RETARDED OF

Internat. J. Math. &

Math. Sci.

Vol. I0 No. (1987)89-92 ⁸⁹

ASYMPTOTIC BEHAVIOR OF RETARDED DIFFERENTIAL ^EQUATIONS

CHEH-CHIH YEH Department

of Mathematics

Central University Chung-Li, Taiwan Republic of China

(Received November 15, 1985)

ABSTRACT.

Some

integral criteria for the asymptotic behdvior of oscillatory solutions of higher order retarded differential equations are given.

KEY

WORDS AND

PHRASES.

Retarded differential equations, oscillation.

1980

AMS SUBJECT

CLASSIFICATION CODE. 5K15.

1. INTRODUCTION.

Recently,

Tong [I

proved the following interesting result.

Theorem. Let

f(t,u)

be continuous on

R x

R. If there are

two

non- +

negative continuous functions

v(t), p(t)

for

t 0,

^and a continuous function

g(u)

for u 0 such that

(a) /.v(t)p(t)dt ^<

(b) g(u)

is positive and nondecreasing for u >

0,

(c) If(t,u) v(t)p(t)g(t-’ lul)

for t

1,

u e

R,

then the equation

u"+f(t,u) 0

has solutions which are asymptotic to

a+bt,

where

a,

^b are

constant

and b 0.

In

this note we generalize

Tong’s

result

to

a more general case which improves also the results of Chert and Yeh

[2]

^and

Kusano

and Singh

[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) LnY(t)+f(t,y(g(t)) ^h(t),

^t

^0,

ⁿ

²

where

L

is an operator defined by n

90 C.C. YEH

ro ^()’= _"%- ^()’ ’"’"

Loy(t)

"=

LiY

^dt i-’

(t) :: .

n

cn-i

Here

ri(t

^e

^{[R ,R]}

^with

ri(t ^>

^{0 for i}

0,1,’’-,n-1.

+

Sufficient smoothness to guarantee the existence of solutions of

(2.1)

on an infinite subinterval of R will be assumed without mention. The following

+

conditions are assumed

to

hold in

thSs note.

(i)

f

C[R xR,R]

and there exist two positive functions

p(t),

+

/

that

Iz(t,u) -< ,(t)( lu ^I),

(it)

g, h

CIR.,R], ^{g(t) <} t,

^lira

g(t)= , ..(t,u)

(iil)

lim inf >

0,

lim sup

(t,L ^<

^{oo, i}

^{1,2," ,n-2,}

whvre

w(,u) ^s

^{defined by}

s,

t(t,u) ^:= [ ^,(, r(,)’’" t(s)as...asa,.

Ju Theorem

1.

Let

(2.2) Wn_, ^(t)p(t)dt

^<

(2.5) Ih(t) ldt ^<

hold. If

y(t)

is a solution of

(2.1),

^then

y(g(t)) 0(Wn_,(t,T))

^for

some

T >

^0.

Proof. Let

y(t)

be a solution of

(2.1)

on an interval

[To,oO), T >0.

It

follows from

(il)

^and

(ill)

that there exist a

T > To

^{and a positive}

constant m such that

and

g(t) _{> To}

for t

>

T

inf

=

^m

tT

By (tlt),

there is a positive constant c such that

wi(t,T ^<

^c.

^(t,T),

ⁱ

1,2,’’" n-2

n-1

Now

a simple

argument

shows that

lY(g(t))l

n-1

.< ILoy(g(t))l .< E ILiy(T Iwi(g(t),T

m i=O

r(t)

^{s s}

/T ^s’ /T

^n-2

)/T

r,(s,) ra(s2)--"

^rn-1

(Sn_ ILnY(S) ^idsds

n-1

^...as,

ASYMPTOTIC BEHAVIOR OF RETARDED DIFFERENTIAL EQUATIONS 91

Hence

n-1 t

.<

cw^n-

(t,T)

i=0

Z iY(T):+Wn_ ^(t T)/T ^IgnY(S)

wtere

.ly(g(t)) ^n-1

mfTt mfTt

w- (t T) ^"<

^cm

^Z ILiy(T)I* ^{Ih(s) [as.} p(s)H(y(g(s)))ds

n- i=O

fvt ^(.(s))

.< M+ Wn_ (s,T)p(s)H _n’t (s’T)? ^s’

M ::

cm

Z [LiY(T)[+m; ^lh(s)Ids.

i=0

JT

By

Biharl’s inequality

[]

^or

^LaSalle’s

^inequality

[5]

^we have

wlY(l{tl)!

n-’ t

T; ^.< G-’iG(M)+/Tt Wn- ^{s,T )p(}

^s

^)ds fx

^dt

where

O(x) :=

]T

^{and O}

^(x)

^is the inverse function of

G(x).

^{is and}

[y(g(t))]

is bounded. is completes the prof.

(2.2)

imply

(t T)

Remark

. ^For

ⁿ

^{2, r0(t) r(t)}

^and

^{g(t) t, eorem}

^improves

Tong’s

result

[ ].

Remark

2. For H(u) [u[ r,

where r e

(0,1], eorem

improves the results of Chert and Yeh

[2, eorem

^and Singh and

Kusano [5,

^{Theorem which}

require the condition

f= ^{ri (t)dt}

^{=, for i}

1,2--’,n-1.

Using

eorem 1,

^{we can prove} the foiiolng theorem which extends

eorem

of

vtos [6].

eorem2.

Let

(2.2)

and

(2.5)

hold.

Assume

that for some

T 0

(2 ) ,(s,)

^Js

(,)-..

^n-2rn--I

_< (s,_)

^n-I

^,(s -- ^{(s T))asa, n- -.-a,}

for

any constant

c >

O,

and

>/s? /s /s

(2.5) ,(s, (s)" rn_ (Sn_ ^{l(s) ]asas}

n_,..’ds

<

n-2 n-I

hold. Then

every

oscillatory solution

y(t)

of

(2.1)

satisfies lim

Liy(t

^{0 for}

^t 1,2,’-’,n-1.

t

e

proof of

eorem 2 Is

essentially the same as that of

eorem In [6],

so we omit the details.

Exampie

1. e

differential equation

(t’(t)).(t)=

^t

92 C.C. YEH

has an oscillatory solution

y(t) +sin(Int)

In this example, condition

(2.2)

^and

(2.4)

are not satisfied, while and

(2.5)

are valid.

Example

2.

Consider the differential equation

(e-ty’) "+e-3t-y(t-) e- [sin t+7cos

^t-e

-atsin

^t

],

for t

>

0. All conditions of Theorem

2

are satisfied. It has

y(t)

e-t as an oscullatory solution which approaches zero as t -->

ACKNOWLEDGEMENT.

birthday.

but lim

y(t)

does not exist.

(2.3)

sin t

This

paper

^is dedicated to Professor Shlh-Ming Lee on his 70th

REFERENCES

I. Tong, J.

^The asymptotic ^behavior of a class of nonlinear differential equations of second order,

P._roc. Ame__.__r. Mat___h. So___c. _ ^{(1982), 235-236.}

2. Chen,

^{L. S. and} Yeh, C. C.

Necessary

and sufficient conditions for asymptotic decay of oscillations in delayed functional equations,

Pro.___c.

^{Royal Soc.}

Edinburgh,

9A (1981), ’135-’145.

3. 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.}