The stability of linear equations on a compact set

SHORT COMMUNICATWNS THE STABILITY OF LINEAR EQUATIONS ON A COMPACT SET* R.

DENCREV Dubna

‘neceived

14

November

1966)

The equation Ax = y

is

considered

subject

to the condition I( Br 11s .M

where A and B are selfconjugate commutative operators space H. The stability of the problem is investigated Let

SW(Y) ={ssH:

in the Hilbert for varying y.

IIBrIl SMM)

be a compact set in H. Let A be a continuous and A-’ a singlevalued (possibly unbounded) operator. Let % = B!Jk be the image of the set 9R . We denote by R( R) the contraction of the operator A-l on w It is known [II, that th e operator R(R ) is continuous. In this paper a method is proposed for calculating the modulus of continuity of the operator R( 951. (This method is to some extent a generalization of a method used by 1.1. Lavrent’ev [21 . ) We denote the modulus of continuity by O(E, M). It is easy to see that 0 (6 M) =

max{li z

that is, O(E. M) is the diameter in Hilbert space.

l

Zh.

vychisl.

Mat.

mat.

Fiz.

‘I,

II : IIAZ II 5

E, IIBX II d M},

of the intersection

6,

1367

-

1376,

of two ellipsoids

1967.

(1)

202

R.

Dencheu

We apply the method of Lagrange multipliers.

We form the functional

(D(z, h) = 11z II*+ hi (e2 - II Ax II") + hz(M2- II Bx 11’) It can be shown (as in the case of convex functions [3, 41), that if 2 is the point at which the maximum is attained in (11, il and ^hz exist satisfying the relations n x - “hiA”;- h&L; = o,, (2) !IBz”Ilshf,

IIAill S e,

(3)

~i(e-_AA^zII)+~a(M-~i~211)=o, L, iiB 0

(2)

We will - (4).

determine II ^xII for any vector ;, satisfying We will determine the maximum of the values

We first

(4)

the relations of 112 II obtained.

assume that both hl and A2 are non-zero.

Then (1 A”x11 = E, and I( ark (( = M. Multiplying

(2) by 2, we obtain (5)

11; 112 = Li,ez+ Ld42. Since A and B commute a spectral such that m A =

family

(6)

operators

E exists

m

f(6)&,

B=

‘~(0)dEe.

(‘31

s[1--.;2(e)-_~2(e)]dEe~=o.

(7)

s

-m

Substituting

of projective

in (2),

s --m

we obtain

OD

-m Let us consider

the expression

P(6) = I-

i#(6)

- ,\P$J~(~). We denote

by

N the set of its zeros; N is not empty, since otherwise it would follow from (‘7) that 2 = 0. Let us assume that N consists of a finite number of points. Let N contain

at least

two points

f2(elbw2)

61 and 62 such that

- f~(e2)cpvl)

z 0.

Then from the equations

2 - i,jyei)

- l;z(p2(&) = 0:

I . &;?(e2) - &(~z(f3~)= 0

we determine

A hl and ^hg, substitute Il,^ll =

them in (5)

[cp2(62)- cpz(e,)i e* + [fz(e,)-

f2(e,p+wz)--

fv32hw3i)

and obtain td(e2)i M2

The

We

denote and 02. If

for

stability

by ~I(E,

any

of

I)

the

two points

linear

equations

on

maximum of this

a;,

8j

E.

a compact

expression

203

set

with

respect

to aI

W

fz(0i)cp2(a,)

-f’(Oj)@(ei)

=

0.

we have

Ii = 2

- Eell _&

(‘y

= Zik,

i9

B&E ‘V

-2 = zf'(ek) bAl12 = &‘, l!Axl!

ilBsit

-2

~~2(~A)~i~k~~2

=

M2.

b9)

Therefore, II2 II =

11 2k I) and ok satisfy

where

We denote

II Xh 112),

iw

(9).

:U) the

by 02(s,

1(X

maximum of

(10)

subject

to the

condition

(9). Finally,

let

N consist

of only

From this

we obtain

11f(e,)X

that

u -=

81

one Point

81.

'Then

h (Ee, - E4,._0)2,

;= I!.& 11-

the

E,

!I A

5FI*,isfies

11 =

the

cp(el)

ii (p(e:)X

II = M.

relation

M

l-1f(&)

(11)

= e

and M

E ;I;Ij=_=-.

We

for

M) the

IfW

hew

I

maximum of

(12)

1

subject

(12) to the

condition

denote

by 03 (e,

We now

consider the case ., where ,I the,* numbers XI, k~ equals 0, one of M. Let the equation x2 = 0. Then 11A2 11 = e, 11Bz II

(11). zero,

example,

I .- hlf2(ei) = c have one root

81.

Then

11;= l/f”(01), II&~.

Hence

from

(3)

we obtain

(23)

for

81 the

if (w

I

condition

(141

204

R.

If

equation

one of

(13)

of

them will

(141

subject

has more than satisfy to

The modulus

of

the

numbers

01.

Let

us consider

of

one root

(14)

condition

it

and (15).

is

easy

Let o~(E,

to

see

that

M) denote

at least

the

maximum

(15).

continuity 02,

Dencheu

of

the

operator

R( 3)

equals

the

greatest

03 and 04,

an example.

Let

8

H = L2(0, I),

s

Bx=ig,

x(t) dt,

Ax =

0

that

is,

the

equation

*

c

s(t)dt

=

y(t)

n

is

considered,

In this

subject

case

to

equation

the

(31

condition

is

of

1

d2;

ds

: Fourier

form

s s&)dz+ iz-

4t)-d, By the

the I

==o.

dt2

0

transformation

* x(t) =

“,

1

_I..- 1 ‘)/Pn)_m

ectei(e)dtl

we obtain

ti2e4 - 82 + L) y (e) = 0. Let iI,

chg # 0.

AI = 0 exist

such

At least that

it=----, and

!I21 = Equating

to

zero

two roots

814 # 8~~.

81,

82 of

the

Then

ei2e22 82 + e22

L-L-

82 + e22

ei2e2w + ~2 ei2 + e22

Jf /at?, and af / a&, we f hd

=

f(e,, 0,).

equation

Age4 - O2 +

The

stability

of

linear

equations

mxf(thCbj=

on

a compact

205

set

y(eM)

01%

112 1) = ~0. From this If & = 0, from (14) and (15) we obtain to the condition 04 G .H*/e* we have rnaaxllZll = )((s, W . Therefore,

for

the

example

subject

considered e)(s, Mj = l(e, M).

by

Translated

J.

Berry

REFEREWCSS TIKHONOV, A.N. On the stability IVauk SSSR 39, 5, 195 - 198,

2.

LAVRENT’EV. M.M. Some Incorrect problems of mathematical Akad. Nauk SSSR, Novosibirsk, 1963. Izd-vo

physics.

3.

ROCKAFELLAR. R.T. functions. Duke

for

4.

KHENZ, H.P.

Extension (Math.

J.

of inverse 1944.

DokZ.

1.

problems,

of Fenchel’s duality 33, 1, 86 - 89, 1966.

and CRELLE, V. Non- linear

programming,

theorem

Mir,

Akad.

convex

MOSCOW,

1965.