v(t)p(t)dt · 2019. 8. 1. · (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

Internat. J. Math. & Math. Sci.Vol. I0 No. (1987)89-92

89

ASYMPTOTIC BEHAVIOR OF RETARDED DIFFERENTIAL EQUATIONS

CHEH-CHIH YEH

Department of MathematicsCentral UniversityChung-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 byn

90 C.C. YEH

ro ()’= "%- ()’ ’"’"Loy(t) "= LiY dt i-’

(t) :: .n

cn-iHere 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= mtT

By (tlt), there is a positive constant c such that

wi(t,T < c. (t,T), i 1,2,’’" n-2n-1

Now a simple argument shows that

n-1lY(g(t))l .< ILoy(g(t))l .< E ILiy(T Iwi(g(t),Tmi=O

r(t) s s

/Ts’ /Tn-2

)/Tr,(s,) ra(s2)--" r (Sn_ ILnY(S) idsds ...as,n-1 n-1

ASYMPTOTIC BEHAVIOR OF RETARDED DIFFERENTIAL EQUATIONS 91

Hence

n-1 t.< cw (t,T) Z iY(T):+Wn_ (t T)/T IgnY(S)n-

i=0

wtere

.ly(g(t)) n-1 mfTt mfTtw- (t T) "< cm Z ILiy(T)I* Ih(s) [as. p(s)H(y(g(s)))dsn- 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)! .< G-’iG(M)+/Tt s,T sn-’

t T; Wn- )p( )ds

fx dtwhere 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,) (,)-.. r (s,_) ,(s (s T))asa, -.-a,n--I -- n-

Jsn-2 n-I

<

for any constant c > O, and

>/s? /s /s(2.5) ,(s, (s)" rn_ (Sn_ l(s) ]asasn_, ..’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.

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v(t)p(t)dt · 2019. 8. 1. · (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