(Krakôw)On continuity of iteration semigroups on metric spaces

AN N ALES SOCIETATIS M A T H E M A TIC A E P O LO N AE Series ï: C O M M E N T A T IO N E S M A T H E M A TIC A E X X IX (1989) R O C Z N IK I PO LS K IE G O T O W A R ZY S T W A M A T E M A T Y C Z N E G O

Séria I: PRACE M A T E M A T Y C ZN E X X IX (1989)

Ma r e k Ce z a r y Zd u n

(Krakôw)

O n con tin u ity o f iteration sem igroups o n m etric spaces

In this note we are concerned with iteration semigroups and groups defined in a metric space X. We give some conditions which imply the continuity of measurable iteration semigroups. This problem has been inves­

tigated before in papers [6] and [5]. First under the assumption that X is a closed and finite interval in R it has been proved in [6] that every measurable iteration semigroup is continuous. Next Smajdor in paper [5] generalized this result for iteration semigroups of compact-valued functions defined in a com­

pact metric space. In the present paper we give a new short proof of a generalization of Smajdor’s theorem for a single-valued functions.

First we introduce the following definitions.

A family of continuous functions {/*, t > 0} ({/*, teR}) defined in a metric space X is said to be an iteration semigroup (group) in X if /*: X -* X and f lo f s = f t+s for t, s > 0 (t, se R).

A function h: R~^X will be called measurable if for every Borel set G <= X the set h- 1 [G] is measurable in Lebesgue’s sense.

Let (Y, | | U) be a Banach space and Y* be its conjugate space. A vector function h : Y is said to be strongly measurable if there exists a sequence of simple functions h„ such that hn-^h a.e.

We will call the set P c Y* norming if there exist c > 0 and C > 0 such that sup{|x*(x)|: x*eP, ||

*|| ^ C} ^ c||x|| for xeY.

Now, we will prove the following theorem.

Th e o r e m

1. If X is a compact metric space and { / ', t > 0} is an iteration semigroup in X such that for every x e X the function is measurable, then the mapping (t, x)->/*(x) is continuous in (0, oo)xX.

Proof. The space X is compact and metric, so is separable. Hence in view of Urysohn’s theorem it follows that there exist a subset К of the Hilbert cube and a homeomorphism h defined on X such that h[X~\ = K. The set К is compact as a continuous image of the compact space X. Put gl := hof *oh~l for t > 0. Note that {g\ t > 0} is an iteration semigroup defined in К such that for every x e K the mapping 1-*дг(х) is measurable.

8 — Comment. Math. 29.1

114 Mar e k Ce z ar y Z d un

Denote by s the linear space of all sequences a = (an) in R such that

where nk((an)) : = ak, k = 1 ,2 ,...

It is easy to verify that (s, | | ||s) is a separable Banach space.

Note that K cz s and the topology in К generated by the metric of the Hilbert cube coincides with topology generated by the norm | | ||s.

Let C(K, s) be the linear space of all continuous functions defined on К with values in s. In C(K, s) we define the following norm

Since К is compact and s is separable and complete, C(K, s) is a separable Banach space (see [3], p. 120 and 315).

Now we shall show that the vector-valued function (0, oo) э t ->g1 e C(K,

) is strongly measurable.

Let L be a set of all sequences r = (e1} s2, ...), £f = ±1. Consider the family P: = x e K , reL} of linear functionals on C(K, s) defined by the following formula

P is a norming set with C = c = 1. In fact, for every x e K and r e L E 5;KI < œ -

In the space s we define the following norm

00 1

/II := sup ||/(x)||s = sup X —|тг„(/(х))|.

x e K x e K n =x e K n = 1 1^

so

^ 1. Let f e C { K , s ) . Put

and r(x):= (e/x), e2(x), ...). We have the inequality

sup{|x*(/)|: x*e P, ||x*|| ^ 1} = sup{|c5(JCtr)(/)|: x e K , reL}

> sup{|<5(,,r(x))(/)|, x e K } O O 1

=

{ Z ^ k ( / (*))!’ x e K} = II/Ц.

Thus P is a norming set.

Continuity of iteration semigroups on metric spaces

For every fixed x e X and r e L the function

<W)(0')= Z

n= 1Z

is measurable, since the function

is measurable and the projections

1, 2, . . . are continuous. Therefore for every

the function t->x*(gf) is measurable. Hence in view of Pettis’ theorem (see [1], p. 252) it follows that the function (0, со)эt ->дгeC(K, s) is strongly measurable.

Fix t0, a and b such that 0 < 3a < b < t0. Put a:= t0 + a — b and /? : = t0 — 2a. Let |e| < a and a ^ s ^ fi. By the definition of the norm | | | | we have the following estimation:

\\gt0+E- g t0\\ =

\\gt0+E~sogs- g t0~sogsW = ||(^0+£- 5- ^ ° - 5)о/||

^

Hence

(1) Il

1*0 11 _

ds ^

a

t o + e -

I ds

Define

t o ~ a b — a

J

= J ||gf*+«-^||dr.

t o - p 2 a

x( t

) : = g\

0,

t e ( a, b}, Ьф(а, b}.

Note that the function t->|lx(OII is measurable and J \\x(t)\\dt < oo, since

— 00

| | < 7 * | | < 1. Hence in view of Bochner’s theorem (see [2], p. 80) it follows that x is integrable in the sense of Bochner.

Let te(2a, Ь — аУ. Then t, t + eef a, by for |e| < a, so

b — a b — a oo

(2) J \\gt+B- g t\\dt = j M t + e)~x(t)\\ dt < J ||x(t + e)-x(OII dt.

2 a 2 a — oo

The integrability of x implies that 00

lim j \\x(t + s)~x(t)\\dt = 0

£ — ► 0 — O O

(see [2], p. 86). Hence by (1) and (2) it follows that \im\\gto+e — gto\ \ = 0.

116 Mar e k Ce z ar y Z d un

Thus g* converges uniformly in К to gt0 as t-* t0, s o / 1 converges uniformly in X to f t0 as t-* t0, since f t = h~1ogtoh for t > 0 and h-1 is uniformly continuous in К and consequently the function (t, х )-> /((х) is continuous.

Theorem 2.

Let X be a metric space and {/*,

>

be an iteration semigroup in X such that every set p[X~\ for t >

is compact. If T is such a subset of X that

T and for every x e T the mapping

is

s > 0

measurable, then for every s > 0, f* -> /s uniformly in X for t^ s .

Proof. Put M : = (J f l [X~\. Let x e M and xn->x, x„eT. The mappings t > о

t-+P{xf) me measurable and so is also the function f - > /‘(x) as a limit of measurable mappings (see [4], p. 447).

Since f* are continuous and / ' o / s = / so / ' for t, s > 0 we have f i r m ] m ] = г [ г т ] ^ Т т а -

Hence { f lysjpq, t > 0} for every s > 0 is an iteration semigroup.

Denote by

the metric of the space X. For |s —1\ < t/2 we have the following estimation:

(3) su p e(/s(x),/'(x)) = s u p e (/5- ''2( / « 2(x )),/"2( / « 2(x)))

x e X x e X

= SUp Q(fs~tl2( x), ftl2(x)).

x e f ‘ /2 [ X l

According to Theorem 1 , f s~t/2 converges uniformly in/ ' /2[Х ] t o / l/2 for s ^ t and further by (3) we infer that / s-> /' uniformly in X.

References

[ 1 ] A. A l e x i e w ic z , Analiza funkcjonalna, Monografie M at. 49, P W N , Warszawa 1969.

[2 ] E. H i lie , R. P h i l l i p s , Functional analysis and semi-groups, Amer. M ath. Soc. Coll. Publ.

Vol. 31, Providence, Rhode Island 1971.

[3 ] K . K u r a t o w s k i , Topologie, vol. I, Monografie M at. 20, P W N , Warszawa 1958.

[4 ] L. S c h w a r t z , Kurs analizy matematyeznej, t. I, P W N , Warszawa 1979.

[5 ] A . S m a j d o r , Iterations o f multi-valued functions, Prace Naukowe Uniwersytetu Slqskiego w Katowicach nr 759, Katowice 1985.

[6 ] M . C. Z d u n , Continuous and differentiable iteration semigroup, Prace Naukowe Uniwersytetu Sl^skiego w Katowicach nr 308, Katowice 1979.

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