Internat. J. Math. & Math. Sci.
(1987)89-92
89
Vol. I0 No.
ASYMPTOTIC BEHAVIOR OF RETARDED DIFFERENTIAL EQUATIONS
CHEH-CHIH YEH
Department of
Mathematics
Central University
Chung-Li, Taiwan
Republic of China
(Received November 15, 1985)
Some integral criteria for the asymptotic behdvior of oscillatory
ABSTRACT.
solutions of higher order retarded differential equations are given.
KEY WORDS AND PHRASES.
1980
1.
Retarded differential equations, oscillation.
AMS SUBJECT CLASSIFICATION CODE.
5K15.
INTRODUCTION.
Recently, Tong
Theorem.
Let
[I proved
f(t,u) be
negative continuous functions
g(u)
function
for
/.v(t)p(t)dt
(a)
(b) g(u)
R x R.
v(t), p(t)
+
If there are two non-
0,
t
for
and a continuous
such that
<
u >
is positive and nondecreasing for
If(t,u)
(c)
0
u
the following interesting result.
continuous on
v(t)p(t)g(t-’ lul)
1,
u e
R,
where
a,
b
t
for
0,
then the equation
u"+f(t,u)
has solutions which are asymptotic to
b
0
a+bt,
are constant and
0.
In this note we generalize Tong’s result to
improves also the results of Chert and Yeh
[2]
a
more general case which
Kusano and Singh
and
[5].
Using this result, we establish an asymptotic behavior of oscillatory solutions
of retarded differential equations.
2.
MAIN RESULTS.
Consider the following retarded differential equations
(2.1)
where
LnY(t)+f(t,y(g(t))
L
n
is an operator defined by
h(t),
t
0,
n
2
C.C. YEH
90
Loy(t)
(t)
n
ri(t
Here
"=
::
ro
.
LiY ()’=
cn-i [R+ ,R]
e
"%-
dt i-’
ri(t
with
> 0
for
’"’"
()’
0,1,’’-,n-1.
i
Sufficient smoothness to guarantee the existence of solutions of
R
an infinite subinterval of
will be assumed without mention.
+
conditions are assumed to hold in
(i)
C[R+ xR,R]
f
(2.1)
on
The following
thSs note.
p(t),
and there exist two positive functions
/
that
Iz(t,u)
(it)
g, h
(iil)
CIR.,R], g(t) < t,
> 0,
lim inf
w(,u)
whvre
s
..(t,u)
lim sup
T
>
1,2,"
,n-2,
Let
<
<
y(g(t))
is a solution of
(2.1),
y(t)
(il)
that there exist a
then
0(Wn_,(t,T))
for
0.
Let
Proof.
It follows from
constant
i
t(s)as...asa,.
r(,)’’"
Ju
Ih(t) ldt
some
oo,
s,
[ ,(,
:=
(2.5)
y(t)
<
(t,L
Wn_, (t)p(t)dt
If
,
g(t)=
lira
(2.2)
hold.
lu I),
defined by
t(t,u)
Theorem 1.
-< ,(t)(
m
(2.1)
be a solution of
(ill)
and
on an interval
T >
To
[To,oO),
T
such that
g(t) > To
for
t
> T
and
inf
tT
By
(tlt),
=
there is a positive constant
m
c
< c.n-1 (t,T),
wi(t,T
such that
i
1,2,’’" n-2
Now a simple argument shows that
lY(g(t))l
m
n-1
.<
ILoy(g(t))l
.<
E
i=O
r(t) r,(s,)
/T
s’
ILiy(T Iwi(g(t),T
s
ra(s2)--"
/T
s
n-2
r
n-1
(Sn_
)/T
>0.
and a positive
ILnY(S) idsds
n-1
...as,
ASYMPTOTIC BEHAVIOR OF RETARDED DIFFERENTIAL EQUATIONS
t
n-1
.< cw
(t,T)
n-
91
(t
iY(T):+Wn_
i=0
Z
T)/T IgnY(S)
Hence
n-1
"< cm Z
.ly(g(t))
w-n-
(t T)
i=O
.< M+
ILiy(T)I*
fvt
mfTt
Ih(s) [as.
mfTt
p(s)H(y(g(s)))ds
(s,T)p(s)H(.(s))
n’t (s’T)? s’
Wn_
wtere
i=0
[LiY(T)[+m;
JT
[]
or LaSalle’s inequality
M :: cm Z
By Biharl’s inequality
wlY(l{tl)!
where
O(x)
(2.2)
imply
Remark
:=
.
fx
dt
]T
For
[ ].
Remark 2.
For
(x)
O
and
[y(g(t))]
(t T)
result
Tong’s
G-’iG(M)+/TtWn-
.<
t T;
n-’
lh(s)Ids.
[u[ r,
H(u)
G(x).
is and
is completes the prof.
g(t)
and
r e
where
[2, eorem
results of Chert and Yeh
s,T )p( s )ds
r(t)
r0(t)
2,
we have
is the inverse function of
is bounded.
n
[5]
eorem
t,
(0,1], eorem
Kusano
and Singh and
improves
improves the
[5,
which
Theorem
require the condition
f=
ri (t)dt
Using
of vtos
eorem 1,
1,2--’,n-1.
i
we can prove the foiiolng theorem which extends
(2.2)
Let
,(s,)
(2.5)
and
(,)-..
Js
r
n--I
Assume that for some
hold.
-
,(s
(s,_)
n-I
n-2
<
for any constant
(2.5)
hold.
c >
,(s,
O,
>/s?
t
proof of
(s)"
/s
rn_ (Sn_
/s
Liy(t
eorem 2 Is
e
(s T))asa, n- -.-a,
y(t)
of
n-2
0
for
t
l(s) ]asas n_, ..’ds
<
n-I
(2.1)
satisfies
1,2,’-’,n-1.
essentially the same as that of
so we omit the details.
Exampie 1.
T 0
and
Then every oscillatory solution
lim
e
eorem
[6].
eorem2.
(2 )
for
=,
differential equation
(t’(t)).(t)=
t
eorem
In
[6],
C.C. YEH
92
In this example, condition
and
(2.5)
+sin(Int)
y(t)
has an oscillatory solution
(2.2)
and
lim
t > 0.
e-
[sin t+7cos
t-e
All conditions of Theorem 2 are satisfied.
_
REFERENCES
],
y(t)
e
-t
sin t
The asymptotic behavior of a class of nonlinear differential
equations of second order,
Chen, L. S. and Yeh, C. C.
P._roc. Ame__.__r. Mat___h. So___c.
(1982), 235-236.
Necessary and sufficient conditions for asymptotic
decay of oscillations in delayed functional equations,
Edinburgh,
t
It has
t -->
birthday.
Tong, J.
-atsin
This paper is dedicated to Professor Shlh-Ming Lee on his 70th
ACKNOWLEDGEMENT.
3.
(2.3)
are not satisfied, while
as an oscullatory solution which approaches zero as
2.
does not exist.
Consider the differential equation
(e-ty’) "+e-3t-y(t-)
I.
y(t)
are valid.
Example 2.
for
(2.4)
but
Pro.___c.
Royal Soc.
9A (1981), ’135-’145.
Kusano, T. and Slngh, B.
Asymptotic behavior of oscillatory solutions of
a differential equation with deviating arguments, J. Mat____h.
Anal. ADD1.
83 (1981 ), 395-407
4.
Bihari, I.
A generalization of a lemma of Bellman and its application to
uniqueness problems of differential equations,
Acta Math. Acad. Sc__i. Hungar.
.7. (1956), 81-94.
5.
LaSalle, J. P.
Uniqueness theorems and successive approximations,
An__.p.n. Mat____h.
50 (1949), 722-730.
6.
Philos, C. G.
Nonoscillation and damped oscillations for differential
equations with deviating arguments, Math. Nachr.
I06 (1982),
109-119.