JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation32042110.1155/2012/320421320421Research ArticleAn Iterative Algorithm for a Hierarchical ProblemYaoYonghong1ChoYeol Je2YangPei-Xia1MarinoGiuseppe1Department of MathematicsTianjin Polytechnic UniversityTianjin 300387Chinatjpu.edu.cn2Department of Mathematics Education and the RINSGyeongsang National UniversityChinju 660-701Republic of Koreagnu.ac.kr201229122011201229092011111120112012Copyright © 2012 Yonghong Yao 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.

A general hierarchical problem has been considered, and an explicit algorithm has been presented for solving this hierarchical problem. Also, it is shown that the suggested algorithm converges strongly to a solution of the hierarchical problem.

1. Introduction

Let H be a real Hilbert space with inner product ·,· and norm ·, respectively. Let C be a nonempty closed convex subset of H. The hierarchical problem is of finding x̃Fix(T) such thatSx̃-x̃,x-x̃0,xFix(T), where S,T are two nonexpansive mappings and Fix(T) is the set of fixed points of T. Recently, this problem has been studied by many authors (see, e.g., ). The main reason is that this problem is closely associated with some monotone variational inequalities and convex programming problems (see ).

Now, we briefly recall some historic results which relate to the problem (1.1).

For solving the problem (1.1), in 2006, Moudafi and Mainge  first introduced an implicit iterative algorithm:xt,s=sQ(xt,s)+(1-s)[tS(xt,s)+(1-t)T(xt,s)] and proved that the net {xt,s} defined by (1.2) strongly converges to xt as s0, where xt satisfies xt=projFix(Pt)Q(xt), where Pt:CC is a mapping defined by Pt(x)=tS(x)+(1-t)T(x),xC,t(0,1), or, equivalently, xt is the unique solution of the quasivariational inequality 0(I-Q)xt+NFix(Pt)(xt), where the normal cone to Fix(Pt), NFix(Pt), is defined as follows: NFix(Pt):x{{uH:y-x,u0},if  xFix(Pt),,otherwise.

Moreover, as t0, the net {xt} in turn weakly converges to the unique solution x of the fixed point equation x=projΩQ(x) or, equivalently, x is the unique solution of the variational inequality 0(I-Q)x+NΩ(x).

Recently, Moudafi  constructed an explicit iterative algorithm:xn+1=(1-δn)xn+δn(σnSxn+(1-σn)Txn),n0, where {δn} and {σn} are two real numbers in (0,1). By using this iterative algorithm, Moudafi  only proved a weak convergence theorem for solving the problem (1.1).

In order to obtain a strong convergence result, Mainge and Moudafi  further introduced the following iterative algorithm:xn+1=(1-δn)Qxn+δn[σnSxn+(1-σn)Txn],n0, where {δn} and {σn} are two real numbers in (0,1), and proved that, under appropriate conditions, the iterative sequence {xn} generated by (1.8) has strong convergence.

Subsequently, some authors have studied some algorithms on hierarchical fixed problems (see, e.g., ).

Motivated and inspired by the results in the literature, in this paper, we consider a general hierarchical problem of finding x̃Fix(T) such that, for any n1,Wnx̃-x̃,x-x̃0,xFix(T), where Wn is the W-mapping defined by (2.3) below and T is a nonexpansive mapping, and introduce an explicit iterative algorithm which converges strongly to a solution x̃ of the hierarchical problem (1.9).

2. Preliminaries

Let C a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping Q:CC is said to be contractive if there exists a constant γ(0,1) such that Qx-Qyγx-y,x,yC.

A mapping T:CC is called nonexpansive ifTx-Tyx-y,x,yC.

Forward, we use Fix(T) to denote the fixed points set of T.

Let {Ti}i=1:CC be an infinite family of nonexpansive mappings and {ξi}i=1 a real number sequence such that 0ξi1 for each i1.

For each n1, define a mapping Wn:CC as follows:Un,n+1=I,Un,n=ξnTnUn,n+1+(1-ξn)I,Un,n-1=ξn-1Tn-1Un,n+(1-ξn-1)I,Un,k=ξkTkUn,k+1+(1-ξk)I,Un,k-1=ξk-1Tk-1Un,k+(1-ξk-1)I,Un,2=ξ2T2Un,3+(1-ξ2)I,Wn=Un,1=ξ1T1Un,2+(1-ξ1)I.

Such Wn is called the W-mapping generated by {Ti}i=1 and {ξi}i=1.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B18">20</xref>]).

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Ti}i=1 be an infinite family of nonexpansive mappings of C into itself with n=1Fix(Tn). Let ξ1,ξ2, be real numbers such that 0<ξib<1 for each i1. Then one has the following results:

for any xC and k1, the limit limnUn,kx exists;

Fix(W)=n=1Fix(Tn).

Using Lemma  3.1 in , we can define a mapping W of C into itself by Wx=limnWnx=limnUn,1x for all xC. Thus we have the following.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B19">21</xref>]).

If {xn} is a bounded sequence in C, then one has limnWxn-Wnxn=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B20">22</xref>]).

Let C be a nonempty closed convex of a real Hilbert space H and T:CC be nonexpansive mapping. Then T is demiclosed on C, that is, if xnxC and xn-Txn0, then x=Tx.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B21">23</xref>]).

Assume {an} is a sequence of nonnegative real numbers such that an+1(1-γn)an+δnγn+ηn,n1, where {γn} is a sequence in (0,1) and {δn},{ηn} are two sequences such that

n=1γn=;

limsupnδn0 or n=1|δnγn|<;

n=1|ηn|<.

Then limnan=0.

3. Main Results

In this section, we introduce our algorithm and give its convergence analysis.

Algorithm 3.1.

Let C be a nonempty closed convex subset of a real Hilbert space H and {Tn}n=1 be infinite family of nonexpansive mappings of C into itself. Let Q:CC be a contraction with coefficient γ[0,1). For any x0C, let {xn} the sequence generated iteratively by xn+1=αnWnxn+(1-αn)T(βnQxn+(1-βn)xn),n0, where {αn},{βn} are two real numbers in (0,1) and Wn is the W-mapping defined by (2.3).

Now, we give the convergence analysis of the algorithm.

Theorem 3.2.

Let C be a nonempty closed convex subset of a real Hilbert space H and {Tn}n=1 be an infinite family of nonexpansive mappings of C into itself. Let Q:CC be a contraction with coefficient γ[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.9) is nonempty. Let {αn},{βn} be two real numbers in (0,1) and {xn} the sequence generated by (3.1). Assume that the sequence {xn} is bounded and

limnαn=0 and limn(βn/αn)=0;

n=0βn=;

limn(1/βn)|(1/αn)-(1/αn-1)|=0 and limn(i=1n-1ξi/αnβn)=limn(1/αn)|1-(βn-1/βn)|=0.

Then limn(xn+1-xn/αn)=0 and every weak cluster point of the sequence {xn} solves the following variational inequalityx̃Ω,(I-Q)x̃,x-x̃0,xΩ.

Proof.

Set yn=βnQxn+(1-βn)xn for each n0. Then we have yn-yn-1=βnQxn+(1-βn)xn-βn-1Qxn-1-(1-βn-1)xn-1=βn(Qxn-Qxn-1)+(βn-βn-1)Qxn-1+(1-βn)(xn-xn-1)+(βn-1-βn)xn-1. It follows that yn-yn-1γβnxn-xn-1+(1-βn)xn-xn-1+|βn-βn-1|(Qxn-1+xn-1)=[1-(1-γ)βn]xn-xn-1+|βn-βn-1|(Qxn-1+xn-1). From (3.1), we have xn+1-xn=αnWnxn+(1-αn)Tyn-αn-1Wn-1xn-1-(1-αn-1)Tyn-1=αn(Wnxn-Wnxn-1)+(αn-αn-1)Wnxn-1+αn-1(Wnxn-1-Wn-1xn-1)+(1-αn)(Tyn-Tyn-1)+(αn-1-αn)Tyn-1. Then we obtain xn+1-xnαnWnxn-Wnxn-1+(1-αn)Tyn-Tyn-1+|αn-αn-1|(Wnxn-1+Tyn-1)+αn-1Wnxn-1-Wn-1xn-1αnxn-xn-1+(1-αn)yn-yn-1+|αn-αn-1|(Wnxn-1+Tyn-1)+αn-1Wnxn-1-Wn-1xn-1. From (2.3), since Ti and Un,i are nonexpansive, we have Wnxn-1-Wn-1xn-1=ξ1T1Un,2xn-1-ξ1T1Un-1,2xn-1ξ1Un,2xn-1-Un-1,2xn-1=ξ1ξ2T2Un,3xn-1-ξ2T2Un-1,3xn-1ξ1ξ2Un,3xn-1-Un-1,3xn-1ξ1ξ2ξn-1Un,nxn-1-Un-1,nxn-1M1i=1n-1ξi, where M1 is a constant such that supn1{Un,nxn-1-Un-1,nxn-1}M1. Substituting (3.4) and (3.7) into (3.6), we get xn+1-xnαnxn-xn-1+(1-αn)[1-(1-γ)βn]xn-xn-1+|βn-βn-1|(Qxn-1+xn-1)+|αn-αn-1|(Wnxn-1+Tyn-1)+αn-1M1i=1n-1ξi=[1-(1-γ)βn(1-αn)]xn-xn-1+|βn-βn-1|(Qxn-1+xn-1)+|αn-αn-1|(Wnxn-1+Tyn-1)+αn-1M1i=1n-1ξi. Therefore, it follows that xn+1-xnαn[1-(1-γ)βn(1-αn)]xn-xn-1αn+|βn-βn-1|αn(Qxn-1+xn-1)+|αn-αn-1|αn(Wnxn-1+Tyn-1)+αn-1M1i=1n-1ξiαn=[1-(1-γ)βn(1-αn)]xn-xn-1αn-1+[1-(1-γ)βn(1-αn)](xn-xn-1αn-xn-xn-1αn-1)+|βn-βn-1|αn(Qxn-1+xn-1)+|αn-αn-1|αn(Wnxn-1+Tyn-1)+αn-1M1i=1n-1ξiαn[1-(1-γ)βn(1-αn)]xn-xn-1αn-1+(|1αn-1αn-1|+|αn-αn-1|αn+|βn-βn-1|αn+i=1n-1ξiαn)M=[1-(1-γ)βn(1-αn)]xn-xn-1αn-1+(1-γ)βn(1-αn)×{M(1-γ)(1-αn)(1βn|1αn-1αn-1|+1βn|αn-αn-1|αn+1βn|βn-βn-1|αn+i=1n-1ξiαnβn)}, where M is a constant such that supn1{M1,xn-xn-1,(Wnxn-1+Tyn-1),(Qxn-1+xn-1)}M. From (iii), we note that limn(1/αn-1)|αn-αn-1/βnαn|=0, which implies that limn1βn|αn-αn-1|αn=0. Thus it follows from (iii) and (3.11) that limn(1βn|1αn-1αn-1|+1βn|αn-αn-1|αn+1βn|βn-βn-1|αn+i=1n-1ξiαnβn)=0. Hence, applying Lemma 2.4 to (3.9), we immediately conclude that limnxn+1-xnαn=0. This implies that limnxn+1-xn=0. Thus, from (3.1) and (3.14), we have limnxn-Tyn=0. At the same time, we note that yn-xn=βn(Qxn-xn)0. Hence we get limnyn-Tyn=0. Since the sequence {xn} is bounded, {yn} is also bounded. Thus there exists a subsequence of {yn}, which is still denoted by {yn} which converges weakly to a point x̃H. Therefore, x̃Fix(T) by (3.17) and Lemma 2.3. By (3.1), we observe that xn+1-xn=αn(Wnxn-xn)+(1-αn)(Tyn-yn)+(1-αn)βn(Qxn-xn), that is, xn-xn+1αn=(I-Wn)xn+1-αnαn(I-T)yn+βn(1-αn)αn(I-Q)xn. Set zn=(xn-xn+1)/αn for each n1, that is, zn=(I-Wn)xn+1-αnαn(I-T)yn+βn(1-αn)αn(I-Q)xn. Using monotonicity of I-T and I-Wn, we derive that, for all uFix(T), zn,xn-u=(I-Wn)xn,xn-u+1-αnαn(I-T)yn-(I-T)u,yn-u+1-αnαn(I-T)yn,xn-yn+βn(1-αn)αn(I-Q)xn,xn-u(I-Wn)u,xn-u+βn(1-αn)αn(I-Q)xn,xn-u+(1-αn)βnαn(I-T)yn,xn-Qxn=(I-W)u,xn-u+(W-Wn)u,xn-u+βn(1-αn)αn(I-Q)xn,xn-u+(1-αn)βnαn(I-T)yn,xn-Qxn. But, since zn0,  βn/αn0 and limnWnu-Wu=0 (by Lemma 2.2), it follows from the above inequality that limsupn(I-W)u,xn-u0,uFix(T). This suffices to guarantee that ωw(xn)Ω. As a matter of fact, if we take any x*ωw(xn), then there exists a subsequence {xnj} of {xn} such that xnjx*. Therefore, we have (I-W)u,x*-u=limj(I-W)u,xnj-u0,uFix(T). Note that x*Fix(T). Hence x* solves the following problem: x*Fix(T),(I-W)u,x*-u0,uFix(T). It is obvious that this is equivalent to the problem (1.9) since WnW uniformly in any bounded set (by Lemma 2.2). Thus x*Ω.

Let x̃ be the unique solution of the variational inequality (3.2). Now, take a subsequence {xni} of {xn} such thatlimsupn(I-Q)x̃,xn-x̃=limi(I-Q)x̃,xni-x̃. Without loss of generality, we may further assume that xnix¯. Then x¯Ω. Therefore, we have limsupn(I-Q)x̃,xn-x̃=(I-Q)x̃,x¯-x̃0. This completes the proof.

Theorem 3.3.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Tn}n=1 be infinite family of nonexpansive mappings of C into itself. Let Q:CC be a contraction with coefficient γ[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.9) is nonempty. Let {αn},{βn} be two real numbers in (0,1) and {xn} the sequence generated by (3.1). Assume that the sequence {xn} is bounded and

limnαn=0, limnβn/αn=0 and limnαn2/βn=0;

n=0βn=;

limn(1/βn)|(1/αn)-(1/αn-1)|=0 and limni=1n-1ξi/αnβn = limn(1/αn)|1-(βn-1/βn)| = 0;

there exists a constant k>0 such that x-TxkDist(x,Fix(T)), where

Dist(x,Fix(T))=infyFix(T)x-y. Then the sequence {xn} defined by (3.1) converges strongly to a point x̃Fix(T), which solves the variational inequality problem (3.2).

Proof.

From (3.1), we have xn+1-x̃=αn(Wnxn-Wnx̃)+αn(Wnx̃-x̃)+(1-αn)(Tyn-x̃). Thus we have xn+1-x̃2αn(Wnxn-Wnx̃)+(1-αn)(Tyn-x̃)2+2αnWnx̃-x̃,xn+1-x̃(1-αn)Tyn-x̃2+αnWnxn-Wnx̃2+2αnWnx̃-x̃,xn+1-x̃(1-αn)yn-x̃2+αnxn-x̃2+2αnWnx̃-x̃,xn+1-x̃. At the same time, we observe that yn-x̃2=(1-βn)(xn-x̃)+βn(Qxn-Qx̃)+βn(Qx̃-x̃)2(1-βn)(xn-x̃)+βn(Qxn-Qx̃)2+2βnQx̃-x̃,yn-x̃(1-βn)xn-x̃2+βnQxn-Qx̃2+2βnQx̃-x̃,yn-x̃(1-βn)xn-x̃2+βnγ2xn-x̃2+2βnQx̃-x̃,yn-x̃=[1-(1-γ2)βn]xn-x̃2+2βnQx̃-x̃,yn-x̃. Substituting (3.30) into (3.29), we get xn+1-x̃2αnxn-x̃2+(1-αn)[1-(1-γ2)βn]xn-x̃2+2βn(1-αn)Qx̃-x̃,yn-x̃+2αnWnx̃-x̃,xn+1-x̃=[1-(1-γ2)βn(1-αn)]xn-x̃2+2βn(1-αn)Qx̃-x̃,yn-x̃+2αnWnx̃-x̃,xn+1-x̃=[1-(1-γ2)βn(1-αn)]xn-x̃2+(1-γ2)βn(1-αn)×{21-γ2Qx̃-x̃,yn-x̃+2(1-γ2)(1-αn)×αnβnWnx̃-x̃,xn+1-x̃}. By Theorem 3.2, we note that every weak cluster point of the sequence {xn} is in Ω. Since yn-xn0, then every weak cluster point of {yn} is also in Ω. Consequently, since x̃=projΩ(Qx̃), we easily have limsupnQx̃-x̃,yn-x̃0.

On the other hand, we observe thatWnx̃-x̃,xn+1-x̃=Wnx̃-x̃,projFix(T)xn+1-x̃+Wnx̃-x̃,xn+1-projFix(T)xn+1. Since x̃ is a solution of the problem (1.9) and projFix(T)xn+1Fix(T), we have Wnx̃-x̃,projFix(T)xn+1-x̃0. Thus it follows that Wnx̃-x̃,xn+1-x̃Wnx̃-x̃,xn+1-projFix(T)xn+1Wnx̃-x̃xn+1-projFix(T)xn+1=Wnx̃-x̃×Dist(xn+1,Fix(T))1kWnx̃-x̃xn+1-Txn+1. We note that xn+1-Txn+1xn+1-Txn+Txn-Txn+1αnWnxn-Txn+(1-αn)Tyn-Txn+xn+1-xnαnWnxn-Txn+yn-xn+xn+1-xnαnWnxn-Txn+βnQxn-xn+xn+1-xn. Hence we have αnβnWnx̃-x̃,xn+1-x̃αn2βn(1kWnx̃-x̃Wnxn-Txn)+αn(1kWnx̃-x̃Qxn-xn)+αn2βnxn+1-xnαn(1kWnx̃-x̃). From Theorem 3.2, we have limnxn+1-xn/αn=0. At the same time, we note that {(1/k)Wnx̃-x̃Wnxn-Txn}, {(1/k)Wnx̃-x̃Qxn-xn}, and {(1/k)Wnx̃-x̃} are all bounded. Hence it follows from (i) and the above inequality that limsupnαnβnWnx̃-x̃,xn+1-x̃0.

Finally, by (3.31)–(3.38) and Lemma 2.4, we conclude that the sequence {xn} converges strongly to a point x̃Fix(T). This completes the proof.

Remark 3.4.

In the present paper, we consider the hierarchical problem (1.9) which includes the hierarchical problem (1.1) as a special case.

From the above discussion, we can easily deduce the following result.

Algorithm 3.5.

Let C be a nonempty closed convex subset of a real Hilbert space H and S a nonexpansive mapping of C into itself. Let Q:CC be a contraction with coefficient γ[0,1). For any x0C, let{xn} the sequence generated iteratively by xn+1=αnSxn+(1-αn)T(βnQxn+(1-βn)xn),n0, where {αn},{βn} are two real numbers in (0,1).

Corollary 3.6.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S:CC be a nonexpansive mapping. Let Q:CC be a contraction with coefficient γ[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.1) is nonempty. Let {αn},{βn} be two real numbers in (0,1) and {xn} the sequence generated by (3.1). Assume that the sequence {xn} is bounded and

limnαn=0, limnβn/αn=0 and limnαn2/βn=0;

n=0βn=;

limn(1/βn)|(1/αn)-(1/αn-1)|=0 and limn(1/αn)|1-(βn-1/βn)|=0;

there exists a constant k>0 such that x-TxkDist(x,Fix(T)), where

Dist(x,Fix(T))=infyFix(T)x-y. Then the sequence {xn} defined by (3.39) converges strongly to a point x̃Fix(T), which solves the hierarchical problem (1.1).

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).

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