I nternat. J. ,ath. & h. Sci.
Vol. 2 #3 (1979) 523-530
523
RESEARCH NOTES
ON SLOW OSCILLATION AND NONOSCILLATION
IN RETARDED EQUATIONS
BHAGAT SINGH
Department of Mathematics
University of Wisconsin Center
54220
Manitowoc, Wisconsin
U.S.A.
(Received September ii, 1978 and in revised form June i0, 1979)
ABSTRACT.
Sufficient conditions have been found to ensure that all oscillatory
solutions of
(r(t)y’(t))’ + a(t)y(t-(t))
are slowly oscillating.
KEY WORDS AND HRASES.
Slowly Oscing.
This behaviour is further linked to nonoscillation.
Oson, Nonosillation, Moderately Oscillating,
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
i.
f(t)
34ci0.
INTRODUCTION.
In studying the asymptotic nature of oscillatory solutions of the equation
(r(t)y’(t))’ +a(t)y(t-(t))
this author in
(i)
f(t),
[9, Theo. 2] showed that a nontrivial oscillatory solution y(t)
of (i) satisfies
lim
y(t)
o
if
f la(t)Idt
<
f
if(t)Jdt
<
and
ft
<
.
B. SINGH
524
Idt
In theorem 4 in [9], it was observed that
solutions which do not approach zero as t
/
lead to slowly oscillating
.
In this work, we give sufficient conditions which cause all oscillatory
solutions of the equation
(r(t)y’(t))’ + a(t)h(y(g(t)))
(2)
f(t)
to oscillate slowly.
Even though voluminous lirature exists about various types of oscillatory
and nonoscillatory criteria for such equations, the asymptotic nature of oscillatory
solutions of these equations has not been so extensively studied.
For a good
literature study of related results Graef [i] and Graef and Spike
[2] have in-
The work of Travis [ll] (c.f[8]) shows that
cluded an exhaustive reference list.
common techniques found for ordinary differential equations fail on retarded
differential equations even when the retardations are small.
[4] and this author [7].
criteria see T. Kusano and H. Onose
2.
For oscillation
ASSUMPTIONS AND DEFINITIONS
The entire study in this work is subject to the following assumptions:
a(t), r(t), f(t), h(t) and g(t) are
(i)
r(t) > o, g(t) > o, g’(t) > o
(ii)
g(t) < t
(iii)
and
g(t)
/
h is odd, sign (h(t))
(iv)
such that
o <
<
as
t
cO(R),
where R is the real line,
-
sign(t) and there exists positive constants ,m
h(t) <
t
m
on some positive half llne.
In order to reckon the half line we shall assume that it would mean for
t > t
O
for some positive t O
tO
will be referred to without further mention.
All functions considered are real valued.
We call a function
large zeros in
[t0,
H(t)
C0[t0,)
oscillatory if
H(t) has arbitrarily
otherwise we call it nonoscillatory.
In order to be more
precise we shall use the term "solution" only for those nontrivial solutions (of
equations under consideration) which can be extended continuously for
t > t 0.
OSCILLATION AND NONOSCILLATION IN RETARDED EQUATIONS
We define a function
Hl(t
3.
to be slowly oscillating if
Y0
x0
> x0
Y0
and
x
0
Hi(t
are
x0 > t o }
Hl(t)
ZHI
If
is unbounded.
{Y0
ZHI
is oscillatory and the set:
consecutive zeros of
C0[t0,oo
g
525
HI
is bounded,
is called
"moderately oscillating".
ON SLOW OSCILLATION
Theorem 4 in [9] states that moderately oscillatory solutions of (i) approach
fla(t) Idt
zero if
<
If(t) Idt
and
<
.
Our next Theorem gives sufficient
conditions for all oscillatory solutions of (2) to be slowly oscillating.
THEOREM (i).
is bounded for large
rlj,f(t),dt
<
and
where
al(t)
> o
and
a2(t)
a.l(t)
Further suppose that
t.
lim inf
t-o
of equation (2) satisfy
lira sup
al(t) + a2(t)
a(t)
Suppose
ly(t)
>
If(t)
> o
alt
Then all oscillatory solutions
y(t)
(3)
O
and
y(t)
(4)
is slowly oscillating.
y(t)
be an oscillatory solution of (2).
PROOF.
Let
(ry’)’
+ h(y(g(t))) + a2(t) h(y(g(t)))
Rewriting (2) we
have
a (t)
I
Suppose to the contrary that
h(x)
/
o
as
(i +
lira inf
t-o
f(t)
x
/
o
lira
t-o
y(t)
(5)
al(t
al(t)
a2(t)
o.
Since
a1()
/
Since
is bounded and
we get that
a2(t)/al(t )) h(y(g(t)))
(If(t) / al(t)) > o (t)
/
o
as
t
reveals that
y’(t)
assumes a constant
526
B. SINGH
y(t) nonoscillatory.
sign making
This contradiction shows that
> o
sup lyit)
t-o
Now
fT
la(t)Idt
fal(t)dt
<
f la2(t)ldn
+
T
T
fT
al(t)dt +
al(t)
al(t)dt
<
since
a2 (t)
is bounded
al(t)
y"(t) +
Here
al(t)
/
and
f
T
If(t) Idt
y(t) is slowly oscillating.
4 of this author [9],
EXAMPLE (i).
as t
<
By Theorem
The proof is now complete.
Consider the equation.
5
i
A-._ y(t)
5
a(t)
t
t3/2
> o
(6)
All conditions of Theorem 3 are satisfied.
Hence
all oscillatory solutions of (6) are slowly oscillating and satisfy
lira sup
t-o
ly(t)l
> o
In fact
(1 + 2 sin(%nt)) is one such
y(t)
solution.
EXAMPLE (2).
Let
y(t)
y"(t) + i + 2 sin
t
t3
Taking
al(t)
i/t 3
a2(t)
this theorem are satisfied.
be a solution of
y(t
)= 2
+ Cos
t4
t
t
>
(7)
2 sin t / t 3 we find that all conditions of
Hence either
y(t) is nonoscillatory or slowly
oscillating with no limit at
Example i suggests the following theorem:
THEOREM (2).
a2(t) / al(t)
Suppose
a(t)
al(t) + a2(t)
is bounded for large t
where
al(t)
Further suppose that
> o
and
OSCILLATION AND NONOSCILLATION IN RETARDED EQUATIONS
If(t)[dt
<
and
solution of (2)
PROOF.
as t
/
a()
y(t)
Then
y(t)
(rY’)’/al(t)
> o,
al(t
t
5
y’’ (t) +
t >
be an oscillatory
lim sup
lira sup
t-o
ly(t)]
]y(t)]
Suppose to the
Then (5) immediately reveals that
/
]y’(t) l>
o
eventually.
nonoscillatory, a contradiction.
EXAMPLE (3).
y(t)
is slowly oscillating and
is bounded.
as
/
Let
/
We only need to show that
contrary that
Since
]f(..t.)l
527
This forces
y(t)
to be
This proves the theorem.
The equation
+ 6 sin.(%nt)
1
y(t)
+
6
in(Zn t) + 2
sin2<n
(8)
t
O
satisfies all conditions of Theorem 2 by choosing
al(t)
and
4t 2
yt)
6 sin Int)
t4
a2(t
/ i + 2
a I and a 2 as
This equation has
sin
as an oscillatory solution satisfying the conclusion
of this theorem.
Our next theorem gives conditions for boundedness of all moderately oscillating
solutions of equation 2.
where
al(t)
f la (t)Idt
/
<
L > o as t
It is well known that if
alt)
/
a(t)
al(t) + a2(_t
is of bounded variation and
then all solutions of
y"(t) + a(t)y(t)
(9)
o
are bounded with bounded derivatives.
See Cesari [3, p.85].
It is not true
for retarded equations
y"(t) + a(t)h(y(g(t)))
as the following example shows.
o
(IO)
528
B. SINGH
EXAMPLE (4).
y" +
has
The equation
2e/2y(t
e t sin t
y(t)
(ii)
o
However our next theorem gives the partial
as a solution.
We prove it in more generality for equation 2.
result.
THEOREM (3).
f lam(t) Id t
<
x2
sup
y(t)
If(t) Idt
(4-))/c2m
Y(X2)
Xl
al(t)
L > o as t
/
Further suppose that there exists a
y of (31)
where for any oscillatory solution
Y(Xl)=
Let
T
y(x)
o
<
xe(xl, x2)
o,
be large enough so that for
g(T1) >
be large enough so that
la2(t) Idt
< /2
fT
and
If(t)
Suppose to the contrary that
TI’ > T2
<
where
is bounded.
PROOF.
T
J’
and
al(t) + a2(t)
a(t)
Suppose
L + )t <
such that
then
-/2)
> T I such that
> T
y(T I)
and
al(t)
< L + %
Let
o
dt < /2
lira sup
Y(T2)
T
t
o and
smallest closed interval containing
ly(t)
PI
TI’.
Then there is a
max
ly(t)
T
<_
It is clear that
t < T
v
I.Y(T ’)I
T 2 < Xl,
(12)
and
lY(g(t)
[Xl, x2].
for te
MI
-jXOy ’(t)dt
xI
< M
I
Let
Xog[xl, x2]
f
x
xo
y’(t)dt
2
we have
2Ml
<
f
x2
Xl
lY’(t) Idt
(13)
such that
MI
lY(XO)
Since
> %.
OSCILLATION AND NONOSCILLATION IN RETARDED EQUATIONS
f
x
x2
lY’(t) 11/21y
I
f
4MI <
f
dt
<
From (14)
(x2
y’(t)
y’(t)dt
Integrating by parts and using (2)
by Schwarz’s inequality.
4
f
y(t)a(t)h(y(g(t)))dt-
Xl)
x2
f
x
<
since
x2
4 <
m
I
a(t) h(y(g(t))) y(g(t))dt +
y(g(t))
x2
(al(t) + la2(t) l)dt
2
+f
X
_< e2m(L + %) +
fxx2 la2(t) Idt
m
I
or
4 <
a2m(L
f
x
_
(14)
x2
I
(12) and (13)
This yields in view of
which gives
+ %) + %
x2
If(t
(15)
fxx2 If(t)
(16)
1
-
+a
"’i
I
(17)
x
since
ii
mfx 2 la2(_t) Idt
< i
REMARK.
L
o,
2m
and
(L + )
conclusion of the theorem.
x2
f
x
If(t) Idt <_ l/2
I
completes the proof.
Coming back to example (4)
e t sin t
>
< %/2
I
This contradiction, in (17)
far any
y(t)f(t)dt
xI
xI
xI < e
xI
y
we get
we have
4
4
529
2e
/2,m 1
2(2 e/2
we see that for the solution
f(t)
+ ) > 4
o and
a2(t)
Thus
o
,/satisfying
the
530
B. SINGH
EXAMPLE (5).
y"(t) + 2
e-3n/2y(t
-)
(18)
o
e -t sin t as a solution.
y
-3z
-4"7
m
e
l
has
L
The equation
Here
e---Z
%
Therefore for
m2(L
+ %)
.01
2(e-4"7
+ .01)
< 4
once again, satisfying this theorem.
REFERENCES
i.
Graef, John R.
2.
Graef, John R. and Paul W. Spikes.
Oscillation, Nonoscillation and Growth of Solutions of
Nonlinear Functional Differential Equations of Arbitrary Order, J.
Math. Anal. Appl. 60 (1977) 398-409.
Asymptotic Properties of Solutions of
Functional Differential Equations of Arbitrary Order, J. Math. Anal.
Appl. 60 (1977) 339-348.
3.
Cesarl, L. "Asymptotic Behavior and Stability Problems in Ordinary
Differential Equations," Sprlnger-Verlag, Berlin, 1959.
4.
Kusano, T. and H. Onose.
5.
Londen, S.
6.
Norkln, S.B.
7.
Singh, B. Oscillation and Nonoscillatlon of Even-order Nonlinear Delaydifferential Equations, Quart. App!. Math. (1973) 343-349.
8.
Singh, B. Asymptotic Nature of Nonoscillatory Solutions of nth Order Retarded
Differential Equations, SlAM J. Math. Anal. 6 (1975) 784-795.
9.
Singh, B. Asymptotically Vanishing Oscillatory Trajectories in Second Order
Regarded Equations, SlAM J. Math. Anal. 7 (1976) 37-44.
Oscillations of Functional Differential Equations
with Retarded Arguments, J. Differential Equatigns, 15 (1974) 269-277.
Some Nonoscillation Theorems for a Second Order Nonlinear
Differential Equation, SlAM J. Math. Anal. 4 (1973) 460-465.
"Differential Equations of Second Order with Retarded Arguments,’
American Mathematical Society, Providence, Rhode Island (1972) 1-8.
I0.
Singh, B. Forced Oscillations in Second Order Functional Equations, Hiroshima
Math. J. 7 (1977) to appear.
ii.
Travls, C.C.
12.
Wallgren, T.
Oscillation Theorems for Second Order Differential Equations
with Functional Arguments, Proc. Amer. Math. Soc. 30 (1972) 199-201.
Oscillation of Solutions of the Differential Equation
f(x), SIAM J. Math. Anal. 7 (1976) 848-857.
y’’ + p(x)y