On a functional-differential equation related to Golomb's self-described sequence

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

Y.-F. S. P

ÉTERMANN

J.-L. R

ÉMY

I. V

ARDI

On a functional-differential equation related to

Golomb’s self-described sequence

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{1 (1999),}

p. 211-230

<http://www.numdam.org/item?id=JTNB_1999__11_1_211_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

On

a

functional-differential

equation

related

to

Golomb’s self-described

_sequence

par Y.-F.S.

PÉTERMANN,

J.-L.

_RÉMY,

et I. VARDI

RÉSUMÉ.

L’équation

différentielle fonctionnelle

_f’(t)

a des liens étroits avec la suite auto-décrite F de

Golomb,

1, 2, 2, 3, 3, 4, 4, 4 5, 5, 5, 6, 6, 6, 6, ....

1, 2, 2, 3, 3, 4,

Nous décrivons les solutions croissantes de cette _équation. Nous montrons

_qu’une

telle solution

_possède

nécessairement un _point

fixe non

négatif,

_{et que pour}

chaque

nombre _p ~ 0 il _y a

ex-actement une solution croissante _{ayant p pour point} fixe. Nous

montrons

_{également qu’en}

_général

une condition initiale ne déter-mine _pas une solution _unique: les courbes représentatives de deux

solutions croissantes distinctes se croisent en effet une infinité de

fois. En

_fait,

nous _{conjecturons que} la différence de deux solutions

croissantes se _comporte de

_façon

très similaire au terme d’erreur

E(n)

dans

_{l’expression}

_asymptotique

_{F(n) =}

_~2-~n~-1

+

_E(n)

(où ~

est le nombre

_d’or).

ABSTRACT. The functional-differential _equation

_f’(t)

_=1/f(f(t))

is

_closely

related to Golomb’s self-described sequence F, 1, 2, 2, 3, 3, 4, 4, 4 4 5, 5, 5,

1, 2, 2, 3, 3, 4,

We describe the

_increasing

solutions of this _equation. We show that such a solution must have a

_nonnegative

fixed _point, and

that for _every number _{p ~} 0 there is

_exactly

one

_increasing

so-lution with _p as a fixed _point. We also show that in

_general

an

initial condition doesn’t determine a unique solution: indeed the

graphs

of two distinct _increasing solutions cross each other

infin-itely

many times. In fact we conjecture that the difference of two

increasing

solutions behaves very

similarly

as the error term

_E(n)

in the _{asymptotic expression}

_~2-~n~-1

+

_{E(n) (where ~}

is the

_golden

_number).

1. INTRODUCTION.

For a real number _p we are

considering

the functional differential

_equation

with initial condition

_{f (p)}

_{= p.} We came across this

_equation

while

study-ing

a self-described _sequence of

_positive

_{integers usually}

referred to as

"Golomb’s

_sequence",

_by

the name of S.W. Golomb who rediscovered it

in 1966

_[Go].

We _say that a _sequence

("word")

W of

_positive

_integers

is

self-described

or

self-generating

if

T(W)

=

W,

where

T(W)

is the _sequence

consisting

of the numbers of consecutive

_equal

entries of W. Golomb’s

- . - - - - . "’.... - - -

-sequence P’

is the _sequence

,

w i m J m

the

_only

_{nondecreasing}

self-described sequence

taking

all

_{positive integral}

values.

In

_[Go]

Golomb asks for a

proof

of the

asymptotic

equivalence

where 0

denotes the

_golden

number. One such

_proof

can be found in

[Fi].

Another _one, based on a heuristic

_argument

of D. Marcus

[Ma]

and later

completed

in

_~Pe~,

establishes in

_particular

the fact that

where

_f

is _any

_positive

solution of

_(1).

In his

_argument

Marcus mentions the solution of

_{(1) f+(t) _}

_~2-~t~-1;

note that this function has two fixed

points,

p = 0 and

p =

_0.

It seems natural to ask whether Marcus’ solution is the

_only

one of

₍₁₎

with initial condition

_{f (0)}

= 0. The _{answer to} this

question

is yes, and

proving

it is

_proposed

as

_problem

10573 of the A.M.S.

_Monthly.

But it

also seems natural then to ask whether other solutions of

₍₁₎

than Marcus’

exist,

and if so how

they

behave. We

eventually

realised that answers to

these

_questions

were too voluminous for a

"problems

section" such as in the

A.M.S.

_Monthly,

and that at least a

large

note was _{necessary to treat} them.

In this _paper we describe all the

increasing

solutions of

(1).

We _prove

that for each

_{p &#x3E;}

0 there is

_exactly

one

(increasing)

solution

f

of

(1)

with

f (p)

=

p

(Theorem 1),

and that there are no other

_increasing

solutions than

those

_having

_{(at least)}

one

_nonnegative

fixed

_point

(Theorem 2).

We

long

thought

(and

tried to

_prove)

that solutions of

₍₁₎

satisfied a more

_general

unicity condition,

i.e. that we could have

f (p)

=

q for at most one solution

of

_(1),

for _{every real} numbers _p and _q, and not

_{only for q}

=

p &#x3E; 1. This

would have

_implied

of course that two distinct solutions could never cross

each other. But in fact the

_unicity

condition is not satified in a _very

_strong

sense: indeed each

_pair

of distinct

_increasing

solutions cross each other

infinitely

often

_{(Theorem 3).}

But are there other solutions than

increasing

ones? We shall see below

("Preliminary

remark

_3.3")

that a solution is

_{necessarily strictly}

monotone

on an interval on which it is defined. We first

thought

there were no

de-creasing

solutions,

until we discovered

f _ (t)

=

-~-~-1 (-t)-~;

note that this function has the fixed

_point

_p

_{= -1~~.}

It is not our _purpose to also

treat

_decreasing

_solutions;

we shall nevertheless see that the existence of

many other such solutions is not difficult to establish: see the last section

of the _paper.

Note added in

_May,

1999. Until well after the _{present paper} was written we

were unaware of the existence of

[McK],

which was

brought

to our attention

by

Berthold

_Schweizer,

of the

_University

of Massachusetts at Amherst. In this work M.A. McKiernan establishes the existence of a

complex

solution

of

₍₁₎

_satisfying

_{f (p)}

=

p, for every

complex

number p

0,

and

announces the

_{presentation,}

in a future article

(which

to our

knowledge

has

not been

_published),

of the extension of this result to all _p

_{with Ipl}

&#x3E; 1. He doesn’t address the

_problem

of the

_unicity

of a solution with a

_given

initial

condition.

Last remark on Golomb’s _sequence. The sequence

F(n)

was

_recently

stud-ied in _{several papers,} and a much more

precise asymptotic

estimate than

Golomb’s

₍₂₎

is now known. I. Vardi

[Va]

first

_proved

that the error term

associated with

₍₂₎

is «

n~-1/

log

n. He also

proposed

two

_conjectures,

which were

_{proved respectively by}

J.-L.

_Remy

and

_by

Y.-F.S. P6termann

and J.-L.

_Remy,

and which can be summarized as follows

(putting

F(t)

=

F([t])

if t is _{not an}

integer).

Theorem 0

_([R6]

and

_[P6R6]).

We have

_{for t &#x3E;}

2

where the real

_functions

h is

_continuous,

not

_identically

_zero, and

_satisfies

h(x) = -h(x

+

₁₎

_{for x}

&#x3E; 0.

It is

_interesting

to note

_that,

while the main term is a solution of the functional differential

_equation

_(1),

the error term associated with

₍₂₎

(i.e.

the second and third terms on the

_right

of

(3))

satisfies an

_approximate

functional-integral

equation

As we

_mentioned,

the function

f+(t) _ ~2-~t~-1,

which is the main term

on the

_right

of

(3),

is an

_increasing

solution of

(1).

We also announced

solutions of

₍₁₎

cross each other an

infinity

of times. We think these

cross-ings

occur _very

_regularly,

in a

_pattern

that can be described

_similarly

as

the error term in

_(3).

Conjecture

0.

_{If f is}

an

_increasing

solution

_of

(1)

which is

_different

_from

where the real

_functions

_{1] is}

_continuous,

_satisfies

_{~(x) _ -q(x}

+

_1),

and vanishes

_exactly

once on

[0,

1).

Acknowledgements.

This work was

partly

done while the first author was

visiting

the second and

_{supported by}

the Université Henri Poincar6

_Nancy

1 and

_by

the INRIA Lorraine. The authors are _very

_grateful

to these insti-tutions for their

_help.

2. STATEMENT OF THE RESULTS.

We

_adopt

the convention that

_f

is a maximal solution

_(MS

for

_short)

of

₍₁₎

above if the

_following

conditions are satisfied.

(a)

It is defined and continuous on some interval I

_containing

_p, but not

only

p.

(b) f(I)

C I.

(c)

Equation

(1)

is satisfied for

_{every t}

in I

_(where

_by

convention we write

f’(t)

=00

if f ( f (t)) = 0).

(d)

f

cannot be extended to a

_larger

interval J

_containing

I and where

Conditions

_{(a), (b)}

and

_(c)

are satisfied.

The main result of this paper is

Theorem 1. For _every real

_{number p &#x3E;}

0 there is

_exactly

one MS

f of

(1)

with

_{f (p)}

_{= p.}

The

_proof

_splits

_up into three cases

depending

on whether _p

_1,

_p =

1,

or _p &#x3E; 1. For _p &#x3E; 1 the

_general

idea is

_{fairly simple:}

First one notes that the functional differential

_equation

₍₁₎

_implies

that all

_higher

derivatives

f(k)(t)

of

_{f (t)}

can be

_expressed

as rational functions of iterates of

f (t).

It

then follows that if p is a fixed

point

of

f (t),

then is a

uniquely

defined function of _p. This then

_implies

that the

_Taylor

series of

_{f (t)}

at p is

uniquely

defined in terms of

_(computable)

rational functions of _p. To show that a solution exists and is

_unique

one then has to _prove

(2)

The

_Taylor

series defined in this _{way converges in} some

_{neighbourhood}

of _p.

(3)

A solution of

₍₁₎

is

_analytic

in this

_{neighbourhood.}

Lemma 2 below shows that the case _p 1 can be reduced _{to p} &#x3E; 1. For _p = 1 _we have not been able to prove

(2)

directly,

_so another method is

_required.

It is not difficult to

_verify

_(see

_Preliminary

remark 3.3 next

section)

that all the solutions of

₍₁₎

with

_{p &#x3E;}

0 are

_increasing.

But are

all the

_increasing

solutions of the functional differential

_equation

₍₁₎

the functions described

_by

Theorem 1 and its

_proof?

It is a

_priori

conceivable

that some other

_increasing

solution

(i.e.

with no fixed

point)

_might

exist. We show that this is not the case.

Theorem 2.

_{If f}

is a solutions

of

(1)

which is

increasing,

and rraaximal

(in

the sense that its interval

_{of defcnition}

I

satisfies

the conditions

(b), (c),

and

_(d)),

then

_f

has a

(nonnegative)

fired

point.

We shall see in the

proof

of Theorem 1 that if 1 _q

q’,

and if

fq (t)

is the solution of

₍₁₎

with _p = _q, with

Iq

its domain of

definition,

then for

0 t q’

and t E

_Iq

we have

fq(t)

But this doesn’t remain true

for all t &#x3E;

_q’,

In fact the

_graphs

of two

_arbitrary

distinct

_increasing

solutions

cross each other

infinitely

_many times.

Theorem 3.

_{if f}

and _g are

_increasing

maximal solutions

of

(1),

then there

is a _sequence

of

numbers _si -3 00

(i

-~

_oo)

with

_{f (si)}

=

g(si).

3. PRELIMINARY REMARKS.

3.1. A maximal solution of

₍₁₎

with _p = 0 _or with

p =

~? where 0

denotes the

_golden

number and I =

[0,

+oo)

is

A maximal solution of

₍₁₎

with _p

_{= -1~~}

and I =

(-oo, 0)

is

The solution

_f-

is

_clearly

_maximal;

to see that

f+

is

_maximal,

see Remark

3.5.

3.2. If

_f

is a MS of

(1)

then

f’(t) ~

0 for

_{every t}

in I.

(Otherwise f ( f (t))

would not be

_defined).

3.3. If

_f

is a MS of

(1)

then

f’

/ remains of the same

_sign

on

I,

whence

the function

_f

o

f

cannot

_change

_sign

on I. It also follows that

_f

is

_strictly

monotone on I.

Suppose

on the

_contrary

that

f’

takes

_positive

and

_negative

values on

such that

_{f (tl) - f (t)}

and

_{f (t2) - f (t)}

are of the same

_sign.

So for

in-stance we have &#x3E;

_{f (t)}

and

_{f (t2)}

&#x3E;

f (t).

Let T be in the interval

and define I- .

f -1(T)

fl and

I+

:=

f -1 (T)

n

(t,

t2l.

The sets I- and

_I+

are not

_empty,

as

f

is

con-tinuous. And we can choose T with 0 since

f’

cannot vanish

_by

Remark 3.2. Now we

put t’

_{:= sup} I-

and t2

=

inf I+. Again

by

using

the

continuity

of

_f

we

have ti

E I-

_{and t2}

E

_I+.

This means

that t’

t

_t2,

that =

f (t2)

=

T,

and that

f (s)

T for

every s in the interval

(~2)-And,

again

since

_f’

cannot

_vanish,

we must have

_{f’(t[ )}

0 and

_f’(t2)

&#x3E; 0. But this is in contradiction with

_f’(ti)

=

f’(t2)

=

1/ f (T),

since

f (T) ~

0.

3.4.

_{If f}

is a MS

of (1)

and

if f’(to)

= _oo for _some real number _to in

I,

then

to is the

_largest

lower bound or the smallest _upper bound of the interval I.

Indeed if a

_{neighbourhood}

of _to is included in

_I,

then with the

_help

of

Remark 3.3 we infer that a

neighbourhood

N of

To

= is included in

I,

that

_{f (To)}

=

0,

and that

f (T)

has _a constant

sign

in N. The relation

f’(To)

=

0,

being

excluded

by

Remark

3.2,

_{we now} reach _a contradiction very

similarly

as in the

proof

of Remark 3.3.

3.5. A MS

_f

of

₍₁₎

is

_indefinitely

differentiable in I.

This will follow from

_expression

₍₄₎

in Section 4

_below,

where

_by

conven-tion we write = _oo

if f, (t) :

_:= = 0 for some t with 2 ~ With the

_help

of Remarks 3.3 and 3.4 we see that this can in fact

_possibly

occur

_only

at a _to

satisfying

=

0,

that is at _to = inf I or _to =

sup 7.

3.6.

_{If f}

is a MS of

(1)

and

if f (a) &#x3E;

0 and

f’(a)

&#x3E; 0 for some _a, then

f (t)

is

_positive

and

_{strictly increasing}

for all t in I

_exceeding

_a, so that

f’(t)

is

positive

and

_(strictly)

_decreasing

for all t in I

_exceeding

a.

3.7.

_{If f is}

an

_increasing

MS of

(1)

and if I is unbounded

above,

then so

is f (I).

Indeed the

_assumption

0

_{f (f (t))}

A for all t leads to a direct

contra-diction,

as then

f’ (t) 2:

1/A

for all t.

3.8. Let a solution

_f

of

(I)

have a fixed

_point

_p. Then we have the

follow-ing.

(i)

0 then

_f

can have no fixed

point

of the other

_sign.

Indeed the derivatives of

_f

at fixed

_points

of

_{opposite signs}

are of

opposite signs.

(ii)

If p

0,

then

_f’(p)

=

1/p

0,

f

is

_decreasing

on

I,

and thus

f

can

have no other fixed

_point.

(iii)

If p &#x3E;

1,

then

_f’(p)

=

l/p

1,

f’(t)

is

_strictly

_decreasing

for t &#x3E; _p,

(iv)

and

_{if f}

has a

larger

fixed

_point

_P2, then _p2 &#x3E; 1. Indeed since

_f’(p)

=

I/p

_&#x3E;

1,

if _we

suppose that p2 is the next fixed

point

of

_f

we must have

_{f (t)}

&#x3E; t for _p t _p2, whence

_f’(p2)

1 and

_{p2 &#x3E;}

1. And _p2 = 1 is ruled _out since then

f’(1)

= 1 and

by

Remark 3.6

_f’(t)

&#x3E; 1 for _p t

_1,

which is in contradiction with

f(p)=pand f(1)=1.

(v)

If p = 0

then

_f

can have no

_negative

fixed

_point.

Note that

_f’(0)

=

oo, so that

by

Remark 3.4

f (t)

is defined either

on some

_nonnegative,

or on some

_nonpositive,

values of t. If it is

on

_nonnegative

values

(like

_f+

of Remark

3.1),

then

_f

can have no

negative

fixed

_point.

If it is on

_{nonpositive values,}

then

_by

Condition

(b) f (t)

is

_always

_negative,

and so is

f’(t) ( f’(t)

= _oo

(or "-oo")

being

allowed),

so that

_f

is

_{strictly decreasing}

and

_negative.

But this is not

compatible

with

_{f (0)}

= 0.

(vi)

From all this it follows that

_{if p}

is

_negative

or

if p

=

1,

then

_f

has

no other fixed

_point,

that if 0 _p 1 then

_f

can have at most one

other fixed

_point

_p2, which must be

_larger

than

_1,

and that

then

_f

can have at most one other fixed

_point

_pl, which must

_satisfy

0pl1.

3.9. If

_f

is a MS of the differential

equation

(1),

then

(by

Remark

3.3)

the inverse function

_{f -1}

is defined on the interval

f (I).

So for t E

_{f (I)}

the function

_f

satisfies the indefinite

_integral

_equation

To see

_this,

note that in

_general,

if a and b

_belong

to

_{f (I),}

we have

where

_k(t)

is an

integrable

function.

Equation

(1’)

follows

by putting

k - 1. But also note that

_(1’)

is not

_{quite equivalent}

to

_(1),

since in

general,

for a solution

f

of

(1)

the inverse function

f -1

is not

necessarily

defined

_everywhere

in I

_{(i.e f (I)}

can be

strictly

included in

I).

4. PROOF OF THEOREM 1.

With the two first lemmas we

_{begin by}

_showing

that a MS of

₍₁₎

with

some _p

_satisfying

0 _p 1 is also a MS of

(1)

for some _P2 &#x3E; 1. First

we show that I contains no t _p, which

_by

Condition

_(a)

_implies

that it contains some t &#x3E; _p.

Lemma 1. If

_f

is a MS of

(1)

with 0

_p

_1,

then I contains no number

Proof.

If _p = 0 this follows from Remark 3.8

(see

the

proof

of

(v)).

If 0 p 1 and to p

belongs

to

I,

then

f (to)

to since

_by

Remark 3.8

_f

has no fixed

_point

smaller than _p. So if we note ao = _to and =

f (ai)

for i &#x3E;

_0,

the _{sequence ai} is

_decreasing

and

_entirely

in I. If it

_eventually

becomes

_negative,

then there is a

_{To &#x3E;}

0 with

f (To)

=

0,

and thus there is also a

Tl

with

To

T1

_p,

f (Tl )

=

To,

and

f’(Ti)

=

oo, which is in

contradiction with Remark 3.4.

So ai

&#x3E; 0 for all

_i,

and a :=

0 exists. If a is in I then

f (a)

=

limi-+oo

f (ai)

=

a, and a is a fixed

point

smaller than _p, a contradiction. If a is not in I we define

f(a) =

limi-+oo

f (ai),

and on the one hand we have a =

f (a)

=

f ( f (a)),

and _on the other

_hand,

since

_f’

is

_decreasing

on

(a, oo),

f’(a)

exists and is

_equal

to

_f’(ai).

It follows that we can extend the definition of

_f

at a in

such a _way that Conditions

(a), (b)

and

(c)

be satisfied on J := I U a

contradiction to Condition

_(d).

D

We are now in

_position

_{to prove} the

following.

Lemma 2.

_{If f}

is a MS

_of

(l~

with 0 _p 1 there is

_exactly

one other

fixed

point

P2

of f ,

which is

larger

than 1.

Proof. By

Remark 3.8 above it is sufficient to show that

_f

has another fixed

point.

By

Lemma 1 there is some

_to

&#x3E; _p such that

_f

is defined on

[p, to].

We

_put

as before _ao = _to and =

f(ai)

for i &#x3E; 0. Then I contains

IP, ai]

for every i. If no such interval contains a fixed

_point

_larger

than _p,

then _ai+1 =

f (ai)

_{&#x3E; ai} for

every i. This

implies

that the sequence

of ai

is bounded

_(if

it

_weren’t,

then

_f’(ai)

would tend to

_0,

and

_{f (ai)/ai}

could

not remain

_larger

than

_1).

Thus a :=

limi,,,.

_ai exists. If a is in

_I,

then

f (a)

= _a and _{we are} done. And the

assumption

that a is not in I leads to

a contradiction as in Lemma 1: we

defines

f (a)

=

ai and we infer

that

_f

satisfies

₍₁₎

at t = a. This concludes the

proof.

0

Now we _prove Theorem

_1,

_{considering separately}

the three cases _p &#x3E;

_1,

0

Case 1: _p &#x3E; 1. In the next three lemmas we first establish the existence of

a solution and describe its domain of definition I.

Lemma 3.

_{If p}

&#x3E; 1 then there is a

_function

f (t)

_defined

in the interval

(p -

P2(p -

1)~2, p

+ p2(p -

1)/2)

and

_satisfying

₍₁₎

there.

Proof.

Let

_{f (t)}

_satisfy

the

_equation

₍₁₎

and define

_f1(t)

=

f (t)

and

_fk+1(t)

= _1.

Also,

one has

and

_similarly

for

_{any k}

&#x3E; 1

where the sum runs over a finite collection

_Sk

of sets of

_exponents

a2, ~ ~ ~ , ak+1, and where the

C(a2, ... , ak+1)

are

_positive

constants.

_Now,

provided t

is such that

_{fk (t)}

&#x3E; 0 for

_{every k &#x3E;}

2

_(it

follows from Remark 3.5 that this will be the case if

f2(t)

&#x3E;

_0),

we have

Now note that when a term under the sum of

(4)

is

_{differentiated,}

the

exponent

bi

of

_{f i (t)}

_appearing

in one of the

resulting

terms

_always

satisfies

ai

b2

ai + 2: it follows that max ai 2k. Hence if we

_{put t}

= _p in the

expression

above we obtain

that is

So the

_Taylor

series for

_f

has a radius of _convergence at least

_{P2(p - 1)/2.}

Now consider

the formal _power series solution of +

_s)

= ₊

s)),

with

h(p)

(p)

=

p. Then

_h(p)

(p

+

_s)

is also its own formal

Taylor

series at s = 0. Note that the above

_computation

is valid at t =

p if we

_replace

_{f by}

_h~p~.

It

follows that this

_Taylor

series has a radius of _convergence at least

_p2(p-1)/2,

and thus

_represents

a solution of

(1)

within that radius.

Lemma 4.

_{If p}

&#x3E; 1 then there is a

function

f (t)

defined

in the interval

Proof.

Consider the function

_f

of Lemma

_2,

defined and

_satisfying

₍₁₎

on

Ko

:=

~p, Po~,

where

_Po

_p

-I- p2(p -

_1)/2,

_and

_put

for t in

_Ko

Since

_f

is

_{strictly increasing}

its inverse function

_{f -1}

is well defined on

f (Ko).

So for t in

_{f (Ko)}

we have

.. 1 ;, - ., ...

The inverse

_{function g-1}

is thus well defined on

Ko,

=

f (v) .

there. But since

_{f (t)}

&#x3E; 1 for t &#x3E; _p we see from

(6)

that in fact the inverse

function

_g-1

is well defined on

g(Ko)

=

f-1(Ko)

_=:

Kl,

where

Kl

is _an interval

_{containing Ko}

and

_larger

than

_{Ko. Moreover,}

since for t in

_Ko

we

infer from

₍₆₎

that

_g’(v)

=

f (v),

_we have for t in

K,

Thus the

_{function g-1}

coincides with

_f

on

Ko,

and satisfies

(1)

on

_Kl.

We

write

_f(t)

:= for t in

Ki.

Continuing

in this _way we obtain a _sequence of intervals

_Ki

=

~p, Pi]

(i

=

0,1,2,... ),

with

Pi+1

_&#x3E;

Pi,

on each of which the function

_f

of Lemma

3 can be extended and still

_satisfy

(1).

And we have

f (Pi+1)

=

Pi.

To conclude the

_proof

of the lemma we have to ensure that the sequence of

Pi

is unbounded. If it is bounded we can do as in Lemmas 3 and

_4,

that

is _put P =

limi-+oo

Pi

and

define

f (P)

=

f (Pi)

=

limi-+oo Pi

= P. We then

_easily

see that

f

still satisfies

(1)

at t = P. But

f (P)

=

P,

_a

contradiction to Remark 3.8. 0

Lemma 5.

_{If p}

&#x3E; 1 then there is a

function

f (t)

defined

in an interval

[Q, p]

and

_satisfying

₍₁₎

_there,

where either

_Q

is a

_fixed

_point

_{of f}

with

satisfies

f(Q)

= Z and

f ( f (Q))

=

f(Z)

= 0

for

_some Z with

_Q

Z 0.

Proof.

This time we use the fact that the function

f

of Lemma 3 is defined

and satisfies

₍₁₎

on

_Jo

:=

(Qo, p~,

where

Qo

&#x3E;

p - p2(p -

_1)/2.

As above

we define

_g(t)

_by

₍₆₎

for t in

_Jo.

_Similarly

as before we obtain

_recursively

a

sequence of intervals

Ji

=

[Qi, p]

(i

=

0,1,2,...

),

with

Qi+1

Qi,

_on each of which the function

_f

of Lemma 2 can be extended and still

_satisfy

(1),

and with

_{f (Qi+1 )}

=

Qi ~

But here the construction

might

end after _a finite number of

_steps:

indeed if for some i we obtain

f (Qi)

0 the

_expression

(6)

does not

_produce

_anymore a well defined function on

Ji.

So we consider

I. At each

_step

we have

f (Qi)

&#x3E;

_0,

so that the construction never ends. Since

_f’(p)

=

1 j p

1 _we

may suppose that

f (Qo)

&#x3E;

_Qo.

Then if for some i &#x3E; 0

f (Qi)

&#x3E;

_Qi,

we have

f-1(Qi) _

_Qi+1

_Qi

whence

I(Qi+1) =

Qi

&#x3E;

_Qi+,.

Thus

_{f (Qi)}

&#x3E;

_Qi

for each i.

_(Note

that this will also be true for each i

_io

in the case where the construction

ends at

_Jio).

Now since

_{f"(t) =}

_{-(f3(t)f23(t))-’}

is

_negative

at least for all t for which

_{f (t)}

&#x3E;

_0,

that is in

_particular

on _every

.li,

f’

is

decreasing

on _every

_Ji,

and thus the _sequence of

Qi

cannot

_diverge.

So

put

Q

:=

limi-4oo Qi.

_Again

we define

f (Q)

=

limi_

f(Qi) .

We then

_easily

see that

f (Q)

=

Q

and that

f

still satisfies

(1)

at t

_{= Q.}

And

_by

Remark 3.8 we must have

_{Q &#x3E;}

0.

II. At the i-th

_step

of the construction we reach a

_Qi

with

f (Qi)

0. So there is a Z &#x3E;

Qi

with

f (Z)

= 0. Note that Z is

negative

since

Qi Z

Qi-l

= 0. Since

f

is well defined and

nonnegative

on

~Z, p~,

_expression

(6)

produces

a

function g

well defined on

~Z, p~,

and whose inverse

_coinciding

with

_f

on

[Z,p],

is well defined on

[Q

:=

g(Z),p]

and satisfies

(1)

there. So we _may extend the definition

of

_f

on

(Q, p~

_{by putting}

_f

= there. And _we have

f (Q)

=

Z,

_as claimed.

0 This concludes the

_proof

of Lemma 5. So if _p &#x3E; 1 we established the

existence of a

_{function f satisfying}

(1)

in an interval

~Q, oo)

where

_Q

is either a fixed

_point

for

f

with 0

Q

_1,

or is such that

f’(Q)

= _oo. With Lemma 1 and Remark 3.4 this shows that

_f

is a MS of

(1).

To _prove the

_unicity,

we first note that the

_proof

of Lemma 2 shows that any solution

h(t)

of

(1)

must have at t =

p the same

Taylor

series as

_{f .}

If h is

_analytic

in some interval

[Qo, Po]

with

_Qo

_p

_Po

then h =

f

there. In order to

_verify

_this,

note in addition that there is some _po with

p &#x3E; p - po &#x3E;

_1,

such

_{that if p &#x3E; t &#x3E; p - po}

then

_h(t)

&#x3E; t &#x3E; 0 and

_h,’(t)

&#x3E; 0: this follows from the facts that 0

_h’(p)

=

1/p

1,

that h’

is

_continuous,

and that h has no other fixed

_point

p’ &#x3E;

1. It follows that if

hk-1(t)

&#x3E; t

_{&#x3E; 0 for p &#x3E;}

t _{&#x3E; p - po then}

_hk(t)

&#x3E;

h(t)

&#x3E; t &#x3E; 0 for _{p &#x3E; t} &#x3E; _{p - po.}

_Thus,

for some _po &#x3E;

_0,

the functions

_hk

all

_satisfy

hk (t)

&#x3E; t if _{p &#x3E; t} &#x3E; _{p - po.} It follows in turn that estimate

₍₅₎

in the

proof

of Lemma 3 remains true if we

_{replace f by h}

and _p

_by

some _to with

Thus

Also note that from estimate

₍₄₎

in the same

_proof

is

_positive

and

decreasing

if k is

_odd,

and

_negative

and

_increasing

if k is even, whence

modulus of the

_Taylor

_expansion

of order k of

_h(t)

_{at p} does not exceed

And estimate

₍₇₎

shows that if t is close

_enough

to p the last

_expression

tends to 0 as k - oo. Thus h is

_analytic

in some

[Qo, Po]

as claimed.

Now we show that h is

_{f ,}

with I =

[Q,

oo)

its domain of definition.

Suppose

it is not the case and let I’ =

[Q~)

or I’ =

(Q’, w~

be the

do-main of definition of h.

_Suppose

first that h =

_f

on I fl I’. Then I = I’

would contradict the

_hypothesis,

I’ C I or

Q’

Q

would contradict the

maximality

of

_{f ,}

and I C I’ oo would contradict the

maximality

of

h. To see this recall that

Io

C

_I’,

and recall the

_proofs

of Lemmas 4 and 5. So there must be some t E I rl I’ with

_{f (t) ~ h(t).}

Now

_by

Remark 3.8 and Lemma

_1,

for

_{every t}

in I which is not a fixed

_point

of

f

we have

0

_{( f (t) - p)/(t - p)}

_1,

and for

_{every t}

in I’ which is not a fixed

_point

of h we have 0

(h(t) -

p)/(t - p)

1’

_Suppose

for instance there is

some t _p with

_{f (t)}

_#

_{h,(t) (the}

other case is treated

similarly).

If we

_put

to :=

sup~t

p, f (t)

_~

h(t)},

we have h -

_f

on

[to, p]

and _to Note

that _to cannot be a fixed

_point

of

f

or

h,

as it then would be the infimum

of I or I’

_(by

Remark

_3.8),

which would contradict the

_maximality

of

_f

or

of

_{h, f}

and h

_being

defined

_{(and distinct)}

on some t _to. Thus

_{h(to) &#x3E;}

_to and

_{f (to)}

&#x3E; to. Now

_{by hypothesis}

_{f (t)}

and

_h(t)

are defined for values of

t

_to;

moreover

_f

and h are

_{strictly increasing.}

So the inverse functions

f-1(t)

and

_h-1(t)

are defined for values of t

h (to)

=

f (to).

It follows that there is some _so with

_to

_so

h(to)

=

f (to),

and such that

_f-1

and

h-1

are both defined on

~so, p~.

Thus

by

Remark 3.9 we

_have,

for t &#x3E; _so,

and

_{f -1 - h-1}

on

(so, p~.

The strict

monotonicity

of

f

and h now

imply

that

_{f -}

h on

[ti,p]

for some _{t1 to}

(satisfying f (tl)

=

h(tl)

=

so),

contradicting

the definition of to. This concludes the

_proof

of Theorem 1 in

case _p &#x3E; 1.

Case 2: 0 _p 1. We first _prove the

_unicity.

Consider two MS of

₍₁₎

with 0 _p 1

_f

and h. Then

_by

Theorem 1

_f

has another fixed

_point

p2 &#x3E;

_1,

and h has another fixed

_point

_p3 &#x3E; 1.

_Suppose

_they

are

unequal,

say p2 P3. Let c be the

_largest

number not

_exceeding

_p3 such that

f (c)

=

h(c).

This number exists since

_{f (p)}

=

h(p)

and since

f

and h are continuous. Note that c _p2 and that

f (t)

h(t)

for t E

_(c,P3~,

and recall that

_{f (t)}

&#x3E; t and

_h(t)

&#x3E; t for t E

_(p,P2).

It follows that

_{f’(c) h’(c).}

It also follows that if c &#x3E; _p then

_{f ( f (c))}

_{h( f (c))}

=

h(h(c)),

_a contradiction.

So c =

p. But then

So _p2 =

P3, and

by

Case 1 h =

f

_on

[p,

oo).

Now we _{prove the existence. For q} =

p2 &#x3E; 1 denote

_by

_fq

the MS of

₍₁₎

and

_by

_Iq

=

[Qq,

oo)

its domain of definition. Recall that the fixed

_points

of

_fp

_are

and 0. So from Remark 3.1 and the

_argument

used to _prove

the

_unicity

we infer that for _{every q} with 1 q

0

we have

fq (t)

for t E

_~max{0,

_0],

and that for every q

&#x3E; 0

we have

fq (t)

&#x3E; for t E

_[max{0,

_{Q9~, q~.}

It follows that each

_fq

with 1 _q

₀

has a fixed

_point

p =

p(q)

with 0 _p 1

_(and

that no

_fq

_{with q &#x3E; 0}

has such a fixed

point).

Moreover the same

_argument

shows that if 1 _q

q’

_0,

then

p(q’)

p(q)

and

_fq(t)

for t E

_{[p(q), q] -}

So the function _{p :}

_(1,~~

-

_(0,1)

is

_decreasing.

What we need to show is that it is

_continuous,

and that

p

To _prove the

_continuity

we consider _{some q} = _p2 with 1 _q

~

and let

q’

--&#x3E; _q with 1

_q’

_0.

Without loss of

_generality

we _{may suppose} that

q’ -

q remains of the same

_{sign. Suppose}

for instance that

q’

increases to _q;

the other case is treated

_similarly.

We

_{put f}

for the MS of

₍₁₎

with fixed

point

q, and h for the MS of

(1)

with fixed

point

q’.

Let s be a real number.

For some s’ and s" between 0 and s we have

If,

for some

_negative

real number _so we have

so s

_-so, and

(this

will hold for instance if _q

_{:= (q +}

_1)/2

and

_{q - q’ (q -}

_1)/2)

we see

with the

_help

of estimate

₍₇₎

that the last term does not exceed in modulus

provided

so+q &#x3E; 1. The last

_expression

tends to 0 as K ~ oo if -so is small

enough.

Now from

₍₂₎

it is clear that for every

k,

h,~~~ (q’) -

-a 0 as

q’

~ _q. So the sum tends to 0 as

q’

_{2013~ and}

h(q’+s)

_converges

uniformly

to

f (q

+

_s)

in

_{[so, -so~.}

It

_easily

follows that

_h(t)

_converges

_uniformly

to

_{f (t)}

in

_{[Qo, q]}

for some

Qo

_q. Now we shall

essentially

follow the construction

of Lemma

_5,

after the

_following.

Remark. Let _{p =}

_p(q)

and

_p’

=

p(q’)

be the other fixed

points

of

_f

and

_h,

and let c &#x3E; 0. Note that

_p’

decreases if

_q’

increases. There is a

j3

_

such that the derivatives of

_{f (t)}

in

_[p

+

_{E, q~}

and of the functions

_h(t)

in

(p’ + 6,g]

are all within the interval

_{~1/~, 1/~].}

Now from

for t E

_{(Qo, q),}

we see _converges

uniformly

to

_{f -1 (t)}

for t E

_{[Qo, q~.}

For E &#x3E; 0 as in the Remark

_put

_p, =

inffpl

_{+ E.} Then if 6 &#x3E;

_0,

_h(t)

_converges

_uniformly

to

_{f (t)}

for t E

(f -1(Qo)

+ 6 :=

provided

Q1

&#x3E; _p~. And we can as well fix 9 = _E.

Recursively

_{we see} that if

_h(t)

_converges

_uniformly

to

_{f (t)}

for t E

_[Qi,

_q],

then it does also for t E + E

:= QZ+i, Q~~

provided

Qi+1

&#x3E; _Pf.

So for t E

_[Q’,q],

and for _every

_Q’

&#x3E;

_{max{liminfi-tooQi}

=:

Q; pE},

_h(t)

converges

uniformly

to

_{f (t).}

We have

_{f (Q)}

=

limsuPi-too

+ E)

Q + e /p

Q+6.

So

_h(t)

_converges

_uniformly

to

_{f (t)}

for t E

_q~,

where

_{[p, Qf]}

is the

_largest

interval

_beginning

_{at p} where 0

_{f (t) -}

t E.

It now

_easily

follows that

inf p’

=

p.

Finally,

if _p2 = _q = 1 _+,E where E

1/9,

we show that

p(q) &#x3E;

_{q -} 9E. If

not _{it is easy} to

_verify

that

_f’(t)

&#x3E; _{q -} 2E if t

_q

and that

_{f’(t) &#x3E;}

_{q +} c

if t

_{q -}

_3E,

whence

_{f (q - 9E)}

_{q -}

_9E,

a contradiction. This shows that

p(q)

~ 1 _{as q} ~ 1 and concludes the

_proof

of Theorem 1 in Case 2. Case 3: _p = 1. We first

prove the existence of a solution. Let

and be the two _sequences of functions

_{[1, oo)}

-3

_{[I, cxJ)}

defined as

follows.

We see

recursively

that

they

are well defined.

Indeed,

is well defined

with 1 for

_{every t &#x3E; 1,}

then is

_strictly

_increasing

and

unbounded,

whence

_{f n}

is well defined with

_{f n (t) &#x3E;}

1 for

_{every t}

&#x3E; 1. Now

we

_have,

for

_{every t}

&#x3E;

_1,

And if

_h+

and h- are two

_increasing

and unbounded functions

_{(1, oo)}

-with

_h+(1)

=

h_(1)

= 1 and

h+(t) &#x3E; h- (t)

for

every t

&#x3E;

_1,

then we

have

-+ -I-lOt

whence

_k-’(t)

for

_{every t &#x3E;}

1. Thus an induction

_argument

starting

with

₍₈₎

shows that the _sequences and

t &#x3E;

_1),

and that in addition

_{f2k(t) &#x3E;}

for

_{every k &#x3E;}

0. Now we

put,

for t &#x3E;

_1,

_F(t)

= and

G(t)

= and _we shall _prove that F - G and that F is a

(maximal)

solution of

(1)

for _p = 1. First we

verify

that F and G are

_{strictly increasing}

and

_unbounded,

and

that their inverse functions

_satisfy

J 1 J 1

whence in

_particular

_they

are

indefinitely

differentiable functions. We _prove

the first

_equation

in

_(9);

the second one is treated

_similarly.

Since

f2~(t) &#x3E;

1 and is

_decreasing,

F is

_integrable

and

Thus we see that the inverse function

H-1

exists. And we know that the

function _g2~ is

_increasing

for

_{every k &#x3E;}

_0,

and that the _sequence

is

_decreasing

for

_{every t}

&#x3E; 1. Now if E &#x3E; 0

_{let 77}

=

H(y) - H(y - E).

Then for k

_{large enough}

we have

whence t :=

H(y)

e

_{(g2k(y - 6)~2A;(~/))}

and

_{(y - e, y).}

It follows that

_{g2~ (t)}

converges to

H-1 (t).

But also _converges to

_G(t),

and the first

_equality

of

₍₉₎

is

_proved.

Now we show that F * G.

Lemma 6. Let F and G be two

_increasing

and

_{unbounded functions}

e

_{indefinitely differentiable,}

and

_satisfying

Then

_if

the

_{equations (9)}

hold we have P’ =- G.

Proof.

First note that F and G must be

_strictly

_increasing,

unbounded

functions,

and that their inverse functions

F-1

and

G-1

are well-defined.

From _ , , _ , ,

we see that

Now let

_H(u)

:=

maxIF(G(u)) -

G(F(u)), 0~.

Then from the above

ex-pression

and since

_{F(v) &#x3E;}

1 and

_{G(v) &#x3E;}

1 for all v &#x3E; 1 me _{may write}

Since on the other hand

F(F(u)) &#x3E;

F(G(u))

and . we have

F(F(u)) - G(F(u))

&#x3E;

_H(u)

whence

There is some

(necessarily nonnegative)

constant C such that

F(t) -G(t)

C for

_{every t}

with 1 t

_3/2.

Let C be the smallest such constant. Then since

_{G(t~ F(t~ t}

we have

if 1 t

_3/2.

Thus C = 0 and F - G _on

(1, 3/2~.

In order _{to see} that F = G on

(1, oo)

we

_recursively

use the

_expression

which we

_{just proved}

holds when t

_belongs

to the interval

_{~1, 3/2~.}

If we

put to

=

3/2

and ti

=

F-1(ti-1)

(i

&#x3E;

0),

then the sequence

Itil

is

strictly

increasing

since

_F-1(t)

&#x3E; t for

_{every t}

&#x3E; 1

_{by hypothesis.}

And if

₍₁₀₎

holds for t

_ti,

then

_F-1(t)

=

G-1(t)

for t

ti,

whence

_F(t)

=

G(t)

for t

ti+1

and

₍₁₀₎

holds for t

_ti+1.

So we

only

need to ensure that the _sequence

{ti~

cannot _converge. If it

_did,

to some T _{oo, say,} then we would have

T =

limi-too

_ti =

F-1(ti)

=

F-1(T),

in contradiction _{to one} of the

hypotheses.

This concludes the

_proof

of the Lemma. 0 Now

_{by differentiating}

₍₁₀₎

and

_{putting v}

=

F(t)

_{we see} that F is _a solution of

₍₁₎

for _p = 1

(and

it is a MS

by

Lemma

1).

Finally

we show that the solution is

_unique.

First note that

_by

Lemma 1

a MS

f

of

(1)

for _p = 1 is defined _{on an} interval I whose lower bound is

1,

that

_by

Remark 3.3

_f

is

_strictly

_increasing,

and that we thus have

f (u) &#x3E;

1,

whence

_f’(u)

_1,

for _{every u} in I. In

_particular

we also have for

every u in I. Also note that

_similarly

as we

_proved

above that F - G on

[1, oo),

we can _{put ti} =

f -1(ti-1)

(i

2:

1)

where _to &#x3E; 1 is some number in

I,

and see that in fact I =

[1, oo).

If F is the solution _we constructed

above,

then we have

Now there is some

_nonnegative

constant C such that

_{IF(t) - f (t)~}

C for

every t

in

_{~l, 5/4~.}

Let C be minimal with this

_property.

Then for some

function v =

v(v,)

where

v(v,)

is between

_{f (u)}

and

_F(u),

we have

" .J.

Thus C = 0 and F -

f

_on

[1,5/4].

And in fact F -

f’

_on

[1,

oo):

again

this

can be seen

_similarly

as in the

proof

of F - G above. This concludes the

proof

of Theorem 1.

5. PROOFS OF THEOREMS 2 AND 3.

We now show that there are no other

increasing

MS of

(1)

than those

described

_by

Theorem 1.

Theorem 2.

_{If f}

is a solutions

_of

_f’(t)

=

1/f (f (t))

which is

_increasing,

and maximal

_(in

the sense that its interval

of definition

I

satisfies

the conditions

(b), (c),

and

_(d~),

then

_f

has a

(nonnegative)

fixed

_point.

Proof.

Let A be the

_largest

lower bound of I. If A &#x3E;

_0,

then

_considering

limits we see that

f (A)

exists and A E

_I;

and we must have

_{f (A) &#x3E;}

A. If A 0 then since

_f

takes

_positive

values and I is an

interval,

f

is defined on

t = 0.

Suppose

f (0)

0. Then since

f

is

increasing,

f (t)

0 if t

0,

and thus

_f’(0)

=

1/ f ( f (0))

0,

which is not true. So we must have

_{f (0)}

&#x3E; 0. Hence there is a

_{to &#x3E;}

0 with

_{f (to) &#x3E;}

to. If

_{f (to)}

= _to the theorem is

proved.

So _suppose

_{f (to)}

&#x3E; to. Then

_f

is defined at least on the interval

[to,

f (ta)

=:

tl].

Now define _tn =

f(tn-1) (n

=

1, 2, ~ ~ ~ ).

Then either for _{every n,} or for some n. In the second case

there is some t &#x3E; 0 with

_f(t)

= _t, and

again

the theorem is

proved.

In the first _case, we first note that the

_increasing

_{sequence tn} cannot

_diverge:

if it did then we would have

f’(tn)

e

_0,

which is

_incompatible

with the

assumption

that for _every n.

Thus tn

_converges to some number

T as n e oo.

Again

_considering

limits we see that

_f(T)

exists and T E I.

We have

_{f (T)}

=

limn-too

= =

T,

and

_again

the theorem is

proved.

D

Remark. It is conceivable that some solution

_f

of

₍₁₎

with

_{f (t)}

&#x3E; 0 if

derivatives of

_{opposite signs}

for

_arguments

of

_{opposite signs,}

and could thus

not be continuous on an interval

_containing

0. In other

_words,

it could not

be an MS on _any interval I.

Now we _prove that the

_graphs

of two

_arbitrary

distinct

_increasing

solu-tions cross each other

_infinitely

_{many times.}

Theorem 3.

_{If f and g}

are

_increasing

maximal solutions

_of

(1),

then there

is a _sequences

of

numbers si

~ oo

(i

e

_oo)

with

_{f (si)}

=

g(si).

Proof.

First note that _any

_increasing

solution of

_f’(t)

= satisfies the

_asymptotic

_equivalence

_(t

2013~

_oo):

this follows from

Proposition

2 in

_[P6]

_(The

condition

_{f :}

_{[0, oo)}

~

[0,

oo)

of this

_general

result can

clearly

be removed for the solutions of

(1)

we

consider).

Let

_{[r, oo)}

be the interval of

_nonnegative

_arguments

on which both

func-tions

_f

_{and g}

are defined. Now first assume that

f and g

never cross each

other,

for instance that

_{f (t)}

&#x3E;

_g(t)

for all t &#x3E; T. It follows that

f’(t)

g’(t)

for all

_{these t,}

and that if we

_put

f (to) - g(to)

=: C for some _to &#x3E; T to be

fixed later _on, then

_{f (t) - g(t) &#x3E;}

C for T t G _to. Now for T t to let

t1 =

tl (t)

=

f -1 (t).

Then _we

have,

if _to is

large enough,

where C’ =

C~~ ~/2.

Note that _we have to = and choose _to

large

enough

to ensure that

f (tp) &#x3E;

T. Then

by integrating

we

_get

from

(11)

But

₍₁₃₎

holds in

_particular

for

_{ti (to)}

=

f-1(to) ~

0’-"to 0

as _to e _oo, and it follows that

_l(t1(t))

cannot remain

_negative

for all t _tp.

So there is at least one

_crossing.

Now _suppose the set S of

_crossings

is bounded. Then s =

sup S

is itself _a

crossing.

(Note

in

passing

that the

assumption

f - g

on an interval leads to a contradiction: i.e.

_crossings

are

crossings

in the strictest

_sense).

_Suppose

that for instance

_{f (t)}

&#x3E;

_g(t)

for

all t &#x3E; s =:

_To.

Let

_Tl

be such that

g(T1)

=

To;

then for _{t &#x3E;}

T1

_we have

f ( f (t))

&#x3E;

f (g(t))

&#x3E;

_g(g(t)),

whence

_g’(t)

&#x3E;

_f’(t).

So if we

_put,

for some

to &#x3E;

_Tl,

&#x3E;

f (tp) - g(to)

_:

C,

then

f (t) - g(t)

&#x3E; C for

Tl

t to.

Now for

Tl t

to let ti = =

f -1 (t).

We

may suppose that

T1

is

large enough

to ensure that _t1 &#x3E;

_t;

and also that _to is

_{large enough}

to ensure

that

_{f (to)}

be

_larger

than

_Tl.

Then the estimates

_{(11), (12)}

and

₍₁₃₎

hold if _to is

_large

_enough,

and it follows as before that

g(t1(t)) - f (tl(t))

cannot

remain

_negative

for all t _to. This concludes the

_proof

of the theorem. 0 6. LAST REMARK ON DECREASING SOLUTIONS.

Although

it is not our _purpose to also treat

_decreasing

solutions of

₍₁₎

(like

f-(t)

=

2013~’"~"~(2013~)~),

_{we note} that it is _now

fairly

easy to establish the existence of

_infinitely

_many such functions.

Indeed,

for every p -1 the

argument

in the

proof

of Case 1 of Theorem

1

_helps

_verify

that there is

_exactly

one MS

fp

of

(1)

with

fP(p)

= _p. Consider the formal _{power series solution}

_{h(p) (t)}

of

₍₁₎

with fixed

_point

_p. From

₍₄₎

it is clear that at t = _p this series has _a radius of _{convergence at} least as

_large

as that of the power series

expansion

of the solution of

₍₁₎

with fixed

point

at t =

lp 1,

which is

_{positive by}

Lemma 3. Thus _represents within that radius a solution of

(1),

which we _may call

fp(t) .

We can then

extend

_fP

to a MS of

(1)

_by

_using

if the fixed

_point

_p is stable

_(as

in Case 1 of Theorem

_1),

_respectively

unstable.

_(In

view of the function

_f-

it seems

_likely

that _p is

unstable).

Unicity

is obtained

_similarly

as in Case 1 of Theorem

1,

and

by using

the

analyticity

of

_flpl(t).

We mention the fact

_that,

_by

_comparing

the coefficients of the

_(formal)

solution

_h(p)(t)

written as a

(formal)

Taylor

series at t =

p with those of

the solution

_{f_1~~(t)}

:=

f- (t) (also

at t =

p),

one can in fact establish the

existence of a solution for _every

p -1/~.

We have no

_opinion

to offer

_regarding

the existence of solutions when

-1/~ p 0.

REFERENCES

[Fi] N.J. Fine. Solution to _{problem 5407.} Amer. Math. Monthly 74 _(1967), 740-743.

[Go] S.W. Golomb. Problem _5407. Amer. Math. Monthly 73 _(1966), 674.

[Ma] Daniel Marcus. Solution to problem 5407. Amer. Math. Monthly 74 _(1967), 740.

[McK] M.A. McKiernan. The functional differential equation D f = _{Proc. Amer.} Math.

Soc. 8 _(1957), 230-233.

[Pé] Y.-F.S. Pétermann. On Golomb’s self describing sequence II. Arch. Math. 67 (1996), 473-477.

[PéRé] Y.-F.S. Pétermann and Jean-Luc Rémy. Golomb’s self-described sequence and functional differential equations. Illinois J. Math. 42 _(1998), 420-440.

[Ré] Jean-Luc _Rémy. Sur la suite autoconstruite de Golomb. J. Number Theory 66 _(1997), 1-28.

[Va] Ilan Vardi. The error term in Golomb’s sequence. J. Number Theory 40 _(1992), 1-11. Y.-F.S. PGTERMANN Université de Geneve Section de Math6matiques 2-4, rue du Lievre, C.P. 240 CH-1211 Geneve 24 SUISSE

E-mail: _{petermanos c2a . unige . ch}

Jean-Luc REMY

Centre de Recherche en _Informatique de Nancy

CNRS et INRIA Lorraine

B.P. 239

F-54506 Vandoeuvre-les-Nancy Cedex FRANCE

E-maid : _{Jean-Luc . Remyoloria . fr} Ilan VARDI IHES 35 route de Chartres F-91440 Bures-sur-Yvette FRANCE E-mail : ilaneihes . fr