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VOL. 19 NO. 2 (1996) 335-342

SOMERESULTS ON BOUNDARY VALUE

PROBLEMS

FOR

FUNCTIONAL DIFFERENTIAL

EQUATIONS

P.CH. TSAMATOSandS. K.NTOUYAS L’ivc.itvf

DcIartn’t

f

451 I0 Icmzmina, G,’’c,’

(Received July 15, 1993 and in revised form December 29, 1993)

ABSTRACT.Existencer,sdtsfirasecond orderbo,mdary vahw p,l)l,nz firfiuwtiCmal differential

(’qati()IlS,aregiwn. The rcsfltsar(})asc(t tmth(’ nmlin(’ar Alterna.tiv, ()fLeray-S(’haaMcrandrely

.m a

lmni

t>mlson slut,ims. Th’s’ rcstltsarcgeu’alizationsfreceipt rr,sdts flom wdinary differential equationsaIllctmil)lcteCmr,ar]ier rcstltsCm thesame

KEY

WORDS AND

PHRASES. BomMary

vahw

l)r()l)l(,ms,

flmcti()naldifferentiale(luati()ns,al)ri,,ri b(mn(ts, L(’ray-Schauderalternatiw,.

1992 AMS SUBJECT

CLASSIFICATION

CODES. 34K10

1.INTRODUCTION.

The purpose ofthis paper is t() provide existen(’e results fi)r second ()r(h’r t)omdary value problems (BVP fi)rshort) fi)rflm(’tional differential (,qmtions. B(’fi)rc w(’ rcfi,r(nr BVP, fi)r the

c,)Ivenien(’eof the rea(l(,,w(,cmt)loytimstan(lat setting forflm(’tional (tifft,r,,ntia.lequations,

[3].

Let r 0 [)egiwn an(tlet C

C([-r, 0],R

) d(,notc thespace ofc(mtinuousflmcti(ms that map tim interval

I-r,

0]

intoR". For 0 E

C,

thenormof isdefine(t t)y

I1

p{lo(O)l-

-,._< o_<o},

where

I-O

d,’n,,tes anyc(,nvenient n(,rmin R". Ifx"

[-r, T]

R",

T

>

0 is c,,ntinu,)us, then

each E

[0, T],

xtEC is defined|)y a’t(O) x(t

+

O),-r

<

0

<

O. We considerthefolh)wing BVP

X

x"(t)

+

f(t,x,

(t)) O, G

[O,T]

(E)

aos’o

a

(0)

3ox(T)

+

3x’(T)=

’1 (BC)

where

f"

[0,

T]

x C x R"

R"

is a contimmusflmcti()n, 0 E C, q E R" an(t 0,’a,30,3, ar(’
(2)

It silently[ ]w nto[ that

I’1’ tmsicoxistenco

I)’vn

lnw’[

recently in

[4]

anal it is nt givon horn’, kY,

’mlfimsi/’

h’n,that. i tl,

lnff

ff 1"

,xistonce tho)r,m in

[4]

w(,havo

[0.7]

x C’ x

R"

int imtolsots inR".

() nmin Intrl)o in tlis lml)Ot is t giw’ comtitims m

f

with inHfly tlw n’,h’l a l)ii

lnnls. Tt,’s, cmditims

fiwtho sohtims,r anlitsterivativcs

.r’

arebtainct viaLZ-ostinates{S,

[C]

ada

Nagmu

type cmlitiCm anahgCmstothat usel in

[2]

fiwordinarytifft,rontial t’quatims.

Ther,sultsffthisl}tl)er aron)t ctmqmraih,withthose tff

[4]

anlseems tbouewow’nwhen

(E) isan rdinary differential oquation,i.e. r=0.

m

wh,,t f,,,,w

II.II

,,,dfi,r th,,

L-norm

defined by

7-’

II.,II

(j,,

I.,(,)l-’dt)’

anal < > sta(ls fi)rthe’Ewli(l(,an innerI)r)hu’t in R".

F)t"

sl)se(lwnt

s(’ woshallstat(, t,’r(,thefi)lh)wing in’(0mlitios. LEIlkIA 1.1. (a) F)ranyfim(’tim,r G

(7’[0. T]

(t)

Fr

anyfimctionx, G

C[0.

T]

T’

"-’

+

g

v@

II*II’-

<_

7T

._,_

T

II*’II

+

-_

[I*(o)I"

+

I*(T)I

r-2.

Theaboveinequalities follows by essentially thesaxnereasoningas in lexnmas 2.3 and2.4d

[].

Obviouslyinthecase T 2we cm me theinequMit,y () insteadof(b).

blAIN

RESULTS

Nowwepresentourtnain remtlt ontheexistenceofsolutionsof the B

VP

(E)-(BC).

THEOREM

2.1. Let

f: [0, T]

xCx

R"

R"

be a c(mtinuousflm(’tion. Assumethat:

H

There exiat nonnegat,ve con,stants

A

and B ,,,th

mu’h that

AT

2

+

v/

2r 4 a

A

v/

B <

mtn{

(1

r

_--’"--)

2 4T

+ rrx/’

V(a,

v/2

)"

4

30

A

< u(0),f(t,u,e)

><_ AI,,(0)I

+

BI,,(0)II,’I

forall E

[0,

T],

u EC an(1

,

ER".
(3)

,’.f(.

,,.

,’) >< ,(

I,’1"

)l,’l-’

’\" d.’,

J_tz

>

2M"

b(,

Io(01

+

I’____LI

31-- ,,,..r

Tc

-.

+2.."

2.’

r,- ’2 T rr 2

c,,=

"’’

.4v@

1Bv

v/2

4

Then tile BVP (E)-(BC)has at least

PROOF.

To prow’ the existence ofsolutionsofthe B\;P (EI-(BC) we apl4y the Nmliwat Alternativeof Leray-Schauderin the mannerappliedin

[4].

Tohthisweneect toverify that the

set ffallposiblesolutions of thefamilytffBVP (Ex}-(BC), where

(t)+,/(t,..t,

(t))=

o,

[o,T]

(E)

is a priori boundedt)y aconstant independent of

I.

Letx beasolutionof(E

)-(BC).

By takingtheinner

prMwt

of (E)withx(t),integrating byi)artsover

[0, T]

andusingtilefact that

weobtain |)y(H)

o

So

le(o)l

I,I

I1,’11" +

--I=(o),

+

7[l=(r)

,,,<

I(o)1

+

I(T)I

+

Jo

<

.r(t),f(t,.r,,x’(t)

>

dt

_<

Io(q.)l

i,.

(o)1

+

I,I

r

,,,

I,’tT}I

+

j,,

< ,’,(Ol..f(.,,.," tl >,It
(4)

(7.(m.,ie(

tmnt

ly

<

[(0)11(0)1

+

I,1

or

The last inequality implies that

I1’11

_<

M,

Ix(O)l <

M,

and

I*(T)I

5 M (*)

Therefore forevery E

[0, T]

I(t)l

5

I(o)1

+ljo

x’()dl

5

I(o)1

+

vll’ll

<

M

+

VM

(1

+ v/-)M,

whichis the required apriori boundonx ontheinterval

[0,

T].

Nextweshall prove thatx isbounded on

[-r, 0].

Fromthe firstbomtdaryconditionwehave:

<,

Ix’(o)l

_<

I(o)1

+

aoM _<1o1+ oM

andconsequently

9

,’,

_<

--[11

+

<,1.’(o)1]

_<

----

I1 +

M.
(5)

Therefore

Als(), (*) implies, t)yth(, meanvahu"the()r(,m, hat{h(’(’(,xists t( t

[0.

T]

s,whtlmt

Tl.r’((,

ll;

<

31;

()r

3I-’

Ir’(t.)l

<

--.

N()w, takingth(’inn(’r i)r()(hwt()f

E

with

.r’(t

w(, hay(,I)y

He

al,,()l

,"

’(

I

< 2h(I ()1)1,

or

d

Integratingtheaboveinequality weget

1

<

21-’()1

h(.,)

I*’(t)la d.’

<

J(,

I’

HellCe

I*’(t)l

_<

v4-,

[O,T].

Consequently the required a priorit)ounds areestablished and theresultsfoll()ws.

THEOREM

2.2. Let

f:

[0,

T]

x

C

x

R

--+

R"

bea continuous function. Assume that (Hi) holds andmoreover

lttt2

There extst a continuous

f

unctton h"

R

+

R

+ a constant N

>

0 and nonnegative constants q,, 1,...,6 such that

<

v,

f(t,

u,v)

>_<

h(Ivl2)(q,

lu(0)l

-t-

q2lvl -t-q.lu(o)ll,,I

+

q4lu(0)l

for.lit

[0,

r],

v

e

R"

and u

e

Cwith

I.I-<

i

,,,a

N ds

j

>

2Q

t()

where

+

q,lvl

+

q.

Q

q,(1

+ V/)2

M +qM

+qa(1

+

v@)M

+q,(1

+

V/)V/M

+qsvf-M

+q6.
(6)

l-luct

f

EA

with a’( w,haw’ by

H

--I

<-

2q,

l.,ltll-’

+

q,l.,"{

t,)l" +

q.,I.,tt}ll.,"ltl

+

q,l.,/tl

+

q,l.,"{tll

+

Integratingthenl)}w,ineqtmlity, nnd truing thoCatwhy-Schwarz in’qtmlitywe got

Jo

Hell(’("

whichcompletesthe proof.

Now we present some exalnples to illustrate how the above results may be tme(i to yield

existenceof solutionsof specific b(mndary value problems.

EXAMPLE

2.3. We(’onsiderthefollowing BVP

x"(t)

+

g(t)z(t)F(t,x,,x’(t))

O,

e [0, 1]

x0

x’(0)

x(1)

+

x’(1)

2 (be)

whereg"

[0,1]

R

is a continuousand positive flmction nd F:

[0,

]]

xC xR"

R"

a.bounded

fimction with boundK.

Here f(t,u,v) g(t}u(O)F(t,u,v),T 1,0 ,30 ;3 1 andq 2.

Thenwehave

< u(O), f(t,u,,,)>=

g(t)u’-’(O)F(t,.,,,)<

g,,ffl.(O)l"

i.e(H) holdswith

A

goK,

go

max{g(t)"

E

[0, 1]}

ndB 0. Withoutloss ,,fgeneralitywe canchoose the fimctionsg an(tFinsuch away thatA 1.
(7)

"i’Ve remarkaim()that

"’

t;

""

This meansthat (H2) holdswith b(,)

.

(The cotditio

Therefore the BVP (c}-{bc)has at l<,astonesolutionby TheCnCl 2.1.

EXAMPLE

2.4. It iseasy to see. asin the l)revios exa.mplc, that thc BVP

.r

.r’

0

.r(1)

+

"(I)

where

q’[0,

]

R

isacontinuousand I)ositiv(’flnt’tion,hasatl(’ast an(l

B

I.

3.

CONCLUDING REMARKS

In

[4]

the

BVP (E)-(BC)

has beenstudied under thefollowing conditions, which are briefly reproducedhere.

(A)) Ther(’ exist a (’(>nstant

M >

0 such that

[u(0)]

> M

and

<

u(0),(,

>=

0 implies

< )(O),.[(I,u,v) ><

0 for all

[0, T]

andv

<

,,,

f(t,

,,,)

>)5

(",

l,,l

+

")l,,l

for

(.4)

lf(,

,

’)l 5

q()(["[)

for

a

e [0, ],

,

e

Let usaddinthe abovelistthe assuml)tions (Hi) and (H2) ofThtx)r(’n 2.1.

(H,)

< (0),f(,

,

,)

>

A[,,(0)[

+

nl(0)[[,’l

for

e [0, ],

,,

C,,,<i

,,

".

We

alsoremindtha.t:

The BVP

(E)-(BC)

hasat leastonesolution if (.4))and

(A)

h,)d,

[4.

Th.

4.11

(,)

d

(.4,)

hod,

[4.

Th.

4.21

(H)

and

(Ha)

(or

(H’2))

hold, Th. 2.1(or Th. 2.2) Thefollowingquestions azeimmediately isen.

Hasthe BVP

(E)-(BC)

asolution if

I

(A,)

and

(Ha) (or

(H’a)

hold? 2)

(H,)

,d

(A,)hod

?

3)

(H,)

nd

(A,)

hod?

The answerinall of the above questions ispositive. Indeed the cases 2) and 3)areobvious,

since everyoneof conditions

(H), (A2)

and

.43

gives independentlya priori boundonx or x

.

Somecomments areneeded forthe case 1). By takingtheinner product of
(8)

<

It

o)I

-k

I,1:

Thisimplies theexistenceofheImd M nd the rest ff thelr[isesseially he ane a

We

sunmarizethe abwediscussion in thefollwing

THEOREM

3.1. TheBVP (E)-(BC) hasat leastneslutirn ifneofthefilhwing pairs cnditionshlds:

1 (A) and(A) 2) (A,)and (A.I

3) (A)and (H)(or

(H’e)

4) (H)and (Ae}

5)(H) and(A.) 6) (H) and(He) (or

(Hz)

REFERENCES

[1]

J. Duglmdji and A. Granas. Fixed Point Theory

V.I.

I,

Ionographie Iatematyczne,

Warsawa,

1982.

[2]

D.Hd,Existenceand uniquenessofsolutins foranonlinearsecondorder differential equation

in Hiltert. space, Proc. EdinburghMath.

Sot’.

33(1990), 89-95

[3]

J. Hale, "’The.oryfFmctional Differential Equations" Springer-,rlag, New York, 1977.

[4]

S. Ntouyas,

Y. Sficas and P. Tsamats, Boundary value problemsfor fimctional differential eqtmtions, Technical

Report

No.

187, (1991), University of Ioannina,Greece.

References

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