I nternat. J. Math. & Mah. Sci.
Vol.
269
(1978)269-283
NECESSARY AND SUFFICIENT CONDITIONS FOR EVENTUALLY
VANISHING OSCILLATORY SOLUTIONS OF
FUNCTIONAL EQUATIONS WITH SMALL DELAYS
BHAGAT SINGH
Department of Mathematics
University of Wisconsin Center
705 Viebahn Street
Manitowoc, Wisconsin 54220 U.S.A.
(Received March 28, 1978 and in Revised form June 30, 1978)
ABSTRACT.
Necessary and sufficient conditions are found for all oscillatory
solutions of the equation
(rn_l(t) (rn_2(t) (---(r2(t) (rl(t)y’ (t)) ’) ’) ’---)
to approach zero.
+ a(t)h(y(g(t)))
b(t)
Sufficient conditions are also given to ensure that all
solutions of this equation are unbounded.
KEY WORDS AND PHRASES. Oory, Nonoiatory, DZay, Funonal.
AMS(MOS) SUBJECT CLASSIFICATION (1970) CODES.
i.
INTRODUCTION.
Recently, T. Kusano and H. Onose [6] studied the equation
(rn_l (t) (rn_2 (t) (--- (r2 (t) (rl(t)y’(t))’)’---)’)’)’
+ a(t)h(y(g(t)))
b(t)
(i)
and found sufficient conditions which force all bounded nonoscillatory solutions
of (I) to approach zero when a(t) is oscillatory.
For positive a(t), the same
270
B. SINGH
conditions ensured that all nonoscillatory solutions of (I) approached zero.
Kartsatos [4, Theo. I] also found sufficient criteria for all bounded non-
oscillatory solutions of (i) to, asymptotically, vanish generalizing results
of this author and Dahiya [7, Theo. I].
In fact, since the work of Hammett
[3] such asymptotic results about the nonoscillatory solutions of ordinary
and retarded differential equations have been obtained by many authors such
as Kartsatos [4], Kusano and Onose [5, 6], this author and Dahiya [7], this
author [9, I0, ii] and many others.
A fairly exhaustive list of references
on oscillation can be found in Graef [2].
nonoscillation properties of solutions.
Most of these results relate to
Very little has been said about the
asymptotic nature of the corresponding oscillatory solutions of these equations.
This author’s work [8, 9, 12] is devoted to this type of study about the
oscillatory solutions of such equations.
Our purpose.in this paper is to further the study initiated by Kusano
and Onose [6] and find necessary and sufficient conditions to ensure that all
oscillatory solutions of equation (I) tend to zero as t
/
.
In the last
section, we give sufficient conditions which cause all solutions of (i) to
then [i] studied a similar problem but our results are
be unbounded,
different and more extensive.
In what follows, we shall restrict our study to those solutions of (I)
which can be continuously extended on some positive half line, say for
t
_>
t O > o.
We shall, therefore, assume the point
of this paper.
solutions on
t
O
fixed for the rest
The term "solution" applies only to continuously extendable
R+
[to,
).
DEFINITIONS AND ASSUMPTIONS.
2.
The following conditions hold for the rest of this paper:
(i)
a (t), b(t), g (t), r (t), ---,
I
rn_ 1 (t)
are real valued and continuous
OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
on
[to,
), h
C (-, );
g(t) < t, g(t) +
(ii)
t h(t) > o, t
(iii)
+
t
as
/
;
o;
m
there exists a number
(iv)
271
h(t___) <
such that
(2)
m
t
ri(t) _>
(v)
> o,
i, 2, ---, n-l; for large
i
t
on
+.
R
A solution is said to be oscillatory if it has arbitrarily large zeros, other-
wise it is said to be nonoscillatory.
To further shorten notations we
designate:
ZlY(t
(rl(t)y,(t)), Z2Y(t)
ZiY(t)
(ri (t) (ri_l (t)
(rl(t)y’
(t))’)’---)’)’)’,
---, n-l.
i, 2,
i
(--- (r2 (t)
(r2(t) (rl(t)y,(t)),),
(3)
MAIN RESULTS.
3.
be a bounded oscillatory solution of (I).
Then
ri+l (ZiY (t)
(4)
tn-i-2
as
t +
,
I, 2, ---, n-2.
i
PROOF.
2, ---, n-l.
Zn_2Y(t I)
y(t)
Since
Let
E
> o
is oscillatory,
is oscillatory for
ZiY(t)
be arbitrary and let
t
I
> t
o
i
be so large that
o,
mMla(t) Idt
< /2
(5)
tI
and
(6)
where
Y(g(t))
< M
for
t
>_. t I.
I,
B. SINGH
272
Integrating equation (i) we have
Irn_l(t)Zn_2Y(t)
Thus
Zn_2Y(t)
Let now
for
/
t2 > t
t > t2
.
tI
be a zero of
Zn_3Y(t)
t
as
o
I
mMftla(x)Idx
<
/
Ib(x) Idx
+
t
<
I
Irn_l (t)Zn_2Y(t)
so that
(7)
< E
Now
ftt Zn_2Y(s)ds
rn_2(t)Zn_3Y(t)
2
which readily gives
Irn_2(t)Zn_3Y(t)
(t-t 2)
<
(9)
t
t
Proceeding this way, the proof is completed.
LEMMA (3 2)
8 e [0,
for some
la(t) Idt
r I (t)
tn-8
i).
1
rn_ 1 (t)
,
Ib(t) Idt
<
and
t > T, g(t) > t O
be large enough so that for
Integrating (I) (for t > T) over
Zn_ 2(t)y(t)
<
Then oscillatory solutions of (i) are bounded.
T > tO
Let
PROOF.
Suppose
[t
O
t]
we have
rn_ 1 (t)
1
+
ftt a (x) h (y (g (x)) )dx
1
rn_l(t0)Zn_2Y(t0)
rn_ 1 (t)
0
(x)dx.
ftb
t
(i0)
O
On repeated integration (I0) yields
r l(t0)y’ (t o
r l(t)y’ (t)
+
ftl/r2(x)dx
t
r2(t0)ZlY(t 0)
o
+
+--- +
r3(t0)Z2(Y(t0))
rn_l(t0)Zn_2Y(t 0)
;tl/r2(x2) tf
to
;tl/r2 (x2)
to
I/r3 (x3)
o
fx3___
;
tO
tO
xn-2
)axdx 2 +---
I/r3(x
O
ftl/r2(x2 tfx21/r3
t
o
to
x2
1
ftxn-2 i/rn_ 1 (x)dx
dx 2
O
fXn-la (x) h (y (g (x )))dx
rn-i (Xn-l) t O
dx 2
273
OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
+
;tl/r2 (x 2)
t
O
i/r 3 (x 3)
to
tO
tO
rn-i (Xn- 1
]Xn-lb (x)dx
dx 2
(Ii)
to
to
Dividing (Ii) by
rl(t) and integrating between
y(g(t0) + rl(t0)Y’ (to) fg(t)i/rl(X)dx
y(g(t))
t
(t)
r3(t0) Z2Y(t0)
+
+
ll/r2
f(t) I/r I (x I)
t
to
0
rn-l(t0)Zn-2Y(t0)
g(t)
we have
r2(t0)ZlY(t0)
o
tO
+
+
and
i/rl (Xl)
ll/r2 (x) dxdxl
/x21/r3 (x)dxdx2dx 1
(x 2)
to
Xl
(t) i/r l(x I)
t0
/
i/r 2(x 2)
to
tO
Xn-2
t
xI
fg(t)i/l(Xl
t
/ i/r2(x 2) f
f
t
+
fg (t)i/rl (Xl)
to
/Xl i/r 2(x 2)
t
o
ft
x2
1
I/r(t) <
i
xo
Xn-2
<
+
ly(g(t0))I
+
f
t
o
Xn-I b
1
Xn- 2
I/rn_ l(xn- I)
O
(x)dxdxn_IdXn_2dXn_ 3
dx
1
(12)
o
2, 3,
to
dx
O
Irl(t0)Y’ (t0)
Ir2 (to)ZlY(t O)
i/rn_l(X 0)
o
n-i
and
g(t) < t
above
ly(g(t))l
dXl
a(x)h(y(g(x)))dxdx 0
i/r 3(x 3)
f
t
Since each
f
t
+
i/r3(x 3)
to
tO
o
x2
dXdXn-2
rn_ 1 (x)
o
ftl/rl(X)dx
to
(X-to)/r I (x)dx
we have from
274
B. SINGH
1
irn_ 1 (t0)Zn-2Y(t0)
(n-2) !n-2
(n-2) !n-2
+
0rl
n-2 tf
Due to conditions on
n-2/r I (x)dx
0
(y (g (s)))Idsdx
to
1
r 1 (x’
0
r (t)
(x-t 0)
;x (x_s)n-2 la (s)I lh
(x)
t
I
1
(n-2)!
ft
t
IX (x-s) n-2 Ib(s) Idsdx.
j
to
we find that each term on the right hand side of
I
(13) except possibly last two are bounded, and since
constants
KI, K and K
2
3
h(t)/t < m, there exist
such that
(x-s)n-2
ft fx
x n-8
to t O
+ K3
(13)
ft sX
t
O
t
(X-s)n-2 Ib(s)
xn_ 8’
dsdx.
o
Rearranging constants still further we get
IY(g(t))
t
S Sx
< K + K
1
4
t
+
+ Ks
:4
x2-8
t0 t0
t
0
la(s) llY(g(s))l dsdx
x2-’
0
t
o
xh_ x
ax
lll,gs))lds
Ib(s
by change of order of integration.
Now
(14)
OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
itx 21
275
dx < C
(15)
s
C > o
since
Ib(s) Ids
<
for some constant
bounded, since
o <
8 <
Thus the last term in (14) is
i.
From (14) and (15), there exists a positive constant
By Gronwall’s inequality
THEOREM (3.1).
y(g (t))
K6
such that
is bounded and the proof is complete.
Subject to the conditions of Lemma 3.2 all oscillatory
solutions of equation (I) approach zero.
PROOF.
Suppose to the contrary that some oscillatory solution
y(t)
of (i) is such that
im sup
t /
for some number
d.
Let
T > t
be large enough so that for
o
t > T
we
have (from lemma 3.1)
ri+ 1 (t) (Ziy (t)
d
<
ileitn-i-2
i
n
I, 2, ---, n-2
(17)
It follows from (4) that
r I (t) y’ (t)
t
as
/o
(18)
/.
tn-2
Let now
TO > T
be a zero f
y(t)
so that for
r i+l (t) Z (y (t)
i
I n-i-2
ile t
t >
TO,
(17) and (18) imply
d
<
i
I, 2, ---, n-2,
n
and
r (t)y’ (t)
I
t n-2
Integrating (I) between
d
<
[T 0, t]
(19)
n
we have
276
B. SlNGH
y(t)
r l(T0)y’
+
(To)
T
r3 (T0)Z2Y(T0)
+
I/r l(x)dx +
TO
;
I/rl (Xl)
TO
i/r2 (x2)
rn_l(T0)Zn_2Y(T 0) "jtl/r 1 (x I)
+
;Xll/r2(xl) TfXll/r2(x)dXdXl
r2(T0)Z I(T0)
T
O
TO
O
O
3
TO
")Xll/r 2 (x2)---J,Xn-2 I/rn_ 1
TO
TO
(x)dxdxn_ 2
ftl/rl
T
(x
I
Xn- 2
fTXl i/r 2 (x 2)
f
TO
O
0
TO
T
Since
C
1.
ly(t)
x2
tl/rl(xI) Ix I I/r 2(x 2)
TO
O
y(t)
T
xn
f
T
i/r 3(x 3)
O
ly(g(t))l
is bounded, let
I
fXn-la (x) h (y (g (x))
I/rn-i -i
dx
+
dx
< CI
dXdXn_ 1
I
ib(x)dxdXn_l
dx I.
(20)
O
for some positive constant
From (20) we have
<
[rl(T0)Y,(T0)l Tftl/rl(X)dx + 1 Ir2(T0)ZIY(T0) Tft (x_T0)dx
O
+
ft
(T
21u2 r3(T0 Z2Y O
.i
(n-2) len-2
+
0
1
elm
(n-2) lun-2
(n-2)
!en-2
TO
(x-T 0)
2/r I (x)dx
;t (X-To) n-2/rl (x)dx
T
O
;tl/rl fx
(x)
T
T
O
(x-s)
n-2
la (s)Idsdx
O
(21)
TO
TO
Now there exists a constant
D
i
> o
such that for each i
i-2
ui
rlx
T
O
r (x)
I
dx
OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
<I
r
Di ; x-0
xn-8
i+l (To) ZiY (To)
(:l
ri+l (T0)ZiY(T0)
ei!
<
Di
i-2
I,
STt xn-i+2-8
dx
O
Di..
Iri+I(T0)ZiY(T0)I
.11
.i
1
n-i+l-8
1
+
T-i+l-
tn-i+l-8
81
d
n
<
277
(22)
in view of (17), conveniently large enough choice of
TO
and the fact that
8<1.
Similarly, as it was shown in the later part of inequality (14)
(by
changing the order of integration) it is easily shown that a large choice of
results in
T0
Clm
S
T
n’2
n-2) lC
t
l/r l(x)
O
sX(x-s)n-21a(s) Idsdx
T
< d_
(23)
O
and
i/r (x)
I
(n-2) len-2
0
(x-s)
TO
n-21h(s) Idsdx
<
d.
n
(24)
From (21), (22), (23) and (24) we get
<_
+
g+
(25)
d
From (25) we see that if we choose a large enough
ly(t)
< d.
T
then for all
O
But this contradicts (16) for any positive
d.
t >
TO,
The proof is now
complete.
EXAMPLE (3. i).
(t(ety
(t))’)’ +
The equation
e-t-2y(t-)
+ 4te -t cos t
2e-tt
sin t +
3e -t cos t
e-tsin
t
e -3t sin t,
(26)
278
t >
B. SINGH
e-2t sin t
y
has
as an oscillatory solution approaching zero.
conditions of Theorem 3.1 are satisfied.
Hence all oscillatory solutions of
.
(26) vanish at
All
Our next theorem leads to a necessary and sufficient criteria for all
oscillatory solutions of equation (1) to vanish at
THEOREM (3.2).
o <
< 1
f a(t)dt
and
finite limit as
t
a(t) > o,
Suppose
/
<
.
1
for
rlt
b(t)/a(t) approaches a
Further suppose that
Then a necessary and sufficient condition for all
oscillatory solution of (I) to approach zero is
lim
t
PROOF.
/a(t)dt
<
a (t)
,
Ib(t) Idt
we have
b(t___)
Suppose that
(SUFFICIENCY).
<
.
as
o
/
a(t)
t
/
.
Since
By Theorem 3.1 all oscillatory
solutions approach zero.
(NECESSITY).
a (t)
Dividing (i) by
be an oscillatory solution of (1).
y(t)
Let
we have
(rn_ l(rn_2(___(rly, (t)) ’) ’) ’--)
Now
y(t)
/
o
t
as
/
.
im
t
Since
h(y(g(t)))
such that for
contradiction.
t
/
(t))’ + 1
(28)
Suppose to the contrary that
Ib()l
>
(29)
> o.
a (t)
o, (28) from (29) reveals that there exists a large
_> T, Zn_lY(t)
> o.
But then
T
is nonoscillatory, a
y(t)
The proof is now complete.
EXAMPLE (3.2).
(t2y
b(t)
+ h(y(g(t)))
y(t)
Consider the equation
5
j (sin(Ent)
coS(nt))
+
Here all conditions of Theorem 3.2 are satisfied.
solutions approach zero.
In fact
y(t)
sin
(nt
t5
(30)
Hence all oscillatory
sin(nt)/t
3
is an oscillatory
OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
279
solution of (30).
The necessity part of Theorem 3.2 leads us to the following theorem.
THEOREM (3.3).
o
_< 8
< I.
Suppose
rl(t)
Further suppose that
-0(I/t n-8)
fa(t)dt
,
<
8
for some
a > o
such that
b(t)/a(t)
and
is
Then all oscillatory solution of equation (i) approach zero as
bounded.
PROOF.
Ib(t) Idt
Since
< ".
fa(t)
<
b(t)/a(t)
boundedness of
Conditions of Theorem 3.1 hold.
implies
The proof is complete.
Our next Theorem gives conditions when oscillatory solutions do not
approach limits.
THEOREM (3.4).
a(t) > o
Suppose
lira
and
inflb(t) I/a(t)
> o.
t/
y(t)
Let
be an oscillatory solution of equation (1).
Then
o.
PROOF.
h(y(g(t)))
/
Suppose to the contrary that
o.
/.
o
as
t
/
.
Then
From equation (i)
a(t)l l(rn_l(rn_2 (--(r 2(t) (rl(t)y’(t))’)’--)’
This shows that
y (t)
y(t)
Zn_lY(t
+
lh(y(g(t)))l _> Ib(t) I/a(t).
is eventually positive contradicting the fact that
is oscillatory.
REMARK.
It is to be noted that the conditions
;a(t)dt
tn_
rl(t)
<
and
Ib(t) Idt
<-
are not needed here.
EXAMPLE (3.3).
All oscillatory solutions of the equation
y’’ (t) + y(t-2)
satisfy
lira suply(t)
t +
Theorem 3.4.
y(t)
> o
2
(31)
2
since this equation satisfies all conditions of
+ 2 cos (t)
is one such solution.
B. SlNGH
280
Next theorem gives nonoscillation criterion.
THEOREM (3.5).
8 < I.
o <
r I (t)
b(t)/a(t)
and
fa(t)dt
a(t) > o,
Suppose
<
and
Further suppose that
lira
t
inflb(t)I/a(t)
> o
Then all solutions of equation (1) are nonoscil-
is bounded.
latory.
Suppose to the contrary that
PROOF.
/
.
---!--i
a(t)
is an oscillatory solution
Since all conditions of Theorem 3.3 are satisfied,
of (1).
t
y(t)
h(y(g(t)))
Thus
/
o.
is oscillatory.
> o
Zn_lY(t)
/
as
o
From equation (i)
l(rn_l(rn_2(t) (---(r l(t)y’ (t)) ’) ’) ’---)
(32) suggests that
y(t)
-
eventually,
> b(t)
I/a(t)
lh(y(g(t)))l
contradicting the fact that
(32)
y(t)
The proof is wow complete.
EXAMPLE (3.5).
The equation
1
( t 2y ’(t))
+
l--(t
t2
1
+
i"
(33)
t-
solution of (33).
THEOREM (3 6)
a(t)
Suppose
Id
1/t 2
y(t)
satisfies all conditions of this theorem,
<
and
is a nonoscillatory
fb(t)dt
.
Then all
oscillatory solutions of (1) are unbounded.
PROOF.
satisfies
for
Suppose to the contrary that some oscillatory solution
ly(t) <_
C
O
for some
O
> o.
From equation (i) on integration
t > T.
> Iftb(s)dsl.
Irn_l(T)Zn_2Y(T)l + C0m ftla(s)Ids
T
T
Irn_ l(t)zn_2y(t)
+
(34) yields that
Zn_2Y(t)
that
C
y(t)
y(t)
is oscillatory.
(34)
assumes a constant sign eventually, contradicting
The proof is now complete.
The following example
shows that under the conditions of Theorem 3.6, it is possible to have
bounded nonoscillatory solutions.
OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
EXAMPLE (3.6).
281
The equation
1
1
1
y(t)
It has
satisfies all conditions of Theorem 3.6.
i/t
as a bounded
nonosc illatory solution.
THEOREM (3.7).
suppose that
Sla(t) Idt
Suppose
Sb(t)dt
and
1, 2, ---, n-1.
i
is bounded
ri(t)
<
.
+_
Further
Then all solutions of
equation (i) are unbounded.
Due to Theorem 3.6 we only need to prove it for a nonoscillatory
PROOF.
solution.
Let
the proof of Theorem 3.6 it follows that
Zn_2Y
From inequality (34) in
be nonoscillatory and bounded.
y(t)
(rn_2Zn_3Y(t))’
and
rn_ 2
as
/
/
+
y(t)
forcing
t
/
Zn_3Y(t)
is bounded, we have
y’ (t)
Proceeding this way we find that
IZn_2Y(t)
/
-+
.
.
.
Since
/
-+
The
proof is now complete by contradiction.
Theorem 3.1 improves our main result in [9]
FINAL REMARK.
(c.f [11])
here it was shown that oscillatory solutions of
(r(t)y’(t)) (n-l) + a(t)h(y(g(t)))
(35)
f(t)
approach zero subject to:
la(t) Itn-2at
tn-2at
<
<
and
o<8<1.
r(t)
r(t)
The restriction on
or equal to
1
(t))"
8 cannot
be greater than
as the following example shows.
EXAMPLE (3.7).
(t2y
cannot be weakened i.e.
The equation
1
+
t
(nt)
2
y(t)
Cos(n(nt)
t
t) 3
(n
+
3 s in (n
t
nt
(nt)
3
t > o
Co s
n nt
t (nt) 2
(36)
282
has
at
B. SlNGH
y
sin(n(nt))
.
as an oscillatory solution which does not have a limit
Only the condition on
even though
1
1
r(t)
1
dt <
r(t)
.
r (t)
is violated.
We see that for
n
3
so that
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2.
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3.
Hammett, M. E. Nonoscillation Properties of a Nonlinear Differential
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4.
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6.
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7.
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8.
Singh, B. Asymptotically Vanishing Oscillatory Trajectories in Second
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9.
Singh, B. A Correction to "Forced Oscillations in General Ordinary
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i0.
Singh, B. Nonoscillation of Forced Fourth Order Retarded Equations, SIAM
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OSCILLATORY SOLUTIONS OF FUNCTIONAL EQUATIONS
283
12.
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Forced Oscillations in General Ordinary Differential Equations
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