fn(x)--g(x) By JAMES C. LILLO

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A R K I V F O R M A T E M A T I K B a n d 5 n r 26

1.64032 C o m m u n i c a t e d 12 F e b r u a r y 1964 b y O . FROSTMAN a n d L . CARLESON

The functional equation fn(x)--g(x)

B y JAMES C. LILLO

l . Introduction and notation

We are interested in studying the real functional equation/n(x)

= g(x)

o n a n interval [a, b] of the real line. I n particular we wish to obtain conditions on g which will assure one t h a t solutions / of the given equation possess certain properties. I f one insists only t h a t / be a pointwise solution, t h e n the problem for n = 2 has been solved [3]. If one insists t h a t / be continuous, only v e r y limited results are known [1], [2], [5]. I n Theorem 2.1 we obtain results which suggest studying the problem in a certain subclass

M[a, b] of

the class of continuous functions. I n example 1 we show t h a t there exists a continuous function g defined on a closed interval [a, b] for which the equation/~(x) =g(x) does not possess a n y continuous solutions / b u t does have a solution / which possesses the D a r b o u x property. Theorem 2.4 gives sufficient conditions to insure t h a t if g is continuous then a n y solution / of the equation/n(x)

= g(x),

which possesses the D a r b o u x property, will also be continuous. I n Theorem 2.5 we consider the special equation/n(x)

=/n+P(x).

To facilitate m a t t e r s we introduce the following notation. L e t [a,b] denote a n y closed interval of the real line where the endpoints § ~ a n d - ~ are allowed. The set of all functions defined on [a, b] with values in [a, b] will be denoted b y

R[a, b].

A function is said to possess the D a r b o u x p r o p e r t y if it takes connected sets into connected sets.

D[a, b]

will denote those functions of

R[a, b]

which possess the Dar- boux property.

C[a,b]

will denote those functions of

R[a,b]

which are continuous on [a, b]. We denote b y

M[a, b]

those functions of

C[a, b]

which are piecewise monotone (written p.m.) on

[a,b].

Here, / is said to be piecewise monotone on [a,b] if there exists a finite partition P = [P0 .... p~] of [a, b] such t h a t on each subinterval [Pi,Pi+l]

the function / is strictly monotone (written s.m.). I f every partition P * which possesses this p r o p e r t y with respect to / is a refinement of P, then P is said to be the partition asssociated with / and will be denoted b y

P(/).

We define

/~

and

/n+l(x) =](/~(x))

for n ~>0. Finally, we define the set

S(n,g)= {/eR[a,b] I/~(x)=g(x)

for all x e [a, b] }.

2. The general equation

fn(x)= g(x)

I t is clear t h a t if /E

M[a, b]

t h e n / ~ E

M[a, b]

for a n y i. We now establish the con- verse. I f / E

D[a, b]

and

f E M[a, b]

t h e n / E

M[a, b].

Theorem 2.1.

I / g E M[a, b] then S(n, g) N D[a, b] c M[a, b] and P(g) is a refinement o/P(/)/or every/eS(n,g) N D[a,b].

357

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J. C. LILLO,

The functional equation fn(x) = g(x)

Proo/.

W e first n o t e t h a t i f / E D [ a , b ] is s.m. on each subinterval [Pi,Pi+l] of

P(g)

t h e n / E C [ a , b ] a n d s o / E M [ a , b ] . Thus, it suffices to show t h a t a n y

[ES(n,g) N D[a,b]

is s.m. on e v e r y subinterval [P*,P~+I] of P(g). Assume f is n o t s.m. on [Pi,P~+I], t h e n s i n c e / E

D[a,b]

it follows easily t h a t there are a t least t w o points

x, yE[pi,pi+l]

for which

/(x)=/(y).

B u t t h e n

g(x)=In(x)=/~(y)=g(y)

which contradicts the fact t h a t g is s.m. on [P,,Pi+I]- This completes the proof of T h e o r e m 2.1.

W e shall see later t h a t there are g E

M[a, b]

such t h a t

D[a,

b] N S(2, g) is e m p t y while

R[a,b]

N S(2,g) is n o t e m p t y . W e shall also see, b y means of a n example, t h a t there are

gEC[a,b]

for which S(2,q) N

C[a,b]

is e m p t y b u t S(2,g) N

D[a,b]

is n o t e m p t y . To facilitate the c o n s t r u c t i o n of this e x a m p l e we n o w o b t a i n several results which are needed here a n d later in the d e v e l o p m e n t . T h e first result is closely related [2]

to the case

g(x)=[(x)=/~(x).

Theorem 2.2.

I f /E D[a,b], /(p) = p and S is a nondegenerate maximal connected set containing p, such that/'(x) = x / o r x E S, then S = [c, d] and (a) / I S is a homeomorphism o/ S onto S, (b) /(x) = x on S or/(c) =d, /(d) =c and/] [c,d] is a reflection o/[c,d] about pe(c,d).

9 Proo/.

S i n c e / ~ = g is s.m. on S a n d

]ED[a,b],

it follows, as in T h e o r e m 2.1, t h a t f is s.m. on S, i = 1,2 .... , n - 1 . I t t h e n follows t h a t f is continuous a n d s.m. on t h e closure ~q = [c,d] of S, i ~ 1 ... n - 1 . Consider first the case where / is increasing on [c,d]. I f

p~=d

t h e n there exists

qE(p,d)

such t h a t

f(q)E(p,d)

for i = 0 ... n. E i t h e r

f+l(q)

> f ( q ) , / ( q )

=q

or

f+l(q)

< f ( q ) for i = 0 , ...,n since / is s.m. in

[p,d].

B u t

[~(q) =q

a n d so/(q) = q. I t n o w follows t h a t / ( x )

-- x

on [p, d]. I f p :~c a similar t r e a t m e n t shows t h a t / ( x ) = - - x on

[c,p].

Thus, if / is increasing on

[c,d]

t h e n / ( x ) = - x on

[c,d].

L e t ] be decreasing on

[c,d]

a n d assume t h a t p =~c. T h e n / is s.m. on

[p,/(c)].

I f / is increasing on [p,/(c)] there is a p o i n t

we(p,l(c))

such t h a t

f(w)e(p,/(e))

for i = 1 , 2 ... n. L e t

qE(c,p)

be such t h a t / ( q )

=w,

t h e n

]~(q)~=q.

Thus, / is decreasing in

[p,](c)] U [c,p].

If/2(c)

4 c

t h e n either

p(c)e (c,/(c))

or there is a t t E (c,p) such t h a t / n - J ( t t ) E [p,/(c)] U

[ p , c ) j = l

... n - 1 a n d

p(tt)=c.

I n t h e first case,

f(c)E(c,/(c))

for all i~>2 a n d so /n(c) =~c. I n the second case, we h a v e t t E S for which /~(/z)=~# which is impossible.

Thus,/2(c)

=c, n

is even, a n d / 2 is a n increasing function on

[c,p].

Thus, / is a reflec- t i o n of

[c,/(c)]

a b o u t p. I n t h e same w a y one m a y show t h a t / is a reflection of

[/(d),d]

a b o u t p. Since S is m a x i m a l / ( d )

=c

a n d / ( c )

=d.

This completes the proof of Theo- r e m 2.2.

Corollary 2.1.

I / / s a t i s f i e s the hypothesis o/Theorem 2.2,/EC[a,b] and S=[c,d] :#

[a, b] is a ray, then / - - x on S.

Proo/.

E i t h e r

cE(a,b)

a n d

d = b = + oo

or

dE(a,b)

a n d

c = a = - o o .

S i n c e / E C [ a , b ] it is clear t h a t in b o t h cases we m a y n o t h a v e / ( e ) = d a n d / ( d )

=c

a n d the result follows.

I f

gER[a,b]

we define

~(g)=(xlxE[a,b ]

a n d

g(x)=x}, y(g)

is called the set of fixed points of g. I f / E S ( n , g ) t h e n one m a y say a great deal a b o u t / 1 7 ( g ) " One of these results is contained in the following theorem.

T h e o r e m 2.3.

I] gER[a,b] and ]ES(n,g) then /lY(g) defines a one to one map o/

7(g)

onto 7(g).

Proo].

Assume x e~(g), b u t t h a t y =/(x) r T h e n

g(y) :#y

and/n+l(x)

=/(/n(x))=

/(x) =y=/(/n(x))-g(y)=~y.

Thus, f ( ~ ( g ) ) c ? ( g ) for a n y i. L e t x ET(g ), t h e n

x = g ( x ) =

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ARKIV FOR MATEMATIK. Bd 5 nr 26

/(/n--l(x)) C/(~(g))

and so ?(g)c/(7(g))" Thus, / defines a m a p of ?(g) onto itself.

Since for a n y x, y ET(g)x ~:y implies/n(x) #/n(y), it follows t h a t the m a p is one to one.

Corollary 2.2. I / g E R [ - o o , oo], 7(9) is a ray, and / E S ( n , g ) N C [ - o o , oo], then /(x) - x / o r x ET(g ).

Proo/. Since g E C[ - o% oo], ~(g) is a closed interval a n d / defines a homeomorphism of 7(g) onto itself. Because ] E C [ - oo, oo] the finite endpoint of 7(g) m u s t be m a p p e d onto a finite point so it m u s t be m a p p e d onto itself since/]y(g) is a homeomorphism.

Our result now follows from Corollary 2.1.

I t is possible to obtain information concerning the existence of solutions /E R[a, b]

for/n(x) =g(x) b y studying the sets 7(g~). Thus, for example, the fact t h a t the function g(x) = - x, x E [0, - 1], a n d g(x) = - x 2, x E [0,1], possesses only one cycle of order 2, n a m e l y [1, - 1 ] , implies t h a t S(2,g) is e m p t y . I n fact, Isaacs [5] h a s stated necessary a n d sufficient conditions for S(2,g) to be non e m p t y in terms of the cycles of g.

Unfortunately, these results give no information a b o u t S(2,g)N D[a,b] except, of course, in the case where S(2,g) is e m p t y .

We now display a function g E C [ - oo, oo] for which S(2,g) rl C [ - o% oo] is e m p t y b u t S(2,g) N D [ - oo, ~ ] is not e m p t y .

E x a m p l e 1. We first define the functions h , / , g.

We define h on [0, 1]: h ( 1 / n ) = ( - 1 ) n n = 1, 2, ...;

h'(x) = ( - 1)n 2n(n + 1 ) x E ( 1 / n + 1, 1/n), n = 1, 2 ....

We define / on [ - oo, oo ]:/(x) = x, x ~ 0;/(x) = 0, x >/2 a n d 0 ~< x ~ 1;

/(x) = x - 1, 1 < x < ~; /(x) = (x - 88 h(4x - 5)/2 + ~, ~ < x < 3;

l ( x ) = - ~ + 1 ~(x ~),-~<x~<2. _ 3

We define g(x)=/2(x) for x E [ - o% oo]. C l e a r l y / E D [ - ~ , ~], g ~ C [ - ~ , ~ ] , and it remains only to prove t h a t S(2,g)N C [ - ~ , ~ ] is e m p t y . Assume / E S ( 2 , g ) N C [ - ~ , ~ ] . Then b y Corollary 2 . 2 / ( x ) - x for - ~ ~<x ~<0. Then for ~11 x, such t h a t /(x)<O, we h a v e g ( x ) = / ( / ( x ) ) = / ( x ) . Thus, /(x)>~O in [0,1]. We assert t h a t there exists ~ > 0 such t h a t / ( x ) ~ 0 for x E[0,~]. Assume / ( x ) ~ 0 on [0,1] and define a = maxt0.1J /(x). Since g ( x ) = / 2 ( x ) = O for x e [ 0 , 1 ] it is clear t h a t / ( x ) = O for xE[0,a].

Thus, if g ( x ) = / ( f ( x ) ) > O t h e n / ( x ) > ~ . Since /((~)=0 a n d g ( x ) < x for a~l x > O , it is clear t h a t / ( x ) < x for all x > 0.

We define a ( n ) = l + 8 8 1 8 8 Then if n is odd we have g(x)>g(a(n)) for all 0 ~<x <a(n). Thus, for n o d d / ( a ( n ) ) =g((~(n)), a n d it follows t h a t / ( x ) =- g(x) whenever /(x) or g(x) is negative. Thus,/(a(n) < 0 for n odd. B u t for n even ](/(a(n))) =9~a(n)) > 0 a n d so/(a(n)) >~. Since limn_~a(n) = 1 + 88 it follows t h a t / is discontinuous at 9 = 1 + ~.

This completes E x a m p l e 1.

Consideration of the above example suggests the restrictions on g E C[a, b] -which will insure t h a t the solutions of/~(x) =g(x) also belong to C[a,b]. This result is contained in the following theorem.

Theorem 2.4. I~ g e C[a, b] and i / e i t h e r (a) or (b) below are satisfied then S[n, ff] N D[a, b] = S[n, g] N C[a, b].

(a) Range o / g = [ a , b ] .

(b) g is not constant on a n y non degenerate interval.

359

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J. C. LILLO, The functional equation fn(x) = s

Proo/. Assume (a) is satisfied a n d / E S ( n , g ) ~ D[a,b]. T h e n t h e range o f / = [ a , b ] . L e t h(x) d e n o t e / ~ - l ( x ) . T h e n range h=[a,b] a n d hED[a,b]. L e t / be d i s c o n t i n u o u s at z. Thus, there exists a sequence {Xl} tending to z such t h a t no subsequence of {/(x~)} converges t o / ( z ) . One m a y also assume t h a t I x ~ - z l > IX~+l-Zl for all i a n d t h a t t h e sign of ( x ~ - z ) is i n d e p e n d e n t of i, s a y negative. W e n o w define a sequence {yj} converging to a p o i n t y such t h a t h ( y j ) = z for ?" odd, a n d for ] even (h(yj)} is a subsequence of {xi}. Since [a,b] = r a n g e of h there exist Yl a n d Y2 such t h a t h ( y l ) = z a n d h ( y 2 ) = x 1. I f Yl a n d Y2 are b o t h finite define (~=(yl+Y2)/2. I f either Yl or Y2 is infinite, let ~ be a n y p o i n t in (Yl,Y2) for which l y l - a I ~>1 a n d l y 2 - a I >~1. I f h ( a ) = z set y3=(~ a n d let Y4 be a n y p o i n t in [Ya, Y2] for which h ( y 4 ) = x 2. I f h(~)>z~let Ya be a n y p o i n t in [0, Y2] for which h(y3) = z, a n d Y4 be a n y p o i n t in [Y3, Y~] for which h(y4) = x 2.

I f h ( a ) < z let Y3 =Yl. Since h ( a ) < x k for some k ~>2 let y4E [y3,a] be a n y p o i n t for which h(ya)=x k. Using Y3, Y4 in place of Yl,Y2 a n d x2 or xk in place of x 1 we r e p e a t the procedure. I n this w a y we o b t a i n a sequence (yj} with t h e s t a t e d properties.

B u t t h e n / ( z ) =limk._)~/(h(y~+l) ) =g(y) =limk_,:r

g(Y2k) ~/(z).

T h u s / ( x ) e C[a, b].

Assume n o w t h a t (b) is satisfied. Since g is n o t c o n s t a n t on a n y interval a n d /n(x)=g(x) we h a v e t h a t f ( x ) , i = 1 ... n is n o t c o n s t a n t on a n y interval. L e t / be discontinuous at z a n d set r =/(z). Thus, there exists a ~ > 0 such t h a t either for a n y w E [ r , r + a ] or for a n y w E [ r , r - a ] there is a sequence (xi}-->z such t h a t /(x~)=w.

Since h is n o t c o n s t a n t in a n y interval we m a y choose in [ r , r + a ] or in [ r , r - a ] , whichever is necessary, a w such t h a t h(w) :~h(r). B u t t h e n h(w) =limg(xi) =g(z) = h ( r ) which is n o t possible. Thus, / is continuous.

Corollary

2.3. Let g(x) be a real analytic/unction in ( - c~, c~) which is not the con- stant/unction. L e t / be defined on ( - ~ , ~ ) into ( - c o , ~ ) , possess t h e D a r b o u x p r o p e r t y there, a n d satisfy/n(x) =g(x). T h e n / is continuous in ( - ~ , oo).

Proo/. Clearly g satisfies condition (b). H o w e v e r , g need n o t belong to C[ - ~ , ~ ] n o r / t o D [ - ~ , ~ ] . I t is, however, easily verified t h a t T h e o r e m 2.4 is valid for t h e open interval ( - ~ , ~ ) .

W e n o w consider the e q u a t i o n / n ( x ) =/m(x), m = n + p , for / E D[a,b]. L e t R ~ d e n o t e the range of f a n d R ~ = [a,b]. R i is c o n n e c t e d a n d so is a n interval. Also R i+ 1 = R ~ for all i, a n d since/n(x) =/m(x), it is clear t h a t R n = R ~+1 = R ~+~ for i = 1 , 2 ... I f R n is a p o i n t c t h e n / ~ ( x ) =/n+l(x) = C a n d we h a v e one of t h e exceptional cases of Theo- r e m 2.4. I f n = 1, we h a v e t h e case studied b y E w i n g a n d U t z [2]. T h e following t h e o r e m extends their results.

Theorem

2.5. A necessary and su//icient condition that/E D[a, b/satis/y In(x) =/n+P(x) /or all x E [a, b], where p and n are minimal, is that either (a) or (b) is saris/led:

(a) p = 1, there exists a sequence of intervals R ~,i = 1 ... n such t h a t / I R~ = R~ + 1, ] I Rn = R n , / ( x ) = x for x e R ~, R ~ #= R i + 1 for i = 0 ... n - 1 a n d R ~ = [a, b].

(b) p = 2 , the sequence R ~ is as a b o v e except t h a t / I R ~ is a reflection.

Proo/. A s s u m e / e D [ a , b ] a n d satisfies the e q u a t i o n / n ( x ) = / n + P ( x ) for all xE[a,b].

Set R ~ [a, b / a n d R t = r a n g e of f . T h e n if in T h e o r e m 2.2 we set S = R ~, a n d a s s u m e S non-degenerate our result follows. Conversely, if / satisfies either (a) or (b) it is clear t h a t / n + ~(x) =/n(x) for all x e [a, b].

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ARKIV FOR MATEMATIK. B d 5 n r 2 6 A C K N O W L E D G E M E N T

This research was sponsored b y National Science F o u n d a t i o n Grant G18915.

Mathematics Dept., Purdue University, Lafayette, Indiana, U.S.A.

R E F E R E N C E S

1. BODEWADT, U. T., Zur I t e r a t i o n reeller Funktionen. Math. Z. 49, 497-516 (1944).

2. EwI~G, G. M., and UTZ, W. R., Continuous solutions of fn(x)=f(x). Can. J. Math. 5, 101-3 (1953).

3. ISACCS, R., I t e r a t e s of fractional order. Can. J. )~lath. 2, 409-16 (1950).

4. LOJAS~EWmS, S., Solution g6n6rale de l'6quation fonctionelle f ( f ( . . . f i x ) . . . ) ) = g(x). Ann. Soc.

Polon. Math. 24, 88-91 (1952).

5. ~ASSERA, J . L., a n d PETR~CCA, A., On t h e functional equation f(f(x))= l/x. Revista Union Mat. Argentina 11, 206-2II (1946).

Tryckt den 3 september 1964 Uppsala 1964. Almqvist & Wiksells Boktryekeri AB

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