Hölder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces

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by  Andrzej Wiśnicki

HO¨LDER CONTINUOUS RETRACTIONS AND
AMENABLE SEMIGROUPS OF UNIFORMLY
LIPSCHITZIAN MAPPINGS IN HILBERT SPACES
3 ANDRZEJ WIS´NICKI
1
0 Dedicated to Professors Kazimierz Goebel and Lech G´orniewicz
2 on the occasion of their 70th birthdays

n
a
Abstract. Suppose that S is a left amenable semitopological semi-
J
group. We prove that if = Tt :t S is a uniformly k-Lipschitzian
5 S { ∈ }
semigroup on a bounded closed and convex subset C of a Hilbert space
1
and k < √2, then the set of fixed points of is a Ho¨lder continu-
S
] ous retract of C. This gives a qualitative complement to the Ishihara-
A
Takahashi fixed point existence theorem.
F
.
h
t
a
m 1. Introduction
[
Let C be a nonempty bounded closed and convex subset of a Banach

2 space X. A mapping T : C C is said to be nonexpansive if
v →
4 Tx Ty x y
6 k − k ≤ k − k
4 for x,y C. The study of the structure of the fixed-point set FixT =
6 x C :∈Tx = x was initiated by R. Bruck (cf. [3,4]) who proved that if
. { ∈ }
4 T has a fixed point in every nonempty closed convex subset of C which
0
is invariant under T, and C is convex and weakly compact or separable,
2
1 then FixT is a nonexpansive retract of C (i.e., there exists a nonexpansive
:
v mapping R : C FixT such that R|FixT = I). Notice that the set of fixed
→
i points of a k-Lipschitz mapping can be very irregular for any k > 1. Indeed,
X
let F be a nonempty closed subset of C, z F, ε > 0, and define
r
a ∈
Tx = x+εdist(x,F)(z x), x C.
− ∈
Then the Lipschitz constant of T tends to 1 if ε 0, but FixT = F (see,
→
e.g., [17]).
In 1973, Goebel and Kirk [7] introduced the class of uniformly Lips-
chitzian mappings and proved a fixed point theorem which was later stud-
ied by several authors. A mapping T : C C is said to be k-uniformly
→
Lipschitzian if
Tnx Tny k x y
k − k ≤ k − k
for every x,y C and n N. Recall that a Banach space X is uniformly
∈ ∈
convex if δ (ε) > 0 for ε > 0, where δ : [0,2] [0,1] is the modulus of
X X
→
convexity of X defined by
2010 Mathematics Subject Classification. 47H10; 47H20; 54C15.
Key words and phrases. Amenable semigroup, Uniformly Lipschitzian mapping,
Ho¨lder continuous retraction, Fixed point.
1
2 A.WIS´NICKI
x+y
δ (ε) = inf 1 : x 1, y 1, x y ε .
X
{ −(cid:13) 2 (cid:13) k k ≤ k k ≤ k − k ≥ }
(cid:13) (cid:13)
(cid:13) (cid:13)
Theorem 1 (cf. [7]). Le(cid:13)t X be(cid:13)a uniformly convex Banach space with the
modulus of convexity δ and let C be a bounded, closed and convex subset
X
of X. Suppose T : C C is uniformly k-Lipschitzian and
→
1
k 1 δ < 1. (1)
X
(cid:18) − (cid:18)k(cid:19)(cid:19)
Then T has a fixed point in C.
It was shown in [17] that under the assumptions of Theorem 1, the fixed-
point set FixT is a (continuous) retract of C. Recently, Go´rnicki (cf. [8,9])
proved several structural results concerning uniformly Lipschitzian map-
pings but many questions remain open. In [16], P´erez Garc´ıa and Fetter
Nathansky gave conditions under which FixT is aH¨older continuous retract
and applied them to the study of n-periodic mappings in Hilbert spaces.
Notice that in a Hilbert space a solution to (1) gives k < √5/2. Lifs-
chitz [13] improved this estimation and showed that in a Hilbert space a
uniformly k-Lipschitzian mapping with k < √2 has a fixed point. The
Lifschitz theorem was generalized to uniformly k-Lipschitzian semigroups
in [6,10,15]. The aim of this note is to give a qualitative complement to
the above results in the case of left amenable (semitopological) semigroups
which partially extends a result of Go´rnicki (cf. [9, Cor. 14]). We show
that if = T : t S is a uniformly k-Lipschitzian semigroup on C and
t
S { ∈ }
k < √2, then Fix = x C : T x = x , the set of (common) fixed
S t∈S { ∈ t }
points of , is a H¨olderTcontinuous retract of C.
S
2. Fixed point theorem
We start with the following variant of a well known result (see, e.g., [2,
Prop. 1.10], [16, Lemma 2.1]).
Lemma 1. Let (X,d) be a complete bounded metric space and let L : X
→
X be a k-Lipschitz mapping. Suppose there exist 0 < γ < 1 and c > 0
such that d(Ln+1x,Lnx) cγn for every x X and n N. Then Rx =
≤ ∈ ∈
lim Lnx is a H¨older continuous mapping on X.
n→∞
Proof. We may assume that diamX = 1. Notice that for every x X and
n,m N, ∈
∈ γn
d(Ln+mx,Lnx) c .
≤ 1 γ
−
Hence, for every x,y X,
∈
γn
d(Rx,Ry) d(Rx,Lnx)+d(Lnx,Lny)+d(Lny,Ry) 2c +knd(x,y).
≤ ≤ 1 γ
−
Take α < 1 such that k γ1−α−1 and fix x,y X, x = y. Then, there
exist n N and 0 < r 1≤such that γn−r = d(x,∈y)α. Fur6thermore, kn−1
∈ ≤ ≤
UNIFORMLY LIPSCHITZIAN SEMIGROUPS 3
(γ1−α−1)n−r. It follows that
γn−r 2c
d(Rx,Ry) 2c +k(γn−r)1−α−1d(x,y) = ( +k)d(x,y)α.
≤ 1 γ 1 γ
− −
(cid:3)
Let S bea semigroup andℓ∞(S) the Banach space of bounded real valued
functions on S with the supremum norm. For s S and f ℓ∞(S), we
∈ ∈
define elements l f and r f in ℓ∞(S) by
s s
l f(t) = f(st), r f(t) = f(ts)
s s
for every t S. An element µ of (ℓ∞(S))∗ is said to be a mean on X if
∈
µ = µ(I ) = 1, where I (t) = 1 for all t S. It is well known that µ is a
S S
k k ∈
mean if and only if
inf f(t) µ(f) supf(t)
t∈S ≤ ≤ t∈S
for each f ℓ∞(S). A mean µ on ℓ∞(S) is said to be left (resp. right)
∈
invariant if µ(l f) = µ(f) (resp. µ(r f) = µ(f)) for each s S and f
s s
∈ ∈
ℓ∞(S).ABanachlimit isa special case ofaninvariant mean, where S equals
the semigroup of natural numbers. A semigroup S is called left (resp. right)
amenable if there exists a left (resp. right) invariant mean on ℓ∞(S). The
term “amenable semigroup” was coined by M. M. Day in his celebrated
paper [5].
Let C be a bounded closed convex subset of a Hilbert space H. For sim-
plicity, we consider only a real Hilbert space. A family = T : t S of
t
S { ∈ }
mappings from C into C is said to be a uniformly k-Lipschitzian semigroup
on C if T x = T T x and
ts t s
T x T y k x y
t t
k − k ≤ k − k
for all t,s S and x C.
∈ ∈
The following construction is well known (see, e.g., [1,18]). Let µ be a
meanonℓ∞(S),x C,andconsiderafunctionalF(y) = T x,y dµ(t),y
t
∈ h i ∈
H,where T x,y dµ(t)denotesthevalueofµatthefunRctiont T x,y .
t t
h i → h i
It is not dRifficult to see that F is linear and continuous since
T x,y dµ(t) sup T x,y sup T x y .
t t t
(cid:12)Z h i (cid:12) ≤ |h i| ≤ k kk k
(cid:12) (cid:12) t∈S t∈S
(cid:12) (cid:12)
By the Riesz theorem, there exists a unique element x¯ such that
(cid:12) (cid:12)
T x,y dµ(t) = x¯,y (2)
t
Z h i h i
for every y H. Furthermore, by the separation theorem, x¯ C. Thus
∈ ¯ ¯ ∈
we obtain a mapping T : C C by putting T x = x¯. Notice that if a
µ µ
→
semigroup = T : t S is uniformly k-Lipschitzian, then
t
S { ∈ }
T¯ x T¯ y,v = T x T y,v dµ(t) sup T x T y v k x y v
µ µ t t t t
− Z h − i ≤ k − kk k ≤ k − kk k
(cid:10) (cid:11) t∈S
for every x,y C,v H, and hence
∈ ∈
T¯ x T¯ y k x y
µ µ
− ≤ k − k
(cid:13) (cid:13)
(cid:13) (cid:13)
4 A.WIS´NICKI
¯
for every x,y C, i.e., T is k-Lipschitz.
µ
∈
Theabovenotionsmaybegeneralized tothecaseofsemitopologicalsemi-
groups. Recall that S is a topological semigroup if there exists a Hausdorff
topology on S such that the mapping S S (s,t) st is (jointly) con-
× ∋ →
tinuous. A semigroup S is semitopological if for each t S the mappings
∈
S s ts and S s st are continuous. Notice that every semigroup
∋ → ∋ →
can be equipped with the discrete topology and then it is called a discrete
semigroup. Let X be a linear closed subspace of ℓ∞(S) containing I such
S
that l (X) X and r (X) X for each s S. Let X∗ be its topological
s s
⊂ ⊂ ∈
dual. Then X is said to be left (resp. right) amenable if there exists a left
(resp. right) invariant mean on X.
The following theorem is a qualitative version of (a slight generalization
of) Theorem 1 in [10].
Theorem 2. Let C be a nonempty bounded closed convex subset of a Hilbert
space H and = T : t S a uniformly k-Lipschitzian semigroup on C.
t
S { ∈ }
Suppose that X is a left amenable subspace of ℓ∞(S) such that the func-
2
tions g(t) = T x,y and h(t) = T x y belong to X for each x C
t t
h i k − k ∈
and y H. If k < √2, then has a fixed point in C and Fix =
∈ S S
x C : T x = x is a H¨older continuous retract of C.
t∈S { ∈ t }
T
Proof. Let µ be a left invariant mean µ on X. Then there exists a mapping
T¯ : C C such that
µ
→
T x,y dµ(t) = T¯ x,y
t µ
Z h i
(cid:10) (cid:11)
for every y H (see (2)). Following [10, Th. 1], fix x C and put
0
x = T¯ x ,∈n = 0,1,.... Then ∈
n+1 µ n
2 2 2
T x y = T x x + x y +2 T x x ,x y ,
t n t n n+1 n+1 t n n+1 n+1
k − k k − k k − k h − − i
T x x ,x y dµ(t) = T¯ x x ,x y = 0,
t n n+1 n+1 µ n n+1 n+1
Z h − − i − −
(cid:10) (cid:11)
and hence
2 2 2
T x y dµ(t) = T x x dµ(t)+ x y
t n t n n+1 n+1
Z k − k Z k − k k − k
for every y H and n = 0,1,.... It follows (writing y = x ) that
n
∈
2 2
T x x dµ(t) T x x dµ(t). (3)
t n n+1 t n n
Z k − k ≤ Z k − k
Furthermore, taking y = T x , we have
s n+1
2 2 2
T x x = T x T x dµ(t) T x x dµ(t)
s n+1 n+1 t n s n+1 t n n+1
k − k Z k − k −Z k − k
2 2
= T x T x dµ(t) T x x dµ(t)
st n s n+1 t n n+1
Z k − k −Z k − k
2 2
(k 1) T x x dµ(t)
t n n+1
≤ − Z k − k
UNIFORMLY LIPSCHITZIAN SEMIGROUPS 5
foranys S,sinceµisleftinvariantand T x T x k T x x .
st n s n+1 t n n+1
∈ k − k ≤ k − k
Thus
2 2 2
T x x dµ(s) (k 1) T x x dµ(t). (4)
s n+1 n+1 t n n+1
Z k − k ≤ − Z k − k
Combining (3) with (4) yields
2 2 2
T x x dµ(t) (k 1) T x x dµ(t).
t n+1 n+1 t n n
Z k − k ≤ − Z k − k
for n = 0,1,.... Hence
T¯n+1x T¯nx 2 = x x 2 2 x T x 2dµ(t)
µ 0 − µ 0 k n+1 − nk ≤ Z k n+1 − t nk
(cid:13) (cid:13)
(cid:13) (cid:13)
+2 T x x 2dµ(t) 4 T x x 2dµ(t) 4(k2 1)ndiamC
t n n t n n
Z k − k ≤ Z k − k ≤ −
for every x C and n = 0,1,.... Since T¯ is k-Lipschitz and k < √2, it
0 µ
follows from ∈Lemma 1 that Rx = lim T¯nx is H¨older continuous on C.
n→∞ µ
We show that R is a retraction onto Fix . It is clear that if x Fix , then
Rx = T¯ x = x. Furthermore, it follows fSrom the generalized p∈aralleSlogram
µ
law that for every x,y,z H,
∈
2 2 2 2 2 2 2
x+y +z = x+y + y +z + z +x x y z
k k k k k k k k −k k −k k −k k
2 2 2 2 2 2
2( x + y )+2( y + z )+2( z + x )
≤ k k k k k k k k k k k k
2 2 2 2 2 2
x y z = 3( x + y + z ).
−k k −k k −k k k k k k k k
Therefore, for every x C and n = 0,1,...,
∈
T Rx Rx 2dµ(t) 3 ( T Rx T T¯nx 2 + T T¯nx T¯nx 2
Z k t − k ≤ Z t − t µ t µ − µ
(cid:13) (cid:13) (cid:13) (cid:13)
+ T¯nx Rx 2)dµ(t) 3(k2 +(cid:13) 1) Rx T¯nx(cid:13) 2 +(cid:13)3µ T T¯nx (cid:13)T¯nx 2.
µ − ≤ − µ t t µ − µ
Let(cid:13)(cid:13)ting ngo to(cid:13)(cid:13)infinity, TtRx Rx(cid:13)(cid:13)2dµ(t) = 0.(cid:13)(cid:13)Therefo(cid:13)(cid:13)re, for each x (cid:13)(cid:13)C
k − k ∈
and s S, R
∈
2 2 2
T Rx Rx 2 T Rx T Rx dµ(t)+2 T Rx Rx dµ(t)
s s t t
k − k ≤ Z k − k Z k − k
2 2 2
= 2 T Rx T Rx dµ(t) 2k Rx T Rx dµ(t) = 0.
s st t
Z k − k ≤ Z k − k
Thus Rx Fix for every x C and the proof is complete. (cid:3)
∈ S ∈
Remark. Notice that a detailed analysis of the proof of Lemma 1 gives the
estimation α = 1/(1 log k) for the exponent and c = k+8diamC/(2
− k2−1 −
k2) for the constant of the H¨older continuous retraction R, 1 < k < √2.
Inparticular, the abovetheorem is applicable to commutative semigroups
since every commutative semigroup is amenable.
Example 1. Let G be an unbounded subset of [0, ) such that t+s,t s
G for all t,s G with t > s (e.g., G = [0, ) or G∞= N). Suppose that− =∈
∈ ∞ T
T : t G is a family of uniformly k-Lipschitzian mappings from C into
t
{ ∈ }
C such that T x = T T x for all t,s G and x C. If sup T < √2,
t+s t s ∈ ∈ t∈G| t|
6 A.WIS´NICKI
where T = inf c > 0 : T x T y c x y for all x,y C denotes the
t t t
| | { k − k ≤ k − k ∈ }
Lipschitz constant of a mapping T , then Fix = x C : T x = x
t T t∈G{ ∈ t }
is a H¨older continuous retract of C. Notice that we Tdo not assume that the
semigroup is continuous in t (i.e., strongly continuous in the terminology
T
of C -semigroups).
0
Now consider a more general case.
Example 2. Let S be a semitopological semigroup and let CB(S) de-
note the closed subalgebra of ℓ∞(S) consisting of continuous functions. Let
LUC(S) (resp. RUC(S)) be the space of left (resp. right) uniformly contin-
uous functions on S, i.e., all f CB(S) such that the mapping S s l f
s
∈ ∋ →
(resp. s r f) from S to CB(S) is continuous when CB(S) has the sup
s
→
norm topology. It is known (see [14]) that LUC(S) and RUC(S) are left
and right translation invariant closed subalgebras of CB(S) containing con-
stants. Noticethat if auniformlyk-Lipschitzian semigroup = T : t S
t
S { ∈ }
is continuous int onC (i.e., the mapping S t T x is continuous for each
t
∋ → 2
x C), then the functions g(t) = T x,y and h(t) = T x y belong to
t t
∈ h i k − k
RUC(S) for every x C and y H. Indeed (see [11]),
∈ ∈
r g r g = sup (r g)(t) (r g)(t) = sup g(ts) g(tu)
s u s u
k − k | − | | − |
t∈S t∈S
= sup T T x T T x,y k T x T x y
t s t u s u
|h − i| ≤ k − kk k
t∈S
and
2 2
r h r h = sup h(ts) h(tu) = sup T x y T x y
s u ts tu
k − k | − | k − k −k − k
t∈S t∈S (cid:12) (cid:12)
= sup ( T x y + T x (cid:12) y )( T x y T x (cid:12)y
ts tu ts tu
| k − k k − k k − k−k − k|
t∈S
2sup T x y sup T T x T T x
t t s t u
≤ k − k k − k
t∈S t∈S
2ksup T x y T x T x
t s u
≤ k − kk − k
t∈S
for any s,u S,x C and y H. Thus Theorem 2 is applicable with
∈ ∈ ∈
X = RUC(S) if we assume that = T : t S is (separately) continuous.
t
S { ∈ }
Recall that a semitopological semigroup S is said to be left reversible if
any two closed right ideals of S have a non-void intersection. In this case
(S, ) is a directed set with the relation a b iff a aS b bS.
≤ ≤ { } ∪ ⊃ { } ∪
In general, the conditions “RUC(S) has a left invariant mean” and “S
is left reversible” are independent of each other. However, if RUC(S) has
sufficiently manyfunctionstoseparateclosedsets, thentheformercondition
implies the latter (see, e.g., [12, Lemma 3.1]).
Recently, Go´rnicki (cf. [9, Cor. 14]) proved that if S is left reversible and
= T : t S is a uniformly k-Lipschitzian semigroup on C, then the set
t
S { ∈ }
of fixed points of is a retract of C. It is not clear how to extend Theorem
S
2 to left reversible semigroups. There are many other open problems in this
area. Perhaps the most natural is the following.
Problem. Under the assumptions of Theorem 1, is the fixed-point set Fix
S
a Lipschitz or H¨older continuous retract of C?
UNIFORMLY LIPSCHITZIAN SEMIGROUPS 7
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Andrzej Wi´snicki, Institute of Mathematics, Maria Curie-Skl odowska
University, 20-031 Lublin, Poland
E-mail address: [email protected]