(Katowice)Fractional iteration oî differentiable functions with multiplier zero§ 1. Let g(x) be a continuously differentiable function on an inter­val^) <

ROCZNIKI POLSKIEGO TO W AR ZYSTW A MATEMATYCZNEGO Séria I : PRACE MATEMATYCZNE X I V (1970)

M

K

(Katowice)

Fractional iteration oî differentiable functions with multiplier zero

§ 1. Let g(x) be a continuously differentiable function on an inter­

val^) <

, a) such that g' (x)

in (

, a) and (

)

< g(x) < x in (

, a).

In [3] we have proved that if, moreover, #'(0) Ф 0, then every solution (p(x) of the functional equation (2)

which is of class G1 in <

, a) must satisfy a certain differential equation and thus, in most cases, it is unique. The purpose of the present note is to show that in the case, where g1 (

) =

, the situation is different.

At first we note the following result (cf. [3]).

L

. Let g(x) be a continuously differentiable function on < 0 , a) fulfilling inequality (1) and the condition g'(x) > 0 in (

, a). Further, let x 0, У о be arbitrary two points of (0, a) such that g{x0) < y 0 < x 0, and let

<p0(x) be an arbitrary function of class C1 on < t /0, x f) and such that щ{х) >

in <y0, x 0} and <p0(x0) = y 0, (p0(y0) = д{я0), <Ро(Уо) = 9'(x o)l<Po(xo)- Then there exists a unique solution <p{x) of equation (2) in <

, a) such that <p(x)

= (p0{x) in <

/0, x 0y. This solution <p(x) is of class C1 in (0, a).

We shall prove the following

T

. Let the function g(x) fulfil the hypotheses of the lemma and suppose that there exists an x0e(0, a) such that the sequence

(b For simplicity the fixed point of g has been placed at zero, hilt this choice is not essential. Zero can he replaced by any real number.

(2) g?2 denotes the second iterate of cp. More generally, for any function f , f n denotes its n-th iterate: f ° ( x) — x , f n + l (x) — f ( f n(x)), n = 0 , 1, 2 , . . .

( 2 ) <p*{x) = g{x)

(3)

converges to zero almost uniformly (3) in the set

(4) о? g ( x ) < y < x .

Then every solution cp(x) of equation (

) in <0, a) which is of class C1 in (

, a), is also of class G1 in <

, a).

P r o o f . Let (p(x) be a solution of equation (

) in <

, a) which is of class C

in (0,u). As is known (cf., e.g., [4]), <p(x) fulfils the inequality (5) g { x ) < ( p ( x ) < x i n ( 0 , a ) ,

and, of course,

(

) = g(0) = 0. For the proof of the theorem it is enough to show that

(

) lim cp’ (x) =

.

гс->0+0

Differentiating (

) we obtain

(7) <p'((p{x))(p'{x) = g'{x),

whence, on replacing x by

?2l(x), we obtain

<p'[<p2i+1{x)]<p'\_<p2i(x)-\ = g'[<p2i(x)l, i.e. by (

)

(

) ¥\А<р{х))\ 9?'(У И ] = g 'W W l - Eeplacing in (

) x by <p(x) we get

(9) = 9 ' И ? И )].

We divide (9) by (

) (it results from (7) that <p'(x) Ф 0 in (

,a)):

cp’ [,gi+l{x)1 = д'\д1{<р{х))\

<р'[дЧх)] g'igHx)]

and, taking the product over i from

to n —

, we get

We fix an e > 0, and put M = sup

o’ (x). The set

<0{Щ),хо>

д(х0) ^ х ^ х о, у = (p(x) ,

is a compact subset of (4) (cf. (5)); consequently, there is an N such that

( 11 ) 0 < r j - g ' H p W )

e

1 J д'[дЧх )1 M for xe <g(x0), xf) and n ^ N.

(3) I.e., uniformly on every compact subset of set (4).

Let us take an arbitrary a?e(0, gN(x0)). Then there exists an n > N and an x* e ( g( x 0), x 0y such that x — gn{x*). It follows by (10) and (11) that 0 < < p ' ( x ) < s . This proves relation (

).

In view of the lemma we get hence the following

C

. Under the conditions of Theorem 1 equation (2 ) has in

<

, a) a C1. solution depending on an arbitrary function.

Eelation (10) allows us to draw one more conclusion. It follows from (7) with x = 0 that <p' (0) = 0 and, since limgn(x) = 0, (cf., e.g., [2],

П—>СО

p.

), we have Um<p'[gn(x)] =

for every a?e(

, a) and every C

solution

тг—>oo

<p of equation (

). Thus we obtain the following

T

2 . I f the function g(x) fulfils the hypotheses of the Lemma, g’ (

) =

, and if equation (

) has a C1 solution p(x) in <

, a), then sequence

(3 ) must converge to zero for у = p(x), xe(0, a).

§

. Now we are going to give some examples of classes of functions that fulfil the conditions of Theorem 1. We shall restrict ourselves to functions which are asymptotically comparable with xa as x -»

. Namely, for any a > 0, we define Ua as the class of those functions f(x) which are defined in a right neighbourhood of zero and for which the limit lim x~af(x)

x-*o+o

exists, is finite and positive (cf. [1]). In the sequel we shall consider only such functions g that g' € t7a-1, a >

. This implies ge Ua, but the converse implication is not true.

T

3 . Let the function g{x) fulfil the hypotheses of the Lemma and suppose that g'aUa~1, a > 1. Then each of the following conditions is sufficient for product (3 ) to converge to zero almost uniformly in set (4 ),

for any ж

е

, a):

(i) The function

(

) h(x) = x~ag(x)

is increasing in (

, a).

(ii) Function (1 2 ) fulfils the inequality

\logh[gi {x)'] — logh[gi {y)'\\ < (a—

) \logx—logy\

for i = 0 , 1 , 2 , . . . , and x, у belonging to set (4).

P r o o f . The condition g' e Ua~l implies that g'(x) may be represented as g' (ж) = xa- Yf { x ) ,

where f(x) has a finite, positive limit as x -> 0 + 0 . Then the general term of product (3) may be written in the form

g' IУ(я)] 1У(®)]а-71У(®)] *

(13)

(14)

We shall show that each of conditions (i), (ii) implies even that

r дЧу) л

lim —;---- =

gl{x)

almost uniformly in set (4), which in view of (13) is quite sufficient ta prove our theorem.

(i) We have (induction)

г —1

(15) gl{x) = xa% f j {h[gj (a>)]}° i = 1 , 2 , . . .

1 = 0

The function h(x) is positive in ( 0 , # 0> provided x 0 has been chosen small enough, and the functions g‘ {x ) are strictly increasing. Thus у < x imphes

and consequently

0

< i % ) < / l дг{х) \x Hence (14) results immediately.

(ii) W e fix a compact set A contained in (4). At first let us note that for large i, say г > i 0, we have

| lo g W (^ )]-lo g ü [jr % )]| < i ( a —l)|logæ—logy| for { x , y ) e A , since the left-hand side converges to zero (uniformly in A), whereas the right-hand side has a positive lower bound in A. Consequently we can find a positive constant $ <

such that

(16) l o g h l g ^ x ) ] - logJijg^y)]

logo;—log y ^ ê ( a —

)

for ( x , y) e A and i = 0 , 1 , 2 , . . . We consider the expression (cf. (15))

(17) l o g ^ M ё дг(%)

a* (log У — log x) + a1 1 1 {logh[g> {yft — loghig1 {x)] } .

1—0

In view of (17) and (16) we have for (x, y ) e A lo g -C M

дг{х) log —

l o g i M - ё 9г(х) log —

^ a1— &(a— 1) a

a—1 (1 — д')аг-{-'д'1

whence

and

Hence (14) results.

ifly) . y

l o g ^ ^ < [ ( l - # K + t f ] l o g * -

g {so) sc

о < Ü M < ( i L g'(x) \æ

(

-

)аг+£

References

[1] M. K u c z m a , Sur Véquation fonctionnelle de Bôttcher, Mathematica, Cluj, 8 (31) (1966), pp. 279-28 5.

[2] — Functional equations in a single variable, Monografie Matematyczne 46, Warszawa 1968.

[3] — Fractional iteration of differentiable functions, Ann. Polon. Math. 22 (1969), pp. 217-227.

£4] J. W o j c i k - G e r , On convex solutions of the functional equation cp2(x) = g(x),

Zeszyty Naukowe Uniwersytetu Jagiellonskiego, Prace Mat. (to appear).