘+𝛽2𝛼2π‘›β€–β€–π‘§π‘›β€–β€–βˆ’π‘2ξ€·+21βˆ’π›Όπ‘›ξ€Έξ€·π‘§βŸ¨π‘“π‘›ξ€Έξ€·π‘§βˆ’π‘“(𝑝),𝑗𝑛+1ξ€ΈβŸ©ξ€·βˆ’π‘+21βˆ’π›Όπ‘›ξ€·π‘§ξ€Έξ«π‘“(𝑝)βˆ’π‘,𝑗𝑛+1β‰€ξ€·π›Όβˆ’π‘ξ€Έξ¬2π‘˜2𝑛+2π›Όπ›½π‘˜π‘›+𝛽2𝛼2π‘›β€–β€–π‘§π‘›β€–β€–βˆ’π‘2ξ€·+21βˆ’π›Όπ‘›ξ€Έπ›Ύβ€–β€–π‘§π‘›β€–β€–β€–β€–π‘§βˆ’π‘π‘›+1β€–β€–ξ€·βˆ’π‘+21βˆ’π›Όπ‘›ξ€·π‘§ξ€Έξ«π‘“(𝑝)βˆ’π‘,𝑗𝑛+1βˆ’π‘ξ€Έξ¬β‰€π›Ό2π‘›π‘˜2π‘›β€–β€–π‘§π‘›β€–β€–βˆ’π‘2ξ€·+𝛾1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘§π‘›β€–β€–βˆ’π‘2+‖‖𝑧𝑛+1β€–β€–βˆ’π‘2+21βˆ’π›Όπ‘›ξ€Έξ€·π‘§βŸ¨π‘“(𝑝)βˆ’π‘,𝑗𝑛+1ξ€Έβˆ’π‘βŸ©,(3.33) so that ‖‖𝑧𝑛+1β€–β€–βˆ’π‘2≀𝑑2𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ›Ύξ€»ξ€·1βˆ’1βˆ’π›Όπ‘›ξ€Έπ›Ύβ€–β€–π‘§π‘›β€–β€–βˆ’π‘2ξ€·+21βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’1βˆ’π›Όπ‘›ξ€Έπ›Ύξ«π‘“ξ€·π‘§(𝑝)βˆ’π‘,𝑗𝑛+1=ξƒ©ξ€Ίξ€·βˆ’π‘ξ€Έξ¬1βˆ’1βˆ’21βˆ’π›Όπ‘›ξ€Έπ›Ύβˆ’π‘‘2𝑛1βˆ’1βˆ’π›Όπ‘›ξ€Έπ›Ύξƒͺβ€–β€–π‘§π‘›β€–β€–βˆ’π‘2ξ€·+21βˆ’π›Όπ‘›ξ€Έξ€·1βˆ’1βˆ’π›Όπ‘›ξ€Έπ›Ύξ«ξ€·π‘§π‘“(𝑝)βˆ’π‘,𝑗𝑛+1.βˆ’π‘ξ€Έξ¬(3.34) Let πœ†π‘›=ξ€Ίξ€·1βˆ’21βˆ’π›Όπ‘›ξ€Έπ›Ύβˆ’π‘‘2𝑛1βˆ’1βˆ’π›Όπ‘›ξ€Έπ›Ύ.(3.35) Consequently, following the lines of the proof of [4, Theorem 3.3], Theorem 3.3 is easily proved.

From the lines of the proof of Theorem 3.3, we can obtain the following corollary.

Corollary 3.4. Let 𝐸 be a real Banach space with a uniformly Gateaux differentiable norm possessing uniform normal structure, 𝐾 a nonempty closed convex and bounded subset of 𝐸, π‘‡βˆΆπΎβ†’πΎ an asymptotically nonexpansive mapping with sequence {π‘˜π‘›}π‘›βŠ‚[1,∞), and π‘“βˆΆπΎβ†’πΎ a contraction with constant π›Ύβˆˆ[0,1). Let {𝑑𝑛}βŠ‚(0,πœ‰π‘›) be such that limπ‘›β†’βˆžπ‘‘π‘›βˆ‘=1,βˆžπ‘›=1𝑑𝑛(1βˆ’π‘‘π‘›)=∞, and limπ‘›β†’βˆž((π‘˜π‘›βˆ’1)/(π‘˜π‘›βˆ’π‘‘π‘›))=0, where πœ‰π‘›=min{(1βˆ’π›Ύ)π‘˜π‘›/(π‘˜π‘›βˆ’π›Ύ),1/π‘˜π‘›}. For an arbitrary 𝑧0∈𝐾, let the sequence {𝑧𝑛}𝑛 be iteratively defined by (1.12). Then(i) for each integer 𝑛β‰₯0, there is a unique π‘₯π‘›βˆˆπΎ such that π‘₯𝑛=𝑑1βˆ’π‘›π‘˜π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+π‘‘π‘›π‘˜π‘›π‘‡π‘›x𝑛;(3.36) if 𝑇 satisfies limπ‘›β†’βˆžβ€–π‘‡π‘›+1π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖=0 and limπ‘›β†’βˆžβ€–π‘‡π‘›+1π‘§π‘›βˆ’π‘‡π‘›π‘§π‘›β€–=0 then(ii) the sequence {𝑧𝑛}𝑛 converges strongly to the unique solution of the variational inequality: ξ«ξ€·π‘βˆˆπΉ(𝑇)suchthat(πΌβˆ’π‘“)𝑝,π‘—π‘βˆ’π‘₯βˆ—ξ€Έξ¬β‰€0,βˆ€π‘₯βˆ—βˆˆπΉ(𝑇).(3.37)

Remark 3.5. Since every nonexpansive mapping is asymptotically nonexpansive, our theorems hold for the case when 𝑇 is simply nonexpansive. In this case, the boundedness requirement on 𝐾 can be removed from the above results.

Remark 3.6. Our results can be viewed as a refinement and improvement of the corresponding results by Shahzad and Udomene [4], Chidume et al. [5], and Lim and Xu [13].

Example 3.7. Let π‘‡βˆΆπΆβ†’πΆ be a nonexpansive mapping. Let the iterative sequence {π‘₯𝑛} be defined by π‘₯𝑛+1=1𝑛1𝑒+1βˆ’π‘›ξ‚π‘‡π‘₯𝑛.(3.38) It is easy to see that {π‘₯𝑛} converges strongly to some fixed point of 𝑇.
In particular, let 𝐻=𝑅2 and define π‘‡βˆΆπ‘…2→𝑅2 by π‘‡ξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ=π‘Ÿπ‘’π‘–(πœƒ+πœ‹/2),(3.39) and take that 𝑒=π‘’π‘–πœ‹ is a fix element in 𝐢. It is obvious that 𝑇 is a nonexpansive mapping with a unique fixed point π‘₯βˆ—=0. In this case, (3.38) becomes π‘₯𝑛+1=1π‘›π‘’π‘–πœ‹+ξ‚€11βˆ’π‘›ξ‚π‘Ÿπ‘›π‘’π‘–(πœƒπ‘›+πœ‹/2).(3.40) It is clear that the complex number sequence {π‘₯𝑛} converges strongly to a fixed point π‘₯βˆ—=0.

Acknowledgment

Y.-C. Liou was partially supported by NSC 101-2628-E-230-001-MY3.

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