Some Krasnonsel'skiĭ-Mann Algorithms and the Multiple-Set Split Feasibility Problem

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Volume 2010, Article ID 513956,12pages doi:10.1155/2010/513956

Research Article

Some Krasnonsel’ski˘ı-Mann Algorithms and

the Multiple-Set Split Feasibility Problem

Huimin He,

1

_{Sanyang Liu,}

1

_{and Muhammad Aslam Noor}

2, 3

1_{Department of Mathematics, Xidian University, Xi’an 710071, China}

2_{Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan} 3_{Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia}

Correspondence should be addressed to Huimin He,[email protected]

Received 3 April 2010; Revised 7 July 2010; Accepted 13 July 2010

Academic Editor: S. Reich

Copyrightq2010 Huimin He 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.

Some variable Krasnonsel’ski˘ı-Mann iteration algorithms generate some sequences{xn}, {yn},

and{zn}, respectively, via the formulaxn1 1−αnxnαnTN· · ·T2T1xn,yn1 1−βnyn βnNi1λiTiyn, zn1 1−γn1zn γn1Tn1zn, where Tn TnmodN and the mod function

takes values in{1,2, . . . , N},{αn},{βn}, and{γn}are sequences in0,1,and{T1, T2, . . . , TN}are

sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence{xn}, {yn}, {zn} generated by the above formulas converge weakly to the common

fixed point of{T1, T2, . . . , TN}, respectively. These results are used to solve the multiple-set split

feasibility problem recently introduced by Censor et al.2005. The purpose of this paper is to introduce convergence theorems of some variable Krasnonsel’ski˘ı-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem.

1. Introduction

The Krasnonsel’ski˘ı-Mann K-M iteration algorithm 1, 2 is used to solve a fixed point equation

Txx, 1.1

whereTis a self-mapping of closed convex subsetCof a Banach spaceX. The K-M algorithm generates a sequence{xn}according to the recursive formula

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where{αn}is a sequence in the interval0,1and the initial guessx0 ∈Cis chosen arbitrarily. It is known3that ifXis a uniformly convex Banach space with a Frechet diﬀerentiable norm in particular, a Hilbert space, ifT :C → Cis nonexpansive, that is,Tsatisfies the property

_Tx₋_Ty_≤_x₋_y _∀_{x, y}_∈_C _1.3

and if T has a fixed point, then the sequence{xn} generated by the K-M algorithm 1.2

converges weakly to a fixed point ofT provided that{αn}fulfils the condition

∞

n0

αn1−αn ∞. 1.4

See4,5for details on the fixed point theory for nonexpansive mappings.

Many problems can be formulated as a fixed point equation1.1with a nonexpansive

T and thus K-M algorithm 1.2 applies. For instance, the split feasibility problem SFP introduced in6–8, which is to find a point

x∈C such thatAx∈Q, 1.5

whereCandQare closed convex subsets of Hilbert spacesH1andH2, respectively, andAis a linear bounded operator fromH1toH2. This problem plays an important role in the study of signal processing and image reconstruction. Assuming that the SFP1.5is consistenti.e., 1.5has a solution, it is not hard to see thatx ∈ Csolves1.5if and only if it solves the fixed point equation

xPC

I−γA∗I−PQ

Ax, x∈C, 1.6

wherePC andPQ are theorthogonalprojections ontoCandQ, respectively,γ > 0 is any

positive constant andA∗denotes the adjoint ofA. Moreover, for suﬃciently smallγ >0, the operatorPCI−γA∗I−PQAwhich defines the fixed point equation1.6is nonexpansive.

To solve the SFP1.5, Byrne7,8proposed his CQ algorithmsee also9which generates a sequence{xn}by

xn1PC

I−γA∗I−PQ

Axn, n≥0, 1.7

whereγ ∈0,2/λ withλbeing the spectral radius of the operatorA∗A. In 2005, Zhao and Yang10considered the following perturbed algorithm:

xn1 1−αnxnαnPCn

I−γA∗I−PQn

Axn, 1.8

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by1.8, Zhao and Yang 10,12also studied the following more general algorithm which generates a sequence{xn}according to the recursive formula

xn1 1−αnxnαnTnxn, 1.9

where {Tn} is a sequence of nonexpansive mappings in a Hilbert space H, under certain

conditions, they proved convergence of1.9essentially in a finite-dimensional Hilbert space. Furthermore, with regard to1.9, Xu13extended the results of Zhao and Yang10in the framework of fairly general Banach space.

The multiple-set split feasibility problem MSSFP which finds application in intensity-modulated radiation therapy 14 has recently been proposed in 15 and is formulated as finding a point

x∈C

N

i1

Ci such thatAx∈Q M

j1

Qj, 1.10

whereNandMare positive integers,{C1, C2, . . . , CN}and{Q1, Q2, . . . , QM}are closed and

convex subsets ofH1 andH2, respectively, andAis a linear bounded operator fromH1 to H2.

Assuming consistency of the MSSFP1.10, Censor et al.15introduced the following projection algorithm:

xn1PΩ ⎛ ⎝_x_n₋_γ

⎛

⎝N

i1

αixn−PCixn

M

j1 βjA∗

Axn−PQjAxn

⎞

⎠ ⎞

⎠_, _1.11

where Ω is another closed and convex subset of H1, 0 < γ < 2/L with L Ni1αi

ρA∗A_jM₁βj andρA∗Abeing the spectral radius ofA∗A, andαi > 0 for alliandβj >0

for allj. They studied convergence of the algorithm1.11in the case where bothH1andH2 are finite dimensional. In 2006, Xu13demonstrated some projection algorithms for solving the MSSFP1.10in Hilbert space as follows:

xn1

PCN

I−γ∇q· · ·PC1

I−γ∇qxn, n≥0,

yn1

N

i1 λiPCi

⎛

⎝_y_n₋_γM

j1 βjA∗

I−PQj

Ayn

⎞

⎠_, _n_≥₀_,

zn1 PCn1

⎛

⎝_z_n₋_γM

j1 βjA∗

I−PQj

Azn

⎞

⎠_, _n_≥₀_,

1.12

whereqx 1/2M_j₁βjPQjAx−Ax 2_,_∇

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study the following more general algorithm which generate the sequences{xn},{yn}, and

{zn}, respectively, via the formulas

xn1 1−αnxnαnTN· · ·T2T1xn, 1.13

yn1

1−βn

ynβn

N

i1

λiTiyn, 1.14

zn1

1−γn1

znγn1Tn1zn, 1.15

whereTn TnmodN,{αn},{βn}, and {γn} are sequences in0,1, and {T1, T2, . . . , TN}are

sequences of nonexpansive mappings. We will show, in a fairly general Banach spaceX, that the sequences{xn},{yn}, and{zn}generated by1.13,1.14, and1.15converge weakly to

the common fixed point of{T1, T2, . . . , TN}, respectively. The applications of these results are

used to solve the multiple-set split feasibility problem recently introduced by15.

Note that, letting C be a nonempty subset of Banach space X and A, B are self-mappings ofC, we useDρA, Bto denote sup{Ax−Bx: x ≤ρ}, that is,

DρA, B:sup

Ax−Bx: x ≤ρ. 1.16

This paper is organized as follows. In the next section, we will prove a weak convergence theorems for the three variable K-M algorithms1.13,1.14, and1.15 in a uniformly convex Banach space with a Frechet diﬀerentiable normthe class of such Banach spaces include Hilbert space and Lp _and _lp _{space for 1} _{< p <} _∞_{. In the last section, we}

will present the applications of the weak convergence theorems for the three variable K-M algorithms1.13,1.14, and1.15.

2. Convergence of Variable Krasnonsel’ski˘ı-Mann Iteration Algorithm

To solve the multiple-set split feasibility problemMSSFPin Section 3, we firstly present some theorems of the general variable Krasnonsel’ski˘ı-Mann iteration algorithms.

Theorem 2.1. LetX be a uniformly convex Banach space with a Frechet diﬀerentiable norm, letC be a nonempty closed and convex subset ofX, and letTi : C → Cbe nonexpansive mapping,i

1,2, . . . , N. Assume that the set of common fixed point of{T1, T2, . . . , TN},

_N

i1FixTi, is nonempty.

Let{xn}be any sequence generated by1.13, where0< αn<1satisfy the conditions

i∞_n₀αn1−αn ∞;

ii∞_n₀αnDρTN· · ·T1, Ti < ∞for everyρ > 0andi 1,2, . . . , N, whereDρTN· · ·T1, Ti sup{TN· · ·T1x−Tix: x ≤ρ}.

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Proof. SinceTi :C → Cis nonexpansive mapping, fori1,2, . . . , N, then, the composition

TN· · ·T2T1is nonexpansive mapping fromCtoC. LetU:TN· · ·T2T1. Takex∈ Nj1FixTj x∈FixUto deduce that

xn1−x ≤1−αnxn−xαnUxn−x

≤ xn−x.

2.1

Thus,{xn−x}is a decreasing sequence, and we have that limn→ ∞xn−xexists. Hence,

{xn}is bounded, so are{Tixn},i1,2, . . . , N, and{Uxn}. Letρsup{xn,Uxn−Tixn:

n≥0, i1,2, . . . , N}<∞,and letr2ρx<∞.

Now sinceXis uniformly convex, by16, Theorem 2, there exists a continuous strictly convex functionϕ, withϕ0 0, so that

_λx 1−λy2≤λx2 1−λy2−λ1−λϕx−y, 2.2

for all x,y ∈ X such that x ≤ r and y ≤ r and for all λ ∈ 0,1. Let Uxn −Tixn,i

1,2, . . . , N, be replaced byen,inote thaten,i ≤ DρU, Ti, and taking a constantMso that

M≥sup{2xn−xαnen,i: n≥0}, by the above2.2, we obtain that

xn1−x21−αnxn−xαnen,i αnTixn−xαnen,i2

≤1−αnxn−xαnen,i2αnTixn−xαnen,i2

−αn1−αnϕxn−Tixn

≤1−αn

xn−x22αnxn−xen,iα2nen,i2

αn

Tixn−x22αnen,iTixn−xα2nen,i2

−αn1−αnϕxn−Tixn

≤ xn−x2MαnDρU, Ti−αn1−αnϕxn−Tixn.

2.3

It follows that

αn1−αnϕxn−Tixn≤ xn−x2− xn1−x2MαnDρU, Ti. 2.4

Since limn→ ∞xn−xexists, by conditioniiand2.4, it implies that

∞

n1

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which further implies that byilim infn→ ∞ϕxn−Tixn 0, hence,

lim inf

n→ ∞ xn−Tixn0. 2.6

On the other hand, it is not hard to deduce from1.13that

xn1−Tixn11−αnxnαnUxn−Tixn1

1−αnxnαnUxn−TixnTixn−Tixn1

≤1−αnxn−TixnαnUxn−Tixnxn1−xn

1−αnxn−TixnαnUxn−Tixnαnxn−Uxn

≤1−αnxn−TixnαnUxn−Tixn

αnxn−TixnαnTixn−Uxn

xn−Tixn2αnUxn−Tixn

≤ xn−Tixn2αnDρTi, U.

2.7

Since∞_n₁αnDρU, Ti < ∞, we see that limn→ ∞xn−Tixnexists. This together with2.6

implies that

lim

n→ ∞xn−Tixn0. 2.8

The demiclosedness principle for nonexpansive mappingssee5,17implies that

ωwxn⊂ N

i1

FTi, 2.9

whereωwxn {x:∃xnj x}denotes the weakω-limit set of{xn}.

To prove that{xn}is weakly convergent to a common fixed pointpof{T1, T2, . . . , TN},

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Indeed, if there are x, x ∈ ωwxnxni x, xmj x, since limn→ ∞xn −xand

limn→ ∞xn−xexist, ifx /x, then

lim

n→ ∞xn−x

2 _lim

j→ ∞

xmj−x

x−x2

lim

j→ ∞

xmj−x

2x−x22 lim

j→ ∞

xmj−x, x−x

lim

j→ ∞

xmj−x

2x−x2

> lim

i→ ∞xmi −x 2 _lim

i→ ∞xni−x 2

lim

i→ ∞xni−x x−x

2

lim

i→ ∞xni−x

2

x−x22 lim

j→ ∞xni −x, x−x

lim

i→ ∞xni−x

2 x−x2

> lim

i→ ∞xni−x

2_lim

n xn−x

2 .

2.10

This is a contradiction.

The proof is completed.

Theorem 2.2. LetX be a uniformly convex Banach space with a Frechet diﬀerentiable norm, letC be a nonempty closed and convex subset ofX, and letTi : C → Cbe nonexpansive mapping,i

1,2, . . . , N, assume that the set of common fixed point of{T1, T2, . . . , TN},

_N

i1FixTi, is nonempty.

Let{yn}be defined by1.14, where0< βn<1satisfy the following conditions

i∞_n₀βn1−βn ∞;

ii∞_n₀βnDρNi1λiTi, Ti < ∞ for every ρ > 0 and i 1,2, . . . , N, where

Dρ

_N

i1λiTi, Ti sup{

_N

i1λiTix−Tix: x ≤ρ}.

Then{yn}converges weakly to a common fixed point q of{T1, T2, . . . , TN}.

Proof. SinceTi:C → Cis a nonexpansive mapping,i1,2, . . . , N, then, it is not hard to see

thatN_i₁λiTiis a nonexpansive mapping fromCtoC.

The remainder of the proof is the same asTheorem 2.1. The proof is completed.

Theorem 2.3. LetXbe a uniformly convex Banach space with a Frechet diﬀerentiable norm, letCbe a nonempty closed convex subset ofX, and letTi:C → Cbe nonexpansive mapping,i1,2, . . . , N,

assume that the set of common fixed point of{T1, T2, . . . , TN},

_N

i1FixTi, is nonempty. Let{zn}be

defined by1.15, where0< γn<1satisfy the conditions

i∞_n₀γn1−γn ∞;

ii∞_n₀γnDρTn1, Ti < ∞for everyρ > 0 andi 1,2, . . . , N, whereDρTn1, Ti

sup{Tn1x−Tix: x ≤ρ}.

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Proof. SinceTnTnmodNand{T1, T2, . . . , TN}is a sequence of nonexpansive mappings from

CtoC, so, the proof of this theorem is similar to Theorems2.1and2.2. The proof is completed.

3. Applications for Solving the Multiple-Set Split

Feasibility Problem (MSSFP)

Recall that a mappingT in a Hilbert spaceHis said to be averaged ifT can be written as 1−λIλS, whereλ∈0,1andSis nonexpansive. Recall also that an operatorAinHis said to beγ-inverse strongly monotoneγ-ismfor a given constantγ >0 if

x−y, Ax−Ay≥γAx−Ay2, ∀x, y∈H. 3.1

A projectionPKofHonto a closed convex subsetKis both nonexpansive and 1-ism. It is also

known that a mappingT is averaged if and only if the complementI −T isγ-ism for some

γ >1/2; see8for more property of averaged mappings andγ-ism.

To solve the MSSFP 1.10, Censor et al. 15 proposed the following projection algorithm 1.11, the algorithm 1.11 involves an additional projection PΩ. Though the MSSFP,1.10includes the SFP1.5as a special case, which does not reduced to1.7, let alone 1.8. In this section, we will propose some new projection algorithms which solve the MSSFP1.10and which are the application of algorithms1.13,1.14, and1.15for solving the MSSFP. These projection algorithms can also reduce to the algorithm1.8when the MSSFP1.10is reduced to the SFP1.5.

The first one is a K-M type successive iteration method which produces a sequence {xn}by

xn1 1−αnxnαn

PCN

I−γ∇q· · ·PC1

I−γ∇qxn, n≥0. 3.2

Theorem 3.1. Assume that the MSSFP1.10is consistent. Let{xn}be the sequence generated by

the algorithm3.2, where0< γ <2/LwithLA2Mj1βjand0< αn<1satisfy the condition:

_∞

n0αn1−αn ∞. Then{xn}converges weakly to a solution of the MSSFP1.10.

Proof. LetTi:PCiI−γ∇q,i1,2, . . . , N.

Hence,

UTN· · ·T1

PCN

I−γ∇q· · ·PC1

I−γ∇q. 3.3

Since

∇qx

M

j1 βjA∗

I−PQj

Ax, x∈C, 3.4

andI−PQjis nonexpansive, it is easy to see that∇qisL-Lipschitzian, withLA 2M

j1βj.

Therefore, ∇q is1/L-ism18. This implies that for any 0 < γ < 2/L, I−γ∇qis averaged. Hence, for any closed and convex subsetK ofH1, the compositePKI−γ∇qis

averaged.

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By the position 2.28, we see that the fixed point set ofU, FixU, is the common fixed point set of the averaged mappings{TN· · ·T1}.

By Reich3, we have {xn} converges weakly to a fixed point ofUwhich is also a

common fixed point of{TN· · ·T1}or a solution of the MSSFP1.10. The proof is completed.

The second algorithm is also a K-M type method which generates a sequence{yn}by

yn1

1−βn

ynβn

N

i1 λiPCi

⎛

⎝_y_n₋_γM

j1 βjA∗

I−PQj

Ayn

⎞

⎠_, _n_≥₀_. _3.5

Theorem 3.2. Assume that the MSSFP1.10is consistent. Let{xn}be any sequence generated by

the algorithm3.5, where0< γ <2/LwithLA2M_j₁βjand0< βn<1satisfy the condition:

_∞

n0βn1−βn ∞. Then{yn}converges weakly to a solution of the MSSFP1.10.

Proof. From the proof ofTheorem 3.1, it is easy to know thatTi :PCiI−γ∇qis averaged,

so, the convex combinationS:N_i₁λiTiis also averaged.

ThusSis nonexpansive.

By Reich3, we have{yn}converges weakly to a fixed point ofS.

Next, we only need to prove the fixed point ofS is also the common fixed point of {TN· · ·T1}which is the solution of the MSSFP1.10, that is, FixS

_N

i1FixTi.

Indeed, it suﬃces to show thatN_n₁FixTi⊃FixNi1λiTi.

Pick an arbitraryx∈FixN_i₁λiTi, thus

_N

i1λiTixx. Also pick ay ∈Fix

_N

n1Ti,

thusTiyy,i1,2, . . . , N.

WriteTi 1−βiIβiTi,i1,2, . . . , Nwithβi∈0,1andTiis nonexpansive.

We claim that ifzis such thatTiz /z, thenTix−y<x−y,i1,2, . . . , N.

Indeed, we have

_T_i_z₋_y2 1−βi

z−yβi

Tiz−y

2

1−βiz−y2βiTiz−y

2 −βi

1−βiz−Tiz

2

≤z−y2−1−βi

z−Tiz2

<z−y2, asz−Tiz>0.

3.6

If we can show thatTixx, then we are done. So assume thatTx /x. Now since

_N

i1λiTix

x /Tx, we have

_x₋_y

N

i1

λiTix−y

≤N

i1

λiTix−y

<x−y.

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This is a contradiction. Therefore, we must haveTixx,i1,2, . . . , N, that is,Nn1FixTix

x.

This proof is completed.

We now apply Theorem 2.3 to solve the MSSFP 1.10. Recall that the ρ-distance between two closed and convex subsetsE1andE2of a Hilbert spaceHis defined by

dρE1, E2 sup

x≤ρ

{PE1x−PE2x}. 3.8

The third method is a K-M type cyclic algorithm which produces a sequence{zn}in

the following manner: apply T1 to the initial guess z0 to get z1 1−γ1z0 γ1PC1z0 − γMj1βjA∗I−PQjAz0, next applyT2toz1to getz2 1−γ2z1γ2PC2z1−γ

_M

j1βjA∗I−

PQjAz1, and continue this way to get zN 1−γNz0 γNPCNzN−1 −γ

M

j1βjA∗I −

PQjAzN−1; then repeat this process to getzN1 1−γN1z0γNPC1zN−γ

_M

j1βjA∗I−

PQjAzN, and so on. Thus, the sequence{zn}is defined and we write it in the form

zn1

1−γn1

z0γn1PCn1

⎛

⎝_z_n₋_γM

j1 βjA∗

I−PQj

Azn

⎞

⎠_, _n_≥0, 3.9

whereCnCnmodN.

Theorem 3.3. Assume that the MSSFP1.10is consistent. Let{xn}be the sequence generated by

the algorithm3.9, where0 < γ <2/LwithLA2M_j₁βjand0< γn<1satisfy the following

conditions:

i∞_n₀γn1−γn ∞;

ii∞_n₀γndρCn1, Ci < ∞ and ∞n0γndρQn1, Qi < ∞ for each ρ > 0, i

1,2, . . . , N.

Then{zn}converges weakly to a solution of the MSSFP1.10.

Proof. From the proof of application3.2, it is easy to verify thatTi:PCiI−γ∇qis averaged,

so,Tn1:Tn1 modNis also averaged.

ThusTn1is nonexpansive.

The projection iteration algorithm3.9can also be written as

zn1

1−γn1

znγn1Tn1zn. 3.10

Givenρ >0, let

ρsupmaxAx,x−γA∗I−PQ

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We compute, forx∈H1, such thatx ≤ρ, _T_n₁_x₋_T_i_x

≤PCn1

x−γA∗I−PQn1

Ax−PCn1

x−γA∗I−PQi

Ax

PCn1

x−γA∗I−PQi

Ax−PCi

x−γA∗I−PQi

Ax

≤PCn1

x−γA∗I−PQi

Ax−PCi

x−γA∗I−PQi

Ax

γA∗PQn1Ax−PQiAx

≤dρ

Cn1, Ci

γAdρ

Qn1, Qi

.

3.12

This shows that

Dρ

Tn1, Ti

≤dρ

Cn1, Ci

γAdρ

Qn1, Qi

. 3.13

It then follows from conditioniithat

∞

n0 γnDρ

Tn1, Ti

≤∞

n0 γndρ

Cn1, Ci

∞

n0 γndρ

Qn1, Qi

<∞. 3.14

Now we cam apply Theorem 2.3 to conclude that the sequence {zn} given by the

projection Algorithm3.9converges weakly to a solution of the MSSFP1.10. The proof is completed.

Remark 3.4. The algorithms3.12,3.13, and3.15of Xu13are some projection algorithms for solving the MSSEP1.10, which are concrete projection algorithms. In this paper, firstly, we present some general variable K-M algorithms1.13,1.14, and1.15, and prove the weak convergence for them in Section 2. Secondly, through the applications of the weak convergence for three general variable K-M algorithms 1.13,1.14, and 1.15, we solve the MSSEP1.10by the algorithms3.2,3.5, and3.9.

Acknowledgments

The work was supported by the Fundamental Research Funds for the Central Universities, no. JY10000970006, and the National Nature Science Foundation, no. 60974082.

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