non-linear difference equation

8/9/2019 Non-linear difference equation

1/42

O n

nonlinear

d i f ference

equations

By

L. J.

G R I M M *

and W. A.

H A R R I S , Jr.**

Introduction.

We consider a system of nonlinear dif ference equations of the

form

(0.1) y(x +

l~)=f(x,y(x-)-),

x\

where x is a complex variable,y an ^-dimensional vector, and / an

n-dimensional vector.

/^

Each component

of the

^-dimensional vector

/ is

assumed

to be

holomorphic in a region

R =

S

Q

X C7

0

,where

fo r

some positive constants a, d

Qy

p

Qy

where the norm of a vector u is

n

given by W =S|w,-|.

Let

(0.2)

be

the

expansion

of / in

powers

ofy

ly

-',y

n

,

where

p is a set of

non-

negative

integersp

lf

--, p

n

, B(#) an nXn

matrix,

/

0

and /p

^-dimensional

vectors, and

(0.3)

Received November 17 , 1965.

*

Department

of Mathematics,

University

of Utah, U. S. A.

**

School

of

Mathematics, University

of Minnesota, U. S. A. and

Research Insti-

tute

for

Mathematical Sciences,

Kyoto

Universi ty,

Japan. U. S.-Japan

Cooperative

Science Program,N SFGrant

GF-214.


2/42

We

shall suppose that f

0

, fy, B

are

holomorphic

in

S

Q

and

have

the

21 2 L.

/. Grimm

and W. A. Harris, Jr.

We

shall suppose that

asymptotic expansions

(0.4)

as

x

approaches

infinity

through

the

sector

5

0

.

In order to construct solutions of (0. 1) it is important to

obtain their first approxim ation ; in general

this

is

difficult.

However,

if

a solution y(x) has a limit

y

Q

as

x

approaches infinity and

/(, 3 > o ) is

defined, then y

Q

must satisfy

the

equation

J^/C

00

,^)-

O n

the other hand, if there exists a

y

Q

which satisfies thisequation,

y

Q

=/(,yo), then, und er suitable stability con ditions, it may b e

expected that every solution

in a

neighborhood

of

y

Q

will approach

y

Q

as x approaches

infinity.

The purpose of this paper is to show

how, un der suitable hypotheses, to con struct the gen eral solution of

the system (0 . 1) in a region of the

form

(0.5)

/


3/42

On nonlinear difference equations

213

(0. 4) as x approaches infinity

through

the sector Si. Suppose

x\

that B

Q

and B

0

I are

nonsingular.

Further

assume

/ o o = 0 , and

also

that if A

i9

i =l,---,n, are the

eigenvalues

of B

Q

,

then

0^arg

(

log^-)- Then

if the

positive constants a >

2

and p^

1

are

sufficiently

large,

there

exists a solution

(0.7)

y=*W

of (0. 1)

such that < ( # )is analytic and admits the asymptotic

expansion

(0.8) ^OO^SAJr"

i / = i

as

x

tends

to

infinity

in the

domain,

S

2

:

|

arg

(#^

-tf

2

)l


4/42

21 4

L. /. Grimm and W. A.

Harris,

Jr.

S

3

: arg(*0-

f

'-0

8

) 0, p

3

>0,

Z i

and let A

possess

the

asymptotic

expanion

as x approaches infinity through the sector

S

3

.

Suppose

that

vi,

-,&

are the

distinct eigenvalues

of A

0y

that none

of

these

is

zero,

and

that

0 = a r g (

log

(/,-//*/)),

i j.

Suppose

further

that

A

Q

has the

block-diagonal (Jordan) form A

Q

=

diag(Ai,>-,A?), where

A^jujlj + Nj, with Ij the m

r

dimensional identity matrix and Nj a

nilpotent matrix.

Then thereexistsamatrix TOOwithcomponents

holomorphic

in some sector,

S

4

: largC^-^-flO|


5/42

On

nonlinear difference

equations

215

that

th e

matr ix A(x")

has the

block-diagonal

form

of

J3(#).

Let

(0.17) z = P(x,u')

be a transformation of the vector

z

such that

P(x,u)

can be repre-

sented

by a

uniformly convergent

series of the

form

(0.18)

in a

region J?

4

= S

4

X

C 7

4

given

by

U

t

:\\u\\0,p

4

>0,

with

5

4

>0 sufficiently

small, with

coefficients

/\

holomorphic

for

x^S

4

,

an d

admitting asymptotic expansions

x

as

^

approaches infinity through

5

4

.

W e are now in a

position

to

prove

our

main theorem.

Theorem

3. Suppose that the

matrix

A

Q

has eigenvalues

satisfying

(0.19) 0


6/42

21 6 L. J. Grimm and W. A. Harris, Jr.

where 2

5

>0 and 0


7/42

On nonlinear difference equations 21 7

In particular,

if the

/l/s

are all

distinct,

the

system

(0. 21)

becomes scalar equations and the system can be solved recursively.

If ,

in

addition, ^yp^l

for all

indices j

and p, the

system (0.22)

has

diagonal homogeneous

fo rm

(0.25) w,(* + )= 0u00 w,00-

Af te r

obtaining the general solution of the system (0. 22) , we

can construct

the

general solution

of (0. 10) by

substituting

the

solution

of (0.22)

into

the

transformation

(0.17).

In

doing

so it is

necessary

to

estimate

the

magnitude

of the

solution

of the

system

(0.22).

If the

reduced system

(0. 22) is

normal

in an

extended

sense (we

shall give a precise def in i t ion of this concept in Section 7) we shall

find a region of the type

in

which

the

solution

is

uniformly

bounded

and

approaches zero

as x

tends

to

in f in i ty

in this

region. Choosing 0consistent with Theorems

1-3, the general solution of the original system (0. 1) is given in

this

region by

(0.26) X*)=000+P(*, t/(*,C(*)),

where U(x,C(#))

is the

general solution

of the

reduced equation

(0. 22) and COO is an arbitrary bounded periodic vector with period

one.

Thus

we have attained our main objective.

The

scalar case, n = l,

has

been treated

by J.

Horn

[9]

under

x

the assumption that f(jx, y) is holomorphic for I%I^>R

0

, \ y \


8/42

218 L. J. Grimm and W. A.

Harris,

Jr.

an

upper half-plane, whi le

w e

have established

th e

convergence

of

this

series. General results

of

essentially

th e

same nature

as

ours

have been obtained by W. A. Harris , Jr. and Y. Sibuya [7] in half-

planes of the for m ] Im x \ > a u n de r the condi t ions

U i | = l , and h U , | * ' = M , |

j

= l

fo r

all

fa.

O urresults for

non l ineardifference equations parallel similar

results

in the

theory

of

ord inary

differential

equations

for

systems

of

th e form

fo r

which

th e

corresponding linear system

--*.


9/42


219

(1.2) X*)=*(*)+*(*)-

Then

X* + )=/(#, X*))

becomes

(1.3) *(* + ) + *(* + )

=/(*,* (*)+(*))

and,

since 000 is a solution, (1. 3) becomes

(*)[*(*)+0(*)]P

which can be written as

(1 .

4) *(#4-i)

=400*00 +/(*, *00) =400*00

where

th e series is convergent for x0, d

z

sufficiently

small, and where the coefficients A(x) and

/p(^)

are

holomorphic for x^S

z>

and have asymptotic expansions

(1. 5)

as x

tends

to in f in i ty

through

the

sector

S

2

.

Equate

the

linear terms

in

(1.3)and

(1.4),

and

using

the

fac t that 0(^)=0(^"

1

)

we

obtain

(1.6) A =

fio(=/,(~,0)).

b) Remarks on Theorem 1. The region of validity of solutions

obtained

in

Theorem

1 is the

sector

S

2

of the

fo rm

(1.7)

where

0 an d p may be

described

as follo w s: In the

complex

d raw all the points

l o g U i l iarg^-,

i =l,'~,n. Then draw rays through each

of

these points extending


10/42

220

L. J. Grimm and W. A. Harris, Jr.

from

th e

origin

to

infinity.

These

rays will divide

th e

plane into

a

countable number

of

sectors,

Si, Sa, The

conculsion

of

Theorem

1 holds for all choices of 0 and p , p>0 , such that the sector

is contained in some sector Sy-

Exactly

one of the

sectors

S/,

call

it

X

0

, will have

one of the

following

properties:

i ) The

positive real axis will

be

interior

to S-

ii) The positive real

axis

will be the lower boundary of S

0

,

i. e., S

will

be a

sector

of the

form 0 < a r g C < C o

fo r

some Co>0.

It

is

clear that case

ii )

will hold

if, and

only

if, at

least

one of the

eigenvalues h satisfies 0


11/42


221

Case ii) 6 ^ 0

B

Figure

1.


12/42

22 2

L.J. Grimm

and

W.A, Harris, Jr.

inf in i ty.

However, the restrictions in Theorem 3, |0|+p


13/42


223

Let

A, B, Q

have

the

parti t ioning

A

=

induced by the parti t ioning of A

0

. If there is a solution of the

desired

form

then ,

'^11 A

X\ x\

:

A

n

A

r

B=

Bl1

.

0

0 D

n

rr

j

, G

=

L ?

1 2

'

Q l r

, G

r t

b ,

and the equat ion for determining

Q

becomes

(2.

7)

AQuA,=AMu-Qu

If Q is determined in

this

m a n n e r , th e n 5 and T are also determined.

Equ ation (2. 7) is a system of no nl in ear

difference

equations of the

form

(2.8) ^(^)=^(^,^^)=^o*W+C * ( ^ ) 3;+A*(^^^)

where

the

components

of the

vector h*(x,y,Ay)

are

polynomials

in

y an d

Ay

w i th

coefficients

that are

O C ^ r

1

) -

Hence, for

\x \ sufficiently

large,

#eS

3

, i.e.,

in some sector

fo r

a

e

>a

3

, we may rewri te th e system (2. 8) in the

form

(2 .9 )

4y

where


14/42

224 L.

/.

Grimm and W, A. Harris, Jr.

are & /

&,

i,j =l,--,r. Thus the problem of block-diagonalization

has been reduced to that of finding a solution of a system of nonl inear

difference equations

of a

form

to

which

Theorem 1 is

applicable.

Applying Theorem 1 w e obtain a solution Q(#) of equation (2. 7)

/x

in a sector S

4

cS

6

,

arg(xe~*

- a*)\


15/42

O n

nonlinear difference equations 225

500=2.8,*-'

= 0

(3.2)

as x approaches infinity

through

the sector S

B

. Suppose further

that BQ is

nonsingular

and that the eigenvalues of Bo

l

A

0

have

absolute value less than 1. Then there exists a unique bounded

holomorphic solution

y of

(3. 1)

in

some sector

S

Q

: \

arg(xe~

ie

-

0

8

) |.a$,

and possessing

there

the

asymptotic expansion

as

x


the

sector S

6

. Further,

there

exists a

constant

C depending only upo n the matrices B(x) and

A(x) ,

such

that for

(3.3) b

Proof:

Since

B

Q

is nonsingular , for

% ^S

5

, \x\ sufficiently

large,

x)

wi l l

exist.

Wri te

(3. 1) as

(3.4)

Since by hypothesis the eigenvalues of B^Ao have mo dulus less than

1, there exists

a

nons ingular cons tant matr ix P such that

(3.5) \ \P-*B^A*P\\


16/42

226


Define

an d let y = Pz.

Then (3.4) becomes

(3 .7 ) z(x)=R(

Let

L

=

sup||/00||,

S6

an d

M K L

V L

1

r

Let $ be the

fami ly

of all

m-dim ens ional vec tor fun c t ions

q >

(x) holo-

morphic forx S

6

, such

that

| |0>(#) | |


17/42

On

nonlinear 'difference equations

227


18/42

228 L. J. Grimm and W. A.

Harris,

Jr.

of

the magni tude of the r>. Write

Then

(4.3) || S

rqM || , /= , - , # , | t > [ =&. We

apply Theorem

1 as

above

t o

obtain

P(k, #) and G(k , x ) for kN

0

in a

sector

fo r some constants 0

9

( A ) an d

I t is im por tan t to show th at there is a single region of this form

in

which all the P(k, #) exist.

This

will be the case if we can show

that

\\A~\k,

x) 'C(k,x) ||


25/42

On nonlinear

difference

equations 235

in the uniform region S

9

(Af

2

) which we write as

Further, there exists

a

constant depending only

on A(x)

(with

0


26/42

236 L. J.

Grimm


to

yield

(6.

3)

1 0 1 = 2

where

g(x,y)

and

h(x,y)

are analytic for #e


27/42

On nonlinear difference equations 23 7

(6.7)

l

Suppress

the

a rg u ment

of w and

rear range

to

obtain

(6 .8) S Q

p

(^+l)(^(^)u;)P-^U)(S QpW^)

I P I ^ ^ V |p | ^JV

,R(x,

After

substi tut ing

the

representat ion

for

R(x,w)

given

in

( 6 . 4 ) ,

w e

obtain th e following formal

representat ions:

W e notice in part icular that th e system of equations for the Qp

obtained

by

equating coeffic ients

of w% is

precisely

the

same

as the

system

for the Pp

except that

th e

nonhom ogenous te rm

is different.

Since [ p |

>N>N

2

,

th e

p's

can be determined in a un i form region

(6.9) S

2

:|arg(*^"-\


28/42

238 L. I Grimmand W. A. Harris, Jr.

(6.10) S3HCP(*)|I


29/42

On nonlinear difference equations 239

an d

from

R(x

y

w^=w-\-

2 Q$(x}w$,

defined

the

func t ions

(6.

12)

R?(X,W)=WJ

+

S |QX*)#, and

R(x,W =W

1

+'-'+W

n

Define (^, v)=1?(^,5) with # as above.

Then

Then

th e

vector

(6.

13)

has

i'ih component

Then

(6 . 14)

By definition

i. e., the

coefficients

of z

k

in the

multiple power series

fo r gf are

positive

and not

less than

the

absolute values

of the

corresponding

coefficients

of the

series

for

gj. Similarly,

for all

j,

*(x, z)

for ail ;,

Thus

m,(x, w)= g> (x, R(x,

where R(x,w)

= (R,

,

i? )

r

. Since

all

terms

of

gf

are

positive,


30/42

24 0

L. j.

Grimm

and W. A.

Harris,

Jr.

Sum

over

j to

obtain

(6.15) m(x,0))=(*,

Let Wi

= v, i =

l,---,n.

Then (6.15) becomes

K#,

*0

as s

A(#,

C 6 = N

Notice that,

because

of the form ofR(x,w}, Jh

a

(x)=

| P =

for oj= 2, ,

J V 1 ,

and that m

N

(x)=0. Hence S maWv is a

Q 5 =

JV

+ 1

majorant fo r m(x,

v} g(x, v), since g(x

9

v)

is a

polynomial

of

degree

at most

N 1

in

v,

and the terms in

m

of degree less than

N+1

ar e

independent of the Q p,and are

hence equal

to the

corresponding^'s.

Recall that

R x,M ; ) =

an d

that


31/42

O n nonlinear

difference

equations

241

Since ||.4( )ll


32/42

242

L. J.

Grimm


W e

shall show

that

(6. 19) has a uniqu e fo rmal solution of the form

(6.20)

f

Then define g

k

an d

h

k

by

(6.21)

Substitute

(6. 20)

into

(6. 19) to

obtain

the

fo rmal equation

Since

this is to be a

fo rmal ident i ty

in

v, equate coefficients

of

v

k

to obtain

(6.

22)

f

4

*

=C

[

G

k

n

k

+

n

k

R

k

(O -

G

k

n

k

+

n

k

S

k

(f

a

) +

f

VA +n

k

T

k

(f

a

) ] ,

where

j ? ? * ,

S*, and T

fe

are d efined by

S n*

T

t

v"

=

=AT * =

JV

w = 2 fe =

JV

Notice that R

k

, S

k

, and T

fe

are all polyn om ials in the

a

(


33/42

O n

nonlinear

difference equations. 243

W e now

show that

(6. 20 ) is a

ma jo ran t

for

Q

y

i. e.,

that

The

proof proceeds

by

induction. First, notice that

for

k = N,

R

N

=

0

S

S

N

= h

N

, T

N

= 0. Since 0^00=0,

n

N

R

N

^m

N

.

Also

n

N

h

N

^

00^ ^

I N ( X ^ ) ,

since ^H

k

t

k

^>h(

9

), and the term h

N

is independent of the

k

= N

~ X\

f's, while /jv is independent of

p.

Since the Q's were determined in

( 5

2 j

,

and


34/42

244 L. J. Grimmand W. A. Harris, Jr.

by

subtracting the terms in the components of Q

a

, \a\=m

f rom

^ ). Recall that

N

1 N

If weexpand SftG^w+ S G

a

v

a

y in powers of #, weobtain

k

= N 05

= 2 =A T

where #

A

is a polynomial in f

a

for a


35/42

O n nonlinear

difference

equations 245

However,

since by the induction hypothesis,

S I I Qp(# +1)11


36/42

246

L.

/. Grimm

and W. A.

Harris,

Jr.

in an

essentially similar fashion

to

yield

%

m

J?

w

I>w

m

(:r )=

Thus

th e

estimate (6. 10 ) yields for all

S \\QvW \\^C(n

n

R

n

+n

m

S

m

+n

n

T

n

)


37/42

On nonlinear

difference

equations 247

CO=

Clearly the mapping is welldefined if 8


38/42

248 L. /.

Grimm


with

Ni th e

n i lpotent matr ix defined

in

( 2 . 2 ) .

The

nex t

k

2

k i

systems, corresponding

to the

indices

j =ki+l,ki + 2,,k

z

, arenon-

homogeneous systems

of the

form

(7. 2)

where

the

components

of g

j

ar e

polynomials

in the

components

of

u

l

,---,u

kl

.

Hence

the

general solution

of

(7.2)

can be

obtained

by

obtaining

the

general solutions

of all of the

systems

(7. 1) and

utilizing these

to

evaluate

the

func t ions

g

j

. Let g

j

(x} be the g

j

evaluted

in this way. The

remaining systems

forj = k

z

+ l, - are of

form

analogous

to

that

of (7. 2) , and we

proceed

in the

m a n n e r

described above to find the general solution of the redu ced system

(0.22).

Thus

the problem of solving (0 . 22) falls n atura lly into tw o parts,

the solving of l inear homogeneous equations and of l inear

nonhomo-

geneous of the forms

(7.3) (# + ) =4(*)(#), and

(7.

4) (* + )

=4(*)M(*)+(*),

respectively, where

A(#)=jul+N+f]A'x-,

s=l

where N

has the

form

of

N

{

above.

We consider the homogenous case (7. 3) first : A system of the

form (7 . 3) is called normal if there exist a formal

fund amen ta l

mat r ix

of the

form

where

R

is a cons tant m atr ix . O therwise the system (7. 3) is called

anormal.

If all of the

corresponding l inear homogeneous systems

are

normal ,


39/42

On nonlinear difference equations 249

we

m ay

assume that

all of the

nilpotent matrices N

are

zero, since

Harris [4] has shown that this may be effected by a linear trans-

formation

which

is a

polynomial

in X'

1

with determinant

n ot

identi-

cally

zero. Further,

it is

known [1] ,

[5 ]

that

in the

normal case

there

exist

analytic

fundamenta l

matrices which have the

formal

fundamenta l

matrices as asymptotic representations in right

half-

planes. Hence the behavior of the fu n da m en tal m atrix as x tends to

infinity is essentially determ ine d by //, but since 0


40/42

250

L.

/. Grimm


are

normal;

ii) if j u 7 * #

for

some r/ and integer 0/ for all j\

iii) there

exists

a fo rmal particular solution of the form

(log )

7

^00> where

Hence by the

results

of

Harris

an d

Sibuya

[5 ]

there

exists an

analytic

solution asymptotic to this formal solution in a sector of the form

(7. 5)

o This

particular solution has the same rate of growth as the

solution

of the

corresponding homogeneous equation.

By

induction

the

general solution

of the

reduced equation

(0. 22) can be

thus con-

structed in a region of the form (7 . 5) , if the reduced equation is

normal

in the extended sense, an d will have properties similar to

those

of the

general solutions when

th e

reduced equation

w as

linear.

W e

m ay

summarize

our

results

in the

following:

Theorem

4

0

Let the reduced equation

(0. 22)

beeither linear

or normal in the extended sense. Then the general solution of

(0.22) can bewritten in the form

(7 .6)

M 00 =*/(*, COO)

in a sector R of the form

where U(x, COO) is holomorphic in R, tends uniformly to

zero

as

x


R, and

possesses

an asymptotic

expansion

in

this region,

and

COO

is an

arbitrary bounded

periodic

vector of period 1 .

Hence, if 0 is chosen sufficiently small 0;>0, an d compatible with

the

hypotheses

of

Theorem

1

9

2, and 3,

(using,

for

instance,

the

sector

S defined in

Section

1 to

choose

0)we can

combine Theorems


41/42

On

nonlinear difference equations 251

1 , 2

9

3

9

and 4 to

obtain

in

this case

the

general solution

of (0. 1) in

th e form

X* ) = 0 0 0 + /> ( * , OX*, C O O ) ) .

8.

General remarks-

If the eigenvalues /I/ of the m atrix A

0

satisfy K |^ | , s imilar

results corresponding to Theorem 3 are available in sectors which

cover a region of the

form

0 < < a r g ( ^ 4 - ^ ) < 2 7 r ,

a>0 .

If we assume the existence of a particular solution, or

/(#,0)= -

0 ,

an d

then

by

choosing either

Ui=--=u

p

=

Q, or u

k

+i=

m

-=u

n

= Q, similar

results are available where now C OO will be either an n p or k

dimensional arbitrary periodic vector .

The possibility of obtaining the uni form asymptotic expansion

fo r the t ransformat ion P(x, u) has been demonstrated by Harr is and

Sibuya [7 ] und er more restrictive hypothesis including

the

uniform

asymptotic expansion

We shall treat this question in a subsequent paper.

O ne

would

expect that

the

results embodied

in

Theorm

4 are

valid wi thout

th e

restriction

:

n o r m a l

in the

extended sense.

BIBLIOGRAPHY

[ 1 ] G. D.

Bi rkho ff ,

General

Theory

of

Linear Difference Equat ions .

Trans.

Arner.

Math. Soc.

12,

243-284

(1911).

[ 2 ] G. D. Bi rkho ff and W. J. T r j i t z i ns ky , Analytic

Theory

of S ingula r D i f fe rence

Equations,

Acta

Math. G , 1-84

(1933).

[ 3 J W. J. A. C u l m e r and W . A. Harris, Jr., Convergent Solutions of Ordinary

Linear

Homogeneous D ifferen ce Equat ions , Paci f ic

J.

Math.

13,

1111-1138

(1963).

[ 4 ] W. A.

Harris,

Jr., Equiv alent C lasses of D i f fe ren ce Equa tio ns , C o nt r ibu tio ns

to

Different ia l Equat ions

1,

253-264

(1963).


42/42

252


[ 5 ] W. A.H arr is , Jr. and Y.Sibuya, Asymptotic Solutions of Systems of Nonlinear

Difference

Equations, Arch. Rational Mech. Anal. 15, 377-395 (1964).

[6] W. A. Harris, Jr. and Y. Sibuya, Note on Linear

Difference

Equations,

Bull.

Amer, Math. Soc. 70 , 123-127 (1964).

[ 7 ] W. A. Harris, Jr. and Y. Sibuya, Gene ral Solution of Nonlinear

Difference

Equations, Trans.

Amer. Math. Soc. 115, 62-75 (1965).

[ 8 ] W. A. Harris, Jr. and Y. Sibuya, O n Asymptotic Solutions o f Systems of

Nonl inear Difference Equations,J. Reine Angew. Math, ( to appear).

[ 9 ] J. Horn , Uber e ine n icht l ineare D ifferenzengleichung, Jber . D eutsch. Math.-

Vere in . 26 , 230-251 (1918).

[10] M. Hu kuh ara, Integ ration form elle d'un systeme d'eq uations non lineaires dans

le

voisinage d'un point singulier, Ann. Mat. Pura Appl.

IV ,

1 9,

35-44 (1940).

[11]

M .

Iwano, Integration analytique d'un systeme d'equations differen tielles

non

lineaires d ans le voisinage d'un point singulier I , II , Ann. Mat. Pura. Appl.,

44,

261-292 (1957);

47 ,

91-150 (1959).

[12] J. Malm quist, Sur 1 'etude analytiqu e des solutions d'un systeme d'equ ations

di ffe r

entielles dans

le

voisinage d'un point singulier d 'indetermination,

I, II,

III, Acta M ath. 73, 87-129 ( 19 40 ); 74, 1-64 (1941); 74, 109-128 (1941).

[13] W. J. Trj itzinsky, No n-line ar

Difference

Equations, Compositio Math. 5, 1-66

(1938).

[14] W. J. Tr j i tz insky, Analyt ic Theory of Non-Linear Singular

Different ial

Equa-

tions, Memor. Sci. Math., N o. 90 (1938).

[15]

A.

Tychonoff ,

Ein

Fixpunktsatz, Math. Ann. Ill, 767-776 (1935).

non-linear difference equation

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