Internat. J. Math. & Math. Sci.
47
(1987)47-50
Vol. I0 No.
ON THE BOUNDEDNESS AND OSCILLATION OF SOLUTIONS TO
(rn(t)x,), + a(t)b(x) = 0
ALLAN J. KROOPNICK
Division of Retirement and
Survvor
Studies
I-B-IO Annex
Social Security Administration
6401 Security Boulevard
Baltimore, Maryland 21235
(Received December 4, 1985 and revised form March 31, 1986)
In this paper, we discuss under what conditions the solutions to (m(t)x’)’
ABSTRACT.
Furthermore, some applications of
a(t)b(x)=O are bounded, oscillatory, and stable.
these results are given during the discussion and proofs of the theorems.
Bounded, oscillatory, stable, periodic, monotonic.
KEY WORDS AND PHRASES.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
34CII, 34C15.
INTRODUCTION.
theorems concerning the second
In this article, we shall present some general
order nonlinear differential equation,
a(t)b(x)
(m(t)x’)’
(i.l)
0
For some related results concerning (l.i) see Bhatia
[I]
and Kroopnick
[2].
A general
boundedness theorem is now proven.
Suppose a(t) and m(t) are
THEOREM i.
a
> O and m(t)> m
O
Ix’l
x(O)
x
O
R
existence theory, (i.I) has at
O
such solution of (i.I) on
to t
and existing on some interval
[O,T).
m’(s)a(s))ds
a(O)m(O)B(x
all
terms
in
-[tB(x(s))(a’(s)m(s)
O
Also, let m’(t) < O,
=fo
b(u)du
+
then
Ixl
least one solution satisfying
[O,T), T >
O.
Considering any
Multiply (i) by m(t)x’ and integrate by parts from O
< T obtaining,
2
(m(t)x (t))
Now
Ixl/=
t/(R).
By standard
and x’(O)
=). Finally, if lim B(x)
C(-=,
are bounded as
PROOF.
and, furthermore, suppose a(t)
> O for some positive constants a and m
O
O
and let b(x)
a’(t)iO,
and
O
clio,=)
O
(1.2)
a(t)m(t)B(x(t))
_t B(x(s))(a’(s)m(s)
a(O)m(O)B(x(O))
1/2(m(O) O
are
1/2(m(O)x’(O))
2
2
positive
m’(s)a(s))ds
O
(1.2)
for
Ixl
large
=-[toB(X(s))d(a(s)m(s)).
except,
perhaps,
for
A. J. KROOPNICK
48
We shall show that this
term
-fotB(x(s))d(a(s)m(s)) > d2/
2
By hypothesis, B(x) > -d so
bounded from below.
is
d(am) > -a m
O
O
d2
Ixl
Consequently, should
or
Ix’
become
large, the LHS of (1.2) becomes infinite which is impossible since the RHS of (1.2) is
That
finite.
extended on
is,
all
terms
(1.2)
in
[O,=) and, therefore,
EXAMPLE i.
must
Ix
stay bounded.
and
Ix’l
So the
is well known that when k
solution may
be
are bounded.
The above theorem shows that all solutions to x"
Here we have m(t)
bounded when n is an odd integer.
REMARK i.
both
a(t)
x
n
O are
k
and b(x)
It
k.
x
0 all solutions are periodic.
When a forcing term f(t) is introduced in (1.1) all solutions to (i.i)
are still bounded provided
[ob(s)ds > Mlxla(M >0,
a> i)
and
Ixl
sufficiently large.
Specifically, consider
(m(t)x’)’ + a(t)b(x)
f(t)
(1.3)
Multiplying (1.3) by m(t)x’ and integrating from 0
LHS of (2)
to t as before we get,
fotf(s)m(s)x’ (s)ds
RHS of (2)
o
(I 4)
Now by the mean value theorem, the term tf(s)m(s)x’(s)ds
p < t). So long as f(t) is bounded and continuous the
unounded
f(p)m(p)(x(t)
result
x(O)) (O <
follows
since
any
solutions of (1.3) would approach infinity faster on the LHS of (1.4) than
the RHS of (1.4) which is impossible.
[3].
The next theorem utilizes the following lemma of Utz
LEMMA i.
Suppose w(t) is a real function for which w(t) is defined for
constant.
(i) If for all t
T, w’(t) < 0 and w"(t)
O, then lim w(t)
if for all t
T, w’(t) > O and w"(t)
O, then lim w(t)
+=
....
> a, a
(ii)
t
With the help of this lemma the following result may now be proven.
2. MAIN RESULTS
THEOREM 2.
The hypotheses are the same as Theorem I. In addition, suppose xb(x)
O, then all solutions to (I.i) are oscillatory.
PROOF.
Suppose x(t) does not oscillate, then x(t) must be of fixed sign, say
x(t) > 0 for
T >_O (a similar argument works for x(t) < 0). x’(t) must also be of
> O for x
-a(t)b(x) < O, we have x"(t)
fixed sign. Otherwise, since (m(t)x’(t))’
m(t) <0
at points where
x’(t)
-a(t)b(x)
However, this implies x(t) has an infinite number
Also, x’(t)> 0 for if x’(t) were negative,
Thus, by our lemma, we would have lim x(t)
which is impossible
t
O.
of relative maxima which is impossible.
then x"(t) > O.
since
x(t) is bounded.
Upon integrating (I.I) from O to t we obtain,
m(t)x’(t)
Since
x’(t) > O,
ultimately
lx(t)
monotonic.
contradiction.
From
Therefore, x(t)
m(O)
a
O
fta(s)b(x(s))ds
O
>0 and
(2.1),
we
im/ x’(t)
see,
must oscillate.
(2 i)
O
however,
x(t)is bounded and
since
that
lim
t
x’(t)
-=,
a
Furthermore, (1.2) implies that all
solutions are stable since the LHS is positive and its size is determined by the RHS
of (1.2).
BOUNDEDNESS AND OSCILLATION OF SOLUTIONS
all
Actually,
bounded
occurs
[I],
(i.i)
must
be
oscillatory
under
somewhat
in his paper showed that oscillation of solutions
0 whether x(t)
long a b’(x)
as
of
solutions
weaker conditions (cf. Bhatia
49
bounded or not).
is
These conditions are
discussed in the next theorem.
Suppose a(t)@ C[O,=), a(t) > O and
hl, m(t)GC
THEOREM 3.
m(t) > h
o
> 0 for some constants h
o
let h
Furthermore,
), xb(x)
and let b(x>C(-%
O, then any bounded solution to (1.1) is necessarily oscillatory (note:
for x
f=b(u)du
/a(t)du
I[o,=)
<
possible).
is
Suppose x(t) does not oscillate, then using the same reasoning employed
Theorem II, we have 1/2i+mx(t)
O.
k > O and
x’(t)
However, from (2.1) we
PROOF.
in
i+m=
have lim x(t) =-which is impossible.
Thus, x(t) must oscillate.
t/=
EXAMPLE 2.
Consider the differential equation
x"
[4],
(Petrovski
+o
1/2x’2
b(x)x’
+
(2.2)
O
O. Let V(x,x’)
=), xb(x) > O for x
x’(-b(x)) + b(x)x’
O, we must have
where b(x)=C(-=,
x’x"
b(x)
by
b(u)du.
Since
dV
theorem
Liapunov’s
p. 151) that any solution with initial conditions near the origin is
stable and oscillatory by Theorem III.
+
In fact, if
=,
b(u)du
then all solutions
are periodic.
EXAMPLE 3.
The differential equation
xexp(-x
x"
has as its general solution x’
0 and bounded for k < O.
origin of the
2
2
(2.3)
O
exp(-x
2
k.
The solutions are unbounded when k>
The latter case corresponds to initial conditions near the
(x,x’) plane.
So
the
trivial
solution
stable
is
in
the
sense
of
Liapunov.
With the help of Theorem III,
the following general oscillation theorem may now
be proven.
Suppose a(t)-C[O,=),
THEOREM 4.
O,
f,.
andlxl(R)
b(u)du
o
constants
ao, mo’ ml’
PROOF.
for t
m(t)l[o,=),
+=. Furthermore,
if
ao_>O
and
ml>_
m(t) >_
mo
for positive
Suppose x(t) does not oscillate. Then
II, assume x(t) > O
must be of fixed sign. Reasoning as in Theorem
(a similar argument works for x(t) < 0),
then x’(t)
is
of
fixed sign
so
x(t)
Multiply (I.I) by 2x’(t)m(t) obtaining,
2m(t)x’(m(t)x’)’
Integrating from T to
2m(t)a(t)b(x)x’
(2.4)
O
we have,
(m(t)x (t)) 2 +
so
b(x)C(-=, +=), xb(x) > O for x
then all solutions to (I.I) are oscillatory.
Let x(t) be a solution of (I.I).
_>T _>O, x(t)
monotonic.
a(t)_>
f$ 2a(s)m(s)b(x)x
ds
(m(T)x (T))
2
(2.5)
(2.5) implies
x(t)
2a m
o o
fb(x)ds !(m(T)x’(T)) 2
x(t)
That is, x(t) must remain bounded.
Therefore, by Theorem III, x(t) oscillates.
(2.6)
is
50
A.J.
KROOPNI(
REFERENCES
I.
BHATIA, N. P.
Some
Jour. Math. Anal.
Theorems For Second Order Differential Equations,
15 (1966), 442-446.
Oscillation
AI.,
(m(t)x’)’ + a(t)b(x)
2.
KROOPNI(I(, A. J., Oscillation Properties of
Ana.!. Appl., 63 (1978), 141-144.
3.
UTZ, W. R., Properties of Solutions of u" + g(t)u
0, Monatsh. Math., 66
(1962), 55-60.
PETROVSKI, I. G., Ordinar Differential Equations, Dover, New York, 1973.
4.
2n-I
0, Jour. Math.