Generalized dissipativeness in a Banach space

(1)

VOL. 19 NO. (1996) 25-32

25

GENERALIZED DISSIPATIVENESS IN A

BANACH

SPACE

DAVID R. GURNEY

SoutheasternLouisianaUniversity

SLU541,Hammond,

LA

70402

U.S.A.

(Received December 12, 1994)

ABSTRACT.

Suppose X

is arealorcomplexBanachspacewithdualX*andasemiscalarproduct

].

Forkarealnumber,asubset

B

of

X

will becalledk-dissipativeifforeach pair of elements

(za,

V), (z2,

V2)in B,

there exists

such that

Rely1

Y2,

h] <

1

212.

This definitionextendsanotionofdissipativenesswhich isequivalenttohavingkequalzerohere.

Anumber of definitions andtheorems relatedtothisoriginal disspativenotion aregeneralizedinthe presentpapertofitthek-dissipativesituation, andproofsaregivenfor thenewtheorems.

KEYWORDS AND PHRASES. Dissipative,hyperdissipative,semi-scalarproduct,Banachspace,

multi-valued mappings,contractionsemi-groups.

1992AMS SUBJECTCLASSIFICATION CODES. 47H15,351L20.

1. INTRODUCTION.

The basic outline ofthispaperfollows Yosida[5],and results stated thereareexpandedtofit the moregeneralsituationpresentedhere.

Suppose X

isarealorcomplexBanachspaceendowed withasemi-scalarproduct such that for

a,/5

realnumbersand a:,F,zelements of

X,

[

+

u,

]

[,]

+

[,

],

I[x,

y]l _<

Ixl-

lyl and

(2)

D(B)

{x:

(x,

y)6Bforsome

R(B)

{y

(x,

y)6Bfor some

x}.

B

,

-

{(u,

x):

(x, u)

B

}.

{(x,

u):

(x, u)

e

}.

B

+

C

{ (x,

y

+

z): (x,

y)

e B

and

(x, z)

e

C}.

Ba

{(x-

Ay,

y):

(x,

y)

e

B}.

Bx

{y: (x,

y)

e

B)

wherex

D(B).

]Bx]

inf{]y: y

e

Sx}.

Bff

(I

AB)

-

where

A

issuch that the statedinverse is

uque.

A

simpleconsequenceof thisnotationis the following.

COROLLARY1.1.

ABx

B

-I. PROOF.

B

I

{(x-

Ay,

x-

(x-

Ay))"

(x,

y)

e

B}

{(x-

Ay,

Ay)"

(x,

y)

B}

=ABe.

[3

DEFINITION1.2. Thedualitymap from

X

intoX*isthemulti-valuedmapping

F

definedfor

ach

xin

X

by

F(x)

{f

6

X*"

[x,f]

Ixl

[/[}.

(1.2)

(1.3)

Accordingtothe Hahn-BanachTheorem,

F(x)

isnon-void. If

X

is aI-Iilbertspace,then

F(x)

xbythe Riesz Representation Theorem and

[y,

F(x)]

istheinnerproductofx andy.

DEFINITION1.3. Forareal numberk,asubset

B

of

X

x

X

willbe calledk-dissipativeiffor eachpair ofelements

(Xl, Yl)

and

(x2,

y2)

in

B,

there exists anelement

f

in

F(Xl

x.)

suchthat

Rely1-

u2,

f] <

klxl- X2I2.

(1.4)

DEFINITION1.4. Let

D

beasubset ofX. Themapping

T

from

D

into

X

isLipschitzwith Lipschitzconstantk

>

0iffor each pair of elementsXl,x2from

D,

(1.5)

]Txl

Tx21

<_

]C,[Xl

X2I.

LEMMA

1.5. Letxand y be elements of

X

andsupposek isareal number. Thereisanelement

f

ofF(x)

such that

Rely,

f] <_

klxl

ifandonlyif

Ix-

Ay]

_> (1- Ak)lx]

foreachpositive realnumber

A

such that

kl

< 1/A.

PROOF. ff

ixl

0,the lemmaholds;so assume

Ixl

0.

ffRe[y,f]

<_

klxl

for some

f

e

F(x)

and

A

isapositivenumbersuch that

Ixl

< 1/,x,

then

< Re[x, f]

ARe[y,

f]

(1.6)

Re[x-

AV, f]

_<

Ix-Since

f

F(x),

II

Ifl

andhence

(1

,X)ll _<

I-Now suppose

(1

)II _<

I-

ul

or

each positive

A

such that

II

<

I/.

Let

fx

(3)

BANACH SPACE 27

Re[x

Ay,

hal

[, ]

[,,

<

Ixl

ARe[y,

hx].

(1.7)

Hence

Rely,

h] _<

klzl

and

Rely,

hal

>_ ]x]

Aklx

+

ARe[y,

I1- kll- lYllhl

(1.8)

Ife

>

O and

A

<

/(Iklll

+

I1

+

1),

then

I1-

< Re[x,

hal

<

Ixllhl

Ixl.

Thus

limMo

Re[x, h;,]

Ix[.

Sincethe closed unitsphereofX*iscompact in the weaktopology of

X’,

thesequence

(hi

hasa

weak"

accumulationpointh X*suchthat

Ihl

<

1. Therefore

Re[x, h]

Ixl,

Re[y,

hi

<_

k,

andsince

I1

Re[,hi <

Ixllhl

<

Ixl,

(I.I0)

[hi

I. Consequently,

f

Ixlh

F(x).

I"1

COROLLARY1.6. Forarealnumberk,asubset

B

of

X

isk-dissipativeifandonlyif for each positive real number

A

suchthat

[k

< l/A,

and elements

(xl,

yl

),

(x2,

y2)

of

B,

PROPOSITION1.7. If kis arealnumber,

B

isak-dissipative subset of

X

X,

and

A

isa

positivereal number such that

[kl < l/A,

then

Bx

and

Bf

areboth single-valued mappings and satisfy,

respectively,thefollowingtwoinequalities:

2-Ak

IB,wl

BAw21

<_

lw

w21

forw,w

D(Bx),and

(1.12)

A(1-

Ak)

Moreover,

B

is

(k/(1

Ak))-dissipativeand also satisfies both of thefollowing:

BA

w

(B

B)w

B(Bw)

for w

D(B),

and

(1.14)

IB

wl

<_

1 1

A--’-

IB

wl

forallw

D(B)

f3

D(B).

(1.15)

PROOF.

Suppose

Xl,X2

D(B),

y

Bxl

and y2

Bx2.

By

Corollary 1.6,

(1.16)

(4)

1

IB,w,

B,wl

I(B

I)w,

-(B

I)w=

1 1

<-

+1

Iw

wl

1-k

2-Ak

A(1-

A)

proving(I.

12).

Toshow

B

and

B

aretingle-valued,suppose

z

A

z2

A2.

By

CoroIl

1.6again,0

(1

A)z

z=[.

Thus

z

z2,andhccfo[e _2. Now suppow, cinhcdomn of

BA.

Suppose

also lhal

(1.17)

f

_

F(wl

w2)

{f

G

X*

"[wl

w2,

I]

I

w=l

=

Then

(1.19)

Hence

Bx

is

(k/(1

Ak))-dissipative. If

(z, V)

G

B,

B,(=- A!/)

1

e B

=

B(B(x-

!,1))

(B

B)(z-proving

(1.14).

Forw G

D(B)

N

D(B)

and eachz

B

w,

(1.20)

(1.21)

Thus since

[B

wl

inf{

Izl"

z

B

w},

(1.15)

isproved, vI

LEMMA

1.8. Let

B

beak-dissipative subset of

X

X. If

D(B)

X

forsomepositive real number

A

such that

1/A >

[kl,

then

D(B#u)

X

for every positivereal

number/

suchthat

1 2-Ak

II

<

#

A

(1.22)

PROOF. First notethe following. Since

A[k[ <

1,theinequality

[kl < 1/A < (2

A[k[)/A

holds.

Also,

(1.22)

leadsto

<

1

II

Now

supposexG

X.

Foreachz G

X,

definethemapping

T

by

(5)

GENERALIZED DISSIPATIVENESS IN A BANACH SPACE 29

Asaresult of(1.13),

Hence Tis aLipschitzmappingwithLipschitzconstant

1 1--Ak #

1 1

#-A

#

<1.

(1.26)

Forn

<

rnand eachpointzEX,

IT"z

T z

<_

a

T’-’z

T

zl

O

(IT

z-

z[

+

[Z2z-

Zz[

+...)

a

(1

+

a

+

c

+...)ITz

zl

=(a-)--lZz-zl"

(1.27)

Hence,

bythecompletenessof thespace

X,

y lim,_ooT

’z

existsinX. SinceaLipschitz mapis

continuous

Ty

T(

lim

T’z

lim

T(T’z)

lim

T"+z

y. (1.28) Consequently,

Ax

#-(y

A

)).

(1.29)

Thusz

(1/#)(y

x)

EBy and y #z x. Therefore

Bx

y. Since x wasarbitrary,

D(B)

X. !"!

THEOREM1.9. Suppose

B

is ak-dissipative subset ofX X.

IfD(B)

X

forsome

positivenumber

A

such that

Ikl

<

l/A,

then

D(B:)

X

foreachpositivereal number#such that

I1

<

1/#.

PROOF. Constructasequenceasfollows. Let,1=’. If bothi)apositive

An

has beenchosen

sothat

Ikl

< 1/m,

andii)

D(B)

X

foreachpositive#such that

Ik[ <

1/

< 1/A,

then let

A,+I

be the average of

An

and

A,/(2

A,

Ikl);

thatislet

A,+I

A,(3

A,

lkl)/(4

2,11).

Then

D(B)

X

for eachpositive#suchthat

I1

<

lh, _<

x/A+.

CLAIM. lirn__,oo

A,

0.

The claimholds if k 0,sosuppose k 0. Theclaim isnow equivalenttosaying%

,X,,Ikl

approacheszero asnincreases. Notethat0

<

% A

llkl

<

1and

:

3-7n

"

1(

7,

+7,,).

(1.30,

%+I=%\4_2%/

=

-%

IfTn

<

1, then 0

<

%+1

<

%

<

1. Thus

(%)

isastrictly decreasing sequence,andassuchhasalimit

3’-E

0,1)

which isthegreatest lowerbound ofthe

%’s.

Suppose

7

>

0. Foreachreal numberxless

than 2, let

f(x)

x(3

x)/(4

21).

Then

f

is acontinuous functionon

(-oo,

2).

Since

f(’y)

<

7,

thereis a6

>

0 such_{that for 7}

<

r/< 7

+

6,

f(r/) <

7. Fornlarge enough, however,7

<

%

<

7

+

6

and7,,+1

I"(7,)

<

7,contradicting the fact that7 isthe greatest lower bound

ofthe%’s.

Thus7 0,

(6)

Hencefor#apositive number such that

]k]

< 1/#,

thereis apositive integernsuchthat

A, <

#and

D(B)=

X. !-I

DEFINITIONI. 0. Ak-dissipative subsetBofXx

X

willbe calledk-hyperdtssipativeif

D(B)

X

forsome

(,

and hence for

each,)

positive real number

A

such that

Ikl

<

1/A.

PROPOSITIONI.11. Ak-hyperdissipativesubsetBofXxXismaximallyk-hyperdissipative

inthesensethatthere doesnotexist ak-dissipative subsetCof

X

x

X

such that

B

is apropersubset ofC.

PROOF. Assumesomek-dissipativesubsetCof

X

x

X

contains

B

as asubset,andsuppose

(xo, Yo)

5C. SinceBisk-hyperdissipative,thereexists anelement

(x,

y)of

B

such that

1 1

xo

Y0 x y.

(l.31)

I:1

+

1

Ikl

+

1

Having

B

as asubset ofCimplies

(x,

y)EC. ApplyingCorollary1.6givesx0 xandY0 Y. !-I

2. CONTINUOUS FAMILIES WITH A BOUNDING FUNCTION

Letacontinuousfamily

{Tt

> 0}

beacollectionof possibly non-linear mappings from

X

into

X

which arestronglycontinuous in (i.e.foreach x E

X,

Ttx

is continuous in

),

andwhichsatisfy

Tox

7xforsomepositive number7. Finally,suppose that forsome continuous function gfromthe non-negative real numbers backintothemselves,

i)

(0)

7,

ii)

limg(t)’"

g(O)

exists, and

tt0

iii)

ITtx-

Ttyl

<

g(t)lx-

yl

for each

>

0and all x,y inX.

(2.1)

Suchafunctiong will be calledabounding

function.

A

continuous family

{Tt

> 0}

with abounding functiong is acontractionsemigroup ifthe

followingthree conditionsaresatisfied:

i) 7=1,

ii)

g(t) <

1for each

>

0, and

iii)

Tt

Ts

x

Ts+t

x for eachx

X,

and all non-negativesandt.

(2.2)

Contraction semigroupsarediscussedbyKato ],Kmura[2], [3],Crandalland Liggett[4],

Yosida

[5],

Miyadera

[5]

andmanyothers. One goalofthispaperis toshow thatevenwithoutthe

properties

(2.2),

continuousfamilieswith aboundingfunctionhave many characteristics whichparallel thoseof contraction semigroups.

The infinitesimal generatorAofacontinuous family

{Tt

> 0}

isgivenby

Tz-

T0z

A

x lim

(2.3)

tl0 t

ifthelimit onthe fightexists. Let

D(A)

denote the domain of

A.

In

this situation,anoperator

B

fromasubset of

X

into

X

willbe calledk-dissipativeifk isa

realnumbersuchthat for eachxand yinthedomainof

B,

Re(B

x By,x

y>

<

klx

yl

2.

(2.4)

THEOREM2.1. The infinitesimal generator ofacontinuousfamily

{Tt

> 0}

withabounding

(7)

31 PROOF.

1 1

Re(-(Ttx

Tox)-

-(Tty

Toy),

x-1 1

1. 1

:Re(x

y,

x-

y>

:Re(Tox- Toy,

x-

y>

t

1 1

9(0)

I-

1

.

Thus forxand y elements of

D(A),

taking thelimitof thefirstand lasttermsas decreasestozerogives

Re<Ax

A

y,x

y)

<_ g’(O)lx

yl

2.

El

(2.5)

One consequenceof Theorem 2.1 isthe following.

COROLLARY2.2. IfAis apositivenumber such that

[g’(0)[ < l/A,

thentheoperatorI

AA

from

D(A)

intoXhasauniqueinverse.

PROOF. Supposexl

AAxl

z x2

AAx2.

If

x

=fi

x2,then

0

((x

AAx)

(x2

AAx2),

x

[xl

x2[

A

(Re(ax

Ax2,Xl

X2)

--

Im(Axl

Ax2,xl

x2))

Ix

z21

ARe<Az

Ax2,

z

z2)

>0.#

ThusXl X2and

I

A A

hasauniqueinverse, l-!

3. EXAMPLES.

Findinggeneralsolutionmethodsfor thenonlinearevolutionequation

du(t)

dt

A

u(t)

for

>

0 with

u(0)

(xo,Yo)

E

D(A),

(3.1)

in thissettingis anopenareaforresearch,butsolutionsdoseem to exist asshownbythefollowingtwo

examples.

Yosida presentsanexample given by Kmura

[2].

Thisexampleis nowmodifiedto fitthecurrent circumstances. Let

R

be theEuclideanplanewiththeusualinnerproduct,let

>

0, and for each element

(z,

y)in

R

R,

let

(max{4x

t,

0}, (t

+

2)2y)

if x

>

0,

(3.2)

Tt(x,

y)

(4x,

(t

+

2)2y)

if x

<

0.

Then

{Tt

t

> 0}

isa continuousfamily ofnon-linearoperators from

R

x

R

intoitselfwith bounding function g given for each t

>

0by

g(t)

(t

+

2)

and_3’ 4.

By

definition, the infinitesimal generator

A

of

{Tt

t

> 0}

is givenby

(-1,4y)

ifx>0,

(3.3)

A(x,y)=

(0,4y)

ifx<0.

(8)

(max{a-

t,

0},

(t

+

2)2b)

u(t)

(a, (t

+

2)2b)

ifa

>

O,

(3.4)

ifa<O.

Since

u(O)

(xo,

Yo),

thesolution canbe pinned downto:

(t +

2)Uo)

u(t)

(max{xo-

t,O},

(o,

(t

+

)o)

if

xo

>

O,

(3.5)

if

xo <

O.

Asanotherexampleconsiderthefollowing. Still in

R

x

R,

for

_>

0let

(Sz-t2-5t,

(t+2)3y)

ify>O,

St(x,

y)

(8x-

5t,8y) ify

_<

O.

(3.6)

then

{St

>

0}

isanother continuousfamilywith aboundingfunctionh definedby h

(t)

(t

+

2)3.

Inthisexample7 8and the infinitesimalgeneratorBisgiven by

f(-5,

Sy) ify>O,

B(x, Y)

(-5,

(3.7)

0)

ify

<

0.

Again, solvingthe evolutionequation

(3.1)

requiresalittleguesswork, but duetothecharacteristics

of the continuous family,onemighttryasolutionof the form

(a-

5t,

(t

+

2)3b)

(t)

(-

5t,

8b)

if b

>

0,

(3.8)

if b<0.

Theinitial conditionsthen leadto anactualsolution:

(t

+ 2)3y

O)

if

>

O,

u(t)

(xo

t 5t, g Yo

(xo

t 5t,

Yo)

if_Yo

-<

O.

(3.9)

Inboth of theseexamples,knowing how theinfinitesimalgeneratorarises is abighelpinsolving the equation. Forthisapproachtobe veryuseful,alistofconditions whichleadto certaintypes of continuous families should bedeveloped. Also,thereisthe question of whether the solutionsareunique. Both of these topicsseemworthyoffurtherinvestigation.

ACKNOWLEDGEMENT. JohnNeubergerof the University of NorthTexaspointedmetoward

this areaofresearch,and

Troy

Hicksof the University of Missouri, Rollaurgedme toconsiderthe Banachspaceapproach.

REFERENCES

1.

KATO,

T. NoteontheDifferentiabilityof NonlinearSemigroups,

Proc. Symp.

inPureMath.

16

(1970) AMS,

Providence, RhodeIsland,91-94.

2.

KOMURA,

Y. Differentiability ofNonlinearSemi-groups,J.Math.Soc.

Japan, 21(3) (1969),

375-402.

3. NonlinearSemi-groupsin HilbertSpace, J.Math.Soc.

Japan, 19(4)

(1967),493-507.

4.

CRANDALL,

M.G.

&

T.M. LIGGETT Generationof Semi-groups ofNon-linear Transformations onGeneral Banach

Spaces,

Amer. J.Math.93

(1971),

265-298.

5.

YOSIDA,

K. FunctionalAnalysis 6th edition, Springer-Verlag,NewYork 1980,250-51

&

445-54.