NONLINEAR ERGODIC THEOREMS FOR
A SEMITOPOLOGICAL SEMIGROUP OF
NON-LIPSCHITZIAN MAPPINGS
WITHOUT CONVEXITY
G. LI AND J. K. KIM
Received 2 February 1999
Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H ,
and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)
inf t∈G Tt x −z}
= {z ∈ H : inf s∈G supt∈G Tts x −z =
for each
x ∈ C and L() = x∈C L(x). In this paper, we prove that s∈G conv{Tts x :
t ∈ G} L() is nonempty for each x ∈ C if and only if there exists a unique
nonexpansive retraction P of C into L() such that P Ts = P for all s ∈ G and
P (x) ∈ conv{Ts x : s ∈ G} for every x ∈ C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without
convexity.
1. Introduction and preliminaries
Let H be a Hilbert space with norm · and inner product (·, ·). Let G be a semitopological semigroup, that is, a semigroup with a Hausdorff topology such that for
each s ∈ G the mappings s → s · t and s → t · s of G into itself are continuous. Let
C be a nonempty subset of H and let = {Tt : t ∈ G} be a semigroup on C, that is,
Tst (x) = Ts Tt (x) for all s, t ∈ G and x ∈ C. Recall that a semigroup is said to be
(a) nonexpansive if Tt x − Tt y ≤ x − y for x, y ∈ C and t ∈ G.
(b) asymptotically nonexpansive [6] if there exists a function k : G → [0, ∞) with
inf s∈G supt∈G kts ≤ 1 such that Tt x − Tt y ≤ kt x − y for x, y ∈ C and t ∈ G.
(c) of asymptotically nonexpansive type [6] if for each x in C, there is a function
r(·, x) : G → [0, ∞) with inf s∈G supt∈G r(ts, x) = 0 such that Tt x − Tt y ≤ x −
y + r(t, x) for all y ∈ C and t ∈ G.
It is easily seen that (a)⇒(b)⇒(c) and that both the inclusions are proper
(cf. [6, page 112]).
Baillon [1] proved the first nonlinear mean ergodic theorem for nonexpansive mappings in a Hilbert space: let C be a nonempty closed convex subset of a Hilbert space
H and T a nonexpansive mapping of C into itself. If the set F (T ) of fixed points of T
Copyright © 1999 Hindawi Publishing Corporation
Abstract and Applied Analysis 4:1 (1999) 49–59
1991 Mathematics Subject Classification: 47H09, 47H10, 47H20
URL: http://aaa.hindawi.com/volume-4/S1085337599000056.html
50
Nonlinear ergodic theorems
is nonempty, then for each x ∈ C, the Cesáro means
n−1
Sn (x) =
1 k
T x
n
(1.1)
k=0
converge weakly as n → ∞ to a point of F (T ). In this case, putting y = P x for each
x ∈ C, P is a nonexpansive retraction of C onto F (T ) such that P T = T P = P and
P x ∈ conv{T n x : n = 0, 1, 2, . . .} for each x ∈ C, where convA is the closure of the
convex hull of A. The analogous results are given for nonexpansive semigroups on C by
Baillon [2] and Breźis-Browder [3]. In [10], Mizoguchi-Takahashi proved a nonlinear
ergodic retraction theorem for Lipschitzian semigroups by using the notion of submean.
Recently, Li and Ma [8, 9] proved the nonlinear ergodic retraction theorems for nonLipschitzian semigroups in a Banach space without using the notion of submean. Also,
in 1992, Takahashi
[13] proved the ergodic theorem for nonexpansive semigroups on
condition that s∈G conv{Tst x : t ∈ G} ⊂ C for some x ∈ C.
In this paper, without using the concept of submean, we prove nonlinear ergodic
theorem for semitopological semigroup of non-Lipschitzian mappings without convexity in a Hilbert space. We first prove that if C is a nonempty subset of a Hilbert space
H, G a semitopological semigroup, and = {Tt : t ∈ G} a representation
of G as
asymptotically
nonexpansive
type
mappings
of
C
into
itself,
then
conv{T
ts x :
s∈G
t ∈ G} L() is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L() such that P Ts = P for all s ∈ G and P x is
in the closed convex hull of {Ts x : s ∈ G}, where L(x) = {z : inf s∈G supt∈G Tts x −
z = inf t∈G Tt x − z} and L() = x∈C L(x). By using this result, we also prove
the ergodic convergence theorem for semitopological semigroup of non-Lipschitzian
mapping without convexity. Our results are generalizations and improvements of the
previously known results of Brézis-Browder [3], Hirano-Takahashi [4], MizoguchiTakahashi [10], Takahashi-Zhang [14], and Takahashi [11, 12, 13] in many directions.
Further, it is safe to say that in the results [1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14], many key
conditions are not necessary.
2. Ergodic convergence theorems
Throughout this paper, we assume that C is a nonempty subset of a real Hilbert space
H, G a semitopological semigroup, and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup on C. For each x ∈ C, define L(x) and L() by
L(x) = z : inf sup Tts x − z = inf Tt x − z ,
L() =
L(x),
(2.1)
s∈G t∈G
t∈G
x∈C
respectively. We denote F () by the set {x ∈ C : Ts (x) = x for all s ∈ G} of common
fixed point of . We begin with the following lemma.
Lemma 2.1. Let C be a nonempty subset of a Hilbert space H and = {Tt : t ∈ G} an
asymptotically nonexpansive type semigroup on C. Then F () ⊂ L().
G. Li and J. K. Kim
51
Proof. Let x ∈ C and f ∈ F (). Since is asymptotically nonexpansive type, for an
arbitrary ε > 0, there exists s0 ∈ G such that for all t ∈ G
r ts0 , f < ε.
(2.2)
Hence, for each a ∈ G,
inf sup Tts x − f ≤ sup Tts0 a x − f ≤ sup Ta x − f + r ts0 , f
s∈G t∈G
t∈G
t∈G
≤ Ta x − f + ε.
(2.3)
Since ε > 0 is arbitrary, we have inf s∈G supt∈G Tts x − f ≤ inf t∈G Tt x − f . Therefore, f ∈ L(x). This completes the proof.
Remark 2.2. It is not easy to prove that F () is nonempty when C is not a convex
subset. However, we can show that L() is nonempty under some conditions and it is
important for the ergodic convergence theorem.
The following proposition plays a crucial role in the proof of our main theorems in
this paper.
Proposition 2.3. Let G be a semitopological semigroup, C a nonempty subset of a
Hilbert space H , and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup
on C. Then, for every x ∈ C, the set
L(x),
(2.4)
conv Tts x : t ∈ G
s∈G
consists of at most one point.
Proof. Let u, v ∈
sume that
s∈G conv{Tts x
: t ∈ G}
L(x), without loss of generality, we as-
2
2
inf Tt x − u ≤ inf Tt x − v .
t∈G
(2.5)
t∈G
Now, for each t, s ∈ G, since
2 2
u − v2 + 2 Tts x − u, u − v = Tts x − v − Tts x − u ,
(2.6)
we have
2
2
u − v2 + 2 inf Tts x − u, u − v ≥ inf Tts x − v − sup Tts x − u
t∈G
t∈G
t∈G
2
2
≥ inf Tt x − v − sup Tts x − u .
t∈G
t∈G
From u ∈ L(x), we have
2
2
u − v2 + 2 sup inf Tts x − u, u − v ≥ inf Tt x − v − inf sup Tts x − u
s∈G t∈G
(2.7)
t∈G
s∈G
t∈G
t∈G
t∈G
(2.8)
2
2
= inf Tt x − v − inf Tt x − u ≥ 0.
52
Nonlinear ergodic theorems
Therefore, for ε > 0 there is an s1 ∈ G such that
u − v2 + 2 Tts1 x − u, u − v > −ε
∀t ∈ G.
(2.9)
From v ∈ conv{Tts1 x : t ∈ G}, we have
u − v2 + 2(v − u, u − v) ≥ −ε.
(2.10)
This inequality implies that u−v2 ≤ ε. Since ε > 0 is arbitrary, we have u = v. This
completes the proof.
Remark 2.4. In the Takahashi-Zhang’s result [14], it is assumed that C is a closed convex
subset, G a reversible semigroup, and an asymptotically nonexpansive semigroup.
Proposition 2.3 shows those key conditions are not necessary.
Let m(G) be the Banach space of all bounded real-valued functions on a semitopological semigroup G with the supremum norm and let X be a subspace of m(G)
containing constants. Then, an element µ of X ∗ (the dual space of X) is called a mean
on X if µ = µ(1) = 1. Let µ be a mean on X and f ∈ X. Then, according to time and
circumstances, we use µt (f (t)) instead of µ(f ). For each s ∈ G and f ∈ m(G), we
define elements ls f and rs f in m(G) given by (ls f )(t) = f (st) and (rs f )(t) = f (ts)
for all t ∈ G, respectively.
Throughout the rest of this section, let X be a subspace of m(G) containing constants
invariant under ls and rs for each s ∈ G. Furthermore, suppose that for each x ∈ C and
y ∈ H, a function f (t) = Tt x −y2 is in X. For µ ∈ X ∗ , we define the value µt (Tt x, y)
of µ at this function. By Riesz theorem, there exists a unique element µ x in X such
that
µt Tt x, y = µ x, y ∀y ∈ H.
(2.11)
Lemma 2.5. Suppose that X has an invariant mean µ. Then we have
conv Tts x : t ∈ G
L(x) = µ x
for every x ∈ C.
s∈G
Further, if Tt is continuous for each t ∈ G and
x ∈ C, then µ x ∈ F ().
s∈G conv{Tst x
(2.12)
: t ∈ G} ⊂ C for some
Proof. Since µ is an invariant mean, it is easy to show that µ x ∈ s∈G conv{Tts x :
t ∈ G} for each x ∈ C. By Proposition 2.3, it is enough to prove that µ x ∈ L(x) for
each x ∈ C. To this end, let ε > 0, since is an asymptotically nonexpansive type
semigroup, for each t ∈ G there is an ht ∈ G such that for each h ∈ G,
(2.13)
r hht , Tt x < ε.
Put M = supt,s∈G Tt x − Ts x, then we have
Thh t x − µ x 2 − Tt x − µ x 2 = µs Thh t x − Ts x 2 − Tt x − Ts x 2
t
t
2 2 (2.14)
= µs Thht t x − Thht s x − Tt x − Ts x ≤ 2Mε
for each h ∈ G.
G. Li and J. K. Kim
53
Hence, we have
2 2
inf sup Ths x − µ x ≤ Tt x − µ x + 2Mε
s∈G h∈G
∀t ∈ G.
(2.15)
Since ε > 0 is arbitrary, we have µ x ∈ L(x). Finally, suppose that s∈G conv{Tst x :
t ∈ G} ⊂ Cand each Tt is continuous from C into itself. Then, we can easily prove
that µ x ∈ s∈G conv{Tst x : t ∈ G} and hence we have µ x ∈ C. For each h ∈ G and
ε ∈ (0, 1), there exists 0 < δ < ε such that Th y − Th µ x < ε whenever y ∈ C and
y − µ x ≤ δ. Since is an asymptotically nonexpansive type semigroup, there is
s0 ∈ G such that
1
δ 2 ∀t ∈ G,
(2.16)
r ts0 , µ x < 2 M1 + 1
where M1 = supt∈G Tt x − µ x. Then for each t, s ∈ G, we have
Tss µ x − µ x 2 + 2 Tt x − µ x, µ x − Tss µ x
0
0
2 2
= Tt x − Tss0 µ x − Tt x − µ x 2 2 2 2
= Tss0 t x − Tss0 µ x − Tt x − µ x − Tss0 t x − Tss0 µ x + Tt x − Tss0 µ x 2 2
≤ δ 2 − Tss0 t x − Tss0 µ x + Tt x − Tss0 µ x .
(2.17)
It follows that
Tss µ x − µ x ≤ δ
0
∀s ∈ G.
(2.18)
This implies that
Th µ x − µ x ≤ Th µ x − Th Tss µ x + Thss µ x − µ x < 2ε.
0
0
(2.19)
Since ε > 0 is arbitrary, we have Th µ x = µ x. This completes the proof.
Now, we prove a nonlinear ergodic theorem for asymptotically nonexpansive type
semigroups without convexity. Before doing this, we give a definition concerning
means. Let {µα : α ∈ A} be a net of means on X, where A is a directed set. Then
{µα : α ∈ A} is said to be asymptotically invariant if for each f ∈ X and s ∈ G,
µα (f ) − µα ls f −→ 0,
µα (f ) − µα rs f −→ 0.
(2.20)
Theorem 2.6. Let C be a nonempty subset of a Hilbert space H , X an invariant
subspace of m(G) containing constants, and = {Tt : t ∈ G} an asymptotically nonexpansive type semigroup on C. If for each x ∈ C and y ∈ H , the function f on G
defined by f (t) = Tt x − y2 belong to X, then for an asymptotically invariant net
{µα : α ∈ A} on X, the net {µα x}α∈A converges weakly to an element x0 ∈ L(x).
54
Nonlinear ergodic theorems
Further, if Tt is continuous for each t ∈ G and
x0 ∈ F ().
s∈G conv{Tst x
: t ∈ G} ⊂ C, then
Proof. Let W be the set of all weak limit points of subnet of the net {µα x : α ∈ A}.
By Proposition 2.3, it is enough to prove that
W⊂
conv Tts x : t ∈ G
L(x).
(2.21)
s∈G
To show this, let z ∈ W and let {uαβ x} be a subnet of {µα x} such that {µαβ x}
converges weakly to z. Now, without loss of generality, we can suppose that {µαβ x}
converges weakly* to µ ∈ X ∗ . It is easily
seen that µ is an invariant
mean on X and then
Lemma 2.5 implies that z = µ x ∈ s∈G conv{Tts x : t ∈ G} L(x). This completes
the proof.
Let C(G) be the Banach space of all bounded continuous real-valued functions on
G and let RUC(G) be the space of all bounded right uniformly continuous functions on
G, that is, all f ∈ C(G) such that the mapping s → rs f is continuous. Then RUC(G)
is a closed subalgebra of C(G) containing constants and invariant under ls and rs .
As a direct consequence of Theorem 2.6, we obtain the following corollary.
Corollafry 2.7 (see [13]). Let C be a nonempty subset of a Hilbert space H and
let G be a semitopological semigroup such that RUC(G) has an invariant mean. Let
= {T
t : t ∈ G} be a nonexpansive semigroup on C such that {Tt x : t ∈ G} is bounded
and s∈G conv{Tst x : t ∈ G} ⊂ C for some x ∈ C. Then, F () = ∅. Further, for
an asymptotically invariant net {µα }α∈A of means on RUC(G), the net {µα }α∈A ,
converges weakly to an element x0 ∈ F ().
Remark 2.8. For the proof of Corollary 2.7, Takahashi [13] used the condition s∈G
conv{Tst x : t ∈ G} ⊂ C. But, from Theorem 2.6, we can prove the result without this
condition except proving the fact that the weak limit of {µα x} is in F ().
3. Nonexpansive retractions
In this section, we prove an ergodic retraction theorem for a semitopological semigroup
of asymptotically nonexpansive type mappings without convexity.
Theorem 3.1. Let C be a nonempty subset of a Hilbert space H and let = {Tt : t ∈ G}
be a semitopological semigroup of asymptotically nonexpansive type mappings on C
such that
statements are equivalent:
L() = ∅. Then the following
(a) s∈G conv{Tts x : t ∈ G} L() = ∅ for each x ∈ C.
(b) There is a unique nonexpansive retraction P of C into L() such that P Tt = P
for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C.
Proof. (b)⇒(a). Let x ∈ C, then P x ∈ L(). Also P x ∈ s∈G conv{Tts x : t ∈ G}. In
fact, for each s ∈ G, P x = P Ts x ∈ conv{Tt Ts x : t ∈ G} = conv{Tts x : t ∈ G}.
G. Li and J. K. Kim
55
(a)⇒(b). Let x ∈ C. Then by Proposition 2.3, s∈G conv{Tts x : t ∈ G} L()
contains exactly one point P x. For each a ∈ G, we have
{P Ta x} =
conv Ttsa x : t ∈ G
L()
s∈G
⊇
conv Tts x : t ∈ G
L() = {P x}
(3.1)
s∈G
and hence we have P Ta = P for every a ∈ G.
Finally, we have to show that P is nonexpansive. Let x, y ∈ C and 0 < λ < 1. Then
for any ε > 0, there exists s1 ∈ G such that
(3.2)
sup Tts1 x − P y ≤ inf Tt x − P y + ε,
t∈G
t∈G
from P y ∈ L(). Hence, we have
λTtss x + (1 − λ)P x − P y 2
1
2
= λ Ttss1 x − P y + (1 − λ)(P x − P y)
2
2
= λTtss1 x − P y + (1 − λ)P x − P y2 − λ(1 − λ)Ttss1 x − P x 2
2
≤ λ Tab x − P y + ε + (1 − λ)P x − P y2 − λ(1 − λ) inf Tt x − P x ,
t∈G
(3.3)
for each t, s, a, b ∈ G. Since ε > 0 is arbitrary, this implies
2
inf sup λTts x + (1 − λ)P x − P y s∈G t∈G
2
2
≤ λTab x − P y + (1 − λ)P x − P y2 − λ(1 − λ) inf Tt x − P x t∈G
2
2
2
= λTab x+(1−λ)P x−P y + λ(1 − λ) Tab x − P x − λ(1 − λ) inf Tt x − P x .
t∈G
(3.4)
Then it is easily seen that
2
2
inf sup λTts x + (1 − λ)P x − P y − λ(1 − λ) inf sup Tab x − P x s∈G t∈G
b∈G a∈G
2
2
≤ sup inf λTab x + (1 − λ)P x − P y − λ(1 − λ) inf Tt x − P x .
t∈G
b∈G a∈G
Since P x ∈ L(), we have
2
2
inf sup λTts x + (1 − λ)P x − P y ≤ sup inf λTts x + (1 − λ)P x − P y .
s∈G t∈G
Let
(3.5)
s∈G t∈G
2
h(λ) = inf sup λTts x + (1 − λ)P x − P y .
s∈G t∈G
(3.6)
(3.7)
56
Nonlinear ergodic theorems
Then for any ε > 0, there exists s2 ∈ G such that for all t ∈ G,
λTts x + (1 − λ)P x − P y 2 ≤ h(λ) + ε
2
and hence
1/2
λTts2 x + (1 − λ)P x − P y, P x − P y ≤ h(λ) + ε
P x − P y
(3.8)
∀t ∈ G.
From P x ∈ conv{Tts2 x : t ∈ G}, we have
1/2
λP x + (1 − λ)P x − P y, P x − P y ≤ h(λ) + ε
P x − P y.
(3.9)
(3.10)
Since ε > 0 is arbitrary, this yields that
That is,
P x − P y2 ≤ h(λ).
(3.11)
2
P x − P y2 ≤ inf sup λTts x + (1 − λ)P x − P y .
(3.12)
s∈G t∈G
Now, one can choose an s3 ∈ G such that Tts3 x − P x ≤ M for all t ∈ G, where
M = 1 + inf t∈G Tt x − P x. Then, we have
λTtss x + (1 − λ)P x − P y 2
3
2
= λ Ttss3 x − P x + (P x − P y)
(3.13)
2
= λ2 Ttss3 x − P x + P x − P y2 + 2λ Ttss3 x − P x, P x − P y
≤ M 2 λ2 + P x − P y2 + 2λ Ttss3 x − P x, P x − P y .
It then follows from (3.6) and (3.12) that
2λ sup inf Tts x − P x, P x − P y
s∈G t∈G
≥ 2λ sup inf Ttss3 x − P x, P x − P y
s∈G t∈G
2
≥ sup inf λTtss3 x + (1 − λ)P x − P y − P x − P y2 − M 2 λ2
s∈G t∈G
2
= sup inf λTts Ts3 x + (1 − λ)P Ts3 x − P y − P x − P y2 − M 2 λ2
(3.14)
s∈G t∈G
2
≥ P Ts3 x − P y − P x − P y2 − M 2 λ2
= −M 2 λ2 .
Hence, we have
1
sup inf Tts x − P x, P x − P y ≥ − M 2 λ.
2
s∈G t∈G
(3.15)
Letting λ → 0, then we have
sup inf Tts x − P x, P x − P y ≥ 0.
s∈G t∈G
(3.16)
G. Li and J. K. Kim
Let ε > 0, then there is s4 ∈ G such that
r ts4 , x < ε
∀t ∈ G.
57
(3.17)
For such an s4 ∈ G, from (3.16), we have
sup inf Tts Ts4 x − P Ts4 x, P Ts4 x − P y ≥ 0
(3.18)
and hence there is s5 ∈ G such that
inf Tts5 Ts4 x − P Ts4 x, P Ts4 x − P y > −ε.
(3.19)
Then, from P Ts4 x = P x, we have
inf Tts5 s4 x − P x, P x − P y > −ε.
(3.20)
Similarly, from (3.16), we also have
sup inf Tts Ts5 s4 y − P Ts5 s4 y, P Ts5 s4 y − P x ≥ 0,
(3.21)
and there exists s6 ∈ G such that
inf Tts6 s5 s4 y − P Ts5 s4 y, P Ts5 s4 y − P x ≥ −ε,
(3.22)
s∈G t∈G
t∈G
t∈G
s∈G t∈G
t∈G
that is,
inf P y − Tts6 s5 s4 y, P x − P y ≥ −ε.
(3.23)
On the other hand, from (3.20)
inf Tts6 s5 s4 x − P x, P x − P y > −ε.
(3.24)
t∈G
t∈G
Combining (3.23) and (3.24), we have
−2ε < Tts6 s5 s4 x − Tts6 s5 s4 y, P x − P y − P x − P y2
≤ Tts6 s5 s4 x − Tts6 s5 s4 y · P x − P y − P x − P y2
≤ r ts6 s5 s4 , x) + x − y · P x − P y − P x − P y2
≤ ε + x − y · P x − P y − P x − P y2 .
(3.25)
Since ε > 0 is arbitrary, this implies P x −P y ≤ x −y. The proof is completed.
Using Lemma 2.1, we have the following ergodic retraction theorem for asymptotically nonexpansive type semigroups.
Theorem 3.2. Let C be a nonempty subset of a real Hilbert space H and let =
{Tt : t ∈ G} be a semitopological semigroup of asymptotically nonexpansive type mappings on
C such that F () = ∅.Then the following statements are equivalent:
(a) s∈G conv{Tts x : t ∈ G} F () = ∅ for each x ∈ C.
(b) There is a unique nonexpansive retraction P of C onto F () such that P Tt =
Tt P = P for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C.
58
Nonlinear ergodic theorems
We denote by B(G) the Banach space of all bounded real-valued functions on G with
supremum norm. Let X be a subspace of B(G) containing constants. Then, according
to Mizoguchi-Takahashi [10], a real-valued function µ on X is called a submean on X
if the following conditions are satisfied:
(1) µ(f + g) ≤ µ(f ) + µ(g) for every f, g ∈ X;
(2) µ(αf ) = αµ(f ) for every f ∈ X and α ≥ 0;
(3) for f, g ∈ X, f ≤ g implies µ(f ) ≤ µ(g);
(4) µ(c) = c for every constant c.
The following corollaries are immediately deduced from Theorem 3.2.
Corollafry 3.3 (see [10]). Let C be a closed convex subset of a Hilbert space H
and let X be an rs -invariant subspace of B(G) containing constants which has a
right invariant submean. Let = {Tt : t ∈ G} be a Lipschitzian semigroup on C with
2 ≤ 1 and F () = ∅, where k is the Lipschitzian constants. If for each
inf s supt kts
t
x, y ∈ C, the function f on G defined by
2
f (t) = Tt x − y ∀t ∈ G
(3.26)
and the function g on G defined by
g(t) = kt2
∀t ∈ G
(3.27)
belongto X, then the following
statements are equivalent:
(a) s∈G conv{Tts x : t ∈ G} F () = ∅ for each x ∈ C.
(b) There is a nonexpansive retraction P of C onto F () such that P Tt = Tt P = P
for every t ∈ G and P x ∈ conv{Tt x : t ∈ G} for every x ∈ C.
Corollafry 3.4 (see [7]). Let C be a nonempty closed convex subset of a Hilbert
space H and let = {Tt : t ∈ G} be a continuous representation of a semitopological
semigroup as nonexpansive
mappings from C into itself. If for each x ∈ C, the set
conv{T
x
:
t
∈
G}
F () = ∅, then there exists a nonexpansive retraction P of
ts
s∈G
C onto F () such that P Tt = Tt P = P for every t ∈ G and P x ∈ conv{Tt x : t ∈ G}
for every x ∈ C.
Remark 3.5. By Theorem 3.2, many key conditions, in Corollaries 3.3 and 3.4, such
as C is convex closed subset and is continuous Lipschitzian semigroup, are not
necessary.
Acknowledgement
The authors wish to acknowledge the financial support of the Korea Research Foundation made in the program year of 1998.
G. Li and J. K. Kim
59
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
J.-B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 22, 1511–1514 (French).
MR 51#11205. Zbl 307.47006.
, Quelques propriétés de convergence asymptotique pour les semi-groupes de contractions impaires, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 3, 75–78 (French).
MR 54#13655. Zbl 339.47028.
H. Brézis and F. E. Browder, Remarks on nonlinear ergodic theory, Advances in Math. 25
(1977), no. 2, 165–177. MR 57#1218. Zbl 399.47058.
N. Hirano and W. Takahashi, Nonlinear ergodic theorems for nonexpansive mappings
in Hilbert spaces, Kodai Math. J. 2 (1979), no. 1, 11–25 (English). MR 80j:47064.
Zbl 404.47031.
H. Ishihara and W. Takahashi, A nonlinear ergodic theorem for a reversible semigroup of
Lipschitzian mappings in a Hilbert space, Proc. Amer. Math. Soc. 104 (1988), no. 2,
431–436 (English). MR 90g:47120. Zbl 692.47010.
W. A. Kirk and R. Torrejón, Asymptotically nonexpansive semigroups in Banach spaces,
Nonlinear Anal. 3 (1979), no. 1, 111–121 (English). MR 82a:47062. Zbl 411.47035.
A. T. M. Lau, K. Nishiura, and W. Takahashi, Nonlinear ergodic theorems for semigroups
of nonexpansive mappings and left ideals, Nonlinear Anal. 26 (1996), no. 8, 1411–1427
(English). MR 97b:47074. Zbl 880.47048.
G. Li, Weak convergence and non-linear ergodic theorems for reversible semigroups of
non-Lipschitzian mappings, J. Math. Anal. Appl. 206 (1997), no. 2, 451–464 (English).
MR 98k:47139. Zbl 888.47046.
G. Li and J. Ma, Nonlinear ergodic theorem for semitopological semigroups of nonLipschitzian mappings in Banach spaces, Chinese Sci. Bull. 42 (1997), no. 1, 8–11
(English). MR 98e:47110. Zbl 904.47063.
N. Mizoguchi and W. Takahashi, On the existence of fixed points and ergodic retractions
for Lipschitzian semigroups in Hilbert spaces, Nonlinear Anal. 14 (1990), no. 1, 69–80
(English). MR 91h:47071. Zbl 695.47063.
W. Takahashi, A nonlinear ergodic theorem for an amenable semigroup of nonexpansive
mappings in a Hilbert space, Proc. Amer. Math. Soc. 81 (1981), no. 2, 253–256 (English).
MR 82f:47079. Zbl 456.47054.
, A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings
in a Hilbert space, Proc. Amer. Math. Soc. 97 (1986), no. 1, 55–58. MR 88f:47051.
, Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups
without convexity, Canad. J. Math. 44 (1992), no. 4, 880–887 (English). MR 93j:47091.
Zbl 786.47047.
W. Takahashi and P.-J. Zhang, Asymptotic behavior of almost-orbits of reversible semigroups
of Lipschitzian mappings, J. Math. Anal. Appl. 142 (1989), no. 1, 242–249 (English).
MR 90g:47121. Zbl 695.47062.
G. Li: Department of Mathematics, Yangzhou University, Yangzhou 225002, China
E-mail address: ligang@cims1.yzu.edu.cn
J. K. Kim: Department of Mathematics, Kyungnam University, Masan, Kyungnam 631701, Korea
E-mail address: jongkyuk@kyungnam.ac.kr