On Difference Approximation of Time Dependent Nonlinear Evolution Equations in Banach Spaces

(1)

NII-Electronic Library Service

M]MolRs oF SAGAMI

INsTITuTE oF T-cENoLoGy Vol.17,No. 1,1983

On

Difference

Approximation

of

Time

Dependent

Nonlinear

Evolution

Equations

in

Banach

Spaces

Kazuo

KoBAyASI"

This paper proves the eonvergence of solutions of a certain approximate differenceequation associated with the nonlinear time

dependent

evolution equation: du(t)ldteA(t)u<t)+f<t),u(O)=xo,

t20; where

{A(t)}

is

a _given

family

of operators ina Banach space. The existence of such

dif-fereneeapproximations isalso investigated.

1. Introduction

In

this paper we treat with a nonlinear abstract

Cauchy

_problem

in

a

Banach

space

X

of

the

form:

(cp;

f,

e)

I:?oStL/dgte

A(t)u(t)

+f(t)

, osts T6 ,

where u(t)

is

an

X-valued

unknown

function

on

[O,

T6];

{A(t)}

isa _given family of operators

in

X

defined

for

a.e.t

in

[O,

To];

f(t)

is

a _given

funetion

which

belongs

to Li(O,To;X).

A

large class of nonlinear partial

differential

equations

that

appear in mathematical _physies is

reduced

to

the

abstract

Cauchy

_problem

(CP;f;

e),

whieh

has

been

studied

by

many authors, e.g.,

Kato

[6],

Crandall

and

Pazy

[4],

Crandall

and Evans

[2]

and

Evans

[5].

Our

main _purpose of

this

paper

is

to give a convergenee theorem

(Theorem

2.1 below)･

for

solutions of a certain approximate

difference

equation assoeiated with

(CP;f,e).

A

similar convergence

theorem

was obtained

by

Evans

[5].

However,

for the _proof we shall

use a technique of

Kobayashi

[8]

whieh

is

more elementary than that of

[5].

The

limits

of solutions of the approximate

difference

equations are regarded as _generalized solutions

to

(CP;f,

e)

and we shall call

them

DS-limit

solutions of

(CP;f,

e).

In

Seetion

2

we _prove the convergenee theorem stated above and

in

seetion 3 we state

some properties of DS-limit solutions of

(CP;f,

e).

Seetion

4 is devoted to the existence

of

DS-limit

solutions

(Theorems

4.1

and 4.2

below).

Let

X

be

a

Banach

space with norm

ll

II. We

mean

by

an operator A

in

X a subset of

XxX.

The

domain

D(A)

and range

R(A)

of A are

defined

by

D(A)={xeX;

[x,

y]eA

for

some yeX}

and

R(A)=:{yGX;

[x,

y]eA

for

some meX} .

For

operators

A

and

B

in

X we set:

A+B={[x,

y+z];

[x,

y] e

A,

[x,

z]c B} _;

Ari

={[y, x];

[x,

y]e

A};

1[IAxHl=inf{HyH;[x,y]eA}

for xeD(A).

* _#Xscli

_asen

₁₉₈₂ tP ₁₁

_fi

₂₅ _Hecve

(2)

Shonan Institute of Technology

ShonanInstitute ofTechnology

NecIIIecit\reva

eg

17

#

M

1e

An

operator A issaid to

be

dissipative

if

II(x-)Iy)-(u-Av)]llllx-ull

tor

all 2>O, and all

[x,y],[u,v]eA.

The

resolvent ,1} and the

Yosida

approximation

A2

of

A

are

defined

by

.Jl!=(I-2A)rt and

Aa=2-i(Jh-l)

where

l

denotes

the

identity.

Finally

we set:

TA(x, y)=(ll_x+2y

]l-Hx

lD/a

;

T.(x, y)=lim Ta(x, _y)=inf r2(x, _y) ;

ale

i>e

T"(X, Y)=-T+(X, -Y) .

It

is

easy to see that A

is

dissipative

if

and only

if

T-(u-x, v-y)SO forall

[x,

_y],

[u,

v] E A,

The

following

lemma

will

be

easily _proved

(e.g.,

see

[5]):

Lemma

IJ.

Let

x,y,zeX.

(i)

T_±

(nx+y,x)=rpllx[1+T.(y,x)

fbrneRi.

(ii)

T±

(17X,

t!Y)=nyStT±

(X,

Y) .fo7',7t!>O･

(iii)

T±

(x,y)Sllyll･

(iv)

r-(x-y,x-z)Z-Ilx-yll-11y-g]1･

(v)

r+(･, ･):XxX->Ri

is

vpper semicontinuous.

2. Convergence

of

finite

difference

approximation

In what

follows,

let

TG(O,

To)

and

{A(t)}

be

a family of operators in

X

defined

for

a.e.te

(O,

Te).

We

denote

by

Nthe

null set off _which

A(t)

is

defined,

_and set

D=

U

{D(A(t));

te[O,

Te]KN}.

We shall centstruet a eontinuous funetion u(t) assoeiated with

(CP;f,e)

through

sequenee

{x:}

solving an approximate

differenee

_problem:

<2.1)

(x:-x:.i)f(t,n-t,"+t)eA(t:)+z:,

k=

1,

2,

･-･,

IVL,

n=1,

2,

･･･,

<2.2)

lim

xon=e,

n-oo

where

{tr}

isa _partitionri.={O==to"<tr< ･･･_<tx.=T.} such that T571,$To and

<2.3)

limld.l==O with

IA.I=

max h." and hC:=t:-t:r,,

n-co ISkSN"

<2.4)

lim

[1

_f"-f

HLt(o,T.tx)

==O

n--eo

where

f"

isthe step

funetion

whieh takes eonstant value zC on

(tk"-i,

t:],

k=1,2,

･･･,

Definition

2.1. An

approximate

differenee

seheme

(2.1)

satisfying

(2.2)-(2.4)

is called

a

flnite

diference

approximation associated _with

(CP;f,

e)

_on

[O,

T] or _simply _a

DS

of

(CP;f,

g).

The

function

defined

by

<2.s)

v"(t)=IE'

l･if

::?t:-,,t.n]

(3)

-60-NII-Electronic Library Service

On DitrlerenceAppromimation

_of

TioneDepe?uientNbn"near MvotutionEquatiens in Banach Spaees

is

ealled an n-th approximate solution of

(CP;.L

e).

If

the

limit

(2.6)

u(t) ==lim u"(t)

n-ee

exists uniformly

in

te

[O,

T] _and

is

_continuous, _then

it

is

expeeted that the funetion u(t>

will

beeome

a solution of

(CP;f,

e)

in

a _generalized _sense.

For

this

_reason

the

limit

u(t>

iscalled a DS-timit solution of

(CP;f;

e)

on

[O,

T].

To obtain the convergenee of u"(t) we

introduee

the following

type

of eondition

for

{A(t)}.

(A)

There

exist sequenees

{tu.}

and

{e.}

of nonnegative

functions

on

[O,

tTlo]

anda

func-tion

M:

[O,

71,]xXxX->R+

such that

A(t)c{[x,y]

eXxX;

M(t,

x,y)<oo},

tEIO,

T,]XN,

and

for each p, w.

is

nondecreasing,

lim.iow.(r)==O

and

(2.7)

ru

(u-x,

v-y)$

[M(s,

x, _y)+M(t, u, v)]

[w.(l

s-t

i)+e.(s)+e.(t)]

for

all s,tG

[O,

Tb]XN,

[x,

_y] e

A(s)

and

[tt,

v] G

A(t).

Definition 2.2.

We

say that a

DS

of

(CP;f;

_e)

_is compatible

with

condition

(A)

(or

simply

(A)-compatibte)

if

(2.s)

1,igm.

(lltllmi.

!g"

e;(r)dr) -o

where e;

is

the

Riemann step

function

defined

by

e}(T)=e,(tk") on

(t:-i,

t:],

ic=1,2,

..･,

_IVh.

and

if

(2.9)

{M(t:,

x:,mC, ye)}

is

bounded

where y:==(x:-x:-i)lh,"-z,".

Theorem

2.1. Let

eeD,

fGL'(O,

To;.X)

and

TG(O,

T6).

Assume

that a

fZtmity{A(t)}

satisfies condition

(A).

ILf

there

is

an

(A)-compatibte

DS

of

(CP;f,e)

on

[O,T],

then

there

exists a

DS-timit

solution

of

(CP;.L

e)

on

[O,

T].

For

the

proof of this theorem consider another

Cauchy

_problem

(CP;.fl

e)

with

fG

A A

Lt(O,

To;X)

and

ecD.

And

let

t:,di,g:,･･･ express an

(A)-compatible

DS

of

(CP;f;E),

that

is

(2.1)-(2.4),

(2,8)

and

(2.9)

are valid with

t:,x:,

_y:,･･･ replaced

by

tA,",di:,

_{g:,･･･.}

In

what

follows,

superscript n of

t:,

t'k",x:,

e2,

･.･ may

be

omitted

if

there

is

no confusion.

Now

choose sequenees

{.flo}

and

{i}

from

C([O,

To];

X)

_so

that

f}

-).L

L->fi'n

L'(O,

7ts;

X)

as _p.oo.

For

eaeh e>O

take

a t,G[O,Te]NN and an x,eD(A(t,)) so that

[IE-x,[ke.

_By

the

properties of

M

and

(2.9)

there

is

a eonstant

C=C(e)

such

that

M(t:,

xc,y:)+M(tAm,,

diy.

,

g?)+

M(t.,

x,, y,>S.C for all k=1, 2,･,･,IVL,_j'=1,

2,

..･,

IVI.,

n, m).1, where y. isa

fixed

_element

belonging

to

A(t,)x,.

Then

we set:

E=

C{(e.(To)+e.(t,)}+

H

fl,

]]

oe+

I]

L,

ll-+1]

y,

[l

;

e:=

Ce.(tC)+

[1

z:-.f},(t2)

]1

_, e}t==

Ce.(t"m,

)+

ll

E]t-jtil,(t"m,

)

ll

;

a(t) ==

Cto.(t)+sup

{11

fl,(s)-.f;(r)

[1;

s, re

[O,

To],

[s-rlS.t}

for k=1, 2,･･･,_N;,,_J'--1,

_2,

･･･,

_IV;.,

_n, _m).1 _and _te

_[O,

_T6].

_(li

･

ll..

denotes

maximum norms.>

We

note that

E,

e:, E7･and o(t) may

depend

onsand _p.

(4)

Shonan 工nstitute of _Teohnology

　　　　　　相模工業大学紀要　第 17 巻　第 1 号

Then

，　assuming 　condition （A ）for ｛A （の｝，　we 　can　state 　the　following　

lemmas

．

Lemma

　2 ．2．　

We

have

　　　 k （

2

．

10

）

1i

田_羅一_{¢ ♂}

Il

_≦

ll

　xtt− x_、

11

＋【

1

£

8

− x、　

ll

＋

1

写衂＋ Σユε讐ん？　　　 ‘旨1 ／

br

k

＝ 1 ，2，…　　，从　an （

1

％，m ≧1．

Proof

．　

It

follows

fro

皿（

2

．

7

）

that

　　　　　　　τ．軌一コ5、，Y、− Y、）≦ σ｛ω。（

T

。）＋θ。（t、）＋θ，（t、）｝

for

乃＝1_，2_，…_，ムln．

By

　u8ing 　the　

identity

　gk＝（xk− xk −₁_）

1h

_，− _zk　_and 　

Lemma

1

．

1

_，　

it

　can 　

be

seen 　that　the　

left

hand

　of　

the

　above 　

inequality

is

larger

　than

　　

τ一（x_厂 x_” x。− x。一、）

1h

、一　［

1

　z，＋y、［

1

≧ （

1

［x厂副一

ll

　x、、一、− x、　

ID

！

h

厂（

11

　z。　−

f

．（t、）

ll

＋

iL

ち

1L

＋

ll

　Y、　］

1

）．

Thus

，　

by

　the　

definition

　of ε歪we 　

have

ll

　x、、− x，］

1

≦

ll

　x、．、− x、

11

＋｛

C

（t・．（

T

。）＋θ，（t、））＋

1

【．

fp

【

1

．．＋

li

　Y、　［

1

｝

h

、＋・轟．

Ilence

，　

by

　the　

definition

　of　

E

　we 　get

　　　　　　　］［x_厂 x_、

ll

_≦

11

　x_、．_，− _x_、_］

1

＋

Eh

、＋ε轟．

Hence

，　an　iteration　yields （2 ．10）．　　　　　　　

Q

．

E

．］D．

　　　

Lemma 　

2

．

3

．　

Set

　ak，_，＝

il

嬬一

ew

1

］，　Pk．丿＝

脅

1

（ん舞十

E7

），　qk，，冨

1

− Pk ，」αnd 　rk，ゴ＝

h2p

店，_丿←

Erqk

．_∫）．

Then

ω θ

have

　　　　　　A ｛2 ．

11

）　　　　　　　αk，ゴ≦Pk，fα k＿t，i十qk，sak ．J＿1十rk．」｛ε量十E｝ t 十σ

G

t

瑟一

t

野

1

）｝プ

b

ゲ

k

＝＝

1

，

2

， …　　，ム乙、，ゴ＝

1

，2，・g・，ハ

r

．　and ％，？n ≧

1

．

Proof

．　

It

follows

from

（

2

．

7

）

that

　　　

・．（x_厂

eJ

，　U_厂

0

，）≦σ｛ω。（

lt

厂

t

，

1

）＋θ。＋θ。（

i

，）｝．

Since

yk

＝（x

彫一xk−1）！

h

，− xk　and 窃＝（£厂

di

，−1）

砺

一

2

，，

1

）y　

Lemma

1

．

1

　we 　

have

　　r、．_」τ．軌一鉱銑一

Pt

，）≧P、，_ゴτ一（Xk一臨 ¢厂 Xk．．、）＋qk，_ゴτ．（Xk一動鰯一

dij

．、）＋r、，fτ．軌一動

2

厂勾

　　　　　　　≧

Pk

，丿（α、，厂 ak ．、の＋

qte

．丿（ak，厂 αk．丿．∂− rre，，τ．（x厂

dif

，　ZiC−

2

丿）． Hence 　we 　have

（2．12）　　　α塵，_丿≦pκ，_ゴak＿1，J十qk．jak，ゴ＿1十？ °

匙，_」｛（7ep（tk）十

Cep

（tJ）十τ＋（Xk 一分J，　Zk−

2

，）十（

7tOp

（

ltk

冖‘，

1

）｝・

But

，　since τ＋（xic一磨，，　zk

一_彡 j）≦

ll

　zk−

fp

（虚訓

1

十［

1

2

_，一

ノ

｝

（

e

_，）

1

【十

1

げ』（

tk

）　−

fp

（診₅）［

1

，（

2

．

11

）follows　from （

2

．

12

），　　　　　　　

Q

●

E

●D ．　　　

Lemma

2

．

4

．　

VVe

h

αve 　　　　　　　 − 62 一 N工工一Eleotronio

(5)

On DUrbrence Approximation of Tiine Depenclent Nedinear Mvetutien Equations in Banach Sipaces

<2.13)

a(t)Sfi-'ia(To)lt-hl+o(26)

fbr

atl te

[O,

Tb],6e

_(O,

To/2)and he

[O,

a].

Proof.

If

te[O,26], a(t)$o(26)

for

o(t)

is

nondecasing.

If

tE<26,7LD], a(t)$a(To)$

fi-'a(To)It-hl

sinee 6fi'It-hlll.

[I]his

shows that

(2.13)

is

valid.

Q.E.D.

Lemma

2.5. Let

OE(O,

To12>

anel

let

n, m

be

integers

such that max{lA.I,1A".I}<6.

Then

we

have

<2.14)

_iL

xc-di?t

[ISII

xg-x,

[1+][

ew-x,

Il+Efl"+

£

]

E:-h?+

:{]

g:ziZ;-z+f?t{a-ta(

Il,)fl,.+a(n)}

l=1 i=1

,fbr

k=1,

2,･･･,

AIh,

o'=1,2,･･･,

IVh,

where .flt,j=={(t:-t"m,

)2+1

d.

I

t:+l

AA.

It"m,

}i/2.

Proof.

This

lemma

will

be

_proved

by

induetion as inKobayashi

[8].

First,

(2.10)

shows

that ak.J="xk-diJll satisfies

(2,14)

if

J'=O. By symmetry ak.j also satisfies

(2.14)

if'

ic==O.

Induetively,

assume

both

ak-i.j and ak.J-t satisfy

(2.14).

Then,

by

(2.11)

we have

k i. .

ak.,$II xo-x,

]l+

[1eo-x.

[I+ELf;t,j+

2]

eihi+

Z

eiht+pk.it,{6'ia(To)fle.-i,J+o(2fi)}

i=1

i=tl A

+qk,Jtimi{briu(To)flt,,･-i+a(2e)}+rk,N(1tk-tj

l)

.

Here

we used the

faet

that _pk,j+qt,j=1 and _{pk,dflt-t.s+qk,sA.J-iS.fl,j}_whieh

is

_proved

by

the

Cauchy-Sehwarz

inequality.

Moreover,

it

follows

from

(2.13)

that

rk,Ja(ltic-fA)Srk.j{6-ia(To)ilt,-t'jl-E"+o(2o)}

S-qk.,hj{6-ia(To)fL,,-i+a(26)}

.

Combining

these

ineqmalities,

we see easily that ak.J satisfies

(2.14)

again. Thus the

_proof

iseompleted

by

induction.

Q.E.D.

Preof

of

Theorem

2.1. If

we set

f=Jf)

e=g

and t."--f:,x::==dik",･･･in the _previous

lemmas,

tihen

(2.14)

gives

- -Nn

(2,15)

L=

lim

_{]Iza"(t)-um(t)1[$211e-x,Il+21im}

_Z

E?h?+

l}r(2S)

nlm--ee n--co t=1

for all

te[O,

T],

6e(O,

Th12),

e>O and p).1.

In

thiscase, we note that a(ee)-O _as

6JO

for

each

p).

1

and

ttm....

Z

X"i

ethi =:

C

lim.-.

Z2

."i ep(ti)hi +

ll

f-fl

IILt

(e, T,;r) -O as p-->oo

by

(2.4)

and

(2.8).

Therefore,

letting

6JO

first

and

in

turn

_p.oo

in

(2.15)

we obtain

LS

2[le-x,ll.

LettingeJOhere,

we

have

L=O,

whieh means

that

_the

limit

in

(2.6)

exists uniformly on

[O,

T].

(2.14)

alse _gives

- Nn

lim

ll

u(s)-u(t)

il52Il

6-x,

il+2

lim

Z

eihi+

Ta(2b)

s-t n-co i=:1

for

3,te[O,

T],

6e(O,

Ih!2),

e>O and p2.1.

Henee

we see that

lim,-tu(s)=u(t)

_as observed above, so that u(t) iscontinuous on

[O,

T].

Thus

the proof

is

eompleted.

Q.E.D.

(6)

ShonanInstitute ofTechnology

reecrec±

\vaet

ng

17

8 ig

1 e

Remark

2.1. The

inequality

(2.14)

is

a _{generalization} of that obtained

by

Kobayashi

[8]

to the time

dependent

case.

(2.14)

has

been

obtained

by

the author in

1976.

3.

DS-Iimit solutions

Our

first

result of

this

seetion

is

the

following

Theorem

3.1. Let

Te(O,

fts),

_e,geb

andLfeLi(O,

llo;X).

Letus

assume

that

{A(t)}

satishes condition

(A)

and there exist

(A)-compactible

DS's

of

(CP;f,

e)

and

(CP;

£

e)

on

[O,

T].

Let

u cand

ab

be

DS-limit

solutions

of

(CP;f;

g)

and

(CP;.

£

g),

respectivety.

7Vien

we

have

(3.1)

1[

u(t)-de(t)

[1

.<.

11

u(s)-de(s)

II+!i

T+

(u(rp)-a(rp),

f(n)

-Rv))dn

fbr

05sSt$T,

To

prove the theorem we use the same notation as

in

Seetion

2. For

R>O

define:

6:=e:+

H

z:-f},(tc)

l]

+TA(xc-a(t,"),

z:-.ll,(t:));

A A

6,M･

=EYt+22-iH

e,,.n-a(t,m.

)

]1

;

p(t)=

Ccv.(t)

+ _sup

{22-i

l]

de(s)-a(r)

H

+

il

L(s)

-L(r)

11

;

1

s-r

]

< t}.

Then we can _prove an estimate similar to

(2.14),

i.e.,

with the same 6,n, m and

flt,j

as

in

Lemma

2.5 we have

(3.2)

l]

x,n-di,m

]ISI]

pttt-x,

II+

lidi:-m,

]1+2ElfT,,,+

JE]

fi?h?+

;g]

Sm,

E?+t",m{6-tp(7laM,,+p(2b)}

±==1 ittt

for

k=1,

2,

･･･,

N;,,

d=1,

2,

･･･,

Nin.

Indeed,

adding

Z:t.i

{I[

z{-fl,(tf)

ll+

11 fl,

ll..+llJl,

ll.,+ra(xt-a(ti),z,-i(ti))}ht).O

to the right

hand

of

(2.10),

we see easily that ak.d=11xk-diJ[1 satisfieg

(3.2)

for

e'=O.

Similarly,

ak,j satisfies

(3.2)

for

k=O.

On

the

other

hand,

since T+(xk-es, z,-2,)$T2(x,-a(t,),z.-.f;(t,))+lle,-Jl,(tAj)ll+22-i]ldiJ-a(ti)]l+H.iS(t,)-.il,(tj)jl+22-illa(t,)-a(f,)".

we obtaine

from

(2.12)

and the

definitions

of 6:,

S,"-

and p(t)

that

- A

ak.fSPk,fak- ±.v+qk.Jak,j-i+rk.J{bk+6j+p(1 tk-tJ

D}

.

Here

we used the

fact

that T+(y,z)STa(y,z)$T2(u,v)+11z-vil+22-`lly-ull.

The

above

estimate shows that

(3.2)

is

valid for

k==1,2,-･･,IV;,,

j'--1,2,･･･,ATin

by

induction

as in

the

proof of

Lemma

2.5. Proof

of

Theorem

3.1. Without

loss

of _generality we may show

(3.1)

fors=O.

Pass-ing tCand

tT

to te

[O,

T] in

(3.2)

ag n, m-oo, we _get

h

(3･3)

ll

u(t)-a(t)

11$lle-x,n+]L

eA-x,

l[

-+-

II/iiim

Nz"

fi,h,+ lim

"z"

s,E,+Tp(2ti)

n-oo i=1 m-o] ttl

for

all

6e(O,

Tof2),

s>O and _p).1.

By

hypothesis

we

have

li-m..

l.l4.",

6ihi->jjTz(u(o-a(o,f(o-fto)ac and

ttt.2",s,h,.o

as _p.oo.

(7)

-64-NII-Electronic Library Service On DitTizrenceApproximation

_of

Time Dependent Nbntinear ElvotcationEquatio7vsinBanach Spaces

Henee,

Letting

let

6So

e,2LO

and

in

turn

_p->

oo

in

<3.3)

to

obtain

ilu(t)-iZ(t)11$ll6-x,]1+I[e"-x,[1+!ITa(u(C)-a(C),.f<C)-flC))clC

here, we obtain

(3.1)

with s=O.

Q.E.D.

Remark

3.1.The

inequality

(3.1)

was

derived

by

Evans

[5].

Theorem

3.2.

Let

a

fZtmity

{A(t)}

satiefy condition

(A).

Let

fin,feL'(O,

71,;X).

g,.,eEDand.f;,.fin

L'(e,Te;X),

_e.-e

as m-oo. Smppose

fbr

ectch m).1 _there

is

an

(A)tompatible

DS

_of

_(CP;.L,,e.)

on

[O,

T],

O<T<To.

Theua

there exists aDS-limit solu-･

tion

of

(CP;Se)

on

[O,

T]. '

Proof.

In

virtue of

Theorem

2.1,for each m).1 there isa DS-limit solution u.(t) of'

(CP;Je;.,e.)

on

[O,

T]

sueh

that

u.(t)=lim....u".(t) uniformly

tor

te[O,

T],

where u:(t)

is･

the n-th approximate solution of

(CP;.fin,

e.).

But,

by

Theorem

3.1 we

have

U

u.(t)-u.(t)

1]

S-

[]

e.-e.

[I+

[I

Ln-Si

llL'

(o,ro;x)

for

all

te[O,T]

and n,m).1, which

implies

that

u(t)=lim.-.u.(t) exists uniformly on

[O,

T]. Hence there isa subsequenee

{n}

of

{m}

sueh that u"(t)

is

a n-th approximate

solution of

(CP;f;

e)

and u(t)=lim.... za"(t) _uniformly on

[O,

T].

Therefore

_u(t)

is

a

DS-!imit

solution of

(CP;.L

e)

on

[O,

T].

Q.E.D.

4. Existence

Ourresults

of

of DS-limit solutions

thissection are the next three theorems.

Theorem

4.1.

Let

{A(t)}

be

a _.frvmily

of

operatoTs

in

X

dojined

.for a.e. t

in

[O,

T6].

Let

feL'(O,

To;X)

and

eGD,

Assume

that the .fottowing

two

conditions

holel:

(R)

R(l-XA(t))==X

fbT

alt sufiicienty smatt R>O and a.e. te

[O,

To].

(Hl)

There

are nondecreasing

functions

w,

L:R.-R

÷ and an

heLi(O,

To;X)

sucla

that

lim.io

tu(r)=O and

(4.1)

t(u-x, v-y)SL(IlxED

[a)(1

s-t

1)

÷

1[

h(s)-h(t)

ll]

.fbr a,e. s,te

[O,

Te]

and all

[x,

y]eA(s),

[u,

v] eA(t).

'Then

(CP;f,

e)

has

a

DS-Mmit

sotution on

[O,

Tb).

Proof.

Let

N

be

a null set off which

A(t),

f(t),

h(t)

and

(4.1)

are

defined.

Put

M(t,x,y)==L(lixii)

for

[x,y]eA(t)

and te[O, T6], and to,(t)==w(t)+sup[,ur.]st

[1h.(3)-h.(r)[1

and e.(t)=:

"h(t)-h.(t)

[I

for

te

[O,

T6]

and p=1,

2,

･･･, where

{h.}

is

a sequenee in

C([O,

TD];X)

satisfying h.->h in Lt(O,Te;X) as _p.co. Then itiseasy to see that

(Hl)

implies

(A).

Therefore,

by

virtue of

Theorem

3.2

we may show the existenee of an

(A)-compatible

DS

of

(CP;f,

g)

on every

[O,

T][

[O,

T6).

Now,

it

follows

frorr}

[5;

Lemma

4.11 (ef.

[7;

Lemma

Al])

that there exists a _partition

A.=:{O==t8<tT<

'''

<'tk.=

IL,}

sueh that t:eN,

TSTASTo,

id.1-->O

as n.oo, and

iff"

and

h"

are step

functions

defined

by

f"(t)=f(t:),

h"(t)=h(t:)

for

te(t:-i,t:]then

lim..-.

(8)

-65-Shonan Institute of Technology

ShonanInstitute of Technology

NecX*Jit\raet rg17 # ng1 ll

fllLice.T.;r)=lirnn-op

Uh"(t)neh[ILi(o,r.:x)=O･

Then

R(I-2A(t))=:X

implies that there exigts a

sequenee

{[xk",yk"]}

satisfying

(2.1)

and

(2.2)

with z:=flt,") and x:=e,

Since

e;(t)$llh"(t)-h(t)Il+llh;(t)-h.(t)ll+Ilh,(t)-h(t)H,

we have

(2.8).

Hence,

_to show that these tC,x:, yC, z,

form

an

(A)-compatible

DS

of

(CP;.L

e)

on

[O,

T]

we may _prove

that

{x:}

is

bounded.

It

now follows

from

(4.1)

_that

{4.2)

Il

x:-u

ll$IIx:"i-t`

li+hc[L(II

zt

ll)(to(l

t,"-t

l)+Il

h(t,n)-h(t)

ll)+

ll

_.fKt:)+v

[1]

for

k=1,2,

･･･,M, nll and

[u,v]eA(t),

te[O, To]NN.

(4.2)

implies that

{xtr}

is

bounded.

Q.E.D.

Theorem

4.2. Let

{A(t)}

be

a

fdmily

qf operatoTs

in

X

defined

fbr

a.e. t in

_[O,

TID].

Let

_feLt(O,

Tb;X)

and

eeD.

Assume

that

(R)

and the

foltowing

condition

hold:

(H2)

There

are a nondecreasing _.fiLnction

L:

R.-R.

and an

hEBV([O,

Tb];X)

such

that

<4.3)

TL(ze-x, v-y)SL(l]x]1)11

h(s)-h{t)

1I(1+[1y1I)

.fbr a.e. s,te

[O,

To]

_and all

[x,

_y]eA(s),

[u,

v] eA(t),

Then

(CP;f;

e)

has a DS-limit sotution on

[O,

tTts>.

Preof.

Put

M(t,x,y)=L(jlxll)(1+11yl[)

for

[x,y]eA(t)

and te[O,llD], and w.(t)=

sups,-.ist11h,(s)-h.(r)U and e.(t)=IIh(t)-h.(t)11

for

te[O,

To] and p==1,2, ･･-, where

h.

is

defined

as

in

the _proof of

Theorem

4.1. Then

(H2)

implies

(A).

Therefore,

by

virtue of

Theorem

3.2

we may show the existenee of

(A)-compatible

DS of

(CP;f;e)

on every

[O,

T]c[O,

rla)

for

6GD

and

feBV({O,

T6];X).

Now,

lett:,x:, _yC, zC

be

the same objeet as

in

the proof of

Theorem

4.1. Then,

as

seen in the proof of Theorem 4.1we may show that

{x:}

and

{y:}

are

bounded.

It

follows

from

(4.3)

that

<4･4)

_l]

xc-u

ll$11

xk"-i-u

l]+hc[L([l

u

[D(1+

[l

v

U)I]

h(t:)-h(t)

ll

+

ll

f(t:)+v

ll]

for

k=1,2,

･･･,.N;,,_nll _and

_[u,v]eA(t),

_tE[O,

_To]XIV;

_where

_N

_is

_a null set off which

A(t),

_.f(t),

h(t)

and

(4.3)

are

defined.

(4.4)

implies that

{x,"}

is

bounded.

Next,

by

applying

<4･4)

with t=t:Ji,u=xk"-i and v=y:-i we obtain

that

dte=ll

y:+f(tk")

II=ll

x:-xk"-iillh: satisfies

the recursive _egtimate

dk

S. (1+C1]

h(t,")-h(tk"r

i)

]Dd,T,+C

]1

h(t,")-h(t:-t)

ll

+

ll

.ilt,")-f(tr-,)

11

where

C=L(supilx:ll)(1+11fH.).

This

implies

that

dtSexp

(C

Var

h)(di+C

Var

h+Varf)

.

Var

means

total

variations. However, since

di

is

majorized

by

a eonstant

independent

of

ei,yC as well as

dk

is

bounded

independently

ofnand

k.

Q.E.D.

Remark

4.1. If

eaeh

A<t)

ig

dissipative

and

<5･4)

Utl}(t)u-Jh(s)xllS-11x-ull+aL(llxll)Ilh(t)-h(s)ll(1+IIAa(s)xll)

holds

for

all s,te[O,

fk]NN,

xeD(,11(s)), uED(,11(t)) and 2>O, then

(H2)

is

true.

Here

Jl(t)

(9)

-ca-NII-Electronic Library Service

On DQ7larenceAppromi?nation of Time Dependent Nontinear Ovolvtion Bquations in Banach Sipaces

and Aa(t)

denote

_the _resolvent _and the Yosida approximation _of A(t) respectively.

In

fact,

sinee y=Aa(s)xa, x2==x-Ry, and v!=Aa<t)ua, ua==u-2v

for

all 2>O,

Ix,

y]eA(s) and

[u,

v] e

A(t),

it

follows

that

T-(ua- xa, v-y)

SR-i([I

JI(s)xx-Jl(t)ua

H-

II

_x2- _tta

ll)

5L(II

xa

ll)[1

h(s)-h(t)

Il

(1+jl

y

H)

.

Thus,

letting

ZJO,

we

have

(4.3).

This

type of condition

has

been

introduced

by

Evans

[5]

Crandall-Pazy

_[4].

Next,

let

q:

[O,

To]xX-)

[O,

oo]

be

a

lewer

semieontinuous

funetion

with

domain

D(q)=

{(t,

x);q(t, x)<oo}.

And

define

f?1=:{xeX; _q(t,x)<oo} foreach te

[O,

To].

Let

a,

b:

D(q)

->R.

be

functions

such that

for

eaeh

(t,

x) eD(q) and each

K>O

there exist r,

h>O

and ao,

boe

Li(O,

t+h)

satisfying thata(s, _y)Sao(s) and b(s,_y)Sbo(s) for a.e. sE

(t,

t+h) and _ye

B(x;T)=

{y

e

X;

]I

y-x

11 (.

r} with _q(s,_y)

SK.

Theerem

4.3.

Let

a

fomily

{A(t)}

be

dojined

eveTywhere on

[O,

To] and satisLfly

D(A(t))c･EZ _.foT all t. Assume

that

the

fbltowing

three

conditions

hold:

(H3)

There

are sequences

{tu,}

aved

{e,}

of

_positive _.ftenetions

dqltned

on

[O,

To]

and

a nonclecreasing

f:te7wtien

L:

R.->R.

szach that _.foTeach p to.(t)

is

nondecreasing

in

t,

e.(t) is Riemann integrable, a).(t)-,O as teO, and limp-..

11epIILio,T,)==O,

<4.6)

t(2e-x, v-y)

SL(11

x11){a,.(1s-t

1)+e.(s)+e.(t)}(1+q(t,

x))

for

atl s, tG

[O,

T,],

[x,

_ylGA(s) and

[u,

v] EA(t), p=1,

2,

･･-;

(Cl)

lf

t.E

[O,

To),

X.eD(A(t.)), t.tt and x.-x,

then

xeD(A(t));

(C2)

Fbr any e>O,

tEEO,

T,) and xED(A(t))neq there aTe a

fi>O

and a

{xb,yfi]e

A(t+fi)

satiefying

"

xe-tiyo-x1]S6E ,

q(t+6, xo)S(exp

!i'O

a(s, x)ds)

[q(t,

x)+!i'"

_b(s,

x)dsl .

Then,

fbT

each

eeD(A(O))n9h

there

exists a positive number

Tt

sueh that

(CP;O,e)

has

a DS-limit solution on

[O,

Te].･

Moreover,

if

a and

b

are constant _.functions, then

we

have

Te=:To.

Proof. For K>q(O,g) take r,T>O and ao<s),bo(s)eLi(O,T') so that a(s,x)S.aD(s),

b(s,x)$bo(s)

for a.e. sG(O,

T')

and xGB(e;r) with _q(s,x)SK.

Fix

[u,v]eA(O)

so that ueB(E,r/4).

Then

(4.6)

implies that T-(x-u,y-v)$M for all

[x,y]

eA(t) with a eonstant

P{=M(u).

Choose

a

_Te(O,T']

so that

(1+]1v[L+M)Tgr!2

and

(exp!:ao(s)ds)(q(O,e)+

!,

bo(s)ds)SK･ Now,

lete.e(O,1], E.JO･

Set

t:=O and x:=g-, _="d

define

t:,xc, _yc,

_k=

1,2,･･･ as follows:

(Sperscript

n of t:,xk",･･･will

be

omitted below.)

(i)

tk=tk-i+hic, O<hkSmin

{en,

TLtkHi},

(ii)

]1xk-hkyk-xk-i"$hf.,

[xk,yk]eA(tk),

(iii)

q(tic,xk)$(exp

!il.,

ae($)

ds)(q(t,-i,

x,-i)

+

!li-,

bo(s,

x)ds),

(iv)

pt,12<h,,where _pt,

is

the supremum of

hk

satisfying

(i)

to

(iii).

It

follows

frorn

T-(xk-u, yk-v)$M that

(10)

ShonanInstitute of Technology

reptx*vc\rcet

m

17

g

ee

1e

ll

xk-e

ll

S211

e-u

]1

+

(s.+

]l

v

[l

+M)tkSr

･

And

<iii)

deduees

q(tk,xk)

S-

(exp

I:k

ao(s)

_ds)(q(o,

e)

_+!:k

bo(s)ds)

_<K .

Henee

_(C2)

assuresthat

(i)

to

(iv)

_are valid

for

hll.

We

new show that

limk-.tk==T.

For

contradiction assume

limre-.

tk= t<T.

By

a very similar _way

in

Seetion

2we

_ean _prove

that

II

xkHxy

ll

${C(]1

top

]lco+

11

_ep

l]

to)+

ll

vt

]]}(ti-tJ')+(En+

Cll

ep

l]

eo)(tk-tt)+

(en+Cll

ep

lle.)(t,-tt)

+CtJ{6":w.(T)(tk-t,)+to.(2a)}

for

al1

k).aL2i).1,

_p).1 and 6e(O,T/2), where

C=LGIell+r)(1+K).

This

shows that

z==limk-... xk exists and zeD(A(t))

by

(Cl).

Furthermore,

sinee

]Iz-ell$rand

_q(t,2)$K,

by

(C2)

again there exist 6>O and

[xe,

_ye]eA(t+6) sueh

that

6Smin

{e.12,

T-t},

[1xo-6yb-z[1$

be.12 and _{q(t+fi,xe)$(exp!i'Oao(s)els)(q(t,z)+!i""bo(s)dB).}

_Setting

_2k=t-tkri+6, we see

easily that

(iii)

holds

true

with tkand xk replaced

by

tk-t+2k

and xb.

However,

ginee

2k>

t-tk-i+hk>2hk>pk

for

all suMeiently

large

k,

it

follows

from

the

definition

of

_ptk

that

IIxo-Ricye-xk-ill>2ken.

Letting

k-oo,

we obtain

lixe-ayb-zlllSe.

which contradiets

to

]lxe-Dya-zll$6e.12.

Henee

we

have

limtk=

T,

so that these tk,x,,

_yk

form a

DS

_of

(CP;O,

e)

on

[O,

T].

On

the

other

hand,

since

(A2)

implies

(A)

with

M(t,x,

_{y)=L(IIxil)(1+}

q(t,

x)),

it

is

easy to see that this

DS

is

(A)-compatible.

Q.E.D.

Corollary

4.4. Let

A

be

a

dissipative

opeTator

in

X. Set

E={zEX;

ljtg[!moieb-idist

(x,

R(I-i(A+a)))==Ofbrall

xeD(A)}.

LffeLi(O,

To;X)

and

f(t)eE.for

a.e. t6(O,

To),

then

theTe

exists a

DS-limit

solution on

[O,

tlD)

of

(4.7)

du(t)fdtGAec(t)+.f<t),

oSt<T6,

u(o)=eeD(A).

Proof.

We

first

assume that

_f

is

in

addition _right _continuous,

Riemann

integrable

_and

f<t)eE

for all

te[O,

Th].

Define

A(t)=A+.f(t).

In

thiseage,

(H3)

and

(Cl)

are obviously

satisfied with _q(t,m)=O.

To

see

(C2)

we may show

that

(4.8)

lirn

ti"

dist

_(x,

R(I-6A(t+6)))

==O

olO

for

all te[e,

T6]

and xeD(A).

However,

it

followB

that

6-i

dist

(x,

R(I-6(A+.f<t+6)))S6-i

dist

(x,

R(I-6(A+f(t)))+ll

.f<t+fi)-.ftt)

II

.

Sinee

_f<t)eE,

this

implies

(4.8).

Henee,

by

Theorem

4.3

there

is

an

(A)-compatible

DS

of

(4.7)

on

[O,

71D)･

Now,

by

virtue of

[1,

Lemma

1.3]

we ean take a sequence

{.f;}

of right eontinuous _and

Riemann

integrable

functions on

[O,

To]

with yalues

in

E

so

that

f;-f

in

Li(O,

T,;

.X) as n-Do.

Therefore

the conclusion

follows

from

Theorem

3.2. Q.E.D.

References

[1]

Ph,

Benilan:

Equationg

d'evolution

dans

un espace

de

Banach _quelconque et applicatiens, Thesis.

U. ParigXI, Orgay, 1or2.

(11)

NII-Electronic Library Service On

[2]

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[9]

[10]

[11]

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T･imeDependent Nbnlinear Evotution Equations in Banach SPaces M.

G,

Crandall

anel

L,

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in

Banach spaces, IsraelJ.Math., 11

(1972),.

57-94.

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(1977),

1-42,

T. Kato: Nonlinearsemigroups and evolution equations, J.Math, Soc.Japan, 19

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Cauehy

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