JAMSAJournal of Applied Mathematics and Stochastic Analysis1687-21771048-9533Hindawi Publishing Corporation32082010.1155/2009/320820320820Research ArticleModified Iterative Algorithms for Nonexpansive MappingsYaoYonghong1NoorMuhammad Aslam2Mohyud-DinSyed Tauseef2XuHong Kun1Department of MathematicsTianjin Polytechnic UniversityTianjin 300160Chinatjpu.edu.cn2Mathematics DepartmentFaculty of SciencesCOMSATS Institute of Information TechnologyIslamabadPakistanciit.edu.pk200903062009200922112008020320092009Copyright © 2009This 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.

Let H be a real Hilbert space, let S, T be two nonexpansive mappings such that F(S)F(T), let f be a contractive mapping, and let A be a strongly positive linear bounded operator on H. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as xn+1=βnxn+(1βn)Syn, yn=αnγf(xn)+(IαnA)Txn, n0 is a real number and {αn}, {βn} are two sequences in (0,1) satisfying the following control conditions: (C1) limnαn=0, (C3) 0<liminfnβnlimsupnβn<1, then xn+1xn0. We also discuss several special cases of this iterative algorithm.

1. Introduction

Let H be a real Hilbert space. Recall that a mapping f:HH is a contractive mapping on H if there exists a constant α(0,1) such that

f(x)-f(y)αx-y,x,yH. We denote by Π the collection of all contractive mappings on H, that is,

={f:HHisacontractivemapping}.

Let T:HH be a nonexpansive mapping, namely,

Tx-Tyx-y,x,yH.

Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems (see  and the references therein).

A typical problem is to minimize a quadratic function over the closed convex set of the fixed points of a nonexpansive mapping T on a real Hilbert space H:

minxC12Ax,x-x,b, where C is a closed convex set of the fixed points a nonexpansive mapping T on H, b is a given point in H and A is a linear, symmetric and positive operator.

In  (see also ), the author proved that the sequence {xn} defined by the iterative method below with the initial point x0H chosen arbitrarily

xn+1=(1-αnA)Txn+αnb,n0, converges strongly to the unique solution of the minimization problem (1.4) provided the sequence {αn} satisfies certain control conditions.

On the other hand, Moudafi  introduced the viscosity approximation method for nonexpansive mappings (see also  for further developments in both Hilbert and Banach spaces). Let f be a contractive mapping on H. Starting with an arbitrary initial point x0H, define a sequence {xn} in H recursively by

xn+1=(1-αn)Txn+αnf(xn),n0, where {αn} is a sequence in (0,1), which satisfies some suitable control conditions.

Recently, Marino and Xu  combined the iterative algorithm (1.5) with the viscosity approximation algorithm (1.6), considering the following general iterative algorithm:

xn+1=(I-αnA)Txn+αnγf(xn),n0, where 0<γ<γ̅/α.

In this paper, we suggest a new iterative method for finding the pair of nonexpansive mappings. As an application and as special cases, we also obtain some new iterative algorithms which can be viewed as an improvement of the algorithm of Xu  and Marino and Xu . Also we show that the convergence of the proposed algorithms can be proved under weaker conditions on the parameter {αn}. In this respect, our results can be considered as an improvement of the many known results.

2. Preliminaries

In the sequel, we will make use of the following for our main results:

Lemma 2.1 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let {sn} be a sequence of nonnegative numbers satisfying the condition sn+1(1-αn)sn+αnβn,n0, where {αn}, {βn} are sequences of real numbers such that

{αn}[0,1] and n=0αn=,

limnβn0 or n=0αnβn is convergent.

Then limnsn=0.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B9">9</xref>, <xref ref-type="bibr" rid="B10">10</xref>]).

Let {xn} and {yn} be bounded sequences in a Banach space X and {βn} be a sequence in [0,1] with 0<lim infnβnlim supnβn<1. Suppose that xn+1=(1-βn)yn+βnxn for all n0 and lim supn(yn+1-yn-xn+1-xn)0. Then limnyn-xn=0.

Lemma 2.3 (see [<xref ref-type="bibr" rid="B2">2</xref>] (demiclosedness Principle)).

Assume that T is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. If T has a fixed point, then I-T is demiclosed, that is, whenever {xn} is a sequence in C weakly converging to some xC and the sequence {(I-T)xn} strongly converges to some y, it follows that (I-T)x=y, where I is the identity operator of H.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Let {xt} be generated by the algorithm xt=tγf(xt)+(I-tA)Txt. Then {xt} converges strongly as t0 to a fixed point x* of T which solves the variational inequality (A-γf)x*,x*-x0,xF(T).

Lemma 2.5 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Assume A is a strong positive linear bounded operator on a Hilbert space H with coefficient γ¯>0 and 0<ρA-1. Then I-ρA1-ργ¯.

3. Main Results

Let H be a real Hilbert space, let A be a bounded linear operator on H, and let S, T be two nonexpansive mappings on H such that F(S)F(T). Throughout the rest of this paper, we always assume that A is strongly positive.

Now, let fΠ with the contraction coefficient 0<α<1 and let A be a strongly positive linear bounded operator with coefficient γ̅>0 satisfying 0<γ<γ̅/α. We consider the following modified iterative algorithm:

xn+1=βnxn+(1-βn)Syn,yn=αnγf(xn)+(I-αnA)Txn,n0, where γ>0 is a real number and {αn}, {βn} are two sequences in (0,1).

First, we prove a useful result concerning iterative algorithm (3.1) as follows.

Lemma 3.1.

Let {xn} be a sequence in H generated by the algorithm (3.1) with the sequences {αn} and {βn} satisfying the following control conditions:

limnαn=0,

0<lim infnβnlim supnβn<1.

Then xn+1-xn0.

Proof.

From the control condition (C1), without loss of generality, we may assume that αnA-1. First observe that I-αnA1-αnγ̅ by Lemma 2.5.

Now we show that {xn} is bounded. Indeed, for any pF(S)F(T), yn-p=αn(γf(xn)-Ap)+(I-αnA)(Txn-p)αnγf(xn)-γf(p)+αnγf(p)-Ap+(1-αnγ̅)Txn-pαnγαxn-p+αnγf(p)-Ap+(1-αnγ̅)xn-p=[1-(γ̅-γα)αn]xn-p+αnγf(p)-Ap. At the same time, xn+1-p=βn(xn-p)+(1-βn)(Syn-p)βnxn-p+(1-βn)Syn-pβnxn-p+(1-βn)yn-p. It follows from (3.2) and (3.3) that xn+1-pβnxn-p+(1-βn)[1-(γ̅-γα)αn]xn-p+αn(1-βn)γf(p)-Ap=[1-(γ̅-γα)αn(1-βn)]xn-p+(γ̅-γα)αn(1-βn)γf(p)-Apγ̅-γα, which implies that xn-pmax{x0-p,γf(p)-Apγ̅-γα},n0. Hence {xn} is bounded and so are {ATxn} and {f(xn)}.

From (3.1), we observe that yn+1-yn=αn+1γf(xn+1)+(I-αn+1A)Txn+1-αnγf(xn)-(I-αnA)Txn=αn+1γ(f(xn+1)-f(xn))+(αn+1-αn)γf(xn)+(I-αn+1A)(Txn+1-Txn)+(αn-αn+1)ATxnαn+1γf(xn+1)-f(xn)+(1-αn+1γ̅)Txn+1-Txn+|αn+1-αn|(γf(xn)+ATxn)αn+1γαxn+1-xn+(1-αn+1γ̅)xn+1-xn+|αn+1-αn|(γf(xn)+ATxn)=[1-(γ̅-γα)αn+1]xn+1-xn+|αn+1-αn|(γf(xn)+ATxn). It follows that Syn+1-Syn-xn+1-xnyn+1-yn-xn+1-xn=(γ̅-γα)αn+1xn+1-xn+|αn+1-αn|(γf(xn)+ATxn), which implies, from (C1) and the boundedness of {xn}, {f(xn)}, and {ATxn}, that lim supn(Syn+1-Syn-xn+1-xn)0. Hence, by Lemma 2.2, we have Syn-xn0asn. Consequently, it follows from (3.1) that limnxn+1-xn=limn(1-βn)Syn-xn=0. This completes the proof.

Remark 3.2.

The conclusion xn+1-xn0 is important to prove the strong convergence of the iterative algorithms which have been extensively studied by many authors, see, for example, [3, 6, 7].

If we take S=I in (3.1), we have the following iterative algorithm:

xn+1=βnxn+(1-βn)yn,yn=αnγf(xn)+(I-αnA)Txn,n0. Now we state and prove the strong convergence of iterative scheme (3.11).

Theorem 3.3.

Let {xn} be a sequence in H generated by the algorithm (3.11) with the sequences {αn} and {βn} satisfying the following control conditions:

limnαn=0,

limnαn=,

0<lim infnβnlim supnβn<1.

Then {xn} converges strongly to a fixed point x* of T which solves the variational inequality (A-γf)x*,x*-x0,xF(T).

Proof.

From Lemma 3.1, we have xn+1-xn0.

On the other hand, we have xn-Txnxn+1-xn+xn+1-Txn=xn+1-xn+β(xn-Txn)+(1-βn)(yn-Txn)xn+1-xn+βnxn-Txn+(1-βn)yn-Txnxn+1-xn+βnxn-Txn+(1-βn)αn(γf(xn)+ATxn), that is, xn-Txn11-βnxn+1-xn+αn(γf(xn)+ATxn), this together with (C1), (C3), and (3.13), we obtain limnxn-Txn=0.

Next, we show that, for any x*F(T), lim supnyn-x*,γf(x*)-Ax*0.

In fact, we take a subsequence {xnk} of {xn} such that lim supnxn-x*,γf(x*)-Ax*=limkxnk-x*,γf(x*)-Ax*. Since {xn} is bounded, we may assume that xnkz, where “” denotes the weak convergence. Note that zF(T) by virtue of Lemma 2.3 and (3.16). It follows from the variational inequality (2.3) in Lemma 2.4 that lim supnxn-x*,γf(x*)-Ax*=z-x*,γf(x*)-Ax*0. By Lemma 3.1 (noting S=I), we have yn-xn0. Hence, we get lim supnyn-x*,γf(x*)-Ax*0.

Finally, we prove that {xn} converges to the point x*. In fact, from (3.2) we have yn-x*xn-x*+αnγf(x*)-Ax*. Therefore, from (3.16), we have xn+1-x*2=βn(xn-x*)+(1-βn)(yn-x*)2βnxn-x*2+(1-βn)yn-x*2=βnxn-x*2+(1-βn)αn(γf(xn)-Ax*)+(I-αnA)(Txn-x*)2βnxn-x*2+(1-βn)[(1-αnγ̅)2xn-x*2+2αnγf(xn)-Ax*,yn-x*]=[1-2αnγ̅+(1-βn)αn2γ̅2]xn-x*2+2αnγf(xn)-γf(x*),yn-x*+2αnγf(x*)-Ax*,yn-x*[1-2αnγ̅+(1-βn)αn2γ̅2]xn-x*2+2αnγαxn-x*yn-x*+2αnγf(x*)-Ax*,yn-x*[1-2αn(γ̅-γα)]xn-x*2+(1-βn)αn2γ̅2xn-x*2+2αn2γαxn-x*γf(x*)-Ax*+2αnγf(x*)-Ax*,yn-x*. Since {xn}, f(x*) and Ax* are all bounded, we can choose a constant M>0 such that 1γ̅-γα{(1-βn)γ̅22xn-x*2+γαxn-x*γf(x*)-Ax*}M,n0. It follows from (3.23) that xn+1-x*2[1-2(γ̅-αγ)αn]xn-x*2+2(γ̅-αγ)αnδn, where δn=αnM+1γ̅-γαγf(x*)-Ax*,yn-x*. By (C1) and (3.17), we get lim supnβn0. Now, applying Lemma 2.1 and (3.25), we conclude that xnx*. This completes the proof.

Taking T=I in (3.1), we have the following iterative algorithm:

xn+1=βnxn+(1-βn)Syn,yn=αnγf(xn)+(I-αnA)xn,n0.

Now we state and prove the strong convergence of iterative scheme (3.28).

Theorem 3.4.

Let {xn} be a sequence in H generated by the algorithm (3.28) with the sequences {αn} and {βn} satisfying the following control conditions:

limnαn=0,

limnαn=,

0<lim infnβnlim supnβn<1.

Then {xn} converges strongly to a fixed point x* of S which solves the variational inequality (A-γf)x*,x*-x0,xF(S).

Proof.

From Lemma 3.1, we have xn-Syn0. Thus, we have xn-Sxnxn-Syn+Syn-Sxnxn-Syn+yn-xnxn-Syn+αn(γf(xn)+Axn)0. By the similar argument as (3.17), we also can prove that lim supnyn-x*,γf(x*)-Ax*0. From (3.28), we obtain xn+1-x*2=βn(xn-x*)+(1-βn)(Syn-x*)2βnxn-x*2+(1-βn)Syn-x*2βnxn-x*2+(1-βn)yn-x*2=βnxn-x*2+(1-βn)αn(γf(xn)-Ax*)+(I-αnA)(xn-x*)2βnxn-x*2+(1-βn){(1-αnγ̅)2xn-x*2+2γf(xn)-Ax*,yn-x*}. The remainder of proof follows from the similar argument of Theorem 3.3. This completes the proof.

From the above results, we have the following corollaries.

Corollary 3.5.

Let {xn} be a sequence in H generated by the following algorithm xn+1=βnxn+(1-βn)yn,yn=αnf(xn)+(1-αn)Txn,n0, where the sequences {αn} and {βn} satisfy the following control conditions:

limnαn=0,

limnαn=,

0<lim infnβnlim supnβn<1.

Then {xn} converges strongly to a fixed point x* of T which solves the variational inequality (I-f)x*,x*-x0,xF(T).

Corollary 3.6.

Let {xn} be a sequence in H generated by the following algorithm xn+1=βnxn+(1-βn)Syn,yn=αnf(xn)+(1-αn)xn,n0, where the sequences {αn} and {βn} satisfy the following control conditions:

limnαn=0,

limnαn=,

0<lim infnβnlim supnβn<1.

Then {xn} converges strongly to a fixed point x* of S which solves the variational inequality (I-f)x*,x*-x0,xF(S).

Remark 3.7.

Theorems 3.3 and 3.4 provide the strong convergence results of the algorithms (3.11) and (3.28) by using the control conditions (C1) and (C2), which are weaker conditions than the previous known ones. In this respect, our results can be considered as an improvement of the many known results.

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