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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 472935, 14 pages
http://dx.doi.org/10.1155/2012/472935
Research Article

Implicit Iterative Method for Hierarchical Variational Inequalities

1Scientific Computing Key Laboratory of Shanghai Universities, Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
5Center for General Education, Kaohsiung Medical University, Kaohsiung 80708, Taiwan

Received 1 February 2012; Accepted 10 February 2012

Academic Editor: Yonghong Yao

Copyright © 2012 L.-C. Ceng 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

We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product , and norm and 𝐶 a nonempty closed convex subset of 𝐻. For a given nonlinear operator 𝐴𝐶𝐻, the classical variational inequality problem (VIP) [1] is to find 𝑥𝐶 such that 𝐴𝑥,𝑥𝑥0,𝑥𝐶.(1.1) The set of solutions of VIP is denoted by VI(𝐶,𝐴). If the set 𝐶 is replaced by the set Fix(𝑇) of fixed points of a mapping 𝑇; then the VIP is called a hierarchical variational inequality problem (HVIP). The signal recovery [2], the power control problem [3], and the beamforming problem [4] can be written in the form of a hierarchical variational inequality problem. In the recent past, several authors paid their attention toward this kind of problem and developed different kinds of solution methods with applications; see [2, 511] and the references therein.

Let 𝐹𝐻𝐻 be 𝜂-strongly monotone (i.e., if there exists a constant 𝜂>0 such that 𝐹𝑥𝐹𝑦,𝑥𝑦𝜂𝑥𝑦2, for all 𝑥,𝑦𝐻) and 𝜅-Lipschitz continuous (i.e., if there exists a constant 𝜅>0 such that 𝐹𝑥𝐹𝑦𝜅𝑥𝑦, for all 𝑥,𝑦𝐻). Assume that 𝐶 is the intersection of the sets of fixed points of 𝑁 nonexpansive mappings 𝑇𝑖𝐻𝐻. For an arbitrary initial guess 𝑥0𝐻, Yamada [10] proposed the following hybrid steepest-descent method: 𝑥𝑛+1=𝑇𝑛+1𝑥𝑛𝜆𝑛+1𝑇𝜇𝐹𝑛+1𝑥𝑛,𝑛0.(1.2) Here, 𝑇𝑘=𝑇𝑘mod𝑁, for every integer 𝑘>𝑁, with the mod function taking values in the set {1,2,,𝑁}; that is, if 𝑘=𝑗𝑁+𝑞 for some integers 𝑗0 and 0𝑞<𝑁, then 𝑇𝑘=𝑇𝑁 if 𝑞=0 and 𝑇𝑘=𝑇𝑞 if 1<𝑞<𝑁. Moreover, 𝜇(0,2𝜂/𝜅2) and the sequence {𝜆𝑛}(0,1) of parameters satisfies the following conditions:(i)lim𝑛𝜆𝑛=0;(ii)𝑛=0𝜆𝑛=;(iii)𝑛=0|𝜆𝑛𝜆𝑛+𝑁| is convergent.

Under these conditions, Yamada [10] proved the strong convergence of the sequence {𝑥𝑛} to the unique element of VI(𝐶,𝐹).

Xu and Kim [12] replaced the condition (iii) by the following condition:(iii)′lim𝑛(𝜆𝑛/𝜆𝑛+𝑁)=1, or equivalently, lim𝑛((𝜆𝑛𝜆𝑛+𝑁)/𝜆𝑛+𝑁)=0

and proved the strong convergence of the sequence {𝑥𝑛} to the unique element of VI(𝐶,𝐹).

On the other hand, let 𝐾 be a nonempty convex subset of 𝐻, and let {𝑇𝑖}𝑁𝑖=1 be a finite family of nonexpansive self-maps on K. Xu and Ori [13] introduced the following implicit iteration process: for 𝑥0𝐾 and {𝛼𝑛}𝑛=1(0,1), the sequence {𝑥𝑛}𝑛=1 is generated by the following process: 𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛,𝑛1,(1.3) where we use the convention 𝑇𝑛=𝑇𝑛mod𝑁. They also studied the weak convergence of the sequence generated by the above scheme to a common fixed point of the mappings {𝑇𝑖}𝑁𝑖=1 under certain conditions. Subsequently, Zeng and Yao [14] introduced another implicit iterative scheme with perturbation for finding the approximate common fixed points of a finite family of nonexpansive self-maps on 𝐻.

Motivated and inspired by the above works, in this paper, we propose a new implicit iterative scheme with perturbation for finding the approximate solutions of the hierarchical variational inequalities, that is, variational inequality problem over the common fixed point set of a finite family of nonexpansive self-maps on 𝐻. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme with perturbation. In particular, necessary and sufficient conditions for strong convergence of the sequence generated by the proposed implicit iterative scheme with perturbation are obtained.

2. Preliminaries

Throughout the paper, we write 𝑥𝑛𝑥 to indicate that the sequence {𝑥𝑛} converges weakly to 𝑥 in a Banach space 𝐸. Meanwhile, 𝑥𝑛𝑥 implies that {𝑥𝑛} converges strongly to 𝑥. For a given sequence {𝑥𝑛}𝐸, 𝜔𝑤(𝑥𝑛) denotes the weak 𝜔-limit set of {𝑥𝑛}, that is, 𝜔𝑤𝑥𝑛=𝑥𝐸𝑥𝑛𝑗𝑛𝑥forsomesubsequence𝑗of{𝑛}.(2.1) A Banach space 𝐸 is said to satisfy Opial’s property if liminf𝑛𝑥𝑛𝑥<liminf𝑛𝑥𝑛𝑦𝑦𝐸,𝑦𝑥,(2.2) whenever a sequence 𝑥𝑛𝑥 in 𝐸. It is well known that every Hilbert space 𝐻 satisfies Opial’s property; see for example [15].

A mapping 𝐴𝐻𝐻 is said to be hemicontinuous if for any 𝑥,𝑦𝐻, the mapping 𝑔[0,1]𝐻, defined by 𝑔(𝑡)=𝐴(𝑡𝑥+(1𝑡)𝑦)(forall𝑡[0,1]), is continuous in the weak topology of the Hilbert space 𝐻. The metric projection onto a nonempty, closed and convex set 𝐶𝐻, denoted by 𝑃𝐶, is defined by, for all 𝑥𝐻,𝑃𝐶𝑥𝐶 and 𝑥𝑃𝐶𝑥=inf𝑦𝐶𝑥𝑦.

Proposition 2.1. Let 𝐶𝐻 be a nonempty closed and convex set and 𝐴𝐻𝐻 monotone and hemicontinuous. Then,(a)[1] VI(𝐶,𝐴)={𝑥𝐶𝐴𝑦,𝑦𝑥0,forall𝑦𝐶},(b)[1] VI(𝐶,𝐴) when 𝐶 is bounded,(c)[16, Lemma  2.24] VI(𝐶,𝐴)=Fix(𝑃𝐶(𝐼𝜆𝐴)) for all 𝜆>0, where 𝐼 stands for the identity mapping on 𝐻,(d)[16, Theorem  2.31] VI(𝐶,𝐴) consists of one point if 𝐴 is strongly monotone and Lipschitz continuous.

On the other hand, it is well known that the metric projection 𝑃𝐶 onto a given nonempty closed and convex set 𝐶𝐻 is nonexpansive with Fix(𝑃𝐶)=𝐶 [17, Theorem  3.1.4 (i)]. The fixed point set of a nonexpansive mapping has the following properties.

Proposition 2.2. Let 𝐶𝐻 be a nonempty closed and convex subset and 𝑇𝐶𝐶 a nonexpansive map.(a)[18, Proposition  5.3] Fix(𝑇) is closed and convex.(b)[18, Theorem  5.1] Fix(𝑇) when 𝐶 is bounded.

The following proposition provides an example of a nonexpansive mapping in which the set of fixed points is equal to the solution set of a monotone variational inequality.

Proposition 2.3 (see [6, Proposition  2.3]). Let 𝐶𝐻 be a nonempty closed and convex set and 𝐴𝐻𝐻 an 𝛼-inverse-strongly monotone operator. Then, for any given 𝜆(0,2𝛼], the mapping 𝑆𝜆𝐻𝐻, defined by 𝑆𝜆𝑥=𝑃𝐶(𝐼𝜆𝐴)𝑥,𝑥𝐻,(2.3) is nonexpansive and Fix(𝑆𝜆)=VI(𝐶,𝐴).

The following lemmas will be used in the proof of the main results of this paper.

Lemma 2.4 (see [18, Demiclosedness Principle]). Assume that 𝑇 is a nonexpansive self-mapping on a closed convex subset 𝐾 of a Hilbert space 𝐻. If 𝑇 has a fixed point, then 𝐼𝑇 is demiclosed, that is, whenever {𝑥𝑛} is a sequence in 𝐾 weakly converging to some 𝑥𝐾 and the sequence {(𝐼𝑇)𝑥𝑛} strongly converges to some 𝑦, it follows that (𝐼𝑇)𝑥=𝑦, where 𝐼 is the identity operator of 𝐻.

Lemma 2.5 (see [19, page 80]). Let {𝑎𝑛}𝑛=1, {𝑏𝑛}𝑛=1, and {𝛿𝑛}𝑛=1 be sequences of nonnegative real numbers satisfying the inequality 𝑎𝑛+11+𝛿𝑛𝑎𝑛+𝑏𝑛,𝑛1.(2.4) If 𝑛=1𝛿𝑛< and 𝑛=1𝑏𝑛<, then lim𝑛𝑎𝑛 exists. If in addition {𝑎𝑛}𝑛=1 has a subsequence, which converges to zero, then lim𝑛𝑎𝑛=0.

Let 𝑇𝐻𝐻 be a nonexpansive mapping and 𝐹𝐻𝐻  𝜅-Lipschitz continuous and 𝜂-strongly monotone for some constants 𝜅>0, 𝜂>0. For any given numbers 𝜆[0,1) and 𝜇(0,2𝜂/𝜅2), we define the mapping 𝑇𝜆𝐻𝐻 by𝑇𝜆𝑥=𝑇𝑥𝜆𝜇𝐹(𝑇𝑥),𝑥𝐻.(2.5)

Lemma 2.6 (see [12]). If 0𝜆<1 and 0<𝜇<2𝜂/𝜅2, then 𝑇𝜆𝑥𝑇𝜆𝑦(1𝜆𝜏)𝑥𝑦,𝑥,𝑦𝐻,(2.6) where 𝜏=11𝜇(2𝜂𝜇𝜅2)(0,1).

3. An Iterative Scheme and Convergence Results

Let {𝑇𝑖}𝑁𝑖=1 be a finite family of nonexpansive self-maps on 𝐻. Let 𝐴𝐻𝐻 be an 𝛼-inverse-strongly monotone mapping (i.e., if there exists a constant 𝛼>0 such that 𝐴𝑥𝐴𝑦,𝑥𝑦𝛼𝐴𝑥𝐴𝑦2, for all 𝑥,𝑦𝐻). Let 𝐹𝐻𝐻 be 𝜅-Lipschitz continuous and 𝜂-strongly monotone for some constants 𝜅>0, 𝜂>0. Let {𝛼𝑛}𝑛=1(0,1), {𝛽𝑛}𝑛=1(0,2𝛼], {𝜆𝑛}𝑛=1[0,1), and take a fixed number 𝜇(0,2𝜂/𝜅2). We introduce the following implicit iterative scheme with perturbation 𝐹. For an arbitrary initial point 𝑥0𝐻, the sequence {𝑥𝑛}𝑛=1 is generated by the following process: 𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛𝛽𝑛𝐴𝑥𝑛𝜆𝑛𝜇𝐹𝑇𝑛𝑥𝑛𝛽𝑛𝐴𝑥𝑛,𝑛1.(3.1) Here, we use the convention 𝑇𝑛=𝑇𝑛mod𝑁. If 𝐴0, then the implicit iterative scheme (3.1) reduces to the implicit iterative scheme studied in [14].

Let 𝐴𝐻𝐻 be an 𝛼-inverse-strongly monotone mapping and 𝑠(0,2𝛼]. By Lemma 2.6, for every 𝑢𝐻 and 𝑡(0,1), the mapping 𝑆𝑡𝐻𝐻 defined by 𝑆𝑡𝑥=𝑡𝑢+(1𝑡)𝑇𝜆̃𝑥with̃𝑥=𝑥𝑠𝐴𝑥,(3.2) satisfies 𝑆𝑡𝑥𝑆𝑡𝑦𝑇=(1𝑡)𝜆̃𝑥𝑇𝜆̃𝑦(1𝑡)(1𝜆𝜏)̃𝑥̃𝑦(1𝑡)̃𝑥̃𝑦=(1𝑡)(𝑥𝑠𝐴𝑥)(𝑦𝑠𝐴𝑦)=(1𝑡)(𝑥𝑦)𝑠(𝐴𝑥𝐴𝑦)(1𝑡)𝑥𝑦2𝑠(2𝛼𝑠)𝐴𝑥𝐴𝑦2(1𝑡)𝑥𝑦,𝑥,𝑦𝐻,(3.3) where 0𝜆<1, 0<𝜇<2𝜂/𝜅2, and 𝜏=11𝜇(2𝜂𝜇𝜅2)(0,1). By Banach’s contraction principle, there exists a unique 𝑥𝑡𝐻 such that 𝑥𝑡=𝑡𝑢+(1𝑡)𝑇𝜆̃𝑥𝑡𝑇𝑥=𝑡𝑢+(1𝑡)𝑡𝑠𝐴𝑥𝑡𝑥𝜆𝜇𝐹𝑇𝑡𝑠𝐴𝑥𝑡.(3.4) This shows that the implicit iterative scheme (3.1) with perturbation 𝐹 is well defined and can be employed for finding the approximate solutions of the variational inequality problem over the common fixed point set of a finite family of nonexpansive self-maps on 𝐻.

We now state and prove the main results of this paper.

Theorem 3.1. Let 𝐻 be a real Hilbert space, 𝐴 an 𝛼-inverse-strongly monotone mapping, and 𝐹𝐻𝐻 a 𝜅-Lipschitz continuous and 𝜂-strongly monotone mapping for some constants 𝜅, 𝜂>0. Let {𝑇𝑖}𝑁𝑖=1 be 𝑁 nonexpansive self-maps on 𝐻 with a nonempty common fixed point set 𝑁𝑖=1Fix(𝑇𝑖). Suppose VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴). Denote by 𝑇𝑛=𝑇𝑛mod𝑁 for 𝑛>𝑁. Let 𝜇(0,2𝜂/𝜅2), 𝑥0𝐻, {𝜆𝑛}𝑛=1[0,1), {𝛼𝑛}𝑛=1(0,1), and {𝛽𝑛}𝑛=1(0,2𝛼] be such that 𝑛=1𝜆𝑛<, 𝛽𝑛𝜆𝑛 and 𝑎𝛼𝑛𝑏,forall𝑛1, for some 𝑎,𝑏(0,1). Then, the sequence {𝑥𝑛}𝑛=1, defined by 𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝜆𝑛𝑛̃𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛𝛽𝑛𝐴𝑥𝑛𝜆𝑛𝜇𝐹𝑇𝑛𝑥𝑛𝛽𝑛𝐴𝑥𝑛,𝑛1,(3.5) converges weakly to an element of 𝑁𝑖=1Fix(𝑇𝑖).
If, in addition, 𝑥𝑛𝑇𝑛̃𝑥𝑛=𝑜(𝛽𝑛), then {𝑥𝑛} converges weakly to an element of VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴).

Proof. Notice first that the following identity: 𝑡𝑥+(1𝑡)𝑦2=𝑡𝑥2+(1𝑡)𝑦2𝑡(1𝑡)𝑥𝑦2(3.6) holds for all 𝑥,𝑦𝐻 and all 𝑡[0,1]. Let ̂𝑥 be an arbitrary element of 𝑁𝑖=1Fix(𝑇𝑖). Observe that 𝑥𝑛̂𝑥2=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝜆𝑛𝑛̃𝑥𝑛̂𝑥2=𝛼𝑛𝑥𝑛1̂𝑥2+1𝛼𝑛𝑇𝜆𝑛𝑛̃𝑥𝑛̂𝑥2𝛼𝑛1𝛼𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛2.(3.7) Since 𝐴 is 𝛼-inverse strongly monotone and {𝛽𝑛}𝑛=1(0,2𝛼], we have 𝑥𝑛̂𝑥𝛽𝑛𝐴𝑥𝑛𝐴̂𝑥2=𝑥𝑛̂𝑥22𝛽𝑛𝐴𝑥𝑛𝐴̂𝑥,𝑥𝑛̂𝑥+𝛽2𝑛𝐴𝑥𝑛𝐴̂𝑥2𝑥𝑛̂𝑥2𝛽𝑛2𝛼𝛽𝑛𝐴𝑥𝑛𝐴̂𝑥2𝑥𝑛̂𝑥2.(3.8) By Lemma 2.6, we have 𝑇𝜆𝑛𝑛̃𝑥𝑛=𝑇̂𝑥𝜆𝑛𝑛̃𝑥𝑛𝑇𝜆𝑛𝑛̂𝑥+𝑇𝜆𝑛𝑛𝑇̂𝑥̂𝑥𝜆𝑛𝑛̃𝑥𝑛𝑇𝜆𝑛𝑛+𝑇̂𝑥𝜆𝑛𝑛̂𝑥̂𝑥1𝜆𝑛𝜏̃𝑥𝑛̂𝑥+𝜆𝑛𝜇𝐹(̂𝑥)1𝜆𝑛𝜏𝑥𝑛̂𝑥𝛽𝑛𝐴𝑥𝑛𝐴̂𝑥+𝛽𝑛𝐴̂𝑥+𝜆𝑛𝜇𝐹(̂𝑥)1𝜆𝑛𝜏𝑥𝑛̂𝑥+𝛽𝑛𝐴̂𝑥+𝜆𝑛𝜇𝐹(̂𝑥)1𝜆𝑛𝜏𝑥𝑛̂𝑥+𝛽𝑛𝐴̂𝑥+𝜆𝑛𝜇𝐹(̂𝑥)1𝜆𝑛𝜏𝑥𝑛̂𝑥+𝜆𝑛(=𝐴̂𝑥+𝜇𝐹(̂𝑥))1𝜆𝑛𝜏𝑥𝑛̂𝑥+𝜆𝑛𝜏𝐴̂𝑥+𝜇𝐹(̂𝑥)𝜏.(3.9) It follows 𝑇𝜆𝑛𝑛̃𝑥𝑛̂𝑥21𝜆𝑛𝜏𝑥𝑛̂𝑥2+𝜆𝑛()𝐴̂𝑥+𝜇𝐹(̂𝑥)2𝜏.(3.10) This together with (3.7) yields 𝑥𝑛̂𝑥2𝛼𝑛𝑥𝑛1̂𝑥2+1𝛼𝑛1𝜆𝑛𝜏𝑥𝑛̂𝑥2+𝜆𝑛()𝐴̂𝑥+𝜇𝐹(̂𝑥)2𝜏𝛼𝑛1𝛼𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛2𝛼𝑛𝑥𝑛1̂𝑥2+1𝛼𝑛𝑥𝑛̂𝑥2+1𝛼𝑛𝜆𝑛(𝐴̂𝑥+𝜇𝐹(̂𝑥))2𝜏𝛼𝑛1𝛼𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛2,(3.11) and so, 𝑥𝑛̂𝑥2𝑥𝑛1̂𝑥2+1𝛼𝑛𝜆𝑛𝛼𝑛()𝐴̂𝑥+𝜇𝐹(̂𝑥)2𝜏1𝛼𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛2𝑥𝑛1̂𝑥2+𝜆𝑛(𝐴̂𝑥+𝜇𝐹(̂𝑥))2𝑥𝜏𝑎𝑛𝑥𝑛12.(3.12) Since 𝑛=1𝜆𝑛[(𝐴̂𝑥+𝜇𝐹(̂𝑥))2/𝜏𝑎] converges, by Lemma 2.5, lim𝑛𝑥𝑛̂𝑥 exists. As a consequence, the sequence {𝑥𝑛} is bounded. Moreover, we have 𝑥𝑛𝑥𝑛12𝑥𝑛1̂𝑥2𝑥𝑛̂𝑥2+𝜆𝑛()𝐴̂𝑥+𝜇𝐹(̂𝑥)2𝜏𝑎0as𝑛.(3.13) Therefore, lim𝑛𝑥𝑛𝑥𝑛1=0.(3.14) Obviously, it is easy to see that lim𝑛𝑥𝑛𝑥𝑛+𝑖=0 for each 𝑖=1,2,,𝑁. Now observe that 𝑥(1𝑏)𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛1𝛼𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛=𝑥𝑛𝑥𝑛10as𝑛.(3.15) Also note that the boundedness of {𝑥𝑛} implies that {𝑇𝑛̃𝑥𝑛} and {𝐹(𝑇𝑛̃𝑥𝑛)} are both bounded. Thus, we have 𝑥𝑛1𝑇𝑛̃𝑥𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛+𝑇𝜆𝑛𝑛̃𝑥𝑛𝑇𝑛̃𝑥𝑛𝑥𝑛1𝑇𝜆𝑛𝑛̃𝑥𝑛+𝜆𝑛𝜇𝐹𝑇𝑛̃𝑥𝑛0as𝑛.(3.16) This implies 𝑥𝑛𝑇𝑛̃𝑥𝑛𝑥𝑛𝑥𝑛1+𝑥𝑛1𝑇𝑛̃𝑥𝑛0as𝑛.(3.17) Consequently, 𝑥𝑛𝑇𝑛𝑥𝑛𝑥𝑛𝑇𝑛̃𝑥𝑛+𝑇𝑛̃𝑥𝑛𝑇𝑛𝑥𝑛𝑥𝑛𝑇𝑛̃𝑥𝑛+̃𝑥𝑛𝑥𝑛=𝑥𝑛𝑇𝑛̃𝑥𝑛+𝛽𝑛𝐴𝑥𝑛𝑥𝑛𝑇𝑛̃𝑥𝑛+𝜆𝑛𝐴𝑥𝑛0as𝑛,(3.18) and hence, for each 𝑖=1,2,,𝑁, 𝑥𝑛𝑇𝑛+𝑖𝑥𝑛𝑥𝑛𝑥𝑛+𝑖+𝑥𝑛+𝑖𝑇𝑛+𝑖𝑥𝑛+𝑖+𝑇𝑛+𝑖𝑥𝑛+𝑖𝑇𝑛+𝑖𝑥𝑛𝑥2𝑛𝑥𝑛+𝑖+𝑥𝑛+𝑖𝑇𝑛+𝑖𝑥𝑛+𝑖0as𝑛.(3.19) This shows that lim𝑛𝑥𝑛𝑇𝑛+𝑖𝑥𝑛=0 for each 𝑖=1,2,,𝑁. Therefore, lim𝑛𝑥𝑛𝑇𝑙𝑥𝑛=0,foreach𝑙=1,2,,𝑁.(3.20)
On the other hand, since {𝑥𝑛} is bounded, it has a subsequence {𝑥𝑛𝑗}, which converges weakly to some 𝑥𝐻, and so, we have lim𝑗𝑥𝑛𝑗𝑇𝑙𝑥𝑛𝑗=0. From Lemma 2.4, it follows that 𝐼𝑇𝑙 is demiclosed at zero. Thus, 𝑥Fix(𝑇𝑙). Since 𝑙 is an arbitrary element in the finite set {1,2,,𝑁}, we get 𝑥𝑁𝑖=1Fix(𝑇𝑖).
Now, let 𝑥 be an arbitrary element of 𝜔𝑤(𝑥𝑛). Then, there exists another subsequence {𝑥𝑛𝑘} of {𝑥𝑛}, which converges weakly to 𝑥𝐻. Clearly, by repeating the same argument, we get 𝑥𝑁𝑖=1Fix(𝑇𝑖). We claim that 𝑥=𝑥. Indeed, if 𝑥𝑥, then by the Opial’s property of 𝐻, we conclude that lim𝑛𝑥𝑛𝑥=liminf𝑘𝑥𝑛𝑘𝑥<liminf𝑘𝑥𝑛𝑘𝑥=lim𝑛𝑥𝑛𝑥=liminf𝑗𝑥𝑛𝑗𝑥<liminf𝑗𝑥𝑛𝑗𝑥=lim𝑛𝑥𝑛𝑥.(3.21) This leads to a contradiction, and so, we get 𝑥=𝑥. Therefore, 𝜔𝑤(𝑥𝑛) is a singleton set. Hence, {𝑥𝑛} converges weakly to a common fixed point of the mappings {𝑇𝑖}𝑁𝑖=1, denoted still by 𝑥.
Assume that 𝑥𝑛𝑇𝑛̃𝑥𝑛=𝑜(𝛽𝑛). Let 𝑦𝑁𝑖=1Fix(𝑇𝑖) be arbitrary but fixed. Then, it follows from the nonexpansiveness of each 𝑇𝑖 and the monotonicity of 𝐴 that 𝑇𝑛̃𝑥𝑛𝑦2=𝑇𝑛𝑥𝑛𝛽𝑛𝐴𝑥𝑛𝑇𝑛(𝑦)2𝑥𝑛𝑦𝛽𝑛𝐴𝑥𝑛2=𝑥𝑛𝑦2+2𝛽𝑛𝐴𝑥𝑛,𝑦𝑥𝑛+𝛽2𝑛𝐴𝑥𝑛2=𝑥𝑛𝑦2+2𝛽𝑛𝐴𝑦,𝑦𝑥𝑛+𝐴𝑥𝑛𝐴𝑦,𝑦𝑥𝑛+𝛽2𝑛𝐴𝑥𝑛2𝑥𝑛𝑦2+2𝛽𝑛𝐴𝑦,𝑦𝑥𝑛+𝛽2𝑛𝐴𝑥𝑛2,(3.22) which implies that 10𝛽𝑛𝑥𝑛𝑦2𝑇𝑛̃𝑥𝑛𝑦2+2𝐴𝑦,𝑦𝑥𝑛+𝛽𝑛𝐴𝑥𝑛2=𝑥𝑛+𝑇𝑦𝑛̃𝑥𝑛𝑥𝑦𝑛𝑇𝑦𝑛̃𝑥𝑛𝑦𝛽𝑛+2𝐴𝑦,𝑦𝑥𝑛+𝛽𝑛𝐴𝑥𝑛2𝑥𝑀𝑛𝑇𝑛̃𝑥𝑛𝛽𝑛+2𝐴𝑦,𝑦𝑥𝑛+𝑀2𝛽𝑛,(3.23) where 𝑀=sup{𝑥𝑛𝑦+𝑇𝑛̃𝑥𝑛𝑦+𝐴𝑥𝑛𝑛1}<. Note that 𝜆𝑛0,𝛽𝑛𝜆𝑛,forall𝑛1, and 𝑥𝑛𝑇𝑛̃𝑥𝑛=𝑜(𝛽𝑛). Thus, for any 𝜀>0, there exists an integer 𝑚01 such that 𝑀𝑥𝑛𝑇𝑛̃𝑥𝑛/𝛽𝑛+𝑀2𝛽𝑛𝜀 for all 𝑛𝑚0. Consequently, 0𝜀+2𝐴𝑦,𝑦𝑥𝑛 for all 𝑛𝑚0. Since 𝑥𝑛𝑥, we have 𝜀+2𝐴𝑦,𝑦𝑥0 as 𝑛. Therefore, from the arbitrariness of 𝜀>0, we deduce that 𝐴𝑦,𝑦𝑥0 for all 𝑦𝑁𝑖=1Fix(𝑇𝑖). Proposition 2.1 (a) ensures that 𝐴𝑥,𝑦𝑥0,𝑦𝑁𝑖=1𝑇Fix𝑖;(3.24) that is, 𝑥VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴).

Corollary 3.2 (see [14], Theorem  2.1). Let 𝐻 be a real Hilbert space and 𝐹𝐻𝐻𝜅-Lipschitz continuous and 𝜂-strongly monotone for some constants 𝜅>0, 𝜂>0. Let {𝑇𝑖}𝑁𝑖=1 be 𝑁 nonexpansive self-maps on 𝐻 such that 𝐶=𝑁𝑖=1Fix(𝑇𝑖). Let 𝜇(0,2𝜂/𝜅2), 𝑥0𝐻, {𝜆𝑛}𝑛=1[0,1), and {𝛼𝑛}𝑛=1(0,1) be such that 𝑛=1𝜆𝑛< and 𝑎𝛼𝑛𝑏,forall𝑛1, for some 𝑎,𝑏(0,1). Then, the sequence {𝑥𝑛}𝑛=1, defined by 𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝜆𝑛𝑛𝑥𝑛=𝛼𝑛𝑥𝑛1+1𝛼𝑛𝑇𝑛𝑥𝑛𝜆𝑛𝑇𝜇𝐹𝑛𝑥𝑛,𝑛1,(3.25) converges weakly to a common fixed point of the mappings {𝑇𝑖}𝑁𝑖=1.

Proof. In Theorem 3.1, putting 𝐴0, we can see readily that for any given positive number 𝛼(0,), 𝐴𝐻𝐻 is an 𝛼-inverse-strongly monotone mapping. In this case, we have VI𝑁𝑖=1𝑇Fix𝑖=,𝐴𝑁𝑖=1𝑇Fix𝑖.(3.26) Hence, for any given sequence {𝛽𝑛}𝑛=1(0,2𝛼] with 𝛽𝑛𝜆𝑛(𝑛1), the implicit iterative scheme (3.5) reduces to (3.25). Therefore, by Theorem 3.1, we obtain the desired result.

Lemma 3.3. In the setting of Theorem 3.1, we have(a)lim𝑛𝑥𝑛̂𝑥 exists for each ̂𝑥𝐶,(b)lim𝑛𝑑(𝑥𝑛,𝐶) exists, where 𝑑(𝑥𝑛,𝐶)=inf𝑝𝐶𝑥𝑛𝑝,(c)liminf𝑛𝑥𝑛𝑇𝑛𝑥𝑛=0,where 𝐶=VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴).

Proof. Conclusion (a) follows from (3.12), and conclusion (c) follows from (3.18). We prove conclusion (b). Indeed, for each ̂𝑥𝐶, (𝐴̂𝑥+𝜇𝐹(̂𝑥))2𝐴̂𝑥𝐴𝑥𝑛1+𝐴𝑥𝑛1𝑥+𝜇𝐹(̂𝑥)𝐹𝑛1𝐹𝑥+𝜇𝑛121𝛼𝑥𝑛1+̂𝑥𝐴𝑥𝑛1𝑥+𝜇𝜅𝑛1𝐹𝑥̂𝑥+𝜇𝑛12=1𝛼𝑥+𝜇𝜅𝑛1+̂𝑥𝐴𝑥𝑛1𝐹𝑥+𝜇𝑛1212𝛼+𝜇𝜅2𝑥𝑛1̂𝑥2+2𝐴𝑥𝑛1𝐹𝑥+𝜇𝑛12.(3.27) This together with (3.12) implies that 𝑥𝑛̂𝑥2𝑥𝑛1̂𝑥2+𝜆𝑛()𝐴̂𝑥+𝜇𝐹(̂𝑥)2𝑥𝜏𝑎𝑛1̂𝑥2+𝜆𝑛121𝜏𝑎𝛼+𝜇𝜅2𝑥𝑛1̂𝑥2+2𝐴𝑥𝑛1𝐹𝑥+𝜇𝑛121+𝜆𝑛2((1/𝛼)+𝜇𝜅)2𝑥𝜏𝑎𝑛1̂𝑥2+𝜆𝑛2𝜏𝑎𝐴𝑥𝑛1𝐹𝑥+𝜇𝑛121+𝛾𝑛𝑥𝑛1̂𝑥2+𝛾𝑛,(3.28) and hence, 𝑑𝑥𝑛,𝐶21+𝛾𝑛𝑑𝑥𝑛1,𝐶2+𝛾𝑛,(3.29) where 𝛾𝑛=𝜆𝑛max2((1/𝛼)+𝜇𝜅)2,2𝜏𝑎𝜏𝑎𝐴𝑥𝑛1𝐹𝑥+𝜇𝑛12,𝑛1.(3.30) Since 𝑛=1𝜆𝑛< and both {𝐴𝑥𝑛1} and {𝐹(𝑥𝑛1)} are bounded, it is known that 𝑛=1𝛾𝑛<. On account of Lemma 2.5, we deduce that lim𝑛𝑑(𝑥𝑛,𝐶) exists, that is, conclusion (b) holds.

Finally, we give necessary and sufficient conditions for the strong convergence of the sequence generated by the implicit iterative scheme (3.5) with perturbation 𝐹.

Theorem 3.4. In the setting of Theorem 3.1, the sequence {𝑥𝑛} converges strongly to an element of VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴)if and only if liminf𝑛𝑑(𝑥𝑛,𝐶)=0 where 𝐶=VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴).

Proof. From (3.28), we derive for each 𝑛1𝑥𝑛̂𝑥21+𝛾𝑛𝑥𝑛1̂𝑥2+𝛾𝑛,̂𝑥𝐶,(3.31) where 𝑛=1𝛾𝑛<. Put 𝑀=𝑛=1(1+𝛾𝑛). Then 1𝑀<.
Suppose that the sequence {𝑥𝑛} converges strongly to a common fixed point 𝑝 of the family {𝑇𝑖}𝑁𝑖=1. Then, lim𝑛𝑥𝑛𝑝=0. Since 𝑥0𝑑𝑛𝑥,𝐶𝑛𝑝,(3.32) we have liminf𝑛𝑑(𝑥𝑛,𝐶)=0.
Conversely, suppose that liminf𝑛𝑑(𝑥𝑛,𝐶)=0. Then, by Lemma 3.3 (b), we deduce that lim𝑛𝑑(𝑥𝑛,𝐶)=0. Thus, for arbitrary 𝜀>0, there exists a positive integer 𝑁0 such that 𝑑𝑥𝑛<𝜀,𝐶8𝑀,𝑛𝑁0.(3.33) Furthermore, the condition 𝑛=1𝛾𝑛< implies that there exists a positive integer 𝑁1 such that 𝑗=𝑛𝛾𝑗<𝜀28𝑀,𝑛𝑁1.(3.34) Choose 𝑁=max{𝑁0,𝑁1}. Observe that (3.31) yields 𝑥𝑛̂𝑥21+𝛾𝑛1+𝛾𝑛1𝑥𝑛2̂𝑥2+1+𝛾𝑛𝛾𝑛1+𝛾𝑛𝑛𝑗=𝑁+11+𝛾𝑗𝑥𝑁̂𝑥2+𝑛1𝑗=𝑁+1𝛾𝑗𝑛𝑖=𝑗+11+𝛾𝑗+𝛾𝑛𝑀𝑥𝑁̂𝑥2+𝑛𝑗=𝑁+1𝛾𝑗.(3.35) Note that 𝑑(𝑥𝑁,𝐶)<𝜀/8𝑀 and 𝑗=𝑁𝛾𝑗<𝜀2/(8𝑀). Thus, for all 𝑛,𝑚𝑁 and all ̂𝑥𝐶, we have from (3.35) that 𝑥𝑛𝑥𝑚2𝑥𝑛+𝑥̂𝑥𝑚̂𝑥2𝑥2𝑛̂𝑥2𝑥+2𝑚̂𝑥2𝑀𝑥2𝑁̂𝑥2+𝑛𝑗=𝑁+1𝛾𝑗𝑀𝑥+2𝑁̂𝑥2+𝑚𝑗=𝑁+1𝛾𝑗𝑀𝑥4𝑁̂𝑥2+𝑗=𝑁𝛾𝑗𝑀𝑥<4𝑁̂𝑥2+𝜀28𝑀.(3.36) Taking the infimum over all ̂𝑥𝐶, we obtain 𝑥𝑛𝑥𝑚2𝑀𝑑𝑥4𝑁,𝐶2+𝜀28𝑀𝑀𝜀428𝑀+𝜀28𝑀=𝜀2,(3.37) and hence, 𝑥𝑛𝑥𝑚𝜀. This shows that {𝑥𝑛}𝑛=1 is a Cauchy sequence in 𝐻. Let 𝑥𝑛𝑝𝐻 as 𝑛. Then, we derive from (3.20) that for each 𝑙=1,2,,𝑁, 𝑝𝑇𝑙𝑝𝑝𝑥𝑛+𝑥𝑛𝑇𝑙𝑥𝑛+𝑇𝑙𝑥𝑛𝑇𝑙𝑝𝑥2𝑛+𝑥𝑝𝑛𝑇𝑙𝑥𝑛0as𝑛.(3.38) Therefore, 𝑝Fix(𝑇𝑙) for each 𝑙=1,2,,𝑁, and hence, 𝑝𝑁𝑖=1Fix(𝑇𝑖).
On the other hand, choose a positive sequence {𝜀𝑛}𝑛=1(0,) such that 𝜀𝑛0 as 𝑛. For each 𝑛1, from the definition of 𝑑(𝑥𝑛,𝐶), it follows that there exists a point 𝑝𝑛𝐶 such that 𝑥𝑛𝑝𝑛𝑥𝑑𝑛,𝐶+𝜀𝑛.(3.39) Since 𝑑(𝑥𝑛,𝐶)0 and 𝜀𝑛0 as 𝑛, it is clear that 𝑥𝑛𝑝𝑛0 as 𝑛. Note that 𝑝𝑛𝑝𝑝𝑛𝑥𝑛+𝑥𝑛𝑝0as𝑛.(3.40) Hence, we get lim𝑛𝑝𝑛𝑝=0.(3.41) Furthermore, for each 𝛽𝑛(0,2𝛼], the mapping 𝑆𝛽𝑛𝐻𝐻 is defined as follows: 𝑆𝛽𝑛𝑥=𝑃𝑁𝑖=1Fix(𝑇𝑖)𝐼𝛽𝑛𝐴𝑥,𝑥𝐻.(3.42) From Proposition 2.3, we deduce that 𝑆𝛽𝑛 is nonexpansive and 𝑆Fix𝛽𝑛=VI𝑁𝑖=1𝑇Fix𝑖(,𝐴=𝐶).(3.43) From Proposition 2.2 (a), we conclude that Fix(𝑆𝛽𝑛) is closed and convex. Thus, from the condition VI(𝑁𝑖=1Fix(𝑇𝑖),𝐴), it is known that 𝐶 is a nonempty closed and convex set. Since {𝑝𝑛} lies in 𝐶 and converges strongly to 𝑝, we must have 𝑝𝐶.

Remark 3.5. Setting 𝐴=0 in Lemma 3.3 and Theorem 3.4 above, we shall derive Lemma  2.1 and Theorem  2.2 in [14] as direct consequences, respectively.

Acknowledgment

This paper is partially supported by the Taiwan NSC Grants 99-2115-M-110-007-MY3 and 99-2221-E-037-007-MY3.

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