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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 580158, 18 pages
http://dx.doi.org/10.1155/2012/580158
Research Article

Algorithms for a System of General Variational Inequalities in Banach Spaces

1Department of Mathematics, Yibin University, Sichuan, Yibin 644007, China
2College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 21 December 2011; Accepted 6 February 2012

Academic Editor: Zhenyu Huang

Copyright © 2012 Jin-Hua Zhu 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

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 and the sets of positive integers and real numbers, respectively. We also assume that 𝐸 is a real Banach space, 𝐸 is the dual space of 𝐸,𝐶 is a nonempty closed convex subset of 𝐸, and , is the pairing between 𝐸 and 𝐸.

In this paper, we are concerned a finite family of a general system of nonlinear variational inequalities in Banach spaces, which involves finding (𝑥1,𝑥2,,𝑥𝑛)𝐶×𝐶××𝐶 such that 𝜆1𝐴1𝑥2+𝑥1𝑥2,𝑗𝑥𝑥1𝜆0,𝑥𝐶,2𝐴2𝑥3+𝑥2𝑥3,𝑗𝑥𝑥2𝜆0,𝑥𝐶,3𝐴3𝑥4+𝑥3𝑥4,𝑗𝑥𝑥3𝜆0,𝑥𝐶,𝑁1𝐴𝑁1𝑥𝑁+𝑥𝑁1𝑥𝑁,𝑗𝑥𝑥𝑁1𝜆0,𝑥𝐶,𝑁𝐴𝑁𝑥1+𝑥𝑁𝑥1,𝑗𝑥𝑥𝑁0,𝑥𝐶,(1.1) where {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} is a finite family of nonlinear mappings and 𝜆𝑖(𝑖=1,2,,𝑁) are positive real numbers.

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

(I) If 𝐸 is a real Hilbert space and 𝑁=2, then (1.1) reduces to𝜆1𝐴1𝑥2+𝑥1𝑥2,𝑥𝑥1𝜆0,𝑥𝐶,2𝐴2𝑥1+𝑥2𝑥1,𝑥𝑥20,𝑥𝐶,(1.2) which was considered by Ceng et al. [1]. In particular, if 𝐴1=𝐴2=𝐴, then the problem (1.2) reduces to finding (𝑥1,𝑥2)𝐶×𝐶 such that𝜆1𝐴𝑥2+𝑥1𝑥2,𝑥𝑥1𝜆0,𝑥𝐶,2𝐴𝑥1+𝑥2𝑥1,𝑥𝑥20,𝑥𝐶,(1.3) which is defined by Verma [2]. Furthermore, if 𝑥1=𝑥2, then (1.3) reduces to the following variational inequality (VI) of finding 𝑥𝐶 such that𝐴𝑥,𝑥𝑥0,𝑥𝐶.(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 𝐶 of the VI or onto some related set, in order to iteratively reach a solution. In particular, Korpelevich’s extragradient method which was introduced by Korpelevich [3] in 1976 generates a sequence {𝑥𝑛} via the recursion𝑦𝑛=𝑃𝐶𝑥𝑛𝜆𝐴𝑥𝑛,𝑥𝑛+1=𝑃𝐶𝑥𝑛𝜆𝐴𝑦𝑛,𝑛0,(1.5) where 𝑃𝐶 is the metric projection from 𝑛 onto 𝐶, 𝐴𝐶𝐻 is a monotone operator, and 𝜆 is a constant. Korpelevich [3] proved that the sequence {𝑥𝑛} converges strongly to a solution of 𝑉𝐼(𝐶,A). Note that the setting of the space is Euclid space 𝑛.

The literature on the VI is vast, and Korpelevich’s extragradient method has received great attention by many authors, who improved it in various ways. See, for example, [416] and references therein.

(II) If 𝐸 is still a real Banach space and 𝑁=1, then the problem (1.1) reduces to finding 𝑥𝐶 such that𝐴𝑥,𝑗𝑥𝑥0,𝑥C,(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 𝐴 in a Banach space 𝐸:𝑥𝑛+1=𝛼𝑛𝑥𝑛+1𝛼nΠ𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑛1,(1.7) where Π𝐶 is a sunny nonexpansive retraction from 𝐸 to 𝐶. 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 {𝑥𝑛} by 𝑥𝑛𝑥 and 𝑥𝑛𝑥, respectively.

For 𝑞>1, the generalized duality mapping 𝐽𝑞𝐸2𝐸 is defined by𝐽𝑞(𝑥)=𝑓𝐸𝑥,𝑓=𝑥𝑞,𝑓=𝑥𝑞1(2.1) for all 𝑥𝐸. In particular, 𝐽=𝐽2 is called the normalized duality mapping. It is known that 𝐽𝑞(𝑥)=||𝑥||𝑞2 for all 𝑥𝐸. If 𝐸 is a Hilbert space, then 𝐽=𝐼, the identity mapping. Let 𝑈={𝑥𝐸||𝑥||=1}. A Banach space 𝐸 is said to be uniformly convex if, for any 𝜀(0,2], there exists 𝛿>0 such that, for any 𝑥,𝑦𝑈,𝑥𝑦𝜀implies𝑥+𝑦21𝛿.(2.2)

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space 𝐸 is said to be smooth if the limitlim𝑛0𝑥+𝑡𝑦𝑥𝑡(2.3) exists for all 𝑥,𝑦𝑈. It is also said to be uniformly smooth if the previous limit is attained uniformly for 𝑥,𝑦𝑈. The norm of 𝐸 is said to be Fréchet differentiable if, for each 𝑥𝑈, the previous limit is attained uniformly for all 𝑦𝑈. The modulus of smoothness of 𝐸 is defined by1𝜌(𝜏)=sup2(𝑥+𝑦+𝑥𝑦)1𝑥,𝑦𝐸,𝑥=1,𝑦=𝜏,(2.4) where 𝜌[0,)[0,) is function. It is known that 𝐸 is uniformly smooth if and only if lim𝜏0(𝜌(𝜏)/𝜏)=0. Let 𝑞 be a fixed real number with 1<𝑞2. Then a Banach space 𝐸 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)𝑐𝜏𝑞 for all 𝜏>0. Note the following.(1)𝐸 is a uniformly smooth Banach space if and only if 𝐽 is single valued and uniformly continuous on any bounded subset of 𝐸.(2)All Hilbert spaces, 𝐿𝑝 (or 𝑙𝑝) spaces (𝑝2) and the Sobolev spaces 𝑊𝑝𝑚(𝑝2) are 2-uniformly smooth, while 𝐿𝑝 (or 𝑙𝑝) and 𝑊𝑃𝑚 spaces (1<𝑝2) are 𝑝-uniformly smooth.(3)Typical examples of both uniformly convex and uniformly smooth Banach spaces are 𝐿𝑝, where 𝑝>1. More precisely, 𝐿𝑝 is min{𝑝,2}-uniformly smooth for every 𝑝>1.

In our paper, we focus on a 2-uniformly smooth Banach space with the smooth constant 𝐾.

Let 𝐸 be a real Banach space, 𝐶 a nonempty closed convex subset of 𝐸, 𝑇𝐶𝐶 a mapping, and 𝐹(𝑇) the set of fixed points of 𝑇.

Recall that a mapping 𝑇𝐶𝐶 is called nonexpansive if𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐶.(2.5) A bounded linear operator 𝐹𝐶𝐸 is called strongly positive if there exists a constant 𝛾>0 with the property𝐹(𝑥),𝑗(𝑥)𝛾𝑥2,𝑥𝐶.(2.6) A mapping 𝐴𝐶𝐸 is said to be accretive if there exists 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)0,(2.7) for all 𝑥,𝑦𝐶, where 𝐽 is the duality mapping.

A mapping 𝐴 of 𝐶 into 𝐸 is said to be 𝛼-strongly accretive if, for 𝛼>0,𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)𝛼𝑥𝑦2,(2.8) for all 𝑥,𝑦𝐶.

A mapping 𝐴 of 𝐶 into 𝐸 is said to be 𝛼-inverse-strongly accretive if, for 𝛼>0,𝐴𝑥𝐴𝑦,𝑗(𝑥𝑦)𝛼𝐴𝑥𝐴𝑦2,(2.9) for all 𝑥,𝑦𝐶.

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 𝐷 be a subset of 𝐶, and let Π be a mapping of 𝐶 into 𝐷. Then Π is said to be sunny if Π[]Π(𝑥)+𝑡(𝑥Π(𝑥))=Π(𝑥)(2.10) whenever Π(𝑥)+𝑡(𝑥Π(𝑥))𝐶 for 𝑥𝐶 and 𝑡0. A mapping Π of 𝐶 into itself is called a retraction if Π2=Π. If a mapping Π of 𝐶 into itself is a retraction, then Π(𝑧)=𝑧 for every 𝑧𝑅(Π), where 𝑅(Π) is the range of Π. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. Then following lemma concerns the sunny nonexpansive retraction.

Lemma 2.2 (see [21]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸, let 𝐷 be a nonempty subset of 𝐶, and let Π be a retraction from 𝐶 onto 𝐷. Then Π is sunny and nonexpansive if and only if 𝑢Π(𝑢),𝑗(𝑦Π(𝑢))0,(2.11) for all 𝑢𝐶 and 𝑦𝐷.

Remark 2.3. (1) It is well known that if 𝐸 is a Hilbert space, then a sunny nonexpansive retraction Π𝐶 is coincident with the metric projection from 𝐸 onto 𝐶.
(2) Let 𝐶 be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space 𝐸, and let 𝑇 be a nonexpansive mapping of 𝐶 into itself with the set 𝐹(𝑇). Then the set 𝐹(𝑇) is a sunny nonexpansive retract of 𝐶.

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

Lemma 2.4 (see [22]). Assume that {𝛼𝑛} is a sequence of nonnegative real numbers such that 𝛼𝑛+11𝛾𝑛𝛼𝑛+𝛿𝑛,(2.12) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that(a)Σ𝑛=1𝛾𝑛=,(b)limsup𝑛(𝛿𝑛/𝛾𝑛)0 or Σ𝑛=1|𝛿𝑛|<.Then lim𝑛𝛼𝑛=0.

Lemma 2.5 (see [23]). Let 𝑋 be a Banach space, {𝑥𝑛},{𝑦𝑛} be two bounded sequences in 𝑋 and {𝛽𝑛} be a sequence in [0,1] satisfying 0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1.(2.13) Suppose that 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1𝛽𝑛)𝑦𝑛, for all 𝑛1 and limsup𝑛𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛0,(2.14) then limn𝑦𝑛𝑥𝑛=0.

Lemma 2.6 (see [24]). Let 𝐸 be a real 2-uniformly smooth Banach space with the best smooth constant 𝐾. Then the following inequality holds: 𝑥+𝑦2𝑥2+2𝑦,𝐽𝑥+2𝐾𝑦2,𝑥,𝑦𝐸.(2.15)

Lemma 2.7 (see [25]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝐸, and let 𝐺 be a nonexpansive mapping of 𝐶 into itself. If {𝑥𝑛} is a sequence of 𝐶 such that 𝑥𝑛𝑥 and 𝑥𝑛𝐺𝑥𝑛0, then 𝑥 is a fixed point of 𝐺.

Lemma 2.8 (see [26]). Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝐸. Assume that the mapping 𝐹𝐶𝐸 is accretive and weakly continuous along segments (i.e., 𝐹(𝑥+𝑡𝑦)𝐹(𝑥) as 𝑡0). Then the variational inequality 𝑥𝐶,𝐹𝑥,𝑗𝑥𝑥0,𝑥𝐶,(2.16) is equivalent to the dual variational inequality 𝑥𝐶,𝐹𝑥,𝑗𝑥𝑥0,𝑥𝐶.(2.17)

Lemma 2.9. Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝐸. Let Π𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Let {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} be a finite family of 𝛾𝑖-inverse-strongly accretive. For given (𝑥1,𝑥2,,𝑥𝑛)𝐶×𝐶××𝐶, where 𝑥=𝑥1,𝑥𝑖=Π𝐶(𝐼𝜆𝑖𝐴𝑖)𝑥𝑖+1,𝑖{𝑖,2,,𝑁1},𝑥𝑁=Π𝐶(𝐼𝜆𝑁𝐴𝑁)𝑥1, then (𝑥1,𝑥2,,𝑥𝑛) is a solution of the problem (1.1) if and only if 𝑥 is a fixed point of the mapping 𝑄 defined by 𝑄(𝑥)=Π𝐶𝐼𝜆1𝐴1Π𝐶𝐼𝜆2𝐴2Π𝐶𝐼𝜆𝑁𝐴𝑁(𝑥),(2.18) where 𝜆𝑖(𝑖=1,2,,𝑁) are real numbers.

Proof. We can rewrite (1.1) as 𝑥1𝑥2𝜆1𝐴1𝑥2,𝑗𝑥𝑥10,𝑥𝐶,𝑥2𝑥3𝜆2𝐴2𝑥3,𝑗𝑥𝑥2𝑥0,𝑥𝐶,3𝑥4𝜆3𝐴3𝑥4,𝑗𝑥𝑥3𝑥0,𝑥𝐶,𝑁1𝑥𝑁𝜆𝑁1𝐴𝑁1𝑥𝑁,𝑗𝑥𝑥𝑁1𝑥0,𝑥C,𝑁𝑥1𝜆𝑁𝐴𝑁𝑥1,𝑗𝑥𝑥𝑁0,𝑥𝐶.(2.19) By Lemma 2.2, we can check (2.19) is equivalent to 𝑥1=Π𝐶𝐼𝜆1𝐴1𝑥2,𝑥2=Π𝐶𝐼𝜆2𝐴2𝑥3,𝑥𝑁1=Π𝐶𝐼𝜆𝑁1𝐴𝑁1𝑥𝑁,𝑥𝑁=Π𝐶𝐼𝜆𝑁𝐴𝑁𝑥1.𝑄𝑥=Π𝐶𝐼𝜆1𝐴1Π𝐶𝐼𝜆2𝐴2Π𝐶𝐼𝜆𝑁𝐴𝑁𝑥=𝑥.(2.20) This completes the proof.

Throughout this paper, the set of fixed points of the mapping 𝑄 is denoted by Ω.

Lemma 2.10. Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝐸. Let Π𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Let {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} be a finite family of 𝛾𝑖-inverse-strongly accretive. Let 𝑄 be defined as Lemma 2.9. If 0𝜆𝑖𝛾𝑖/𝐾2, then 𝑄𝐶𝐶 is nonexpansive.

Proof. First, we show that for all 𝑖{𝑖,2,,𝑁}, the mapping Π𝐶(𝐼𝜆𝑖𝐴𝑖) is nonexpansive. Indeed, for all 𝑥,𝑦𝐶, from the condition 𝜆𝑖[0,𝛾𝑖/𝐾2] and Lemma 2.6, we have Π𝐶𝐼𝜆𝑖𝐴𝑖𝑥Π𝐶𝐼𝜆𝑖𝐴𝑖𝑦2𝐼𝜆𝑖𝐴𝑖𝑥𝐼𝜆𝑖𝐴𝑖𝑦2=(𝑥𝑦)𝜆𝑖𝐴𝑖𝑥𝐴𝑖𝑦2𝑥𝑦22𝜆𝑖𝐴𝑖𝑥𝐴𝑖𝑦,𝑗(𝑥𝑦)+2𝐾2𝜆2𝑖𝐴𝑖𝑥𝐴𝑖𝑦2𝑥𝑦22𝜆𝑖𝛾𝑖𝐴𝑖𝑥𝐴𝑖𝑦2+2𝐾2𝜆2𝑖𝐴𝑖𝑥𝐴𝑖𝑦2=𝑥𝑦2+2𝜆𝑖𝐾2𝜆𝑖𝛾𝑖𝐴𝑖𝑥𝐴𝑖𝑦2𝑥𝑦2,(2.21) which implies for all 𝑖{1,2,,𝑁}, the mapping Π𝐶(𝐼𝜆𝑖𝐴𝑖) is nonexpansive, so is the mapping 𝑄.

3. Main Results

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

Algorithm 3.1. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸. Let Π𝐶 be a sunny nonexpansive retraction from 𝐸 to 𝐶. Let {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} be a finite family of 𝛾𝑖-inverse-strongly accretive. Let 𝐵𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0 and 𝐹𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝜌(0,𝛼). For any 𝑡(0,1), define a net {𝑥𝑡} as follows: 𝑥𝑡=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡,𝑦𝑡=Π𝐶𝐼𝜆1𝐴1Π𝐶𝐼𝜆2𝐴2Π𝐶𝐼𝜆𝑁𝐴𝑁𝑥𝑡,(3.1) where, for any 𝑖,𝜆𝑖(0,𝛾𝑖/𝐾2) is a real number.

Remark 3.2. We notice that the net {𝑥𝑡} defined by (3.1) is well defined. In fact, we can define a self-mapping 𝑊𝑡𝐶𝐶 as follows: 𝑊𝑡𝑥=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))Π𝐶𝐼𝜆1𝐴1Π𝐶𝐼𝜆2𝐴2Π𝐶𝐼𝜆𝑁𝐴𝑁𝑥,𝑥𝐶.(3.2)
From Lemma 2.10, we know that if, for any 𝑖,𝜆𝑖(0,𝛾𝑖/𝐾2), the mapping Π𝐶(𝐼𝜆1𝐴1)Π𝐶(𝐼𝜆2𝐴2)Π𝐶(𝐼𝜆𝑁𝐴𝑁)=𝑄 is nonexpansive and ||𝐼𝑡𝐵||1𝑡𝛼. Then, for any 𝑥,𝑦𝐶, we have 𝑊𝑡𝑥𝑊𝑡𝑦=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑄(𝑥)Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑄(𝑦)((𝑡𝐹+(𝐼𝑡𝐵))𝑄)𝑥((𝑡𝐹+(𝐼𝑡𝐵))𝑄)𝑦=𝑡(𝐹𝑥𝐹𝑦)+(𝐼𝑡𝐵)(𝑄𝑥𝑄𝑦)𝑡𝜌𝑥𝑦+𝐼𝑡𝐵𝑄𝑥𝑄𝑦𝑡𝜌𝑥𝑦+(1𝑡𝛼)𝑥𝑦=(1(𝛼𝜌)𝑡)𝑥𝑦.(3.3) This shows that the mapping 𝑊𝑡 is contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸. Let Π𝐶 be a sunny nonexpansive retraction from E to 𝐶. Let {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} be a finite family of 𝛾𝑖-inverse-strongly accretive. Let 𝐵𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0, and let 𝐹𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝜌(0,𝛼). Assume that Ω and 𝜆𝑖(0,𝛾𝑖/𝐾2). Then the net {𝑥𝑡} generated by the implicit method (3.1) converges in norm, as 𝑡0+ to the unique solution ̃𝑥 of VI ̃𝑥Ω,(𝐵𝐹)̃𝑥,𝑗(𝑧̃𝑥)0,𝑧Ω.(3.4)

Proof. We divide the proof of Theorem 3.3 into four steps.(I) Next we prove that the net {𝑥𝑡} is bounded.
Take that 𝑥Ω, we have 𝑥𝑡𝑥=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡Π𝐶𝑥(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥=𝑡𝐹𝑦𝑡𝑥𝐹𝑦+(𝐼𝑡𝐵)𝑡𝑥𝑥+𝑡𝐹𝑥𝑡𝐵𝐹𝑦𝑡𝑡𝑥𝐹𝑦+𝐼𝑡𝐵𝑡𝑥𝐹𝑥+𝑡𝑥𝐵𝑦𝑡𝜌𝑡𝑥𝑦+(1𝑡𝛼)𝑡𝑥𝐹𝑥+𝑡𝑥𝐵𝑦=(1(𝛼𝜌)𝑡)𝑡𝑥𝐹𝑥+𝑡𝑥𝐵𝑄𝑥=(1(𝛼𝜌)𝑡)𝑡𝑥𝑄𝐹𝑥+𝑡𝑥𝐵𝑥(1(𝛼𝜌)𝑡)𝑡𝑥𝐹𝑥+𝑡𝑥𝐵.(3.5) It follows that 𝑥𝑡𝑥𝐹𝑥𝑥𝐵𝛼𝜌.(3.6) Therefore, {𝑥𝑡} is bounded. Hence, {𝑦𝑡},{𝐵𝑦𝑡},{𝐴𝑖𝑥𝑡}, and {𝐹(𝑦𝑡)} are also bounded. We observe that 𝑥𝑡𝑦𝑡=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡Π𝐶𝑦𝑡(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑦𝑡𝐹𝑦=𝑡𝑡𝑦𝐵𝑡0.(3.7)
From Lemma 2.10, we know that 𝑄𝐶𝐶 is nonexpansive. Thus, we have 𝑦𝑡𝑦𝑄𝑡=𝑄𝑥𝑡𝑦𝑄𝑡𝑥𝑡𝑦𝑡0.(3.8) Therefore, lim𝑡0𝑥𝑡𝑥𝑄𝑡=0.(3.9)
(II) {𝑥𝑡} is relatively norm-compact as 𝑡0+.
Let {𝑡𝑛}(0,1) be any subsequence such that 𝑡𝑛0+ as 𝑛. Then, there exists a positive integer 𝑛0 such that 0<𝑡𝑛<1/2, for all 𝑛𝑛0. Let 𝑥𝑛=𝑥𝑡𝑛. It follows from (3.9) that 𝑥𝑛𝑥𝑄𝑛0.(3.10) We can rewrite (3.1) as 𝑥𝑡=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡+(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡.(3.11)
For any 𝑥Ω𝐶, by Lemma 2.2, we have (𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥𝑡𝑥,𝑗𝑥𝑡=(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥,𝑗Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡0.(3.12) With this fact, we derive that 𝑥𝑡𝑥2=𝑥𝑥𝑡𝑥,𝑗𝑥𝑡=𝑥(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥,𝑗𝑥𝑡+(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡ΠC(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥,𝑗𝑥𝑡𝑥(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥,𝑗𝑥𝑡𝐵𝑥+𝑡𝑥𝐹𝑥,𝑗𝑥𝑡𝑥(1𝑡(𝛼𝜌))𝑦𝑡𝑥𝑥𝑡𝐵𝑥+𝑡𝑥𝐹𝑥,𝑗𝑥𝑡𝑄𝑥=(1𝑡(𝛼𝜌))𝑥𝑄𝑡𝑥𝑥𝑡𝐵𝑥+𝑡𝑥𝐹𝑥,𝑗𝑥𝑡𝑥(1𝑡(𝛼𝜌))𝑥𝑡𝑥𝑥𝑡𝐵𝑥+𝑡𝑥𝐹𝑥,𝑗𝑥𝑡𝑥(1𝑡(𝛼𝜌))𝑥𝑡2𝐵𝑥+𝑡𝑥𝐹𝑥,𝑗𝑥𝑡.(3.13)
It turns out that 𝑥𝑡𝑥21𝐵𝑥𝛼𝜌𝑥𝐹𝑥,𝑗𝑥𝑡,𝑥Ω.(3.14)
In particular, 𝑥𝑛𝑥21𝐵𝑥𝛼𝜌𝑥𝐹𝑥,𝑗𝑥𝑛,𝑥Ω.()
Since {𝑥𝑛} is bounded, without loss of generality, 𝑥𝑛̃𝑥𝐶 can be assumed. Noticing (3.10), we can use Lemma 2.7 to get ̃𝑥Ω=𝐹(𝑄). Therefore, we can substitute ̃𝑥 for 𝑥 in (**) to get 𝑥𝑛̃𝑥21𝛼𝜌𝐵(̃𝑥)𝐹(̃𝑥),𝑗̃𝑥𝑥𝑛.(3.15) Consequently, the weak convergence of {𝑥𝑛} to ̃𝑥 actually implies that 𝑥𝑛̃𝑥 strongly. This has proved the relative norm compactness of the net {𝑥𝑡} as 𝑡0+.
(III) Now, we prove that ̃𝑥 solves the variational inequality (3.4). From (3.1), we have 𝑥𝑡=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡+(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥𝑡=Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑥(𝑡𝐹+(𝐼𝑡𝐵))𝑡𝑦𝑡𝑥+𝑡𝐹𝑡𝑥+(𝐼𝑡𝐵)𝑡𝑥𝐹𝑡𝑥𝐵𝑡=1𝑡(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡𝑦(𝑡𝐹+(𝐼𝑡𝐵))𝑡𝑥𝑡.(3.16) For any 𝑧Ω, we obtain 𝐹𝑥𝑡𝑥𝐵𝑡,𝑗𝑧𝑥𝑡=1𝑡(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡Π𝐶(𝑡𝐹+(𝐼𝑡𝐵))𝑦𝑡,𝑗𝑧𝑥𝑡1𝑡𝑦(𝑡𝐹+(𝐼𝑡𝐵))𝑡𝑥𝑡,𝑗𝑧𝑥𝑡1𝑡𝑦(𝑡𝐹+(𝐼𝑡𝐵))𝑡𝑥𝑡,𝑗𝑧𝑥𝑡1=𝑡𝑦𝑡𝑥𝑡,𝑗𝑧𝑥𝑡+𝑦(𝐵𝐹)𝑡𝑥𝑡,𝑗𝑧𝑥𝑡.(3.17)
Now we prove that 𝑦𝑡𝑥𝑡,𝑗(𝑧𝑥𝑡)0. In fact, we can write 𝑦𝑡=𝑄(𝑥𝑡). At the same time, we note that 𝑧=𝑄(𝑧), so 𝑦𝑡𝑥𝑡,𝑗𝑧𝑥𝑡=𝑥𝑧𝑄(𝑧)𝑡𝑥𝑄𝑡,𝑗𝑧𝑥𝑡.(3.18) Since 𝐼𝑄 is accretive (this is due to the nonexpansivity of 𝑄), we can deduce immediately that 𝑦𝑡𝑥𝑡,𝑗𝑧𝑥𝑡𝑥=𝑧𝑄(𝑧)𝑡𝑥𝑄𝑡,𝑗𝑧𝑥𝑡0.(3.19) Therefore, 𝐹𝑥𝑡𝑥𝐵𝑡,𝑗𝑧𝑥𝑡𝑦(𝐵𝐹)𝑡𝑥𝑡,𝑗𝑧𝑥𝑡.(3.20) Since 𝐵,𝐹 is strongly positive, we have 0(𝛼𝜌)𝑧𝑥𝑡2(𝐵𝐹)𝑧𝑥𝑡,𝑗𝑧𝑥𝑡=𝐹𝑥𝑡𝑥𝐵𝑡(𝐹(𝑧)𝐵(𝑧)),𝑗𝑧𝑥𝑡.(3.21) It follows that 𝐹(𝑧)𝐵(𝑧),𝑗𝑧𝑥𝑡𝐹𝑥𝑡𝑥𝐵𝑡,𝑗𝑧𝑥𝑡.(3.22) Combining (3.20) and (3.22), we get 𝐹(𝑧)𝐵(𝑧),𝑗𝑧𝑥𝑡𝑦(𝐵𝐹)𝑡𝑥𝑡,𝑗𝑧𝑥𝑡.(3.23) Now replacing 𝑡 in (3.23) with 𝑡𝑛 and letting 𝑛, noticing that 𝑥𝑡𝑛𝑦𝑡𝑛0, we obtain 𝐹(𝑧)𝐵(𝑧),𝑗(𝑧̃𝑥)0,𝑧Ω,(3.24) which is equivalent to its dual variational inequality (see Lemma 2.8) (𝐵𝐹)̃𝑥,𝑗(𝑧̃𝑥)0,𝑧Ω,(3.25) that is, ̃𝑥Ω 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 𝑥Ω is also a solution of (3.4) Then, we have (𝐵𝐹)𝑥,𝑗̃𝑥𝑥0.(3.26) From (3.25), we have 𝑥(𝐵𝐹)̃𝑥,𝑗̃𝑥0.(3.27) So, (𝐵𝐹)𝑥,𝑗̃𝑥𝑥+𝑥(𝐵𝐹)̃𝑥,𝑗(̃𝑥0𝐵𝐹)̃𝑥𝑥𝑥,𝑗𝑥̃𝑥0(𝐵𝐹)𝑥̃𝑥,𝑗̃𝑥0(𝛼𝜌)𝑥̃𝑥20.(3.28) Therefore, 𝑥=̃𝑥. In summary, we have shown that each cluster point of {𝑥𝑡} (as 𝑡0) equals ̃𝑥. Therefore, 𝑥𝑡̃𝑥 as 𝑡0. This completes the proof.

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

Algorithm 3.4. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸. Let Π𝐶 be a sunny nonexpansive retraction from 𝐸 to 𝐶. Let {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} be a finite family of 𝛾𝑖-inverse-strongly accretive. Let 𝐵𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0, and let 𝐹𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝜌(0,𝛼). For arbitrarily given 𝑥0𝐶, let the sequence {𝑥𝑛} be generated iteratively by 𝑥𝑛+1=𝛽𝑛𝑥𝑛+1𝛽𝑛Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵Π𝐶𝐼𝜆1𝐴1Π𝐶𝐼𝜆2𝐴2Π𝐶𝐼𝜆𝑁𝐴𝑁𝑥𝑛,𝑛0,(3.29) where {𝛼𝑛} and {𝛽𝑛} are two sequences in [0,1] and, for any 𝑖,𝜆𝑖(0,𝛾𝑖/𝐾2) is a real number.

Theorem 3.5. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝐸, and let Π𝐶 be a sunny nonexpansive retraction from 𝐸 to 𝐶. Let {𝐴𝑖𝐶𝐸,𝑖=1,2,,𝑁} be a finite family of 𝛾𝑖-inverse-strongly accretive.Let 𝐵𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝛼>0, and let 𝐹𝐶𝐸 be a strongly positive bounded linear operator with coefficient 𝜌(0,𝛼). Assume that Ω. For given 𝑥0𝐶, let {𝑥𝑛} be generated iteratively by (3.29). Suppose the sequences {𝛼𝑛} and {𝛽𝑛} satisfy the following conditions:(1)lim𝑛𝛼𝑛=0 and Σ𝑛=1𝛼𝑛=,(2)0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛1.Then {𝑥𝑛} converges strongly to ̃𝑥Ω which solves the variational inequality (3.4).

Proof. Set 𝑦𝑛=Π𝐶(𝐼𝜆1𝐴1)Π𝐶(𝐼𝜆2𝐴2)Π𝐶(𝐼𝜆𝑁𝐴𝑁)𝑥𝑛 for all 𝑛0. Then 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1𝛽𝑛)Π𝐶(𝛼𝑛𝐹+(𝐼𝛼𝑛𝐵))𝑦𝑛 for all 𝑛0. Pick up 𝑥Ω.
From Lemma 2.10, we have 𝑦𝑛𝑥=𝑄𝑥𝑛𝑥𝑄𝑥𝑛𝑥.(3.30) Hence, it follows that 𝑥𝑛+1𝑥=𝛽𝑛𝑥𝑛+1𝛽𝑛Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑥=𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑥𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛Π𝐶𝑥𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑥=𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑥+𝛼𝑛𝐹𝑥𝑥𝐵𝛽𝑛𝑥𝑛𝑥+1𝛽𝑛𝛼𝑛𝜌+1𝛼𝑛𝛼𝑦𝑛𝑥+1𝛽𝑛𝛼𝑛𝐹𝑥𝑥𝐵1𝛼𝑛1𝛽𝑛𝑥(𝛼𝜌)𝑛𝑥+𝛼𝑛1𝛽𝑛𝐹𝑥(𝛼𝜌)𝑥𝐵.𝛼𝜌(3.31) By induction, we deduce that 𝑥𝑛+1𝑥𝑥max0𝑥,𝐹𝑥𝑥𝐵𝛼𝜌.(3.32) Therefore, {𝑥𝑛} is bounded. Hence, {𝐴𝑖𝑥𝑖}(𝑖=1,2,,𝑁),{𝑦𝑛},{𝐵𝑦𝑛}, and {𝐹(𝑦𝑛)} are also bounded. We observe that 𝑦𝑛+1𝑦𝑛=𝑄𝑥𝑛+1𝑥𝑄𝑛𝑥𝑛+1𝑥𝑛.(3.33) Set 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1𝛽𝑛)𝑧𝑛 for all 𝑛0. Then 𝑧𝑛=Π𝐶(𝛼𝑛𝐹+(𝐼𝛼𝑛𝐵))𝑦𝑛. It follows that 𝑧𝑛+1𝑧𝑛=Π𝐶𝛼𝑛+1𝐹+𝐼𝛼n+1𝐵𝑦𝑛+1Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝛼𝑛+1𝐹+𝐼𝛼𝑛+1𝐵𝑦𝑛+1𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛=𝑦𝑛+1𝑦𝑛+𝛼𝑛+1𝐹𝑦𝑛+1𝑦𝐵𝑛+1𝛼𝑛𝐹𝑦𝑛𝑦𝐵𝑛𝑦𝑛+1𝑦𝑛+𝛼𝑛+1𝐹𝑦𝑛+1𝑦𝐵𝑛+1𝛼𝑛𝐹𝑦𝑛𝑦𝐵𝑛𝑥𝑛+1𝑥𝑛+𝛼𝑛+1𝐹𝑦𝑛+1𝑦𝐵𝑛+1𝛼𝑛𝐹𝑦𝑛𝑦𝐵𝑛.(3.34) This implies that limsup𝑛𝑧𝑛+1𝑧𝑛𝑥𝑛+1𝑥𝑛0.(3.35) Hence, by Lemma 2.5, we obtain lim𝑛𝑧𝑛𝑥𝑛=0. Consequently, lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛1𝛽𝑛𝑧𝑛𝑥𝑛=0.(3.36) At the same time, we note that 𝑧𝑛𝑦𝑛=Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑦𝑛=Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛Π𝐶𝑦𝑛𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑦𝑛=𝛼𝑛𝐹𝑦𝑛𝑦𝐵𝑛0.(3.37) It follows that lim𝑛𝑥𝑛𝑦𝑛=0.(3.38) From Lemma 2.10, we know that 𝑄𝐶𝐶 is nonexpansive. Thus, we have 𝑦𝑛𝑦𝑄𝑛=𝑄𝑥𝑛𝑦𝑄𝑛𝑥𝑛𝑦𝑛0.(3.39) Thus, lim𝑛𝑥𝑛𝑄(𝑥𝑛)=0. We note that 𝑧𝑛𝑧𝑄𝑛𝑧𝑛𝑥𝑛+𝑥𝑛𝑥𝑄𝑛+𝑄𝑥𝑛𝑧𝑄𝑛𝑧2𝑛𝑥𝑛+𝑥𝑛𝑥𝑄𝑛Π=2𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛Π𝐶𝑥𝑛+𝑥𝑛𝑥𝑄𝑛𝑦2𝑛𝑥𝑛+𝛼𝑛𝐹𝑦𝑛𝑦𝐵𝑛+𝑥𝑛𝑥𝑄𝑛0.(3.40) Next, we show that limsup𝑛𝐹𝑧(̃𝑥)𝐵(̃𝑥),𝑗𝑛̃𝑥0,(3.41) where ̃𝑥Ω is the unique solution of VI(3.4).
To see this, we take a subsequence {𝑧𝑛𝑗} of {𝑧𝑛} such that lim𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛̃𝑥=lim𝑛𝑗𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛𝑗̃𝑥.(3.42) We may also assume that 𝑧𝑛𝑗𝑧. Note that 𝑧Ω in virtue of Lemma 2.7 and (3.40). It follows from the variational inequality (3.4) that lim𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛̃𝑥=lim𝑛𝑗𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛𝑗̃𝑥=𝐹(̃𝑥)𝐵(̃𝑥),𝑗(𝑧̃𝑥)0.(3.43) Since 𝑧𝑛=Π𝐶(𝛼𝑛𝐹+(𝐼𝛼𝑛𝐵))𝑦𝑛, according to Lemma 2.2, we have 𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛Π𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛,𝑗̃𝑥𝑧𝑛0.(3.44) From (3.44), we have 𝑧𝑛̃𝑥2=Π𝐶𝛼𝑛𝐹+I𝛼𝑛𝐵𝑦𝑛𝑧̃𝑥,𝑗𝑛=Π̃𝑥𝐶𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝛼𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑧,𝑗𝑛+𝛼̃𝑥𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑧̃𝑥,𝑗𝑛𝛼̃𝑥𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑧̃𝑥,𝑗𝑛=𝛼̃𝑥𝑛𝐹+𝐼𝛼𝑛𝐵𝑦𝑛𝑧̃𝑥,𝑗𝑛̃𝑥+𝛼𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛̃𝑥1𝛼𝑛𝑦(𝛼𝜌)𝑛𝑧̃𝑥𝑛̃𝑥+𝛼𝑛𝐹𝑧(̃𝑥)𝐵(̃𝑥),𝑗𝑛̃𝑥1𝛼𝑛(𝛼𝜌)22𝑦𝑛̃𝑥2+12𝑧𝑛̃𝑥2+𝛼𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛.̃𝑥(3.45) It follows that 𝑧𝑛̃𝑥21𝛼𝑛𝑦(𝛼𝜌)𝑛̃𝑥2+2𝛼𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛,̃𝑥1𝛼𝑛𝑥(𝛼𝜌)𝑛̃𝑥2+2𝛼𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛.̃𝑥(3.46) Finally, we prove 𝑥𝑛̃𝑥. From 𝑥𝑛+1=𝛽𝑛𝑥𝑛+(1𝛽𝑛)𝑧𝑛 and (3.46), we have 𝑥𝑛+1̃𝑥2𝛽𝑛𝑥𝑛̃𝑥2+1𝛽𝑛𝑧𝑛̃𝑥2𝛽𝑛𝑥𝑛̃𝑥2+1𝛽𝑛1𝛼𝑛𝑥(𝛼𝜌)𝑛̃𝑥2+2𝛼𝑛𝑧𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛=̃𝑥1𝛼𝑛1𝛽𝑛𝑥(𝛼𝜌)𝑛̃𝑥2+𝛼𝑛1𝛽𝑛×2(𝛼𝜌)𝑧𝛼𝜌𝐹(̃𝑥)𝐵(̃𝑥),𝑗𝑛.̃𝑥(3.47) We can apply Lemma 2.4 to the relation (3.47) and conclude that 𝑥𝑛̃𝑥. 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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