HOUSTON
JOURNAL
OF
MATHEMATICS
Volume 17, No. 1, 1991
GENERALIZING
HARTMAN'S
OSCILLATION
RESULT
FOR
+ c(t)lxlP-x_- O,p > 1'
MANUEL
DEL
MANUEL
ELGUETA
RAUL
PINO
MANASEVICH
1. Introduction.
Let us considerthe nonlinear ordinary differential equation
(1.1)
where'= •,d c .[0, +oo)
I• isa continuous
function
andp isa realnumber
greater than one.
By a solutionof (1.1) we understand
a functionx ß CI[0, +oo) such
that Ix' p-2x' ß Cl[0,+oo) andwhichsatisfies
(1.1). Existence,
uniquenessand extensionto [0,+oo) of solutionsto the I.V.P. for (1.1) are well
established.
See,for instance,[3].
A nontrivialsolutionx of (1.1) will be calledoscillatoryif givenany
•- _• 0 thereexistst* > •- suchthat x(t*) = 0. From[4] or [6] the following
proposition follows
PROPOSITION
1.1. Let x be a non-trivialoscillatorysolutionof (1.1) and
y be any other non-trivialsolutionof (1.1). Then either there existsa
constant cr • 0 such that x - cry, or if tl < t2 are any two consecutive
zeroesof x then thereexistsexactlyonet3 ß (tl, t2) suchthat y(t3) = 0.
Proposition 1.1 extendsto any p > I the usually so-calledfirst Sturm
comparison
theoremfor (1.1) whenp - 2. In particularit tell us that if
(1.1) has a non trivial oscillatorysolutionthen any other solutionis also
oscillatory.If this is the case,we will saythat (1.1) is oscillatory.We will
* This work was sponsoredby the CONICYT, by the D.I.B. of Chile, for the first and
third author and by the DIUC, U. Cat61ica de Chile for the second author.
63
64
MANUEL
DEL PINO, MANUEL
ELGUETA
AND RAUL MANASEVICH
alsosaythat (1.1) is non-oscillatory
if it is not oscillatory,i.e. if thereexists
a non-trivialsolutionof (1.1) whichis ultimatelyonesigned.
For p = 2, (1.1) reducesto the linearequation
(1.2)
x" + c(t)x = O.
In this casea very importantcriterionon c sothat (1.2) is oscillatoryis due
to Hartman. To present his result in a convenientform let us first introduce
the linear operatorA: C[0, +c•) • C[0, +c•) by
(1.3)
(nf)(t)= •
f(s)ds,
nf(o)= f(O).
Leta(t)= f• c(s)ds,
thenHartman's
result
[5]canbeestablished
as
THEOREM
1.2.
Iœ
(i)
--c• < liminf•_+• Aa(t) • limsup•_•+•Aa(t) •_+c•, or
(ii) lim•_•+• Aa(t) = -•-•,
then (1.2) is oscillatory.
Hartman's result has been extendedby many authors, seefor instance,
Butler [1], Coles[2] and Willet [8].
In section 2 we will prove a Hartman's type oscillation thoerem for
Eq. (1.1) with p • 1 any real number.Our resultwill show,in particular
that if (i) or (ii) of Th. 1.1 aretrue thenthe nonlineardifferentialequation
(1.1) is oscillatory.More than this our resultwill generalizeTh. 1.2 in
the sense that it will provide sufficient conditions for oscillation even if
liminf•_+• Aa(t) =
2.
Main
result.
Let A• denotethe n• composition
of A with itself.We will prove
THEOREM 2.1. //for some n
(i)
--c• • liminf,_+•A•a(t) • limsup•~+•Aa(t) _• +c•,
or
(ii) lim•_+• Aa(t) =
tl•en any nontrivialsolutionoœ(1.1) is oscillatory.
We note that if liminf•_•+• Aa(t) - -c• but for somen • N, n • 1,
(i) holdsthen (1.1) is oscillatory.Seesection3 for examples
illustratingthis
situation.
GENERALIZING
HARTMAN'S
OSCILLATION
RESULT
65
Pl•oor: Supposethat (i) or (ii) of Th. 2.1 is true and that thereexists
a nontrivialsolutionu(t) of (1.1) whichis not oscillatory.Withoutlossof
generality
wecanassume
that u(t) > 0 forall t E [0,+oc). Let usdefine
w'[0,-l-co)
• I•byw(t)-•,(•)
•,(•). Thenw(t) isa solution
ofthe
followinggeneralized
Ricatti equation
(2.1)
w'(t)+ (p- 1)lw(t)l
•' + c(t)= 0
forall t E [0,+oc). Integrating
(2.1)from0 to t • [0,+•c) weobtain
(2.2)
w(t)- w(O)
+ (p- 1)
Iw(s)P'ds
+ a(t)= O.
ApplyingtheoperatorA'•,n • •1, to (2.2)yields
(2.3)
A'•w(t)
q-(p- 1)A•
Iw(s)lV'ds
+ A•a(t)- w(O)
= O.
Fromhypothesis
(i) or (ii) we knowthat for somem • N thereexista > 0
and t0 > 0 such that
(2.4)
A•a(t) • -•
for all t ) t0. We then claim that
(2.5)
Iw(s)lP'ds
< +•.
In factif (2.5)is not truethenfrom(2.3),withn = m, it follows
that
(2.6)
lira A•w(t)=-•.
Ontheotherhand,bynoting
thatJensen's
inequality,
see[9,pp. 110],can
be appliedto the inte•al termof (2.3) to yield
(2.7)
Iw(X)l•'dX
• IAw(s)l
•' - (1- s)lAw(s)l
•',
66
MANUEL
DEL PINO, MANUEL
ELGUETA
AND RAUL MANASEVICH
for s •_ 0. It followsfrom (2.3), with n - 1, that
(2.8)
Aw(s)l/ds
+Aa(t)
<w(O)
+7
•(t)+•(P-•)
fo•
•fo
•(•- s)+lAw(s)/ds,
where(1 - s)+ = max{1- s,0}. Sincethe latter term on the righthand
sideof (2.8) tendsto 0 as t • +•, thereexistsa constant• suchthat
(2.9)
Aw(t)
+ (p-1• IA•(s)l•'ds
+A•(t)<•.
t
-
By induction on n and from Jensen'sinequality it follows that for each
(2.•0) A•(t)+ (p-•• IA•(s)l•'ds
+A•(t)<•.
t
-
In particularby settingn = m in (2.10) and since(2.4) holds,we have
(2.11)A•w(t)
+(p-1)•t
forallt • t0.If wenowletk(t)= f• IA•(s)l•'dsthen
(2.12)
k'(t)= A•w(t)p'
Now from (2.11) and (2.6) it followsthat
(2.13) (p-1)k(t)
<•_ A•w(t)
<_2A•w(t)
=21A•w(t)
I
for all t greaterthansomef > 0. From(2.12)and (2.13)we getfor t > •
(2.14)
•'(t)
•
(k(t)).•
• •'
where
• = ((p- 1)/2)/. Integrating
(2.14)between
s andt withf < s < t
we obtain
-(2.15) • (k(s))p,_x
- (k(t))p,_x
> Sp•-1
--
tp• -1
'
GENERALIZING
HARTMAN'S
OSCILLATION
RESULT
67
Sincefrom(2.6) k(t) -• -t-•x:ast -• -t-•x:,thenlettingt goto +cx:in (2.15)
yields
(2.16)
k(s)_<
for all s > [, whichcontradicts
the fact that IA(s)l
hence the claim
is true.
Now from the definitionof Aw and H61der'sinequ•ity we find tha½
(2.17)
lAw(t)
I<
fdw(s)
P'ds
-tl/P,
Also(2.3), for n = 1, becomes
(2.18)
Aw(t)+ (p- •)A
+ Aa(t)- w(O)= O.
From(2.5), (2.17) and (2.18)we conclude
that
(2.19)
lim Aa(t)= w(0)-
< +•.
(2.19) contradicts
the fact that (i) or (ii) is true,henceTh. 2.1 foUows.
3. Some examples.
In this sectionwe will give two examplesto which Th. 2.1 apply.
PROPOSITION3.1. Considerthe equation
(3.X)
' + tXg(t)[u[P-•u
=0
whereA is a real number•eater t•an one and g: [0,+•) • • is a Tperiodic continuousfunction with mean value zero. Then any nontrivial
solutionof (3.1) is osciHatory.
The proofof this propositionis a consequence
of the followinglemma
which is stated without proof.
68
MANUEL
DEL PINO, MANUEL
ELGUETA AND RAUL MANASEVICH
LEMMA. Let [A]denotethe greatestpositiveintegerlessthan or equalto
Then
a(t)- f• s•g(s)ds
canbewritten
as
(3.2)
a(t)= Z t•-ipi(t)-t-O(t),
i=0
whereeachPi is a T-periodiccontinuous
functiondefinedon [0,-t-•x•)with
meanvaluezeroand 0 ß[0,-t-•) • R is a boundedcontinuous
function.
PROOf or PROPOSITION
3.1: From (3.2) we can find that for j • N
(3.3)AJa(t)
-_
•)J(t)
if j > [A],
where
thePi,Oj functions
satisfy
thesame
properties
aspi,0 respectively.
Thusfrom(3.3),limsupt_•+•Aa(t) = +c• andforj > [•],
(3.4)
-c• < liminfAJa(t)< k
for a certain constant k. Hence from Th. 2.1, proposition 3.1 follows. !
Note. In theaboveexample
wehavethatliminf•__• AJa(t)- -c•
if j _• [A]thusthe oscillatory
natureof (3.1) doesnot followfromTh. 1.1.
In our secondexamplewe consider(1.1) with c • C[0,+c•) defined
as follows.Let (ai)•l, {bi)•l be two sequences
of real numbersdefined
respectively
by ai -- i - 2-',bi - i -t-2-i i • N. Let {gi)?--1denotea
sequence
of functions
gi: [0,-t-•) -• R of classC2 suchthat gi(t) > 0 if
t • (ai, hi) andgi(t) -0 otherwise.We alsoaskthat
(3.5)
gi(s)ds-i
for eachi • N. Next defineg' [0,-t-•) -• • by
(3.6)
g(t)- •-•,(-1)i
gi(t).
i--1
GENERALIZING
HARTMAN'S
OSCILLATION
RESULT
69
Then g is of classC 2 on its domain.Finally we set
(3.7)
c(t) = (tg(t))" anda(t)=
c(s)ds.
The reasonfor definingc in this form is two-fold. On the one hand, from
(3.7) we find that
(3.8)
Aa(t)= •
c(•')d•'ds
= g(t),
and from (3.5) and the meanvaluetheorem
(3.9)
max gi(s)>
i. 2i-x
-
s½[0,+o•)
for eachi e N. Thusfrom (3.8) and (3.9) we obtainliminf,_•+• Aa(t) =
-o0 and therefore Th.l.1 cannot be used to study the oscillatory character
of (1.1). Let us next take p = 2 in (1.1) and we note that the Lebesgue
measure
oftheset{t[J'jc(s)ds
y•0} isfinite.Since
thisimphes
that
(3.10)
hm approx
c(s)ds= O,
a well known oscillation criterion due to Olech, Opial and Wazewski, see
[7], is alsonot applicable.
In spite of these above remarks and as a consequenceof our Th. 1.2
we have
PROPOSITION
3.2. Any nontrivialsolutionof Eq. (1.1) with p any real
number greater than one and c defined as above is oscillatory.
PROOf:From (3.9) we havelimsupt_•+•Aa(t) = +½x•. Also, for t E
(bi,ai+x)- (i q-2-i,i q-I - 2-(i+1)),i e N],wehave
(3.11) A2a(t)
= •_(,•x)
ieven
i odd.
From(3.11)it canbe easilyshownthat A2a(t)is bounded
for t e (0,+oo).
Hencefrom (i) of Th. 2.1, Eq. 1.1. is oscillatory.!
70
MANUEL
DEL PINO, MANUEL
ELGUETA
AND RAUL MANASEVICH
REFERENCES
1. Butler G. J., Integral averagesand the oscillation of secondorder ordinarg differential equations,SIAM J. Math. 11 (1980), 190-200.
2. Coles W. J., An oscillation criterion for secondorder differential equations,Proc.
Amer. Math. Soc. 19 (1968), 775-759.
3. Del Pino M., Applied Math. Engineering Thesis. U. of Chile
4. Del Pino M., Manasevich R., Elgueta M., Sturm's comparison theorem for
([u'lP-eu') ' +c(t)(lulP-eu),p > 1, TechnicalReport, Dept. of Math., F.C.F.M.,
U. of Chile (1988).
5. Hartman P., On nonoscillalorg linear differential equations of second order, Amer.
J. Math. 74 (1952), 389-400.
6. Mirsow J.D., On Some Analogs of Sturm's and Kneser's Theoremsfor Nonlinear
Systems,Journal of Math. Anal. and Aplic. 53 (1976), 418-425.
7. Olech C., Opial Z. & Wazewski., Sur le probleme d'oscillation des intdgrales de
l'equationy" + g(t)y = 0, Bull. Acad. Polon. Sci. 5 (1957), 621-626.
8. Willeft D. W., On the oscillatory behaviour of the solutions of secondorder linear
differential equations,Ann. Polon. Math. 21 (1969), 175-194.
9. Royden H.L., Real Analysis, The Macmillan Co. (1971).
Manuel Elgueta
Department of Mathematics, F.C.F.M.
U. de Chile
Gasilia 170, Correo 3, Santiago, Chile
M. del Pino, R. Manasevich
Department of Mathematics
U. Catolieu
de Chile
Casilla 6177, Correo 22
Santiago, Chile (M. Elpueta)
Received April 24, 1989
Revised Version Received September 26, 1989