Báo cáo hóa học: "A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES" pot

FOR A FINITE FAMILY OF NONEXPANSIVEMAPPINGS IN BANACH SPACES FENG GU AND JING LU Received 18 January 2006; Revised 22 August 2006; Accepted 23 August 2006 The purpose of this paper is to

FOR A FINITE FAMILY OF NONEXPANSIVE

MAPPINGS IN BANACH SPACES

FENG GU AND JING LU

Received 18 January 2006; Revised 22 August 2006; Accepted 23 August 2006

The purpose of this paper is to study the weak and strong convergence of implicit iter-ation process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002)

Copyright © 2006 F Gu and J Lu 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

1 Introduction and preliminaries

Throughout this paper we assume thatE is a real Banach space and T : E → E is a

map-ping We denote byF(T) and D(T) the set of fixed points and the domain of T,

respec-tively

Recall thatE is said to satisfy Opial condition [11], if for each sequence{ x n }inE, the

condition that the sequencex n → x weakly implies that

lim inf

n →∞ x

n − x< liminf

n →∞ x

for all y ∈ E with y = x It is well known that (see, e.g., Dozo [9]) inequality (1.1) is equivalent to

lim sup

n →∞

x

n − x< limsup

n →∞

x

Definition 1.1 Let D be a closed subset of E and let T : D → D be a mapping.

(1)T is said to be demiclosed at the origin, if for each sequence { x n }inD, the

condi-tionsx n → x0weakly andTx n →0 strongly implyTx0=0

Hindawi Publishing Corporation

Fixed Point Theory and Applications

Volume 2006, Article ID 82738, Pages 1 11

DOI 10.1155/FPTA/2006/82738

2 A new composite implicit iterative process

(2)T is said to be semicompact, if for any bounded sequence { x n } inD such that

 x n − Tx n  →0 (n → ∞), then there exists a subsequence{ x n i } ⊂ { x n }such that

x n i → x ∗ ∈ D.

(3)T is said to be nonexpansive, if  Tx − T y  ≤  x − y , for alln ≥1 for allx, y ∈ D.

Let E be a Hilbet space, let K be a nonempty closed convex subset of E, and let

{ T1,T2, ,T N }:K → K be N nonexpansive mappings In 2001, Xu and Ori [19] intro-duced the following implicit iteration process{ x n }defined by

x n = α n x n −1+ (1− α n)T n(modN) x n, ∀ n ≥1, (1.3)

wherex0∈ K is an initial point, { α n } n ≥1is a real sequence in (0, 1) and proved the weakly convergence of the sequence{ x n } defined by (1.3) to a common fixed point p ∈ F =

N

i =1F(T i)

Recently concerning the convergence problems of an implicit (or nonimplicit) itera-tive process to a common fixed point for a finite family of asymptotically nonexpansive mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces have been considered by several authors (see, e.g., Bauschke [1], Chang and Cho [3], Chang et al [4], Chidume et al [5], Goebel and Kirk [6], G ´ornicki [7], Halpern [8], Lions [10], Reich [12], Rhoades [13], Schu [14], Shioji and Takahashi [15], Tan and Xu [16,17], Wittmann [18], Xu and Ori [19], and Zhou and Chang [20])

In this paper, we introduce the following new implicit iterative sequence{ x n }with errors:

x1= α1x0+β1T1





α1x0+β1T1x1+γ1v1+γ1u1,

x2= α2x1+β2T2





α2x1+β2T2x2+γ2v2+γ2u2,

x N = α N x N −1+β N T N



α N x N −1+βN T N x N+γN v N+γ N u N,

x N+1 = α N+1 x N+β N+1 T1





α N+1 x N+βN+1 T1x N+1+γ N+1 v N+1+γ N+1 u N+1,

x2N = α2N x2N −1+β2N T N





α2N x2N −1+β2N T N x2N+γ2N v2N+γ2N u2N,

x2N+1 = α2N+1 x2N+β2N+1 T1





α2N+1 x2N+β2N+1 T1x2N+1+γ2N+1 v2N+1+γ2N+1 u2N+1,

(1.4)

for a finite family of nonexpansive mappings{ T i } N

i =1:K → K, where { α n },{ β n },{ γ n },

{ α n },{  β n }, and{ γ n }are six sequences in [0, 1] satisfyingα n+β n+γ n =  α n+βn+γ n =1

for alln ≥1,x0is a given point inK, as well as { u n }and{ v n }are two bounded sequences

inK, which can be written in the following compact form:

x n = α n x n −1+β n T n(modN) y n+γ n u n,

y n =  α n x n −1+βn T n(modN) x n+γn v n, ∀ n ≥1. (1.5)

Especially, if{ T i } N

i =1:K → K are N nonexpansive mappings, { α n },{ β n },{ γ n }are three sequences in [0, 1], andx0is a given point inK, then the sequence { x n }defined by

x n = α n x n −1+β n T n(modN) x n −1+γ n u n, ∀ n ≥1 (1.6)

is called the explicit iterative sequence for a finite family of nonexpansive mappings

{ T i } N

i =1

The purpose of this paper is to study the weak and strong convergence of iterative sequence { x n }defined by (1.5) and (1.6) to a common fixed point for a finite family

of nonexpansive mappings in Banach spaces The results presented in this paper not only generalized and extend the corresponding results of Chang and Cho [3], Xu and Ori [19], and Zhou and Chang [20], but also in the case ofγ n =  γ n =0 orβn =  γ n =0 are also new.

In order to prove the main results of this paper, we need the following lemmas Lemma 1.2 [2] Let E be a uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let T : K → K be a nonexpansive mapping with F(T) = ∅ Then

I − T is semiclosed at zero, that is, for each sequence { x n } in K, if { x n } converges weakly to

q ∈ K and {(I − T)x n } converges strongly to 0, then (I − T)q = 0.

Lemma 1.3 [17] Let{ a n } and { b n } be two nonnegative real sequences satisfying the fol-lowing condition: a n+1 ≤ a n+b n for all n ≥ n0, where n0 is some nonnegative integer If

∞

n =0b n < ∞ , then lim n →∞ a n exists If in addition { a n } has a subsequence which converges strongly to zero, then lim n →∞ a n = 0.

Lemma 1.4 [14] LetE be a uniformly convex Banach space, let b and c be two constants with

0< b < c < 1 Suppose that { t n } is a sequence in [b,c] and { x n } and { y n } are two sequence in

E such that lim n →∞  t n x n+ (1− t n)y n  = d, limsup n →∞  x n  ≤ d, and limsup n →∞  y n  ≤

d hold for some d ≥ 0, then lim n →∞  x n − y n  = 0.

Lemma 1.5 Let E be a real Banach space, let K be a nonempty closed convex subset of E, and let { T1,T2, ,T N }:K → K be N nonexpansive mappings with F =N

i =1F(T i)= ∅ Let { u n } and { v n } be two bounded sequences in K, and let { α n } , { β n } , { γ n } , { α n } , {  β n } , and { γ n } be six sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n =  α n+βn+γn = 1, for all n ≥ 1;

(ii)τ =sup{ β n:n ≥1} < 1;

(iii)∞

n =1γ n < ∞ ,∞

n =1γn < ∞

If { x n } is the implicit iterative sequence defined by (1.5), then for each p ∈ F =N

i =1F(T i)

the limit lim n →∞  x n − p  exists.

4 A new composite implicit iterative process

Proof Since F =N

n =1F(T i)= ∅, for any givenp ∈ F, it follows from (1.5) that

x n − p  = 1− β n − γ n

x n −1+β n T n(modN) y n+γ n u n − p

≤1− β n − γ nx n −1− p+β

nT

n(modN) y n − p+γ

nu

n − p

=1− β n − γ nx n −1− p+β nT n(modN) y n − T n(modN) p+γ nu n − p

≤1− β nx n −1− p+β ny n − p+γ nu n − p.

(1.7) Again it follows from (1.5) that

y n − p  = 1−  β n −  γ n

x n −1+βn T n(modN) x n+γn v n − p

≤1−  β n −  γ nx n −1− p+βnT n(modN) x n − p+γnv n − p

=1−  β n −  γ nx n −1− p+βnT n(modN) x n − T n(modN) p+γnv n − p

≤1−  β nx n −1− p+βnx n − p+γnv n − p.

(1.8) Substituting (1.8) into (1.7), we obtain that

x n − p  ≤ 1− β n βnx n −1− p+β n βnx n − p

+β n γnv n − p+γ nu n − p. (1.9) Simplifying we have



1− β n βnx n − p  ≤ 1− β n βnx n −1− p+σ n, (1.10)

whereσ n = β n γn  v n − p +γ n  u n − p  By condition (iii) and the boundedness of the

sequences{ β n },{ u n − p }, and{ v n − p }, we have∞

n =1σ n < ∞ From condition (ii)

we know that

β n βn ≤ β n ≤ τ < 1 and so 1 − β n βn ≥1− τ > 0; (1.11)

hence, from (1.10) we have

x

n − p  ≤  x n −1− p+ σ n

1− τ =x

n −1− p+b

whereb n = σ n /(1 − τ) with∞

i =1b n < ∞ Takinga n =  x n −1− p in inequality (1.12), we havea n+1 ≤ a n+b n, for alln ≥1, and satisfied all conditions inLemma 1.3 Therefore the limit limn →∞  x n − p exists This

2 Main results

We are now in a position to prove our main results in this paper

Theorem 2.1 Let E be a real Banach space, let K be a nonempty closed convex subset of

E, and let { T1,T2, ,T N }:K → K be N nonexpansive mappings with F =N

i =1F(T i)= ∅

(the set of common fixed points of { T1,T2, ,T N } ) Let { u n } and { v n } be two bounded sequences in K, and let { α n } , { β n } , { γ n } , { α n } , {  β n } , and { γ n } be six sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n =  α n+βn+γn = 1, for all n ≥ 1;

(ii)τ =sup{ β n:n ≥1} < 1;

(iii)∞

n =1γ n < ∞ ,∞

n =1γn < ∞ Then the implicit iterative sequence { x n } defined by (1.5) converges strongly to a common fixed point p ∈ F =N

i =1F(T i ) if and only if

lim inf

Proof The necessity of condition (2.1) is obvious

Next we prove the sufficiency of Theorem 2.1 For any given p ∈ F, it follows from

(1.12) inLemma 1.5that

x n − p  ≤  x n −1− p+b n ∀ n ≥1, (2.2)

whereb n = σ n /(1 − τ) with∞

n =1b n < ∞ Hence, we have

d

x n,F

≤ d

x n −1,F

It follows from (2.3) andLemma 1.3that the limit limn →∞ d(x n,F) exists By condition

(2.1), we have limn →∞ d(x n,F) =0.

Next we prove that the sequence{ x n }is a Cauchy sequence inK In fact, for any

posi-tive integersm and n, from (2.2), it follows that

x

n+m − p  ≤  x n+m −1− p+b

n+m ≤x

n+m −2− p+b

n+m −1+b n+m

≤ ··· ≤x n − p+ n+m

i = n+1

b i ≤x n − p+ ∞

i = n+1

Since limn →∞ d(x n,F) =0 and∞

n =1b n < ∞, for any given > 0, there exists a positive

integern0such thatd(x n,F) <  /8,∞

i = n+1 b i <  /2, for all n ≥ n0 Therefore there exists

p1∈ F such that  x n − p1 <  /4, for all n ≥ n0 Consequently, for anyn ≥ n0and for all

m ≥1, from (2.4), we have

x

n+m − x n  ≤  x n+m − p1 +x

n − p1 

≤2x n − p1+ ∞

i = n+1

b i < 2 · 

4+



This implies that { x n } is a Cauchy sequence in K By the completeness of K, we can

assume that limn →∞ x n = x ∗ ∈ K Moreover, since the set of fixed points of a nonexpansive

mapping is closed, so isF; thus x ∗ ∈ F from lim n →∞ d(x n,F) =0, that is,x ∗is a common fixed point ofT1,T2, ,T N This completes the proof ofTheorem 2.1 

6 A new composite implicit iterative process

Theorem 2.2 Let E be a real Banach space, let K be a nonempty closed convex subset of

E, and let { T1,T2, ,T N }:K → K be N nonexpansive mappings with F =N

i =1F(T i)= ∅

(the set of common fixed points of { T1,T2, ,T N } ) Let { u n } be a bounded sequence in K, and let { α n } , { β n } , and { γ n } be three sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n = 1, for all n ≥ 1;

(ii)τ =sup{ β n:n ≥1} < 1;

(iii)∞

n =1γ n < ∞

Then the explicit iterative sequence { x n } defined by (1.6) converges strongly to a common fixed point p ∈ F =N

i =1F(T i ) if and only if lim inf n →∞ d(x n,F) = 0.

Proof Taking βn =  γ n =0, for all n ≥1 in Theorem 2.1, then the conclusion of

Theorem 2.2can be obtained fromTheorem 2.1immediately This completes the proof

Theorem 2.3 Let E be a real uniformly convex Banach space satisfying Opial condition, let

K be a nonempty closed convex subset of E, and let { T1,T2, ,T N }:K → K be N nonex-pansive mappings with F =N

i =1F(T i)= ∅ Let { u n } and { v n } be two bounded sequences

in K, and let { α n } , { β n } , { γ n } , { α n } , {  β n } , and { γ n } be six sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n =  α n+βn+γn = 1, for all n ≥ 1;

(ii) 0< τ1=inf{ β n:n ≥1} ≤sup{ β n:n ≥1} = τ2< 1;

(iii)βn →0 (n → ∞ );

(iv)∞

n =1γ n < ∞ ,∞

n =1γn < ∞ Then the implicit iterative sequence { x n } defined by (1.5) converges weakly to a common fixed point of { T1,T2, ,T N }

Proof First, we prove that

lim

n →∞x n − T n(modN)+ j x n  =0, ∀ j =1, 2, ,N. (2.6) Letp ∈ F Put d =limn →∞  x n − p  It follows from (1.5) that

x n − p  = 1− β n

x n −1− p + γ n

u n − x n −1

+β n

T n(modN) y n − p + γ n

u n − x n −1 d, n −→ ∞ (2.7)

Again since limn →∞  x n − p exists, so{ x n } is a bounded sequence inK By virtue of

condition (iv) and the boundedness of sequences{ x n }and{ u n }we have

lim sup

n →∞

x

n −1− p + γ n

u n − x n −1 

≤lim sup

n →∞

x

n −1− p+ lim sup

n →∞ γ nu

n − x n −1 = d, p ∈ F. (2.8)

It follows from (1.8) and condition (iii) that

lim sup

n →∞

T n(modN) y n − p + γ n

u n − x n −1 

≤lim sup

n →∞

y n − p+ lim sup

n →∞ γ nu n − x n −1

=lim sup

n →∞

y n − p

≤lim sup

n →∞



1−  β nx n −1− p+βnx n − p+γ nv n − p

≤lim sup

n →∞



1−  β nx n −1− p+ lim sup

n →∞



β nx n − p+ lim sup

n →∞ γnv n − p

= d, p ∈ F.

(2.9)

Therefore, from condition (ii), (2.7)–(2.9), andLemma 1.4we know that

lim

n →∞T n(modN) y n − x n −1 =0. (2.10) From (1.5) and (2.10) we have

x n − x n −1 =  β n

T n(modN) y n − x n −1 +γ n

u n − x n −1 

≤ β nT n(modN) y n − x n −1+γ nu n − x n −1 −→0, n −→ ∞, (2.11) which implies that

lim

n −→∞x n − x n −1 =0 (2.12) and so

lim

n →∞x n − x n+ j  =0 ∀ j =1, 2, ,N. (2.13)

On the other hand, we have

x n − T n(modN) x n  ≤  x n − x n −1 +x n −1− T n(modN) y n

+T n(modN) y n − T n(modN) x n. (2.14) Now, we consider the third term on the right-hand side of (2.14) From (1.5) we have

T n(modN) y n − T n(modN) x n

≤y n − x n  =  α n



x n −1− x n



+βnT n(modN) x n − x n+γ nv n − x n

≤  α nx

n −1− x n+β

nT

n(modN) x n − x n+γ

nv

Substituting (2.15) into (2.14), we obtain that

x n − T n(modN) x n  ≤ 1 +αnx n − x n −1+x n −1− T n(modN) y n

+βnT

n(modN) x n − x n+γ

nv

n − x n. (2.16)

8 A new composite implicit iterative process

Hence, by virtue of conditions (iii), (iv), (2.10), (2.12) and the boundedness of sequences

{ T n(modN) x n − x n }and{ v n − x n }we have

lim

n →∞x

n − T n(modN) x n  =0. (2.17) Therefore, from (2.13) and (2.17), for anyj =1, 2, ,N, we have

x n − T n(modN)+ j x n  ≤  x n − x n+ j+x n+ j − T n(modN)+ j x n+ j

+T n(modN)+ j x n+ j − T n(modN)+ j x n

≤2x n − x n+ j+x n+ j − T n(modN)+ j x n+ j  −→0, n −→ ∞

(2.18) That is, (2.6) holds

SinceE is uniformly convex, every bounded subset of E is weakly compact Again since

{ x n }is a bounded sequence inK, there exists a subsequence { x n k } ⊂ { x n }such that{ x n k }

converges weakly toq ∈ K.

Without loss of generality, we can assume thatn k = i(modN), where i is some positive

integer in{1, 2, ,N } Otherwise, we can take a subsequence{ x n k j } ⊂ { x n k } such that

n k j = i(modN) For any l ∈ {1, 2, ,N }, there exists an integerj ∈ {1, 2, ,N }such that

n k+j = l(modN) Hence, from (2.18) we have

lim

k →∞

x

n k − T l x n k  =0, l =1, 2, ,N. (2.19)

ByLemma 1.2, we know thatq ∈ F(T l) By the arbitrariness ofl ∈ {1, 2, ,N }, we know thatq ∈ F =N

j =1F(T j)

Finally, we prove that{ x n }converges weakly toq In fact, suppose the contrary, then

there exists some subsequence{ x n j } ⊂ { x n }such that{ x n j }converges weakly toq1∈ K

andq1= q Then by the same method as given above, we can also prove that q1∈ F =

N

j =1F(T j)

Taking p = q and p = q1 and by using the same method given in the proof of Lemma 1.5, we can prove that the following two limits exist and limn →∞  x n − q  = d1

and limn →∞  x n − q1 = d2, whered1andd2are two nonnegative numbers By virtue of the Opial condition ofE, we have

d1=lim sup

n k −→∞

x n

k − q< limsup

n k →∞

x n

k − q1 = d2

=lim sup

n j →∞

x n

j − q1 < limsup

n j →∞

x n

j − q  = d1. (2.20)

This is a contradiction Henceq1= q This implies that { x n }converges weakly toq This

Theorem 2.4 Let E be a real uniformly convex Banach space satisfying Opial condition, let

K be a nonempty closed convex subset of E, and let { T1,T2, ,T N }:K → K be N nonex-pansive mappings with F =N

i =1F(T i)= ∅ Let { u n } be a bounded sequence in K, and let

{ α n } , { β n } , and { γ n } be three sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n = 1, ∀ n ≥ 1;

(ii) 0< τ1=inf{ β n:n ≥1} ≤sup{ β n:n ≥1} = τ2< 1;

(iii)∞

n =1γ n < ∞

Then the explicit iterative sequence { x n } defined by (1.6) converges weakly to a common fixed point of { T1,T2, ,T N }

Proof Taking βn =  γ n =0, for all n ≥1 in Theorem 2.3, then the conclusion of

Theorem 2.4can be obtained fromTheorem 2.3immediately This completes the proof

Theorem 2.5 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let { T1,T2, ,T N }:K → K be N nonexpansive mappings with F =

N

i =1F(T i)= ∅ and there exists an T j, 1≤ j ≤ N, which is semicompact (without loss of generality, assume that T1is semicompact) Let { u n } and { v n } be two bounded sequences in

K, and let { α n } , { β n } , { γ n } , { α n } , {  β n } , and { γ n } be six sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n =  α n+βn+γn = 1, for all n ≥ 1;

(ii) 0< τ1=inf{ β n:n ≥1} ≤sup{ β n:n ≥1} = τ2< 1;

(iii)βn →0 (n → ∞ );

(iv)∞

n =1γ n < ∞ ,∞

n =1γn < ∞ Then the implicit iterative sequence { x n } defined by (1.5) converges strongly to a common fixed point of { T1,T2, ,T N } in K.

Proof For any given p ∈ F =N

i =1F(T i), by the same method as given in provingLemma 1.5and (2.19), we can prove that

lim

whered ≥0 is some nonnegative number, and

lim

k →∞

x

n k − T l x n k  =0, l =1, 2, ,N. (2.22) Especially, we have

lim

k →∞

x

By the assumption,T1is semicompact; therefore it follows from (2.23) that there exists a subsequence{ x n ki } ⊂ { x n k }such thatx n ki → x ∗ ∈ K Hence from (2.22) we have that

x ∗ − T l x ∗  =lim

k i →∞

x n

ki − T l x n ki  =0 ∀ l =1, 2, ,N, (2.24) which implies thatx ∗ ∈ F =N

i =1F(T i) Take p = x ∗ in (2.21), similarly we can prove that limn →∞  x n − x ∗  = d1, whered1≥0 is some nonnegative number Fromx n ki → x ∗

we know thatd1=0, that is,x n → x ∗ This completes the proof ofTheorem 2.5 

10 A new composite implicit iterative process

Theorem 2.6 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let { T1,T2, ,T N }:K → K be N nonexpansive mappings with

F =N

i =1F(T i)= ∅ and there exists an T j, 1≤ j ≤ N, which is semicompact (without loss

of generality, assume that T1is semicompact) Let { u n } be a bounded sequence in K, and let

{ α n } , { β n } , and { γ n } be three sequences in [0, 1] satisfying the following conditions:

(i)α n+β n+γ n = 1, for all n ≥ 1;

(ii) 0< τ1=inf{ β n:n ≥1} ≤sup{ β n:n ≥1} = τ2< 1;

(iii)∞

n =1γ n < ∞

Then the explicit iterative sequence { x n } defined by (1.6) converges strongly to a common fixed point of { T1,T2, ,T N } in K.

Proof Taking βn =  γ n =0, for all n ≥1 in Theorem 2.5, then the conclusion of

Theorem 2.6can be obtained fromTheorem 2.5immediately This completes the proof

Remark 2.7 Theorems2.3–2.6improve and extend the corresponding results in Chang and Cho [3, Theorem 3.1] and Zhou and Chang [20, Theorem 3], and the implicit it-erative process{ x n }defined by (1.3) is replaced by the more general implicit or explicit iterative process{ x n }defined by (1.5) or (1.6)

Remark 2.8 Theorems2.3–2.6generalize and improve the main results of Xu and Ori [19] in the following aspects

(1) The class of Hilbert spaces is extended to that of Banach spaces satisfying Opial’s

or semicompactness condition

(2) The implicit iterative process{ x n }defined by (1.3) is replaced by the more general implicit or explicit iterative process{ x n }defined by (1.5) or (1.6)

Remark 2.9 The iterative algorithm used in this paper is different from those in [1,8,10,

14,18]

Acknowledgments

The present studies were supported by the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), and the Scientific Research Foundation from Zhejiang Province Education Committee (20051897)

References

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Hilbert space, Journal of Mathematical Analysis and Applications 202 (1996),... 150–159.

[2] F E Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces,

Nonlin-ear Functional Analysis (Proc... iterative processes for asymptotically nonexpansive map-pings, Nonlinear Analysis and Applications: to V Lakshmikantham on His 80th Birthday Vol 1,

2, Kluwer Academic, Dordrecht,