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Volume 2007, Article ID 21972, 18 pagesdoi:10.1155/2007/21972 Research Article Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces Fumiaki Ko

Volume 2007, Article ID 21972, 18 pages

doi:10.1155/2007/21972

Research Article

Block Iterative Methods for a Finite Family of Relatively

Nonexpansive Mappings in Banach Spaces

Fumiaki Kohsaka and Wataru Takahashi

Received 7 November 2006; Accepted 12 November 2006

Recommended by Ravi P Agarwal

Using the convex combination based on Bregman distances due to Censor and Reich, we define an operator from a given family of relatively nonexpansive mappings in a Banach space We first prove that the fixed-point set of this operator is identical to the set of all common fixed points of the mappings Next, using this operator, we construct an iterative sequence to approximate common fixed points of the family We finally apply our results

to a convex feasibility problem in Banach spaces

Copyright © 2007 F Kohsaka and W Takahashi This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

LetH be a Hilbert space and let { C i } m i =1 be a family of closed convex subsets ofH such

thatF =m i =1C iis nonempty Then the problem of image recovery is to find an element

ofF using the metric projection P ifromH onto C i(i =1, 2, ,m), where

P i(x) =arg min

for allx ∈ H This problem is connected with the convex feasibility problem In fact,

if{ g i } m i =1 is a family of continuous convex functions from H intoR, then the convex feasibility problem is to find an element of the feasible set

m



i =1



x ∈ H : g i(x) ≤0

We know that eachP iis a nonexpansive retraction fromH onto C i, that is,

P i x − P i y  ≤  x − y  (1.3)

for allx, y ∈ H and P2

i = P i Further, it holds thatF =m i =1F(P i), whereF(P i) denotes the set of all fixed points ofP i(i =1, 2, ,m) Thus the problem of image recovery in the

setting of Hilbert spaces is a common fixed point problem for a family of nonexpansive mappings

A well-known method for finding a solution to the problem of image recovery is the

block-iterative projection algorithm which was proposed by Aharoni and Censor [1] in finite-dimensional spaces; see also [2–5] and the references therein This is an iterative procedure, which generates a sequence{ x n }by the rulex1= x ∈ H and

x n+1 =

m



i =1

ω n(i)α i x n+

1− α i

P i x n (n =1, 2, ), (1.4)

where{ ω n(i) } m i =1⊂[0, 1] (n ∈ N) with m i =1ω n(i) =1 (n ∈ N) and { α i } m i =1⊂(−1, 1) In particular, Butnariu and Censor [3] studied the strong convergence of{ x n }to an element

ofF.

Recently, Kikkawa and Takahashi [6] applied this method to the problem of finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces LetC

be a nonempty closed convex subset of a Banach spaceE and let { T i } m i =1be a finite family

of nonexpansive mappings fromC into itself Then the iterative scheme they dealt with is

stated as follows:x1= x ∈ C and

x n+1 =

m



i =1

ω n(i)α n,i x n+

1− α n,i

T i x n (n =1, 2, ), (1.5)

where{ ω n(i) } m i =1⊂[0, 1] with m i =1ω n(i) =1 (n ∈ N) and { α i } m i =1⊂[0, 1] They proved that the generated sequence{ x n }converges weakly to a common fixed point of{ T i } m i =1 under some conditions onE, { α n,i }, and { ω n(i) } Then they applied their result to the

problem of finding a common point of a family of nonexpansive retracts ofE; see also

[7–10] for the previous results on this subject

Our purpose in the present paper is to obtain an analogous result for a finite family

of relatively nonexpansive mappings in Banach spaces This notion was originally

intro-duced by Butnariu et al [11] Recently, Matsushita and Takahashi [12–14] reformulated the definition of the notion and obtained weak and strong convergence theorems to ap-proximate a fixed point of a single relatively nonexpansive mapping It is known that if

C is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach

spaceE, then the generalized projection Π C(see, Alber [15] or Kamimura and Takahashi [16]) fromE onto C is relatively nonexpansive, whereas the metric projection P CfromE

ontoC is not generally nonexpansive.

LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive

Banach spaceE, let J be the duality mapping from E into E ∗, and let{ T i } m i =1 be a finite family of relatively nonexpansive mappings fromC into itself such that the set of all

com-mon fixed points of{ T i } m i =1 is nonempty Motivated by the convex combination based

on Bregman distances [17] due to Censor and Reich [18], the iterative methods intro-duced by Matsushita and Takahashi [12–14], and the proximal-type algorithm due to the

authors [19], we define an operatorU n(n ∈ N) by

U n x =ΠC J −1

m

i =1

ω n(i)α n,i Jx +1− α n,i

JT i x

(1.6)

for allx ∈ C, where { ω n(i) } ⊂[0, 1] and{ α n,i } ⊂[0, 1] with m i =1ω n(i) =1 (n ∈ N) Such

a mappingU n is called a block mapping defined by T1,T2, ,T m,{ α n,i }and{ ω n(i) } In

Section 4, we show that the set of all fixed points ofU nis identical to the set of all common fixed points of{ T i } m i =1(Theorem 4.2) InSection 5, under some additional assumptions,

we show that the sequence{ x n }generated byx1= x ∈ C and

x n+1 = U n x n (n =1, 2, ) (1.7) converges weakly to a common fixed point of{ T i } m i =1(Theorem 5.3) This result gener-alizes the result of Matsushita and Takahashi [12] IfE is a Hilbert space and each T iis a nonexpansive mapping fromC into itself, then J is the identity operator on E, and hence

(1.5) and (1.7) are coincident with each other InSection 6, we deduce some results from Theorems4.2and5.3

2 Preliminaries

LetE be a (real) Banach space with norm  · and letE ∗denote the topological dual ofE.

We denote the strong convergence and the weak convergence of a sequence{ x n }tox in E

byx n → x and x n  x, respectively We also denote the weak ∗convergence of a sequence

{ x n ∗ }tox ∗inE ∗byx ∗ n ∗ x ∗ For allx ∈ E and x ∗ ∈ E ∗, we denote the value ofx ∗atx

by x,x ∗ We also denote byRandNthe set of all real numbers and the set of all positive

integers, respectively The duality mapping J from E into 2 E ∗

is defined by

J(x) =x ∗ ∈ E ∗: x,x ∗

=  x 2=x ∗ 2 

(2.1) for allx ∈ E.

A Banach space E is said to be strictly convex if  x  =  y  =1 and x y imply

( x + y)/2  < 1 It is also said to be uniformly convex if for each ε ∈(0, 2], there exists

δ > 0 such that

 x  =  y  =1,  x − y  ≥ ε (2.2) imply( x + y)/2  ≤1− δ The space E is also said to be smooth if the limit

lim

t →0

 x + ty  −  x 

exists for allx, y ∈ S(E) = { z ∈ E :  z  =1} It is also said to be uniformly smooth if the limit (2.3) exists uniformly inx, y ∈ S(E) It is well known that  pandL p(1< p < ∞) are

uniformly convex and uniformly smooth; see Cioranescu [20] or Diestel [21] We know that ifE is smooth, strictly convex, and reflexive, then the duality mapping J is

single-valued, one-to-one, and onto The duality mapping from a smooth Banach spaceE into

E ∗ is said to be weakly sequentially continuous if Jx n ∗ Jx whenever { x n }is a sequence

inE converging weakly to x in E; see, for instance, [20,22]

Let E be a smooth, strictly convex, and reflexive Banach space, let J be the duality

mapping fromE into E ∗, and letC be a nonempty closed convex subset of E Throughout

the present paper, we denote byφ the mapping defined by

φ(y,x) =  y 2−2y,Jx + x 2 (2.4)

for ally,x ∈ E Following Alber [15], the generalized projection from E onto C is defined

by

ΠC(x) =arg min

for allx ∈ E; see also Kamimura and Takahashi [16] IfE is a Hilbert space, then φ(y,x) =

 y − x 2for ally,x ∈ E, and hence Π Cis reduced to the metric projectionP C It should

be noted that the mappingφ is known to be the Bregman distance [17] corresponding

to the Bregman function · 2, and hence the projectionΠC is the Bregman projection

corresponding toφ We know the following lemmas concerning generalized projections.

Lemma 2.1 (see [15]; see also [16]) Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E Then

φx,Π C y +φΠC y, y ≤ φ(x, y) (2.6)

for all x ∈ C and y ∈ E.

Lemma 2.2 (see [15]; see also [16]) Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, let x ∈ E, and let z ∈ C Then z =ΠC x is equivalent to

 y − z, Jx − Jz ≤0 (2.7)

for all y ∈ C.

LetC be a nonempty closed convex subset of a smooth, strictly convex, and reflexive

Banach spaceE, let T be a mapping from C into itself, and let F(T) be the set of all fixed

points ofT Then a point z ∈ C is said to be an asymptotic fixed point of T (see Reich

[23]) if there exists a sequence{ z n }inC converging weakly to z and lim n  z n − Tz n  =0

We denote the set of all asymptotic fixed points ofT by F(T) Following Matsushita and Takahashi [12–14], we say thatT is a relatively nonexpansive mapping if the following

conditions are satisfied:

(R1)F(T) is nonempty;

(R2)φ(u,Tx) ≤ φ(u,x) for all u ∈ F(T) and x ∈ C;

(R3)F(T) = F(T).

Some examples of relatively nonexpansive mappings are listed below; see Reich [23] and Matsushita and Takahashi [12] for more details

(a) IfC is a nonempty closed convex subset of a Hilbert space E and T is a

non-expansive mapping fromC into itself such that F(T) is nonempty, then T is a

relatively nonexpansive mapping fromC into itself.

(b) IfE is a uniformly smooth and strictly convex Banach space and A ⊂ E × E ∗is

a maximal monotone operator such thatA −10 is nonempty, then the resolvent

J r =(J + rA) −1J (r > 0) is a relatively nonexpansive mapping from E onto D(A)

(the domain ofA) and F(J r)= A −10

(c) IfΠC is the generalized projection from a smooth, strictly convex, and reflex-ive Banach spaceE onto a nonempty closed convex subset C of E, then Π C is a relatively nonexpansive mapping fromE onto C and F(Π C)= C.

(d) If{ C i } m i =1 is a finite family of closed convex subset of a uniformly convex and uniformly smooth Banach space E such that m i =1C i is nonempty and T =

Π1Π2···Πmis the composition of the generalized projectionsΠifromE onto

C i(i =1, 2, ,m), then T is a relatively nonexpansive mapping from E into itself

andF(T) =m i =1C i

The following lemma is due to Matsushita and Takahashi [14]

Lemma 2.3 (see [14]) Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E and let T be a relatively nonexpansive mapping from C into itself Then F(T) is closed and convex.

We also know the following lemmas

Lemma 2.4 (see [16]) Let E be a smooth and uniformly convex Banach space and let { x n }

and { y n } be sequences in E such that either { x n } or { y n } is bounded If lim n φ(x n,y n)= 0,

then lim n  x n − y n  = 0.

Lemma 2.5 (see [16]) Let E be a smooth and uniformly convex Banach space and let r > 0 Then there exists a strictly increasing, continuous, and convex function g : [0,2r] → R such that g(0) = 0 and

g x − y  ≤ φ(x, y) (2.8)

for all x, y ∈ B r = { z ∈ E :  z  ≤ r }

Lemma 2.6 (see [24]; see also [25,26]) Let E be a uniformly convex Banach space and let

r > 0 Then there exists a strictly increasing, continuous, and convex function g : [0,2r] → R

such that g(0) = 0 and

tx + (1 − t)y 2

≤ t  x 2+ (1− t)  y 2− t(1 − t)g x − y  (2.9)

for all x, y ∈ B r and t ∈ [0, 1].

3 Lemmas

The following lemma is well known For the sake of completeness, we give the proof

Lemma 3.1 Let E be a strictly convex Banach space and let { t i } m i =1⊂ (0, 1) with m i =1t i = 1.

If { x i } m i =1is a finite sequence in E such that







m



i =1

t i x i







2

=

m



i =1

t ix i 2

then x1= x2= ··· = x m

Proof If x k x lfor somek,l ∈ {1, 2, ,m }, then the strict convexity of E implies that





 t k

t k+t l x k+ t l

t k+t l x l







2

< t t k

k+t l

x k 2 + t l

t k+t l

x l 2. (3.2) Using this inequality, we have







m



i =1

t i x i







2

=







t k+t l  t k

t k+t l x k+ t l

t k+t l x l

 + 

i k,l

t i x i







2

≤t k+t l 



 t k

t k+t l x k+ t l

t k+t l x l







2 + 

i k,l

t ix i 2

<t k+t l  t k

t k+t l  x 2+ t l

t k+t l  y 2

 +

i k,l

t ix i 2

=

m



i =1

t ix i 2.

(3.3)

We also need the following lemmas

Lemma 3.2 Let E be a smooth, strictly convex and reflexive Banach space, let z ∈ E and let

{ t i } ⊂ (0, 1) with m i =1t i = 1 If { x i } m i =1is a finite sequence in E such that

φ

z,J −1

m

j =1

t j Jx j

= φz,x i

(3.4)

for all i ∈ {1, 2, ,m } , then x1= x2= ··· = x m

Proof By assumption, we have

φ

z,J −1

m

j =1

t j Jx j

=

m



i =1

t i φz,x i

This is equivalent to

 z 2−2



z,m

i =1

t i Jx i

 +







m



i =1

t i Jx i







2

=

m



i =1

t i

 z 2−2 z,Jx i

+x i 2

, (3.6)

which is also equivalent to







m



i =1

t i Jx i







2

=

m



i =1

t iJx i 2

SinceE is smooth and reflexive, E ∗is strictly convex Thus,Lemma 3.1implies thatJx1=

Jx2= ··· = Jx m By the strict convexity ofE, J is one-to-one Hence we have the desired

Lemma 3.3 Let E be a smooth, strictly convex, and reflexive Banach space, let { x i } m i =1be a finite sequence in E and let { t i } m i =1⊂ [0, 1] with m i =1t i = 1 Then

φ

z,J −1

m



i =1

t i Jx i

≤

m



i =1

t i φz,x i

(3.8)

for all z ∈ E.

Proof Let V : E × E ∗ → Rbe the function defined by

Vx,x ∗

=  x 2−2 x,x ∗

+x ∗ 2

(3.9) for allx ∈ E and x ∗ ∈ E ∗ In other words,

Vx,x ∗

= φx,J −1x ∗

(3.10) for allx ∈ E and x ∗ ∈ E ∗ We also haveφ(x, y) = V(x,J y) for all x, y ∈ E Then we have

from the convexity ofV in its second variable that

φ

z,J −1

m

i =1

t i Jx i

= V

z,

m



i =1

t i Jx i

≤

m



i =1

t i Vz,Jx i

=

m



i =1

t i φz,x i

. (3.11)

4 Block mappings by relatively nonexpansive mappings

LetE be a smooth, strictly convex, and reflexive Banach space and let J be the duality

mapping fromE into E ∗ LetC be a nonempty closed convex subset of E and let { T i } m i =1

be a finite family of relatively nonexpansive mappings fromC into itself In this section,

we study some properties of the mappingU defined by

Ux =ΠC J −1

m

i =1

ω i

α i Jx +1− α i

JT i x

(4.1)

for allx ∈ C, where { α i } m i =1⊂[0, 1] and{ ω i } m i =1⊂[0, 1] with m i =1ω i =1 Recall that such

a mappingU is called a block mapping defined by T1,T2, ,T m,{ α n,i }and{ ω n(i) }. Lemma 4.1 Let E be a smooth, strictly convex, and reflexive Banach space and let C be a nonempty closed convex subset of E Let { T i } m i =1be a finite family of relatively nonexpansive mappings from C into itself such thatm i =1F(T i ) is nonempty and let U be the block mapping defined by ( 4.1 ), where { α i } ⊂ [0, 1] and { ω i } ⊂ [0, 1] with m i =1ω i = 1 Then

for all u ∈m i =1F(T i ) and x ∈ C.

Proof Let u ∈m i =1F(T i) andx ∈ C Then it holds from Lemmas2.1and3.3that

φ(u,Ux) = φ

u,Π C J −1

m

i =1

ω i

α i Jx +1− α i

JT i x

≤ φ

u,J −1

m

i =1

ω i

α i Jx +1− α i

JT i x

≤

m



i =1

ω i

α i φ(u,x) +1− α i

φu,T i x ≤ φ(u,x).

(4.3)

Theorem 4.2 Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed convex subset of E Let { T i } m i =1be a finite family of relatively nonexpansive mappings from C into itself such thatm i =1F(T i ) is nonempty and let U be the block mapping defined by ( 4.1 ), where { α i } ⊂ [0, 1) and { ω i } ⊂ (0, 1] with m i =1ω i = 1 Then

F(U) =

m



i =1

FT i

Proof Since the inclusion F(U) ⊃m i =1F(T i) is obvious, it suffices to show the inverse inclusionF(U) ⊂m i =1F(T i) Letz ∈ F(U) be given and fix u ∈m i =1F(T i) LetV : E ×

E ∗ → Rbe the function defined by (3.9) Then, as in the proof ofLemma 4.1, we have

φ(u,z) = φ(u,Uz) ≤ φ

u,J −1

m

i =1

ω i

α i Jz +1− α i

JT i z

≤

m



i =1

ω i

α i φ(u,z) +1− α i

φu,T i z ≤ φ(u,z).

(4.5)

Ifk ∈ {1, 2, ,m }, then we have

φ(u,z) =

m



i =1

ω iα i φ(u,z) +1− α i φu,T i z

≤

i k

ω i φ(u,z) + ω k

α k φ(u,z) +1− α k

φu,T k z .

(4.6)

Using (4.6), we have

ω k φ(u,z) =

1−

i k

ω i

φ(u,z) ≤ ω k

α k φ(u,z) +1− α k

φu,T k z . (4.7) Hence we have

ω k

1− α k

φ(u,z) ≤ ω k

1− α k

φu,T k z . (4.8) Sinceω k > 0, α k < 1, and u ∈ F(T k), we have

φ(u,z) ≤ φu,T k z ≤ φ(u,z). (4.9) Thus

φ

u,J −1

m

i =1

ω i

α i Jz +1− α i

JT i z

= φu,T j z = φ(u,z) (4.10)

for allj ∈ {1, 2, ,m }.

Ifm =1, thenω1=1 In this case,

Ux =ΠC J −1 

α1Jx +1− α1

for allx ∈ C If α1=0, thenU = T1, and hence the conclusion obviously holds Ifα1> 0,

then we have from (4.10) that

φu,J −1 α1Jz +1− α1 JT1z = φu,T1z = φ(u,z). (4.12) Then, usingLemma 3.2, we havez = T1z.

We next consider the case wherem ≥2 In this case, it holds that 0< ω i < 1 for all

i ∈ {1, 2, ,m } Let I = { i ∈ {1, 2, ,m }:α i 0} IfI is empty, then we have from (4.10) that

φ

u,J −1

m

i =1

ω i JT i z

= φu,T i z (4.13) for alli ∈ {1, 2, ,m } UsingLemma 3.2, we haveT1z = T2z = ··· = T m z Hence we have

z = Uz =ΠC J −1

m

i =1

ω i JT i z

=ΠC J −1

m

i =1

ω i JT j z

=ΠC T j z = T j z (4.14)

for allj ∈ {1, 2, ,m } Thus z ∈m i =1F(T i)

On the other hand, ifI is nonempty, then we have from (4.10) that

φ

u,J −1



i ∈ I

ω i α i Jz +m

i =1

ω i

1− α i

JT i z

= φu,T i z = φ(u,z) (4.15) for alli ∈ {1, 2, ,m } Then, fromLemma 3.2, we havez = T1z = T2z = ··· = T m z Thus

z ∈m i =1F(T i) This completes the proof 

5 Weak and strong convergence theorems

LetE be a smooth, strictly convex, and reflexive Banach space and let C be a nonempty

closed convex subset ofE Let { T i } m i =1be a finite family of relatively nonexpansive map-pings fromC into itself such thatm i =1F(T i) is nonempty and letU nbe a block mapping fromC into itself defined by

U n x =ΠC J −1

m

i =1

ω n(i)α n,i Jx +1− α n,i

JT i x

(5.1)

for allx ∈ C, where { ω n(i) } ⊂[0, 1] and{ α n,i } ⊂[0, 1] with m i =1ω n(i) =1 for alln ∈ N.

In this section, we study the asymptotic behavior of{ x n }generated byx1= x ∈ C and

x n+1 = U n x n (n =1, 2, ). (5.2)

Lemma 5.1 Let E be a smooth and uniformly convex Banach space and let C be a nonempty closed convex subset of E Let { T i } m i =1be a finite family of relatively nonexpansive mappings from C into itself such that F =m i =1F(T i ) is nonempty and let { α n,i:n,i ∈ N, 1 ≤ i ≤ m }

and { ω n(i) : n,i ∈ N, 1 ≤ i ≤ m } be sequences in [0, 1] such that m i =1ω n(i) = 1 for all n ∈

N Let { U n } be a sequence of block mappings defined by ( 5.1 ) and let { x n } be a sequence generated by ( 5.2 ) Then {Π F x n } converges strongly to the unique element z of F such that

lim

n →∞ φz,x n

=min

 lim

n →∞ φy,x n

Proof If u ∈ F, then we have fromLemma 4.1that

φu,x n+1

≤ φu,x n

(5.4)

for alln ∈ N Thus the limit of φ(u,x n) exists Sinceφ(u,x n)≥(u  −  x n )2for allu ∈ F

andn ∈ N, the sequence { x n }is bounded ByLemma 2.1, we haveφ(u,Π F x n)≤ φ(u,x n)

So, the sequence{Π F x n }is also bounded By the definition ofΠFand (5.4), we have

φΠF x n+1,x n+1

≤ φΠF x n,x n+1

≤ φΠF x n,x n

Thus limn φ(Π F x n,x n) exists We next show that{Π F x n }is a Cauchy sequence Taker > 0

such that{Π F x n } ⊂ B r Then, byLemma 2.5, we have a strictly increasing, continuous and convex functiong : [0,2r] → Rsuch thatg(0) =0 and

gΠF x m −ΠF x n  ≤ φΠF x m,ΠF x n

(5.6)

(a) IfC is a nonempty closed convex subset of a Hilbert space... IfE is a uniformly smooth and strictly convex Banach space and A ⊂ E × E ∗is

a maximal monotone operator such thatA −10... and let { T i } m i =1

be a finite family of relatively nonexpansive mappings fromC into itself In this section,