Journal of Applied Mathematics

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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Volume 2012 |Article ID 580158 | https://doi.org/10.1155/2012/580158

Jin-Hua Zhu, Shih-Sen Chang, Min Liu, "Algorithms for a System of General Variational Inequalities in Banach Spaces", Journal of Applied Mathematics, vol. 2012, Article ID 580158, 18 pages, 2012. https://doi.org/10.1155/2012/580158

Algorithms for a System of General Variational Inequalities in Banach Spaces

Academic Editor: Zhenyu Huang
Received21 Dec 2011
Accepted06 Feb 2012
Published22 Apr 2012

Abstract

The purpose of this paper is using Korpelevich's extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

1. Introduction

Throughout this paper, we denote by โ„• and โ„ the sets of positive integers and real numbers, respectively. We also assume that ๐ธ is a real Banach space, ๐ธโˆ— is the dual space of ๐ธ,๐ถ is a nonempty closed convex subset of ๐ธ, and โŸจโ‹…,โ‹…โŸฉ is the pairing between ๐ธ and ๐ธโˆ—.

In this paper, we are concerned a finite family of a general system of nonlinear variational inequalities in Banach spaces, which involves finding (๐‘ฅโˆ—1,๐‘ฅโˆ—2,โ€ฆ,๐‘ฅโˆ—๐‘›)โˆˆ๐ถร—๐ถร—โ‹ฏร—๐ถ such that ๎ซ๐œ†1๐ด1๐‘ฅโˆ—2+๐‘ฅโˆ—1โˆ’๐‘ฅโˆ—2๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—1๎ซ๐œ†๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,2๐ด2๐‘ฅโˆ—3+๐‘ฅโˆ—2โˆ’๐‘ฅโˆ—3๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—2๎ซ๐œ†๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,3๐ด3๐‘ฅโˆ—4+๐‘ฅโˆ—3โˆ’๐‘ฅโˆ—4๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—3โ‹ฎ๎ซ๐œ†๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,๐‘โˆ’1๐ด๐‘โˆ’1๐‘ฅโˆ—๐‘+๐‘ฅโˆ—๐‘โˆ’1โˆ’๐‘ฅโˆ—๐‘๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๐‘โˆ’1๎ซ๐œ†๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,๐‘๐ด๐‘๐‘ฅโˆ—1+๐‘ฅโˆ—๐‘โˆ’๐‘ฅโˆ—1๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๐‘๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,(1.1) where {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} is a finite family of nonlinear mappings and ๐œ†๐‘–(๐‘–=1,2,โ€ฆ,๐‘) are positive real numbers.

As special cases of the problem (1.1), we have the following.

(I) If ๐ธ is a real Hilbert space and ๐‘=2, then (1.1) reduces to๎ซ๐œ†1๐ด1๐‘ฅโˆ—2+๐‘ฅโˆ—1โˆ’๐‘ฅโˆ—2,๐‘ฅโˆ’๐‘ฅโˆ—1๎ฌ๎ซ๐œ†โ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,2๐ด2๐‘ฅโˆ—1+๐‘ฅโˆ—2โˆ’๐‘ฅโˆ—1,๐‘ฅโˆ’๐‘ฅโˆ—2๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,(1.2) which was considered by Ceng et al. [1]. In particular, if ๐ด1=๐ด2=๐ด, then the problem (1.2) reduces to finding (๐‘ฅโˆ—1,๐‘ฅโˆ—2)โˆˆ๐ถร—๐ถ such that๎ซ๐œ†1๐ด๐‘ฅโˆ—2+๐‘ฅโˆ—1โˆ’๐‘ฅโˆ—2,๐‘ฅโˆ’๐‘ฅโˆ—1๎ฌ๎ซ๐œ†โ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,2๐ด๐‘ฅโˆ—1+๐‘ฅโˆ—2โˆ’๐‘ฅโˆ—1,๐‘ฅโˆ’๐‘ฅโˆ—2๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,(1.3) which is defined by Verma [2]. Furthermore, if ๐‘ฅโˆ—1=๐‘ฅโˆ—2, then (1.3) reduces to the following variational inequality (VI) of finding ๐‘ฅโˆ—โˆˆ๐ถ such thatโŸจ๐ด๐‘ฅโˆ—,๐‘ฅโˆ’๐‘ฅโˆ—โŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ.(1.4)

This problem is a fundamental problem in variational analysis and, in particular, in optimization theory. Many algorithms for solving this problem are projection algorithms that employ projections onto the feasible set ๐ถ of the VI or onto some related set, in order to iteratively reach a solution. In particular, Korpelevichโ€™s extragradient method which was introduced by Korpelevich [3] in 1976 generates a sequence {๐‘ฅ๐‘›} via the recursion๐‘ฆ๐‘›=๐‘ƒ๐ถ๎€บ๐‘ฅ๐‘›โˆ’๐œ†๐ด๐‘ฅ๐‘›๎€ป,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐‘ฅ๐‘›โˆ’๐œ†๐ด๐‘ฆ๐‘›๎€ป,๐‘›โ‰ฅ0,(1.5) where ๐‘ƒ๐ถ is the metric projection from โ„๐‘› onto ๐ถ, ๐ดโˆถ๐ถโ†’๐ป is a monotone operator, and ๐œ† is a constant. Korpelevich [3] proved that the sequence {๐‘ฅ๐‘›} converges strongly to a solution of ๐‘‰๐ผ(๐ถ,A). Note that the setting of the space is Euclid space โ„๐‘›.

The literature on the VI is vast, and Korpelevichโ€™s extragradient method has received great attention by many authors, who improved it in various ways. See, for example, [4โ€“16] and references therein.

(II) If ๐ธ is still a real Banach space and ๐‘=1, then the problem (1.1) reduces to finding ๐‘ฅโˆ—โˆˆ๐ถ such that๎ซ๐ด๐‘ฅโˆ—๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆC,(1.6) which was considered by Aoyama et al. [17]. Note that this problem is connected with the fixed point problem for nonlinear mapping, the problem of finding a zero point of a nonlinear operator, and so on. It is clear that problem (1.6) extends problem (1.4) from Hilbert spaces to Banach spaces.

In order to find a solution for problem (1.6), Aoyama et al. [17] introduced the following iterative scheme for an accretive operator ๐ด in a Banach space ๐ธ:๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผn๎€ธฮ ๐ถ๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘›๐ด๐‘ฅ๐‘›๎€ธ,๐‘›โ‰ฅ1,(1.7) where ฮ ๐ถ is a sunny nonexpansive retraction from ๐ธ to ๐ถ. Then they proved a weak convergence theorem in a Banach space. For related works, please see [18] and the references therein.

It is an interesting problem of constructing some algorithms with strong convergence for solving problem (1.1) which contains problem (1.6) as a special case.

Our aim in this paper is to construct two algorithms for solving problem (1.1). For this purpose, we first prove that the system of variational inequalities (1.1) is equivalent to a fixed point problem of some nonexpansive mapping. Finally, we prove the strong convergence of the proposed methods which solve problem (1.1).

2. Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence {๐‘ฅ๐‘›} by ๐‘ฅ๐‘›โ†’๐‘ฅ and ๐‘ฅ๐‘›โ‡€๐‘ฅ, respectively.

For ๐‘ž>1, the generalized duality mapping ๐ฝ๐‘žโˆถ๐ธโ†’2๐ธโˆ— is defined by๐ฝ๐‘ž๎€ฝ(๐‘ฅ)=๐‘“โˆˆ๐ธโˆ—โˆถโŸจ๐‘ฅ,๐‘“โŸฉ=โ€–๐‘ฅโ€–๐‘ž,โ€–๐‘“โ€–=โ€–๐‘ฅโ€–๐‘žโˆ’1๎€พ(2.1) for all ๐‘ฅโˆˆ๐ธ. In particular, ๐ฝ=๐ฝ2 is called the normalized duality mapping. It is known that ๐ฝ๐‘ž(๐‘ฅ)=||๐‘ฅ||๐‘žโˆ’2 for all ๐‘ฅโˆˆ๐ธ. If ๐ธ is a Hilbert space, then ๐ฝ=๐ผ, the identity mapping. Let ๐‘ˆ={๐‘ฅโˆˆ๐ธโˆถ||๐‘ฅ||=1}. A Banach space ๐ธ is said to be uniformly convex if, for any ๐œ€โˆˆ(0,2], there exists ๐›ฟ>0 such that, for any ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ,โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ฅ๐œ€implies๐‘ฅ+๐‘ฆ2โ€–โ€–โ€–โ‰ค1โˆ’๐›ฟ.(2.2)

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space ๐ธ is said to be smooth if the limitlim๐‘›โ†’0โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–๐‘ก(2.3) exists for all ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ. It is also said to be uniformly smooth if the previous limit is attained uniformly for ๐‘ฅ,๐‘ฆโˆˆ๐‘ˆ. The norm of ๐ธ is said to be Frรฉchet differentiable if, for each ๐‘ฅโˆˆ๐‘ˆ, the previous limit is attained uniformly for all ๐‘ฆโˆˆ๐‘ˆ. The modulus of smoothness of ๐ธ is defined by๎‚†1๐œŒ(๐œ)=sup2(๎‚‡โ€–๐‘ฅ+๐‘ฆโ€–+โ€–๐‘ฅโˆ’๐‘ฆโ€–)โˆ’1โˆถ๐‘ฅ,๐‘ฆโˆˆ๐ธ,โ€–๐‘ฅโ€–=1,โ€–๐‘ฆโ€–=๐œ,(2.4) where ๐œŒโˆถ[0,โˆž)โ†’[0,โˆž) is function. It is known that ๐ธ is uniformly smooth if and only if lim๐œโ†’0(๐œŒ(๐œ)/๐œ)=0. Let ๐‘ž be a fixed real number with 1<๐‘žโ‰ค2. Then a Banach space ๐ธ is said to be ๐‘ž-uniformly smooth if there exists a constant ๐‘>0 such that ๐œŒ(๐œ)โ‰ค๐‘๐œ๐‘ž for all ๐œ>0. Note the following.(1)๐ธ is a uniformly smooth Banach space if and only if ๐ฝ is single valued and uniformly continuous on any bounded subset of ๐ธ.(2)All Hilbert spaces, ๐ฟ๐‘ (or ๐‘™๐‘) spaces (๐‘โ‰ฅ2) and the Sobolev spaces ๐‘Š๐‘๐‘š(๐‘โ‰ฅ2) are 2-uniformly smooth, while ๐ฟ๐‘ (or ๐‘™๐‘) and ๐‘Š๐‘ƒ๐‘š spaces (1<๐‘โ‰ค2) are ๐‘-uniformly smooth.(3)Typical examples of both uniformly convex and uniformly smooth Banach spaces are ๐ฟ๐‘, where ๐‘>1. More precisely, ๐ฟ๐‘ is min{๐‘,2}-uniformly smooth for every ๐‘>1.

In our paper, we focus on a 2-uniformly smooth Banach space with the smooth constant ๐พ.

Let ๐ธ be a real Banach space, ๐ถ a nonempty closed convex subset of ๐ธ, ๐‘‡โˆถ๐ถโ†’๐ถ a mapping, and ๐น(๐‘‡) the set of fixed points of ๐‘‡.

Recall that a mapping ๐‘‡โˆถ๐ถโ†’๐ถ is called nonexpansive ifโ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(2.5) A bounded linear operator ๐นโˆถ๐ถโˆˆ๐ธ is called strongly positive if there exists a constant ๐›พ>0 with the propertyโŸจ๐น(๐‘ฅ),๐‘—(๐‘ฅ)โŸฉโ‰ฅ๐›พโ€–๐‘ฅโ€–2,โˆ€๐‘ฅโˆˆ๐ถ.(2.6) A mapping ๐ดโˆถ๐ถโ†’๐ธ is said to be accretive if there exists ๐‘—(๐‘ฅโˆ’๐‘ฆ)โˆˆ๐ฝ(๐‘ฅโˆ’๐‘ฆ) such thatโŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘—(๐‘ฅโˆ’๐‘ฆ)โŸฉโ‰ฅ0,(2.7) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ, where ๐ฝ is the duality mapping.

A mapping ๐ด of ๐ถ into ๐ธ is said to be ๐›ผ-strongly accretive if, for ๐›ผ>0,โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘—(๐‘ฅโˆ’๐‘ฆ)โŸฉโ‰ฅ๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€–2,(2.8) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ.

A mapping ๐ด of ๐ถ into ๐ธ is said to be ๐›ผ-inverse-strongly accretive if, for ๐›ผ>0,โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘—(๐‘ฅโˆ’๐‘ฆ)โŸฉโ‰ฅ๐›ผโ€–๐ด๐‘ฅโˆ’๐ด๐‘ฆโ€–2,(2.9) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ.

Remark 2.1. Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for instance, [6, 19, 20].
Let ๐ท be a subset of ๐ถ, and let ฮ  be a mapping of ๐ถ into ๐ท. Then ฮ  is said to be sunny if ฮ []ฮ (๐‘ฅ)+๐‘ก(๐‘ฅโˆ’ฮ (๐‘ฅ))=ฮ (๐‘ฅ)(2.10) whenever ฮ (๐‘ฅ)+๐‘ก(๐‘ฅโˆ’ฮ (๐‘ฅ))โˆˆ๐ถ for ๐‘ฅโˆˆ๐ถ and ๐‘กโ‰ฅ0. A mapping ฮ  of ๐ถ into itself is called a retraction if ฮ 2=ฮ . If a mapping ฮ  of ๐ถ into itself is a retraction, then ฮ (๐‘ง)=๐‘ง for every ๐‘งโˆˆ๐‘…(ฮ ), where ๐‘…(ฮ ) is the range of ฮ . A subset ๐ท of ๐ถ is called a sunny nonexpansive retract of ๐ถ if there exists a sunny nonexpansive retraction from ๐ถ onto ๐ท. Then following lemma concerns the sunny nonexpansive retraction.

Lemma 2.2 (see [21]). Let ๐ถ be a closed convex subset of a smooth Banach space ๐ธ, let ๐ท be a nonempty subset of ๐ถ, and let ฮ  be a retraction from ๐ถ onto ๐ท. Then ฮ  is sunny and nonexpansive if and only if โŸจ๐‘ขโˆ’ฮ (๐‘ข),๐‘—(๐‘ฆโˆ’ฮ (๐‘ข))โŸฉโ‰ค0,(2.11) for all ๐‘ขโˆˆ๐ถ and ๐‘ฆโˆˆ๐ท.

Remark 2.3. (1) It is well known that if ๐ธ is a Hilbert space, then a sunny nonexpansive retraction ฮ ๐ถ is coincident with the metric projection from ๐ธ onto ๐ถ.
(2) Let ๐ถ be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space ๐ธ, and let ๐‘‡ be a nonexpansive mapping of ๐ถ into itself with the set ๐น(๐‘‡)โ‰ โˆ…. Then the set ๐น(๐‘‡) is a sunny nonexpansive retract of ๐ถ.

In what follows, we need the following lemmas for proof of our main results.

Lemma 2.4 (see [22]). Assume that {๐›ผ๐‘›} is a sequence of nonnegative real numbers such that ๐›ผ๐‘›+1โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธ๐›ผ๐‘›+๐›ฟ๐‘›,(2.12) where {๐›พ๐‘›} is a sequence in (0,1) and {๐›ฟ๐‘›} is a sequence such that(a)ฮฃโˆž๐‘›=1๐›พ๐‘›=โˆž,(b)limsup๐‘›โ†’โˆž(๐›ฟ๐‘›/๐›พ๐‘›)โ‰ค0 or ฮฃโˆž๐‘›=1|๐›ฟ๐‘›|<โˆž.Then lim๐‘›โ†’โˆž๐›ผ๐‘›=0.

Lemma 2.5 (see [23]). Let ๐‘‹ be a Banach space, {๐‘ฅ๐‘›},{๐‘ฆ๐‘›} be two bounded sequences in ๐‘‹ and {๐›ฝ๐‘›} be a sequence in [0,1] satisfying 0<liminf๐‘›โ†’โˆž๐›ฝ๐‘›โ‰คlimsup๐‘›โ†’โˆž๐›ฝ๐‘›<1.(2.13) Suppose that ๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘ฅ๐‘›+(1โˆ’๐›ฝ๐‘›)๐‘ฆ๐‘›, for all ๐‘›โ‰ฅ1 and limsup๐‘›โ†’โˆž๎€ฝโ€–โ€–๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›โ€–โ€–โˆ’โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–๎€พโ‰ค0,(2.14) then limnโ†’โˆžโ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–=0.

Lemma 2.6 (see [24]). Let ๐ธ be a real 2-uniformly smooth Banach space with the best smooth constant ๐พ. Then the following inequality holds: โ€–๐‘ฅ+๐‘ฆโ€–2โ‰คโ€–๐‘ฅโ€–2+2โŸจ๐‘ฆ,๐ฝ๐‘ฅโŸฉ+2โ€–๐พ๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ธ.(2.15)

Lemma 2.7 (see [25]). Let ๐ถ be a nonempty bounded closed convex subset of a uniformly convex Banach space ๐ธ, and let ๐บ be a nonexpansive mapping of ๐ถ into itself. If {๐‘ฅ๐‘›} is a sequence of ๐ถ such that ๐‘ฅ๐‘›โ‡€๐‘ฅ and ๐‘ฅ๐‘›โˆ’๐บ๐‘ฅ๐‘›โ†’0, then ๐‘ฅ is a fixed point of ๐บ.

Lemma 2.8 (see [26]). Let ๐ถ be a nonempty closed convex subset of a real Banach space ๐ธ. Assume that the mapping ๐นโˆถ๐ถโ†’๐ธ is accretive and weakly continuous along segments (i.e., ๐น(๐‘ฅ+๐‘ก๐‘ฆ)โ‡€๐น(๐‘ฅ) as ๐‘กโ†’0). Then the variational inequality ๐‘ฅโˆ—๎ซโˆˆ๐ถ,๐น๐‘ฅโˆ—๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ฅ0,๐‘ฅโˆˆ๐ถ,(2.16) is equivalent to the dual variational inequality ๐‘ฅโˆ—๎ซ๎€ทโˆˆ๐ถ,๐น๐‘ฅ,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ฅ0,๐‘ฅโˆˆ๐ถ.(2.17)

Lemma 2.9. Let ๐ถ be a nonempty closed convex subset of a real 2-uniformly smooth Banach space ๐ธ. Let ฮ ๐ถ be a sunny nonexpansive retraction from ๐ธ onto ๐ถ. Let {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of ๐›พ๐‘–-inverse-strongly accretive. For given (๐‘ฅโˆ—1,๐‘ฅโˆ—2,โ€ฆ,๐‘ฅโˆ—๐‘›)โˆˆ๐ถร—๐ถร—โ‹ฏร—๐ถ, where ๐‘ฅโˆ—=๐‘ฅโˆ—1,๐‘ฅโˆ—๐‘–=ฮ ๐ถ(๐ผโˆ’๐œ†๐‘–๐ด๐‘–)๐‘ฅโˆ—๐‘–+1,๐‘–โˆˆ{๐‘–,2,โ€ฆ,๐‘โˆ’1},๐‘ฅโˆ—๐‘=ฮ ๐ถ(๐ผโˆ’๐œ†๐‘๐ด๐‘)๐‘ฅโˆ—1, then (๐‘ฅโˆ—1,๐‘ฅโˆ—2,โ€ฆ,๐‘ฅโˆ—๐‘›) is a solution of the problem (1.1) if and only if ๐‘ฅโˆ— is a fixed point of the mapping ๐‘„ defined by ๐‘„(๐‘ฅ)=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†1๐ด1๎€ธฮ ๐ถ๎€ท๐ผโˆ’๐œ†2๐ด2๎€ธโ‹ฏฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘๐ด๐‘๎€ธ(๐‘ฅ),(2.18) where ๐œ†๐‘–(๐‘–=1,2,โ€ฆ,๐‘) are real numbers.

Proof. We can rewrite (1.1) as ๎ซ๐‘ฅโˆ—1โˆ’๎€ท๐‘ฅโˆ—2โˆ’๐œ†1๐ด1๐‘ฅโˆ—2๎€ธ๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—1๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,โŸจ๐‘ฅโˆ—2โˆ’๎€ท๐‘ฅโˆ—3โˆ’๐œ†2๐ด2๐‘ฅโˆ—3๎€ธ๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—2๎€ธ๎ซ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,โˆ—3โˆ’๎€ท๐‘ฅโˆ—4โˆ’๐œ†3๐ด3๐‘ฅโˆ—4๎€ธ๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—3โ‹ฎ๎ซ๐‘ฅ๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ,โˆ—๐‘โˆ’1โˆ’๎€ท๐‘ฅโˆ—๐‘โˆ’๐œ†๐‘โˆ’1๐ด๐‘โˆ’1๐‘ฅโˆ—๐‘๎€ธ๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๐‘โˆ’1๎ซ๐‘ฅ๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆC,โˆ—๐‘โˆ’๎€ท๐‘ฅโˆ—1โˆ’๐œ†๐‘๐ด๐‘๐‘ฅโˆ—1๎€ธ๎€ท,๐‘—๐‘ฅโˆ’๐‘ฅโˆ—๐‘๎€ธ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ.(2.19) By Lemma 2.2, we can check (2.19) is equivalent to ๐‘ฅโˆ—1=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†1๐ด1๎€ธ๐‘ฅโˆ—2,๐‘ฅโˆ—2=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†2๐ด2๎€ธ๐‘ฅโˆ—3,โ‹ฎ๐‘ฅโˆ—๐‘โˆ’1=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘โˆ’1๐ด๐‘โˆ’1๎€ธ๐‘ฅโˆ—๐‘,๐‘ฅโˆ—๐‘=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘๐ด๐‘๎€ธ๐‘ฅโˆ—1.โŸบ๐‘„๎€ท๐‘ฅโˆ—๎€ธ=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†1๐ด1๎€ธฮ ๐ถ๎€ท๐ผโˆ’๐œ†2๐ด2๎€ธโ‹ฏฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘๐ด๐‘๐‘ฅ๎€ธ๎€ทโˆ—๎€ธ=๐‘ฅโˆ—.(2.20) This completes the proof.

Throughout this paper, the set of fixed points of the mapping ๐‘„ is denoted by ฮฉ.

Lemma 2.10. Let ๐ถ be a nonempty closed convex subset of a real 2-uniformly smooth Banach space ๐ธ. Let ฮ ๐ถ be a sunny nonexpansive retraction from ๐ธ onto ๐ถ. Let {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of ๐›พ๐‘–-inverse-strongly accretive. Let ๐‘„ be defined as Lemma 2.9. If 0โ‰ค๐œ†๐‘–โ‰ค๐›พ๐‘–/๐พ2, then ๐‘„โˆถ๐ถโ†’๐ถ is nonexpansive.

Proof. First, we show that for all ๐‘–โˆˆ{๐‘–,2,โ€ฆ,๐‘}, the mapping ฮ ๐ถ(๐ผโˆ’๐œ†๐‘–๐ด๐‘–) is nonexpansive. Indeed, for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ, from the condition ๐œ†๐‘–โˆˆ[0,๐›พ๐‘–/๐พ2] and Lemma 2.6, we have โ€–โ€–ฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘–๐ด๐‘–๎€ธ๐‘ฅโˆ’ฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘–๐ด๐‘–๎€ธ๐‘ฆโ€–โ€–2โ‰คโ€–โ€–๎€ท๐ผโˆ’๐œ†๐‘–๐ด๐‘–๎€ธ๎€ท๐‘ฅโˆ’๐ผโˆ’๐œ†๐‘–๐ด๐‘–๎€ธ๐‘ฆโ€–โ€–2=โ€–โ€–(๐‘ฅโˆ’๐‘ฆ)โˆ’๐œ†๐‘–๎€ท๐ด๐‘–๐‘ฅโˆ’๐ด๐‘–๐‘ฆ๎€ธโ€–โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’2๐œ†๐‘–โŸจ๐ด๐‘–๐‘ฅโˆ’๐ด๐‘–๐‘ฆ,๐‘—(๐‘ฅโˆ’๐‘ฆ)โŸฉ+2๐พ2๐œ†2๐‘–โ€–โ€–๐ด๐‘–๐‘ฅโˆ’๐ด๐‘–๐‘ฆโ€–โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’2๐œ†๐‘–๐›พ๐‘–โ€–โ€–๐ด๐‘–๐‘ฅโˆ’๐ด๐‘–๐‘ฆโ€–โ€–2+2๐พ2๐œ†2๐‘–โ€–โ€–๐ด๐‘–๐‘ฅโˆ’๐ด๐‘–๐‘ฆโ€–โ€–2=โ€–๐‘ฅโˆ’๐‘ฆโ€–2+2๐œ†๐‘–๎€ท๐พ2๐œ†๐‘–โˆ’๐›พ๐‘–๎€ธโ€–โ€–๐ด๐‘–๐‘ฅโˆ’๐ด๐‘–๐‘ฆโ€–โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2,(2.21) which implies for all ๐‘–โˆˆ{1,2,โ€ฆ,๐‘}, the mapping ฮ ๐ถ(๐ผโˆ’๐œ†๐‘–๐ด๐‘–) is nonexpansive, so is the mapping ๐‘„.

3. Main Results

In this section, we introduce our algorithms and show the strong convergence theorems.

Algorithm 3.1. Let ๐ถ be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space ๐ธ. Let ฮ ๐ถ be a sunny nonexpansive retraction from ๐ธ to ๐ถ. Let {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of ๐›พ๐‘–-inverse-strongly accretive. Let ๐ตโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐›ผ>0 and ๐นโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐œŒโˆˆ(0,๐›ผ). For any ๐‘กโˆˆ(0,1), define a net {๐‘ฅ๐‘ก} as follows: ๐‘ฅ๐‘ก=ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก,๐‘ฆ๐‘ก=ฮ ๐ถ๎€ท๐ผโˆ’๐œ†1๐ด1๎€ธฮ ๐ถ๎€ท๐ผโˆ’๐œ†2๐ด2๎€ธโ‹ฏฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘๐ด๐‘๎€ธ๐‘ฅ๐‘ก,(3.1) where, for any ๐‘–,๐œ†๐‘–โˆˆ(0,๐›พ๐‘–/๐พ2) is a real number.

Remark 3.2. We notice that the net {๐‘ฅ๐‘ก} defined by (3.1) is well defined. In fact, we can define a self-mapping ๐‘Š๐‘กโˆถ๐ถโ†’๐ถ as follows: ๐‘Š๐‘ก๐‘ฅโˆถ=ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))ฮ ๐ถ๎€ท๐ผโˆ’๐œ†1๐ด1๎€ธฮ ๐ถ๎€ท๐ผโˆ’๐œ†2๐ด2๎€ธโ‹ฏฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘๐ด๐‘๎€ธ๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ถ.(3.2)
From Lemma 2.10, we know that if, for any ๐‘–,๐œ†๐‘–โˆˆ(0,๐›พ๐‘–/๐พ2), the mapping ฮ ๐ถ(๐ผโˆ’๐œ†1๐ด1)ฮ ๐ถ(๐ผโˆ’๐œ†2๐ด2)โ‹ฏฮ ๐ถ(๐ผโˆ’๐œ†๐‘๐ด๐‘)=๐‘„ is nonexpansive and ||๐ผโˆ’๐‘ก๐ต||โ‰ค1โˆ’๐‘ก๐›ผ. Then, for any ๐‘ฅ,๐‘ฆโˆˆ๐ถ, we have โ€–โ€–๐‘Š๐‘ก๐‘ฅโˆ’๐‘Š๐‘ก๐‘ฆโ€–โ€–=โ€–โ€–ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘„(๐‘ฅ)โˆ’ฮ ๐ถโ€–โ€–(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘„(๐‘ฆ)โ‰คโ€–((๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘„)๐‘ฅโˆ’((๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘„)๐‘ฆโ€–=โ€–๐‘ก(๐น๐‘ฅโˆ’๐น๐‘ฆ)+(๐ผโˆ’๐‘ก๐ต)(๐‘„๐‘ฅโˆ’๐‘„๐‘ฆ)โ€–โ‰ค๐‘ก๐œŒโ€–๐‘ฅโˆ’๐‘ฆโ€–+โ€–๐ผโˆ’๐‘ก๐ตโ€–โ€–๐‘„๐‘ฅโˆ’๐‘„๐‘ฆโ€–โ‰ค๐‘ก๐œŒโ€–๐‘ฅโˆ’๐‘ฆโ€–+(1โˆ’๐‘ก๐›ผ)โ€–๐‘ฅโˆ’๐‘ฆโ€–=(1โˆ’(๐›ผโˆ’๐œŒ)๐‘ก)โ€–๐‘ฅโˆ’๐‘ฆโ€–.(3.3) This shows that the mapping ๐‘Š๐‘ก is contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.3. Let ๐ถ be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space ๐ธ. Let ฮ ๐ถ be a sunny nonexpansive retraction from E to ๐ถ. Let {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of ๐›พ๐‘–-inverse-strongly accretive. Let ๐ตโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐›ผ>0, and let ๐นโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐œŒโˆˆ(0,๐›ผ). Assume that ฮฉโ‰ โˆ… and ๐œ†๐‘–โˆˆ(0,๐›พ๐‘–/๐พ2). Then the net {๐‘ฅ๐‘ก} generated by the implicit method (3.1) converges in norm, as ๐‘กโ†’0+ to the unique solution ฬƒ๐‘ฅ of VI ฬƒ๐‘ฅโˆˆฮฉ,โŸจ(๐ตโˆ’๐น)ฬƒ๐‘ฅ,๐‘—(๐‘งโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘งโˆˆฮฉ.(3.4)

Proof. We divide the proof of Theorem 3.3 into four steps.(I) Next we prove that the net {๐‘ฅ๐‘ก} is bounded.
Take that ๐‘ฅโˆ—โˆˆฮฉ, we have โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–=โ€–โ€–ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–=โ€–โ€–ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’ฮ ๐ถ๐‘ฅโˆ—โ€–โ€–โ‰คโ€–โ€–(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–=โ€–โ€–๐‘ก๎€ท๐น๎€ท๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ท๐‘ฆ๎€ธ๎€ธ+(๐ผโˆ’๐‘ก๐ต)๐‘กโˆ’๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ+๐‘ก๐นโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐‘ก๐ตโˆ—๎€ธโ€–โ€–โ€–โ€–๐น๎€ท๐‘ฆโ‰ค๐‘ก๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธโ€–โ€–โ€–โ€–๐‘ฆ+โ€–๐ผโˆ’๐‘ก๐ตโ€–๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–โ€–โ€–๐น๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–โ€–โ€–๐‘ฆโ‰ค๐‘ก๐œŒ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–โ€–โ€–๐‘ฆ+(1โˆ’๐‘ก๐›ผ)๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–โ€–โ€–๐น๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–โ€–โ€–๐‘ฆ=(1โˆ’(๐›ผโˆ’๐œŒ)๐‘ก)๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–โ€–โ€–๐น๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–โ€–โ€–๐‘„๎€ท๐‘ฅ=(1โˆ’(๐›ผโˆ’๐œŒ)๐‘ก)๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐‘„โˆ—๎€ธโ€–โ€–โ€–โ€–๐น๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–โ€–โ€–๐‘ฅโ‰ค(1โˆ’(๐›ผโˆ’๐œŒ)๐‘ก)๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–โ€–โ€–๐น๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–.(3.5) It follows that โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–โ‰คโ€–โ€–๐น๎€ท๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–๐›ผโˆ’๐œŒ.(3.6) Therefore, {๐‘ฅ๐‘ก} is bounded. Hence, {๐‘ฆ๐‘ก},{๐ต๐‘ฆ๐‘ก},{๐ด๐‘–๐‘ฅ๐‘ก}, and {๐น(๐‘ฆ๐‘ก)} are also bounded. We observe that โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฆ๐‘กโ€–โ€–=โ€–โ€–ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’ฮ ๐ถ๐‘ฆ๐‘กโ€–โ€–โ‰คโ€–โ€–(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’๐‘ฆ๐‘กโ€–โ€–โ€–โ€–๐น๎€ท๐‘ฆ=๐‘ก๐‘ก๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘ก๎€ธโ€–โ€–โŸถ0.(3.7)
From Lemma 2.10, we know that ๐‘„โˆถ๐ถโ†’๐ถ is nonexpansive. Thus, we have โ€–โ€–๐‘ฆ๐‘ก๎€ท๐‘ฆโˆ’๐‘„๐‘ก๎€ธโ€–โ€–=โ€–โ€–๐‘„๎€ท๐‘ฅ๐‘ก๎€ธ๎€ท๐‘ฆโˆ’๐‘„๐‘ก๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฆ๐‘กโ€–โ€–โŸถ0.(3.8) Therefore, lim๐‘กโ†’0โ€–โ€–๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘„๐‘ก๎€ธโ€–โ€–=0.(3.9)
(II) {๐‘ฅ๐‘ก} is relatively norm-compact as ๐‘กโ†’0+.
Let {๐‘ก๐‘›}โŠ‚(0,1) be any subsequence such that ๐‘ก๐‘›โ†’0+ as ๐‘›โ†’โˆž. Then, there exists a positive integer ๐‘›0 such that 0<๐‘ก๐‘›<1/2, for all ๐‘›โ‰ฅ๐‘›0. Let ๐‘ฅ๐‘›โˆถ=๐‘ฅ๐‘ก๐‘›. It follows from (3.9) that โ€–โ€–๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐‘„๐‘›๎€ธโ€–โ€–โŸถ0.(3.10) We can rewrite (3.1) as ๐‘ฅ๐‘ก=ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก+(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก.(3.11)
For any ๐‘ฅโˆ—โˆˆฮฉโŠ‚๐ถ, by Lemma 2.2, we have ๎ซ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’๐‘ฅ๐‘ก๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘ก=๎ซ(๎€ธ๎ฌ๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ท๐‘ฅ,๐‘—โˆ—โˆ’ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ธ๎ฌโ‰ค0.(3.12) With this fact, we derive that โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–2=๎ซ๐‘ฅโˆ—โˆ’๐‘ฅ๐‘ก๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘ก=๎ซ๐‘ฅ๎€ธ๎ฌโˆ—โˆ’(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘ก+๎ซ๎€ธ๎ฌ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’ฮ C(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘กโ‰ค๎ซ๎€ท๐‘ฅ๎€ธ๎ฌ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))โˆ—โˆ’๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘ก๎ซ๐ต๎€ท๐‘ฅ๎€ธ๎ฌ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๐‘ฅ๎€ธ๎ฌโ‰ค(1โˆ’๐‘ก(๐›ผโˆ’๐œŒ))โˆ—โˆ’๐‘ฆ๐‘กโ€–โ€–โ€–โ€–๐‘ฅโˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๎ซ๐ต๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๐‘„๎€ท๐‘ฅ๎€ธ๎ฌ=(1โˆ’๐‘ก(๐›ผโˆ’๐œŒ))โˆ—๎€ธ๎€ท๐‘ฅโˆ’๐‘„๐‘ก๎€ธโ€–โ€–โ€–โ€–๐‘ฅโˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๎ซ๐ต๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๐‘ฅ๎€ธ๎ฌโ‰ค(1โˆ’๐‘ก(๐›ผโˆ’๐œŒ))โˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–โ€–โ€–๐‘ฅโˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๎ซ๐ต๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–๐‘ฅ๎€ธ๎ฌโ‰ค(1โˆ’๐‘ก(๐›ผโˆ’๐œŒ))โˆ—โˆ’๐‘ฅ๐‘กโ€–โ€–2๎ซ๐ต๎€ท๐‘ฅ+๐‘กโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘ก.๎€ธ๎ฌ(3.13)
It turns out that โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅโˆ—โ€–โ€–2โ‰ค1๎ซ๐ต๎€ท๐‘ฅ๐›ผโˆ’๐œŒโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘ก๎€ธ๎ฌ,๐‘ฅโˆ—โˆˆฮฉ.(3.14)
In particular, โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–2โ‰ค1๎ซ๐ต๎€ท๐‘ฅ๐›ผโˆ’๐œŒโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐นโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โˆ’๐‘ฅ๐‘›๎€ธ๎ฌ,๐‘ฅโˆ—โˆˆฮฉ.(โˆ—โˆ—)
Since {๐‘ฅ๐‘›} is bounded, without loss of generality, ๐‘ฅ๐‘›โ‡€ฬƒ๐‘ฅโˆˆ๐ถ can be assumed. Noticing (3.10), we can use Lemma 2.7 to get ฬƒ๐‘ฅโˆˆฮฉ=๐น(๐‘„). Therefore, we can substitute ฬƒ๐‘ฅ for ๐‘ฅโˆ— in (**) to get โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค1๎ซ๎€ท๐›ผโˆ’๐œŒ๐ต(ฬƒ๐‘ฅ)โˆ’๐น(ฬƒ๐‘ฅ),๐‘—ฬƒ๐‘ฅโˆ’๐‘ฅ๐‘›๎€ธ๎ฌ.(3.15) Consequently, the weak convergence of {๐‘ฅ๐‘›} to ฬƒ๐‘ฅ actually implies that ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ strongly. This has proved the relative norm compactness of the net {๐‘ฅ๐‘ก} as ๐‘กโ†’0+.
(III) Now, we prove that ฬƒ๐‘ฅ solves the variational inequality (3.4). From (3.1), we have ๐‘ฅ๐‘ก=ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก+(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโŸน๐‘ฅ๐‘ก=ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ท๐‘ฅโˆ’(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘กโˆ’๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฅ+๐‘ก๐น๐‘ก๎€ธ๎€ท๐‘ฅ+(๐ผโˆ’๐‘ก๐ต)๐‘ก๎€ธ๎€ท๐‘ฅโŸน๐น๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐ต๐‘ก๎€ธ=1๐‘ก๎€บ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ท๐‘ฆโˆ’(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘กโˆ’๐‘ฅ๐‘ก.๎€ธ๎€ป(3.16) For any ๐‘งโˆˆฮฉ, we obtain ๎ซ๐น๎€ท๐‘ฅ๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐ต๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก=1๎€ธ๎ฌ๐‘ก๎ซ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘กโˆ’ฮ ๐ถ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘ฆ๐‘ก๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘กโˆ’1๎€ธ๎ฌ๐‘ก๎ซ๎€ท๐‘ฆ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘กโˆ’๐‘ฅ๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก1๎€ธ๎ฌโ‰คโˆ’๐‘ก๎ซ๎€ท๐‘ฆ(๐‘ก๐น+(๐ผโˆ’๐‘ก๐ต))๐‘กโˆ’๐‘ฅ๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก1๎€ธ๎ฌ=โˆ’๐‘ก๎ซ๐‘ฆ๐‘กโˆ’๐‘ฅ๐‘ก๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก+๎ซ๎€ท๐‘ฆ๎€ธ๎ฌ(๐ตโˆ’๐น)๐‘กโˆ’๐‘ฅ๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก.๎€ธ๎ฌ(3.17)
Now we prove that โŸจ๐‘ฆ๐‘กโˆ’๐‘ฅ๐‘ก,๐‘—(๐‘งโˆ’๐‘ฅ๐‘ก)โŸฉโ‰ฅ0. In fact, we can write ๐‘ฆ๐‘ก=๐‘„(๐‘ฅ๐‘ก). At the same time, we note that ๐‘ง=๐‘„(๐‘ง), so ๎ซ๐‘ฆ๐‘กโˆ’๐‘ฅ๐‘ก๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก=๎ซ๎€ท๐‘ฅ๎€ธ๎ฌ๐‘งโˆ’๐‘„(๐‘ง)โˆ’๐‘ก๎€ท๐‘ฅโˆ’๐‘„๐‘ก๎€ท๎€ธ๎€ธ,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก๎€ธ๎ฌ.(3.18) Since ๐ผโˆ’๐‘„ is accretive (this is due to the nonexpansivity of ๐‘„), we can deduce immediately that ๎ซ๐‘ฆ๐‘กโˆ’๐‘ฅ๐‘ก๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก๎€ท๐‘ฅ๎€ธ๎ฌ=โŸจ๐‘งโˆ’๐‘„(๐‘ง)โˆ’๐‘ก๎€ท๐‘ฅโˆ’๐‘„๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก๎€ธ๎ฌโ‰ฅ0.(3.19) Therefore, ๎ซ๐น๎€ท๐‘ฅ๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐ต๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘กโ‰ค๎ซ๎€ท๐‘ฆ๎€ธ๎ฌ(๐ตโˆ’๐น)๐‘กโˆ’๐‘ฅ๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก๎€ธ๎ฌ.(3.20) Since ๐ต,๐น is strongly positive, we have โ€–โ€–0โ‰ค(๐›ผโˆ’๐œŒ)๐‘งโˆ’๐‘ฅ๐‘กโ€–โ€–2โ‰ค๎ซ๎€ท(๐ตโˆ’๐น)๐‘งโˆ’๐‘ฅ๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก=๐น๎€ท๐‘ฅ๎€ธ๎ฌ๎ซ๎€ท๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐ต๐‘ก๎€ท๎€ธ๎€ธโˆ’(๐น(๐‘ง)โˆ’๐ต(๐‘ง)),๐‘—๐‘งโˆ’๐‘ฅ๐‘ก.๎€ธ๎ฌ(3.21) It follows that ๎ซ๐น๎€ท(๐‘ง)โˆ’๐ต(๐‘ง),๐‘—๐‘งโˆ’๐‘ฅ๐‘กโ‰ค๎ซ๐น๎€ท๐‘ฅ๎€ธ๎ฌ๐‘ก๎€ธ๎€ท๐‘ฅโˆ’๐ต๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก๎€ธ๎ฌ.(3.22) Combining (3.20) and (3.22), we get ๎ซ๐น๎€ท(๐‘ง)โˆ’๐ต(๐‘ง),๐‘—๐‘งโˆ’๐‘ฅ๐‘กโ‰ค๎ซ๎€ท๐‘ฆ๎€ธ๎ฌ(๐ตโˆ’๐น)๐‘กโˆ’๐‘ฅ๐‘ก๎€ธ๎€ท,๐‘—๐‘งโˆ’๐‘ฅ๐‘ก๎€ธ๎ฌ.(3.23) Now replacing ๐‘ก in (3.23) with ๐‘ก๐‘› and letting ๐‘›โ†’โˆž, noticing that ๐‘ฅ๐‘ก๐‘›โˆ’๐‘ฆ๐‘ก๐‘›โ†’0, we obtain โŸจ๐น(๐‘ง)โˆ’๐ต(๐‘ง),๐‘—(๐‘งโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ค0,๐‘งโˆˆฮฉ,(3.24) which is equivalent to its dual variational inequality (see Lemma 2.8) โŸจ(๐ตโˆ’๐น)ฬƒ๐‘ฅ,๐‘—(๐‘งโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ฅ0,๐‘งโˆˆฮฉ,(3.25) that is, ฬƒ๐‘ฅโˆˆฮฉ is a solution of (3.4).
(IV) Now we show that the solution set of (3.4) is singleton.
As a matter of fact, we assume that ๐‘ฅโˆ—โˆˆฮฉ is also a solution of (3.4) Then, we have ๎ซ(๐ตโˆ’๐น)๐‘ฅโˆ—๎€ท,๐‘—ฬƒ๐‘ฅโˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ฅ0.(3.26) From (3.25), we have ๎ซ๎€ท๐‘ฅ(๐ตโˆ’๐น)ฬƒ๐‘ฅ,๐‘—โˆ—โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ฅ0.(3.27) So, ๎ซ(๐ตโˆ’๐น)๐‘ฅโˆ—๎€ท,๐‘—ฬƒ๐‘ฅโˆ’๐‘ฅโˆ—+๎ซ๎€ท๐‘ฅ๎€ธ๎ฌ(๐ตโˆ’๐น)ฬƒ๐‘ฅ,๐‘—โˆ—โŸน๎ซ(๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ฅ0๐ตโˆ’๐น)ฬƒ๐‘ฅโˆ’๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ,๐‘—โˆ—โŸน๎ซ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ฅ0(๐ตโˆ’๐น)โˆ—๎€ธ๎€ท๐‘ฅโˆ’ฬƒ๐‘ฅ,๐‘—โˆ—โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0โŸน(๐›ผโˆ’๐œŒ)โ€–๐‘ฅโˆ—โˆ’ฬƒ๐‘ฅโ€–2โ‰ค0.(3.28) Therefore, ๐‘ฅโˆ—=ฬƒ๐‘ฅ. In summary, we have shown that each cluster point of {๐‘ฅ๐‘ก} (as ๐‘กโ†’0) equals ฬƒ๐‘ฅ. Therefore, ๐‘ฅ๐‘กโ†’ฬƒ๐‘ฅ as ๐‘กโ†’0. This completes the proof.

Next, we introduce our explicit method which is the discretization of the implicit method (3.1).

Algorithm 3.4. Let ๐ถ be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space ๐ธ. Let ฮ ๐ถ be a sunny nonexpansive retraction from ๐ธ to ๐ถ. Let {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of ๐›พ๐‘–-inverse-strongly accretive. Let ๐ตโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐›ผ>0, and let ๐นโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐œŒโˆˆ(0,๐›ผ). For arbitrarily given ๐‘ฅ0โˆˆ๐ถ, let the sequence {๐‘ฅ๐‘›} be generated iteratively by ๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ตฮ ๎€ธ๎€ธ๐ถ๎€ท๐ผโˆ’๐œ†1๐ด1๎€ธฮ ๐ถ๎€ท๐ผโˆ’๐œ†2๐ด2๎€ธโ‹ฏฮ ๐ถ๎€ท๐ผโˆ’๐œ†๐‘๐ด๐‘๎€ธ๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(3.29) where {๐›ผ๐‘›} and {๐›ฝ๐‘›} are two sequences in [0,1] and, for any ๐‘–,๐œ†๐‘–โˆˆ(0,๐›พ๐‘–/๐พ2) is a real number.

Theorem 3.5. Let ๐ถ be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space ๐ธ, and let ฮ ๐ถ be a sunny nonexpansive retraction from ๐ธ to ๐ถ. Let {๐ด๐‘–โˆถ๐ถโ†’๐ธ,๐‘–=1,2,โ€ฆ,๐‘} be a finite family of ๐›พ๐‘–-inverse-strongly accretive.Let ๐ตโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐›ผ>0, and let ๐นโˆถ๐ถโ†’๐ธ be a strongly positive bounded linear operator with coefficient ๐œŒโˆˆ(0,๐›ผ). Assume that ฮฉโ‰ โˆ…. For given ๐‘ฅ0โˆˆ๐ถ, let {๐‘ฅ๐‘›} be generated iteratively by (3.29). Suppose the sequences {๐›ผ๐‘›} and {๐›ฝ๐‘›} satisfy the following conditions:(1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and ฮฃโˆž๐‘›=1๐›ผ๐‘›=โˆž,(2)0<liminf๐‘›โ†’โˆž๐›ฝ๐‘›โ‰คlimsup๐‘›โ†’โˆž๐›ฝ๐‘›โ‰ค1.Then {๐‘ฅ๐‘›} converges strongly to ฬƒ๐‘ฅโˆˆฮฉ which solves the variational inequality (3.4).

Proof. Set ๐‘ฆ๐‘›=ฮ ๐ถ(๐ผโˆ’๐œ†1๐ด1)ฮ ๐ถ(๐ผโˆ’๐œ†2๐ด2)โ‹ฏฮ ๐ถ(๐ผโˆ’๐œ†๐‘๐ด๐‘)๐‘ฅ๐‘› for all ๐‘›โ‰ฅ0. Then ๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘ฅ๐‘›+(1โˆ’๐›ฝ๐‘›)ฮ ๐ถ(๐›ผ๐‘›๐น+(๐ผโˆ’๐›ผ๐‘›๐ต))๐‘ฆ๐‘› for all ๐‘›โ‰ฅ0. Pick up ๐‘ฅโˆ—โˆˆฮฉ.
From Lemma 2.10, we have โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–=โ€–โ€–๐‘„๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘„โˆ—๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–.(3.30) Hence, it follows that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅโˆ—โ€–โ€–=โ€–โ€–๐›ฝ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–=โ€–โ€–๐›ฝ๐‘›๎€ท๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›ฮ ๎€ธ๎€ท๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘ฅโˆ—๎€ธโ€–โ€–โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–ฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’ฮ ๐ถ๐‘ฅโˆ—โ€–โ€–โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–=๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๎€ท๐‘›โˆ’๐‘ฅโˆ—๎€ธ+๐›ผ๐‘›๎€ท๐น๎€ท๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—โ€–โ€–๎€ธ๎€ธโ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๎€ท1โˆ’๐›ฝ๐‘›๐›ผ๎€ธ๎€ท๐‘›๎€ท๐œŒ+1โˆ’๐›ผ๐‘›๐›ผโ€–โ€–๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐›ผ๐‘›โ€–โ€–๐น๎€ท๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–โ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎€ธโ€–โ€–๐‘ฅ(๐›ผโˆ’๐œŒ)๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐น๎€ท๐‘ฅ(๐›ผโˆ’๐œŒ)โˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–.๐›ผโˆ’๐œŒ(3.31) By induction, we deduce that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅโˆ—โ€–โ€–๎ƒฏโ€–โ€–๐‘ฅโ‰คmax0โˆ’๐‘ฅโˆ—โ€–โ€–,โ€–โ€–๐น๎€ท๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅโˆ’๐ตโˆ—๎€ธโ€–โ€–๎ƒฐ๐›ผโˆ’๐œŒ.(3.32) Therefore, {๐‘ฅ๐‘›} is bounded. Hence, {๐ด๐‘–๐‘ฅ๐‘–}(๐‘–=1,2,โ€ฆ,๐‘),{๐‘ฆ๐‘›},{๐ต๐‘ฆ๐‘›}, and {๐น(๐‘ฆ๐‘›)} are also bounded. We observe that โ€–โ€–๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›โ€–โ€–=โ€–โ€–๐‘„๎€ท๐‘ฅ๐‘›+1๎€ธ๎€ท๐‘ฅโˆ’๐‘„๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–.(3.33) Set ๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘ฅ๐‘›+(1โˆ’๐›ฝ๐‘›)๐‘ง๐‘› for all ๐‘›โ‰ฅ0. Then ๐‘ง๐‘›=ฮ ๐ถ(๐›ผ๐‘›๐น+(๐ผโˆ’๐›ผ๐‘›๐ต))๐‘ฆ๐‘›. It follows that โ€–โ€–๐‘ง๐‘›+1โˆ’๐‘ง๐‘›โ€–โ€–=โ€–โ€–ฮ ๐ถ๎€ท๐›ผ๐‘›+1๎€ท๐น+๐ผโˆ’๐›ผn+1๐ต๐‘ฆ๎€ธ๎€ธ๐‘›+1โˆ’ฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โ€–โ€–โ‰คโ€–โ€–๎€ท๐›ผ๐‘›+1๎€ท๐น+๐ผโˆ’๐›ผ๐‘›+1๐ต๐‘ฆ๎€ธ๎€ธ๐‘›+1โˆ’๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โ€–โ€–=โ€–โ€–๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›+๐›ผ๐‘›+1๎€ท๐น๎€ท๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›+1๎€ธ๎€ธโˆ’๐›ผ๐‘›๎€ท๐น๎€ท๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฆ๎€ธ๎€ธ๐‘›+1โˆ’๐‘ฆ๐‘›โ€–โ€–+๐›ผ๐‘›+1โ€–โ€–๐น๎€ท๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›+1๎€ธโ€–โ€–โˆ’๐›ผ๐‘›โ€–โ€–๐น๎€ท๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–+๐›ผ๐‘›+1โ€–โ€–๐น๎€ท๐‘ฆ๐‘›+1๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›+1๎€ธโ€–โ€–โˆ’๐›ผ๐‘›โ€–โ€–๐น๎€ท๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›๎€ธโ€–โ€–.(3.34) This implies that limsup๐‘›โ†’โˆž๎€ทโ€–โ€–๐‘ง๐‘›+1โˆ’๐‘ง๐‘›โ€–โ€–โˆ’โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–๎€ธโ‰ค0.(3.35) Hence, by Lemma 2.5, we obtain lim๐‘›โ†’โˆžโ€–๐‘ง๐‘›โˆ’๐‘ฅ๐‘›โ€–=0. Consequently, lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–=lim๐‘›โ†’โˆž๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ง๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–=0.(3.36) At the same time, we note that โ€–โ€–๐‘ง๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=โ€–โ€–ฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=โ€–โ€–ฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’ฮ ๐ถ๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=๐›ผ๐‘›โ€–โ€–๐น๎€ท๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›๎€ธโ€–โ€–โŸถ0.(3.37) It follows that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–=0.(3.38) From Lemma 2.10, we know that ๐‘„โˆถ๐ถโ†’๐ถ is nonexpansive. Thus, we have โ€–โ€–๐‘ฆ๐‘›๎€ท๐‘ฆโˆ’๐‘„๐‘›๎€ธโ€–โ€–=โ€–โ€–๐‘„๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฆโˆ’๐‘„๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โŸถ0.(3.39) Thus, lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘„(๐‘ฅ๐‘›)โ€–=0. We note that โ€–โ€–๐‘ง๐‘›๎€ท๐‘งโˆ’๐‘„๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ง๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐‘„๐‘›๎€ธโ€–โ€–+โ€–โ€–๐‘„๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘งโˆ’๐‘„๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘งโ‰ค2๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐‘„๐‘›๎€ธโ€–โ€–โ€–โ€–ฮ =2๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’ฮ ๐ถ๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐‘„๐‘›๎€ธโ€–โ€–๎€ทโ€–โ€–๐‘ฆโ‰ค2๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+๐›ผ๐‘›โ€–โ€–๐น๎€ท๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฆโˆ’๐ต๐‘›๎€ธโ€–โ€–๎€ธ+โ€–โ€–๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐‘„๐‘›๎€ธโ€–โ€–โŸถ0.(3.40) Next, we show that limsup๐‘›โ†’โˆž๎ซ๐น๎€ท๐‘ง(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌโ‰ค0,(3.41) where ฬƒ๐‘ฅโˆˆฮฉ is the unique solution of VI(3.4).
To see this, we take a subsequence {๐‘ง๐‘›๐‘—} of {๐‘ง๐‘›} such that lim๐‘›โ†’โˆž๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ=lim๐‘›๐‘—โ†’โˆž๎‚ฌ๎‚€๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›๐‘—โˆ’ฬƒ๐‘ฅ๎‚๎‚ญ.(3.42) We may also assume that ๐‘ง๐‘›๐‘—โ‡€๐‘ง. Note that ๐‘งโˆˆฮฉ in virtue of Lemma 2.7 and (3.40). It follows from the variational inequality (3.4) that lim๐‘›โ†’โˆž๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ=lim๐‘›๐‘—โ†’โˆž๎‚ฌ๎‚€๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›๐‘—โˆ’ฬƒ๐‘ฅ๎‚๎‚ญ=โŸจ๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—(๐‘งโˆ’ฬƒ๐‘ฅ)โŸฉโ‰ค0.(3.43) Since ๐‘ง๐‘›=ฮ ๐ถ(๐›ผ๐‘›๐น+(๐ผโˆ’๐›ผ๐‘›๐ต))๐‘ฆ๐‘›, according to Lemma 2.2, we have ๐›ผ๎ซ๎€ท๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’ฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›๎€ท,๐‘—ฬƒ๐‘ฅโˆ’๐‘ง๐‘›๎€ธ๎ฌโ‰ค0.(3.44) From (3.44), we have โ€–โ€–๐‘ง๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2=๎ซฮ ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+Iโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›๎€ท๐‘งโˆ’ฬƒ๐‘ฅ,๐‘—๐‘›=๎ซฮ โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ๐ถ๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๎€ท๐›ผ๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›๎€ท๐‘ง,๐‘—๐‘›+๐›ผโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ๎ซ๎€ท๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›๎€ท๐‘งโˆ’ฬƒ๐‘ฅ,๐‘—๐‘›โ‰ค๐›ผโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ๎ซ๎€ท๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๐‘›๎€ท๐‘งโˆ’ฬƒ๐‘ฅ,๐‘—๐‘›=๐›ผโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ๎ซ๎€ท๐‘›๎€ท๐น+๐ผโˆ’๐›ผ๐‘›๐ต๐‘ฆ๎€ธ๎€ธ๎€ท๐‘›๎€ธ๎€ท๐‘งโˆ’ฬƒ๐‘ฅ,๐‘—๐‘›โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ+๐›ผ๐‘›๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ(๐›ผโˆ’๐œŒ)๐‘›โ€–โ€–โ€–โ€–๐‘งโˆ’ฬƒ๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ+๐›ผ๐‘›๎ซ๐น๎€ท๐‘ง(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธ(๐›ผโˆ’๐œŒ)22โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+12โ€–โ€–๐‘ง๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐›ผ๐‘›๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.45) It follows that โ€–โ€–๐‘ง๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฆ(๐›ผโˆ’๐œŒ)๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›,โ‰ค๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ(๐›ผโˆ’๐œŒ)๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.46) Finally, we prove ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ. From ๐‘ฅ๐‘›+1=๐›ฝ๐‘›๐‘ฅ๐‘›+(1โˆ’๐›ฝ๐‘›)๐‘ง๐‘› and (3.46), we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๎€ท1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ง๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2โ‰ค๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎‚€๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ(๐›ผโˆ’๐œŒ)๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+2๐›ผ๐‘›๎ซ๎€ท๐‘ง๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›๎‚=๎€ทโˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ1โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๎€ธโ€–โ€–๐‘ฅ(๐›ผโˆ’๐œŒ)๐‘›โ€–โ€–โˆ’ฬƒ๐‘ฅ2+๐›ผ๐‘›๎€ท1โˆ’๐›ฝ๐‘›๎€ธร—๎‚ป2(๐›ผโˆ’๐œŒ)๎ซ๎€ท๐‘ง๐›ผโˆ’๐œŒ๐น(ฬƒ๐‘ฅ)โˆ’๐ต(ฬƒ๐‘ฅ),๐‘—๐‘›๎‚ผ.โˆ’ฬƒ๐‘ฅ๎€ธ๎ฌ(3.47) We can apply Lemma 2.4 to the relation (3.47) and conclude that ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ. This completes the proof.

Acknowledgments

The authors would like to express their thanks to the referees and the editor for their helpful suggestion and comments. This work was supported by the Scientific Reserch Fund of Sichuan Provincial Education Department (09ZB102,11ZB146) and Yunnan University of Finance and Economics.

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