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SPECIAL SOLUTIONS OF CERTAIN DIFFERENCE

BY

F. JOHN

of LEXINGTOIV, KY., U. S. A.

EQUATIONS.

Let f (x) be a solution of the difference equation f ( x + i ) - f ( x ) = g(x)

for

x > o . f(x)

may be uniquely determined by prescribing its values arbitrarily for o < x ~ 1. For certain functions g(x) however the solution f may also be characterized by simple properties, instead of prescribed values in an interval.

A solution may e.g. be uniquely determined by its asymptotic behaviour for large x; this leads to the >)HauptlSsung>) of the difference equation, as defined by N. E. NSrlund in his ~)Vorles.ungen iiber Differenzenrechnung)). In special instances solutions have also been characterized by local properties. Thus it has been proved by H. Bohr, that for

g(x) ~

log x, all strongly convex solutions o f ( i ) are of the form

f ( x ) - ~

const. + log

F(x). 1

Here a function is called ~)strongly convexly, if

(2) f ( ) . X '{- (I - - ~)~/) ~ ~ f ( x ) -{- (I - - Z ) f ( y ) f o r 0 ~ Z ~ I,

whereas ,>convexity>> alone only implies, that

f , x -~ y~ <= f ( x ) +.f(y) , .

2

An analogous result has been derived recently by A. E. Mayer: ~ The only con- vex solution of the functional equation

I / f ( x +

I ) ~ - x f ( x ) is given by

i Cf. e. g. E. A r t i n : E i n f i i h r u n g in die Theorie der G a m m a f u n k t i o n , or Courant: Differential and I n t e g r a l Calculus, vol. I I p. 325.

Convexity ~ boundedness in some finite i n t e r v a l is e q u i v a l e n t to s t r o n g convexity, Cf.

Hardy, Littlewood, Polya: Inequalities, p. 9 I.

Konvexe L5sung der F u n k t i o n a l g l e i c h u n g I / f ( x + I) ~ xf(x), Acta m a t h e m a t i c a 7o, p. 59.

(2)

176 F. John.

i n t h e p r e s e n t p a p e r we shall give s o m e t h e o r e m s c o n c e r n i n g m o n o t o n e or convex s o l u t i o n s of (I) f o r c e r t a i n g e n e r a l classes of f u n c t i o n s g(x). As special cases we shall o b t a i n t h e t h e o r e m of B o h r , a n d also a s o m e w h a t w e a k e r f o r m of t h e t h e o r e m of A. E. M a y e r , w i t h *convex>> r e p l a c e d by >)strongly convex,).

Also an a n a l o g o u s t h e o r e m f o r m o r e g e n e r a l difference e q u a t i o n s will be derived.

U n d e r s u i t a b l e r e s t r i c t i o n s f o r g(x) t h e special s o l u t i o n s o b t a i n e d here, c a n be p r o v e d to be i d e n t i c a l w i t h t h e ~HauptlSsungen>> m e n t i o n e d above.

T h e o r e m A. L e t g(x) be defined f o r x > o, a n d let g. 1. b. g ( x ) = o. 1

T h e n every t w o m o n o t o n e n o n - d e c r e a s i n g s o l u t i o n s of (I) differ a t m o s t by a c o n s t a n t .

P r o o f : L e t f ( x ) = 9 ( x ) a n d f ( x ) - ~ W(x) be t w o m o n o t o n e n o n - d e c r e a s i n g s o l u t i o n s of (I) f o r x > o. T h e n

is a f u n c t i o n of p e r i o d I a n d is u n i f o r m l y b o u n d e d , as t h e m o n o t o n e f u n c t i o n s q~(x) a n d V 2(x) are c e r t a i n l y b o u n d e d f o r I ~ x ~ 2. L e t

L e t p ( x ) n o t be a c o n s t a n t .

M ~ - I . u. b. p (x) m - ~ g. 1. b. p (x).

T h e n M > m . L e t ~ be a n u m b e r w i t h o < ~ <

M - - m

< - - - - T h e r e is a n x 0 > o such t h a t g(xo)<= ~ . As ~0(x) is n o n - d e c r e a s i n g

2

__> g ( x 0 ) = ~ ( x 0 + i) - ~ ( x 0 ) >= ~ ( b ) - ~ ( a )

f o r all a a n d b w i t h

(3) x0 ~ a _--< b ~< x0 + I.

H e n c e , as W is n o n - d e c r e a s i n g

>-- w (b) - ~ (a) + ~ (~) - p

(.) __

p (b) - p (a) f o r all a, b s a t i s f y i n g (3). T h u s in p a r t i c u l a r f o r b = x 0 + I

>-- p (Xo + i) - p (~) = ~ (~o) - p

(-)

f o r all a in x 0 ~ a_--< x o + I, a n d c o n s e q u e n t l y

1 g. 1. b. d e n o t e s t h e g r e a t e s t l o w e r b o u n d , 1. u. b. t h e l e a s t u p p e r b o u n d .

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Special Solutions of Certain Difference Equations.

_>_ 1. u. b. (p ( . % ) - - p ( a ) ) = p (Xo)--m.

Similarly for a =

x o

177

=> p

(b)

- v

(~o)

for all b in Xo <= b ~ x o + I, and therefore

_>_ 1. u. b. ( p (b) - - p (Xo)) = M - - ~ (x0).

Adding we obtain the contradiction

2 e > ~ M - - m ,

which proves the theorem.

Theorem A does not assert the existence of a monotone solution. This is guaranteed under more restrictive conditions by

Theorem A': I f

g(x)

is non-increasing and lim g ( x ) = o, all non-decreasing solutions of (I) are given by

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l(x) = c - g (~) + F, (g ( ~ ) - g (~ + ,)),

~=1

where C is a constant; in case ~ g(~) converges,

f(x)

may simply be written

cx=l

in the form C ' - - ~ g ( x + # ) .

/z~O

Proof: According to theorem A, it only remains to be shown, t h a t the

f ( x )

given by (4) represents a non-decreasing solution of (I). Now the infinite series in (4) converges; for if o < x < n, where n is an integer, the expression

IV N

Y I g (,) - ~ (x + , ) l --< Y~ (g (,) - g (,~ + ~)) =

#~1 #71

= ~ g (,) - g ( N +

,~,) _<

q

(,)

~ I ~ i /z~l

is bounded uniformly in N. Moreover it is obvious, t h a t

f(x)

is non-decreasing and t h a t it is a solution of (I).

23--3932. A c t a mathematica. 71. Imprim6 le 4 juilleb 1939.

(4)

178 F. John.

Example: Let g (x) = l o g - -

I x + -

2 Then, according to theorem A'

X + - ~ I t + - X + ~ t +

f ( x ) : C - - l o g - - + ~ log 2 log

x # x + [ ~

,u=l

:

C + log 1-[

- - - i

, = ~ x + re+

: C + log

H x

~ (, +;)V,+ ~ ~+i 1

x + L ~ = l x + I -- tt~l 9

2 2

1 + - -

#

I

| I - { - - -

L e t C ' : C + l o g l ~ 2~ . Then

f ( x ) = C' + log

IX ~ ('+xlI'+ ;l'+~l 9 -~-H (, + ~-I" (;~x + '~-~.//

(5)

f (x) = C' + log F (x)

It is evident from theorem A', t h a t even every solution of

I x + - -

(6) f ( x + I) - - f ( x ) = log 2 ,

x

that is non-decreasing f o r sufficiently large x, will have to be of the form (5).

Let now F(x) be a strongly convex solution of the functional equation

(5)

Special Solutions of Certain Difference Equation. 179

_ i

(7) F (x -[- I) X F(X)

treated by A. E. Mayer. Then

F(x)

is either monotone non-decreasing or mono- tone non-increasing for sufficiently large x. Besides

F(x)

is of constant sign, as F ( x ) is continuous and according to (7)

F ( x ) ~

o. F(x) satisfies the simpler functional equation

(7') F(x +

2 ) - - x F ( x ) .

x - h i

Therefore, if F ( x ) > o, then F ( x ) is non-increasing for sufficiently large x, and if

F(x)< o, F(x)

is non-decreasing for sufficiently large x. Hence

f ( x ) =

= - - l o g ] F ( 2 x ) l is a non-decreasing solution of (6) for sufficiently large x.

Thus

F(x)

is of the form C. As

F(x)

is convex, C > o, and using

the functional equation (7), it follows that C--- ~ g 2

Theorem B. Let g(x) be defined for x > o and let lira inf g ('~) = o.

X ~ X

Then every two strongly convex solutions of (I) differ at most by a constant.

Proof: I f

f(x)

is a strongly convex solution of (I), replacing ~ by I -

f ( ( I -- Z)x + Zy) ~ (I --,~)f(x)

+

Zf(y);

we derive from (2),

adding this inequality to (2) it follows that

f ( Z x + (i -

z)y) + f ( ( i - Z)x § zy)

<=f(x) +/(y)

for o < ~ =< I. Let a and h be arbitrary positive numbers, then for Z - - a a + h ' I - - k - - h

-- - - - - , y = x + a + h the last inequality yields

a+h

f ( ~ + h + ~ ) - f ( x + ~) >=f(x + h ) - / ( x ) .

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180 F. J o h n .

H e n c e t h e f u n c t i o n

u ( x ) - ~ f ( x + h ) " f ( x )

is m o n o t o n e n o n - d e c r e a s i n g f o r every positive h. Besides

O ~ U (X -~" 1) - - U (X) = g (X + h) - - g (x).

W e h a v e

g. 1. b. (g (~ + h) - - g (x)) = o .

For, o t h e r w i s e t h e r e would exist a positive e such t h a t g ( x + h) - - e(x) >

f o r all x > o a n d h e n c e f o r positive i n t e g e r s n g ( . h ) > ( . - - i ) ~ + o ( h ) ; by a s s u m p t i o n t h e r e are a r b i t r a r i l y large Xo, such t h a t

g (Xo) ~ . Xo

let n h = < x o < ( n + I ) h ; then,

g (xo)

g (n h) ( n , I) ~ .q (h) n - - I .q (h)

2 ~ >

X 0 - - (n -{- I ) h > - - - - > = Xo + " Xo

> _ - - - - e + - -

n + I x o

w h e r e a s t h e e x p r e s s i o n on t h e r i g h t is c e r t a i n l y > 2 h f o r sufficiently large x o.

A c c o r d i n g to t h e o r e m A

u ( x ) ~ f ( x + h ) - - f ( x )

is t h e n u n i q u e l y d e t e r m i n e d up to an additive c o n s t a n t C ~ Ch. H e n c e f o r two c o n v e x solutions ~0 (x) a n d

(x) of (i)

p (x) =- ~ (x) - ~ (x)

is a f u n c t i o n of p e r i o d I, f o r w h i c h

(x + h) - - p (x) = Ch

is i n d e p e n d e n t of x . L e t again

M = l . u . b .

p (x), m = g . l. b. p (x)

a n d o < ~ < M - - , - - m in case p ( x ) is n o t a c o n s t a n t . L e t 3>----Xo > 2 be such, 3

t h a t p ( X o ) > = M - - ~ , a n d let xl be such t h a t p ( x l ) = < m + e a n d x o < x 1<_-4.

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Special Solutions of Certain Difference Equations. 181 T h e n f o r h : x ~ - - x 0 ( o < h ~ 2 ) a n d a n y x > o

p (x + h ) - - p (x) = p (xl) - - p (Xo) = c h _-< m - - • + 2 ~ .

P u t x = x o - - h ( > o). T h e n

m - i + 2 ~ >__ p (Xo) - p (~o - h) >= ( M - - ~) - - i = - -

~,

as M = I. u. b. 19 (x). C o n s e q u e n t l y

3 e > = M - - m ,

w h i c h leads to a c o n t r a d i c t i o n . T h u s p ( x ) is c o n s t a n t a n d t h e o r e m ]3 proved.

Theorem B': L e t g(x) have a c o n t i n u o u s d e r i v a t i v e f o r x > o . L e t g'(x) be m o n o t o n e n o n - i n c r e a s i n g and lira

g'(x)~ o.

T h e n all s t r o n g l y c o n v e x solu-

x - - ~c

t i o n s of (I) are of the f o r m

(s) / ( x ) = c - z x - 9 (~) - ~ (g ( x + t,) - ~ (t,) - x g '

(,))

te~l

w h e r e C : f ( I ) a n d

7 = lim (re)-- g(~ .

P r o o f : A c c o r d i n g to t h e o r e m B we only h a v e to show, t h a t t h e

f(x)

g i v e n by (8) is a convex solution of (I). I n o r d e r to p r o v e t h e c o n v e r g e n c e of t h e infinite series, it is sufficient to note, t h a t it obviously converges f o r x = o a n d t h a t t h e series ~, [g' (x +/z) -- g' (u)], o b t a i n e d by f o r m a l d i f f e r e n t i a t i o n , converges absolutly a n d h e n c e u n i f o r m l y f o r u n i f o r m l y b o u n d e d x owing to t h e same ar- g u m e n t as used in t h e p r o o f of t h e o r e m A'). M o r e o v e r

f ( x + I ) - - f ( x ) = - - 7 - - g ( x + I) + g(x)

o v

- - Z [g(x -~- , "F I ) - - g ( x -~- , ) - - g ' ( , ) ] tt~l

(8)

]82 F. John.

as

besides

9 g( f

lim [g (x + n + I) -- n)] -= lim g' (n 0

+ y) dy =

o;

f ( I ) = C - - y - - g(1)-- Z [g(I -~- ~t) - - g ( ~ ) - - g'(~$)]

/~,=1

= C - - 7 + l i m g ' ( ~ ) - - g(n + I = ( 2 .

T h e c o n v e x c h a r a c t e r of

f(x)

is obvious.

Note: T h e a s s u m p t i o n of

strong convexity

of

f(x)

c a n n o t be w e a k e n e d to

convexity

alone. T h e r e are u n d e r t h e same a s s u m p t i o n s f o r

g(x)

as in t h e o r e m B' always m a n y o t h e r >>convex>> solutions, at least on t h e basis of Zermelo's axiom. F o r w i t h g(x) also g ( x ) - - I satisfies t h e assumptions. T h u s the e q u a t i o n F (x + I) - - F (x) -~ g (x) - - I has a s t r o n g l y c o n v e x s o l u t i o n F ( x ) . L e t n o w ~ (x) be a

discontinuous

solution of t h e f u n c t i o n a l e q u a t i o n ~ (x + y) ---- ~ (x) + ~ (y), w h i c h m a y be c h o s e n in such a way, t h a t ~ ( I ) - ~ I. 1 ~ ( x ) is c o n v e x a n d q~(x + I ) = ~ (x)-~ I. T h e n

f ( x ) = F(x)+ q~(x)

is a d i s c o n t i n u o u s c o n v e x solu- tion of o u r e q u a t i o n (I).

E x a m p l e : F o r g ( x ) - ~ log x t h e a s s u m p t i o n s of t h e o r e m B' are satisfied. All s t r o n g l y c o n v e x solutions of (I) will t h e n be g i v e n by

f ( x ) ~ - / ( I ) - - T X - - ~

log I + - - - - l o g x

~ 1

w h e r e 7 = lira - - l o g n is E u l e r ' s c o n s t a n t . H e n c e

+ log r(x)

a n d we regain t h e t h e o r e m of H . Bohr.

I t seems possible to e x t e n d t h e p r e c e d i n g t h e o r e m s in various ways to e q u a t i o n s of m o r e g e n e r a l ~ype t h a n e q u a t i o n (I). An e ~ t e n s i o n to difference e q u a t i o n s shall be given here.

1 Such solutions are constructed by Hamel: Mathematische Annalen 60, p. 459--462.

(9)

Special Solutions of Certain Difference Equations. 183 T h e o r e m C, Given the difference equation

(9)

aof(X + n) + a J ( x + n - - I) +... + ant(x) = g(x)

with c o n s t a n t real coefficients ak for a f u n c t i o n

f(x)

defined for x > o. L e t all roots of the characteristic e q u a t i o n

(IO) F ( O ) = ao O n + gtl e n - 1 § " " "[- ( t n = 0

be simple a n d of absolute value I (real or imaginary).

I f Q - - + I is a root of (1o), and if lira

g(x)= o,

t h e n the difference of a n y

x ~ av

two m o n o t o n e non-decreasing solutions of (9) is a constant.

I f e = + I is n o t a root of (IO) a n d if lim'q(x) - - = o, t h e n there exists at

x ~ oo 3C

most one m o n o t o n e non-decreasing solution of (9).

Proof: L e t

f(x)=q~(x)

a n d

f ( x ) = ~p(x)

be two m o n o t o n e non-decreasing solutions of (9). Then

9~(x)--~p(x)~P(x)

is a solution of the corresponding homogeneous equation. H e n c e

P(x)

is of the f o r m

P ( x ) =

b e ~ l

where the

p~(x)

are functions of period I, d e t e r m i n e d by the system of linear equations

P ( x § h ) = (h ---- o , . . . , , - -

! ~ 1

the d e t e r m i n a n t of these equations is ((~l .. 9 Qn)~'II (Q~- (~k) ~ o. As difference

l > k

of two m o n o t o n e f u n c t i o n s

P(x)

is b o u n d e d in e v e r y positive finite interval;

hence all

pi(x)

are b o u n d e d everywhere.

W e c a n n o t assert, t h a t

P(x)

is u n i f o r m l y almost periodic in t h e o r d i n a r y sense, at it is n o t necessarily continuous. I n spite of t h a t the existence of a relatively dense set of t r a n s l a t i o n numbers of

P ( x ) f o r

every e > o is easily established. L e t

e~=e~

and let B - ~ M a x (I § According to K r o n e e k e r ' s

k

i Cf. e. g. P. M. Batchelder: An introduction to linear difference equations, Chapt. I, w 4, (the proofs given there have to be modified slightly for t h e case of non-analytic solutions) or 1NSr- lund 1. c. p. 29~--6.

(10)

184 F. John.

t h e o r e m I t h e r e is for given ~ > o a n u m b e r 1 such t h a t every interval of l e n g t h l contains a solution x of t h e n + I inequalities

I ~ x l ~ ~ , '~ I z~ xl =< ~ g ( m o d 2 7~) (k = I , . . . , ~);

if y is t h e integral n u m b e r n e a r e s t to x , l i l y I < ~(I + H ) <

B (k = I, ...', n).

Thus every interval of l e n g t h L = l + 4 contains an i n t e g r a l n u m b e r y, for which all I Zkyl-<_d (rood 2~). T h e n

Ip(x + y ) - P(x)l = ~ b~(x + ~,,,~o~+~, - ~(~)~)

l = l

__< ~ b,(~,'/I. I ~ ' - ~1--< ~

l = l

for sufficiently small d----d (s).

L e t P(x) n o t be a constant. T h e n there are two values x 0 a n d xl, such t h a t I P ( x o ) - - P ( x l ) l = 3 ~ > o.

Hence, if y is a t r a n s l a t i o n n u m b e r p e r t a i n i n g to e, [ ~ ) (X 0 + ~]) __ .~O (X 1 + ~)] ~ $.

Thus t h e t o t a l v a r i a t i o n of P ( x ) is >_--e in every interval of sufficient length.

L e t N be such, t h a t t h e t o t a l variation of P (x) ~--- 99 (x) - - ~) (x) is ~ e in every interval of l e n g t h N. As t h e t o t a l v a r i a t i o n of /)(x) is at m o s t equal to the sum of the t o t a l variations of ~0(x) a n d ~ ( x ) a n d as those f u n c t i o n s are m o n o t o n e , we have

(x + iv) - ~ (x) + ~ (x + iv) - ~

(x) >_

~;

hence for every positive i n t e g e r rn

1 Cf. e. g. J. F a v a r d : L e c o n s s u r l e s f o n c t i o n s p r e s q u e p ~ r i o d i q u e s , p. I 8 - - 2 I .

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Special Solutions of Certain Difference Equations. 185 (x + m N) + ~ (x + ~ ;v) _>_ , , ~ + ~ (x) + ~ (~)

a n d c o n s e q u e n t l y

(, 2) l i m i n f 9~ (x) + ga (x) => ~ > o, unless

.P(x)

is a c o n s t a n t .

T h e s o l u t i o n s of (9) m a y be f o u n d by t h e m e t h o d of v a r i a t i o n s of con- s t a n t s . 1 f ( x ) is of t h e f o r m

f ( x ) = ~ q~ c, (x),

I = i

w h e r e

cz(x)

is a s o l u t i o n of t h e e q u a t i o n

~, (x + ~) - ~, (z) =

T h i s implies

( I 3 )

r (*)

z + l P

k--1

~, ( . + k) -=- ~, (x) + . ,

I f lim g(x) = o, we h a v e

x ~

k--1

I I , ~ g(~ + ~) l i m k I z.~ ~+~

a n d h e n c e

k--I

- ~ _ ~ k Z I~(x + ~)1= o,

c~ (x + k)

lira - - ~ o,

k ~ x + k

l i m

f ( x

§ k) _~ o . k - - ~ x + k

I f

f ( x )

is m o n o t o n e , this implies

I n p a r t i c u l a r

f ( ) lira

~, ,x,~ = o.

x ~ X

1 Cf. B a t c h e l d e r loc. cir. p. I 3.

24--3932. A c t a mathematica. 71. Imprim6 le 4 juil!et 1939.

(12)

186 F. John.

This constitutes a contradiction to (I2) and consequently

P(x)is

a constant.

Thus the first part of theorem C is proved.

If only lim g(X)=o," it follows from (I3) , t h a t at least

x ~ r X

lim - - -- o.

~_~

(x + k) ~

Hence for a solution

f(x)

of (9)

lim

f (x + k)

k - ~ (x + k) ~

and if f is also monotone

f ( x )

(I4)

}ira x~ - - = O .

The function E ( x ) = ~ ( x ) + ~ (x) is a monotone non-decreasing solution of the difference equation

3,

2 ] a . ~ ( x + ~) = 2 ~(z);

ke=0

according to

(I2)

and (14)

I X2

(I5) 2 ~ x < ~(x) <

for sufficiently large x.

Let now Q-- + I not be a root of (Io): Then

Let ~ [ a ~ [ : A .

(~6)

I f

n

.~ a ~ d ~ o .

/z=O

x (~ + ~) - ~ (x) < ~ ] I2 (x) l,

then also

I~! ~ x

~.(x+ i ) - - ~ . ( x ) < ~ - ~ [ ( )1 ( i = o, . . . , ,), and

(13)

Special Solutions of Certain Difference Equations. 187

C o n s e q u e n t l y

! I1 o !

I # = o =

--< ~. I~,1 [4 (z + ,) - z (x)] __< I,)1-14 (x) l.

# = 0 :2

I,q. IZ ( x ) l - I= g (~)1 =< 12 g ( x ) - ,~ 4 (x)l-_< I,~1.14 (x)l,

2

B u t f o r sufficiently large x

14(~)1

<

4 Ig (x)l

Ig(x)l < I ~ .x.

T h u s [Z(x)[ < I - ' ~ - x , a n d we are led to a c o n t r a d i c t i o n with (I5), unless

P(x)

2

is a c o n s t a n t or (I6) does n o t hold.

T h e r e f o r e , if

P(x)

is n o t a c o n s t a n t , we h a v e f o r all sufficiently l a r g e x

4 (x + ,) - z (x) > I~l Ix (x) l, f ~ ' l Z(x)

4(X "{- n) >~ I 4- 2 A /

as 4 ( x ) > o f o r l a r g e x a c c o r d i n g t o (I5). B u t t h e n 4(x) would increase ex- p o n e n t i a l l y w i t h x , w h i c h also c o n t r a d i c t s (I5). H e n c e P ( x ) = - c o n s t . ~ - C . As C would h a v e to be a solution of t h e h o m o g e n e o u s e q u a t i o n b e l o n g i n g to (9), C . 6 ~ o a n d h e n c e C ~ o. T h i s completes t h e p r o o f o f t h e o r e m C.

Example:

T h e p r e v i o u s l y c o n s i d e r e d e q u a t i o n f ( x + ~) + f ( x ) = log x

c a n have at m o s t one m o n o t o n e n o n - d e c r e a s i n g solution. T h a t g i v e n by

f(x)

= log

r(;)

s o l u t i o n is

(14)

188 F. John.

Theorem D. L e t g (x) be defined and continuously differentiable for x > o, and let all roots o f equation (Io) be simple and of absolute value I.

I f lira

g' (x)= o,

every two strongly convex solutions of (9) differ at most

x ~ oo

by a constant.

I f + I is not a root of equation (Io) and if lira

g' (x)= o,

equation (9)has

x ~ X

at most one strongly convex solution.

Proof: I f

f(x)

is a convex solution of (9), t h e n for

h > o u (x) ~ f ( x + h ) - - - f ( x )

is a monotone non-decreasing solution of the difference equation

ao u (x + ~) + al u (x + . - i) + . . . + an u (5) = g (x + h) - - ~ (x).

As g ( x + h ) - - g ( x ) = h ' g ' ( ~ ) we have

l i m [ g ( x + h ) - - g ( x ) ] ~ - o or lim g ( x + h ) - - g ( X ) = o

X ~ z ~ O o X

respectively. Hence, according to theorem C,

f ( x + h ) - - f ( x )

is uniquely deter- mined except for an additive constant. Let

f(x)----9~ (x)

and

f(x)--~ tp (x)

be two convex solutions of (9). Then

P(x)-~q~(x)--~p(x)

is a solution of the homo- geneous equation. Besides for h > o

(I7) (x + h) - - ~ ( x ) = c o n s t . = CA.

As difference of two convex functions

P(x)

is continuous. Thus

P(x)

is representable in the form (I I) with continuous functions 2 , ( x ) . The

p,(x)

being continuous and of period I are uniformly bounded. Therefore

P(x)

is bounded.

But from (I7)

P(x + kh) = P(x) + k CA

for integers It. The boundedness of

P(x)

implies, t h a t Ch----o for every h > o. Thus

P(x)

is a constant.

I s )

Similar uniqueness theorems may be exspected for equations of the form

1

f f ( x + y) K(y) dy = g (x),

0

(15)

w h i c h c o r r e s p o n d s t o s o l u t e v a l u e I.

S p e c i a l S o l u t i o n s of C e r t a i n Dii~erence E q u a t i o n s . 189 u n d e r s u i t a b l e c o n d i t i o n s f o r K a n d g.l O f p a r t i c u l a r i n t e r e s t i n t h i s connec~

t i o n s e e m s t o b e t h e c a s e , i n w h i c h a l l r o o t s o f t h e e q u a t i o n

1

f e~ K (y) d y =

o~

0

t h e c h a r a c t e r i s t i c e q u a t i o n (IO), a r e s i m p l e a n d o f a b -

1 For equations of type (I8) see F. John: Bestimmung einer Funktion aus ihren Integralen fiber gewisse Mannigfaltigkeiten, Mathematische Annalen IO9, p. 488--520, and J. Delsarte: Les fonc- tions ,,moyenne-p6riodiques,,, Journal de Math6matiques pures et appliqu6es, vol. Ioo, p. 4o3--453 .

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