I nternat. J. Math. & Math. Si.
Vol. 6 No. 3 (1983) 559-566
559
ASYMPTOTIC RELATIOHIPS BETWEEN TWO HIGHER
ORDER ORDINARY DIFFERENTIAL EQUATIONS
TAKA KUSANO
laculty of Science
Hrosh tlnlersty
1982)
ABSTRACT.
Some asymptotic relationships betveen the two ordinary differential equations
(n) +
()
x
(2)
y(n)
are studied.
+
Pl (t)x
(n-l) +
Pl(t)y(n-l)
+
+
Pn(t)x
0,
+
Pn(t)Y
f(t,y),
Conditions are given that lead to an asymptotic equivalence between cer-
taln of the solutions of (i) and certain of the solutions of (2).
The case where the
perturbation f(t,y) depends on a functional argument is also discussed.
KEY ORDS AND PHRASES. Ordinary differential equations, aymptotic relations, function differential equations.
980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 34A.
1.
INTRODUCT ION
We are concerned with some asymptotic relationships between the following two
differential equations
x(n)
+
Pl(t)x(n-1) +
(n)
+
Pl (t)y
_<
_< n,
Y
where
Pi: [t0’)
R,
i
i
(n-l)
and
f:
+
+
Pn(t)x
0,
(1.1)
+
Pn(t)Y
f(t,y),
(1.2)
[t0,oo) XR
R are continuous functions.
We obtain conditions that lead to an asymptotic equivalence between certain of the
solutions of the linear equation (i.I) and certain of the solutions of the perturbed
linear equation (1.2).
(i.i)
No restriction is placed on the behavior of solutions of
which may be oscillatory pr onoscillatory.
The example given at the end of
this note deals with the case where (i.I) has both oscillatory and nonoscillatory so-
lutions.
2.
Our results generalize those of Hallam [i] for second order equations.
MAIN RESULTS.
In what follows we denote by
W(l’’’’"m)(t)
the Wronskian of
I (t)
m (t).
T. KUSANO
560
Let a fundamental system of solutions of (I.i)
,xn (t)}, be fixed and sup-
Xl(t)
pose that there exist positive continuous functions
xi(t), i(t),
I
_< i _<
n, which
satisfy the inequalities
Ixi(t)
W(x
<_ xi(t)
n) (t)
x
Xi_l,Xi+I
I
W(x
n)
x
I
<
(t)
I < i < n,
[t0,),
on
*
i(t)
on
[t0,oo),
i
<_ i <_ n.
(2.1)
(2.2)
t
Note that W(x
n) (t)
is a constant multiple of exp(-
x
I
I Pl(S)ds)"
t
0
With regard to (1.2) we assume that f(t,y) satisfies the inequality
If(t,y)
< (t,
[t0,)X R+ R+, R+--
where
IYl)
on
[t0,o)XR,
(2.3)
[0,oo), is continuous and nondecreasing in the second
variable for each fixed t.
THEOREM i.
Suppose conditions (2.1)o (2.2). and (2.3) are satisfied. Also suppose
that, for some k, i < k < n, and some constant c > i,
(s)(s,c_(s))ds <
t
,
i < i < n,
(2.4)
0
and
x (t)
i(s)(s’cxk(S))ds
I
Xk(t)
*
*
i
,
o(I)
as t
/
(2.5)
t
for
I <
i < n
with i
k.
Then there exists a solution y(t) of equation (1.2) such that
y(t)
Xk(t)
+
o(xk(t))
l_n_n addition, for any solution
lYk(t)
<
c(t)
on
x(t)
[t0,)
Yk(t)
Yk(t)
a.s
O(Xk(t))
.
(2.6)
of (1.2) satis.fying the inequality
there exist_s
+
t
a__
a__s
solution x(t) of (I.I) such that
t
.
(2.7)
ASYMPTOTIC RELATIONSHIPS BETWEEN DIFFERENTIAL EQUATIONS
PROOF.
In view of (2.4) and (2.5) we can choose t
*(s)(s
k
t
c(s))ds
561
> t so large that
O
I
< C-1
(2.8)
1
and
xi*(t)
r
J
*
Xk(t)
#
for 1 < i < n with $
l,=o)
C--i
t > t I,
2(n-l)
(2.9)
t
C[tl,OO)
Let
k.
be the locally convex space of continuous func-
defined by
F
{y e
Define the operator
y(t)
<
[t, oo) with the compact open topology and consider the closed convex subset F
tions on
of C[t
*
*
i(s)#(s’cxk(S))ds
Xk(t
+
7.
i=l
:
C[tI,) ly(t) < CXk(t)
F- C[tl, by the formula
(-l)n-i-Ix i (t)
f
(x
t
_>
Xn) (s) f (s,y(s))ds,
Xi_l, xi+ I
I
tl}.
(2 I0)
t
where
(Xl,... ,xi_ l,xi+1
n
I
i--0
(-l)n-ix i %-’J’(t)W
(xI,.
,xi+I
(-l)n-ix.I (n-l) (t)W(xi
maps F into F.
in F.
is
n) (t)
x
(2.11)
n) (t)
x
Using the identities
,Xn) (t)
xi_ l,xi+1
we easily see that a fixed point of
i)
,Xi_l,Xi+1
W(xI
y(t) of
We seek for a fixed point y
n
7.
i--0
W(xI
Xn) (t)
0,
x
n) (t)
0 < j < n- 2
(x I
(2.12)
Xn) (t),
(2.13)
a solution of equation (1.2).
This follows immediately from (2.1), (2.2),
(2.3), (2.8),
(2.9) and (2.10).
ii)
i__s continuous.
Let
{yg}
be a sequence in F converging uniformly to
y e F
562
T. KUSANO
[tl,CO).
on every compact subinterval of
[tl,t2].
form
Let
g
> 0
*
be given and choose t > t
3
2
*
so that
i < i < n,
(2.14)
3
,
where M
max
l<i<n
max
*
9
>_
N;
<
f(s,y(s))
s e
2nld(t3-t I)
[tl,t3]
i < i < n,
this is possible since f(t,y) is continuous and
{yg}
(2.15)
converges uni-
[tl,t3].
formly to y on
Using (2.14) and (2.15)
y(t)
7y(t) _<
Y.
i=l
we have
t
,
n
f 3
2]
and all
t
9
xi(t)
>_
,
i (s) If(s,Y (s))
f(s,y(s))Ids
I
,
Z
2
t=1
for all t g [t l,t
J
xi(t)
n
+
iii)
Take an integer N such that
xi(t).
t[tl,t2]
i(s) If(s,yg(s))
for all
e
4nM
<
i(s)(s,cxk(S))ds
t
Consider any compact subinterval of the
N.
F i__s relatively compact.
f
,
,
i (s)(s’cxk(S))ds
t
<
3
This shows that
]
is continuous on
F.
From (2.10), (2.11) and (2.12) (with
J
0)
we obtain
n
l<y)’(=)l _< li(t)[ + z lxi(t)
i=l
On any compact subinterval of
[t0,=)
[tl,m)
*
*
i(s),(s,cxk(S))ds.
j
t
the right-hand side of the above inequality is
bounded by a constant independent of y e F.
compact subinterval of
r
Therefore
F is equicontinuous on every
and its relative compactness follows from the Ascoli-
Arzela theorem.
Applying now the Schauder-Tychonoff fixed-point theorem, we see that the operator
has a fixed point y
y(t)
in F.
As remarked earlier, y(t) is a solution of (1.2).
A’rtr1c RELATIONSHIPS
BETWEEN DIFFERENTIAL EQUATIONS
563
That y(t) satisfies the order relation (2.6) follows from the inequality
n
,
i-I
,
[
t
with the help of hypotheses (2.4) and (2.5).
To prove the opposite relationship between the solutions of (i.I) and (1.2),
let
be a solution of (2.2) satisfying the inequality
yk(t)
[to,).
<
,
CXk(t)
on
Then it is esy to verify that the function x(t) defined by
+
Yk(t)
x(t)
lYk(t)
?.
i-i
(-l)n-ixi(t)
,Xn)(S)f(s,Yk(S))ds
(Xl,...,Xi_l,Xi+1
(2.16)
t
is a solution of (i.i) which satisfies the order relations (2.7).
Thus the proof of
Theorem 1 is complete.
Let conditions (2.1), (2.2) and (2.3) be satisfied.
THEOREM 2.
Suppose that,
for some integer k, 1 < k < n, and some constant c > i,
k(S)(s,cxk(S))ds
t
(2.17)
<
o
and
,
i(t)
Xk(t)
x
for I < i < n
It gi(s)$(s’cxk(S))ds
Jo
with i
o(1)
(2.18)
as t
t
k.
Then there exists a solution y(t) of (I,2) such that (2.6) is satisfied. In add--
Yk(t)
ition, if
is
an
solution of (1.2) satisfying the inequality
,
[Yk(t)[ _< CXk(t),
then there exists a solution x(t) of (i) such that (2.7) is satisfied.
PROOF.
It suffices to proceed as in the proof of Theorem i by replacing (2,10)
and (2.16), respectively, by
y(t)
Xk(t)
n
+
i=l
iSk
/
(-ll,n-k-1 Xk(t)
f
(-x)n-xx(t)J
t
t
.
1
(Xl,
Xk_l,Xk+1
xn)(s)f(s,y(s))ds
t
(Sl,...,Xi_l,Xil,...,xn)(s)f(s,y(s))ds
(2.19)
564
T. KUSANO
and
Yk(t)
x(t)
(-l)n-kxk(t)
+
(x
Xk_l,Xk+l,...,Xn)(S)f(s,Yk(S))ds
I
t
t
n
+
I
Z (-i
i=l
)n-i-lxi (t)
(xI
Xn)(S)f(s,Yk(S))ds.
Xi_l,Xi+1
(2.20)
tl
i+k
The details will be left to the reader.
It is not hard to see that the above arguments can be applied to establish sireilar asymptotic relationship between (i.i) and the functional differential equation
y
where
(n)
(t) +
Pl(t)y
(n-l)
(t) +
+
[t0,)
and f(t,y) are as before and g
Pi(t)
.
that lira g(t)
t
(2.21)
f(t,y(g(t)),
Pn(t)y(t)
R is a continuous function such
For example, we have the following analogue of Theorem i.
Suppose that,
Let conditions (2.1), (2.2), ad (2.3) be satisfied.
THEOREM 3.
for some k, I < k < n, and some constant c > I,
<
i(s)(s,cxk(g(s)))ds
t
,
1 <
i_< n,
(2.22)
as t
(2.23)
0
,
Xk(t)
J
,
,
x (t) ,f
i
i(s)(s,cxk(g(s)))ds
o(1)
t
for I < i < n with i
k.
Then (2.21) has a solution y(t) which satisfies (2.6).
a solution
o_
(2.21) satisfying
lYk(t)
,
_< CXk(t),
l_n addition, i__f
Yk(t)
i_s
then (1.1) has a solution x(t) such
that (2.7) is satisfied.
3.
EXAMPLE.
Consider the third order differential equations
where
Xl(t)
y > 0
x"’ + x
0,
y’’’ + y
b(t)[y[sgn
is a constant and b
(2/3e-t
x2(t)
e
t/2
(3.1)
[0, =o)
(3.2)
y,
R is a continuous function.
cos2)t,
mental system of solutions of (3.1) such that
x
3(t)
st/2
sinO2)t
W(Xl,X2,X3)(t)
i.
The functions
form a funda-
We can take
ASYMPTOTIC RELATIONSHIPS BETWEEN DIFFERENTIAL EQUATIONS
*
*
x2(t)
(2/)e-t
Xl(t)
(213)e
*
x3(t)
e
t/2
*
$1(t)
(72)e t
*
2(t)
565
*
3(t)
-t/2
Suppose that
e(l-y)tlb(t Idt
=.
<
(3.3)
0
Theorem 1 (with k-- i) then ensures that (3.2) has a solution
Yl(t)
x
+ o(e -t)
l(t)
as t
Yl(t)
such that
(3.4)
0%
Suppose next that
e
(l+(y/2))t
Ib(t) Idt
(3.5)
<
0
Then, applying Theorem i (with k-- 2 and k
Y2(t)
and
Y3(t)
3), we see that (3.2) has solutions
such that
Y2(t)
x
2(t)
+ o(e t/2
as t
oo,
(3.6)
Y3(t)
x3(t)
+ o (e t/2
as t
oo.
(3,7)
and
Obviously,
Since
Yl(t)
is nonoscillatory, whereas
Y2(t)
and
y3(_t)
are oscillatory.
(3.5) is stronger than (3.3), (3.5) guarantees the existence of all the three
solutions
Yl(t), Y2(t)
and
Y3(t)
listed above.
From Theorem 1 it also follows that, in case (3.5) holds, if y(t) is a solution
of (3.2) satisfying
ly(t)
< c e
t/2
(3.8)
for some constant c > i, then there exists a solution x(t) of (3.1) such that x(t)
y(t) + o(e
estimate.
y(t)
e
3t
t/2)
Note that not all solutions of (3.2) are subject to this
6t
28e
3 and b(t)
In fact, equation (3.2) with y
has a solution
as t
o%
even though (3.5) is satisfied.
Finally, consider the functional differential equation
y’’’(t) + y(t)
b(t)ly(t +
sin
t)l
sgn y(t + sin t),
(3.9)
566
T. KUSANO
where
y
and
b(t)
are as above.
Appealing to Theorem i
(3.5) holds, (3.9) has three solutions
(3.6),
Yl(t), Y2(t
and
Y3(t)
we conclude that if
with properties
(3.4),
and (3.7), respectively.
ACKNOWLEDGEMENT.
University.
This work was done while the author was visiting the Iowa State
The author wishes to express his sincere thanks to Professor R. S.
Dahiya for this invitation and hospitality.
I.
HALLAM, T.G.
Asymptotic relationships between the solutions of two second order
differential equations, Ann. Polon. Math. 24 (1971), 295-300.