Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 4, November 1995, pp. 417-423. �9 Printed in India.

Oscillation of higher order delay differential equations

P DAS, N MISRA* and B B MISHRA Department of Mathematics, Indira Gandhi Institute of Technology, Satang, Talcher, Orissa, India * Department of Mathematics, Berhampur University, Berhampur 760 007, India

MS received 25 October 1994

Abstract. A sufficient condition was obtained for oscillation of all solutions of the odd-order delay differential equation

x~')(t) + ~, pi(t)x(t -- ~rl) = 0, (.) i = 1

where pi(t) are non-negative real valued continuous function in [ T, oo] for some T i> 0 and trlE(0 , oo) (i = 1, 2 . . . . . m). In particular, for pi(t) = pie(0, oo) and n > 1 the result reduces to

1 ( i = ~ 1 (Pla~)'/2)2>(n--2)'(n~,

implies that every solution of( , ) oscillates. This result supplements for n > 1 to a similar result proved by Ladas et al [J. Diff. Equn., 42 (1982) 134-152] which was proved for the case n = 1.

Keywords. Odd order; delay equation; oscillation of all solutions.

1. Introduction

This paper was motivated by certain results of the paper [7] and [8] due to Ladas et al. In [7] authors proved that all solutions of the odd-order delay differential equation

xtn)(t) + ~ pix(t - ai) = 0, (1) i=1

oscillates (i.e., every solution x(t) has zeros for arbitrarily large t) if and only if the associated characteristic equation

2" + ~, pie - ~ ' = 0 (2) i=1

has no real roots, where p~ and tr~e(0, 00) for i = 1, 2 . . . . . m. Further, it was proved that (2) has no real roots if and only if

n (p l ) l / 'a l > - . e

In the literature, it was observed that the odd-order differential equations of the form

xt")(t) + ~ pi(t)x(t - tri) = 0, (3) i=1

where pieC([ T, 00), (0, 00)), T >f 0 and a/e(0, 00), is least studied. In this connection, we may refer, in particular, to [4], [5], [9] and the references therein. For n = 1, (3) is almost well-studied. In this case there are several results associated with its characteristic

417

418 P Das et al

equation (see [3-1 and [7-1) as well as conditions on coefficients and deviating arguments which ensures that every solution of (3) oscillates. In [8-1, authors proved that if pieC([ T, oo), (0, ~) ) , aie(O, ~ ) (i = 1, 2 , . . . , m) and n = 1 then

C lim inf pi(s)ds > 0 (i = I, 2 . . . . . m) (4) t ~ d t - a d 2

and

f ) 1 lira inf pi(s)ds m i = l \ t~oo t - a i

f )c )]' + - lim inf pi(s)ds lim inf (s)ds > - (5) m i < j \ t ~ o o t - a j k t ~ o o e

i , j= 1

then every solution of (3) oscillates. If pi(t)= pie(O, oc)(i = 1, 2 . . . . . m) then the above result becomes

( p i a i ) 1/2 > -

" " \ i = 1 e

implies that every solution of (3) oscillates. In this paper an attempt has been made to obtain a similar result which shows that every solution of(3) oscillates. Our result fails to hold when n = 1. Indeed, when pi(t) = pie(0, oo), the main result of this paper shows that if

- > (."(n - 2)!) -I m i=l e

then every solution of(3) oscillates. Although our result does not generalize the result of Ladas et al [8-1, but certainly supplements for higher order equations.

2. Main results

In the beginning of this section we prove a lemma for its use in the sequel.

Lemma 1. Let f eC(")([ T, oo), (0, oo)), T >~ 0 such that f(")(t) <<. O, t >/ T. I f n is odd and (re(O, o~) then there exists T O >~ T such that

f (t - a) tr "-1 f ( " - 1)(0 >1 (n - 1)~' t/> T o. (6)

Proof. Since f ( t ) >1 0 and f(")(t) <~ 0 for t ~> T, there exists T 1 t> T and 0 ~< k ~< n - 1 such that

f~ > 0 for j ~< k and

f (~)(t) f ~ 1)(0 <~ 0 for k ~<j ~< n - 1.

Expanding f( t ) by Taylor's theorem, there exists xe( t - a, t) such that

k~l o.j O.k f ( t ) = ~ f O ) ( t - a) + ~.f(k)(x)

j=O O-k

>~ -i-;f(k)(t), t >~ T 1 + a. (7) g~

Oscillation o f higher order delay differential equations 419

Similarly, expanding f(k)(t) by Taylor 's theorem we get

I ~n-k- 1 , - , - o., r ( . - m,~ (8) f(k)(t -- O.) ~ -(~_--~ ~ ' ~ . j I,,/, t >~ T 1 + o..

Replacing t by t - o. in the inequality (7) we get

o.k f ( t -- o.) >t ~ . f (k ) ( t -- O.), t/> T 1 + 2o'. (9)

Further , using (8) in (9) along with the fact that k!(n - k - 1)! ~< (n - 1)! and setting T O = T 1 + 2o. we have our proposed inequality.

This completes the p roof of the l emma.

Theorem 1. Suppose that p i~C([ T, oo), (0, oo)), T > 0 and o.ie(O, oo) (i = 1, 2, 3 . . . . . m). Further i f

lira inf pds)ds > 0 (i = 1,2 . . . . . m) t ~ (20 t - - OJ O t

and

where

and

l i ._ 1 2 ~ - - o.i P i i + (Pijpji(o.io.j)n- 1)1/2 > (n -- 1)! 1,

(n) n -

m i = 1 m i<j e i , j = 1

(10)

f t

Pij = pi(s) ds t - - O~u j

then every solution o f (3) oscillates.

Proof. On the contrary, suppose that x(t) > 0 for t/> t o. Dividing (3) th roughout by x t"- 1)(t) we get

_ _ x ( t - o.,) x(')(t) + i Pi(t) x(---~-l)~ = O, (11)

i=1 x(.- l~(t ) that is,

x("~(t) x( t - o.i) x("- 1)( t - t~ x(,-1)(t-----~ + i pi(t)x(,---ii~t-_-~oo.i) x(n-l)( t) = 0 .

i = !

By Lemma 1, there exists t 1 >f t o such that

x(t - o.i) x ( t - r - o . i /n) (o . i /n )"- 1 . . . . . ~ - - - - , t>~t l ,

x t"- 1)(t -- coo.i) x r 1)(t - too.i) (n - 1)!

and the use of this inequali ty in (12) results

x("~(t) ~ x ("- l~(t - o2o.i) + Kips(t) <. O,

x (, - 1)(0 ~ x ( . - 1)(0 i = 1

where

Ki = (n - 1)!

(12)

(13)

420 P Das et al

Integrat ing bo th sides of (13) f rom t - toe k to t we get

Setting

and

[X(. - 1)(t __ COak)" ~ f t X ( n - 1)(S - - (Off/) log \ X(.- t)(t ) /1 >1 ~ Ki Pi(S) ~ ds.

i= 1 t -- ta~k

x t . - 1)(t - ogai) e~ = lira inf ,-, ~ X (n- l )( t )

f t

Pig = lira inf pi(s)ds i, j = 1, 2 . . . . . m

we see that

log(a k) ~> ~ KdqPik- i = 1

Suppose tha t ak < ~ for k = 1, 2, 3 . . . . . m. In this case, dividing bo th sides of the above inequality by a k and using the fact tha t

! for >1 1, a k e

and ~k ~> 1 (since x t"- 1)(0 is positive decreasing) it follows tha t

1>1e i=1 Ki~kkPik, k = l , 2 . . . . . m.

Summing the above inequality for k = 1, 2 . . . . . m we obta in

m ~ Ki~ pik" e k = l i = 1

m ~1 am e >>" Kl-~lPll + K2~--~2P21 + "'" al + Km~-~l Pml

~ 2 a m + K1 ~-~-! p12 + + + a 2 K2~-22P22 "'" Krupp.2

+ . . . . . . . . .

that is,

+ .........

+ K1 al a 2 a m --am Plm + g2~mP2m + "" + Krn~mPmm"

Rearranging the right hand side elements of the above inequali ty first a long the diagonal then above and below the d iagonal respectively, we get

m a i a i K i ; p o + ~ K i ; p O. (14)

e i = 1 i > j i < j i , j = 1 i , j = 1

tha t is, m

m ~ KiPii + E e i = 1 i < j

i , j= 1

a i a j (15)

Oscillation of hi#her order delay differential equations 421

Since the ar i thmetic mean is greater than the geometr ic mean

O~ i O~j K , p . ~J + K j p j , ~ >>. 2~/(popji)(K, Kj). (16)

I n view of (16) , ( 1 5 ) reduces to

m >~ ~ KiPu + 2 ~ ((pqpn)(KiKj)) 1/z. e i= 1 i< j

i,j= 1

Put t ing the value of K, and Kj in the above inequali ty we obta in

1 " 2 ~ n " - I m ,~=1 f in - l~pi i . . 1 _ ( ( P i j p j i ) ( f i f f j ) n - 1)1/2 ~ (n -- 1)! ,

"= i<j e i,j = 1

which is a cont radic t ion to our assumpt ion . Next, assume tha t e~ = oo for some i = 1, 2 . . . . . m. Tha t is,

l im inf xt" - l ~ ( t - C~ i ) t - .oo x~._l)(t ) = 0% (17)

for some i = 1, 2 . . . . . rn. F r o m (3) it follows that

x~")(t) + pi(t)x( t - fi) <<- O,

for the value of i for which (17) holds. F r o m the inequali ty above (13) it follows tha t

x(t- f3 (f,/n)"- t> - - ( 1 8 )

x ~"- x)(t - COal) (n - 1)! "

F r o m (17) and (18) it follows that

x~")(t) + _ ,t, ( f i/n),- 1 x ~" _ 1)( t _c~ <<. O. /-'it ) ( n - 1)! (19)

In tegra t ing bo th sides of(19) f rom t - coai/2 to t and using the fact tha t x ~"- 1)(0 > 0 and decreasing we get

(a , /n)"- 1

f ' p(s)ds <~ O. (20) x{,- 1)(0 - x ~"- x)(t - o)fi/2 ) + -~ : ~ . _,- ,~,/2

Dividing bo th sides of (20) first by x ~"- x)(t) and then by x ~"- 1)(t -~o f i /2 ) we have the following inequalities respectively:

1 x ' " - l ) ( t - t ~ (ai/n)"-I x ' " - l ) ( t - m f i ) f ' pi(s)ds <:0 - x t"- 1 ) (0 { (n - 1)[ x t " - 1)(t) Jr- o~r

(21)

and x~"-l)(t)

x~.- 1)(t - (of i/2) (ai/n).- 1 x t . - 1j(t _mai ) f t pi(s)ds < O.

1 4 (n - 1)! x (~- 1)(t - o ) a i / 2 ) t - . , , , , /2

In view of (10), (17) and (21) we obta in

(22)

x~.- 1j(t - o~th/2 ) lim = o0. (23) t-, ~ x ~ . - 1~(t)

422 P Das et al

Using (23) in (22) along with (10) we see that

x (n- 1)(t - coa~) lim x( n_ 1)(t_ oJai/2) < o% t "* O0

Replacing t by t + anrJ2 in the above inequality we get

lira x(n- 1)(t - o~ai/2) < ~ , ,-. ~o x(n- 1)(t)

which is a contradiction to (23). This completes the proof of the theorem.

C O R O L L A R Y 1.

I f pi(t) = pie(O, ~ ) and ale(O, oo) then

- - > ( n ' ( n - 2)l)- m i=1 e

implies that every solution of (3) oscillates.

Proof. In this particular case

/ n - - l ~p, a p,j= j

and hence (10) reduces to

1 ~Pi + 2 (pipj~aT) 1/2 > ( n - 1)I m i = 1 i < j e

i,j = 1

that is, (24) holds. Hence the proof follows from Theorem 1.

Example 1. The equation

x ( a ) ( t ) + ( 4 + ~ ) x ( t - 1 ) + ( 1 6 + - ~ ) x ( t - 2 ) = O ,

satisfies the hypotheses of Theorem 1 and hence every solution of it oscillates. But Theorem 5.2 of [8] is not applicable to it.

Example 2. Consider the equation

x ' ( t ) + ( 4 + ~ ) x ( t - - 1 ) + ( 1 6 + ~ ) x ( t - 2 ) = O .

By Theorem 5.2 of [8], every solution of it oscillates. But Theorem 1 of this paper is not applicable to this equation. This is due to the fact that Theorem 1 holds only for n > 1 and is an odd integer.

Acknowledgement

Research of the first author was supported by the National Board for Higher Mathematics (Deptt. of Atomic Energy), Government of India.

Oscillation o f higher order delay differential equations 423

References

[1] Das P• •sci••ati•n •f •dd••rder d••ay equati•ns•Pr•c.•ndian Acad.Sci.( M ath.Sci.) • •3 ( • 993) 34 •-347 [2] Das P, Oscillation criteria for odd-order neutral equations, J. Math. Anal. Appl. 188 (1994) 245-257 [3] Das P, Necessary and sufficient conditions for oscillation of neutral equations with periodic coeffi-

cients, Czech. Math. J. 44 (1994) 281-291 [4] Das P, Oscillation of odd-order neutral equations, Czech. Math. J. (to appear) [5] Gopalsamy K, Lalli B S and Zhang B G, Oscillation of odd-order neutral differential equations, Czech.

Math. J. 42 (1992) 313-323 [61 Ladas G, Philos Ch G and Sficas Y G, Oscillation in neutral equations with periodic coefficients, Proc.

Am. Math. Soc. 113 (1991) 123-133 [7] Ladas G, Sficas Y G and Stavroulakis I P, Necessary and sufficient conditions for oscillations of higher

order delay differential equations, Trans. Am. Math. Soc. 285 (1984) 81-90 [8.] Ladas G and Stavr•u•akis • P• •s•il•ati•ns caused by severa• retarded and advanced arguments. J. Di•.

Equn., 44 (1982) 134-152 [9.] Ladde G S, Lakshmikantham V and Zhang B G, Oscillation theory of differential equations with

deviating arguments, (New York and Basel: Marcel Dekker) (1987) [10] Philos Ch G, Oscillations of some delay differential equations with periodic coefficients, J. Math. Anal.

Appl. 162 (1992) 452-475