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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 658905, 11 pages
http://dx.doi.org/10.1155/2012/658905
Research Article

Two General Algorithms for Computing Fixed Points of Nonexpansive Mappings in Banach Spaces

School of Mathematical Sciences, Yancheng Teachers University, Jiangsu, Yancheng 224051, China

Received 16 January 2012; Accepted 23 February 2012

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Shuang Wang. 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

Recently, Yao et al. (2011) introduced two algorithms for solving a system of nonlinear variational inequalities. In this paper, we consider two general algorithms and obtain the extension results for computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points solve the same system of nonlinear variational inequalities.

1. Introduction

Let 𝑋 be a real Banach space and let 𝐢 be a nonempty closed convex subset of 𝑋. Recall that a mapping π‘‡βˆΆπΆβ†’πΆ is said to be nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–, for all π‘₯,π‘¦βˆˆπΆ. We denote by Fix(𝑇) the set of fixed points of 𝑇.

Recently, Yao et al. [1] considered the following algorithms:π‘₯𝑑=Π𝐢(πΌβˆ’π‘‘πΉ)Π𝐢(πΌβˆ’πœ†π΄)Π𝐢(πΌβˆ’πœ‡π΅)π‘₯𝑑,(1.1) and for an arbitrary point π‘₯0∈𝐢,π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€ΈΞ πΆξ€·πΌβˆ’π›Όπ‘›πΉξ€ΈΞ πΆ(πΌβˆ’πœ†π΄)Π𝐢(πΌβˆ’πœ‡π΅)π‘₯𝑛,𝑛β‰₯0,(1.2) where Ξ πΆβˆΆπ‘‹β†’πΆ is a sunny nonexpansive retraction, πΉβˆΆπΆβ†’π‘‹ is a strongly positive bounded linear operator and 𝐴,π΅βˆΆπΆβ†’π‘‹ are 𝛼-inverse-strongly accretive and 𝛽-inverse-strongly accretive operators, respectively. They proved that the {π‘₯𝑑} defined by (1.1) and {π‘₯𝑛} defined by (1.2) converge strongly to a unique solution π‘₯ of the variational inequality ⟨𝐹(π‘₯),𝑗(π‘₯βˆ’π‘§)βŸ©β‰€0. Furthermore, they proved that the above algorithms converge strongly to some solutions of a system of nonlinear inequalities, which involves finding (π‘₯βˆ—,π‘¦βˆ—)βˆˆπΆΓ—πΆ such thatξ«πœ†π΄π‘¦βˆ—+π‘₯βˆ—βˆ’π‘¦βˆ—ξ€·,𝑗π‘₯βˆ’π‘₯βˆ—ξ«ξ€Έξ¬β‰₯0,βˆ€π‘₯∈𝐢,πœ‡π΅π‘₯βˆ—+π‘¦βˆ—βˆ’π‘₯βˆ—ξ€·,𝑗π‘₯βˆ’π‘¦βˆ—ξ€Έξ¬β‰₯0,βˆ€π‘₯∈𝐢.(1.3) For related works, please see [2–5] and the references therein.

In this paper, we introduce two general algorithms (3.3) and (3.22) (defined below) and prove that the proposed algorithms strongly converge to π‘₯βˆ—βˆˆFix(𝑇) which solves the variational inequality ⟨𝐹π‘₯βˆ—,𝑗(π‘₯βˆ—βˆ’π‘’)βŸ©β‰€0,π‘’βˆˆFix(𝑇), where πΉβˆΆπΆβ†’π‘‹ is a 𝛽-Lipschitzian and πœ‚-strongly accretive operator. It is worth pointing out that our proofs contain some new techniques.

2. Preliminaries

Let 𝑋 be a real Banach space with norm β€–β‹…β€– and let π‘‹βˆ— be its dual space. The value of π‘“βˆˆπ‘‹βˆ— and π‘₯βˆˆπ‘‹ will be denoted by ⟨π‘₯,π‘“βŸ©. For the sequence {π‘₯𝑛} in 𝑋, we write π‘₯𝑛⇀π‘₯ to indicate that the sequence {π‘₯𝑛} converges weakly to π‘₯. π‘₯𝑛→π‘₯ means that {π‘₯𝑛} converges strongly to π‘₯.

Let πœ‚>0, a mapping 𝐹 of 𝐢 into 𝑋 is said to be πœ‚-strongly accretive if there exists 𝑗(π‘₯βˆ’π‘¦)∈𝐽(π‘₯βˆ’π‘¦) such that𝐹π‘₯βˆ’ξ‚­πΉπ‘¦,𝑗(π‘₯βˆ’π‘¦)β‰₯πœ‚β€–π‘₯βˆ’π‘¦β€–2,(2.1) for all π‘₯,π‘¦βˆˆπΆ. A mapping 𝐹 from 𝐢 into 𝑋 is said to be 𝛽-Lipschitzian if, for 𝛽>0,‖‖𝐹π‘₯βˆ’β€–β€–πΉπ‘¦β‰€π›½β€–π‘₯βˆ’π‘¦β€–,(2.2) for all π‘₯,π‘¦βˆˆπΆ. From the definition of 𝐹 (see [1]), we note that a strongly positive bounded linear operator 𝐹 is a ‖𝐹‖-Lipschitzian and 𝛾-strongly accretive operator.

Let π‘ˆ={π‘₯βˆˆπ‘‹βˆΆβ€–π‘₯β€–=1}. A Banach space 𝑋 is said to be uniformly convex if for each πœ–βˆˆ(0,2], there exists 𝛿>0 such that for any π‘₯,π‘¦βˆˆπ‘ˆ,β€–β€–β€–β€–π‘₯βˆ’π‘¦β€–β‰₯πœ–impliesπ‘₯+𝑦2‖‖‖≀1βˆ’π›Ώ.(2.3) It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space 𝑋 is said to be smooth if the limitlim𝑑→0β€–π‘₯+π‘‘π‘¦β€–βˆ’β€–π‘₯‖𝑑,(2.4) exists for all π‘₯,π‘¦βˆˆπ‘ˆ. It is said to be uniformly smooth if the limit (2.4) is attained uniformly for π‘₯,π‘¦βˆˆπ‘ˆ. Also, we define a function 𝜌∢[0,∞)β†’[0,∞) called the modulus of smoothness of 𝑋 as follows:1𝜌(𝜏)=sup2(.β€–π‘₯+𝑦‖+β€–π‘₯βˆ’π‘¦β€–)βˆ’1∢π‘₯,π‘¦βˆˆπ‘‹,β€–π‘₯β€–=1,‖𝑦‖=𝜏(2.5) It is known that 𝑋 is uniformly smooth if and only if limπœβ†’0𝜌(𝜏)/𝜏=0. Let π‘ž be a fixed real number with 1<π‘žβ‰€2. Then a Banach space 𝑋 is said to be π‘ž-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)β‰€π‘πœπ‘ž for all 𝜏>0.

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

Lemma 2.1 (see [6]). Let π‘ž be a given real number with 1<π‘žβ‰€2 and let 𝑋 be a π‘ž-uniformly smooth Banach space. Then β€–π‘₯+π‘¦β€–π‘žβ‰€β€–π‘₯β€–π‘žξ«+π‘žπ‘¦,π½π‘žξ¬(π‘₯)+2β€–πΎπ‘¦β€–π‘ž,(2.6) for all π‘₯,π‘¦βˆˆπ‘‹, where 𝐾 is the π‘ž-uniformly smooth constant of 𝑋 and π½π‘ž is the generalized duality mapping from 𝑋 into 2π‘‹βˆ— defined by π½π‘žξ€½(π‘₯)=π‘“βˆˆπ‘‹βˆ—βˆΆβŸ¨π‘₯,π‘“βŸ©=β€–π‘₯β€–π‘ž,‖𝑓‖=β€–π‘₯β€–π‘žβˆ’1ξ€Ύ,(2.7) for all π‘₯βˆˆπ‘‹.

Lemma 2.2 (see [7]). Let 𝐢 be a closed convex subset of a smooth Banach space 𝑋, let 𝐷 be a nonempty subset of 𝐢 and Ξ  be a retraction from 𝐢 onto 𝐷. Then Ξ  is sunny and nonexpansive if and only if βŸ¨π‘’βˆ’Ξ (𝑒),𝑗(π‘¦βˆ’Ξ (𝑒))βŸ©β‰€0,(2.8) for all π‘’βˆˆπΆ and π‘¦βˆˆπ·.

Lemma 2.3 (see [8]). Let 𝐢 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝑋 and let 𝑇 be a nonexpansive mapping of 𝐢 into itself. If {π‘₯𝑛} is a sequence of 𝐢 such that π‘₯𝑛⇀π‘₯ and π‘₯π‘›βˆ’π‘‡π‘₯𝑛→0, then π‘₯ is a fixed point of 𝑇.

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,(2.9) 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‘›<∞, and (iii) 𝛾𝑛β‰₯0(𝑛β‰₯0), βˆ‘βˆžπ‘›=0𝛾𝑛<∞. Then limπ‘›β†’βˆžπ‘ π‘›=0.

Lemma 2.5 (see [11]). Let {π‘₯𝑛} and {𝑧𝑛} be bounded sequences in 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.(2.10) 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.

In addition, we need the following extension of Lemma 2.5 in Wang and Hu [2] in a 2-uniformly smooth Banach space.

Lemma 2.6. Let 𝐢 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let πΉβˆΆπΆβ†’π‘‹ be a 𝛽-Lipschitzian and πœ‚-strongly accretive operator with √0<πœ‚β‰€2𝛽𝐾 and 0<𝑑<πœ‚/2𝛽2𝐾2. Then 𝑆=(πΌβˆ’π‘‘πΉ)βˆΆπΆβ†’π‘‹ is a contraction with contraction coefficient πœπ‘‘=√1βˆ’2𝑑(πœ‚βˆ’π‘‘π›½2𝐾2).

Proof. By Lemma 2.1, we have ‖𝑆π‘₯βˆ’π‘†π‘¦β€–2=β€–β€–ξ‚€(π‘₯βˆ’π‘¦)βˆ’π‘‘πΉπ‘₯βˆ’ξ‚β€–β€–πΉπ‘¦2=β€–π‘₯βˆ’π‘¦β€–2ξ‚¬βˆ’2𝑑𝐹π‘₯βˆ’ξ‚­πΉπ‘¦,𝑗(π‘₯βˆ’π‘¦)+2𝑑2𝐾2‖‖𝐹π‘₯βˆ’β€–β€–πΉπ‘¦2≀‖π‘₯βˆ’π‘¦β€–2βˆ’2π‘‘πœ‚β€–π‘₯βˆ’π‘¦β€–2+2𝑑2𝛽2𝐾2β€–π‘₯βˆ’π‘¦β€–2=ξ€Ίξ€·1βˆ’2π‘‘πœ‚βˆ’π‘‘π›½2𝐾2ξ€Έξ€»β€–π‘₯βˆ’π‘¦β€–2,(2.11) for all π‘₯,π‘¦βˆˆπΆ. From √0<πœ‚β‰€2𝛽𝐾 and 0<𝑑<πœ‚/2𝛽2𝐾2, we have ‖𝑆π‘₯βˆ’π‘†π‘¦β€–β‰€πœπ‘‘β€–π‘₯βˆ’π‘¦β€–,(2.12) where πœπ‘‘=√1βˆ’2𝑑(πœ‚βˆ’π‘‘π›½2𝐾2)∈(0,1). Hence 𝑆 is a contraction with contraction coefficient πœπ‘‘.

3. Main Results

Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let π‘‡βˆΆπΆβ†’πΆ be a nonexpansive mapping with Fix(𝑇)β‰ βˆ…. Let πΉβˆΆπΆβ†’π‘‹ be a 𝛽-Lipschitzian and πœ‚-strongly accretive operator with √0<πœ‚β‰€2𝛽𝐾. Let π‘‘βˆˆ(0,πœ‚/2𝛽2𝐾2) and πœπ‘‘=√1βˆ’2𝑑(πœ‚βˆ’π‘‘π›½2𝐾2), consider a mapping 𝑆𝑑 on 𝐢 defined by𝑆𝑑π‘₯=Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯,π‘₯∈𝐢,(3.1) where Π𝐢 is a sunny nonexpansive retraction from 𝑋 onto 𝐢. It is easy to see that 𝑆𝑑 is a contraction. Indeed, from Lemma 2.6, we have‖‖𝑆𝑑π‘₯βˆ’π‘†π‘‘π‘¦β€–β€–β‰€β€–β€–Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯βˆ’Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚β€–β€–β‰€β€–β€–ξ‚€π‘‡π‘¦πΌβˆ’π‘‘πΉξ‚ξ‚€π‘‡π‘₯βˆ’πΌβˆ’π‘‘πΉξ‚β€–β€–π‘‡π‘¦β‰€πœπ‘‘β€–π‘‡π‘₯βˆ’π‘‡π‘¦β€–β‰€πœπ‘‘β€–π‘₯βˆ’π‘¦β€–,(3.2) for all π‘₯,π‘¦βˆˆπΆ. Therefore, the following implicit method is well defined:π‘₯𝑑=Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑,π‘₯π‘‘βˆˆπΆ.(3.3)

Theorem 3.1. The net {π‘₯𝑑} generated by the implicit method (3.3) converges in norm, as 𝑑→0+ to the unique solution π‘₯βˆ—βˆˆFix(𝑇) of the variational inequality: 𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—ξ€Έξ‚­βˆ’π‘’β‰€0,π‘’βˆˆFix(𝑇).(3.4)

Proof. We first show that the solution set of (3.4) is singleton. As a matter of fact, we assume that π‘₯βˆ—βˆˆFix(𝑇) and Μƒπ‘₯∈Fix(𝑇) both are solutions to (3.4), then 𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—ξ€Έξ‚­ξ‚¬βˆ’Μƒπ‘₯≀0,(3.5)𝐹̃π‘₯,𝑗̃π‘₯βˆ’π‘₯βˆ—ξ€Έξ‚­β‰€0.(3.6) Adding (3.5) to (3.6), we get 𝐹π‘₯βˆ—βˆ’ξ€·π‘₯𝐹̃π‘₯,π‘—βˆ—ξ€Έξ‚­βˆ’Μƒπ‘₯≀0.(3.7) The strong accretive of 𝐹 implies that π‘₯βˆ—=Μƒπ‘₯, and the uniqueness is proved. Below we use π‘₯βˆ—βˆˆFix(𝑇) to denote the unique solution of (3.4).
Next, we prove that {π‘₯𝑑} is bounded. Taking π‘’βˆˆFix(𝑇), from (3.3) and using Lemma 2.6, we haveβ€–β€–π‘₯𝑑‖‖=β€–β€–Ξ βˆ’π‘’πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’Ξ πΆπ‘’β€–β€–β‰€β€–β€–ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘’βˆ’π‘‘β€–β€–β‰€β€–β€–ξ‚€πΉπ‘‡π‘’πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚β€–β€–β€–β€–π‘‡π‘’+π‘‘β€–β€–πΉπ‘’β‰€πœπ‘‘β€–β€–π‘₯π‘‘β€–β€–β€–β€–βˆ’π‘’+𝑑‖‖,𝐹𝑒(3.8) that is, β€–β€–π‘₯π‘‘β€–β€–β‰€π‘‘βˆ’π‘’1βˆ’πœπ‘‘β€–β€–β€–β€–.𝐹𝑒(3.9) Observe that lim𝑑→0+𝑑1βˆ’πœπ‘‘=1πœ‚.(3.10) From 𝑑→0+, we may assume, without loss of generality, that π‘‘β‰€πœ‚/2𝛽2𝐾2βˆ’πœ–, where πœ– is an arbitrarily small positive number. Thus, we have 𝑑/(1βˆ’πœπ‘‘) to be continuous, for all π‘‘βˆˆ[0,πœ‚/2𝛽2𝐾2βˆ’πœ–]. Therefore, we obtain 𝑀1𝑑=sup1βˆ’πœπ‘‘ξ‚΅πœ‚βˆΆπ‘‘βˆˆ0,2𝛽2𝐾2βˆ’πœ–ξ‚Ήξ‚Ό<+∞.(3.11) From (3.9) and (3.11), we have {π‘₯𝑑} bounded and so is {𝐹𝑇π‘₯𝑑}.
On the other hand, from (3.3), we obtainβ€–β€–π‘₯π‘‘βˆ’π‘‡π‘₯𝑑‖‖=β€–β€–Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’Ξ πΆπ‘‡π‘₯π‘‘β€–β€–β‰€β€–β€–ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’π‘‡π‘₯𝑑‖‖‖‖=𝑑𝐹𝑇π‘₯π‘‘β€–β€–ξ€·βŸΆ0π‘‘βŸΆ0+ξ€Έ.(3.12)
Next, we show that {π‘₯𝑑} is relatively norm-compact as 𝑑→0+. Assume that {𝑑𝑛}∈(0,πœ‚/2𝛽2𝐾2) such that 𝑑𝑛→0+ as π‘›β†’βˆž. Put π‘₯π‘›βˆΆ=π‘₯𝑑𝑛. It follows from (3.12) thatβ€–β€–π‘₯π‘›βˆ’π‘‡π‘₯π‘›β€–β€–βŸΆ0(π‘›βŸΆβˆž).(3.13)
For a given π‘’βˆˆFix(𝑇), by (3.3) and using Lemma 2.2, we haveπ‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑π‘₯,π‘—π‘‘ξ€Έξ‚­βˆ’π‘’β‰€0.(3.14) By (3.14) and using Lemma 2.6, we have β€–β€–π‘₯π‘‘β€–β€–βˆ’π‘’2=π‘₯𝑑π‘₯βˆ’π‘’,𝑗𝑑=π‘₯βˆ’π‘’ξ€Έξ¬π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑π‘₯,𝑗𝑑+βˆ’π‘’ξ‚¬ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑π‘₯βˆ’π‘’,π‘—π‘‘ξ€Έξ‚­β‰€βˆ’π‘’ξ‚¬ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑π‘₯βˆ’π‘’,π‘—π‘‘ξ€Έξ‚­β‰€βˆ’π‘’ξ‚¬ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚ξ€·π‘₯𝑇𝑒,π‘—π‘‘ξ€Έξ‚­ξ‚¬βˆ’π‘’+𝑑𝐹𝑒,π‘—π‘’βˆ’π‘₯π‘‘ξ€Έξ‚­β‰€πœπ‘‘β€–β€–π‘₯π‘‘β€–β€–βˆ’π‘’2+𝑑𝐹𝑒,π‘—π‘’βˆ’π‘₯𝑑,(3.15) that is, β€–β€–π‘₯π‘‘β€–β€–βˆ’π‘’2≀𝑑1βˆ’πœπ‘‘ξ‚¬ξ€·πΉπ‘’,π‘—π‘’βˆ’π‘₯𝑑≀𝑀1𝐹𝑒,π‘—π‘’βˆ’π‘₯𝑑.(3.16) In particular, β€–β€–π‘₯π‘›β€–β€–βˆ’π‘’2≀𝑀1𝐹𝑒,π‘—π‘’βˆ’π‘₯𝑛.(3.17)
Since {π‘₯𝑑} is bounded, without loss of generality, we may assume that {π‘₯𝑛} converges weakly to a point Μƒπ‘₯. Noticing (3.13) we can use Lemma 2.3 to get Μƒπ‘₯∈Fix(𝑇). Therefore we can substitute Μƒπ‘₯ for 𝑒 in (3.17) to getβ€–β€–π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯≀𝑀1𝐹̃π‘₯,𝑗̃π‘₯βˆ’π‘₯𝑛.(3.18) Consequently, the weak convergence of {π‘₯𝑛} to Μƒπ‘₯ actually implies that π‘₯𝑛→̃π‘₯. This has proved the relative norm compactness of the net {π‘₯𝑑} as 𝑑→0+.
We next show that Μƒπ‘₯ solves the variational inequality (3.4). Observe thatπ‘₯𝑑=Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘₯𝑑+ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑+π‘₯π‘‘βˆ’π‘‘πΉξ€·π‘₯π‘‘ξ€ΈβŸΉπΉξ€·π‘₯𝑑=1π‘‘ξ‚ƒΞ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘₯𝑑+ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑.(3.19) For any π‘’βˆˆFix(𝑇), we have ⟨𝐹π‘₯𝑑π‘₯,𝑗𝑑1βˆ’π‘’βŸ©=π‘‘ξ‚¬Ξ πΆξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑π‘₯,π‘—π‘‘ξ€Έξ‚­βˆ’1βˆ’π‘’π‘‘ξ‚¬ξ‚€πΌβˆ’π‘‘πΉξ‚π‘₯π‘‘βˆ’ξ‚€πΌβˆ’π‘‘πΉξ‚π‘‡π‘₯𝑑π‘₯,𝑗𝑑1βˆ’π‘’β‰€βˆ’π‘‘ξ«π‘₯π‘‘βˆ’π‘‡π‘₯𝑑π‘₯,𝑗𝑑+ξ‚¬βˆ’π‘’ξ€Έξ¬πΉπ‘₯π‘‘βˆ’πΉπ‘‡π‘₯𝑑π‘₯,𝑗𝑑1βˆ’π‘’β‰€βˆ’π‘‘ξ«(πΌβˆ’π‘‡)π‘₯𝑑π‘₯βˆ’(πΌβˆ’π‘‡)𝑒,𝑗𝑑‖‖π‘₯βˆ’π‘’ξ€Έξ¬+π›½π‘‘βˆ’π‘‡π‘₯𝑑‖‖‖‖π‘₯π‘‘β€–β€–βˆ’π‘’β‰€π›½π‘€2β€–β€–π‘₯π‘‘βˆ’π‘‡π‘₯𝑑‖‖,(3.20) where 𝑀2=sup{β€–π‘₯π‘‘βˆ’π‘’β€–,π‘‘βˆˆ(0,πœ‚/2𝛽2𝐾2)}.
Now replacing 𝑑 in (3.20) with 𝑑𝑛 and letting π‘›β†’βˆž, we have𝐹̃π‘₯,𝑗(Μƒπ‘₯βˆ’π‘’)≀0.(3.21) That is, Μƒπ‘₯∈Fix(𝑇) is a solution of (3.4), hence Μƒπ‘₯=π‘₯βˆ— by uniqueness. In summary, we have shown that each cluster point of {π‘₯𝑑} (at 𝑑→0) equals π‘₯βˆ—. Therefore, π‘₯𝑑→π‘₯βˆ— as 𝑑→0.

Theorem 3.2. Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋 with a weakly sequentially continuous duality mapping 𝑗. Let πΉβˆΆπΆβ†’π‘‹ be a 𝛽-Lipschitzian and πœ‚-strongly accretive operator with √0<πœ‚β‰€2𝛽𝐾. Suppose that π‘‡βˆΆπΆβ†’πΆ is a nonexpansive mapping with Fix(𝑇)β‰ βˆ…. Let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let {𝛼𝑛} and {𝛽𝑛} be two real sequences in (0,1) and satisfy the conditions:(A1)limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=0𝛼𝑛=∞,(A2)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1.
For given π‘₯1∈𝐢 arbitrarily, let the sequence {π‘₯𝑛} be generated by 𝑦𝑛=Ξ πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯𝑛,π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›,𝑛β‰₯0.(3.22) Then the sequence {π‘₯𝑛} strongly converges to π‘₯βˆ—βˆˆFix(𝑇) which solves the variational inequality (3.4).

Proof. We proceed with the following steps.
Step 1. We claim that {π‘₯𝑛} is bounded. From limπ‘›β†’βˆžπ›Όπ‘›=0, we may assume, without loss of generality, that 0<π›Όπ‘›β‰€πœ‚/2𝛽2𝐾2βˆ’πœ– for all 𝑛. In fact, let π‘’βˆˆFix(𝑇), from (3.22) and using Lemma 2.6, we have ‖‖𝑦𝑛‖‖=β€–β€–Ξ βˆ’π‘’πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’Ξ πΆπ‘’β€–β€–β‰€β€–β€–ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘’βˆ’π›Όπ‘›β€–β€–πΉπ‘’β‰€πœπ›Όπ‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖‖‖‖,𝐹𝑒(3.23) where πœπ›Όπ‘›=1βˆ’2𝛼𝑛(πœ‚βˆ’π›Όπ‘›π›½2𝐾2)∈(0,1). Then from (3.22) and (3.23), we obtain 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‘’β‰€π›½π‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘’1βˆ’π›½π‘›ξ€Έξ‚€πœπ›Όπ‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖‖‖‖≀𝐹𝑒1βˆ’1βˆ’π›½π‘›ξ€Έξ€·1βˆ’πœπ›Όπ‘›β€–β€–π‘₯𝑛‖‖+ξ€·βˆ’π‘’1βˆ’π›½π‘›ξ€Έπ›Όπ‘›β€–β€–β€–β€–ξƒ―β€–β€–π‘₯𝐹𝑒≀max𝑛‖‖,π›Όβˆ’π‘’π‘›β€–β€–β€–β€–πΉπ‘’1βˆ’πœπ›Όπ‘›ξƒ°.(3.24) By induction, we have β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘’β‰€max1β€–β€–βˆ’π‘’,𝑀3‖‖‖‖𝐹𝑒,(3.25) where 𝑀3=sup{𝛼𝑛/(1βˆ’πœπ›Όπ‘›)∢0<π›Όπ‘›β‰€πœ‚/2𝛽2𝐾2βˆ’πœ–}<+∞. Therefore, {π‘₯𝑛} is bounded. We also obtain that {𝑦𝑛} and {𝐹𝑇π‘₯𝑛} are bounded.Step 2. We claim that limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–=0. Observe that ‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–=β€–β€–Ξ πΆξ‚€πΌβˆ’π›Όπ‘›+1𝐹𝑇π‘₯𝑛+1βˆ’Ξ πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›β€–β€–β‰€β€–β€–ξ‚€πΌβˆ’π›Όπ‘›+1𝐹𝑇π‘₯𝑛+1βˆ’ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯𝑛‖‖≀‖‖𝑇π‘₯𝑛+1βˆ’π‘‡π‘₯𝑛‖‖+𝛼𝑛+1‖‖𝐹𝑇π‘₯𝑛+1β€–β€–+𝛼𝑛‖‖𝐹𝑇π‘₯𝑛‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+𝛼𝑛+1‖‖𝐹𝑇π‘₯𝑛+1β€–β€–+𝛼𝑛‖‖𝐹𝑇π‘₯𝑛‖‖.(3.26) Therefore, we have limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(3.27) From (3.22), (3.27), and using Lemma 2.5, we have limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘¦π‘›β€–=0.Step 3. We claim that limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘‡π‘¦π‘›β€–=0. Observe that β€–β€–π‘¦π‘›βˆ’π‘‡π‘¦π‘›β€–β€–=β€–β€–Ξ πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’Ξ πΆπ‘‡π‘¦π‘›β€–β€–β‰€β€–β€–π‘‡π‘₯π‘›βˆ’π‘‡π‘¦π‘›β€–β€–+𝛼𝑛‖‖𝐹𝑇π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘¦π‘›β€–β€–+𝛼𝑛‖‖𝐹𝑇π‘₯𝑛‖‖.(3.28) Hence, from Step 2 and limπ‘›β†’βˆžπ›Όπ‘›=0, we have limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘‡π‘¦π‘›β€–β€–=0.(3.29)Step 4. We claim that limsupπ‘›β†’βˆžβŸ¨πΉπ‘₯βˆ—,𝑗(π‘₯βˆ—βˆ’π‘¦π‘›)βŸ©β‰€0, where π‘₯βˆ—=lim𝑑→0π‘₯𝑑 and π‘₯𝑑 is defined by (3.3). Since 𝑦𝑛 is bounded, there exists a subsequence {π‘¦π‘›π‘˜} of {𝑦𝑛} which converges weakly to πœ”. From Step 3, we obtain π‘‡π‘¦π‘›π‘˜β‡€πœ”. From Lemma 2.3, we have πœ”βˆˆFix(𝑇). Hence, using Theorem 3.1, we have π‘₯βˆ—βˆˆFix(𝑇) and limsupπ‘›β†’βˆžξ‚¬πΉπ‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­=limπ‘˜β†’βˆžξ‚¬πΉπ‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›π‘˜ξ€Έξ‚­=𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—ξ€Έξ‚­βˆ’πœ”β‰€0.(3.30)Step 5. We claim that {π‘₯𝑛} converges strongly to π‘₯βˆ—βˆˆFix(𝑇). From (3.22) and using Lemma 2.2, we have ξ‚¬Ξ πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯𝑛𝑦,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ‚­β‰€0.(3.31) Observe that β€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–2=ξ‚¬Ξ πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’π‘₯βˆ—ξ€·π‘¦,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ‚­=ξ‚¬Ξ πΆξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯𝑛𝑦,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ‚­+ξ‚¬ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’π‘₯βˆ—ξ€·π‘¦,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ‚­β‰€ξ‚¬ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’π‘₯βˆ—ξ€·π‘¦,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ‚­β‰€ξ‚¬ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯βˆ—ξ€·π‘¦,π‘—π‘›βˆ’π‘₯βˆ—ξ€Έξ‚­+𝛼𝑛𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­β‰€β€–β€–ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯π‘›βˆ’ξ‚€πΌβˆ’π›Όπ‘›πΉξ‚π‘‡π‘₯βˆ—β€–β€–β€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­β‰€πœπ›Όπ‘›β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­β‰€πœ2𝛼𝑛2β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+12β€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛼𝑛𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­,(3.32) that is, β€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–2β‰€πœπ›Όπ‘›β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+2𝛼𝑛𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­.(3.33) By (3.22) and (3.33), we have β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2≀𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘¦π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+ξ€·1βˆ’π›½π‘›ξ€Έξ‚ƒπœπ›Όπ‘›β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+2𝛼𝑛𝐹π‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έβ‰€ξ€Ίξ€·ξ‚­ξ‚„1βˆ’1βˆ’π›½π‘›ξ€Έξ€·1βˆ’πœπ›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯βˆ—β€–β€–2+2𝑀3ξ€·1βˆ’π›½π‘›ξ€Έξ€·1βˆ’πœπ›Όπ‘›ξ€Έξ‚¬πΉπ‘₯βˆ—ξ€·π‘₯,π‘—βˆ—βˆ’π‘¦π‘›ξ€Έξ‚­=ξ€·1βˆ’πœ†π‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+πœ†π‘›π›Ώπ‘›,(3.34) where πœ†π‘›=(1βˆ’π›½π‘›)(1βˆ’πœπ›Όπ‘›), 𝛿𝑛=2𝑀3⟨𝐹π‘₯βˆ—,𝑗(π‘₯βˆ—βˆ’π‘¦π‘›)⟩. It is easy to see that πœ†π‘›β†’0, βˆ‘βˆžπ‘›=1πœ†π‘›=∞ and limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0. Hence, by Lemma 2.4, the sequence {π‘₯𝑛} converges strongly to π‘₯βˆ—βˆˆFix(𝑇). From π‘₯βˆ—=lim𝑑→0π‘₯𝑑 and Theorem 3.1, we have π‘₯βˆ— to be the unique solution of the variational inequality (3.4).

Taking 𝑇=Π𝐢(πΌβˆ’πœ†π΄)Π𝐢(πΌβˆ’πœ‡π΅) and 𝐹=𝐹, where 0<πœ†β‰€π›Ό/𝐾2 and 0<πœ‡β‰€πœ‚/𝐾2, we obtain the following theorems immediately.

Corollary 3.3 (see [1, Theorem 3.5]). The net {π‘₯𝑑} generated by the implicit method (1.1) converges in norm, as 𝑑→0+, to the unique solution Μƒπ‘₯ of variational inequalityΜƒπ‘₯∈Ω,⟨𝐹(Μƒπ‘₯),𝑗(Μƒπ‘₯βˆ’π‘§)βŸ©β‰€0,π‘§βˆˆΞ©.(3.35)

Corollary 3.4 (see [1, Theorem 3.7]). Let 𝐢 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋 and let Π𝐢 be a sunny nonexpansive retraction from 𝑋 onto 𝐢. Let the mappings 𝐴,π΅βˆΆπΆβ†’π‘‹ be 𝛼-inverse-strongly accretive and 𝛽-inverse-strongly accretive operators, respectively. Let πΉβˆΆπΆβ†’π» be a strongly positive linear bounded operator with coefficient 𝛾>0. For given π‘₯0∈𝐢, let the sequence {π‘₯𝑛} be generated iteratively by (1.2). Suppose that the sequences {𝛼𝑛} and {𝛽𝑛} satisfy the conditions (A1) and (A2), then {π‘₯𝑛} converges strongly to Μƒπ‘₯∈Ω which solves the variational inequality (3.35).

Acknowledgments

Supported by the Natural Science Foundation of Yancheng Teachers University under Grant (11YCKL009) and Professor and Doctor Foundation of Yancheng Teachers University under Grant (11YSYJB0202).

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