Pn0BI,EUS Ar nEgONAtICE
FoR
81ffiT
Al{D
SECOIiID
OnDEn
DlF-
FERnNtlAl EqUAtrOAs
WA
LTAPUNOV-LIKE
rwcTr0Ns
by
J. J.
Nleto*
Dtperttcnt
of
ll,athenatlce
fhc UnLverrtty
of
Texee
at
Atllngotn
Atllngton,
Te:€l
76019
Technl.eal
RePott
#181
Apttl'
1981
rf,csearch
petttrlly
aupportcd
by
u.
9.
Arny
Rcrserch
Grent
#DaAG29-80-C-0060.
1,
INT&ODUCITON
Recently
[7r8r10]
an attempt
{g made
6uec€saful1y
to
coobtnt
thr
tttu
baalc
technlqueg
nanely,
Lyapunov-Sehtuldt
nethod
ard
the
ncthod
of
uppet
end loser
golutlonel
to Lnvaatlgete
the
exlatence
of
perlodlc
aolutlonr
of
dlfferentlal
equatLone.
when
t{e
wlgh to
extend
such
regulta
fot
8y3t€ml
of dtfferentlal
equettone there
are
ttro
poeaLbLlltles
to
follor.
One
ls
to
generellze
the
ft€thod of upper
and lower
solutlons
ueing
a sultebl€
Gotl€1
and the other
le to utlll.ae
the
concept
of
Lyepunov-llke
functlons
6nd
ths
theory
of
dlffetentlal
Lnequalltlec.
In
the
pepef,r
ve
rhal1
dlscusl
the
ex-
Lstelee of
ptoblems
et r€aonence
fot
f{rst and
recond
order
dlffetentiel
By6ten6
by the
aecond
approach
develop{ng
neccseary
theory
of
dlffetcntlrl
{nequalltles
fot
problemr
et fesonance.
2, AlT
ASSMACT
EXISTEHCE RESI'LT
Let
E be
a real
Htlbert
Space.
Consldet
the
ncnllnotr
oPcretot
equetlon
(2.1)
Lu-Nu
rhere
L: D(t)CE-tE
lsallneaf
opef,atotsnd
Nl
D(lt)CE'--+B
t
nonllneer.
operator wlth O(t)
n
D(H)
f t.
Let EO
.
N(t) be ruch
thet
dln
Eo
<
-
and E - E0
(E
91.
Furthermorer
ile
aesune
thet
Et
lr elro
the
range of
L.
$uppose
thet
P;
E
-t
E0 le
the
ldeupotent
ptoJectlon
opare-
tor and Ht
El
-t
E1
the conpact
tnvefse
of L
on 61.
then 1t
le re1l
knorn
[31
that
the
problen
(2.1]
{a
equlvalent
to the
coupled
eyatetn of
opeta-
tof,
equstlonE
(2,2)
ut
.
H(I-P)l[(uOtut)
(2,tj
0
-
FN(uO+ur)
ConcernLng
the
problern
(2.1),
t*e
have
the
following
reeult
Ile_orem
2.1:
Suppoae that
(r)
i$ul
<
Jo
for
all
u
€
D(N)
({t;
There
exl.sts
t'r
R0
>
0
auch that
(N(uo+ut)ru')
I
0
(or <
0) $henever
luol
-
.snd
flurfl
- r'r where
t0
*
E0
and ul
*
81.
?hen
the
problern
(2.1)
edrnlts
et
Leaet one
solution'
3.
COWARTSOfi
PRIfiEIPLES
Gtven
s
E
cfto,rnt
,.
*,*]
and
v
€
cfto,tnt,
xr]
the
functlon
S(tru)
ls
s
modlfled
functlon
relatlve
to
(3.1)
E(trtr)
-
g{trp(tru)1
.r
[Gr-d-u
l+u2
'
r-
rr
s€cllo,znl*n
xIRr
we deflne
,
p(r,u)
,u')
*
g[(+ll.
t*u-
[41.
Ro
r
lf€
vr
lf
eay
that
.]
s(t
Thle
funct{on
rltll be called
the modlftcetlon
of
g(trurut}
telatlve
to
'rrr
Here,
ln
(3'1)
and
(3.2), p(t,u)
o
maxtv(t)ru]"
. Let
us conetder
the
petlodlc
boundaty
value
problem
(PBVP
fot
short)
'
(3.2i
(3.3)
E(trurur
)
-
ut
-
g(tru)
t'le
have
the
following
comparlgon
u(0)
- u(2t).
resultg:
3l,eorg$_3ri.
il)
r-m(ti
r
o
{:
C
[
[Or:i''r
l,
L--
Let
g€
-
l-Lnr
sup
h+0"
-T
ni
and
I
-i
r
I and
assume
that
rljo,?nl
x
fr,
Rl
MIi*Itt
J
e(t,m(t))
ror at1
I
€ (0,2nJ, where
{
1r}
there
exlate
B
€
D-8(r)
>
g(r19(
L{!:ggr*e'!il:.1.
l'et
g
€
holde.
Aleo
*esumel
euch
that
€
(0r2nl
and
m(U)
!
n(Zn)
r-1
cIIo,2nLIR
i
L-J
t))
fot
s1l-
t
g(0)
>
B(2n)
{fff}
g
16 atrlctly
decreaelng
ln u
The*
ft(t)
<
g{t}
on
[or?nJ.
foreach
ge[0r2nJ.
L,:ggl"
$uppoe*
tlrat
ffi{t)
<
s{ti
otl
[0'2fi1
ie
not
true.
Then
thete
exiete
*
?S
n
[t:r]Tri
end
an e
>
0
euch
ilrst
i3.4)
m(to)
-F(to)+e
and
m(t)
1
B(t)+€,
t€
[0'2n1'
If
N+
€
iSn?nJ.
we
have
D-mit')
t
U-$(tO)'
Hence'
gitfi,*itoli
?
D-rrr(ts)
>
D-F(to)
:
g(t0'F(to))
an*j
becdtrre
cf
{rr.r}
we
eee
that
B(to)
>
tn(to)
whtch
le
a contradlctlon
to
i3.4)"
Ii:
t0
=
9,
,;e
heve
fcr
sftell
h
>
t),
rn(2n-h)
-
rn(?n)
<
m(24-h)
il{2n*h}
*. c
-
BiQ} -
c
<
S{Zn-h} -
B(?r},
trhtch
lnpltes
D-nr(2rr)
,
i.
ti;+:;r
fol.iaws
t:lrrri
3i1,r)
>
rn{?n),
and
$e
obtal"n B(0)
>
3(2tt)
,,-,'rJ:,rh
agr:1.n
l.a
s cr:ntra<!J'ctian.
Thue,
tde
have
estebllehed
that
**a
[{iu2n].
-
n(o)
.
>
D-g(ztr).
:
m(2*)
>
n(o)
m(t)
<.
B(t)
.[u,
Znl
x
*,
*]
and
eupPose
that
({)
of
Theorem
3'1
-l
(a)
rn(O)
'
m(Zr)
(b)
The
PBVP
(3.3)
hae a
tnaxl"mal
aolutton
r(t)
(c)
For
evefy
lowet
eolutlon
of
(3.3)
v1
the
modlfled
PBI|P
(3.5)
ut
r
S(tpu)
rr(0)
- u(2t)
shere
E
ls
defined
by
(3.1.),
hae
a
golutlcn
$(t).
Then
rtl(t)
<
r(t)
on
[0,2n1
.
Proof.
Let
E
the
modtfled
functlon
relctLve
to
m(t)
snd
leb
u(t)
be
a eolutton
of
the
rnodlfied
pgw
(3'5)
Budranteed
by
(c)'
t{e
ahell
ftrgt
ehow
that
m(t)
<
u(t)
on
[0,2n]
.
\f
thls
fa faloen
then
there
exLetg
a
ennller
e>0
euchthat
m(t)
5
u(t)
*
e,
x
e
[O'Zrtl
Ead
et leaet
one
t0
€
[0rZnJ
aatlsfylng
t(aO)'u(t')*e'
If
to
c
(0r2nl,
we
have
m(to)
t
u(to)
and
n-rn(to)
t
ut
(t0). on the
other
hand,
tre
herre
ln
vlew
of
the
deftnltlon
of
f
and
p(tlu)
t
(lq'11trt
)
-u.(-:C
0<D-rn(t0)-u'(to}:e(to,m(t0)i.g(tCI'p(t0,u(eo)))-ffi
u(to)
-m(
tn)
"
d1%)"-
<
0
whtch
Le
a
eontradlct{on'
lf'
t0
-
o,
we
obtal'n
D.m(2rr}
>
u'(zn},
and
from
(a),
re
get
p(2rr,u(2n))
*
max(n(Zn),u(2n))
-
nax(m(0),
u(0))
-
m(O)1
and
then