An approximation property for abstract differential equations

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

M. A. M ^ALIK

An approximation property for abstract differential equations

Rendiconti del Seminario Matematico della Università di Padova, tome 55 (1976), p. 45-48

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Approximation Property

for Abstract Differential Equations

M. A. MALIK

(*)

1. - Let X be a reflexive Banach and A be a closed linear

operator

with domain

D,

^{dense in X. Let} A* be its

adjoint

with domain

7 the dual space of .X. ^Let also

[a, b] c R

^{be an}

interval;

R

represents

^the real line.

By 5)[a,63(X*)

^{we mean} the space of ^all

infinitely

differentiable

(X*-valued)

functions defined on

[a, b]

^with

compact support

^and

9)’ Ea,63(X)

the space of X-valued distributions

on

[a, b]. Similarly

^{we define}

1‘~R(x)

^and

0,(D,.).

^{Note that}

DA

and

D~*

^are also Banach spaces under their

graph

^norms.

Consider a

homogeneous

abstract differential

equation

For

convenience,

^{we write}

DEFINITION 1.

By

^{mean the set of all those U E}

~)~~(~)

which are weak solutions of

(1 )

^{on i.e.}

(~Z*~==0

^{for all}

wED’[a,q](DA*).

(*)

Indirizzo dell’A.: Concordia

University

of Montreal.

Research

supported by

^{N.R.C. of Canada.}

46

DEFINITION 2.

By Nra,bJ

^{we mean the set} of all those ^u E

which are solutions of

(1 )

^{i.e. Lu = 0 on a c t c b.}

’

Similarly,

^{we define}

YR

^and

NR .

2. - In this _paper we prove the

following.

THEOREM.

I f

the abstract

differential operator

^E

satisfies

Hyp.

^{I. Let d} &#x3E; 0 be

f ixed.

^E and supp

[a, b],

then supp 99 ^c

[a - d,

^b

+ d],

j

⁼

0, 1, 2,

^{.... The « const o}

depends on j

^acnd

support of

99. Then

VR

is dense in under the

topology o f

^·

REMARK. This kind of results has been studied

by

S. Zaidman

[5]

for where

Hyp.

^{I holds}

weakly

^{with L1 =}

0 ;

H is a Hilbert

Space.

These results are related to the

problem

of existence of

global

solution of Lu ⁼

f .

^In

[4]

^Zaidman

proved

^an

approximation

prop-

erty

^{to ensure} the existence of ^a

global

solution whereas in

[2]

^and

[3]

the author used Hahn-Banach theorem after

establishing Hyp.

^{I and II}

under suitable condition on the resolvent of A*.

In section

3,

^we

present

^a variation of ^an

example

^of

Agmon

^and

Nirenberg [1]

^to show that the result is the best

possible.

PROOF OF THE THEOREM. Let q ^E such that

for all x E

YR .

^To prove the

Theorem,

^{it is}

enough

^{to show}

h, q)

^{= 0}

for all h E Extend _(p ⁼ 0 outside the interval

[a + L1,

^{b -}

L1].

We first observe that 99 ^E

Z* ( 1~R(D~* ))

where the closure is

being

^taken

in

9),(X*).

^In

fact,

^if there exists UE

9),(X) , (recall

^X

is

reflexive)

^{such that}

for all k E

9,,(D,.)

^and

From

(3)

and the definition of

TTR,

^{one has}

^{U E YR}

^{but then}

(4)

contradicts

(2)

^{and so} the choice of _99. Thus ₉₉ E

L* ( ~R(DA* )) .

Now consider a sequence

5),(D.,.)

^{such in}

~R(X * ).

From

Hyp.

^{I and}

II, kn

^{is a}

Cauchy

sequence ⁱⁿ

~R(X*).

^Since

0,(X*)

is

complete,

there exists k such that in

2)~(~*).

^{It is easy to}

verify

that A* and so L* =

(lji)(dldt)

^{- A* is a} closed linear

operator

with domain dense in

~R(X* ).

^{Thus and}

from

Hyp.

^I

supp k

^c

[a, b]. Consequently

^{k E}

5)r,,,63(DA.)

^{and L* k = 99.}

Thus for an

arbitrary

choice of h E

Vr,,,,b3

^{one has}

This

completes

^the

proof.

If is a weak solution of

(1 ),

^then

= L(D[a,b](R) ; DA) .

^In

fact, consider w = y X x

^where

D[a,b](R)

^and

As U is a weak solution of

(1)

from where

represents

^the

duality

between X and X*. Hence

DA (A**

^{= A as} the space is

reflexive)

^{and so}

for

To conclude that U e we observe if

u, 1jJn)

^{converges in}

X,

in view of

(8) 1p n)

^{also converges in X.}

It

clearly implies

^that

Nra,bJ.

^{So we have}

^proved

^{the fol-}

lowing.

COROLLARY. Under the

Hyp.

^{I and II}

o f

^{the dense}

in the

topology o f

^·

3. - An

example.

Let X be the Banach space

consisting

^{of all}

continuous

complex

valued functions defined on 0~1 ^and

vanishing

48

at the

origin.

Define A =

(i/4 ) (d/dx)

the closed linear

operator

^{on X}

with domain

D~ consisting

^{of all} c’-functions in

X; 4

&#x3E; 0. Consider the

equation

Let u c- is a solution of

(9)

^on

a c t c b.

^{It is} obvious that

x)

^{is also a} solution of

(9)

^{and so}

x)

is constant

along

the direction

(d,1 )

^{as its} directional derivative

along

that direction is zero. Since

(u*«)(t, x) = 0,

^{one has}

(u*«)(t, x)

^{= 0 for J. As}

« is

arbitrary

^we conclude that

By using

^{a sim-}

ilar

argument

^{one can show that if u E}

~R (DA )

^{is a} solution of

(9)

^then

u - 0.

Thus,

^both

NR

^and when restricted to

are identical. In fact both vanish.

REFERENCES

[1] S. AGMON - L. NIRENBERG,

Properties of

^solutions

of ordinary differential equations

ⁱⁿ Banach space, Comm. ^Pure

Appl.

Math., 16 (1963), pp. ^121-239.

[2] ^{M. A.} MALIK, Existence

of

^solutions

of

^abstract

differential equations

^{in a}

local _space, Canad. Math. Bull., ¹⁶ (1973), pp. 239-244.

[3] ^{M. A.} MALIK, Weak

generalized

^solutions

of

^abstract

differential equations,

J. Math. Anal. Math., ⁴⁰ (1972), pp. 763-768.

[4] ^S. ZAIDMAN, ^Un teorema i esistenza

globale

per alcune

equazioni differen-

ziali astratte, Ricerche di Matematica, ¹³ (1964), pp. 57-69.

[5] ^S. ZAIDMAN, ^{On a certain}

approximation

property

for first-order

^abstract

differential equations,

Rendiconti del Seminario Matematico dell’Univer- sità di Padova, ⁴⁶

(1971),

pp. 191-198.

Manoscritto

pervenuto

in redazione 1’8

gennaio

^1975.