Internat. J. Math. & Math. Sci.
35
(1987) 35-45
Vol. i0 No.
OSCILLATION THEOREMS FOR SECOND ORDER NONLINEAR
DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS
S.R. GRACE
Department of Mathematics
P. O. Box 1682
University of Petroleum & Minerals
Dhahran, Saudl Arabia
and
B.S. LALLI
Devartment of Mathematics
University of Saskatchewan
Saskatoon, Saskatchewan
S7N OWO Canada
(Received November 28, 1984 and in revised form March 28, 1986)
ABSTRACT,
New oscillation criteria for the oscillatory behaviour of the differential
(a(t) x(t))
+ p(t)x(t) + q(t) f(x[g(t)])
("
0
d
-t)
and
(a (t) (x
(t)) (t))
+ p(t)x(t) + q(t) f (x [g (t) ])
0
are established
KEY WORDS AND PHRASES. Oscillation theorems, differential equations, non-oscillatlons.
1980 AMS SUBJECT CLASSIFICATION CODE 34CI0.
I.
INTRODUCTION.
This paper is concerned with the oscillatory behavior of solutions of second
order nonlinear damped differential equations with deviating argument of the form
(a(t)x(t))
+ p(t)x(t) + q(t) f(x[q(t)])
0
(I.I)
and
(a (t) ,(x (t) x (t)
where a, g, p, q:
q(t)
not
lim g(t)
t->
on
.
[tO,
identically
-)
+ p(t)x(t) + q(t) f(x[g(t)])
->
zero
,
f: R-> R
(-", ") are continuous, a(t) > 0,
and
any ray of the form It*, ) for some t* _> t
[0, ),
on
(1.2)
0
o
exist
We restrict our attention to those solutions of equations(.T_l)and(l.)which
of
neighborhood
and which are nontrivial in any
[t -), t _> t
some ray
o
S. R. GRACE AND B. S. LALLI
36
Such a solution is called oscillatory if it has arbitrarily large zeros,
infinity.
An equation is called oscillatory if all its
otherwise it is called nonoscillatory.
solutions are oscillatory.
In the study of the second order sublinear differential equation
q(t)Ix(t)lasnx(t)
x (t) +
where
q:
[tO,
->
)
R
0 < ,< i
0,
is continuous,
(1.3)
there are many criteria for oscillation which
involve the behavior of the integral of
In particular, Belohorec [I] has shown
q.
that the condition
/=sSq(s) ds
for some
8
is sufficient for the oscillation of equation
[0, ]
(1.3), and Kamenev [6] has established
that equation (3) is oscillatory if
t
i
lira sup [
s)q(s)ds
(t
t->
to
Recently, Kura [7] has presented a new criterion for the oscillation of equation (1.3)
which improves upon those of Belohorec and Kamenev.
Kura proved that a sufficient
condition for the oscillation of equation (3) is that
lira sup
t->
t
i
s)
(t
sq(s)ds
for some
S
[0,]
to
These results have been further extended by Philos [8] to a more general equation
x (t) + p(t) x(t) + q(t) f(x(t))
p, q: [t
where
and
f
O
(R))
->
R, f: R-> R
is strongly sublinear
i.e.
0
0
are continuous, xf(x) > 0, f’(x) > 0 for x
du
The above results can be applied
f+/-0 f-< -.
only to ordinary differential equations.
The purpose of this paper is to establish some new oscillation criteria for the
differential
function
f
equations
(I.I) and (1.2).
xf(x) > 0
other that
In fact, we impose no conditions on the
for x
Thus our
0 and nondecreasing.
results can be applied to superlinear, linear and sublinear differential equations.
We also like to mention that we do not stipulate that the function
(I.I) and (1.2) is either retarded or advanced.
retarded, advanced and equations of mixed-type.
2.
THE
EUATION
assmae that
g
in equations
Hence our theorems hold for ordinary,
SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS
xf(x) > 0
f’(x) > 0
and
37
x # 0
for
(2.1)
s
a-
T>_ t o
exp(
]a (u) du) ds
T
T
(2.2)
Suppose further that there is a dlfferentiable function
=)-> (0, (R))
[t o
o’:
such that
o(t) > 0
o(t) < g(t)
for
t
Let conditions (2.1)
THEOREM 1.
>__ t O and
(2.3)
limo(t)
(2.3) hold and
asste
that there exists a
twice differentiable function
p:
-> (0, (R))
[t0,)
such that
-
0 (t) < 0
If
lira sup
t->
1
p (t)
and
0(t)
t
f
t
(t-s)
+ a (t)
<
p (t)
h
a--/
n o(s) q(s)ds
< p(t)
a(t)
for
for some
(2.4)
t > t
O
n
>_
(2.5)
i
o
then equation (l. I) is oscillatory.
PROOF.
for
t
_> t I.
that
Let x(t) be a nonoscillatory solution of equation (I.I), say x(t) > 0
such
By a Lemma in [5] and condition (2.2), there exists a t 2 _> t
x(t) > 0
x[g(t)] > 0
and
for all t
(2.6)
>__ t 2
Now, define
t
(t)
0(t)
[
t2
x(s)
ds
f(x [d(s)])
t>_t 2
Then it is easy to verify that
] {(x[)])
t
f(x [(t)])
\a(t)/
f[(t)])
2
+
a
(a(t) x(t)
f"(x[ o(t) ])
(t)o (t)
x (t) X[o (t) f’ (x[o(t)
2(x
o(t)
(2.7)
S.R. GRACE and B.
38
t
$.
LALLI
2
oCt)Ct) x(t) x[o(tl]f’ (x[oCt)])
Using conditions (2.1), (2.3) and (2.6) ue obtain
t>_t 2
for
f(x[ o(t) ]) <_
and by conditions (2.1), (2.3), (2.4) and (2.6) we have
for
= >_ t 2
Thus, for
ever
t >__ t 2
(2.8)
we have
t
(s) ds
2
(t-t2)n= (t2)
t
+ nCt-t 2) n-I
(t-s) n-2 u(a) de
t2
<_
(t-t2)n .(t2) + n(t-t2)n’l(t 2)
eli2.
tn t 2
the other hand, for
0
-
t
> t2
O (-sln
e have
qlslds
2
a(s) q(s)d=
i-
0
SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS
-
and consequently
1
itto
-5
S)
(t
n
p(s)
a- q(s) ds
tit2
<I- )n
/
ds,
a(s) q(s)
0
n’"(t2)
n
<I t2)
<
(t 2)
+
for
t
>__
t
<
39
t2’-I
i
---u/
2
This gives,
t
lira sup 1
t->
s)
(t
to
which contradicts (2.5).
n 0(s)
a(s) q(s)ds
t2
_< (t 2) +
t
0(s) q(s)ds
a
o
This completes the proof of the theorem.
For illustration we consider the following examples.
EXAMPLE I.
x (t) +
The differential equation
(-
l)x(t)
t(in[t+sint])
< 1
0 <
x(t)
has a nonoscillatory solution
negative for
>
t
EXAMPLE
e,
2.
1
([ x(t))
Ix[t+sint]
0
sgnx[t+sint]
Only the damping coefficient
+ 2 x(t)
2
4
I
+----_Ix[t
+ cost]
a
sgnx[t + cost]
>_t0 >_
t
and
0
(2.10)
2.
are satisfied and
and
1
x (t) + x(t)
has
a
t3a -3 Ix[t3]l
solution
nonoscillatory
satisfied
for
p(t)
q(t) is negative for
x (t)
i
+
is
Consider the differential equation
(t)- t- I. The conditions of Theorem
so, all solutions of equation (2.10) are oscillatory.
EXAMPLE 3. The differential equation
p(t)-
P(t)
violating the assumptions of Theorem I.
> 0
We let
(2.9)
t > e
and
In t.
a
x(t) +
t and
t
t
(t)
0
x(t)
All
t
.
3
except
the
and t
conditions
(2.5),
condition
1
>_ t o
of
(2.11)
Theorem
since
are
the function
On the other hand, the differential equation
o
t3a-3 Ix[t3] a sgnx[ t 3
is oscillatory by Theorem
>_i
sgnx[t 3]
for
p(t)
t
I/2
0 ,a
and
i
>_
(t)
and t
t
>__ t o
1
(2.12)
3.
[9] can describe the
One can check that none of the oscillation criteria of If]
character of either equation (2.10) for a > 0 or equation (2,12) for
oscillatory
The following theorem is concerned with the case when condition (2.4) fails.
We assume that
S.R. GRACE and B. S. LALLI
40
It
will
be
du
f(u)
]
and
(2.13)
convenient to make use of the following notation:
for any
t
t
we let
(t) + a(t)
(t, c)
0 c
where
C2[[t o
THEOREM 2.
t
>
t
Let
>_ 0
(t)
0
ca(t)
(0,, )].,
)
(2.1)
conditions
and the function
O
a(t)
O
p
be as in Theorem
>_
]’(t, i)
(2.3) and (2.13) hold and let
0
7
(t, !)
o(t)
>
t
for
such that
<_ 0
t
for
>_ t o
(2.14)
and
t
lira sup
<
If
0(t)
t->
a(u) q(u)duds
to tO
(2.15)
then equation (I. I) is oscillatory.
PROOF.
for
t
_> o
t
fact that
Let x(t) be a nonoscillatory solution of equation (1.1), say x(t) > 0
As in the proof of Theorem
we get (2.7).
Now, using (2.6) and the
O(t)
ct) <
>_
for
t
t
>_
a(t) q(t) +.
t
we obtain
2
t
"it
x(t)
ds + Y(t, i)
t fcx(
2
fcx
By condition (2.13), we have
C"
(t) <- pt) q(t) +
a(t)
where
C
Ct)
du
{(t2)f(U
< (t 2)
-
t
a(s)
2
q(s)ds +
By the Bonnet theorem, for any
t
ft
y(s, I)
2
l)
x(t)
f(x(t))
Integrating the above inequality from
).
t
"
(t) + (t
xCs)
f(x(s)) ds
t
C0(t 2) +
there exists a
{
vCt 2, I)
2
t
C’0(t)
> t2,
t
t
2
f(x(s))
ds
y(s
i)
;
[t2,
to
t
we get
x (s) ds
f(x(s))
t]
such that
SECOND ORDER NONLINEAR
DIFFERENTIAL
EQUATIONS
41
<
Thus, for every
t
>’t
(t 2
i)
Y(t2,
i)
du
f(u)
x(t 2)
du
f(u)
/
x(t
2)
2
t
(t) < K
q(s) ds +
C’(t)
t2
where
(2.16)
(t2) C(t2
K
+
Integrating (2.16) from
t
a
s
ft ft
2
2
t
q(u)duds
y(t2
to
2
I)
x(t
t
f(u"
2
we have
_< C-(t) + Kt -(t) +
((t2)
C0(t2)
,
Dividing by O(t) and taking limit superior of both
side as t ->
we obtain a
contradiction to (2.15). This completes the proof of the
theorem.
It is easy to check that Theorem 2 is not applicable
to equation (2.12)
if
u
I. On the other hand, Theorem 2 can be applied
in some cases in which
Theorem
is not applicable. Such a case is described
in Example 4 below.
EXAMPLE 4. Consider the differential equation
IREMARK.
x (t) +
where
c
>
0, a
x(t) + --ix[g,t)]
t
>
and
o(t)
lasgnx[g(t)]
g(t)
with
t
The conditions of Theorem 2 are satisfied for
0
t
o(t)
p(t)
0
t
L to
for
> 0,
t
(2.17)
t
o
and hence equation (2.17) is
oscillatory.
3.
TRE EQUATION.
In order to obtain results for equation (1.2) similar to those in section 2 we
assume
0 <
C
_< (x) <_ C1
S
T
a exp(
T
THEOREM 3.
in Theorem
-
x
(3.1)
p [u) du) ds
ca(u)
for all T > t o
Let conditions (2.1), (2.3), (3.1) and (3.2) hold and let
such that
(t) < 0
for all
and
(t)
+ a_)/(_h
(t)\a.()/
<
1
c
p__)
a(t)
for all t > t 0
If condition (2.5). holds, then equation (1.2) is oscillatory.
(3.2)
p
be as
(3.3)
42
S.R. GRACE and B. S. LALLI
>
0
x(t)
Let
PROOF
x(t)
for
be a nonoscillatory solution of equation (I2).
>_ t o
t
There exists a
t
_>
t
O
>
x[o(t)]
so that
0
The hypotheses of Lemma in [5] are satisfied and hence there exists a
Assume that
for
t
2
_>
t
t
_> t I.
such
that
>_
for all t
> 0
x[ (t)
and
x (t) > 0
t
2
Now, we define
Then for every
m
t
t
t
(t) + a(t)
@(x (s)) (s) ds
-f(x[o(S)])
(t)
a(t)
+
>_ t 2
we obtain
2
(t) q(t) f(xg(t)]) +
f(x[o(t)])
(t)
t
for
ds
f(x[)-[i
t2
t2
1
a(t)
a(t)
(x (t)) (t)
f(x[o(t)])
(x (t)
o(t)(t) *lx(t))f’(x[o(tI])x(t)x[o(t)]
f2(x[ o(t)
It is easy to verify that
(t)
<-a(t)
q(t) +
<---
q(t)
a(t)
(t) +
t
>__
a(t)a(t
t
cla(t
f(x[o(t)])
2
The rest of the proof is similar to that of Theorem
and hence is omitted.
Next, we present an interesting result, where condition on
we replace condition (3.1) by the following one.
%(x) >
C >
0
for all
is weakened, i.e.,
x
(3.4)
The result is an immediate consequence of Theorem 3, so we omit the proof.
THEOREM 4.
Let conditions (2oi), (2.3), (3.2) and (3.4) hold and assume that
there exists a function 0
[t0, "), (0, ’)] such that
(t)
<_ 0
and
<
(t) + a(t) Pa(t)
)
< 0
for
t > to
(3.5)
If condition (2.5) holds, then equation (1.2) is oscillatory.
The following examples are illustrative.
EXAMPLE 5.
Consider the differential equation
(i
)
x
0
for
t >_ t O
e
(3.6)
SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS
(2.1), (2.3), (2.5) and
The conditions
The upper bound
Equation (3.6) has a nonoscillatory solution
(3.1) and (3.3) fail.
EXAMPLE 6.
>
8
o(t)
8t
x(t)
p(t)
t.
conditions
Int.
Consider the differential equation
+
where
(3.2), of Theorem 3 are satisfied for
elXl is undefined and hence both
of the function
c
43
>
81
-82
0,
0
H2
82 > 0,
and
a
>
--+ 2
sgnx[,t
sint]
and
0
t
> to
0
sint]
I.
We
(3.7)
J
p(t)
take
All the conditions of Theorem 4 are satisfied and so,
and
every
solution of equation (3.7) is oscillatory.
[9]
We note that the results in [I]
cannot be applied to equation (3.7) since,
conditions of the form
some of the
f--) > K
(x)
> 0
o=+ lu<
(u)
x / 0
for
(u)
,--=..u<
or
+0
required in these papers, are not satisfied.
EXAMPLE 7.
sinx))
((2
a
where
(1)
g
(ii)
We
>
0
+
+
g(t)
and
t-7/61x[g(t)]imsgnx[g(t)]
81
8t
o(t)
let
t
+/-
82
g(t)
0
t
>_ t O
(3.8)
1
satisfies either (i) or (ii):
is a nondecreasing continuous functlm for
g(t)
P(t)
Consider the differential equation
cost, 8
in
case
>
0,
81 >
(i)
and
0
and
(t)
t
8281
> to
with
lim g(t)
=-.
t-
_>0.
B2
8t
in
case
(ii)
and take
I/6.
The conditions of Theorem 3 are satisfied and so,
every solution of equation
(3.8) is oscillatory.
It is easy to check that Theorem 4 is not applicable to equation (3.8) because
condition (3.5) is violated.
Next, we consider the differential equation
((I +
where
a
>
x2))
0
Theorem 4 for
and
p(t)
i
i
Ix[g(t)]
+[
x + E
g(t)
i sgnx[g(t)
is as in equation
(3.8).
0
t
>_ t O
1
(3.9)
Equation (3.9) is oscillatory by
1.
It is easy to verify that Theorem 3 fails to apply to equation (3.9), since
condition (3.1) is not satisfied.
REMARK.
The above examples illustrate that our results apply to superlinear,
linear or sublinear damped differential equations.
Moreover, since we impose no
restrictions
applicable
on
the function
g
in equations
(I.I) and (1.2), our results are
to ordinary, retarded, advanced and equations of mixed type.
S.R. GRACE and B. S. LALLI
44
We
that
believe
the
of
behavior
oscillatory
equation
(3.9)
(3.7)
is
not
deducible from any other known oscillation criteria.
Finally, we give the following oscillation criterion which is similar to the one
Here we omit the proof.
in Theorem 2.
Let
THEOREM 5.
t
t
conditions
(2.1), (2.3), (3.1) and (3.2)
o(t)
hold,
>
for
t
and
O
du
f(u)
and
<
f(u)
-((t,
>_ 0
(t)
I) >__
C
C2[[t0,(R)),
p
Assume that there exists a function
cu <
7Ct, c-I) < 0
0
(3. i0)
(0,(R))]
such that
t > t
for
o
(3.11)
and
t
lira sup
t->
<
condition (2.15) holds, than equation (1.2) is oscillatory.
The following example is illustrative.
EXAMPLE 8.
Consider the differential equation
i
+1 x +--IxCgCt)]lsgnx[g(t)]
sinx))
((2
0
t
t
>
t
and
It
O
P(t)
I.
>
a
where
is any nondecreasing continuous function with
check
to
that
g(t)
>
t
for
the conditions of Theorem 5 are satisfied with
and hence equations (3.12) is oscillatory.
t
p(t)
If
g(t)
easy
is
t > tO > 0
0,
(2.2)
then conditions
and (3.2) take the form
fas) ds
and condition (3.2) can be replaced by condition (3.4).
2.
(I.I) and (1.2)
In that case, we can introduce a continuous,
The results of this paper can be applied to equations of the form
when
f
is not a monotonic function.
nondecreasing function
xF(x) > 0
ar.]
F
f()
on
R
such that
for
C
>_
x # 0
(3.13)
For illustration we can consider the following differential equation.
(t.Cx))
where
g
+ t
f(x[gCt)])
is as in Example
7,
0
f
t
>_ t o
> 0
1
0 < e <
(3.14)
is any continuous, nondecreaslng function on
R
SECOND ORDER NONLINEAR
DIFFERENTIAL
II
EQUATIONS
45
=>
xf(x)> 0 for x # 0 e.g. f(x)
sgnx,
0 or f(x)
sinhx
or f is any continuous function on R satisfying condition (3.13) e.g.
sinx
x e
f(x)
sgnx, e
0...etc. and
is any continuous function on
with
>
satisfying condition (3.4) e.g.
If
take
we
/,
p(t)
the
+ x 2 or e x or In(e + x 2)
(x)
or
2
+/-
R
sinx.
of Theorem 4 are satisfied and thus all
conditions
solutions of equation (3.14) are oscillatory.
In
the
case
(x)
I,
oscillatory solution x(t)
3.
e --0, f(x)
x
and
g(t)
t,
equation
(3.14) has the
sinlnt.
The results of this paper are presented in a form which is essentially new.
REFERENCES
I.
BELOHOREC, S., Two
Remarks on the
Equation. Acta. Fac.
Differential
Properties of
Re num. Natur.
Solutions of a
Comenian.
Univ.
Nonlinear
Math.
22
(1969), 19-26.
2.
GRACE, S. R. and LALLI, B. S.,
Functional
Differential
Oscillation Theorems for Nonlinear Second Order
Equations with Damping. Bull. Inst. Math. Acad. Sinica,
13 (1985), 192-193.
3.
GRACE, S. R. and LALLI, B. S., Oscillation Theorems for
Differential Equations. J. Math. Anal. Appl. (to appear).
4.
GRACE, S. R. and LALLI, B. S., Oscillation of Solutions of Damped Nonlinear Second
Order
5.
Functional
Differential
Equations.
Second Order Nonlinear
Bull. Inst. Math. Acad. Sinica,
12
(1984), 5-9.
GRACE, S.R., LALLI, B.S.
and YEH, C.C., Oscillation Theorems for Nonlinear Second
Order Differential Equations With a Nonlinear Damping Term. SlAM J. Math. Anal.,
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