Internat. J. Math. & Math. Sci.
VOL. 13 NO.
(1990) 151-158
151
STRONG OSCILLATIONS FOR SECOND ORDER NONLINEAR FUNCTIONAL
DIFFERENTIAL EQUATIONS
JURANG YAN
Department of Mathematics
Shanxi University
Talyuan, Shanx i
People’s Republic of China
(Received July 28, 1986 and in revised form May 5, 1987)
ABSTRACT.
In this paper, we establish some strongly oscillation theorems for
nonlinear second order functional differential equation
x"(t) + p(t) f(x(t), x(g(t)))
0
without assuming that g(t) is retarded or advanced.
KEY WORDS AND PHRASES.
Strong oscillation, nonlinear, functional differential
equations.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
Primary 34K15.
We consider the second order nonlinear functional differential
equation
x"(t) + p(t) f(x(t)
where
x(g(t)))
p(t), g(t) e C([t O ), R)
g(t)
0
as t
sign of u and v when they have the same sign.
and f(u,v)
(R,R) and has the
We shall restrict our attention to
solutions of (I.I) which exist on some positive half-line
A nontrlvial solution x(t)
is called oscillatory if x(t) has an unbounded set of zeros, and otherwise it is
called nonoscillatory.
Equation (I.I) is said to be oscillatory if every solution of
(I.I) is oscillatory
Bradley [I],
[6], Waltman [7], Wong [8] and
wellknown theorem of Wintner [9] and Leighton [I0] that the
Oscillation theory for (I.I) has been developed by many authors.
Chiou
[2], Erbe [3] Gollwitzer [4], Ladas [5],
references therein
It is
Travis
linear equation
x"(t) + p(t) x(t)
is oscillatory if
p(t)dt
0
even p(t) is not assumed nonnegatlve
Bradley [I]
152
J. YAN
and Waltman [7] demonstrated that the equation
x"(t) + p(t) x(g(t))
p(t)
is oscillatory if
(1.2)
0
p(t)
0 and
Travis
dt
[6] constructed a
counterexample showing that the Le[ghton-Wintner oscillation theorem can not be
extended to Equation (1.2) unless
O. The author [II] extended Bradley-Waltman
p(t)
0 and p(t)dt
then (1.I) is
p(t)
,
oscillation theorem to (1.I) i.e. if
ocillatory.
The purpose of this paper is to establish some strongly oscillation criteria for
(1.1).
We are primarily interested in the case when
p(t)
are satisfied.
Considering the equation
x"(t) + %p(t) x(t)= 0,
We shall call p(t) a strongly oscillatory coefficient if (1.3) is oscillatory for all
positive
%.
p(t)
If
lira
O, Nehari [12] shows that
supt/t
p(s)ds
is a necessary and sufficient condition for
p(t) to be a strongly oscillatory
Fzluatlon (I.I)
In general, motivated by Nehari, we define as follows:
coefficient.
(I.l)
is said to be strongly oscillatory if the related equation of
x"Ct) + Xp(t) fCx(t), x(g(t)))
(1.4)
0
is osillatory for all positive
2.
MAIN RESULTS.
For Equation (I.I) the following conditlons are assumed to hold throughout the
paper:
p(t)
(i)
0 and there exists
h(t)
min
(g(t), t) such that
0
<
k
h’(t) where k
is a constant.
(il) there exists m
>
0 such that
e
lira inf
where
where
(v)
I
C’(R), v(v)
and
lul
>
implies
>
0 and ’(v)
O,
0 for v
#
0, and
lira
6 are constants.
We begin with a Lemma which needed in establishing our results.
’(v)
)
>
0
STRONG OSCILLATION FOR NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
LEMMA 2.1.
I
,
o
t w2(s)
ds +
0 Equation (l.t) has a nonoscillatory
loe p(s)ds,
x’(t)/@(x(h(t))), o and
w(t)
>
o
Then the following inequality holds for all large t,
w(t)
where
I
Suppose that for
solution x(t).
(2.1)
are positive constants.
I
PROOF. Assume that Equation (1.4)at I
has a nonoscillatory solution
o
x(t) > 0 for t
t > 0. A similar proof will hold if x(t) < 0 for t ) t
o
o
It is easy to verify that x"(t) < 0 and x’(t) >0 for all large t. Let
w(t)
x’(t)/(x(h(t))), then
’
w’(t)
Since
If
>
x’(t)
llm
-hoP(t)
0 for large t,
x(t)
f(x(t), x((t)))
(x(’h(-t)
#’(x(h(t)))x’(h(t))h’(t) w(t).
q(x( h(f)
llm x(t) exists either as a finite or infinite limit.
e is finite, then
[(x(t), x(g(t)))
f(=,=)
t
If
lira
==,
x(t)
then by (ii) we have that
f(x(t), x(g(t)))
(x( g (t)
)
In either case, for sufficiently large t, we have that
for large t.
f.(x(t)_, x(g(t)))
e
(x(g(t)))
e
where
rain (e
(2.2)
e2).
Since x(t) is increasing, for large t we have that
x(.g(t)))
p(t) f(x(t),
(x(h(t)))
<
and in view of x"(t)
Thus for
)
P (t)
(x(h(t)))x’ (h(t))h’ (t)
w(t)
@(x(h(t)))
t
)
)
t
#(x(g(t)))
logP(t)
0 for large t and (ii) we see that
’
t
f(x(t), x(g(t)))
)
’(x(h(t)))x’(t)h’(t)
#(x(h(t)))
, k6w2(t) OW2(t)
o
w’(t) + o
w2(t)
w’(t) + o
w2(t)
+ Io e p(t)
0
(2.3)
and
O.
From (2.4), we have that
d
(- -I’’) + ot)
--:r
wrd
O.
(2.4)
153
J. YAN
154
Integrating the above inequality from
<
0
b
where
w(t)
From (2.5) we have that
w(t)
lira
(2.5)
+ b
ot
kt
W(tl
to t we obtain that
t
Integrating (2.3) from
(2.6)
O.
to t, then letting
t
we obtain that (2.1) holds for
t
all large.
[Pn(t)},
We introduce the function sequence
n
0,1,2,...,
defined as follows:
p(t),
Po(t)
Pn+i (t)
sup
lim
t
(12 there
Pm(t) does
(s) ds,
Pn
1,2,3
n
.....
(2.7)
Assume that one of the following conditions holds,
THEOREM 2.2.
(I I) there is an integer
but
It Po (s)ds’
Pl(t)
/
Pn(t)
such that
m
;
tPm(t)
is an integer
is defined for n- 1,2,...,m, and
(2.8)
Pn(t)
2 such that
m
is defined for n
1,2,..., m-l,
not exist, i.e.
"
Pm-12
(t) dt
(R).
hen equation (1.1) is strongly oscillatory.
Assume to the contrary that Equation (1.4)at X-Xo > 0, has a non0. A similar argument holds when x(t) < 0
t
oscillatory solution x(t) > 0 for t
o
x’(t)/(x(h(t))). As in the proof of Lemma 2.1, we can
O. Let w(t)
t
for t
PROOF.
o
obtain
I.
Suppose m
w’(t) +
ow2(t)
Define
u (t)
o
+
ut(t)
O
where
k Go.
o
o
u2(t)
O
However,
+ X
o
Po (t)
0,
t
t
t
(2.9)
o
ow(t); then
+
oPo(t)
0
t
t
(2.10)
t
o
by a well-known theorem of Wintner
[13] this implies the
equation
y"(t) +
is nonosclllatory.
Po(t)
goPo(t)y(t)
(2.11)
0
This contradicts the fact that the condition
is a strongly oscillatory coefficient.
If m
>
I, integrating (2.10) from
t
to t we obtain
(I
I)
implies
EQUATIONS
STRONG OSCILLATION FOR NONLINEAR FUNCTIONAL DIFFERENTIAL
,J
Since
u
lim
o
U
(t) +
2
Uo(t)
Ul(t)
Hence
u(t)
where
u21(t)
+
u (s)ds +
o
Op
rt
o
Jt Po(S)ds
0
(2.12)
from (2.12) we get
(),
(2.13)
t )t t,
Then (2.13) implies that
Jt u-(s)ds’o-
Ul(t)-
2
ftt
ow(t)=0,
lim
(t)
O
+
u(t
(t)
(t)
U
where
o
155
+
-
u(t)- 2oUl(t)p
2
.2 2
op l(t).
(t)
(2.14)
; 0,
iPl(t)
2o
51
(I[)
2, then by the Neharl theorem, the condition
If m
2
y"(t) +
IPi
(t)y(t)
is oscillatory, contradicting
(1
If the condition
implies equation
(2.15)
0
(2.14).
is satisfied, then by the Leighton-Wintner theorem we have
2
that equation (2.15) is oscillatory, again a contradiction.
When m
>
2, we obtain inductively that
2
u’ (t) + u m-I (t) +
m-I
m-I
(I I) or (1
2
REMARK.
0
(2.16)
t
Pm_l(t)
is
Applying the Wintner theorem again, it follows that equation
y"(t) +
is nonosctllatory.
(t)
is a constant and
m-2
defined by (2.7).
2
m-lPm-I
2
Ce_tPm_t (t)y(t)
(2.17)
0
But this contradicts again the fact that the condition
implies that equation (2.17) is oscillatory.
The proof is thus complete.
Theorem 2.2 includes Theorem 2.2 in [6] as a special case, i.e.
t.
(u)=u and m
Consider the function sequence
{qn(t,,)},
n--l,2,..., which is defined as
follows:
qo(t,)
qn (t’’q)
where
$ and
r
"
p(s)ds,
ql(t,,)
2
(s,,q)ds +
qn-I
qo2(s,q)ds
qo(t,D),
q are positive constants.
t
+
qo(t
q),
to,
n
2,3
(2.18)
156
J. YAN
For all positive constants
THEOREM 2.3.
and q assume that any one of following
conditions is satisfied:
(II
I)
there is a positive integer m such that
qn(t,,n)
is defined for
n-l,2,..., m-l, but qm(t,,) does not exist;
(II 2) qn(t,,n) is defined for n=l,2,..., but the function sequence (2.18) is
not convergent for all large t.
(If
3)
the sequence (2.18) is convergent and
llm qn(t,,n)
q(t,,n), but
n /
,’).
0
Then equatlon (I.4) is strongly oscillatory.
PROOF. ssume that Equatlon(1.Z) atA=A > 0, has a solutlon x(t) > 0 for
O
0. A similar proof will hold if x(t) < 0 for t
t
t
Let
t
O
O
w(t)=x’(t)/(x(h(t))). From Lemma 2.1 we can obtain (2.1). It follows that
w(t)
Hence
qo(t,o), where n
L2(t
q(t,,n)
O
2
qo(t,no)
o
ql(t’o’no
If m
ft
o.
>
t
(lI I) holds.
Suppose that
O
O
2
w (t)
where
=.
to.
t
(2.19)
If m--I from (2.1), (2.19) implies tha:
qo(S’no )ds
/
qo(t’no
w(t),
t
This is in contradlclton to the nonexistence of
I, from (2.1) and (2.19) we get that
qm-1(s’No)
w2(s)
ds
oo
qm (t
ds
<
o..
Applying (2.1) we have that
2
(I12)
t
w(t). Hence
qm_l
ft qm-1 (s o no)dS + qo (t
and we arrive at a contradiction of
Suppose that
".
)
U
O
(Ill).
From (2.18) and (2.19), we conclude that for all
holds.
large t
qn-1 (t’o’no)
Therefore
qn(t,o,no)
llm
the fact that
qn(t’o’no
qn(t,
Suppose that
llm
(If
no
3)
w(t), n-l,2
......
exists and has a flnte limit.
(2.20)
But this constradlcts
is not convergent.
By (2.20),
holds.
qn(t’o’no)
q(t,o,no)
(2.21)
w(t).
n
Using (2.21), we have that
condition
(113).
ft
q
2
(S’o’o)dS
This completes the proof.
2
w (s)ds
<
which contradicts the
STRONG OSCILLATION FOR NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
As se that
THEOREM 2.4.
du
(u)- <
and
du
f_
<
(u)
’
>
a
(2.22)
0.
Further assume that sequence (2.18) for all positive constants
and
satisfies any
one of the following conditions:
(III
I)
qn(t,,)
there is a positive integer m such that
is defined for
n=0,1,2...m, and
Jt qm
;
(s’ ,rl)ds
o
(1112 qn(t,,n)
n=0,1,2,...,and
is defined for
llm
n
exists and satisfies
q(s,E,r)ds
qn(t,,n)
q(t,,q)
(R).
0
Then Equation (I.I) is strongly osillatory.
x(t)
>
> 0, has a nonoscillatory solution
o
x’(t)
0 is handled similarly. Let w(t)
Assume that Equation (1.4)at k=k
PROOF.
0 for
t
t
o
<
The case x(t)
(x(h(t)
By Lemmma 2.1, we find that (2.1) holds.
(III I) holds, then, as proof of Theorem 2.3,
Suppose that
qm (t’o’no)
o t q2(s’o’ qo)dS
qm(t’o’o
x’ (t)
#(x(h(t)))
+
qo(t,o ’qo
w(t)
or
(h(t)h’ (t)
x’k#(x(h(t)))
(2.23)
From (2.22) and (2.23), we have that
t
qm(t’o ’no)ds
lim
t
o
This contradicts condition
(1112
Suppose that
lira
m
q
=t
o
,n o)
lira
x(h(t))
fx(h(t
))
o
du
k(u)
(llll).
holds, then it follows from (2.23) that
x’ (h(t)h’ (t)
k(x(h(t)))
namely
q(t6o,n o)
x’ (h(t)h’ (t)
k(x(h(t)))
He nce
I im
157
ft q(S’o’no)dS
o
lira
t
x(h(
fx(h(t o ))
du
(u)
which is again a contradiction, and the proof of the theorem is complete.
Equation (I.i) is said to be strongly bounded oscillatory if all bounded
solutions of Equation (1.4)
for any
%e(O, =) are oscillatory.
J. YAN
158
F.ow he proof of Therem (2.4), we see that the following result holds.
holds, then Equation
COROLLARY 2.5. Assane that the condition (Ill 1) or
([112
(I.I) is strongly bounded oscillatory.
ACKNOWLEDGEMENT.
The author would like to thank the referee for his valuable
s ugges ions.
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ERBE, L.
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GOLLWITZER, H.E. On nonlinear oscillations for a second order delay equation,
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LADAS, G.
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WONG, J.S.W.
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