I nte. J. Ma.
{1978)1-12
Vo1.
Math.
BEHAVIOR OF SECOND ORDER NONLINEAR
DIFFERENTIAL EQUATIONS
RINA LING
Department of Mathematics
California State University
Los Angeles, California 90032
(Received November 29, 1977 and in revised form March 31, 1978)
ABSTRACT.
Qualitative behavior of second order nonlinear differential equations
It includes properties such as posltlvlty,
with variable coefficients is studied.
number of zeroes, oscillatory behavior, boundedness and monotoniclty of the
solutions.
i.
INTRODUCTION.
Second order nonlinear differential equations of the form
y(t) + p(t)y(t) +
where n is an integer
>_ 2,
q(t)yn(t)
0
occur in many physical problems, such as the mass-
spring systems and satellite (see Ames
[i], Mclachlan [2] and Struble [3]) and
nuclear energy distribution (see Canosa and Cole
[4, 5]).
In this work, qualitative behavior of real-valued solutions of (i.i) is
studied.
With certain conditions on the coefficients p(t) and q(t), and n,
properties such as posltivlty, number of zeroes, boundedness and monotonlclty
RINA LING
2
are obtained.
It would be assumed that the coefficients and their derivatives
The work is
are continuous real-valued functions on the interval of interest.
divided into four parts; the first part deals with the case of p(t) < 0 and
q(t) < 0, the second part with the case of p(t) < 0 and q(t) > O, the third
part with p(t) > 0 and q(t) < 0 and the fourth part with p(t) > 0 and q(t) > O.
Papers in the past, Skldmore [6], Abramovlch [7], Rankln [8], and Grimmer and
Patula [9] have studied behavior of second order linear differential equations.
Nonllnear differential equations have been investigated in Chen [i0] and Chen,
Yeh and Yu
[11],
the former is on oscillatory behavior of bounded solutlons and
the latter on asymptotic behavior of solutions.
The results here are of a
different nature and are independent of theirs.
2.
CASE OF p(t) < 0 AND q(t) < 0.
THEOREM 2.1.
If (i)
is odd, then either y(t) >
p(t) < 0, t > 0,
O, t
> 0 or
(2)
q(t) < O, t > 0
y(t) < 0, t
> 0.
and
(3)
n
The graph is concave
upward for y > 0, and concave downward for y < 0.
PROOF.
Equation (i.i) can be written in the form
+ (p(t) +
q(t)yn-l)y
(2.1)
O.
Let z(t) be real-valued and satisfy the linear equation
+ p(t)z
(2.2)
0.
By a theorem in Hartman [12, p. 346-347], z(t) has no zero.
Since n
i is
even and q(t) < 0, we have
n-1
< p(t)
p(t) + q(t)y
and by Sturm First Comparison Theorem, z(t) has at least a zero on
(0,=), if y
has a zero on (0,), a contradiction.
The concavity of the graph follows from (I.i) written in the form
NONLINEAR DIFFERENTIAL EQUATIONS
3
-p(t)y- q(t)y n
y
The case of n being even is considered in the following theorem.
THEOREM 2.2.
is even, then
PROOF.
If (I)
q(t) < 0, t > 0
(2)
p(t) < 0, t >0,
and
(3)
n
and
(3)
n
y(t) < O, for all t > 0.
Since q(t) < 0 and n is even, we have
+ p(t)y + q(t)yn
+ p(t)y,
<
therefore
0 <
and by Bellman and Kalaba
+ p(t)y
[13, p. 67], y(t)
< 0 for all t > 0.
CASE O p(C) < 0 AND q(=) > O.
THEOREM 3.1.
If (I)
is even, then y(t) >
PROOF.
METHOD i.
(2)
p(t) < 0, t > 0,
q(t) > 0, t > 0
0, for all t > 0.
If in (I.i), we let y(t)
-z(t), then the equation be-
comes
-E- p(t)z + q(t)(-l)nz n-- 0,
.
since n is even,
+ p(t)z
and by Theorem 2.2, z(t) < 0 for all t > 0.
METHOD 2.
q(t)z n
0
Therefore y(t) > 0 for all t > 0.
Since q(t) > 0 and n is even, we have
y
+ p(t)y
< y
+ p(t)y + q(t)y
therefore
+ p(t)y
and by Bellman and Kalaba
[13, p. 67], y(t)
< 0
> 0 for all t > 0.
RINA LING
4
4.
CASE OF p(t) > 0 AND q(t) < 0.
If n is odd, the following two theorems on the number of zeroes of the
solution can be obtained.
THEOREM 4.1.
(3)
(2)
p(t) > 0, a < t < b,
If (i)
q(t) < 0, a < t < b and
n is odd, then a necessary condition for y to have two zeroes on (a, b] is
that
b
4
p(t)dt >
a
PROOF.
Equation (i.i) can be written in the form
+ (p(t) +
q(t)yn-l)y
0.
Let z(t) be real-valued and satisfy the linear equation
+ p(t)z
0.
Since q(t) < 0 and (n- i) is even,
p(t) + q(t)y n-I < p(t)
and by Hrtman
[12], z(t)
has at least two zeroes on (a, b) if y(t) has two
By Lyapunov Theorem in Hartman [12, p. 346], a necessary
zeroes on (a, b].
condition for z(t) to have two zeroes on
[a, b]
is that
b
p(t)dt >
THEOREM 4.2.
(3)
n is odd,
If (i)
(4)
4
(2)
p(t) > 0, 0 < t < T,
y(t) has N zeroes on (0,T], then
T
N <
p(t)dt)
(T
0
+
i.
q(t) < 0, 0 < t
<
T,
NONLINEAR DIFFTIAL EQUATIONS
PROOF.
5
As in the proof of Theorem 4.1, it can be shown that if z(t) has
M zeroes on (0,T), then N
<_ M. But
by Hartman [12, p. 346-347 ],
T
M <
(T
p(t)dt
+
i
0
and the conclusion follows.
In the next two theorems, n is assumed to be even.
THEOREM 4.3.
is
p(t) > 0, t >_ 0,
If (I)
(2)
q(t) < 0, t
>_
0
and
(3)
n
even, then y(t) < 0, for t > 0.
PROOF.
Since q(t) < 0 and n is even,
+ p(t)y + q(t)yn
+ p(t)y,
<
therefore,
0 <
and by Bellman and Kalaba
;+
p(t)y
[13, p. 67], y(t) <_ 0, for all
t > 0.
If in addition to the hypotheses of Theorem 4.3, we assume that
(0)
then y is negative and monotonic increasing.
THEOREM 4.4.
t > 0 and
PROOF.
(4)
If (i)
(0)
> 0, (2)
p(t) > 0, t > 0,
(3)
q(t) < 0,
n is even, then y is montonic increasing.
Integration of (i.i) from 0 to t leads to
(t)
(0)+
I
p(s)yds +
0
I
q(s)yn
ds
0.
0
Since p > 0 and y < 0 by Theorem 4.3, q < 0 and n is even,
(t)
(0) " r(t)
(0) +
I
0
p(s)yds +
I
0
q(s)yn
ds,
> 0,
RINA LING
6
therefore,
(t)
(t)
>
(0)
>
O,
(o)
>
o
and so y is monotonic increasing.
5.
CASZ 0 p(=) > 0 AD q(t) > 0.
For p(t) > 0, q(t) > 0 and n odd, the following theorems on the oscillatory
behavior and boundedness of the solutions can be obtained.
THEOREM 5.1.
odd and
(4)
(i)
p(t) > 0, t > 0,
If
z(t), for
(3)
q(t) > 0, t > 0,
+ p(t)
z(t) is a real-valued solution to
oscillates more rapidly than
PROOF.
(2)
z
n is
0, then y(t)
t > 0.
Equatlon (I.i) can be written in the form
+ (p(t) +
q(t)yn-1)y
O.
Let z(t) be a real-valued solution to
E + p(t)z
which has been widely discussed.
0
Since q(t) > 0
and
(n
i) is even,
p(t) < p(t) + q(t)y n-1
and the conclusion follows from comparison theorems in Hartman and Sanchez
[12,
14].
THEOREM 5.2.
t > 0,
then
(3)
lY(t2)
PROOF.
If
n is odd and
<
p(t) > 0,
(i)
lY(tl)I.
(4)
(t)
> 0, t
>_ 0,
(2)
q(t) > 0,
y has successive extrema at
tl, t2,
t
I
(The amplitudes of oscillations do not grow.)
The proof is by contradiction, so assume that
Multiplication of (i.i) by
leads to
+ p(t) + q(t)yn
0.
lY(tl)
<
(t)
<
> 0,
t2,
[Y(t2)
NONLINEAR DIFFERENTIAL EQUATIONS
Integrating (5.1) from t
t
t
I to t
t
2
I P(t) I
dt
t
I
we get
t2,
2
q(t)Yn
+
t
7
dt
0.
I
Since
t
t
2
p(t)
t
at
;g (p(t2)y2(t2)
2
(t)y 2
P(ti)y2(ti)
t
I
dt)
1
and
t
t
2
q(t)yn
t
(q(t2) yn+ 1 (t 2)
i
dt
n
+
i
q(t I)
yn+
2
[ (t) yn+
(t I)
t
I
dr),
I
(5.2) becomes
P(t2)Y2(t2) P(tl)Y2(tl)
t
I
t
<
t
2
(t)y2
dt
+
n
2
+
n
+
1
q(t2) yn+ (t2)
12 I (t)yn+1
t
+ I
q(tl )yn+l (tl)
2
dt
+
I
2
n
I
y2(t2)(P(t2 ) P(tl))
+
2
+
n
i
y
n+l
(t2)(q(t2)
q(tl))’
therefore
P(tl)(y2(t2)
y2(tl))
<
2
n
+
i
q(tl) (yn+ (tl)
< 0, since
q(tl)
yn+l (t)
2
> 0 and (n
+ i)
is even,
which is a contradiction since the left hand side is positive.
THEOREM 5.3.
and
(3)
If (i)
p(t) > i,
(t)
<
0, t > 0,
n is odd, then y(t) is bounded for all t.
(2)
q(t) > 0,
(t)
< 0
RINA LING
8
PROOF.
Multiplication of (1.1) by
and integration of the resulting
equation from 0 to t lead to
I
#2(t) + p(t) y2(t)
t
(s)y2
2
2
n+l
+ n+ q() yn+l ()
ds
0
I
t
(s)yn+lds
C,
0
where C is a constant,
therefore
y2(t)(p(t) +
2
q(t)
yn-
I
(t))
t
t
(s)y 2
ds
0
0
Since p > i, q > 0, n is odd,
< 0 and
y2(t)
<
<
0, it follows that
C, for all t
and so y is bounded.
THEOREM 5.4.
(t)
<
0, t > 0,
PROOF.
If
(I)
(3)
and
p(t) > 0,
(t)
(2)
> 0, t > 0,
q(t) > 0,
n is odd, then y(t) is bounded for all t.
As in Theorem 5.3, equation (i.I) is multiplied by
and the
resulting equation integrated from 0 to t, giving,
2
2(t) + P(t)y2(t) + n---
q(t )
yn+
(t)
I
2
n+l
t
0
0
Since q > 0, (n
+ i)
is even and
<_ 0,
p(t)y2(t)
< C
+
I
t
(s)y2
ds
0
so
p(t)y2(t) _< [C[ +
I
0
(s)y 2
ds =C+
t
P(s)Y2
(S)p(s)
ds
ds.
NONLINEAR DIFFERENTIAL EQUATIONS
9
[12, p. 24
and by Gronwall’ s inequality, Hartman
t
p(t)y2(t) _<
ICI
exp
I p(s)(s)
ds
0
p(t)
p(0)
IC
therefore
y2(t)
<
p
for all
t.
In the next theorem, (i.i) is assumed to have constant coefficients
and
P0
q0"
THEOREM 5.5.
If
(i)
P0
> 0,
(2)
q0
> 0 and
(3)
n is odd, then y is
oscillatory.
PROOF.
The equation is
9 + pOy +
q0 yn
0.
By linearization in McLachlan [12, p. 106],
9+ay=0,
where
2
a
I (P0
i
A sin
e + q0
An
n
sin e) sine de,
0
where A is a constant.
Since
P0
> 0,
q0
> 0 and n is odd, the integrand in
2
a
i
I
--w (P0
0
is positive, so a > 0.
sin2 e + q0 An-
s
inn+!
e)de
i0
RINA LING
By Theorem 5.4, y is bounded.
The fact that
ldt=m
0
and y is bounded imply that y is oscillatory, see Hartman
[12, p. 354].
In the next two theorems, n is assumed to be even.
THEOREM 5.6.
is even, then y(t) >
PROOF.
and
(3)
n
0, for t > 0.
The equation is
+ p(t)y + q(t)yn
Let y(t)
q(t) > 0, t > 0
(2)
p(t) > 0, t > 0,
If (i)
0.
-z(t), then
-.
(-l)nq(t)
p(t)z +
z
n
0.
Since n is even,
q(t)z n
+ p(t)z
and by Theorem 4.3, z < 0.
0
Therefore y(t) > 0
for
t > 0.
If in addition to the hypotheses of Theorem 5.6, we assume that
(0)
< 0,
then y is positive and monotonic decreasing.
THEORE .7.
and
PROOF.
(4)
Tf
()
(0) ! 0,
(2)
p() > 0,
(3)
n is even, then y is monotonic decreasing.
Integration of (i.I) from 0 to t leads to
t
(t)- (0)+
f
0
Since p > 0
>_ 0,
and
p(s)y ds +
I
t
q(s)tn
ds
0"
0
y > 0 by Theorem 5.6, q > 0 and n is even,
q() > 0,
NONLINEAR DIFFERENTIAL EQUATIONS
(t)
(t)
so
<
(0)
<
O,
;(o)
<
o
11
and y is monotonic decreasing.
REFERENCES
Nonlinear. Ordln.ary Differential Equatl.o.ns in Tra.nsp.ort
Processes., Academic Press, New York and London, 196’8.
i.
Ames, W. F.
2.
McLachlan, N. W. Ordinary Non-Linear Differential Equation.s. in Engin.ee.ring
and Physlcal Sciences, 2nd ed., Oxford University Press, London and New
York, 1958.
3.
Struble, R. A. Nonlinear Differential Equations, McGraw-Hill Company, New
York and London, 1962.
4.
Canosa, J. and J. Cole. Asymptotic Behavior of Certain Nonlinear BoundaryValue Problems, J. Math. Phys. 9 (1968) 1915-1921.
5.
Canosa, J.
6.
Skidmore, A. On the Differentlal Equation y"
App1. 43 (1973) 46-55.
7.
Abramovich, S. On the Behavior of the Solutions of y"
Math. Anal. Appl. 52 (1975) 465-470.
8.
Rankin, S. M.
9.
Grimmer, R. C. and W. T. Patula. Nonoscillatory Solutions of Forced SecondOrder Linear Equations, J. Math. Anal. Appl. 56 (1976) 452-459.
Expansion Method for Nonlinear Boundary-Value Problems, J. Math.
Phys. 8 (1967) 2180-2183.
+ p(x)y
f(x), J. Math. Anal.
+ p(x)y
f(xi,
J.
Oscillation Theorems for Second-Order Nonhomogeneous Linear
Differential Equations, J. Math. Anal. Appl. 53 (1976) 550-553.
i0.
Chen, L. S.
ii.
Chen, M. P., C. C. Yeh and C. S. Yu.
12.
Hartman, P.
Some Oscillation Theorem for Differential Equations with
Functional Arguments, J. Math. Anal. Appl. 58 (1977) 83-87.
Asymptotic Behavior of Nonoscillatory
Solutions of Nonlinear Differential Equations with Retarded Arguments,
J. Math. Anal. Appl. 59 (1977) 211-215.
Ordinary Differential Equations, John Wiley and Sons, New York
and London, 1964.
RINA LING
12
13.
Bellman, R. E. and R. E. Kalaba. uasilinearizatlon and Nonlinear BoundaryValue Problems, American Elsevler Publishing Company, New York, 1965.
14.
Sanchez, D. A. Ordinary Differential Equations and Stability Theory,
W. H. Freeman and Company, San Francisco and London, 1968.
Nonlin differential equation, osry and
of solons, boundedn and onotonity of olons.
KEF WORDS AND PHRASES.
ymptotie behavior
AMS(MOS) SUBJECT CLASSIFICATIONS (1970).
34C10, 34C15, 34D05.