VOL. 19 NO. 2 (1996) 335-342
SOMERESULTS ON BOUNDARY VALUE
PROBLEMS
FOR
FUNCTIONAL DIFFERENTIAL
EQUATIONSP.CH. TSAMATOSandS. K.NTOUYAS L’ivc.itvf
DcIartn’t
f451 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 thesameKEY
WORDS ANDPHRASES. BomMary
vahwl)r()l)l(,ms,
flmcti()naldifferentiale(luati()ns,al)ri,,ri b(mn(ts, L(’ray-Schauderalternatiw,.1992 AMS SUBJECT
CLASSIFICATION
CODES. 34K101.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 intervalI-r,
0]
intoR". For 0 EC,
thenormof isdefine(t t)yI1
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, theneach E
[0, T],
xtEC is defined|)y a’t(O) x(t+
O),-r<
0<
O. We considerthefolh)wing BVPX
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(’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’ xR"
int imtolsots inR".() nmin Intrl)o in tlis lml)Ot is t giw’ comtitims m
f
with inHfly tlw n’,h’l a l)iilnnls. Tt,’s, cmditims
fiwtho sohtims,r anlitsterivativcs
.r’
arebtainct viaLZ-ostinates{S,[C]
adaNagmu
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,,,,wII.II
,,,dfi,r th,,L-norm
defined by7-’
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, GC[0.
T]
T’
"-’
+
g
v@
II*II’-
<_
7T
._,_
TII*’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. Letf: [0, T]
xCxR"
R"
be a c(mtinuousflm(’tion. Assumethat:H
There exiat nonnegat,ve con,stantsA
and B ,,,thmu’h that
AT
2+
v/
2r 4 aA
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".,’.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
4Then 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 theset 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 takingtheinnerprMwt
of (E)withx(t),integrating byi)artsover[0, T]
andusingtilefact thatweobtain |)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(7.(m.,ie(
tmnt
ly<
[(0)11(0)1
+
I,1
or
The last inequality implies that
I1’11
_<
M,
Ix(O)l <
M,
andI*(T)I
5 M (*)Therefore forevery E
[0, T]
I(t)l
5I(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.Therefore
Als(), (*) implies, t)yth(, meanvahu"the()r(,m, hat{h(’(’(,xists t( t
[0.
T]
s,whtlmtTl.r’((,
ll;
<
31;()r
3I-’
Ir’(t.)l
<
--.
N()w, takingth(’inn(’r i)r()(hwt()f
E
with.r’(t
w(, hay(,I)yHe
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. Letf:
[0,
T]
xC
xR
--+R"
bea continuous function. Assume that (Hi) holds andmoreoverlttt2
There extst a continuousf
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],
ve
R"
and ue
CwithI.I-<
i,,,a
N ds
j
>
2Qt()
where+
q,lvl
+
q.Q
q,(1+ V/)2
M +qM
+qa(1+
v@)M
+q,(1+
V/)V/M
+qsvf-M
+q6.l-luct
fEA
with a’( w,haw’ byH
--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 BVPx"(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.boundedfimction 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,
gomax{g(t)"
E[0, 1]}
ndB 0. Withoutloss ,,fgeneralitywe canchoose the fimctionsg an(tFinsuch away thatA 1."i’Ve remarkaim()that
"’
t;
""
This meansthat (H2) holdswith b(,)
.
(The cotditioTherefore 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(lB
I.3.
CONCLUDING REMARKS
In
[4]
theBVP (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()(["[)
fora
e [0, ],
,
e
Let usaddinthe abovelistthe assuml)tions (Hi) and (H2) ofThtx)r(’n 2.1.
(H,)
< (0),f(,
,
,)>
A[,,(0)[
+
nl(0)[[,’l
fore [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 ifI
(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<
It
o)I
-kI,1:
Thisimplies theexistenceofheImd M nd the rest ff thelr[isesseially he ane a
We
sunmarizethe abwediscussion in thefollwingTHEOREM
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 TheoryV.I.
I,
Ionographie Iatematyczne,Warsawa,
1982.[2]
D.Hd,Existenceand uniquenessofsolutins foranonlinearsecondorder differential equationin 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, TechnicalReport
No.
187, (1991), University of Ioannina,Greece.