NONLINEAR ERGODIC THEOREMS
FOR ASYMPTOTICALLY ALMOST
NONEXPANSIVE CURVES IN
A HILBERT SPACE
GANG LI AND JONG KYU KIM
Received 3 May 2000
We introduce the notion of asymptotically almost nonexpansive curves which include
almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings and study the asymptotic behavior and prove nonlinear ergodic theorems for such
curves. As applications of our main theorems, we obtain the results on the asymptotic
behavior and ergodicity for a commutative semigroup of non-Lipschitzian mappings
with nonconvex domains in a Hilbert space.
1. Introduction
Let H be a real Hilbert space with norm · and inner product (·, ·). Let C be a nonempty
subset of H and G be a commutative semitopological semigroup with identity. In this
case, (G, ) is a directed system when the binary relation “” on G is defined by b a
if and only if there is c ∈ G such that a + c = b. Let = {T (t) : t ∈ G} be a semigroup
acting on C, that is, T (t + s)x = T (t)T (s)x for all t, s ∈ G and x ∈ C. Recall that a
semigroup on C is said to be
(a) nonexpansive if T (t)x − T (t)y ≤ x − y for x, y ∈ C and t ∈ G,
(b) asymptotically nonexpansive, [9], if there exists a function k : G → [0, ∞) with
lim supt∈G kt ≤ 1 such that
T (t)x − T (t)y ≤ kt x − y
(1.1)
for x, y ∈ C and t ∈ G,
(c) of asymptotically nonexpansive type, [9], if for each x ∈ C, there is a function
r(·, x) : G → [0, ∞) with limt∈G r(t, x) = 0 such that
T (t)x − T (t)y ≤ x − y + r(t, x) ∀y ∈ C, t ∈ G,
(1.2)
where limt∈G α(t) denotes the limit of a net α(·) on the directed system (G, ).
Copyright © 2000 Hindawi Publishing Corporation
Abstract and Applied Analysis 5:3 (2000) 147–158
2000 Mathematics Subject Classification: 47H09, 47H10, 47H20
URL: http://aaa.hindawi.com/volume-5/S1085337500000312.html
148
Nonlinear ergodic theorems
It is easily seen that (a)⇒(b)⇒(c) and that both the inclusions are proper (cf. [9,
page 112]).
In 1975, 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 let T be a nonexpansive mapping of C into itself. If the set F (T ) of fixed
points of T is nonempty, then the Cesáro means
n−1
Sn (x) =
1 k
T x
n
(1.3)
k=0
converge weakly as n → ∞ to a fixed point y of T for each x ∈ C. In this case, letting
y = P x for each x ∈ C, P is a nonexpansive retraction of C onto the fixed point
set F (T ) of 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 denotes the closure of the convex hull of A. The analogous
results are given for nonexpansive semigroups by Baillon and Brézis [2] and Brézis and
Browder [3]. In [13], Mizoguchi and Takahashi proved a nonlinear ergodic retraction
theorem for Lipschitzian semigroups by using the notion of submean.
In this paper, we introduce the notion of asymptotically almost nonexpansive curves
which include almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings, and we prove nonlinear ergodic theorems for such curves. As
applications of our main theorems, we obtain the results on the asymptotic behavior
and ergodicity for a commutative semigroup of non-Lipschitzian mappings with nonconvex domains in a Hilbert space. Our results generalize and improve the previously
known results of Baillon [1], Baillon and Brézis [2], Hirano and Takahashi [6], Ishihara
and Takahashi [7], Lau, Nishiura, and Takahashi [10], Li and Ma [11, 12], Mizoguchi
and Takahashi [13], Takahashi [14, 15], Takahashi and Zhang [16], and Tan and Xu
[17] in many directions.
2. Preliminaries and notations
Throughout this paper, let H be a real Hilbert space with norm · and inner product
(·, ·). Let G be a commutative semitopological semigroup with identity and let m(G)
be the Banach space of all bounded real-valued functions on G with the supremum
norm. For each s ∈ G and f ∈ m(G), we define rs f in m(G) given by
rs f (t) = f (t + s) ∀t ∈ G.
(2.1)
Let X be a subspace of m(G) and µ be an element of X ∗ (the dual space of X). Then,
we denote by µ(f ) the value of µ at f ∈ X. To specify the variable t, we write the
value µ(f ) by µ(t)f (t) or f (t)dµ(t). When X contains a constant 1, an element µ
of X ∗ is called a mean on X if µ = µ(1) = 1. Further, let X be invariant under rs for
all s ∈ G. Then, a mean µ on X is said to be invariant if µ(rs f ) = µ(f ) for all s ∈ G
and f ∈ X. For s ∈ G, we can define a point evaluation δs by δs (f ) = f (s) for every
f ∈ m(G). A convex combination of point evaluations is called a finite mean on G.
Recently, the notion of the almost nonexpansive curve was introduced by Rouhani [5]
and Kada and Takahashi [8].
G. Li and J. K. Kim
149
Let u(·) : G → H be a function, in what follows we refer to such u(·) as a curve
in H . A bounded function u is called an almost nonexpansive curve if there exists a
function ε : G × G → R with lims·t∈G ε(s, t) = 0, such that
u(h + s) − u(h + t)2 ≤ u(s) − u(t)2 + ε(s, t) ∀s, t, h ∈ G.
(2.2)
In the case ε(s, t) = 0 for all s, t ∈ G, u is called a nonexpansive curve.
Now, we define the concept of the asymptotically almost nonexpansive curve.
Definition 2.1. The curve u(·) is said to be asymptotically almost nonexpansive if the
following conditions are satisfied:
(1) u(h + t) − u(h + s)2 ≤ u(t) − u(s)2 + ε(t, s, h) for all t, s, h ∈ G, where
ε(t, s, h) ≥ 0 for all t, s, h ∈ G;
(2) for an arbitrary ε > 0 there exists t0 ∈ G, and for each t t0 there exists
ht = h(ε, t) ∈ G such that
ε(t, s, h) < ε
∀t t0 , s t0 , h ht .
(2.3)
Note that, if u(·) is bounded then condition (1) is equivalent to
u(h + t) − u(h + s) ≤ u(t) − u(s) + ε1 (t, s, h)
∀t, s, h ∈ G,
(2.4)
where ε1 (t, s, h) satisfies the same condition (2) as ε(t, s, h). We denote by L(u) the
following subset (possibly empty) of H :
L(u) = z ∈ H : lim u(t) − z exists .
t∈G
(2.5)
Throughout the rest of this paper, u(·) is a bounded asymptotically almost nonexpansive curve and X is a subspace of m(G) containing constants invariant under rs for
each s ∈ G. Furthermore, suppose that for each x ∈ H , the function t → u(t) − x2
is in X. Then by Riesz theorem, there exists a unique element uµ in H such that
µt u(t), x = uµ , x
∀x ∈ H.
(2.6)
We denote uµ by µt u(t). If µ is a finite mean on G,
µ=
n
ai δti
ti ∈ G, ai ≥ 0, 1 ≤ i ≤ n,
i=1
n
ai = 1 ,
(2.7)
i=1
then
n
µt u(t) =
ai u ti .
(2.8)
i=1
We denote by ωw (u) the set of all weak limits of subnets of the net {u(t) : t ∈ G}.
150
Nonlinear ergodic theorems
3. Asymptotic behavior of curves
We begin with the following lemmas and proposition which play an important role in
the proof of our main theorems.
Lemma 3.1. Let u(·) be a bounded asymptotically almost nonexpansive curve. Then
the set L(u) (possibly empty) is closed and convex.
Proof. We can show the closedness from this inequality,
u(t) − x − u(s) − x
= u(t) − x−u(t) − xn +u(t) − xn −u(s) − xn +u(s) − xn −u(s) − x
≤ u(t) − x−u(t) − xn+u(t) − xn−u(s) − xn+u(s) − xn−u(s) − x
≤ 2xn − x + u(t) − xn − u(s) − xn .
(3.1)
And also, the convexity follows from the equality
u(t) − λq1 + (1 − λ)q2 2 = λu(t) − q1 2 + (1 − λ)u(t) − q2 2
2
− λ(1 − λ)q1 − q2 .
Proposition 3.2. The set
s∈G conv{u(t) : t
(3.2)
s}∩L(u) consists of at most one point.
Proof. Suppose that
L(u) = ∅. Let p be the unique asymptotic center of {u(t) : t ∈ G}
in L(u) and x ∈ s∈G conv{u(t) : t s} ∩ L(u). We conclude the proof by showing
that x = p. Since
u(t) − x 2 = u(t) − p2 + x − p2 + 2 u(t) − p, p − x ,
(3.3)
we have
2 lim u(t) − p, p − x + p − x2 ≥ 0.
t∈G
For any ε > 0, there exists t0 ∈ G such that
2 u(t) − p, p − x + p − x2 ≥ −ε
∀t t0 .
(3.4)
(3.5)
Since x ∈ conv{u(t) : t t0 }, it follows that
2(x − p, p − x) + p − x2 ≥ −ε,
(3.6)
that is, p −x2 ≤ ε. Since ε > 0 is arbitrary, we have x = p. This completes the proof.
Since G is commutative, there exists a net {λα : α ∈ I } of finite means on G such that
lim λα − rs∗ λα = 0 ∀s ∈ G,
(3.7)
α∈I
where I is a directed set and rs∗ is the conjugate of rs (see [4]).
G. Li and J. K. Kim
Lemma 3.3. λα (t)u(t +h) converges weakly to an element p in
∩L(u) uniformly in h ∈ G.
151
s∈G conv{u(t) : t s}
Proof. For any {tα } ∈ G, let W be the set of all weak limit points of λα (t)u(t + tα ).
In view of Proposition 3.2, it suffices to show that
conv{u(t) : t s} ∩ L(u).
(3.8)
W⊂
s∈G
To show this, let {tαβ : β ∈ J } be a subnet of {tα : α ∈ B} such that λαβ (t)u(t + tαβ )
converges weakly to some z in H , where J is a directed set. For any ε > 0, there exists
tε ∈ G such that for any t t0 , there exists ht ∈ G such that
ε(t, s, h) < ε
∀t, s t0 , h ht .
(3.9)
Then
u(h + t) − z2 − u(t) − z2 − 2 u(h + t) − u(t), λαβ (s) u s + tαβ + tε − z
= u(h + t) − u(t), u(h + t) + u(t) − 2λαβ (s) u s + tαβ + tε
2 2 = λαβ (s) u(h + t) − u s + tαβ + tε − u(t) − u s + tαβ + tε (3.10)
2 2 ≤ λαβ (s) u(h + t) − u h + s + tαβ + tε
− u(t) − u s + tαβ + tε
+ 4M 2 λαβ − rh∗ λαβ < ε + 4M 2 λα − l ∗ λα β
h
β
for all t t0 and h ht , where M = supt∈G u(t). Note that λαβ (t)u(t + tαβ + tε )
converges weakly to z. For fixed t tε and h ht , taking the limit for β ∈ J , we have
u(h + t) − z2 − u(t) − z2 ≤ ε
∀t tε , h ht .
(3.11)
Therefore,
inf sup u(τ ) − z2 ≤ u(t) − z2 + ε
s∈G τ s
∀t tε ,
(3.12)
and hence
inf sup u(τ ) − z2 ≤ sup inf u(τ ) − z2 + ε.
s∈G τ s
s∈G τ s
(3.13)
Since ε > 0 is arbitrary, wehave z ∈ L(u).
Now, we show that z ∈ s∈G conv{u(t) : t s}. For each s ∈ G, since λαβ (t)u(t +
tαβ + s) ∈ conv{u(t) : t s}, we get z ∈ s∈G conv{u(t) : t s}. This completes
the proof.
Now, we can prove the ergodic convergence theorem for asymptotically almost
nonexpansive curves.
A net {µα : α ∈ A} of continuous linear functionals on X is called strongly regular
if it satisfies the following conditions:
152
Nonlinear ergodic theorems
(a) supα∈A µα < +∞;
(b) limα∈A µα (1) = 1;
(c) limα∈A µα − rs∗ µα = 0 for every s ∈ G.
Theorem 3.4. Let {µα : α ∈ A} be a strongly regular net of continuous linear functional
on X. Then there exists p ∈ s∈G conv{u(t) : t s} ∩ L(u) such that
w − lim u(t + h) dµα (t) = p uniformly in h ∈ G.
(3.14)
α∈A
Moreover, uµ = p for each invariant mean µ.
Proof. By Lemma 3.3, there exists p ∈ s∈G conv{u(t) : t s} L(u) and for any
ε > 0 and y0 ∈ H with y0 = 1, there exists α0 ∈ B such that
ε
λα (t)u(t + h) − p, y0 <
∀h ∈ G.
(3.15)
0
supα∈A µα Suppose that
λ α0 =
n
ai δti ,
ti ∈ G, ai ≥ 0, i = 1, 2, . . . , n,
i=1
n
ai = 1.
(3.16)
i=1
Since {µα : α ∈ A} is strongly regular, there exists α1 ∈ A such that
ε
µα (1) − 1 <
,
(p + 1)
µα − r ∗ µα < ε , 1 ≤ i ≤ n, ∀α α1 ,
si
M
where M = sup{u(t) : t ∈ G}. Since for all α α1 , h ∈ G,
λα0 (t) u(t + s + h) dµα (s) − p, y0 =
λα0 (t) u(t + s + h) − p, y0 dµα (s) − p, y0 µα (1) − 1 (3.17)
≤ sup µα sup λα0 (t) u(t + s + h) − p, y0 + ε ≤ 2ε,
α∈A
s∈G
u(s
+
h)
dµ
−
λ
(s),
y
(t)
u(t
+
s
+
h)
dµ
(s),
y
α
0
α0
α
0 n
u(s + h) −
=
ai u ti + s + h dµα (s), y0 ≤
n
i=1
(3.18)
i=1
ai M µα − rt∗i µα < ε.
Thus, we obtain, for all α α1 , h ∈ G,
u(s
+
h)
dµ
(s)
−
p,
y
α
0 < 3ε.
(3.19)
G. Li and J. K. Kim
153
This completes the proof.
Theorem 3.5. Let u(·) be a bounded asymptotically almost nonexpansive curve. Then
the following conditions are equivalent:
(1) w − limt∈G u(t) exists;
(2) ωw (u) ⊂ L(u);
(3) w − limt∈G (u(h + t) − u(t)) = 0 for every h ∈ G.
Proof. (3)⇒(2). Let ε > 0. Then there exists tε ∈ G and for each t tε there exists
ht ∈ G such that
ε(t, s, h) < ε
∀t, s tε , h ht .
(3.20)
Let z ∈ ωw (u). Then we can take a subnet {u(tα ) : α ∈ J } with tα tε for each
α ∈ J and
w − lim u tα = z.
(3.21)
α∈J
Since L(u) is nonempty by Lemma 3.3, let p ∈ L(u). Since for each t tε and h ht ,
u h + tα − p2 − u tα − p2 + 2 u h + tα − u tα , p − z
2 2
= u h + tα − z − u tα − z
2
2 = u h + tα − u(h + t) + u(h + t) − z
2
+ 2 u h + tα − u(h + t), u(h + t) − z − u tα − z
(3.22)
2
2
≤ u tα − u(t) + ε + u(h + t) − z
2
+ 2 u h + tα − u(h + t), u(h + t) − z − u tα − z
2 2
= u(h + t) − z + u(t) − z + 2 u tα − z, z − u(t)
+ 2 u h + tα − u(h + t), u(h + t) − z + ε,
for fixed t tε and h ht . Taking the limit for α ∈ J , we have
u(h + t) − z2 ≤ u(t) − z2 + ε.
(3.23)
This implies z ∈ L(u) in the same
way as in Lemma 3.3.
(2)⇒(1). Since ωw (u) ⊂ s∈G conv{u(t) : t ≥ s}, ωw (u) is a singleton from
Proposition 3.2. This implies (1) holds.
(1)⇒(3). It is clear.
4. Asymptotic behavior of almost-orbits
In this section, using the main results in Section 3, we prove the ergodic theorems and
weak convergence theorems for almost-orbits of commutative semigroups of asymptotically nonexpansive type mappings with nonconvex domains.
154
Nonlinear ergodic theorems
Let C be a nonempty subset of a Hilbert space H and = {T (t) : t ∈ G} be a family
of mappings from C into itself. Recall that is said to be a commutative semigroup
of asymptotically nonexpansive type mappings on C if the following conditions are
satisfied:
(a) T (t + s)x = T (t)T (s)x for all t, s ∈ G and x ∈ C;
(b) for each x ∈ C and t ∈ G, there exists α(t, x) ≥ 0 such that
T (t)x − T (t)y ≤ x − y + α(t, x) ∀y ∈ C,
(4.1)
with
lim α(t, x) = 0
t∈G
∀x ∈ C.
(4.2)
A function u(·) : G → C is said to be an almost-orbit of = {T (t) : t ∈ G} if
lim sup u(h + t) − T (h)u(t) = 0.
(4.3)
t∈G
h∈G
Throughout the rest of this section, = {T (t) : t ∈ G} is a commutative semigroup
of asymptotically nonexpansive type mappings on C, u(·) : G → C is a bounded
almost-orbit of = {T (t) : t ∈ G}, and X is a subspace of m(G) containing constants
invariant under rs for each s ∈ G. Furthermore, suppose that for each x ∈ H , the
function t → u(t) − x2 is in X. Denote by F () the set of common fixed points of
= {T (t) : t ∈ G}.
We begin with the following lemmas.
Lemma 4.1. Let u(·) be a bounded almost-orbit of the commutative semigroup =
{T (t) : t ∈ G} of asymptotically nonexpansive type mappings on C. Then it is an
asymptotically almost nonexpansive curve.
Proof. Put ϕ(t) = suph∈G u(h + t) − T (h)u(t). Then limt∈G ϕ(t) = 0. Since
u(h + t) − u(h + s) ≤ u(h + t) − T (h)u(t) + T (h)u(t) − T (h)u(s)
+ u(h + s) − T (h)u(s)
≤ ϕ(t) + ϕ(s) + α h, u(t) + u(t) − u(s),
(4.4)
for every h, t, s ∈ G. It is easily seen that u(·) is an asymptotically almost nonexpansive
curve.
Lemma 4.2. If u(·) and v(·) are almost-orbits of , then limt∈G u(t) − v(t) exists.
Furthermore, we have F () ⊆ L(u).
Proof. Set
ϕ(t) = sup u(s + t) − T (s)u(t),
s∈G
ψ(t) = sup v(s + t) − T (s)v(t).
s∈G
(4.5)
G. Li and J. K. Kim
Then, limt∈G ϕ(t) = limt∈G ψ(t) = 0. Since for each t, s ∈ G,
u(s + t) − v(s + t) ≤ u(s + t) − T (s)u(t) + T (s)u(t) − T (s)v(t)
+ v(s + t) − T (s)v(t)
≤ ϕ(t) + ψ(t) + α s, u(t) + u(t) − v(t),
inf sup u(τ ) − v(τ ) ≤ ϕ(t) + ψ(t) + u(t) − v(t).
155
(4.6)
s∈G τ s
It follows that
inf sup u(τ ) − v(τ ) ≤ sup inf u(τ ) − v(τ ),
s∈G τ s
(4.7)
s∈G τ s
which complete the proof of the first part. The second part is obvious.
We can prove the following proposition from Lemma 4.2 and Proposition 3.2.
Proposition 4.3. The set
s∈G conv{u(t) : t
s}∩F () consists of at most one point.
From Theorem 3.4, we can prove the following theorem which is an extension of
the result of Tan and Xu [17] in many directions.
Theorem 4.4. Let C be a nonempty subset of H , = {T (t) : t ∈ G} a commutative
semigroup of asymptotically nonexpansive type mappings on C, and u(·) be a bounded
almost-orbit of . If {µα : α ∈ A} is a strongly regular net of continuous linear functional on X, then
w − lim u(t + h) dµα (t) = p ∈
conv u(t) : t s
L(u)
(4.8)
α∈A
s∈G
uniformly in h ∈ G. Further, if each T (t) is continuous and
C, then p ∈ F ().
s∈G conv{u(t) : t
s} ⊂
Proof. By Lemma
4.1 and Theorem 3.4, we need only to prove that if each T (t) is
continuous and s∈G conv{u(t) : t s} ⊂ C, then p ∈ F (). By assumption, we have
p ∈ C. Let 0 < ε ≤ 1. Then there exists t1 ∈ G such that
ε
ϕ(t) = sup T (h)u(t) − u(h + t) <
,
8d
h∈G
(4.9)
ε
α(t, p) <
,
8d
for each t t1 , where d = 1 + sup{u(t) − p : t ∈ G}. Since
2
T (s)p − p2 + 2 u s + t + t1 − p, p − T (s)p + u s + t + t1 − p 2
2 = u s + t + t1 − T (s)u t + t1 + T (s)u t + t1 − T (s)p (4.10)
+ 2 u s + t + t1 − T (s)u t + t1 , T (s)u t + t1 − T (s)p
2
≤ u t + t1 − p + ε
156
Nonlinear ergodic theorems
for s t1 , this implies that
T (s)p − p2 µα (1) + 2µα (t) u s + t + t1 − p, p − T (s)p
2
≤ µα − rs∗ µα (t)u t + t1 − p + µα (1)ε.
(4.11)
Taking the limsup for α ∈ A, we get
T (s)p − p 2 ≤ ε
∀s t1 .
(4.12)
It follows that T (t)p is convergent strongly to p, therefore, p ∈ F () by the continuity
of {T (t) : t ∈ G}. This completes the proof.
Let AO() be the set of all almost-orbits of . Then for each h ∈ G and u ∈ AO(),
the function v : G → C, defined by v(t) = T (h)u(t), is also an almost-orbit of . In
fact, as before, we set ϕ(t) = sups∈G u(s + t) − T (s)u(t). Since
v(s + t) − T (s)v(t) = T (h)u(s + t) − T (s)T (h)u(t)
≤ T (h)u(s + t) − u(h + s + t)
+ u(h + s + t) − T (s + h)u(t)
(4.13)
≤ ϕ(s + t) + ϕ(t),
the result follows.
Using Theorem 4.4, we have the following ergodic retraction theorem.
Theorem 4.5. Let C be a nonempty bounded subset of a Hilbert space H and let be a commutative semigroup of asymptotically nonexpansive type mappings on C such
that each T (t) is continuous. Then for an invariant mean µ, the mapping P : u → uµ
is a unique retraction from the set AO() onto F () such that
(1) P is nonexpansive in the sense that
P u − P v ≤ lim u(t) − v(t);
t∈G
(4.14)
(2) P T (h)u
= T (h)P u = P u for u ∈ AO() and h ∈ G;
(3) P u ∈ s∈G conv{u(t) : t s} for u ∈ AO().
As a direct consequence of Theorem 3.5, we can prove the following theorem which
is an extension of the Takahashi and Zhang [16]. Note that we do not assume F () to
be nonempty.
Theorem 4.6. Let C be a nonempty subset of a Hilbert space H and let be a
commutative semigroup of asymptotically nonexpansive type mappings on C, and let
G. Li and J. K. Kim
157
u(·) be a bounded almost-orbit of . Then w − limt∈G u(t) exists (in L(u)) if and only
if w − limt∈G (u(h + t) − u(t)) = 0 for all h ∈ G.
Acknowledgement
This work was supported by Korea Research Foundation Grant (KRF-99-041-D00025).
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Gang Li: Department of Mathematics, Yangzhou University, Yangzhou 225002,
China
E-mail address: yzlgang@pub.yz.jsinfo.net
Jong Kyu Kim: Department of Mathematics, Kyungnam University, Masan, Kyungnam
631-701, Korea
E-mail address: jongkyuk@kyungnam.ac.kr