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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 ?? be a real Hilbert space with inner product ?·,·? and norm ?·? and ?? a nonempty closed convex subset of ??. For a given nonlinear operator ??:?????, the classical variational inequality problem (VIP) [1] is to find ??*??? such that ?????*,??-??*?=0,??????.(1.1) The set of solutions of VIP is denoted by VI(??,??). If the set ?? is replaced by the set Fix(??) of fixed points of a mapping ??; 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 ??:????? be ??-strongly monotone (i.e., if there exists a constant ??>0 such that ?????-????,??-???=?????-???2, for all ??,?????) and ??-Lipschitz continuous (i.e., if there exists a constant ??>0 such that ?????-?????=?????-???, for all ??,?????). Assume that ?? is the intersection of the sets of fixed points of ?? nonexpansive mappings ????:?????. For an arbitrary initial guess ??0???, Yamada [10] proposed the following hybrid steepest-descent method: ????+1:=????+1????-????+1?????????+1?????,???=0.(1.2) Here, ????:=????mod??, for every integer ??>??, with the mod function taking values in the set {1,2,,??}; that is, if ??=????+?? for some integers ??=0 and 0=??<??, then ????=???? if ??=0 and ????=???? if 1<??<??. Moreover, ???(0,2??/??2) and the sequence {????}?(0,1) of parameters satisfies the following conditions:(i)lim???8????=0;(ii)?8??=0????=8;(iii)?8??=0|????-????+??| is convergent.

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

Xu and Kim [12] replaced the condition (iii) by the following condition:(iii)'lim???8(????/????+??)=1, or equivalently, lim???8((????-????+??)/????+??)=0

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

On the other hand, let ?? be a nonempty convex subset of ??, and let {????}????=1 be a finite family of nonexpansive self-maps on K. Xu and Ori [13] introduced the following implicit iteration process: for ??0??? and {????}8??=1?(0,1), the sequence {????}8??=1 is generated by the following process: ????=????????-1+?1-?????????????,???=1,(1.3) where we use the convention ????:=????mod??. They also studied the weak convergence of the sequence generated by the above scheme to a common fixed point of the mappings {????}????=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 ??.

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 ??. 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 ??????? to indicate that the sequence {????} converges weakly to ?? in a Banach space ??. Meanwhile, ??????? implies that {????} converges strongly to ??. For a given sequence {????}???, ????(????) denotes the weak ??-limit set of {????}, that is, ???????????:=?????:????????????forsomesubsequence????of{??}.(2.1) A Banach space ?? is said to satisfy Opial’s property if liminf???8????????-??<liminf???8????????-????????,?????,(2.2) whenever a sequence ??????? in ??. It is well known that every Hilbert space ?? satisfies Opial’s property; see for example [15].

A mapping ??:????? is said to be hemicontinuous if for any ??,?????, the mapping ??:[0,1]???, defined by ??(??):=??(????+(1-??)??)(forall???[0,1]), is continuous in the weak topology of the Hilbert space ??. The metric projection onto a nonempty, closed and convex set ?????, denoted by ????, is defined by, for all ?????,????????? and ???-???????=inf????????-???.

Proposition 2.1. Let ????? be a nonempty closed and convex set and ??:????? monotone and hemicontinuous. Then,(a)[1] VI(??,??)={??*???:?????,??-??*?=0,forall?????},(b)[1] VI(??,??)?Ø when ?? is bounded,(c)[16, Lemma??2.24] VI(??,??)=Fix(????(??-????)) for all ??>0, where ?? stands for the identity mapping on ??,(d)[16, Theorem??2.31] VI(??,??) consists of one point if ?? is strongly monotone and Lipschitz continuous.

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

Proposition 2.2. Let ????? be a nonempty closed and convex subset and ??:????? a nonexpansive map.(a)[18, Proposition??5.3] Fix(??) is closed and convex.(b)[18, Theorem??5.1] Fix(??)?Ø when ?? 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 ????? be a nonempty closed and convex set and ??:????? an ??-inverse-strongly monotone operator. Then, for any given ???(0,2??], the mapping ????:?????, defined by ??????:=????(??-????)??,??????,(2.3) is nonexpansive and Fix(????)=VI(??,??).

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 ?? is a nonexpansive self-mapping on a closed convex subset ?? of a Hilbert space ??. If ?? has a fixed point, then ??-?? is demiclosed, that is, whenever {????} is a sequence in ?? weakly converging to some ????? and the sequence {(??-??)????} strongly converges to some ??, it follows that (??-??)??=??, where ?? is the identity operator of ??.

Lemma 2.5 (see [19, page 80]). Let {????}8??=1, {????}8??=1, and {????}8??=1 be sequences of nonnegative real numbers satisfying the inequality ????+1=?1+?????????+????,???=1.(2.4) If ?8??=1????<8 and ?8??=1????<8, then lim???8???? exists. If in addition {????}8??=1 has a subsequence, which converges to zero, then lim???8????=0.

Let ??:????? be a nonexpansive mapping and ??:?????????-Lipschitz continuous and ??-strongly monotone for some constants ??>0, ??>0. For any given numbers ???[0,1) and ???(0,2??/??2), we define the mapping ????:????? by??????:=????-??????(????),??????.(2.5)

Lemma 2.6 (see [12]). If 0=??<1 and 0<??<2??/??2, then ????????-????????=(1-????)???-???,???,?????,(2.6) where ??=1-v1-??(2??-????2)?(0,1).

3. An Iterative Scheme and Convergence Results

Let {????}????=1 be a finite family of nonexpansive self-maps on ??. Let ??:????? be an ??-inverse-strongly monotone mapping (i.e., if there exists a constant ??>0 such that ?????-????,??-???=???????-?????2, for all ??,?????). Let ??:????? be ??-Lipschitz continuous and ??-strongly monotone for some constants ??>0, ??>0. Let {????}8??=1?(0,1), {????}8??=1?(0,2??], {????}8??=1?[0,1), and take a fixed number ???(0,2??/??2). We introduce the following implicit iterative scheme with perturbation ??. For an arbitrary initial point ??0???, the sequence {????}8??=1 is generated by the following process: ????=????????-1+?1-???????????????-???????????-????????°?????????-????????????,???=1.(3.1) Here, we use the convention ????:=????mod??. If ??=0, then the implicit iterative scheme (3.1) reduces to the implicit iterative scheme studied in [14].

Let ??:????? be an ??-inverse-strongly monotone mapping and ???(0,2??]. By Lemma 2.6, for every ????? and ???(0,1), the mapping ????:????? defined by ??????:=????+(1-??)??????with??=??-??????,(3.2) satisfies ????????-????????????=(1-??)????-?????????=(1-??)(1-????)???-???=(1-??)???-???=(1-??)?(??-??????)-(??-??????)?=(1-??)?(??-??)-??(????-????)?=(1-??)???-???2-??(2??-??)?????-?????2=(1-??)???-???,???,?????,(3.3) where 0=??<1, 0<??<2??/??2, and ??=1-v1-??(2??-????2)?(0,1). By Banach’s contraction principle, there exists a unique ??????? such that ????=????+(1-??)??????????????=????+(1-??)??-????????????-??????°????-????????.??(3.4) This shows that the implicit iterative scheme (3.1) with perturbation ?? 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 ??.

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

Theorem 3.1. Let ?? be a real Hilbert space, ?? an ??-inverse-strongly monotone mapping, and ??:????? a ??-Lipschitz continuous and ??-strongly monotone mapping for some constants ??, ??>0. Let {????}????=1 be ?? nonexpansive self-maps on ?? with a nonempty common fixed point set ?????=1Fix(????). Suppose ?VI(????=1Fix(????),??)?Ø. Denote by ????:=????mod?? for ??>??. Let ???(0,2??/??2), ??0???, {????}8??=1?[0,1), {????}8??=1?(0,1), and {????}8??=1?(0,2??] be such that ?8??=1????<8, ????=???? and ??=????=??,forall??=1, for some ??,???(0,1). Then, the sequence {????}8??=1, defined by ????:=????????-1+?1-?????????????????=????????-1+?1-???????????????-???????????-????????°?????????-????????????,???=1,(3.5) converges weakly to an element of ?????=1Fix(????).
If, in addition, ?????-?????????=??(????), then {????} converges weakly to an element of ?VI(????=1Fix(????),??).

Proof. Notice first that the following identity: ??????+(1-??)??2=??????2+(1-??)????2-??(1-??)???-???2(3.6) holds for all ??,????? and all ???[0,1]. Let ?? be an arbitrary element of ?????=1Fix(????). Observe that ????????-??2=??????????-1+?1-???????????????????-??2=??????????-1??-??2+?1-?????????????????????-??2-?????1-???????????-1-??????????????2.(3.7) Since ?? is ??-inverse strongly monotone and {????}8??=1?(0,2??], we have ??????-??-??????????????-????2=????????-??2-2???????????-????,????-???+??2????????????-????2=????????-??2-?????2??-???????????????-????2=????????-??2.(3.8) By Lemma 2.6, we have ????????????????=????-????????????-??????????+??????????=??????-????????????-??????????+??????????????=???-??1-???????????????-??+????=??????(??)?1-??????????????-??-??????????????-????+???????????+????=??????(??)?1-????????????????-??+???????????+????=??????(??)?1-???????????????-??+??????????+????=??????(??)?1-???????????????-??+????(=???????+?????(??)?)1-???????????????-??+???????·??????+?????(??)????.(3.9) It follows ????????????????-??2=?1-???????????????-??2+????·(?)??????+?????(??)2??.(3.10) This together with (3.7) yields ????????-??2=??????????-1??-??2+?1-???????1-???????????????-??2+????·(?)??????+?????(??)2???-?????1-???????????-1-??????????????2=??????????-1??-??2+?1-?????????????-??2+?1-?????????·(??????+?????(??)?)2??-?????1-???????????-1-??????????????2,(3.11) and so, ????????-??2=??????-1??-??2+?1-?????????????·(?)??????+?????(??)2??-?1-???????????-1-??????????????2=??????-1??-??2+????·(??????+?????(??)?)2-??????????-????-1??2.(3.12) Since ?8??=1????·[(??????+?????(??)?)2/????] converges, by Lemma 2.5, lim???8?????-??? exists. As a consequence, the sequence {????} is bounded. Moreover, we have ??????-????-1??2=??????-1??-??2-????????-??2+????·(?)??????+?????(??)2?????0as???8.(3.13) Therefore, lim???8??????-????-1??=0.(3.14) Obviously, it is easy to see that lim???8?????-????+???=0 for each ??=1,2,,??. Now observe that ????(1-??)??-1-??????????????=?1-???????????-1-??????????????=??????-????-1???0as???8.(3.15) Also note that the boundedness of {????} implies that {????????} and {??(????????)} are both bounded. Thus, we have ??????-1-??????????=??????-1-??????????????+??????????????-??????????=??????-1-??????????????+???????????????????????0as???8.(3.16) This implies ??????-??????????=??????-????-1??+??????-1-???????????0as???8.(3.17) Consequently, ??????-??????????=??????-??????????+??????????-??????????=??????-??????????+??????-??????=??????-??????????+??????????????=??????-??????????+???????????????0as???8,(3.18) and hence, for each ??=1,2,,??, ??????-????+????????=??????-????+????+??????+??-????+??????+????+??????+??????+??-????+????????????=2??-????+????+??????+??-????+??????+?????0as???8.(3.19) This shows that lim???8?????-????+???????=0 for each ??=1,2,,??. Therefore, lim???8??????-??????????=0,foreach??=1,2,,??.(3.20)
On the other hand, since {????} is bounded, it has a subsequence {??????}, which converges weakly to some ?????, and so, we have lim???8???????-???????????=0. From Lemma 2.4, it follows that ??-???? is demiclosed at zero. Thus, ???Fix(????). Since ?? is an arbitrary element in the finite set {1,2,,??}, we get ????????=1Fix(????).
Now, let ??* be an arbitrary element of ????(????). Then, there exists another subsequence {??????} of {????}, which converges weakly to ??*???. Clearly, by repeating the same argument, we get ??*??????=1Fix(????). We claim that ??*=??. Indeed, if ??*???, then by the Opial’s property of ??, we conclude that lim???8??????-??*??=liminf???8????????-??*??<liminf???8????????-????=lim???8??????-????=liminf???8????????-????<liminf???8????????-??*??=lim???8??????-??*??.(3.21) This leads to a contradiction, and so, we get ??*=??. Therefore, ????(????) is a singleton set. Hence, {????} converges weakly to a common fixed point of the mappings {????}????=1, denoted still by ??*.
Assume that ?????-?????????=??(????). Let ????????=1Fix(????) be arbitrary but fixed. Then, it follows from the nonexpansiveness of each ???? and the monotonicity of ?? that ????????????-??2=???????????-???????????-??????(??)2=????????-??-????????????2=????????-??2+2???????????,??-?????+??2????????????2=????????-??2+2??????????,??-?????+???????-????,??-??????+??2????????????2=????????-??2+2?????????,??-?????+??2????????????2,(3.22) which implies that 10=?????????????-??2-????????????-??2?+2?????,??-?????+??????????????2=?????????+????-???????????????-??????-????-??????????-??????+2?????,??-?????+??????????????2????=????-??????????????+2?????,??-?????+??2????,(3.23) where ??:=sup{?????-???+?????????-???+????????:??=1}<8. Note that ?????0,????=????,forall??=1, and ?????-?????????=??(????). Thus, for any ??>0, there exists an integer ??0=1 such that ???????-?????????/????+??2????=?? for all ??=??0. Consequently, 0=??+2?????,??-????? for all ??=??0. Since ???????*, we have ??+2?????,??-??*?=0 as ???8. Therefore, from the arbitrariness of ??>0, we deduce that ?????,??-??*?=0 for all ????????=1Fix(????). Proposition 2.1 (a) ensures that ?????*,??-??*?=0,?????????=1???Fix???;(3.24) that is, ??*??VI(????=1Fix(????),??).

Corollary 3.2 (see [14], Theorem??2.1). Let ?? be a real Hilbert space and ??:???????-Lipschitz continuous and ??-strongly monotone for some constants ??>0, ??>0. Let {????}????=1 be ?? nonexpansive self-maps on ?? such that ???=????=1Fix(????)?Ø. Let ???(0,2??/??2), ??0???, {????}8??=1?[0,1), and {????}8??=1?(0,1) be such that ?8??=1????<8 and ??=????=??,forall??=1, for some ??,???(0,1). Then, the sequence {????}8??=1, defined by ????:=????????-1+?1-?????????????????=????????-1+?1-??????????????-???????????????????,???=1,(3.25) converges weakly to a common fixed point of the mappings {????}????=1.

Proof. In Theorem 3.1, putting ??=0, we can see readily that for any given positive number ???(0,8), ??:????? is an ??-inverse-strongly monotone mapping. In this case, we have ?VI?????=1???Fix????=,???????=1???Fix???.(3.26) Hence, for any given sequence {????}8??=1?(0,2??] with ????=????(???=1), 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)lim???8?????-??? exists for each ?????,(b)lim???8??(????,??) exists, where ??(????,??)=inf??????????-???,(c)liminf???8?????-?????????=0,where ???:=VI(????=1Fix(????),??).

Proof. Conclusion (a) follows from (3.12), and conclusion (c) follows from (3.18). We prove conclusion (b). Indeed, for each ?????, (??????+?????(??)?)2=???????-??????-1??+????????-1???????+????(??)-????-1??????????+????-1????2=?1????????-1??+??-????????-1??????+??????-1?????????-??+????-1????2=1?????????+??????-1??+???-????????-1?????????+????-1?????2?1=2???+????2??????-1??-??2???+2??????-1?????????+????-1????2.(3.27) This together with (3.12) implies that ????????-??2=??????-1??-??2+????·(?)??????+?????(??)2=??????????-1??-??2+????·1?2?1???????+????2??????-1??-??2???+2??????-1?????????+????-1????2?=?1+????·2((1/??)+????)2???????????-1??-??2+????·2?????????????-1?????????+????-1????2=?1+???????????-1??-??2+????,(3.28) and hence, ????????,????2=?1+?????????????-1,????2+????,(3.29) where ????=?????·max2((1/??)+????)2,2?????????????????-1?????????+????-1????2?,???=1.(3.30) Since ?8??=1????<8 and both {??????-1} and {??(????-1)} are bounded, it is known that ?8??=1????<8. On account of Lemma 2.5, we deduce that lim???8??(????,??) 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 ??.

Theorem 3.4. In the setting of Theorem 3.1, the sequence {????} converges strongly to an element of ?VI(????=1Fix(????),??)if and only if liminf???8??(????,??)=0 where ???:=VI(????=1Fix(????),??).

Proof. From (3.28), we derive for each ??=1????????-??2=?1+???????????-1??-??2+????,??????,(3.31) where ?8??=1????<8. Put ????=8??=1(1+????). Then ?1=??<8.
Suppose that the sequence {????} converges strongly to a common fixed point ?? of the family {????}????=1. Then, lim???8?????-???=0. Since ???0=?????=????,??????-??,(3.32) we have liminf???8??(????,??)=0.
Conversely, suppose that liminf???8??(????,??)=0. Then, by Lemma 3.3 (b), we deduce that lim???8??(????,??)=0. Thus, for arbitrary ??>0, there exists a positive integer ??0 such that ????????<??,??v8???,???=??0.(3.33) Furthermore, the condition ?8??=1????<8 implies that there exists a positive integer ??1 such that 8???=??????<??2?8????,???=??1.(3.34) Choose ??*=max{??0,??1}. Observe that (3.31) yields ????????-??2=?1+??????1+????-1???????-2??-??2+?1+?????????-1+????=?????=??*+1?1+???????????*??-??2+??-1???=??*+1?????????=??+1?1+?????+????=?????????????*??-??2+?????=??*+1????????.(3.35) Note that ??(????*v,??)<??/8??? and ?8??=??*????<??2?/(8??). Thus, for all ??,??=??* and all ?????, we have from (3.35) that ??????-??????2=?????????+????-???????-??2????=2????-??2????+2????-??2???????????=2??*??-??2+?????=??*+1???????????????????+2??*??-??2+?????=??*+1????????????????=4??*??-??2+8???=??*?????????????<4??*??-??2+??28????.(3.36) Taking the infimum over all ?????, we obtain ??????-??????2??????????=4??*,????2+??28??????????=428???+??28????=??2,(3.37) and hence, ?????-?????=??. This shows that {????}8??=1 is a Cauchy sequence in ??. Let ?????????? as ???8. Then, we derive from (3.20) that for each ??=1,2,,??, ????-????????=????-??????+??????-??????????+??????????-????????????=2????+????-????-???????????0as???8.(3.38) Therefore, ???Fix(????) for each ??=1,2,,??, and hence, ????????=1Fix(????).
On the other hand, choose a positive sequence {????}8??=1?(0,8) such that ?????0 as ???8. For each ??=1, from the definition of ??(????,??), it follows that there exists a point ??????? such that ??????-?????????=?????,??+????.(3.39) Since ??(????,??)?0 and ?????0 as ???8, it is clear that ?????-??????0 as ???8. Note that ????????=????-????-??????+????????-???0as???8.(3.40) Hence, we get lim???8????????-??=0.(3.41) Furthermore, for each ?????(0,2??], the mapping ??????:????? is defined as follows: ????????:=???????=1Fix(????)???-?????????,??????.(3.42) From Proposition 2.3, we deduce that ?????? is nonexpansive and ???Fix??????=VI?????=1???Fix????(,??=??).(3.43) From Proposition 2.2 (a), we conclude that Fix(??????) is closed and convex. Thus, from the condition ?VI(????=1Fix(????),??)?Ø, it is known that ?? is a nonempty closed and convex set. Since {????} lies in ?? and converges strongly to ??, we must have ?????.

Remark 3.5. Setting ??=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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