FOR RELATIVELY NONEXPANSIVE MAPPINGSIN BANACH SPACES SHIN-YA MATSUSHITA AND WATARU TAKAHASHI Received 29 October 2003 We first introduce an iterative sequence for finding fixed points of
Trang 1FOR RELATIVELY NONEXPANSIVE MAPPINGS
IN BANACH SPACES
SHIN-YA MATSUSHITA AND WATARU TAKAHASHI
Received 29 October 2003
We first introduce an iterative sequence for finding fixed points of relatively nonexpansive mappings in Banach spaces, and then prove weak and strong convergence theorems by using the notion of generalized projection We apply these results to the convex feasibility problem and a proximal-type algorithm for monotone operators in Banach spaces
1 Introduction
LetE be a real Banach space and let A be a maximal monotone operator from E to E ∗, whereE ∗is the dual space ofE It is well known that many problems in nonlinear analysis
and optimization can be formulated as follows: find
A well-known method for solving (1.1) in a Hilbert spaceH is the proximal point
algo-rithm:x0∈ H and
x n+1 = J r n x n, n =0, 1, 2, , (1.2) where{ r n } ⊂(0,∞) and J r =(I + rA) −1for allr > 0 This algorithm was first introduced
by Martinet [9] In [16], Rockafellar proved that if lim infn→∞ r n > 0 and A −10= ∅, then
the sequence{ x n }defined by (1.2) converges weakly to an element of solutions of (1.1)
On the other hand, Kamimura and Takahashi [4] considered an algorithm to generate a strong convergent sequence in a Hilbert space Further, Kamimura and Takahashi’s re-sult was extended to more general Banach spaces by Kohsaka and Takahashi [7] They introduced and studied the following iteration sequence:x = x0∈ E and
x n+1 = J −1
α n Jx +1− α n
JJ r n x n , n =0, 1, 2, , (1.3) whereJ is the duality mapping on E and J r =(J + rA) −1J for all r > 0 Kohsaka and
Taka-hashi [7] proved that ifA −10= ∅, lim n→∞ α n =0,∞
n=0α n = ∞, and lim n→∞ r n = ∞, then
the sequence generated by (1.3) converges strongly to an element ofA −10
Copyright©2004 Hindawi Publishing Corporation
Fixed Point Theory and Applications 2004:1 (2004) 37–47
2000 Mathematics Subject Classification: 47H09, 47H05, 47J25
Trang 2On the other hand, Reich [13] studied an iteration sequence of nonexpansive map-pings in a Banach space which was first introduced by Mann [8]:x0∈ C and
x n+1 = α n x n+
1− α n
Sx n, n =0, 1, 2, , (1.4) whereS is a nonexpansive mapping from a closed convex subset C of E into itself and { α n } ⊂[0, 1] He proved that ifF(T) is nonempty and∞
n=0α n(1− α n)= ∞, then the
sequence generated by (1.4) converges weakly to some fixed point ofS.
Motivated by Kohsaka and Takahashi [7], and Reich [13], our purpose in this paper is
to prove weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces which were first introduced by Butnariu et al [3] and further studied by the authors [10] For this purpose, we consider the following iterative sequence:x0∈ C
and
x n+1 =ΠC J −1 α n Jx n+
1− α nJTx n
, n =0, 1, 2, , (1.5) whereT is a relatively nonexpansive mapping from C into itself and Π Cis the generalized projection ontoC Notice that if E is a Hilbert space and S = T, then the sequences (1.4) and (1.5) are equivalent We prove that if F(T) is nonempty and the duality mapping
J is weakly sequentially continuous, then the sequence { x n }converges weakly to a fixed point ofT and if the interior of F(T) is nonempty, then { x n }converges strongly to a fixed point ofT Using these results, we also consider the convex feasibility problem and
a proximal-type algorithm for monotone operators in Banach spaces
2 Preliminaries
LetE be a Banach space with norm · and letE ∗be the dual ofE Then we denote by x,x ∗ the pairing betweenx ∈ E and x ∗ ∈ E ∗ When{ x n }is a sequence inE, we denote
the strong convergence and the weak convergence of{ x n }tox ∈ E by x n → x and x n x,
respectively
A Banach spaceE is said to be strictly convex if ( x + y)/2 < 1 for all x, y ∈ E with
x = y =1 andx = y It is also said to be uniformly convex if lim n→∞ x n − y n =0 for any two sequences{ x n }, { y n }inE such that x n = y n =1 and limn→∞ ( x n+y n)/2 =
1 The following result was proved by Xu [19]
Proposition 2.1 [19] Let r > 0 and let E be a Banach space If E is uniformly convex, then there exists a continuous, strictly increasing, and convex function g : [0, ∞) →[0,∞) with g(0) = 0 such that
λx + (1 − λ)y 2
≤ λ x 2+ (1− λ) y 2− λ(1 − λ)g x − y (2.1)
for all x, y ∈ B r = { z ∈ E : z ≤ r } and λ with 0 ≤ λ ≤ 1.
LetU = { x ∈ E : x =1}be the unit sphere ofE The norm of E is said to be Gˆateaux differentiable if for each x, y ∈ U, the limit
limt→
0
x + ty − x
Trang 3exists In this case,E is called smooth The norm of E is said to be Fr´echet differentiable if
for eachx ∈ U, the limit is attained uniformly for y ∈ U It is also said to be uniformly smooth if the limit is attained uniformly for x, y ∈ U The (normalized) duality mapping
J from E to E ∗is defined by
Jx =x ∗ ∈ E ∗:
x,x ∗
= x 2=x ∗ 2
(2.3) forx ∈ E We say that J is weakly sequentially continuous if for a sequence { x n } ⊂ E, x n
x, then Jx n Jx, where ∗ denotes the weak ∗ ∗convergence We list several well-known properties of the duality mapping:
(1) ifE is smooth, then J is single valued and norm-to-weak ∗continuous;
(2) ifE is Fr´echet differentiable, then J is norm-to-norm continuous;
(3) ifE is uniformly smooth, then J is uniformly norm-to-norm continuous on each
bounded subset ofE.
For more details, see [17] Assume thatE is smooth Then the function V : E × E →Ris defined by
V(x, y) = x 2−2 x,J y + y 2 (2.4) forx, y ∈ E From the definition of V, we have that
x − y 2≤ V(x, y) ≤ x + y 2 (2.5) forx, y ∈ E The function V also has the following property:
V(y,x) = V(z,x) + V(y,z) + 2 z − y,Jx − Jz (2.6) forx, y,z ∈ E The following result was proved by Kamimura and Takahashi [5]
Proposition 2.2 (Kamimura and Takahashi [5]) Let r > 0 and let E be a uniformly convex and smooth Banach space Then
for all y,z ∈ B r = { w ∈ E : w ≤ r } , where g : [0, ∞) →[0,∞) is a continuous, strictly
increasing, and convex function with g(0) = 0.
LetC be a nonempty closed convex subset of E Suppose that E is reflexive, strictly
convex, and smooth Then, for anyx ∈ E, there exists a unique point x0∈ C such that
Vx0,x=min
Following Alber [1], we denote such anx0byΠC x The mapping Π C is called the gen-eralized projection from E onto C It is easy to see that in a Hilbert space, the mapping
ΠCis coincident with the metric projection Concerning the generalized projection, the following are well known
Trang 4Proposition 2.3 (Alber [1]; see also Kamimura and Takahashi [5]) Let C be a nonempty closed convex subset of a smooth Banach space E and x ∈ E Then
x0=ΠC x ⇐⇒x0− y,Jx − Jx0
Proposition 2.4 (Alber [1]; see also Kamimura and Takahashi [5]) Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed convex subset of E, and let x ∈ E Then
Vy,Π C x+VΠC x,x≤ V(y,x) for each y ∈ C. (2.10) LetT be a mapping from C into itself We denote by F(T) the set of fixed points of T.
A point p in C is said to be an asymptotic fixed point of T [12] ifC contains a sequence { x n }which converges weakly top such that the strong lim n→∞(x n − Tx n)=0 The set of asymptotic fixed points ofT is denoted by ˆF(T) We say that the mapping T is called relatively nonexpansive [3,10] ifF(T) = F(T) andˆ
V(p,Tx) ≤ V(p,x) for each x ∈ C, p ∈ F(T). (2.11)
3 Main results
In this section, we discuss the weak and strong convergence of (1.5) To prove our results,
we need the following proposition
Proposition 3.1 Let E be a uniformly convex and smooth Banach space, let C be a non-empty closed convex subset of E, and let T be a relatively nonexpansive mapping from C into itself such that F(T) is nonempty Let { α n } be a sequence of real numbers such that
0≤ α n ≤ 1 Suppose { x n } is the sequence generated by x0∈ C and x n+1 =ΠC J −1(α n Jx n+ (1− α n)JTx n ), n =0, 1, 2, Then {Π F(T) x n } converges strongly to some fixed point of T, whereΠF(T) is the generalized projection from C onto F(T).
Proof We know that F(T) is closed and convex (see [10]) So, we can define the general-ized projectionΠF(T)ontoF(T) Let p ∈ F(T) FromProposition 2.4and the convexity
of · 2, we have
Vp,x n+1
= Vp,Π C J −1
α n Jx n+
1− α n
JTx n
≤ Vp,J −1
α n Jx n+
1− α n
JTx n
= p 2−2
p,α n Jx n+
1− α n
JTx n +α n Jx n+
1− α n
JTx n 2
≤ p 2−2α n
p,Jx n
−2
1− α n
p,JTx n
+α nx n 2
+
1− α nTx n 2
= α n
p 2−2p,Jx n
+x n 2
+
1− α n
p 2−2p,JTx n
+Tx n 2
= α n Vp,x n
+
1− α n
Vp,Tx n
≤ α n Vp,x n
+
1− α n
Vp,x n
= Vp,x n
.
(3.1)
Trang 5Hence, limn→∞ V(p,x n) exists and, in particular,{ V(p,x n)}is bounded Then, by (2.5),
{ x n } is also bounded This implies that{ Tx n } is bounded Letu n =ΠF(T) x n for each
n ∈N∪ {0} Then, we have
Vu n,x n+1
≤ Vu n,x n
It follows from (2.10) that
Vu n+1,x n+1
= VΠF(T) x n+1,x n+1
≤ Vu n,x n+1
− Vu n,ΠF(T) x n+1
Combining this with (3.2), we obtain
Vu n+1,x n+1
≤ Vu n,x n
It follows that{ V(u n,x n)}converges Then, from (3.3),
Vu n,u n+1
≤ Vu n,x n
− Vu n+1,x n+1
By induction, we have
Vu n,u n+m
≤ Vu n,x n
− Vu n+m,x n+m
(3.6) for eachm ∈N UsingProposition 2.2, we have, form, n with n > m,
gu m − u n ≤ Vu m,u n
≤ Vu m,x m
− Vu n,x n
whereg : [0, ∞) →[0,∞) is a continuous, strictly increasing, and convex function with
g(0) =0 Then the properties ofg yield that { u n }is a Cauchy sequence SinceE is
com-plete andF(T) is closed, { u n }converges strongly to some pointu in F(T).
Now, we can prove a weak convergence theorem
Theorem 3.2 Let E be a uniformly convex and uniformly smooth Banach space, let C be
a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself such that F(T) is nonempty, and let { α n } be a sequence of real numbers such that
0≤ α n ≤ 1 and lim inf n→∞ α n(1− α n)> 0 Suppose { x n } is the sequence generated by ( 1.5 ).
If J is weakly sequentially continuous, then { x n } converges weakly to some fixed point of T Proof As in the proof ofProposition 3.1, we know that{ x n }and{ Tx n }are bounded Put
r =supn∈N∪{0} { x n , Tx n } Since E is a uniformly smooth Banach space, E ∗is a uni-formly convex Banach space (see [17,18] for more details) Therefore, byProposition 2.1, there exists a continuous, strictly increasing, and convex functiong : [0, ∞) →[0,∞) with g(0) =0 such that
λx ∗+ (1− λ)y ∗ 2
≤ λx ∗ 2
+ (1− λ)y ∗ 2
− λ(1 − λ)gx ∗ − y ∗ (3.8)
Trang 6for eachx ∗,y ∗ ∈ B r = { z ∗ ∈ E ∗: z ∗ ≤ r }andλ with 0 ≤ λ ≤1 Letp ∈ F(T) We have
Vp,x n+1
= Vp,Π C J −1
α n Jx n+
1− α n
JTx n
≤ Vp,J −1
α n Jx n+
1− α n
JTx n
= p 2−2
p,α n Jx n+
1− α n
JTx n +α n Jx n+
1− α n
JTx n 2
≤ p 2−2α n
p,Jx n
−2
1− α n
p,JTx n +α nx n 2
+
1− α nTx n 2
− α n
1− α n
gJx n − JTx n
= α n Vp,x n
+
1− α n
Vp,Tx n
− α n
1− α n
gJx n − JTx n
≤ Vp,x n
− α n
1− α ngJx n − JTx n,
(3.9)
and hence
α n
1− α n
gJx n − JTx n ≤ Vp,x n
− Vp,x n+1
Since{ V(p,x n)}converges and lim infn→∞ α n(1− α n)> 0, it follows that
lim
n→∞ gJx n − JTx n =0. (3.11) Then the properties ofg yield that
lim
n→∞Jx n − JTx n =0. (3.12) SinceJ −1is uniformly norm-to-norm continuous on bounded sets, we obtain
lim
n→∞x n − Tx n = n→∞limJ −1
Jx n
− J −1
JTx n =0. (3.13) This implies that if there exists a subsequence{ x n i }of{ x n }such thatx n i v for some
v ∈ E, then, by the definition of T, v is a fixed point of T.
Letu n =ΠF(T) x nfor eachn ∈N∪ {0} It follows from (2.9) that
u n − z,Jx n − Ju n
for eachz ∈ F(T) Let { x n i }be a subsequence of{ x n }such that{ x n i }converges weakly
tov Then we have v ∈ F(T) By (3.14), we have
u n i − v,Jx n i − Ju n i
FromProposition 3.1, we know that{ u n }converges strongly to someu ∈ F(T) and J is
weakly sequentially continuous Lettingi → ∞, we have
On the other hand, from the monotonicity ofJ, we have
Trang 7Combining this with (3.16), we have
Using the strict convexity ofE, we obtain u = v Therefore, { x n }converges weakly to
Next, we also consider the strong convergence of (1.5) We can prove the following theorem without the assumption of “weakly sequentially continuous” in the duality map-pingJ.
Theorem 3.3 Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself, and let { α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 and lim inf n→∞ α n(1−
α n)> 0 Suppose { x n } is the sequence generated by ( 1.5 ) If the interior of F(T) is nonempty, then { x n } converges strongly to some fixed point of T.
Proof Since the interior of F(T) is nonempty, there exist p ∈ F(T) and r > 0 such that
whenever h ≤1 By (2.6), we have, for anyu ∈ F(T),
Vu,x n
= Vx n+1,x n
+Vu,x n+1
+ 2
x n+1 − u,Jx n − Jx n+1
This implies
x n+1 − u,Jx n − Jx n+1
+1
2Vx n+1,x n
=1
2
Vu,x n
− Vu,x n+1. (3.21)
We also have
x n+1 − p,Jx n − Jx n+1
=x n+1 −(p + rh) + rh,Jx n − Jx n+1
=x n+1 −(p + rh),Jx n − Jx n+1
+rh,Jx n − Jx n+1. (3.22)
On the other hand, sincep + rh ∈ F(T), as in the proof ofProposition 3.1, we have that
Vp + rh,x n+1
≤ Vp + rh,x n
From (3.21), this inequality is equivalent to
0≤x n+1 −(p + rh),Jx n − Jx n+1
+1
2Vx n+1,x n
Then, by (3.21), we have
rh,Jx n − Jx n+1
≤x n+1 − p,Jx n − Jx n+1
+1
2Vx n+1,x n
=1
2
Vp,x n
− Vp,x n+1
,
(3.25)
Trang 8and hence
h,Jx n − Jx n+1
≤ 1
2r
Vp,x n
− Vp,x n+1
Sinceh with h ≤1 is arbitrary, we have
Jx n − Jx n+1 ≤ 1
2r
Vp,x n
− Vp,x n+1
So, ifn > m, then
Jx m − Jx n = Jx m − Jx m+1+Jx m+1 − ··· − Jx n−1+Jx n−1− Jx n
≤
n− 1
i=
Jx i − Jx i+1 ≤ 1
2r
n− 1
i=
Vp,x i
− Vp,x i+1
= 1
2r
Vp,x m
− Vp,x n
.
(3.28)
We know that{ V(p,x n)}converges So,{ Jx n }is a Cauchy sequence SinceE ∗is complete,
{ Jx n }converges strongly to some point inE ∗ SinceE ∗has a Fr´echet differentiable norm, thenJ −1is continuous onE ∗ Hence,{ x n }converges strongly to some pointu in C As in
the proof ofTheorem 3.2, we also have that x n − Tx n →0 So, we haveu ∈ F(T), where
4 Applications
In this section, using Theorems3.2and3.3, we give some applications We first consider the problem of weak convergence concerning nonexpansive mappings in a Hilbert space Theorem 4.1 (Browder and Petryshyn [2]) Let C be a nonempty closed convex subset of
a Hilbert space H, let T be a nonexpansive mapping from C into itself such that F(T) is nonempty, and let λ be a real number such that 0 < λ < 1 Suppose that { x n } is given by
x0∈ C and
x n+1 = λx n+ (1− λ)Tx n, n =0, 1, 2, (4.1)
Then { x n } converges weakly to u in F(T), where u =limn→∞ P F(T) x n and P F(T) is the metric projection from C onto F(T).
Proof Let α n = λ for each n ∈N∪ {0} It is clear that lim inf n→∞ α n(1− α n)=λ(1 − λ) > 0.
We show that ifT is nonexpansive, then T is relatively nonexpansive It is obvious that F(T) ⊂ F(T) If uˆ ∈ F(T), then there existsˆ { x n } ⊂ C such that x n u and x n − Tx n →0 SinceT is nonexpansive, T is demiclosed So, we have u = Tu This implies F(T) = F(T).ˆ Further, in a Hilbert spaceH, we know that
for everyx, y ∈ H So, Tx − T y ≤ x − y is equivalent toV(Tx,T y) ≤ V(x, y)
There-fore,T is relatively nonexpansive UsingTheorem 3.2, we obtain the desired result
Trang 9We also consider the strong convergence concerning nonexpansive mappings in a Hilbert space For related results, see Moreau [11], and Kirk and Sims [6]
Theorem 4.2 Let C be a nonempty closed convex subset of a Hilbert space H, let T be a nonexpansive mapping from C into itself, and let λ be a real number such that 0 < λ < 1 Suppose that { x n } is given by x0∈ C and
x n+1 = λx n+ (1− λ)Tx n, n =0, 1, 2, (4.3)
If the interior of F(T) is nonempty, then { x n } converges strongly to u in F(T), where u =
limn→∞ P F(T) x n and P F(T) is the metric projection from C onto F(T).
Next, we apply Theorems3.2and3.3to the convex feasibility problem Before giving them, we introduce the following lemma which was proved by Reich [12]
Lemma 4.3 (Reich [12]) Let E be a uniformly convex Banach space with a uniformly Gˆateaux-di fferentiable norm, let { C i } m i=1be a finite family of closed convex subsets of E, and letΠi be the generalized projection from E onto C i for each i =1, 2, ,m Then
Vp,Π mΠm 1···Π2Π1x≤ V(p,x) (4.4)
for each p ∈ F(Πˆ mΠm 1···Π2Π1), x ∈ E, and ˆF(Π mΠm 1···Π2Π1)= ∩ m i=1C i
As direct consequences ofLemma 4.3and Theorems3.2 and3.3, we can prove the following two results
Theorem 4.4 Let E be a uniformly convex and uniformly smooth Banach space, let { C i } m i=1
be a finite family of closed convex subsets of E such that ∩ m i=1C i is nonempty, letΠi be the generalized projection from E onto C i for each i =1, 2, ,m, and let { α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 and lim inf n→∞ α n(1− α n)> 0 Suppose that { x n } is given
by x0∈ E and
x n+1 = J −1
α n Jx n+
1− α n
JΠ mΠm 1···Π2Π1x n
, n =0, 1, 2, (4.5)
If J is weakly sequentially continuous, then { x n } converges weakly to u in ∩ m i=1C i , where
u =limn→∞Π∩ m i =1C i x n andΠ∩ m i =1C i is the generalized projection from E onto ∩ m i=1C i
Proof Put T =ΠmΠm 1···Π2Π1 It is clear thatF(T) ⊂ F(T) andˆ ∩ m i=1C i ⊂ F(T) By
Lemma 4.3, we have that T is a relatively nonexpansive mapping and F(T) = ∩ m i=1C i ApplyingTheorem 3.2,{ x n }converges weakly tou =limn→∞Π∩ m
i =1C i x n
Theorem 4.5 Let E be a uniformly convex and uniformly smooth Banach space, let { C i } m i=1
be a finite family of closed convex subsets of E, let Π i be the generalized projection from E onto
C i for each i =1, 2, ,m, and let { α n } be a sequence of real numbers such that 0 ≤ α n ≤1
and lim inf n→∞ α n(1− α n)> 0 Suppose that { x n } is given by x0∈ E and
x n+1 = J −1
α n Jx n+
1− α n
JΠ mΠm 1···Π2Π1x n
, n =0, 1, 2, (4.6)
If the interior of ∩ m i=1C i is nonempty, then { x n } converges strongly to u in ∩ m i=1C i , where
u =limn→∞Π∩ m C i x n andΠ∩ m C i is the generalized projection from E onto ∩ m i=1C i
Trang 10LetA be a multivalued operator with the domain D(A) = { x ∈ E : Ax = ∅}and the graphG(A) = {( x,x ∗)∈ E × E ∗:x ∗ ∈ Ax } The operator A is said to be monotone if
x − y,x ∗ − y ∗
≥0 for each
x,x ∗ ,
y, y ∗
The operatorA is maximal monotone if A is monotone, and for any monotone operator B
fromE to E ∗withG(A) ⊂ G(B), we have A = B We know that if A is maximal monotone,
thenA −10 is closed and convex The following result is also well known
Theorem 4.6 (Rockafellar [15]) Let E be a reflexive, strictly convex, and smooth Banach space and let A be a monotone operator from E to E ∗ Then A is maximal if and only if R(J + rA) = E ∗ for all r > 0.
LetE be a reflexive, strictly convex, and smooth Banach space and let A be a maximal
monotone operator fromE to E ∗ UsingTheorem 4.6and the strict convexity ofE, we
obtain that for everyr > 0 and x ∈ E, there exists a unique x r ∈ D(A) such that
IfJ r x = x r, then we can define a single-valued mappingJ r:E → D(A) by J r =(J + rA) −1J.
Such a J r is called the resolvent of A We know that J r is relatively nonexpansive (see [10, 12, 14]), andA −10= F(J r) for all r > 0 (see [17,18]) As direct consequences of Theorems3.2and3.3, we also have the following two results
Theorem 4.7 Let E be a uniformly convex and uniformly smooth Banach space, let A be a maximal monotone operator from E to E ∗ such that A −10 is nonempty, let J r be the resolvent
of A, where r > 0, and let { α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 and
lim infn→∞ α n(1− α n)> 0 Suppose the sequence { x n } is given by x0∈ E and
x n+1 = J −1
α n Jx n+
1− α n
JJ r x n , n =0, 1, 2, (4.9)
If J is weakly sequentially continuous, then { x n } converges weakly to u in A −10, where u =
limn→∞ΠA −1 0x n andΠA −1 0is the generalized projection from E onto A −10.
Theorem 4.8 Let E be a uniformly convex and uniformly smooth Banach space, let A be
a maximal monotone operator from E to E ∗ , let J r be the resolvent of A, where r > 0, and let { α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 and lim inf n→∞ α n(1− α n)> 0 Suppose the sequence { x n } is given by x0∈ E and
x n+1 = J −1
α n Jx n+
1− α n
JJ r x n , n =0, 1, 2, (4.10)
If the interior of A −10 is nonempty, then { x n } converges strongly to u in A −10, where u =
limn→∞ΠA −1 0x n andΠA −1 0is the generalized projection from E onto A −10.
References
[1] Y I Alber, Metric and generalized projection operators in Banach spaces: properties and appli-cations, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type
(A G Kartsatos, ed.), Lecture Notes in Pure and Appl Math., vol 178, Marcel Dekker, New York, 1996, pp 15–50.