Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Volume 2012 |Article ID 474031 | 15 pages | https://doi.org/10.1155/2012/474031

Convergence of Implicit and Explicit Schemes for an Asymptotically Nonexpansive Mapping in π‘ž -Uniformly Smooth and Strictly Convex Banach Spaces

Academic Editor: Hong-Kun Xu
Received24 Apr 2012
Accepted23 Jun 2012
Published08 Aug 2012

Abstract

We introduce a new iterative scheme with Meir-Keeler contractions for an asymptotically nonexpansive mapping in π‘ž-uniformly smooth and strictly convex Banach spaces. We also proved the strong convergence theorems of implicit and explicit schemes. The results obtained in this paper extend and improve many recent ones announced by many others.

1. Introduction

Let 𝐸 be a real Banach space. With π½βˆΆπΈβ†’2πΈβˆ—, we denote the normalized duality mapping given by 𝐽(π‘₯)=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–2ξ€Ύ,,‖𝑓‖=β€–π‘₯β€–(1.1) where βŸ¨β‹…,β‹…βŸ© denotes the generalized duality pairing and πΈβˆ— the dual space of 𝐸. In the sequel we will donate single-valued duality mappings by 𝑗. Given π‘ž>1, by π½π‘ž we will denote the generalized duality mapping given by π½π‘žξ€½(π‘₯)=π‘“βˆˆπΈβˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–π‘ž,‖𝑓‖=β€–π‘₯β€–π‘žβˆ’1ξ€Ύ.(1.2)

We recall that the following relation holds: π½π‘ž(π‘₯)=β€–π‘₯β€–π‘žβˆ’2𝐽(π‘₯),(1.3) for π‘₯β‰ 0.

We recall that the modulus of smoothness of 𝐸 is the function 𝜌𝐸∢[0,∞)β†’[0,∞) defined by πœŒπΈξ‚†1(𝑑)∢=sup2(.β€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–)βˆ’1βˆΆβ€–π‘₯‖≀1,‖𝑦‖≀𝑑(1.4)𝐸 is said to be uniformly smooth if lim𝑑→0(𝜌𝐸(𝑑)/𝑑)=0.

Let π‘ž>1. 𝐸 is said to be π‘ž-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌𝐸(𝑑)β‰€π‘π‘‘π‘ž. Examples of such spaces are Hilbert spaces and 𝐿𝑝 (or 𝑙𝑝).

We note that a π‘ž-uniformly smooth Banach space is uniformly smooth. This implies that its norm uniformly FrΓ©chet differentiable (see [1]).

If 𝐸 is uniformly smooth, then the normalized duality map 𝑗 is single-valued and norm to norm uniformly continuous.

Let 𝐸 be a real Banach space and 𝐢 is a nonempty closed convex subset of 𝐸. A mapping π‘‡βˆΆπΆβ†’πΆ is said to be asymptotically nonexpansive if there exists a sequence {β„Žπ‘›}βŠ‚[0,∞) with limπ‘›β†’βˆžβ„Žπ‘›=0 such that ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›ξ€·π‘¦β€–β‰€1+β„Žπ‘›ξ€Έβ€–π‘₯βˆ’π‘¦β€–,π‘₯,π‘¦βˆˆπΆ,𝑛β‰₯1,(1.5) and 𝐹(𝑇) denotes the set of fixed points of the mapping 𝑇; that is, 𝐹(𝑇)={π‘₯βˆˆπΆβˆΆπ‘‡π‘₯=π‘₯}. For asymptotically nonexpansive self-map 𝑇, it is well known that 𝐹(𝑇) is closed and convex (see e.g., [2]).

Theorem 1.1 (Banach [3]). Let (𝑋,𝑑) be a complete metric space and let 𝑓 be a contraction on 𝑋; that is, there exists π‘Ÿβˆˆ(0,1) such that 𝑑(𝑓(π‘₯),𝑓(𝑦))β‰€π‘Ÿπ‘‘(π‘₯,𝑦) for all π‘₯,π‘¦βˆˆπ‘‹. Then 𝑓 has a unique fixed point.

Theorem 1.2 (Meir and Keeler [4]). Let (𝑋,𝑑) be a complete metric space and let πœ™ be a Meir-Keeler contraction (MKC) on 𝑋, that is, for every πœ€>0, there exists 𝛿>0 such that 𝑑(π‘₯,𝑦)<πœ€+𝛿 implies 𝑑(πœ™(π‘₯),πœ™(𝑦))<πœ€ for all π‘₯,π‘¦βˆˆπ‘‹. Then πœ™ has a unique fixed point.

This theorem is one of generalizations of Theorem 1.1, because contractions are Meir-Keeler contractions.

We recall that, given a π‘ž-uniformly smooth and strictly convex Banach space 𝐸 with a generalized duality map π½π‘žβˆΆπΈβ†’πΈβˆ— and 𝐢 a subset of 𝐸, a mapping πΉβˆΆπΆβ†’πΆ is called(1)π‘˜ξ…ž-Lipschitzian, if there exists a constant π‘˜ξ…ž>0 such that ‖𝐹π‘₯βˆ’πΉπ‘¦β€–β‰€π‘˜ξ…žβ€–π‘₯βˆ’π‘¦β€–(1.6) holds for every π‘₯ and π‘¦βˆˆπΆ;(2)πœ‚-strongly monotone, if there exists a constant πœ‚>0 such that ⟨𝐹π‘₯βˆ’πΉπ‘¦,π‘—π‘ž(π‘₯βˆ’π‘¦)⟩β‰₯πœ‚β€–π‘₯βˆ’π‘¦β€–π‘ž(1.7) holds for every π‘₯,π‘¦βˆˆπΆ and π‘—π‘ž(π‘₯βˆ’π‘¦)βˆˆπ½π‘ž(π‘₯βˆ’π‘¦).

In 2010, Ali and Ugwunnadi [5] introduced and considered the following iterative scheme: π‘₯0π‘₯∈𝐻,𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑦𝑛=ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘(𝑛+1)𝑖(𝑛+1)π‘₯𝑛+𝛼𝑛π‘₯𝛾𝑓𝑛,βˆ€π‘›β‰₯1,(1.8) where 𝑇1,𝑇2,…,𝑇𝑁 a family of asymptotically nonexpansive self-mappings of 𝐻 with sequences {1+π‘˜π‘–(𝑛)𝑝(𝑛)}, such that π‘˜π‘–(𝑛)𝑝(𝑛)β†’0 as π‘›β†’βˆž and π‘“βˆΆπ»β†’π» are a contraction mapping with coefficient π›Όβˆˆ(0,1). Let 𝐴 be a strongly positive-bounded linear operator with coefficient 𝛾>0, and 0<𝛾<𝛾/𝛼. They proved the strong convergence of the implicit and explicit schemes for a common fixed point of the family 𝑇1,𝑇2,…,𝑇𝑁, which solves the variational inequality ⟨(π΄βˆ’π›Ύπ‘“)π‘₯,π‘₯βˆ’π‘₯βŸ©β‰€0,forallβ‹‚π‘₯βˆˆπ‘π‘–=1Fix(𝑇𝑖).

Motivated and inspired by the results of Ali and Ugwunnadi [5], we introduced an iterative scheme as follows. for π‘₯1=π‘₯∈𝐢, π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑦𝑛=ξ€·πΌβˆ’πœ‡π›Όπ‘›πΉξ€Έπ‘‡π‘›π‘₯𝑛+𝛼𝑛π‘₯π›Ύπœ™π‘›ξ€Έ,βˆ€π‘›β‰₯1,(1.9) where 𝑇 is an asymptotically nonexpansive self-mapping of 𝐢 with sequences {1+β„Žπ‘›}, such that β„Žπ‘›β†’0 as π‘›β†’βˆž and πœ™βˆΆπΆβ†’πΆ are a Meir-Keeler contraction (MKC, for short). Let 𝐹 is a π‘˜β€²-Lipschitzian and πœ‚-strongly monotone operator with 0<πœ‡<min{(π‘žπœ‚/πΆπ‘ž(π‘˜ξ…ž)π‘ž)1/(π‘žβˆ’1),1}. We will prove the strong convergence of the implicit and explicit schemes for a fixed point of 𝑇, which solves the variational inequality ⟨(π›Ύπœ™βˆ’πœ‡πΉ)𝑝,π½π‘ž(π‘§βˆ’π‘)βŸ©β‰€0, for π‘§βˆˆπΉ(𝑇). Our results improve and extend the results of Ali and Ugwunnadi [5] for an asymptotically nonexpansive mapping in the following aspects:(i)Hilbert space is replaced by a π‘ž-uniformly smooth and strictly convex Banach space;(ii)contractive mapping is replaced by a MKC;(iii)Theorems 3.1 and 4.1 extend the results of Ali and Ugwunnadi [5] from a strongly positive-bounded linear operator 𝐴 to a π‘˜ξ…ž-Lipschitzian and πœ‚-strongly monotone operator 𝐹.

2. Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [6]). Let π‘ž>1 and 𝐸 be a π‘ž-uniformly smooth Banach space, then there exists a constant πΆπ‘ž>0 such that β€–π‘₯+π‘¦β€–π‘žβ‰€β€–π‘₯β€–π‘ž+π‘žβŸ¨π‘¦,π‘—π‘ž(π‘₯)⟩+πΆπ‘žβ€–π‘¦β€–π‘ž,βˆ€π‘₯,π‘¦βˆˆπΈ.(2.1)

Lemma 2.2 (see [7, Lemma 2.3]). Let πœ™ be a MKC on a convex subset 𝐢 of a Banach space 𝐸. Then for each πœ€>0, there exists π‘Ÿβˆˆ(0,1) such that β€–π‘₯βˆ’π‘¦β€–β‰₯πœ€impliesβ€–πœ™π‘₯βˆ’πœ™π‘¦β€–β‰€π‘Ÿβ€–π‘₯βˆ’π‘¦β€–βˆ€π‘₯,π‘¦βˆˆπΆ.(2.2)

Lemma 2.3 (see [8]). Let {π‘₯𝑛} and {𝑧𝑛} be bounded sequences in a Banach space 𝐸 and {𝛾𝑛} be a sequence in [0,1] which satisfies the following condition: 0<liminfπ‘›β†’βˆžπ›Ύπ‘›β‰€limsupπ‘›β†’βˆžπ›Ύπ‘›<1.(2.3) Suppose that π‘₯𝑛+1=𝛾𝑛π‘₯𝑛+(1βˆ’π›Ύπ‘›)𝑧𝑛,𝑛β‰₯0, and limsupπ‘›β†’βˆž(‖𝑧𝑛+1βˆ’π‘§π‘›β€–βˆ’β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖)≀0. Then limπ‘›β†’βˆžβ€–π‘§π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.4 (see [9, 10]). Let {𝑠𝑛} be a sequence of nonnegative real numbers satisfying 𝑠𝑛+1≀1βˆ’πœ†π‘›ξ€Έπ‘ π‘›+πœ†π‘›π›Ώπ‘›+𝛾𝑛,𝑛β‰₯0,(2.4) where {πœ†π‘›}, {𝛿𝑛} and {𝛾𝑛} satisfy the following conditions: (𝑖){πœ†π‘›}βŠ‚[0,1] and βˆ‘βˆžπ‘›=0πœ†π‘›=∞, (𝑖𝑖)limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0 or βˆ‘βˆžπ‘›=0πœ†π‘›π›Ώπ‘›<∞,(𝑖𝑖𝑖)π›Ύπ‘›βˆ‘β‰₯0(𝑛β‰₯0),βˆžπ‘›=0𝛾𝑛<∞. Then limπ‘›β†’βˆžπ‘ π‘›=0.

Lemma 2.5 (see [11]). Let 𝐢 be a nonempty closed convex subset of a uniformly convex Banach space 𝐸 and π‘‡βˆΆπΆβ†’πΈ is an asymptotically nonexpansive mapping with 𝐹(𝑇)β‰ βˆ…. Then the mapping πΌβˆ’π‘‡ is demiclosed at zero, that is, π‘₯𝑛⇀π‘₯ and β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖→0, then π‘₯=𝑇π‘₯.

Lemma 2.6. Let 𝐹 be a π‘˜β€²-Lipschitzian and πœ‚-strongly monotone operator on a π‘ž-uniformly smooth Banach space 𝐸 with π‘˜ξ…ž>0,πœ‚>0,0<𝑑<1 and 0<πœ‡<min{(π‘žπœ‚/πΆπ‘ž(π‘˜ξ…ž)π‘ž)1/(π‘žβˆ’1),1}. Then 𝑆=(πΌβˆ’π‘‘πœ‡πΉ)βˆΆπΈβ†’πΈ is a contraction with contractive coefficient 1βˆ’π‘‘πœ and 𝜏=(π‘žπœ‡πœ‚βˆ’πΆπ‘žπœ‡π‘ž(π‘˜ξ…ž)π‘ž)/π‘ž.

Proof. From (2.1), we have ‖𝑆π‘₯βˆ’π‘†π‘¦β€–π‘ž=β€–π‘₯βˆ’π‘¦βˆ’π‘‘πœ‡(𝐹π‘₯βˆ’πΉπ‘¦)β€–π‘žβ‰€β€–π‘₯βˆ’π‘¦β€–π‘žξ«+π‘žβˆ’π‘‘πœ‡(𝐹π‘₯βˆ’πΉπ‘¦),π½π‘ž(π‘₯βˆ’π‘¦)+πΆπ‘žβ€–βˆ’π‘‘πœ‡(𝐹π‘₯βˆ’πΉπ‘¦)β€–π‘žβ‰€β€–π‘₯βˆ’π‘¦β€–π‘žβˆ’π‘‘π‘žπœ‡πœ‚β€–π‘₯βˆ’π‘¦β€–π‘ž+π‘‘πΆπ‘žπœ‡π‘žξ€·π‘˜ξ…žξ€Έπ‘žβ€–π‘₯βˆ’π‘¦β€–π‘ž=ξ€Ίξ€·1βˆ’π‘‘π‘žπœ‡πœ‚βˆ’πΆπ‘žπœ‡π‘žξ€·π‘˜ξ…žξ€Έπ‘žξ€Έξ€»β€–π‘₯βˆ’π‘¦β€–π‘žβ‰€ξƒ¬1βˆ’π‘‘π‘žπœ‡πœ‚βˆ’πΆπ‘žπœ‡π‘žξ€·π‘˜ξ…žξ€Έπ‘žπ‘žξƒ­π‘žβ€–π‘₯βˆ’π‘¦β€–π‘ž=(1βˆ’π‘‘πœ)π‘žβ€–π‘₯βˆ’π‘¦β€–π‘ž,(2.5) where 𝜏=(π‘žπœ‡πœ‚βˆ’πΆπ‘žπœ‡π‘ž(π‘˜ξ…ž)π‘ž)/π‘ž, and ‖𝑆π‘₯βˆ’π‘†π‘¦β€–β‰€(1βˆ’π‘‘πœ)β€–π‘₯βˆ’π‘¦β€–.(2.6) Hence 𝑆 is a contraction with contractive coefficient 1βˆ’π‘‘πœ.

Lemma 2.7 (see [5, Lemma 2.9]). Let π‘‡βˆΆπΈβ†’πΈ be a uniformly Lipschitzian with a Lipschitzian constant 𝐿β‰₯1, that is, there exists a constant 𝐿β‰₯1 such that ‖𝑇𝑛π‘₯βˆ’π‘‡π‘›π‘¦β€–β‰€πΏβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ.(2.7)

Lemma 2.8 (see, e.g., MitrinoviΔ‡ [12, page 63]). Let π‘ž>1. Then the following inequality holds: 1π‘Žπ‘β‰€π‘žπ‘Žπ‘ž+π‘žβˆ’1π‘žπ‘π‘ž/(π‘žβˆ’1),(2.8) for arbitrary positive real numbers π‘Ž,𝑏.

3. Main Result

Theorem 3.1. Let 𝐸 be a π‘ž-uniformly smooth and strictly convex Banach space, and 𝐢 a nonempty closed convex subset of 𝐸 such that πΆΒ±πΆβŠ‚πΆ and have a weakly sequentially continuous duality mapping π½π‘ž from 𝐸 to πΈβˆ—. Let π‘‡βˆΆπΆβ†’πΆ be an asymptotically nonexpansive mapping with sequences {1+β„Žπ‘›}, such that β„Žπ‘›β†’0 as π‘›β†’βˆž and πΉβˆ—βˆΆ=𝐹(𝑇)β‰ βˆ…. Let 𝐷 be a bounded subset of 𝐢 such that supπ‘₯βˆˆπ·β€–π‘‡π‘›+1π‘₯βˆ’π‘‡π‘›π‘₯β€–β†’0. Let 𝐹 be a π‘˜ξ…ž-Lipschitzian and πœ‚-strongly monotone operator on 𝐢 with 0<πœ‡<min{(π‘žπœ‚/πΆπ‘ž(π‘˜β€²)π‘ž)1/(π‘žβˆ’1),1}, and πœ™ be a MKC on 𝐢 with 0<𝛾<(π‘žπœ‡πœ‚βˆ’πΆπ‘žπœ‡π‘ž(π‘˜β€²)π‘ž)/π‘ž=𝜏. Let {𝛼𝑛} be a sequence in (0,1) satisfying the following conditions:(A1)limπ‘›β†’βˆžπ›Όπ‘›=0;(A2)limπ‘›β†’βˆž(β„Žπ‘›/𝛼𝑛)=0. Let {π‘₯𝑛} be defined by π‘₯𝑛=𝛼𝑛π‘₯π›Ύπœ™π‘›ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯𝑛.(3.1)
Then, {π‘₯𝑛} converges to a fixed point say 𝑝 in πΉβˆ— which solves the variational inequality ⟨(πœ‡πΉβˆ’π›Ύπœ™)𝑝,π½π‘ž(π‘βˆ’π‘§)βŸ©β‰€0,βˆ€π‘§βˆˆπΉβˆ—.(3.2)

Proof. Let π‘βˆˆπΉβˆ—. Since 𝛼𝑛→0 and β„Žπ‘›/𝛼𝑛→0 as π‘›β†’βˆž, then (1βˆ’π›Όπ‘›πœ)(β„Žπ‘›/𝛼𝑛)β†’0 as π‘›β†’βˆž, so βˆƒπ‘0βˆˆπ‘ such that for all 𝑛β‰₯𝑁0, 𝛼𝑛<(π‘˜ξ…ž)βˆ’1 and (1βˆ’π›Όπ‘›πœ)(β„Žπ‘›/𝛼𝑛)<(1/2)(πœβˆ’π›Ύ). Thus, for 𝑛β‰₯𝑁0β€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘ž=βŸ¨π›Όπ‘›ξ€·π‘₯π›Ύπœ™π‘›ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯π‘›βˆ’π‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘=𝛼𝑛π‘₯βŸ¨π›Ύπœ™π‘›ξ€Έβˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€Έξ€·βˆ’π‘βŸ©+βŸ¨πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›ξ€Έπœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘=𝛼𝑛π‘₯βŸ¨π›Ύπœ™π‘›ξ€Έβˆ’π›Ύπœ™(𝑝),π½π‘žξ€·π‘₯π‘›ξ€Έβˆ’π‘βŸ©+π›Όπ‘›βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©ξ€·βˆ’π‘+βŸ¨πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›ξ€Έπœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘β‰€π›Όπ‘›π›Ύβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘ž+ξ€·1βˆ’π›Όπ‘›πœξ€Έξ€·1+β„Žπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘ž+π›Όπ‘›βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©=ξ€Ίβˆ’π‘1βˆ’π›Όπ‘›(ξ€·πœβˆ’π›Ύ)+1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›ξ€»β€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘ž+π›Όπ‘›βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©β‰€βˆ’π‘βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘ξ€·(πœβˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœβ„Žξ€Έξ€·π‘›/π›Όπ‘›ξ€Έβ‰€βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘β‰€β€–β€–π‘₯(πœβˆ’π›Ύ)βˆ’(1/2)(πœβˆ’π›Ύ)2β€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–π‘›β€–β€–βˆ’π‘π‘žβˆ’1.πœβˆ’π›Ύ(3.3) Therefore, β€–β€–π‘₯π‘›β€–β€–β‰€βˆ’π‘2β€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–.πœβˆ’π›Ύ(3.4) Thus, {π‘₯𝑛} is bounded and therefore {πœ™(π‘₯𝑛)} and {πœ‡πΉπ‘‡π‘›π‘₯𝑛} are also bounded. Also from (3.1), we have β€–β€–π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖=𝛼𝑛‖‖π‘₯π›Ύπœ™π‘›ξ€Έβˆ’πœ‡πΉπ‘‡π‘›π‘₯π‘›β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(3.5) From (3.5) and ‖𝑇𝑛+1π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖→0, we obtain ‖‖𝑇𝑛+1π‘₯π‘›βˆ’π‘₯𝑛‖‖≀‖‖𝑇𝑛+1π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖+‖‖𝑇𝑛π‘₯π‘›βˆ’π‘₯π‘›β€–β€–β€–β€–π‘‡βŸΆ0,𝑛+1π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖≀1+β„Ž1‖‖𝑇𝑛π‘₯π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0,(3.6) Thus, ‖‖𝑇π‘₯π‘›βˆ’π‘₯𝑛‖‖≀‖‖𝑇π‘₯π‘›βˆ’π‘‡π‘›+1π‘₯𝑛‖‖+‖‖𝑇𝑛+1π‘₯π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0.(3.7) Since {π‘₯𝑛} is bounded, now assume that 𝑝 is a weak limit point of {π‘₯𝑛} and a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} converges weakly to 𝑝. Then, by Lemma 2.5 and (3.7), we have that 𝑝 is a fixed point of 𝑇, hence π‘βˆˆπΉβˆ—.
Next we observe that the solution of the variational inequality (3.2) in πΉβˆ— is unique. Assume that Μƒπ‘ž,π‘βˆˆπΉβˆ— are solutions of the inequality (3.2), without loss of generality, we may assume that there is a number πœ€ such that β€–π‘βˆ’Μƒπ‘žβ€–β‰₯πœ€. Then by Lemma 2.2, there is a number π‘Ÿ such that β€–πœ™π‘βˆ’πœ™Μƒπ‘žβ€–β‰€π‘Ÿβ€–π‘βˆ’Μƒπ‘žβ€–. From (3.2), we know ⟨(πœ‡πΉβˆ’π›Ύπœ™)𝑝,π½π‘ž(π‘βˆ’Μƒπ‘ž)βŸ©β‰€0,(3.8)⟨(πœ‡πΉβˆ’π›Ύπœ™)Μƒπ‘ž,π½π‘ž(Μƒπ‘žβˆ’π‘)βŸ©β‰€0.(3.9) Adding (3.8) and (3.9), we have ⟨(πœ‡πΉβˆ’π›Ύπœ™)π‘βˆ’(πœ‡πΉβˆ’π›Ύπœ™)Μƒπ‘ž,π½π‘ž(π‘βˆ’Μƒπ‘ž)βŸ©β‰€0.(3.10) Noticing that (πœ‡πΉβˆ’π›Ύπœ™)π‘βˆ’(πœ‡πΉβˆ’π›Ύπœ™)Μƒπ‘ž,π½π‘žξ¬(π‘βˆ’Μƒπ‘ž)=βŸ¨πœ‡πΉπ‘βˆ’πœ‡πΉΜƒπ‘ž,π½π‘ž(π‘βˆ’Μƒπ‘ž)βŸ©βˆ’βŸ¨π›Ύπœ™π‘βˆ’π›Ύπœ™Μƒπ‘ž,π½π‘ž(π‘βˆ’Μƒπ‘ž)⟩β‰₯πœ‡πœ‚β€–π‘βˆ’Μƒπ‘žβ€–π‘žβˆ’π›Ύβ€–πœ™π‘βˆ’πœ™Μƒπ‘žβ€–β€–π‘βˆ’Μƒπ‘žβ€–π‘žβˆ’1β‰₯πœ‡πœ‚β€–π‘βˆ’Μƒπ‘žβ€–π‘žβˆ’π›Ύπ‘Ÿβ€–π‘βˆ’Μƒπ‘žβ€–π‘žβ‰₯(πœ‡πœ‚βˆ’π›Ύπ‘Ÿ)β€–π‘βˆ’Μƒπ‘žβ€–π‘žβ‰₯(πœ‡πœ‚βˆ’π›Ύπ‘Ÿ)πœ€π‘ž>0.(3.11) Therefore 𝑝=Μƒπ‘ž. That is, π‘βˆˆπΉβˆ— is the unique solution of (3.2).
Finally, we show that π‘₯𝑛→𝑝 as π‘›β†’βˆž. From (3.3), we get β€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘žβ‰€π›Όπ‘›βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘π›Όπ‘›ξ€·(πœβˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›=βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘ξ€·(πœβˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœβ„Žξ€Έξ€·π‘›/𝛼𝑛,(3.12) and in particular β€–β€–π‘₯π‘›π‘—β€–β€–βˆ’π‘π‘žβ‰€ξ‚¬π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ‚€π‘₯π‘›π‘—βˆ’π‘ξ‚ξ‚­ξ‚€(πœβˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›π‘—πœβ„Žξ‚ξ‚€π‘›π‘—/𝛼𝑛𝑗.(3.13) Since π‘₯𝑛𝑗⇀𝑝, from the above inequality and π½π‘ž is a weakly sequentially continuous duality mapping, we have π‘₯𝑛𝑗→𝑝 as π‘—β†’βˆž. Next, we show that 𝑝 solves the variational inequality (3.2). Indeed, from the relation π‘₯𝑛=𝛼𝑛π‘₯π›Ύπœ™π‘›ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯𝑛,(3.14) we get (πœ‡πΉβˆ’π›Ύπœ™)π‘₯𝑛1=βˆ’π›Όπ‘›ξ€Ί(πΌβˆ’π‘‡π‘›)π‘₯π‘›βˆ’π›Όπ‘›πœ‡πΉπ‘₯𝑛+π›Όπ‘›πœ‡πΉπ‘‡π‘›π‘₯𝑛.(3.15) So, for any π‘§βˆˆπΉβˆ—ξ«(πœ‡πΉβˆ’π›Ύπœ™)π‘₯𝑛,π½π‘žξ€·π‘₯𝑛1βˆ’π‘§ξ€Έξ¬=βˆ’π›Όπ‘›βŸ¨(πΌβˆ’π‘‡π‘›)π‘₯π‘›βˆ’π›Όπ‘›πœ‡πΉπ‘₯𝑛+π›Όπ‘›πœ‡πΉπ‘‡π‘›π‘₯𝑛,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©1βˆ’π‘§=βˆ’π›Όπ‘›βŸ¨(πΌβˆ’π‘‡π‘›)π‘₯π‘›βˆ’(πΌβˆ’π‘‡π‘›)𝑧,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©βˆ’π‘§+⟨(πœ‡πΉβˆ’πœ‡πΉπ‘‡π‘›)π‘₯𝑛,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©1βˆ’π‘§β‰€βˆ’π›Όπ‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§π‘ž+1𝛼𝑛1+β„Žπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘§π‘ž+⟨(πœ‡πΉβˆ’πœ‡πΉπ‘‡π‘›)π‘₯𝑛,π½π‘žξ€·π‘₯π‘›ξ€ΈβŸ©β‰€β„Žβˆ’π‘§π‘›π›Όπ‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§π‘ž+⟨(πœ‡πΉβˆ’πœ‡πΉπ‘‡π‘›)π‘₯𝑛,π½π‘žξ€·π‘₯π‘›ξ€Έβˆ’π‘§βŸ©.(3.16) Now replacing 𝑛 in (3.16) with 𝑛𝑗 and letting π‘—β†’βˆž, using (πœ‡πΉβˆ’πœ‡πΉπ‘‡π‘›)π‘₯𝑛𝑗→(πœ‡πΉβˆ’πœ‡πΉπ‘‡π‘›)𝑝=0 for π‘βˆˆπΉβˆ—, and the fact that π‘₯𝑛𝑗→𝑝 as π‘—β†’βˆž, we obtain ⟨(πœ‡πΉβˆ’π›Ύπœ™)𝑝,π½π‘ž(π‘βˆ’π‘§)βŸ©β‰€0,βˆ€π‘§βˆˆπΉβˆ—. This implies that π‘βˆˆπΉβˆ— is a solution of the variational inequality (3.2). Every weak limit of {π‘₯𝑛} say 𝑝 belongs to πΉβˆ—. Furthermore, 𝑝 is a strong limit of {π‘₯𝑛} that solves the variational inequality (3.2). As this solution is unique we get that π‘₯𝑛→𝑝 as π‘›β†’βˆž. This completes the proof.

Corollary 3.2. Let 𝐸 be a π‘ž-uniformly smooth and strictly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸 such that πΆΒ±πΆβŠ‚πΆ and have a weakly sequentially continuous duality mapping π½π‘ž from 𝐸 to πΈβˆ—. Let π‘‡βˆΆπΆβ†’πΆ be a nonexpansive mapping. Let {𝛼𝑛} be a sequence in (0, 1) satisfying limπ‘›β†’βˆžπ›Όπ‘›=0. Let πœ™ and 𝐹 be as in Theorem 3.1. For 𝑇, let {π‘₯𝑛} be defined by π‘₯𝑛=𝛼𝑛π‘₯π›Ύπœ™π‘›ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›ξ€Έπœ‡πΉπ‘‡π‘₯𝑛.(3.17) Then, {π‘₯𝑛} converges to a fixed point say 𝑝 in πΉβˆ— which solves the variational inequality (3.2).

4. Explicit Algorithm

Theorem 4.1. Let 𝐸 be a π‘ž-uniformly smooth and strictly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸 such that πΆΒ±πΆβŠ‚πΆ and have a weakly sequentially continuous duality mapping π½π‘ž from 𝐸 to πΈβˆ—. Let π‘‡βˆΆπΆβ†’πΆ be an asymptotically nonexpansive mapping with sequences {1+β„Žπ‘›}, such that β„Žπ‘›β†’0 as π‘›β†’βˆž and πΉβˆ—βˆΆ=𝐹(𝑇)β‰ βˆ…. Let 𝐷 be a bounded subset of 𝐢 such that supπ‘₯βˆˆπ·β€–π‘‡π‘›+1π‘₯βˆ’π‘‡π‘›π‘₯β€–β†’0. Let 𝐹 be a π‘˜ξ…ž-Lipschitzian and πœ‚-strongly monotone operator on 𝐢 with 0<πœ‡<min{(π‘žπœ‚/πΆπ‘ž(π‘˜β€²)π‘ž)1/(π‘žβˆ’1),1}, and πœ™ be a MKC on 𝐢 with 0<𝛾<(π‘žπœ‡πœ‚βˆ’πΆπ‘žπœ‡π‘ž(π‘˜β€²)π‘ž)/π‘ž=𝜏. Let {𝛼𝑛}, {𝛽𝑛} be sequences in (0,1) satisfying the following conditions:(B1)limπ‘›β†’βˆžπ›Όπ‘›=0;(B2)limπ‘›β†’βˆž(β„Žπ‘›/𝛼𝑛)=0;(B3)βˆ‘βˆžπ‘›=1𝛼𝑛=∞;(B4)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1.Then, {π‘₯𝑛} defined by (1.9) converges strongly to a fixed point say 𝑝 in πΉβˆ— which solves the variational inequality (3.2).

Proof. Since 𝛼𝑛→0 and β„Žπ‘›/𝛼𝑛→0 as π‘›β†’βˆž, (1βˆ’π›Όπ‘›πœ)(β„Žπ‘›/𝛼𝑛)β†’0 as π‘›β†’βˆž. Thus, βˆƒπ‘0βˆˆπ‘ such that (1βˆ’π›Όπ‘›πœ)(β„Žπ‘›/𝛼𝑛)<(1/2)(πœβˆ’π›Ύ) and 𝛼𝑛<(π‘˜ξ…ž)βˆ’1, forall𝑛β‰₯𝑁0. For any point π‘βˆˆπΉβˆ— and 𝑛β‰₯𝑁0, ‖‖𝑦𝑛‖‖=β€–β€–π›Όβˆ’π‘π‘›ξ€·ξ€·π‘₯π›Ύπœ™π‘›ξ€Έξ€Έ+ξ€·βˆ’πœ‡πΉπ‘πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘β€–β€–πœ‡πΉβ‰€π›Όπ‘›π›Ύβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘+𝛼𝑛(ξ€·β€–π›Ύπœ™π‘)βˆ’πœ‡πΉπ‘β€–+1βˆ’π›Όπ‘›πœξ€Έξ€·1+β„Žπ‘›ξ€Έβ€–β€–π‘₯𝑛‖‖=ξ€Ίβˆ’π‘1βˆ’π›Όπ‘›ξ€·(πœβˆ’π›Ύ)+1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›ξ€»β€–β€–π‘₯π‘›β€–β€–βˆ’π‘+π›Όπ‘›β€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–.(4.1) But β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘β‰€π›½π‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–.βˆ’π‘(4.2) Therefore, β€–β€–π‘₯𝑛+1β€–β€–β‰€ξ€Ίπ›½βˆ’π‘π‘›+ξ€·1βˆ’π›½π‘›ξ€Έξ€Ί1βˆ’π›Όπ‘›ξ€·(πœβˆ’π›Ύ)+1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›β€–β€–π‘₯ξ€»ξ€»π‘›β€–β€–βˆ’π‘+𝛼𝑛1βˆ’π›½π‘›ξ€Έβ€–=ξ‚Έβ€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξ‚΅ξ€·(πœβˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›π›Όπ‘›β€–β€–π‘₯ξ‚Άξ‚Ήπ‘›β€–β€–βˆ’π‘+𝛼𝑛1βˆ’π›½π‘›ξ€Έβ€–β‰€ξ‚ƒπ›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–1βˆ’π›Όπ‘›ξ€·1βˆ’π›½π‘›ξ€Έξ‚ƒ12β€–β€–π‘₯(πœβˆ’π›Ύ)𝑛‖‖+ξ€·π›Όβˆ’π‘π‘›ξ€·1βˆ’π›½π‘›ξ€Έξ€Έ/2(πœβˆ’π›Ύ)ξ‚»β€–β€–π‘₯(1/2)(πœβˆ’π›Ύ)β€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–β‰€max𝑛‖‖,βˆ’π‘2β€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–ξ‚Ό.πœβˆ’π›Ύ(4.3) By induction, we have β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘β‰€max𝑁0β€–β€–,βˆ’π‘2β€–π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘β€–ξ‚Όπœβˆ’π›Ύ,𝑛β‰₯𝑁0.(4.4) Next we show that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=0.(4.5) From (1.9), 𝑦𝑛+1βˆ’π‘¦π‘›=𝛼𝑛+1ξ€·π‘₯π›Ύπœ™π‘›+1ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›+1ξ€Έπ‘‡πœ‡πΉπ‘›+1π‘₯𝑛+1βˆ’π›Όπ‘›ξ€·π‘₯π›Ύπœ™π‘›ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯𝑛.(4.6) Therefore, ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–=‖‖𝛼𝑛+1π›Ύξ€·πœ™ξ€·π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’πœ™π‘›+𝛼𝑛+1βˆ’π›Όπ‘›ξ€Έξ€·π‘₯π›Ύπœ™π‘›ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›+1ξ€Έπ‘‡πœ‡πΉπ‘›+1π‘₯𝑛+1βˆ’ξ€·πΌβˆ’π›Όπ‘›+1ξ€Έπ‘‡πœ‡πΉπ‘›+1π‘₯𝑛+ξ€·πΌβˆ’π›Όπ‘›+1ξ€Έπ‘‡πœ‡πΉπ‘›+1π‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›+1π‘₯𝑛+ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›+1π‘₯π‘›βˆ’ξ€·πΌβˆ’π›Όπ‘›ξ€Έπ‘‡πœ‡πΉπ‘›π‘₯𝑛‖‖.(4.7) Hence,‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–β‰€π›Όπ‘›+1𝛾‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||𝛼𝑛+1βˆ’π›Όπ‘›||π›Ύβ€–β€–πœ™ξ€·π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›+1πœξ€Έξ€·1+β„Žπ‘›+1ξ€Έβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||𝛼𝑛+1βˆ’π›Όπ‘›||β€–β€–π‘‡β€–πœ‡πΉβ€–π‘›+1π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›πœξ€Έβ€–β€–π‘‡π‘›+1π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖,limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0,(4.8) and by Lemma 2.3limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖=0.(4.9) Thus, from (1.9), β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0asπ‘›βŸΆβˆž.(4.10)
Next, we show that limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖=0.(4.11) Since β€–β€–π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘‡π‘›π‘₯𝑛‖‖=β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+‖‖𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›ξ€·β€–β€–ξ€·π‘₯π›Ύπœ™π‘›ξ€Έβ€–β€–+β€–β€–πœ‡πΉπ‘‡π‘›π‘₯𝑛‖‖.(4.12) Thus, β€–β€–π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖≀11βˆ’π›½π‘›β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝛼𝑛‖‖π‘₯π›Ύπœ™π‘›ξ€Έβ€–β€–+β€–β€–πœ‡πΉπ‘‡π‘›π‘₯𝑛‖‖.(4.13) Hence, limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖=0.(4.14) Since 𝑇 is Lipschitz with constant 𝐿 and for any positive number 𝑛β‰₯1, we have β€–β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖+‖‖𝑇𝑛π‘₯π‘›βˆ’π‘‡π‘›+1π‘₯𝑛‖‖+‖‖𝑇𝑛+1π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘‡π‘›π‘₯𝑛‖‖+‖‖𝑇𝑛π‘₯π‘›βˆ’π‘‡π‘›+1π‘₯𝑛‖‖‖‖𝑇+𝐿𝑛π‘₯π‘›βˆ’π‘₯π‘›β€–β€–βŸΆ0.(4.15) Therefore, limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖=0.(4.16) Next we show that limsupπ‘›β†’βˆžξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›βˆ’π‘ξ€Έξ¬β‰€0,(4.17) where π‘βˆˆπΉβˆ— is the unique solution of inequality (3.2). Let {π‘₯𝑛𝑗} be a subsequence of {π‘₯𝑛} such that limsupπ‘›β†’βˆžξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›βˆ’π‘ξ€Έξ¬=limπ‘—β†’βˆžξ‚¬π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ‚€π‘₯𝑛𝑗.βˆ’π‘ξ‚ξ‚­(4.18) Since {π‘₯𝑛} is bounded, we may also assume that there exists some π‘§βˆˆπΆ such that π‘₯𝑛𝑗⇀𝑧. From (4.11) it follows that π‘₯π‘›π‘—βˆ’π‘‡π‘₯π‘›π‘—βŸΆ0asπ‘—βŸΆβˆž.(4.19) By Lemma 2.5, the weak limit π‘§βˆˆπΆ of {π‘₯𝑛𝑗} is a fixed point of the mapping 𝑇, so this implies that π‘§βˆˆπΉβˆ—. Hence by Theorem 3.1 and π½π‘ž is a weakly sequentially continuous duality mapping, we have limsupπ‘›β†’βˆžβŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘₯π‘›ξ€Έβˆ’π‘βŸ©=βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘ž(π‘§βˆ’π‘)βŸ©β‰€0.(4.20) Finally, we show that β€–π‘₯π‘›βˆ’π‘β€–β†’0. By contradiction, there is a number πœ€0 such that limsupπ‘›β†’βˆžβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘β‰₯πœ€0.(4.21)
Case 1. Fixed πœ€1(πœ€1<πœ€0), if for some 𝑛β‰₯π‘βˆˆβ„• such that β€–π‘₯π‘›βˆ’π‘β€–β‰₯πœ€0βˆ’πœ€1, and for the other 𝑛β‰₯π‘βˆˆβ„• such that β€–π‘₯π‘›βˆ’π‘β€–<πœ€0βˆ’πœ€1.
Let 𝑀𝑛=2π‘žβŸ¨π›Ύπœ™π‘βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›ξ€ΈβŸ©βˆ’π‘ξ€·πœ€0βˆ’πœ€1ξ€Έπ‘ž.(4.22) From (4.20), we know limsupπ‘›β†’βˆžπ‘€π‘›β‰€0. Hence, there is a number 𝑁, when 𝑛>𝑁, we have π‘€π‘›β‰€πœβˆ’π›Ύ. We extract a number 𝑛0β‰₯𝑁 satisfying β€–π‘₯𝑛0βˆ’π‘β€–<πœ€0βˆ’πœ€1, then we estimate β€–π‘₯𝑛0+1βˆ’π‘β€–β€–β€–π‘¦π‘›0β€–β€–βˆ’π‘π‘ž=‖‖𝛼𝑛0ξ€·π‘₯π›Ύπœ™π‘›0ξ€Έ+ξ€·πΌβˆ’πœ‡π›Όπ‘›0𝐹𝑇𝑛0π‘₯𝑛0β€–β€–βˆ’π‘π‘žξ€·=βŸ¨πΌβˆ’πœ‡π›Όπ‘›0𝐹𝑇𝑛0π‘₯𝑛0βˆ’ξ€·πΌβˆ’πœ‡π›Όπ‘›0𝐹𝑝,π½π‘žξ€·π‘¦π‘›0ξ€Έβˆ’π‘βŸ©+𝛼𝑛0π‘₯π›Ύπœ™π‘›0ξ€Έβˆ’π›Ύπœ™(𝑝),π½π‘žξ€·π‘¦π‘›0βˆ’π‘ξ€Έξ¬+𝛼𝑛0βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0ξ€ΈβŸ©β‰€ξ€·βˆ’π‘1βˆ’π›Όπ‘›0πœξ€Έξ€·1+β„Žπ‘›0ξ€Έβ€–β€–π‘₯𝑛0β€–β€–β€–β€–π‘¦βˆ’π‘π‘›0β€–β€–βˆ’π‘π‘žβˆ’1+𝛼𝑛0π›Ύβ€–β€–πœ™ξ€·π‘₯𝑛0ξ€Έβ€–β€–β€–β€–π‘¦βˆ’πœ™(𝑝)𝑛0β€–β€–βˆ’π‘π‘žβˆ’1+𝛼𝑛0ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0<ξ€Ίβˆ’π‘ξ€Έξ¬1βˆ’π›Όπ‘›0ξ€·(πœβˆ’π›Ύ)+1βˆ’π›Όπ‘›0πœξ€Έβ„Žπ‘›0πœ€ξ€»ξ€·0βˆ’πœ€1‖‖𝑦𝑛0β€–β€–βˆ’π‘π‘žβˆ’1+𝛼𝑛0ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0=ξƒ¬βˆ’π‘ξ€Έξ¬1βˆ’π›Όπ‘›0(πœβˆ’π›Ύ)βˆ’1βˆ’π›Όπ‘›0πœξ€Έβ„Žπ‘›0𝛼𝑛0ξ€·πœ€ξƒ­ξƒ­0βˆ’πœ€1‖‖𝑦𝑛0β€–β€–βˆ’π‘π‘žβˆ’1+𝛼𝑛0ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0β‰€ξ‚Έπ›Όβˆ’π‘ξ€Έξ¬1βˆ’π‘›02(ξ‚Ήξ€·πœ€πœβˆ’π›Ύ)0βˆ’πœ€1‖‖𝑦𝑛0β€–β€–βˆ’π‘π‘žβˆ’1+𝛼𝑛0βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0ξ€ΈβŸ©β‰€ξ‚Έπ›Όβˆ’π‘1βˆ’π‘›02ξ‚Ή1(πœβˆ’π›Ύ)π‘žξ€·πœ€0βˆ’πœ€1ξ€Έπ‘ž+π‘žβˆ’1π‘žβ€–β€–π‘¦π‘›0β€–β€–βˆ’π‘π‘ž+𝛼𝑛0ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0β‰€ξ‚Έπ›Όβˆ’π‘ξ€Έξ¬1βˆ’π‘›02ξ‚Ήξ€·πœ€(πœβˆ’π›Ύ)0βˆ’πœ€1ξ€Έπ‘ž+π‘žπ›Όπ‘›0βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0ξ€Έβˆ’π‘βŸ©.(4.23) But β€–β€–π‘₯𝑛0+1β€–β€–βˆ’π‘π‘žβ‰€π›½π‘›0β€–β€–π‘₯𝑛0β€–β€–βˆ’π‘π‘ž+ξ€·1βˆ’π›½π‘›0‖‖𝑦𝑛0β€–β€–βˆ’π‘π‘ž<𝛽𝑛0ξ€·πœ€0βˆ’πœ€1ξ€Έπ‘ž+ξ€·1βˆ’π›½π‘›0‖‖𝑦𝑛0β€–β€–βˆ’π‘π‘ž.(4.24) Therefore, β€–β€–π‘₯𝑛0+1β€–β€–βˆ’π‘π‘ž<𝛽𝑛0ξ€·πœ€0βˆ’πœ€1ξ€Έπ‘ž+ξ€·1βˆ’π›½π‘›0𝛼1βˆ’π‘›02ξ‚Ήξ€·πœ€(πœβˆ’π›Ύ)0βˆ’πœ€1ξ€Έπ‘ž+π‘žπ›Όπ‘›0ξ€·1βˆ’π›½π‘›0ξ€Έξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0=1βˆ’π‘ξ€Έξ¬1βˆ’2𝛼𝑛0ξ€·1βˆ’π›½π‘›0ξ€Έ(ξ‚„ξ€·πœ€πœβˆ’π›Ύ)0βˆ’πœ€1ξ€Έπ‘ž+π‘žπ›Όπ‘›0ξ€·1βˆ’π›½π‘›0ξ€Έξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›0=1βˆ’π‘ξ€Έξ¬1βˆ’2𝛼𝑛0ξ€·1βˆ’π›½π‘›0ξ€Έξ€·(πœβˆ’π›Ύ)βˆ’π‘€π‘›ξ€Έξ‚„ξ€·πœ€0βˆ’πœ€1ξ€Έπ‘ž<ξ€·πœ€0βˆ’πœ€1ξ€Έπ‘ž.(4.25) Hence, we have β€–β€–π‘₯𝑛0+1β€–β€–βˆ’π‘<πœ€0βˆ’πœ€1.(4.26) In the same way, we can get β€–β€–π‘₯π‘›β€–β€–βˆ’π‘<πœ€0βˆ’πœ€1,βˆ€π‘›β‰₯𝑛0.(4.27) It contradicts the limsupπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘β€–β‰₯πœ€0.
Case 2. Fixed πœ€1(πœ€1<πœ€0), if β€–π‘₯π‘›βˆ’π‘β€–β‰₯πœ€0βˆ’πœ€1, for all 𝑛β‰₯π‘βˆˆβ„•, from Lemma 2.2, there is a number π‘Ÿ(0<π‘Ÿ<1) such that β€–β€–πœ™ξ€·π‘₯𝑛‖‖‖‖π‘₯βˆ’πœ™(𝑝)β‰€π‘Ÿπ‘›β€–β€–βˆ’π‘,𝑛β‰₯𝑁.(4.28) It follows (1.9) that β€–β€–π‘¦π‘›β€–β€–βˆ’π‘π‘ž=β€–β€–π›Όπ‘›π›Ύπœ™(π‘₯𝑛)+(πΌβˆ’πœ‡π›Όπ‘›πΉ)𝑇𝑛π‘₯π‘›β€–β€–βˆ’π‘π‘žξ€·=βŸ¨πΌβˆ’πœ‡π›Όπ‘›πΉξ€Έπ‘‡π‘›π‘₯π‘›βˆ’ξ€·πΌβˆ’πœ‡π›Όπ‘›πΉξ€Έπ‘,π½π‘žξ€·π‘¦π‘›ξ€Έβˆ’π‘βŸ©+𝛼𝑛π‘₯βŸ¨π›Ύπœ™π‘›ξ€Έβˆ’π›Ύπœ™(𝑝),π½π‘žξ€·π‘¦π‘›ξ€ΈβŸ©βˆ’π‘+π›Όπ‘›ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›β‰€ξ€·βˆ’π‘ξ€Έξ¬1βˆ’π›Όπ‘›πœξ€Έξ€·1+β„Žπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–β€–β€–π‘¦βˆ’π‘π‘›β€–β€–βˆ’π‘π‘žβˆ’1+π›Όπ‘›π›Ύβ€–β€–πœ™ξ€·π‘₯π‘›ξ€Έβ€–β€–β€–β€–π‘¦βˆ’πœ™(𝑝)π‘›β€–β€–βˆ’π‘π‘žβˆ’1+π›Όπ‘›ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›<ξ€Ίβˆ’π‘ξ€Έξ¬1βˆ’π›Όπ‘›ξ€·(πœβˆ’π›Ύπ‘Ÿ)+1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›ξ€»β€–β€–π‘₯π‘›β€–β€–β€–β€–π‘¦βˆ’π‘π‘›β€–β€–βˆ’π‘π‘žβˆ’1+π›Όπ‘›βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›ξ€ΈβŸ©=ξ‚Έβˆ’π‘1βˆ’π›Όπ‘›ξ‚Έξ€·(πœβˆ’π›Ύπ‘Ÿ)βˆ’1βˆ’π›Όπ‘›πœξ€Έβ„Žπ‘›π›Όπ‘›β€–β€–π‘₯ξ‚Ήξ‚Ήπ‘›β€–β€–β€–β€–π‘¦βˆ’π‘π‘›β€–β€–βˆ’π‘π‘žβˆ’1+π›Όπ‘›ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›β‰€ξ‚ƒπ›Όβˆ’π‘ξ€Έξ¬1βˆ’π‘›2ξ‚„β€–β€–π‘₯(πœβˆ’π›Ύπ‘Ÿ)π‘›β€–β€–β€–β€–π‘¦βˆ’π‘π‘›β€–β€–βˆ’π‘π‘žβˆ’1+π›Όπ‘›βŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›ξ€ΈβŸ©β‰€ξ‚ƒπ›Όβˆ’π‘1βˆ’π‘›2ξ‚„1(πœβˆ’π›Ύπ‘Ÿ)π‘žβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘ž+π‘žβˆ’1π‘žβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘π‘ž+π›Όπ‘›ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›β‰€ξ‚ƒπ›Όβˆ’π‘ξ€Έξ¬1βˆ’π‘›2ξ‚„β€–β€–π‘₯(πœβˆ’π›Ύπ‘Ÿ)π‘›β€–β€–βˆ’π‘π‘ž+π‘žπ›Όπ‘›ξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›.βˆ’π‘ξ€Έξ¬(4.29) Therefore, β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘π‘žβ‰€π›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘π‘ž+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’π‘π‘ž<𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘π‘ž+ξ€·1βˆ’π›½π‘›ξ€Έξ‚ƒπ›Ό1βˆ’π‘›2ξ‚„β€–β€–π‘₯(πœβˆ’π›Ύπ‘Ÿ)π‘›β€–β€–βˆ’π‘π‘ž+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›π‘žβŸ¨π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›ξ€ΈβŸ©=1βˆ’π‘1βˆ’2𝛼𝑛1βˆ’π›½π‘›ξ€Έ(ξ‚„β€–β€–π‘₯πœβˆ’π›Ύπ‘Ÿ)π‘›β€–β€–βˆ’π‘π‘ž+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›π‘žξ«π›Ύπœ™(𝑝)βˆ’πœ‡πΉπ‘,π½π‘žξ€·π‘¦π‘›.βˆ’π‘ξ€Έξ¬(4.30) By Lemma 2.4, we have that β€–π‘₯π‘›βˆ’π‘β€–β†’0 as π‘›β†’βˆž. It contradicts the β€–π‘₯π‘›βˆ’π‘β€–β‰₯πœ€0βˆ’πœ€1. This completes the proof.

The following corollary follows from Theorem 4.1.

Corollary 4.2. Let 𝐸 be a π‘ž-uniformly smooth and strictly convex Banach space, and let 𝐢 be a nonempty closed convex subset of 𝐸 such that πΆΒ±πΆβŠ‚πΆ and have a weakly sequentially continuous duality mapping π½π‘ž from 𝐸 to πΈβˆ—. Let 𝑇,πΉβˆ—,𝐹,πœ™ and {𝛼𝑛} be as in Corollary 3.2. Let {π‘₯𝑛} be defined by π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑦𝑛=ξ€·πΌβˆ’πœ‡π›Όπ‘›πΉξ€Έπ‘‡π‘₯𝑛+𝛼𝑛π‘₯π›Ύπœ™π‘›ξ€Έ,βˆ€π‘›β‰₯0,(4.31) Then, {π‘₯𝑛} converges strongly to a fixed point say 𝑝 in πΉβˆ— which solves the variational inequality (3.2).

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Copyright © 2012 Meng Wen 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.

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