Estimates for the solutions of linear difference and differential equations with variable coefficients

ESTIMATES FOR THE SOLUTIONS OF LINEAR DIFFERENCE AND DIFFERENTIAL EQUATIONS WITR VARIABLE COEFFICIENTS* A.A. KQRNEICHUK Moscow 19 Vovember

(Received

1964)

Introduction WEconsider efficients

the linear

homogeneous difference

i

equation

with variable

(Aj + aj, n+j).?Jn+j= 0,

co-

(0.1)

j=O n=o,

1,

. . . . aj,n+j

-+ 0 as n + 33 for

ak, n+k = 0 for n = 0, 1, . . . . we also constant coefficients

i

j = 0, 1, . . . , k - 1; Ak = 1; consider

Ajun+j

-L

the limit

0.

of the characteristic

with

(0.2)

j=i If h = hi are the roots

equation

equation

k -=I:

xdjhj

then the general

solution

of equation

0,

(0.2)

(0.3) can be written

in the forts

k--I

where ci are -arbitrary

l

Zh, v~chist.

constants;

Vat. mat.

Fiz.

5,

Vi = 0 if

4,

Ai iS a simple rOOt Of (0.3);

758 - 773. 1965.

266

Differential

for

a root

of multiplicity

Let x be the root whi.ch also

urith variable

equations

1 the quantity

multiplicitg

267

Vi takes values

amongst the smallest

has the largest

coefficients

roots

from 0

(in modulus)

G + I. Then, as

it

of

is

to

I- 1.

(0.3)

not

diffi-

cult to verify, the ratio !“;~I / (!f + 1,; t ii l!f remains bounded as M + .o for any UM which is a solution of the limit equation (0.2). And at the same time there is always a solution of (0.2) for which this ratio does not tend to zero as If + -0. The ratio fyql / (.A + 1); 1 x IJf cannot be estimated so simply for the solution of (0.1) with variable coefficients, since there is no exnression similar to (0.4) for the general solution. In Section 2 of this paper we obtain such estimates. It turns out that jy~,f[ f (M t l)“! x Iv remains bounded as M-t a if the variable coefficients of (0.1) tend to constant limits sufficiently quickly. Estimates for the solution of the difference equation (0.1) have been obtained comparatively easily after by subjecting this equation to a equation to a certain transformation; in flf this transformation was applied to a second order equation to estimate the computation errors for Jrcobi polynomials using a well-known recurrence formula (see [II, formula (0.1)). This transformation can naturally be generalized for linear k-th order differential equations with variable coefficients which Sections 3 and 4 of this paper are concerned with tend to constant limits. this problem. In Section 5 we give a comparison with several well-known results.

1. Transformation of the difference equation Let ;a be a solution initial conditions

of the limit

ug =

_ ui

=

.*.

=

equation

iik_2 =

and let yq be some solution of equation (n 2 k, N> n) and form the sum

It follows

N-n+k

k

2

2

from (1 1) that N A--i Yiv+i + 2 l=n

y’

c

(Ajf

(0.1).

i&&i

Nfj_f)gN+j-iiif

3 l=n-k

yl

2 j=o

=

satfsfsing

the

1,

(1.1)

We take the integers

=:

we can tr~sform this 7x-l I-n+&

aj, fUN+j-l f

a-=d

Ui,

0,

(0.2)

0.

n, N

W)

sum to the form

(Aj + aj, f)~N+j_f

=

0;

(1.3)

258

this

transformation

A.A.

Korneichuk

is described

in detail

in El].

Unlike the original equation (0. l), which connects the k + 1 values of y witb successive numerals, in (1.3) the value of yN+l is connected to all the preceding values UP to the (n - k)-th inclusive.

2, An estimate

for

the solution equation

of

the

difference

Let us suppose that the coefficients of the difference tend to constant limits as 1 -, QI so fast that the series

converge.

equation

(0.1)

Let

(2.1) Then there

is au n

such co

that k-l

P.2) Using this

n we find 12.3)

Gw (2.5) and prove the estimate l?f&fldc*(M+1);/x[

M .

f2.6)

For Y = 0, 1, . . . . n the estimate (2.6) follows from (2.3) and (2.5). Suppose that it has been verified for ‘il = 0, 1, . . . , IV; we then Prove (2.2) and (2.6) we have it for ,W= N t 1. From (1.3),

Differential

equations

with

variable

269

coefficients

n-1

x

+

l=n-k

j=O

III the first sum on the right-hand side of (2.7) R: + j - 1 t 1 f N t k IX < N t 2, since k g n; for the same reason in the second sum on the right-hand side of (2.7) N t j - 1 f 1 < A’+ k - n + 1 < iV + 2. Further, from (2.7). (2.4) and (2.5) we obtain

This proves

the following

theorem.

Let x be one of the roots

of the characteristic

sre largest in modulus snd let If the series

l=j

converge,

then for

coefficients

aj, 1 I (I +

it have the greatest

17,

WIY solution

the ratio

equation

Lyle /(II

i=O,

I,.

multiplicity

which 3 + 1.

. . , k-l,

yl of the equation + 1); 1% lz

(0.3)

(0.1)

with variable

remains bounded as 1 + co.

3. Transformation of the differential equation We consider the linear with variable coefficients

.4h = 1; aj(x) have finite clusive. Besides coefficients

k-th order

i[(aj + oj(~))~(s)I’j’ j-0 Uj(Z)+O Uk(x) = 0,

homogeneous differential

= 0,

OGx
equation

(3.1)

as Z-+03, i =O, 1, . . . . k--l;

derivatives with respect to n up to the j-th order in(3.1) we shall consider the limit equation with constant

iAj,lj)(Z)=

0I

(3.2)

o
j-0

If

&l = Mj are the roots

of the characteristic

$i,$ I-O

= 0,

equation (3.3)

270

A .A.

then the general

solution

Korne ichuk

of equation

13.2)

can be written

in the form

k-i O(5)=

2 Cj(Z+'I)'iexp~jZ?T, j=O

(3.4)

where Cj are arbitrary constants; ‘r* = 0 if a simple root of (3.3); for a root of multiplicity 1 the quitit, Tj takes values from 0 to 1 - 1.. Llj

Let u = G t $3 be the root part a, also has the greatest

of

(3.3)

which,

multiplicity

is

having the largest

?.

Then the ratio

(x + l)T exp & will remain bounded as z + co for the limit equation (3.2). Let ;(%I

be a solution

of

(3.2)

satisfying

the initial

U(0) = E’(0) = . . * = iP+2)(0) = 0, and let

y(x)

be a solution

of

(3.1).

any solution

real

lu(x) I I u(x)

of

conditions

$k-Q(0) T=i1,

(3.5)

We take some x0 & 0, x 2 x,, and

form the integral x

k

I(Aj+al(~))y(E)l(j)iI(z.-E)d5=0.

D

(3.3)

X0j=O

Using the initial conditions (3.5), we can tr~sform (3.6) to the following

integrating repeatedly by parts equation for y = y(x):

Y = BY + f,

(3.7)

where x R-i By =

f=

12 z. j=O

h-l

A-j

2

~((~j+~+~j+~(~))YfEf)Ifjh~~(I--11(5-~O)

j=O

I=1

a detailed description is given in Ill,

(3.8)

ai 6) E(i) (z - f) y (f) df,

of the transformation

of

(3.9)

; (3.5)

to the form (3.7)

4. Estiaate for the solutian of the differential equation We define the space RX,, of functions which are continuous with the norm II g IIR = y,mxx1 y(z) I/(x + ii);*exp%z. 50 -0

on [r~, a)

Differential

equations

with

variable

271

coefficients

If u(r) is a solution of (3.2) then, as we see from (3.4), it belongs with all its derivatives to DO. Further, it follows from (3.9) that f belongs to RX0 as a function of x. Suppose that the coefficients of the differential equation as the integrals

(3.1)

tend to constant

limits

as x + ;o as quickly

m*

s

Iaj(E) I (E+f)‘G

f=Qi,...,k-1,

0

He show that

converge.

in this

case By E RX0 if

y E

HzO. From (3.8)

we have 5

BY IIG%

Since finite

k-1

SE1~j(E)IIl~~~~ll~,(~--4:+~~‘~~~~~~~--4~~11~ll~,~4+~)r

d

x

(4.1)

Tgj=O

ai are finite, and, therefore,

from (4.1) Thus By E

the integrand in the right-hand side of (3.8) is By is continuous as a function of x. It follows

that the ratio. Jzo.

lsyl

/ (x + 15

exp ax is bounded for

x > x0.

Using the well-known theorem about the continuity of the limit of a uniformly convergent sequence of continuous functions, we can easily prove that RX0 is complete. Thus, Rx0 is a Banach space for any x0 > 3. Now choose

x0 so that a k-1 sz X0j=O

Then from (4.1) 3 (2.V)

we obtain

\\B/IR < o ( 1. In this case (see [31, Theorem x0 equatio!c (3.7) has a unique solution in RX0 and this

and 1 (3.V))

solution

satisfies

I aj(E) I II u(j) IIaO(E + i)‘dE < o < 1.

the condition II f IIR <-----. 20 l--o

II Y IIR

We have thus proved Theorem

the

following

x0

theorem.

2 .

Let G = z t i? ‘be a root

of the characteristic

equation

(3.3)

which

272

A.A.

has the largest

real

Korne ichuk

part 2 and also the largest

multlpllclty

T + 1. If

the Integrals

s cl converge, eoefflclents

then for

1aj(f) 1 (5 -I- l)‘dE,

any solution

the ratio b IIy(x)

y(x)

i = 0, ‘1,. . . , k - 1, of the equation

/ (x + 1) -r exp &

(3.1)

with variable

remains bounded as x + co.

In order to Illustrate the fact that the convergency requirements for the corresponding Integrals are essential, let us consider an example where they are not satisfied. Let us take the equation k! y(“)(s) - ___ (Z+1)RY(4

The limit characteristic city k. In this example

s

equation

I ao(5)

I (E+

=o:

pk = 0 has the root

l)k-'dS

=

k!

0

It Is not difficult

to verify

that equation

= (X + l)k for which the ratio bounded as x + co.

y(r)

- __ dE soE+'i (4.2)

-_

IA= 0 of multlpll-

00,

has the solution

[y(x) 1 / (z + Ijk-l

= x + I Is not

5. Comparison with known results Polncar6’s theorem (see solution of the difference to one of the roots of the Its roots are different In estimates for y,,+k.

[41 Chap. V, Section 5) states that for any equation (0.1) the ratio yu+k+I / yn+k tends characteristic equation (0.3) as n + co (If al! modulus). Sowever this theorem does not give

The equation i

(-4j + aj(r))sr(‘)(,)

= 0,

(5.1)

j=O

which, to within the accuracy of the factor (-l)j, Is conjugate to (3.1). was considered In [51, (Chap. 3. problems 35.36) on the assumption that the Integrals

Differential

r is

converge, characteristic that

equations

the maximum multiplicity equation (3.3); r > f.

equation

(5.1)

a solution

has

lim(y(z)/(x X-m is

described.

(X + ljT

The

exp

with

&

question

for

In

y(x)

of all the particular, for

the

c)=

boundedness

an arbitrary

solution

roots oP the limit a method of nroving

which

+ I)%(e&pcpkz)-

of

273

coefficients

variable

0

of of

the

lyh) I /

ratio

equation

is

(5.1)

not

con-

sidered. In and

conclusion,

V.P.

Shirikov

the

author

for

a number

wishes

to

of

thank

useful

Ye.P.

Zbidkov,

V. P.

Zbidkov

comments. Translated

by

Weinstein

X.

RSF%RENCES 1.

KORYEICWK, A. A., MrARKDV, A.S. and Jacobi polynomials (Vychisleniye Institute

of

Nuclear

Sciences

OS! SAN !!A The mnogochlenov

(OIYAI)

of

calculation Yakobi),

preurint,

No.

Joint 1733,

Dubna,

1951. 2.

and

differential

reshenii (OIYAI)

4.

XL’FOYD, raznostei)

5.

the with

koeffitsientami),

preprint,

A.@.

Finite

ential

equations,

nenii),

Izd-vo

and in.

linear

Dubna,

difference

(Otsenki

Institute

uravnenii of

Nuclear

s

Sciences

1954.

AKILOV, 9.P. Functional analysis in normalized analiz v normirovannykh prostranstvakh), 1959. difference

Fizmatigz,

CODDINGTON, E.A.

of

coefficients

i differentsial’nykh Joint

P-1372,

Moscow,

solutions variable

raznostnykh

KANTOROVICR, C.V. and spaces (Funktsional’nyi Fizmatgiz,

for

equations

lineinykh

ueremennymi

3.

Estimates

KORNEIC!WK, A.A.

Moscow,

calculus

(Ischisleniye

LEVINSON, N. The theory of ordinary (Teoriya obyknovennykh differentsial’nykh lit.,

konechnykh

19.59.

Moscow,

1958.

dilfer-

urav-