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LIBRARY
Michig 1n Sta”
Unix city

n12.3ll

This is to certify that the
the‘gisentitled
Oscillation Properties of a Delay Differ;

ential Equation of Order 2n

presented by

Raymond D. Terry

has been accepted towards fulfillment
of the requirements for

P_h _-_—_D . degree in W1 C S

 

Date ' l 2

ABSTRACT

OSCILLATION PROPERTIES OF A DELAY DIFFERENTIAL
EQUATION OF ORDER 2n

BY

Raymond D. Terry

The main purpose of this thesis is to provide criteria
for the oscillation of solutions of certain nonlinear delay
differential equations of even order. In the first four sec-

tions we consider the equation
(1.1) D“[r<t)nny]<t> + f(t.yT<t))yT(t) = o.

It is assumed that: (l) yT(t) = y(t-T(t)): (2) the delay
T(t) is nonnegative and bounded: and (3) the function f(t,u)
is continuous, nonnegative, odd and monotone in u.

In chapter one a classification of solutions according
to types Bj(j = O,...,n-l) is introduced. When r(t) E 1
this classification coincides with the one introduced by
Kiguradze (Dokl. Akad. Nauk SSR, 144 (1962), 33-36). It is
first demonstrated that a nonoscillatory solution y(t) of
(1.1) is necessarily of type Bj for some j = O,...,n-l.

Chapter two provides integral criteria for the nonexist-
ence of solutions of type Bj: conditions for the oscillation
of all solutions of (1.1) follow immediately. The major re-

sult of this section is Theorem 2.4.

Raymond D. Terry

In chapter three we let r(t) E 1 in (1.1) and consider
the asymptotic properties of the resulting equation (3.1).
A necessary and sufficient condition is given for the exist-
ence of a solution of (3.1) which is asymptotic to th-l.

In chapter four Lyapunov's direct method is used to ob-
tain nonoscillation criteria when the conditions on f(t,u)
are weakened. The results derived here agree with those ob-
tained in section two.

Chapter five deals with a more general nonlinear delay
equation of order 2n:

n n N

(5-1) D [r(t)D Y](t) + i§1f1(t)Fi[u*(t€2n-Ti(t))] = 0
The results of this chapter depend on the assumption of either
(i) the existence of an index j for which Fj has some de-
gree of superhomogeneity; or (ii) the existence of two indices

j,k for which x]:1Fj has prescribed monotone properties.

Oscillation Properties of 3 Delay Differential

Equation of Order 2n

BY

‘5
(Y‘

I \-

Raymond D?yTerry

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1972

ACKNOWLEDGEMENTS

The author wishes to express his deep gratitude to Dr.

Pui-Kei Wong for his encouragement and patience during the

writing of this thesis.

ii

TABLE OF CONTENTS

Chapter Page
1. Introduction . . . . . . . . . . . . . . . . . 1
2. Integral Criteria for Oscillation. . . . . . . 9

3. The Asymptotic Character of Certain Solutions. 27

4. An Application of Lyapunov's Direct Method . . 32
5. A More General Delay Differential Equation . . 44
List of References. . . . . . . . . . . . . . . . . . . 57

iii

Chapter 1. Introduction

The main purpose of this paper is to discuss the oscil-
latory and nonoscillatory behavior of solutions of the 2n-th

order delay differential equation1
(1.1) Dnrr<t)D“yHt) + yT(t)f(t.y4t>) = o.

where yT(t) = y[t - T(t)], O S rr(t) s T, and O<msr(t) 5M.
Throughout chapters two and three f(t,u) is assumed to sat-

isfy the following three hypotheses:

(i) f(t,u) is a continuous real-valued function on [O,w)><R:
(fiJ for each fixed t e [0,w), f(t,u) < f(t,v) for O<u<v;
and

did) for each fixed t e [0,m), uf(t,u) > O for u # O.

In chapter four, these restrictions on f(t,u) shall be re-
laxed and replaced by others as indicated there.

Existence and uniqueness theorems for solutions of (1.1)
are well known,cf. [3], chapter 1. The basic initial value

problem is usually stated in terms of a first order system

 

1For typographical reasons the operator notation will
be used consistently with the possible exception of an oc-
casional y’ or y”. We have

13.):

(Day)(t) = Dsy<t) = y‘s’(t) = dts and

DSYT (t) = (DSY)(t-T(t)) = Y(8)(t""r(t))-

Y'(t) fltdflt). Y,r (13)]. t 2 t

O

(1.2)

y(t) ¢(t). to-Tstst.

0

where y(t) is an m-vector, f a given continuous m-vector,
and ¢(t) is a given continuous m-vector function on
[to-T,to]. By converting (1.1) into a first order system

of the form (1.2), existence and uniqueness results for (1.2)
may then be applied to (1.1). Briefly, a solution of (1.1)
for t 2 to is uniquely determined by (Zn-l) continuous

, initial functions ¢k(t) satisfying Dky(t) = ¢k(t) for

to - T s t s to, where we usually require that ¢k(to)
Dky(to + O), k = O,l,...,2n-1. Throughout this paper a sol-
ution of (1.1) is understood to mean a solution which can be
continued indefinitely.

Oscillation theory of ordinary differential equations
originated with the fundamental investigations of Sturm in
the nineteenth century. In recent decades, the subject has
been broadened in many directions, and the study of oscilla-
tions of functional differential equations is one of them.

In the paragraphs below we shall state some basic definitions
and notions needed in the sequel.

A solution y(t) of (1.1) is said to be oscillatory on
[0,m) if for each to > 0, there exists a To > to such
that y(To) = 0; it is called nonoscillatory otherwise.
Following Kiguradze [5] we say that a solution y(t) is of

type Aj if

Dky(t) 2 O, k

(")k+1DkY(t) 20, k = 2j+2, ...,2n

O,l,...,2j+l and

for all t sufficiently large. In an analogous manner we
shall say that y(t) is of type Bj if the derivatives of
y and y1 have certain sign properties, where y1(t) =
r(t)Dny(t). Specifically, if n is even and j s (n-2)/2

or n is odd and j s (n-3)/2, we require that

Dky(t) 2 0. k = o,..., 2j+1;

(-)k+1Dky(t) 2 0. k

2j+2,...,n; and

n+k+lbk

(-) Y1(t)2 0: k: O,...,n

(n+1)/2.

NI

If n is even and j 2 n/2 or n is odd and j

we require that

Dky(t) 2 O, k = O,...,n:

Dkylun 2 O, k = O,...,2j-n+1; and

n+k-l k

H D y1(t) 2 o, k

2j-n+2,...,n

Finally, if n is odd and j = (n-l)/2, y(t) will be of

type Bj if

Dky(t) 2 O, k=0,...,n and

(-)kay1(t) 2 O, k 0, . . . ,n.

When r(t) s 1, these definitions reduce to the definition
of an Aj-solution. In [5] Kiguradze proved a fundamental

lemma which.we state as follows:

Lemma 1.1. Let u(t) be a continuous nonnegative function
on (O,n) with continuous derivatives up to order Zn in-
clusive which do not change sign on this interval. If
Dznu(t) s 0, then there exists a number 0 s p s 1 such

that

Dku(t) 2 O, k = O,...,£

(-)k+leu(t) 2 O, k

z+l,...,2n

where z = 2p4-1. Furthermore, O S D‘n(t) s-‘i u(t).
t

In view of this result, all nonoscillatory solutions of
(1.1) with r(t) a 1 are of type Aj, j = O,...,n-1. In the
general case, we argue as follows: Suppose y(t) is a non-
oscillatory solution of (1.1), which we may assume to be non-
negative because of (iii). First of all, no two successive
derivatives of y or y1 can be negative. To see this we

suppose Dky and Dk+1y are negative for large t, then

there are constants Co > O and to > O for which Dky is

a negative decreasing function on [to,w) and Dky(t) < -Co

for t 2 to. Hence,2
t t
k-l k-l k
D y(t) — D y(to) — St D yds < - Co St ds _ C0(t to),{
o 0
which implies that lim Dk-1y(t) = - m. Proceeding inductively
tea
1

and using the fact that Dky and Dk- y are eventually neg-
ative, we conclude that y(t) < O for large t, which is a

contradiction. A similar argument establishes the claim for

 

2During an integration the variable in a differential
expression will be suppressed if there is no resulting ambiguity.

the derivatives Dky1 and Dk+ly1, k 2 1. Of special inte-

rest is the case in which y1 and Dyl are negative. Then
there exist constants Co > O and to > O for which y1
is a negative decreasing function on [to,m) and y1(t)<-Co

for t 2 to. Hence,

t t y (S)
n-1 n-1 _ n _ 1 _ -l _
D y(t) D y(to) — St D y ds — St 7(3) ds< COM (t to),
o 0
which implies that lim Dn-1y(t) = -m. Since r(t) > O,
t-«n
Dny(t) < O for large t. Using this and the fact that
n-l

D y(t) < O for large t, we proceed as in the first part
of the argument to conclude that y(t) is eventually nega-

tive, which is again a contradiction.

Secondly, if two successive derivatives of y or y1
are positive, than all preceding derivatives of y or y1 are

positive. If Dky and Dk+1y are positive for large t,
then there are constants C1 > O and t1 > O for which Dky

is a positive increasing function on [t1,m) and Dky(t) > CI

for t 2 t1. Hence,

k

D -ly(t)-Dk-l

t k
y(tl) = gt D y ds > Cl(t-t1)o

which implies that lim Dk_1y(t) > +m. A similar argument
t*° . k k+l
establishes the claim for the derivatives D yl and D

Y1!
k 2 1. Now consider the case that y1 and Ulyl are even-
tually positive. Then there exist constants C1 > O and

t1 > O for which yl is a positive increasing function on

[t1,m) and y1(t) > CI for t 2 t1. Hence,

__ __ t t y (s) _
Dn 1y (t) -Dn 1y(t1) = S Dny ds g —]-J_;-(-s—)ds > ClM 1(t-t1),

t1 t1

which implies that lim Dn-1y(t) = +m. Since r(t) > O,
t-ooo
Dny(t) > O for large t. Using this and the fact that

Dn-1y(t) > O for large t, we proceed as in the first part
of the argument to conclude that Dky(t) > O, k = O,...,n.

It follows from these two Observations that if y is a
positive nonoscillatory solution of (1.1): then it is of
type Bj for some j = O,u.,n—l.

In chapter two integral criteria are given for the non-
existence of solutions of type Bj as well as for the os-
cillation of all solutions of (1.1). Criteria for the non-
existence of solutions of type Aj then follows as corollar-

ies. Papers presenting integral criteria for the oscillation

of second order delay equations are extensive. The equation
(1.3) y” (t) + p(t)y: (t) = 0

has been the subject of numerous studies. Gollwitzer [4]
separated his study of (1.3) into two cases: y > 1 or y < 1,
see also Wong [10]. Bradley [1] has also recently consideed
the case y = 1. It is of interest to study (1.1) as one
generalization of (1.3).

Chapter three provides a necessary and sufficient condi-
tion for the existence of a nonoscillatory solution of (1.1)
with r(t) a 1 having prescribed asymptotic behavior. Paral-

lel results for the case of fourth order linear equations may

be found in Leighton and Nehari [7] while that of a class of
nonlinear fourth order equations is in Wong [11].

In chapter four Lyapunov's direct method is used to ob—
tain nonoscillation criteria when conditions (ii) and (iii)
are replaced by weaker assumptions. This method was employed
recently by Yoshizawa [12] to study the oscillatory behavior
of a nonlinear second order differential equation. In this
paper we show that his method is applicable to equations of
order 2n with retarded arguments.

Chapter five deals with a more general nonlinear equa-
tion of order Zn in which the function f(t,yT(t)) is re-

placed by a sum of products of functions of the form:

-1
f1 (t) F1 (yT (t) .D (t). . . .,D“ y (t).y1 (t).
. y . 'r .
1,1 Ti 2 1,n 1,n+1

n-l
.,D y (t)).
1Ti,2n

Here the analysis is simplified by the separation of f into
a function of t and a function of the derivatives of y

and y1, ‘where the variables have been retarded. Ostensibly,
the problem is more complicated because f has been replaced
by a function of 2n+l variables and because there are 2n
different delay terms. The major difference is in the assump-
tion of (i) the existence of an index j for which Fj has
some degree of super-homogeneity: or (ii) the existence of

two indices j,k for which x-iFj has prescribed monotone
properties. The second order case (n = l) with r(t) E l

was considered by Staikos and Petsoulas [9] subject to

2

x29” Fj(x.y) for every <x.y) e R . x eR

(i) Fj().x.>.y.) =
and some integer p > 0: (ii) [Fi(x,0)]/x is nonincreasing
on (0,»). we will assume tacitly that n 2 2.

For recent related results, see the papers of Burkowski

[2], Ladas [6], Wong [10], as well as the book by Norkin [8].

Chapter 2. Integral Criteria for Oscillation

In this chapter we prove some results on the nonexist-
ence of solutions of type Bj which in turn give rise to os-

cillation criteria for (1.1).

The following lemmas are useful in obtaining the proof

of such a nonoscillation theorem.

Lemma 2.1. Let y(t) be a solution of (1.1) of type B

n-l'
Then for sufficiently large t, the following estimates are
valid:
(a) t(D“‘1y1)(t) 2 2<D“‘2y1)<t):
(b) t(D“'ky1)(t) s 2k(D“"k‘ly1)(t), k = l,...,n-l:
(c) ‘tyi (t) s 2nM(Dn_1y)(t): and
(d) t(Dn-ky)(t) 5 2mm?1 + k)(Dn-k-1Y): k = l,...,n-l,

‘where Yl(t) = r(t)(Dny)(t).

Proof: Suppose y(t) is a solution of type B Then

n-l'

there is a To > 0 such that Dky, k = O,...,n-l and Dkyl,
k = O,...,n-l, are positive for t > To. Hence yT(t) > O
for t - T(t) > To' i.e., for t > TO + T = T1. From (1.1)
we have
n n

D [r(t)D y](t) = - yT<t)f<t.yT(t))

n n-l . . .
so that D y1 < O for t > T1. Thus D 3fi_ 18 a p031tive

decreasing function on (T1,m) and

10

-. — .— fit —.
(2.1) (Dn 2Y1) (t) 2 (Dn 2y1)<t> - (Dn 2y1) ”‘1’ =31. Dn 1ylds
l
{t n—1 n-1
2 RT (D Y1) (t)ds= (t‘TI) (D Y1) (t).
1

Since t--T1 2-%t: for t 2 2T1, we have (Dn-

%t(Dn-1Y1)(t) for t 2 2Tl which proves (a).

2Y1) (t) 2

To prove (b) we proceed inductively and suppose that

(2.2) (t-Tl) (Dn‘kyl) (t) 2 kwn'k'lyl) (t). t 2 T1

for some k, l s k s n-2. An integration of (2.2) yields

t “t t
[(s--T]_)Dn k lyl] - E Dn k 1yl ds = S (s-Tl)Dn kyl ds
T1 T1 T1
t
gk ‘ an’ly ds.
T 1
1
Hence,
t
(t-Tl) (Dn k lyl) (t) s (k+l) S DIn k 1y1 ds
T
1
—k- -k-2
(2.3) = <k+1)[<D“ 2y1)(t)-D“ Y1) (Tin
_k_
s <k+1)<D“ 2y1> (t)
for t > T1 and %4:(Dn-k-1yl)(t) s (k+l)(Dn_k-2yl)(t). Thus

(2.2) is valid for all k, l s k s n-1, and (b) is proved.

In particular, for k = n-l, (2.2) becomes

(2.2)' <t-T1>(Dy1) (t) s (n-1)y1<t). t 2 T1.

Integrating this, one gets

t dt t t
[(s-T1)yl (8)]T1" X yl(s)ds = S (s--T1)Dyl ds :: (n-l) Q y1(s)ds.
'T1 T1 T1

11

Since r(t) s M and Dn-ly(Tl) > O, we have

t n
(t‘Tl)Yl(t) S n S r(s) D y ds
“Tl

D“‘1y(t) — Dn‘2y<T1>]

IA

an
n-l
5 PM (D Y) (t)

for t 2 T and ty1 (t) s 2nM(Dn-1y) (t) for t 2 2T1,

l
which proves (c).

Furthermore, y1(t) r(t)Dny(t) and r(t) 2 m so that

for t 2 T1

t t n
(s-T )y (s)ds 2 m (s-T )D y(s)ds
ST 1 1 ST 1

n-l

= m(t-T )D y(t)-m St Dn-1y(s)ds.
T

l

1
Combining this with (2.3) one gets
_ t _ t -
m(t--T1)Dn 1y(t) s m 3 Dn 1y(s)ds + nM 8 Dn 1y(s)ds
T T
l

t
(m4—nM) Dn 1y(s)ds
1 T1

(m4-nM) [Dn‘2y(t)-D“‘1y<T1)]

IA

(ma-nM)Dn‘2y(t).

It follows that for t 2 T1

n-l l

(t-T D y(t) s (an— + l)Dn-2y(t)

l)
and for t 2 2T1

l

t D“‘1y(t) 2 2(an' + 1)D““2y(t).

To prove the final assertion, we proceed inductively

and assume that for t 2 T1,

12

k-l

1 + k)Dn— y(t)

(t -T1)Dn-ky (t) s (an-

for some k,l s k s n-2. Integrating (2.5) for t 2 T1 yields

_ _ t t _ _ t _
[(s--T1)Dn k 1y] --3 Dn k 1y ds = 3 (s--T1)Dn kyds
T T T
1 l
-l t n-k-l
s mwn +k)g D yds
T1
so that
— — — t — —
(t-T1)Dn k 1y(t) 3 [mm 1+(k+1)] 3 Dn k lyds
T1
2 [ :erer1 + (k+1) 1 (Dn‘k‘z y (t) -D""“2 y (T1))
5 [an-l+ (k+l)] Dn_k-2y(t)

for t 2 T1, which implies that (d) is valid for t 2 2T1.
The investigation of similar inequalities for solutions
of type Bn-k (k = 2,...,n) is slightly more complicated.
There are two cases.
Suppose n is an even integer and suppose y(t) is a

solution of (1.1) of type Bj' where j s n;2 . Then

D234-2

 

2j + 2 s n and the first negative derivative is y. By

applying the same procedure as in Lemma 1.1, one obtains for

k = 1,...,2j-+1

23+2—k 23+l-k y(t), t 2 T and

y(t) s k D 1

(2.4a) (t-T1)D

2j+2—k 2j+l-k

y(t) s2kl) y(t), t 2 2T1.

(2.4b) t D

If j > £32,, i.e., if j 2 n/2, then 2j+2 2 n+2 and the
first negative derivative is D2344”n yl. We obtain for

t 2 T1 :

13

(2.5a) (t—.--T1)D2j+2"n’k y1(t) s k D23"‘1""‘k yl(t), k=l,"q2j-n+l;
(2.5b) (t-T1) yl(t) s (2j-n)M D“"’1 y(t);

and

(2.5c) (t - T1) D""k y(t) s [<2j-n)Mm‘1+k]D"’k"l y(t).

where k = l,...,n-1. Moreover, for t 2 2T1

(2.6a) tD23+2““‘ky1(t) s 2kD23+1’“"ky1(t), k = 1,...,2j-n—l;
(2.6b) ty1(t) 5 2(2j—n)MD“"1 y(t):

and

(2-60) (t"Tl)Dn-ky(t) s 2[(2j-n)Mm-1 + kph-k.l y(t).

where k = l,...,n-1.

We remark that if n is an odd integer and j s n;3 .

 

then 2j+2 s n-l so that the inequalities (2.5) are valid.

If n is an odd integer and j 2 2%; then 2j+2 2 n+3 and

the inequalities (2.6) and (2.7) are valid. For j =-- ,
2j+2-n = 1, so Dy1 is the first negative derivative. We

obtain for t > T1

(2.7a) (t-Tl) y1(t) s MD“‘1y(t)
and

(2.7b) (t-T1)Dn_ky(t) s (Mm—la-k)Dn-k—l y(t), k = l,...,n-l.

Hence, for t 2 2Tl

n—

(2.8a) t y1(t) s 2MD 1 y(t)

and

(2.8b) tDn-k y(t) s 2 (Mm—li-k)Dn—k-l y(t), k = l,...,n-1.

14

For T1 = O the results of Lemma 2.1 may be improved

to yield

n-k

(a). (b) t D y1<t) “‘k

k D '1 y1(t), k = l,...,n-l7

IA

nMDn-1 y(t): and

V\

(C) tY1(t)

l n—k—l

(d) t D“-k y(t) s (an' +k)D y(t), k=l,...,n-l

for large t. If r(t) a l, r(t) a O and y(t) is a solu-

tion of type Bn on (0,»), then T :0 and m=M=1

-1 1

so that

tnzn-kyvc) s k D2“"1""'1 y(t), k = l,...,2n-l,

which is a special case of Kiguradze's Lemma [5]. Similarly,

if r(t)

l, r(t) E O and y(t) is a solution of type Bj

on (O,¢), then

2j+2—k 2j+l-k

t D y(t) S k D y(t), k = l,...,2j+l.

Lemma 2.2. Let y(t) be a solution of (1.1) that is ulti-

mately positive.
n-2

(a) Suppose n is even and j s —E— or n is odd and
j S Bil“ If y(t) is a solution of type Bj' then there

are constants k > O and to > 0 such that

 

2.
D 3 yT(t)
2j 2 k, t 2 to.
D y(t)
(b) Suppose n is even and j 2'? or n is odd and
j 2.2%l . If y(t) is a solution of type Bj' then there

are constants K > O and t1 > 0 such that

15

2'-
D 3 n Y1: (t)

D2j--n Y1“)

 

Remark 1. Part (a) of this lemma is analogous to one proved

by Bradley [1] for the equation

Y” (t) + p(t)y,r (t) = 0.

Since his proof depends only on the concavity of y, Lemma
2.2 follows easily for in (a), D23 y is concave and in (b),

D23"-n y1 is concave.

Remark 2. The above result may, however, be obtained via

Lemma 2.1 and the observations following it.

Proof: Let y(t) be a solution of type B. where n is

 

 

 

 

3
even and j s 232- or n is odd and j s n;3 . For t 2 T1,
D23+1 y(t) > 0. Since 7(t) 2 o,
2‘ 2' 2'
D Jmi = D Jam-Tm) s D Jy(t)
so that with the help of (2.5b), we have
2' 2' 2' 2' 1
D Jy (t) D Jy(t) -D 3y,r (t) D 3+ y(s)
—-——I-—-2j - 1 = 23. = T(t) 23.
D y(t) D y(t) D Y(t)
2j+1 2j
S T D2j y(s) s 2:1; D2y(s)
D y(t) D 3y<t)
2T
S — I
S

where t-7(t) s s s t. Since 3 tends to infinity with t,

we Obtain

16

2i
lim D YT(t) = 1

tan D23y(t)
If n is odd and j = -—-, we observe that for t:>T1,
Dn_1y(t) and Dny(t) are both positive. Since r(t) > O,

y1(t) > O and Dy1(t) < O, ‘we have by a similar argument

 

 

 

that
D2jy (t) _ 1| = D2jy(t)-D2ij(t) = 7(t) Dny(s)
D2jy(t) Dij(t) Dn'1y<t)
T y1(s) 2MT Dn-1y(s)
Sr<s)D“’1y(t) 5 "TS Dn'lyuz)
ms

where t"T(t) s s s t. The conclusion then follows as in
the previous case.

If n is even and j 2 n/2 or n is odd and j22(n+1)/2,

 

 

 

 

 

 

 

we note that for t > T1: Dzj-n+ly1(t) > 0. Since T(t) 2 O,
2'- 2'- 2'-
D 3 “lem = D 3 “y1(t-w(t)) s D 3 nylon
so that
2j-n 2j-n _ 2j-n 2j-n+1
D 1711“) _ 1 = D yyt) D Y1; (t) = 'r(t) D 3:15)
2._ 2._ ._
D 3 ny1(t) D 3 ny1(t) D23 ny1(t)
._ 2._
D23 n+1Y1(S) 2T D 3 ny1(8)
S T 2"?) :3 ‘g— 2"!)
D 3 y1(t) D 3 y1(t)
2T

S I

where t"T(t) s s s t. Part (b) now follows and the lemma

is proved.

17

Remark 3. Lemma 2.1 is clearly valid for T(t) unbounded
if, in addition, lim (t-T(t)) = +o. It is of interest to
note that Lemma 2.2ais valid also even if T(t) is unbounded
provided 0 s m(t) < pt, where u will be specified below.

Since t"T(t) s s, ‘we Obtain in the first case of Remark 2

2 D23+l

 

 

 

 

 

D jy (t) D y(s) y(s)
fi-l ='r(t) 2] Srr(t) 2.
D y(t) D y(t) D 3 Ms)
s 1ft) 5 r(t)
s-Tl t-T(t)-T1
In the second case,
D2jy (t) T(t)Y (s) y (s)
--2—.-—I—-— -- 1‘ = 1_1 S Ill—LE). ...—1:1...—
D 3y(t) r(s)Dn y(t) m D“ y(S)

 

Spam-11.63). S “In-1412)....

 

 

 

 

 

 

s-T1 t-T(t)-T1
In the third case,
2'- 2'— -l 2'~ -1
D 3 nyl.(t) 1 ( ) D 3 “ Y1(s) ( ) D 3 " y1(s>
.___ - ='rt . sTt .—-f—
.. .. 2..
D23 ny1(t) D23 nyl(t) D 3 ny1(s)
S1112.). S LE)...
s-T1 t-'r(t)-T1
t _
We note that t-T(t)-T1 s 1 k for any 0 < k < 1 pro
vided T(t) s %E%-(t-—Tl). Moreover, for any 0 < E <-%,

there is a O < k < 1 satisfying

1:35-}.-E
2-k — 2

Thus, in the first and third cases, if 0 g 7(t) s (%-e)(t-T1)

18

for some 0 < e <-;: then there is a O < k < 1 for which

 

 

 

 

 

 

 

 

 

2
2j 2j-n
D y (t) D y (t)
—52—1———- - 1 5 1 - k or 2j_n 17 — 1 < l—k
D 3y (t) D y1(t)
respectively, which implies that Dzij(t) 2 kD23y(t) or
2j-n 2j-n
D le (t) 2 kD Y1 (t) .
. . 21(t) EB _
In a Similar manner t"T(t)"T%S M (t T1) for any
0 < k < 1 provided T(t) s 3- é;_ ) (t-T1). Moreover,
[l+fi(1-k)]
for any 0 < 6 < ME; , there is a O < k < l satisfying
m. 2_1-'k = _E_. _ e
M [1+9- (1-k)] mm
M
. . m
Thus, in the second case, 1f 0 s T(t) s M+m e](t T1) for
some 0 < 6 < ME; , there is a O < k < l for which
DZJy.(tJ
—-.-—1-—— - 1 s 1 - k
23 '
D y(t)
2j 2j

which implies that D yT(t) 2 k D y(t).
Using the two lemmas of this section, we may prove the

following result.

Theorem 2.1. Suppose that for some j = O,1,...,n-1 and for

all constants C > O

3 t23f(t,c)dt = +2

Then (1.1) has no solutions of type Bj'

Proof: Suppose y(t) is a solution of type Bj' where n is

even and js (n-2)/2 or n is odd and js (n-3)/2. Let

19

-1
Dn Y1 (t)

23'y(t)

 

w(t) =
D

Then we see from (1.1) that

 

,(t) Dn-1y1(t) D2j+1y(t) y (t) f(t (t)) o
w + . + . ,y = .
[ngy (t) )2 D235“) T
Since Dn-1y1(t) and D2j+ly(t) are positive for t > T1
y (t)
(2.9) w’ (t) + 51— f(t,y (t)) s o.
D Jy(t) 7

From (2.4) we obtain

2j 2'

t D 3 y(t) s 223. (2j+1)! y(t)

for t > 2T1 so that for t > 2Tl + T = T2 0

2' 2' 2' .

(t-T(t)) 3 D 3 yT(t) 2 2 3 (23+1): mi.

2j ‘1

Setting N1 = [2 (2j+l)£] and using the fact that

O S 7(t) s T, ‘we can rewrite this as

2

-23'3'
Y¢(t) 2 N1(t T) D yT(t), t > T

2
Combining this with (2.9), we have

2i
- D y (t)
w’(t)+Nl(t-T)23-Tj—-T-— f(t,yT(t)) so, t>T

D y(t) 2

Since Dy(t) > O for t > T1, there is a to > T2 and a

C > 0 such that y (t) 2 C for t 2 to. Hence
T

2i
- D Y (t)
w’ (t) +Nl(t-T)23 ———T--2j f(t,C) s O, t 2 t

D y(t) °°

20

By Lemma 2.2 (a) there is a constant k > O and a t1 > t

such that Dzij(t) 2 k D23y(t) for t 2 t . Thus

1

2.
(2.10) w’(t) + N1k(t - T) 3 f(t,C) s o, t 2 t1

An integration then yields

t 2.
w(t) — w(tl) + le g (s-T) jf(s,c)ds s o.
't
1

In View of the hypothesis, we must ultimately have w(t) < O,

which implies that Dr'-1

y1(t) < O for large t. This con-
tradicts the assumption that y is a Bj-solution.

Now suppose that n is even and j 2 n/2 or that n
is odd and j 2 (n+l)/2. Let

(t) Dn-1y1(t)
w = . -
DZJ’ny1(t)

 

Then we see from (1.1) that

n-l 2j-n+l

D y1(t) D
y1(t)]2

Y1(t) yT(t)

+
y1(t)

 

 

 

w’ (t) + f(t,y“r (t)) = O.

23-n D23-n

[D

2j-n+1

Since Dn_ly1(t) and D y1(t) are positive for t > T1,

we have

y (t)
(2.11) w’(t) + 2j_; f(t,y (t)) s o, t > T
D y1(t) T

 

1

From (2.6) we obtain

2j ' I n_l - ‘1 .
y1<t) s 2 (2)-n). M -H1 [<23-n)Mm + a] y(t)
J:

tZJDZJ-n

for t > 2T so that for t > 2T1 + T = T

1 2 '

21

. - . ..1
2 2 - 2 . n . -1 .
(t-T<t)) JD 3 nle(t) 2 2 J(2)-n): M n [(2)-n)Mm 4'3]YT(t)-
i=1
Setting
. n-l
N 1 = 223(2j-n)! M H [(2j-n)Mm l + j]

2 j—l

and using the fact that O : w(t) s T, we can rewrite this

as

_ 2j 2j-n
Y¢(t) 2 N2(t T) D y1T(t), t > T2

Combining this with (2.11) we get

2j-n
D y11(t)

ny1(t)

 

2j
I ._ z
'w (t) + N2(t T) D2j_ f(t,yT(t)) 2 0, t > T2

, there is a t > T and a
1 o 2

C > 0 such that yT(t) 2 C for t 2 to. Hence

Since y’(t) > O for t > T

2'-n
2j D J y1T(t)
w’(t) + N2(t-T) 2._n1— f(t,C) s o, t 2 t
D 3 y1(t)

 

0.

By Lemma 2.2 (b), there is a constant K > O and a tl > tO
such that D23"n 23‘“

y1T(t) 2 KD y1(t) for t 2 t . Thus,

1

with N1 and k replaced by N and K, (2.10) holds as

2
in the first part of the proof and we arrive at the same con—

tradiction.

, then letting

Finally, if n is odd and j =51;-l

-1
Dn y1(t)

 

w(t) = _
Dn 1y(t)

we see from (1.1) that
n-l n
D t D t t
y1( ) y( ) y ( )

- + —
[Dn 1y(t)]2 Dn ly(t)

 

W'(t) + f(t.YT(t)) = 0.

22

Since Dn_lyl(t) and Dny(t) are positive for t > T1,
one has
y (t)
(2.12) w'(t)+B—I-l——— f(t,y (t)) s o, t > T1
D y(t) T
From (2.8) we obtain
n-1 n-1
[r(t) 2 N3(t T) D y,r (t), t > T2
n-l
where T 2 2T + T and N 1 = 2n 1 n (Mm 1 + j).
2 1 3 j=l
Since y’(t) > O for t > T1, there is a tO > T2 and a

C > 0 such that yT(t) 2 C for t 2 to. Moreover, by
Lemma 2.2 (a), there is a constant k > O and a tl > to
n-l

such that D yT(t) 2 k Dn-ly(t) for t 2 t1. Thus (2.10)

holds once again (with N replaced by N3), and the con—

1
clusion follows as before.

Corollary 2.1. Suppose for all constants C > O

Sf(t,C)dt = +2,
Then all solutions of (1.1) are oscillatory.

Corollary 2.2. Suppose p(t) > O and
S p(t)dt = + m .
Then all solutions of the equation

2y+1

(2.13) D“[r<t)D“y(t)] + p(t)yT

(t)=0,y20

are oscillatory.
Remark 1. Under the hypothesis 0 < m s r(t) s M, the only

nonoscillatory solutions of (1.1) are of types Bo'°°"Bn-l'

23

If y is nonoscillatory of type Aj for some j = O,...,n-l,

then y is nonoscillatory of type B for some k==O,...,n-l.

k
There are two cases: (i) If n is even and j s (n—2)/2 or
n is odd and j s (n-3)/2, then y, Dy,...,D23+1
D23+2

y are all
eventually positive and y is ultimately negative.
This is precisely the case if y is a Bj-solution. Hence

j = k; (ii) If n is even and j 2 n/2 or n is odd and

N

j (n-l)/2, then an Aj-solution is a Bk-solution for some

k = j,...,n-l.

If Dar: 8 = O,...,n—k, are positive and (-)3D(n_k+3)

r>O
for j = l,...,k-1,' then an Ak-solution is a Bk-solution.
In other words, if Dkr > O for k = O,...,j , j s n - l

and (-)kar > O for k = j + l,...,n-1, then an Aj-solu-

tion is a Bj-solution.

Remark 2. It is clear that the condition

dt
(H) r(t)

= +m
is sufficient to guarantee that all nonoscillatory solutions
of (1.1) are of type Bj for some j.

In view of the previous remarks, we may state without

proof the following results.

Theorem 2.2.
(a) Suppose n is even and k s (n-2)/2 or n is odd and

k s (n-3)/2. If, for all constants C > O,

a:
(2.14) S t2kf(t,C)dt = +2;

then (1.1) has no solutions of type Ak’

24

(b) Suppose n is even and k 2 n/2 or n is odd and
k 2 (n-1)/2. If Djr > 0, j = O,...,(n-k): (-)3Djr > O for
j = n-k+1,...,n—l and if, for all constants C > O, (2.14)

holds, then (1.1) has no solutions of type Ak'

Theorem 2.3.

(a) Suppose n is even and k s (n—2)/2 or n is odd and

k s (n-3)/2. If for some k = O,...,n-l

a
(2.15) S t2k p(t)dt = +o :
then (2.13) has no solutions of type Ak'

(b) Suppose n is even and k 2 n/2 or n is odd and
k 2 (n-1)/2. If Djr > 0, j = O,..., (n-k): (-)ijr > 0,
j = n-k+l,...,n-l; and if (2.15) holds, then (2.13) has no
solutions of type Ak'
Letting y = O and r(t) 2 l in (2.13), we obtain results

for the equation

D2“y(t> + p(t)yT(t) = o

analogous to Bradley's results [1] for the linear second

order equation
Y”(t) + p(t)yT(t) = 0.
An improvement can be made easily in the nonoscillation

criteria of Theorem 2.1. We state this as

Theorem 2.4. Equation (1.1) has no solution of type Bj if,

for all constants C > O,

Q . .
(2.16) S t23f(t,ct23)dt = +2.

25

Proof: Suppose y(t) is a solution of type Bj, where n

is even and j s (n-2)/2 or n is odd and j s (n-3)/2.

Since D23+1y(t) > O for t > T

23

. *
1, there 18 a t1 2 to > T2

* * *
and a C > 0 such that D yT(t) 2 C1 for t 2 t . Thus

1

(t) 2 N c (t—T)2j t 2 t*
YT II ' 1

so that (2.10) becomes
a ' * o
w’(t) + le (t-T)23f(t,N1cl(t—T)23) s o

*-
for t 2 t1. where we may assume that t1 2 t1. Letting
*
C = NIC1 and integrating from t1 to t, we must ultimate-
1

1y have w(t) < O which implies that Dn‘ y1 < O for large

t, ‘which is absurd.
In a similar manner we can modify the second and third

parts of the proof of Theorem 2.1 by observing that in the

2j-n+l

second part D yl(t) > O for t > T and in the third

1

*
part Dny(t) > O for t > T. Hence there are constants C

2!
* * * . 2j-n * , *
, t f
C3, t2 3 for which D y1(t) 2 C2 1 t 2 t2 and
n-1 * *

D y(t) 2 C3 if t 2 t3, respectively. Thus we have

* 2]“ *-
YT(t) 2 N2C2 (t T) , t 2 t2 or

t c* t- Zj t > t*
YT( ) 2 N3 3 ( T) , _ 3

so that (2.10) becomes
' * ' *
w’(t) + NiK(t-T)23 f(t,Nici(t-T)23) s o, t 2 ti

_ * .
where i = 2,3; taking C = Nici and using the hypotheSIS

if follows that in each case an integration from t1 to t

26

implies that w(t) < O for large t which contradicts the

fact that y is a Bj-solution.
We may restate Theorem 2.4 as

ore f

(a) Suppose that n is even and j s (n-2)/2 or n is odd
and j s (n-3)/2. If for all constants C > O (2.16) holds;

then either (1.1) is oscillatory or for sufficiently large

t. y Dzjy < o.
(b) Suppose that n is even and j 2 n/2 or n is odd and
j 2 (n-l)/2. If for all constants C > 0 (2.16) holds: then

either (1.1) is oscillatory or for sufficiently large t,

yDzJ-ny1 < O.

For j = n-1 and r(t) 2 1, (b) reduces to the alterna-

2 -2

tive that either (1.1) is oscillatory or yD n y < O which

is essentially Theorem 3.1 of Ladas [6].

Chapter 3. The Asymptotic Character of Certain Solutions

In this chapter some special results are given on the

asymptotic behavior of solutions of the equation

(3.1) D2“y(t) + f(t.yT(t))yT(t) = o.

where f(t,u) satisfies the three conditions in section one.

Lemma 3.1. Let y(t) be a solution of (3.1) which is even—

tually positive. Then

lim (2n-1):t‘(2n‘1) y(t) = lim D2“‘1y(t).

t-uo t-oo

Proof: Suppose that y(t) is a solution of (3.1) which is

eventually positive. Then there is a T1 > 0 such that y(t)

is positive for t > T1. Hence Y7(t) > O for t"T(t) > T

o * o
i.e., for t > T1 + T = T . Then by Taylor's Theorem with

1'

*
Remainder, for t > T

t
(Zn-1)! y(t) - S *(t-s)2n-1D2ny(s)ds

T

(3.2) (Zn-l)! R(t)

t _
(2n—1): y(t) + S *(t-s)“ lyT<s)f(s.yT(s»ds.

T
where
2n-l k
1 k * *
R(t) = E f.- Dy(T )(t-T)
k=O ’
Since yT(s) and hence — Dzny(s) are positive for s > T*,

O o o * 0
condition three together Wlth (t-T ) > (t-s) > 0 imply
that

27

28

(2n-l)!ElUfl s (2n-1)!y(t) + (t._T*)2n-1[D2n-l 2n-

y(T*)-Ia 1y(t)].

Dividing this by (t--Tm')2n-1 and noting that
'k — — 'k
lim (Zn-1)! (t-T )2“ 1120;) = n2“ 1 y(T ),
t-Dco
it follows upon passage to the limit that
.... * ... .—
(3.3) lim DZ“ 1y(t) 5 lim (2n-l)!(t-T ) (2“ Dy(t).
tau tic

We remark that this inequality could also be Obtained direct-
ly from Lemma 2.1 if y is assumed to be a solution of type

Bn_1.

To prove the reverse inequality, we choose an n for

'k
which T < n < t. By restricting s to lie in the inter-

val [T*,n], (t-s)2n-1 2 (t-n)2n“1 and

— n
(2n—1): R(t) 2 <2n-1):y<t) + (t-n)2n 1 S *yT<s)f<s.yT(s)>ds
T

2n

2n"1[D:’:’r1"11r'(T*) - D ’11! (n) J-

= (Zn-1)3Y(t) + (t-n)

Multiplying this by (t-T*)_(2n-1), keeping n fixed and

letting t-om through a sequence of points for which

-(2n-l)

*
(t-T ) y(t) tends to its upper limit, we have

D2n-1y(T*) 2 lim (Zn-1) s (t - T*)

tam

— (2""1)y(t) + D2n-1y(T*) - Dzmlym)

from which it follows that

TERI (Zn-l)! (t-T*)-(2n-1)y(t) s Dzn—l

tam

y(n).

2n

Since n is arbitrary and lim D _ly(t) exists,

tam

29

Zn-

(3.4) 135 (Zn-l)! (t-T*)‘(2“’1)y(t) s 1im D 1y(t).

tau tan

By combining (3.3) and (3.4) we obtain the desired result.

Theorem 3.1. Equation (3.1) has a solution y(t) > O sat-
isfying

(3.5) y(t) ~ kth-l , O < k

if, and only if, for all C > O

(3.6) S tzn-l f(t,Ct2n_1)dt < o .

Proof: First suppose that (3.6) holds. Choose To > 0 suf-
ficiently large so that

2n

3 tzn-l f(t,Ct ‘1)dt s (Zn-1)! - .
T

O

Nfld

Now consider the solution y(t) = y(t,To) of (3.1) subject to:

Dky(To) = O, k = O,1,...,n-; Dky1(To) = O, k = O,...,n-2;
provided n 2 2; Dn-1y1(To) = 1 and y(t) = O for To - T
s t s To; y(t,To) is positive on some open interval whose

left-hand endpoint is To' Let t = T1 be the first zero

of y(t,To) in (To,m). By Taylor's Theorem with Remainder

t _
l = (2n-1)!y(t,To) + S (t-s)2n lyT(s)f(S.yT(s)).

To

(3.7) (t-To)2“'

Since y(s) 2 O for T -T s s s T , y (s) 2 O for T -T
O 1 'r O
s s - T(s) s Tl' i.e. for s 2 To - T + T(S) and hence
yT(s) 2 O for s > To. A similar argument shows that xjs)20
for s < T1. Thus
1

(3.8) (2n-1):y(t) = (2n-1)!y(t,To) S (t-To)2n- , To<:t < Tl

30

Moreover, letting t = T1 in (3.7)

T

(Tl-To)2“‘1 = g 1 ”‘1‘ 902""1 y,r (s)f(s.y,r<s)>ds
TO
5 (T 'T )2n_1 T1 (S)f(s (3))ds
1 0 ST YT IYT .
0

By condition (iii) and (3.8),
u- -1 —
(2n-1) :yT<s)f<s.yT<s>) s (s—o(s)>2“ lus. <2n-1): (as-0(a))“ 1)

s SZn-l f (s, Cszn—l) ,

-l
where C = (Zn-l)! and 0(3) = w(s) + To .
Substituting this in the previous inequality, we obtain
T1 2n-l 2n-1 °° 2n
8 f(s,Cs )ds s S s

0 TO

_1f(s,Cszn-1

(Zn-1)! S )ds
This contradicts the initial choice of To and demonstrates
the existence of a positive nonoscillatory solution y(t).
The first half of Theorem 3.1 then follows from Lemma 3.1.

To prove the second assertion, suppose that (3.1) has a

positive solution y(t) satisfying (3.5% By Lemma 3.1,

lim Dzn-l y(t) = (2n-1)!k, so that

tau

(3.9) S yT(s)f(s,yT(s»ds = - S Dzny ds = Dzn-1y(T1)-(2n-1)U<.
T1 T1

The hypothesis (3.5) ensures that for e > 0 given there is

* . 2n-l . *
a T > T1 for which y(t) > (k-e)t prOVided t 2 T .
Hence, yT(t) 2 (k-E)(t..T)2n-l. By (ii) we have f(s,yT(s))
2 f(s,(k-e)(s-T)2n-1). Since (3.9) is valid with T1 re-

placed by T*,

31

t — ...
132“"1y('r*) - (Zn-l) 3k 2 (k- 98 *(s-T)2n 1f(s, (k-e) (s-T)2n 1mg.
T

For s-T 2 -2]=-s, i.e. for s 2 2T, we have
a)

Q
S 821.1 1f(s,Cszn-':l')ds s 22“ 1 S (s-T)2n lf(s,22n lC (s-T)2n—])ds

3 N1 ,

where

- 22n’1(k- e)‘1[92“‘1 y(T") - <2n—1):k].

...:
I

c = 21’2n (k-e).

and the lower endpoint of integration is not less than
max(T*,2T). This proves the theorem.

We remark that in the case n = 2 and T(t) E 0, these
results reduce to those of Leighton and Nehari [7] in the lin-

ear case and to those of WOng [11] in the nonlinear case.

Chapter 4. An Application of Lyapunov's Direct Method

In this chapter we use Lyapunov's second method to Ob-
tain nonoscillation criteria for the equation (1.1). We con-

sider the equivalent system:

Yk(t) = Dky(t), k = O,...,n-l:

(4.1) zk(t) Bk[r(t)Dn_ly(t)], k = O,...,n-l: and

Dz 1(t)

n_ - f(t.yT(t))yT(t).

To simplify notation we shall let n = (yo,...,yn_l) and

g = (20,...,zn_1). For the variables (t,n,g) we also de-
fine
R1 = R = (‘w,m);
RT = [T,oo), T 2 O;
*
R = (01”)?

R* = (-°°oo)7

pat * * ~k .
R = R x R x -°- xR , p times;
RP*= R* X R* x --- xR* , p times;
* *
R=RxR'

. * * _ _.
X R<23+1) x (R *)n 1 j X R:

* n—l—j

R(2j+l)* X (R*) X R0

X

333': “T

In the following a scalar function v of the variables

t,n,g will be called a Lyapunov function for (4.1) if it is
32

33

continuous in (t,n,§) in the domain of definition and is
locally Lipschitzian in (n,g). Following Yoshizawa [12],

‘we define

(402) {7(1) (tonog) =Ffi+ %{V(t+h,n(t+ h)IC(t+h)) -V(ttnog)}
40

Theorem 4.1. Suppose that there exist two continuous func-
tions V(t,n,g) and ‘W(t,n,g) which are defined on RT j
l

and ST j respectively for some fixed T. Assume further
I

that V(t,n,g) satisfy:

(i) Both V(t,n,g) and W(t,n,g) tend to infinity as

. * * _ _.
(23+l) x (R *)n l ]

tam uniformly for (n,g) in R x R or

* n-1-' .
R(2j+l)* x (R* ) J x R, respectively:

(ii) 0(1)(t,n,g) s O for all sufficiently large t, where

(n,g) is a solution of (4.1) which for large t lies in the

' *
(23+1) X

* _ _.
region R (R *)n 1 j x R: and -

(iii) W(1)(t,n,g) s 0 for all sufficiently large t, where

(n,g) is a solution of (4.1) which for large t lies in the

* n-l-j

region R(2j+1)* x (R* )

x R.

Then (1.1) has no solutions of type Bj‘

Proof: Let y(t) be a solution of (1.1) of type Bj’ Since
y(t) and y1(t) are positive for large t, there is a pos-
. . . . . (2j+1)*

itive To for which (n(t),g(t)) lies in R x R for

t 2 To’ By (ii), for t sufficiently large, i.e., for t 2

V(t.n(t).g(t)) < V(T1.n(T1).g<T1>).

34

On the other hand, condition (1) implies that there is a

T2 > T1 for which

v(tln(t)l€ (t)) > V(T10T](T1)Ig (T1))

for t 2 T2, ‘which is a contradiction.
By letting y(t) be a negative solution of (1.1) of type
Bj and considering W(t,n(t),g(t)), ‘we Obtain an analogous

contradiction.
Lemma 4.1. For (t,n(t),g(t)) 6 RT n-l assume that there
exists a Lyapunov function v(t,n(t),C(t)) satisfying:

(i) zn_1v(t,n,g) > 0:

(ii) v(1)(t,n,€) s — x(t), where x(t) is a continuous

function defined on RT such that

t
(4.3) lim x(s)ds 2 0
tea T ‘

*
for T 2 T sufficiently large.

Moreover, suppose there exist a T1 and a function

w(tvnvC) WhiCh for (t.n.g) in the region RT x R(Zn-l)*
l

x R*, is a Lyapunov function satisfying:

(iii) zn_1 s w(t,n,g) s b(zn-l);

where b(u) is a continuous function, b(O) = O and b(u) < O

for u # O; and

(iv) 6(1,<t.n(t).g(t)) s -p(t)W(t.n(t).g(t)).

where p(t) 2 O is a continuous function such that

t
(4.4) 8 exp [ - ST p(s)ds )dt = + m ,

35

If (n(t),g(t)) is a solution of (4.1) which lies in the

(Zn-1) *

region R x R for sufficiently large values of t, then

zn_1(t) 2 O for large t.

Proof: Suppose there is a sequence (ti) for which tk 4 m
as R40 and zn_1(tk) < 0. Assume tk 2 T* and that tk
is sufficiently large so that by (4.3),

t
lim_ A(s)ds 2 O, t 2 tk

t-wo t.k

and Y0 (t) p o o o 'yn_1(t)' 20 (t) O o o o ' 211-2 (t) are POSitj-ve'
were we assume n 2 2. For the case n = 1, see Yoshizawa
[2]. Consider the function v(t,n(t),g(t)) for t 2 tk .

t
(4.5) V(t.n(t).g(t)) s v(tk.n(tk).g(tk)) +Stkv(1)(sn(s).c(s))ds

t

S v(tkvn(tk)og(tk))-S X(S)d8.
t
k

Since zn-l(tk) < O, v(tk,n(tk),g(tk)) < 0, so there is a
T1 2 tk for which

t
Sth‘s’ds 2 %'V(tkvn(tk)'€<tk)) .

which implies that for t 2 T1,

VfimfiLUU)S%Nmmfifl4wfl)<m

By (i), zn_1(t) < O for t > T1. By (iii), there is a

T2 > T1 and a Lyapunov function w(t,n(t),g(t)) defined on

RT x R(2n-l)*
2

(iV)

x R*. For this w(t,n(t),g(t)) we have by

36

t

zn-l(t) s w(t.n(t).g(t)) s W(T2.n(T2).g(T2))eXP[-S p(s)ds] .
T
2

where t 2 T2 > T1. BY (iii).

t
zn_l(t) s b(zn_l(T2))exp[- ST p(s)ds] .
2

Substituting this into the above expression, one gets

11
zn_1(u) = [zn_2(u)]' s b(zn_1(T2))exp[- ST p(s)ds] .
‘ 2

Integrating from T2 to t, we arrive at

t t
zn_2(t) s zn_2(T2)-+b(zn_1(T2)) ST exp[- STp(s)ds]dt.
2 2

Letting t4» and using (4.4), it follows that zn_2(t) < O
for sufficiently large t, which is a contradiction.

By the same argument we can prove the following lemma.

Lemma 4.2. For (t,n(t),g(t)) E ST*n-l , assume that there

exists a Lyapunov function v(t,n(t),g(t)) satisfying:

(1) zn-l v(t,n,g) > O; and

(ii) v(l)(t,n,g) s -x(t), where 1(t) is a continuous fumr-

tion defined on RT* and for large T,

t
lim x(s)ds 2 O .
tam T

Moreover, assume that there exists a T1 and a function

. *
w(t,n,§) which for (t,n,g) in the region RT1 x R(Zn-l)*)(R

is a Lyapunov function satisfying

(iii) -zn_ s w(t,n,g) s b(z

1 n-1)'

37

‘where b(u) is a continuous function, b(O) = O and b(u)‘<0

for u # 0: and

(1V) w(l) (tan (1:) 1C (t)) S -9 (t)W(t,T‘| (t) I C (t) ) o

where p(t) 2 O is a continuous function for which

S

If (n(t),g(t)) is a solution of (4.1) which lies in the

t
exp [- ST p(s)ds]dt = +o .

region R(2n-l)* x R for sufficiently large values of t, then

zn_1(t) s O for large t.

Remark 1: Since 0 < m s r(t) s M, condition (4.4) is equi-
valent to

t
(4.6) S ;%t) {exp[- ST p(s)ds]}dt.
1

To see this we merely note that

t 1

MS r(u) exp[-S:p(s)ds]du 2 St

 

u t u
exp[-STp (s)ds]du 2 mg filmexfi-S'Ipadsfiu.

In the case n = l, we have v(t,n,g) = v(t,y,y’) since y==z
and 21 = y’. Condition (4.6) arises naturally in the proof

of Lemma 4.1.

Remark 2: Suppose we let p(t) a o in each of the two lem-
mas. Condition (4.4) is then trivially valid, and the alter-

native condition (4.6) reduces to

(4.7) S tit) = +c .

 

Thus, we may replace condition (iv) by w(1)(t,q,g):so and ob-

tain two easy corollaries whose statements are left to the reader.

38

Remark 3: Let r(t) a l and f(t,yT(t)) be nonnegative.
As already noted, solutions of type B1 are solutions of

type A The lemma asserts that a solution y(t) for which

1'
Dky(t) > O, k = O,l,...,2n-2 must satisfy D2n-1y(t) > O,
i.e., y(t) must be a solution of type An-l’ But this is

obvious from Kiguradze's lemma.

Theggem 42;. Suppose there are continuous functions a(t),

b(t), a(zn_2) and 6(zn_2) satisfying:

a) For large T,

t t
lim a(s)ds 2 0, lim 8 b(s)ds 2 O:
tam T tam T
I _ __£L_.
b) zn_2a(zn_2) > O. a (zn_2) — dzn-Z [a(zn_2)] 2 0:

where yk (k = O,...,n—l) and zk(k = O,...,n-2) are non-

negative for large t,

d
n-2

 

zn-2 3(Zn_2) > 0' B’(zn-Z) = dz [5(Zn_2)] 2 0'

where yk (k = O,...,n-l) and z (k = O,...,n-2) are non-

k
positive for large t; and

L‘\

N

c) a(t) “(Zn-2) f(t.yT(t))yT(t) for large t. y 0.

N

b(t) B(zn_2) f(t,yT(t))yT(t) for large t, y S 0.

If (n(t),g(t)) is a solution of (4.1) which for large t

*
lies in the region R(2n 1) x

IR: then zn-l(t) 2 O for
large t. If (n(t),g(t)) is a solution of (4.1) which for
large t lies in the region R(2n-l)* x R; then zn_1(t) s O

for large t.

39

Pr of: Let x(t) = a(t) or b(t), p(t)

O and define
V(t.n.C) and ‘W(t.n.g) by
zn-l(t)
V(t:n(t)oC(t)) = a[z 2(t)]
n-

 

and
t
w(t.n(t).g(t)) = zn_1(t) + a[2n_2<t>] ST a(s)ds.

Conditions (i), (ii), and (iii) of Lemma 4.1 hold. In par-

 

ticular,
. zi-l(t) .

(i) zn-l v(t,n,§) = ETE;:;TETT' > 0 since zn_2 > 0.
(ii) v (t g) = —L—' {az’ - 22 a’ (Z ) i
1 (1) '“' a2(zn_2 n-1 n-1 n-2

S 23-1
“(Zn-2) '
by (b). Using (1.1)
. -f(t.y7(t))yT(t)
V(1)(t,n(t),g(t)) s “[zn-z (t) 1 s a (t) .

(ii)2 lim t x(s)ds = lim t a(s)ds 2 0,

tea T tau T

for large t by (a).
t
(iii) zn_l s w(t,n,g) s zn_l + a(zn_2) STa(s)ds,

' z 2 0.
Since n-2

Also

t t t
a(zn_2) S a(s)ds 5 S a(zn_2) a(s)ds SS f(s,yT(s))ds
T T T

4O

i.e., w(t,n(t),g(t)) s: zn-l(t) + [zn_1(T)-zn_l(t)]=zn_1(T).

So we may let the function b(u) of Lemma 4.1 to be the con—

stant function b(u) = - zn_l(T).
_ t
(iv) w(l)(t,n,g) = zn_’_1(t)+a(t)a(zn_2(t))+ f8;(s)ds]a(zn_2(t))zn_l(t) < o

for large t by (a) provided zn-l is assumed negative.

Moreover, suppose we let

 

( ( ) ( )) zn’1(t) d
v t,n t ,g t = ——‘ an
B[2n_2(t)]
t
w(t.n<t).g<t)) = — zn—l(t) — B[zn_2(t)] g b(s)ds.
T

Similar routine computations show that v and w satisfy

the three conditions of Lemma 4.2.

Theorem 4.3. Suppose that, in addition to the hypotheses of

Theorem 4.2,
S a(s)ds = S b(s)ds = + m

Then (1.1) has no solutions of type Bn-l'

Proof: Suppose we define

z (t) t
n-l
a[zn_2(t)] + S a(s)ds , y 2 O
V(tIT](t)r€(t)) at
a(s)ds , y < O
)0

 

zn-l(t)

B[zn_2(t)]

 

t
+ S b(s)ds , y < O
O

W(t.n(t).g(t)) t
§b(s)ds, y2O.
‘o

41

Assume that y(t) is a solution of (1.1) of type Bn-l'
Then for large t, yk (k = O,...,n-l) and 2k (k==0,...,n-l)
are positive

t

V(t,n(t),g(t)) 2 So a(s)ds and

t
W(t,n(t),g(t)) 2 S b(s)ds .
0

Thus V and W both tend to infinity as t 4 a uniformly.

Next,

' n-l I

Vtufit,n(t),g(t» =[;;a;:;T-] + a(t) s -a(t) + a(t) = O :
. _ Z _1 ' _ -
w(1,(t.n<t).g<t»- r5727] + b(t) s b(t) + b(t) - 0.

Hence V and W satisfy the three conditions of Theorem 4.1,

and the proof is complete.

Iheerem 4,4. Suppose that there are continuous functions
a(t). b(t). a(y) and a(y) satisfying:

a) S a(s)ds = Sm b(s)ds ;

b) ya(y) > O, a'(y) 2 0, where y and y' are nonnegative
for large t:
yB(y) > O, B'(y) 2 0, where y and y' are nonpositive

for large t; and
c) a(t) a(y) s f(t.yT(t))yT(t) .
b(t) am >. f(t.yT<t))yT(t)

Then (1.1) has no solutions of types B0,...,B

42

Proof:
Let
2 (t) t
_£:JL_.
a(y(t)) + S a(s)ds , y 2 O
V(t.n(t).g(t)) = 2
S a(s)ds , y < O
O

t
S b(s)ds , y > O
O

wct.n(t).g<t)) .
zn-l(t)

t

V and ‘W will then satisfy the three conditions of Theorem
4.1. The details are omitted.

We observe that Theorem 4.4 is only one of a sequence
of similar results. Suppose that n is even and j=S(n-2)/2
or n is odd and j s (n-3)/2. Then we may replace y by
yzj in conditions (b) and (c) and require DFy, k = 0,1,...,
2j+l to have the usual signs. Similarly, if n is odd and
j = 23;; ‘we may replace y by z in (b) and (c). Finally,

if n is even and j 2 n/2 or n is odd and j 2 (n+l)/2,

'we replace y by z in (b) and (c). In each case we con-

23
clude that (1.1) has no solutions of type Bj""'Bn-l'

Theorem 4, . Let p(t) > 0

.Q '
If S t23 p(t)dt = + 2 ;

then there are no solutions of

43

(4.8) D“[r(t)n“y<t)] + p(t)yT<t) = o

of type Bj'

Proef: Let y(t) be a solution of type Bj‘ For equation

(4.8).
f(t.yT(t>)yT(t) = p(t)yT(t) 2 u(t-T)2j 2(t-T) .
223. (t - T)

depending on whether (i) n is even and j s (n-2)/2 or n
is odd and j s (n-3)/2: (ii) n is odd and j = (n-l)/2:

(iii) n is even and j 2 n/2 or n is odd and j 2 (n+l)/2.
We note that u is a known constant (determined in section

two) once case (i), (ii) or (iii) is prescribed. ‘We let
x(t) = a(t) = b(t) = th p(t)

With the choices of V and W 'prescribed by Theorem 4.4
and the remarks following it, it follows that (4.8) has no

solutions of type Bj'

Chapter 5. A More General Delay Differential Equation

Throughout this section vectors in R2n will be denoted

by lower case Greek letters and scalars by lower case Latin

letters. To facilitate the discussion we shall also adopt
the following notation:

u = (xl'x2’ooo'x2n) ’
xkx’k "k

71" (Ti'l(t)oTi'2(t-)o"‘:Ti'2n(t)) 7

(3.11:2. :19.

“k:

t€2n = (t,t,---,t) , 2n times: and

u = (Y I°°'lyn_llzol"’ozn_1)

o
For the vector 0 = (51,...,32n) we shall form the composites:
u(o) = [x1(sl(t)),...,x2n(82n(t))]; and

'k
u (0)

[Yo(81(t) ). . . ..yn_1(sn(t)) .zo(sn+1(t)). . . . .zn_1 (8211(t» )-

The purpose of this section is to present conditions for the
nonexistence of certain types of nonoscillatory solutions of
the even order delay equation:
n n N *
(5.1) D (r(tm y](t) + 23 fi(t)Fifu (t62n-Ti(t))] = 0.
i=1

where O < m s r(t) S M and the delays Ti k(t) satisfy

0 s Ti k(t) S T. It will be assumed throughout that:
I

44

45

(i) fi(t) 2 O: fi(t) and Fi(u) are continuous functions
of the variables t and g respectively:

di) sgn Fim) = sgn x1,

Fi(-u) = -Fi(u): and

afi) r34u> # o if u 2 o.

Lemmas 2.1 and 2.2 are still valid for equation (5.1), as are
certain analogues of the theorems in sections two through
four. However, somewhat different hypotheses will be con-

sidered here.

Theorem 5.1. Suppose there is an index j (l s j s N) and

some q 2 O ‘which for all u E R2n and for all c 6 R sat-
isfy:

(5.2) Fj(cu) 2 c2q+1Fj(u) and

(5.3) S“ t2kfj(t)dt = + 2

for some integer k = 0,1,...,n-l. Then (5.1) has no solu-

tions of type Bk’

Proof: Let y(t) be a solution of type B First suppose

k.
that n is even and k s (n—2)/2 or that n is odd and

k s (n-3)/2. Define

wk<t> = [D2ky(t>1‘1n“‘lzo(t).

Then we see from (5.1) that

(5.4) wk'(t) = [D2ky(t)]‘1D"zo(t)-rnzkwt)J‘2D"’120(t)3k+1y(t).

46

There is a T1 2 0 such that Dsy(t) > o (s = o,...,2k+l)

for t > T1. Beginning with D2k+2y(t) the various deriva-
tives of y and z alternate in sign. Hence yT (t) > 0
1,1

for t - Ti'1(t) > T1, i.e., for t > T1 + T. Thus, for

t > T1
w ’(t) < [D2k (tH'anz (t) - - 2% [1321‘ (t) 1'1f «waft - (t))]
k Y o ‘ i=1 Y i i (5m Ti °
For t > T1 + T,

(5.5) Wk’(t) < -fj(t)[D2kY(t)]-1Fj[u*(t€2n - Tj(t))]-

Since y’(t) is positive on (T1,w), y(t) is an increasing

function for t > T1. Thus, for t > T1 + T

(5.6) y (t) 2y ('r +T).
T3.1 HA 1

which implies that

2q 2q
-fy (12) s - [Y (T + T) .
"’j.1 ] T:‘n.1 1 1

Using this and (5.2), one gets

“2' (t) s -fj (t)[D21‘y (t>1‘1 [yTj 1m 12‘?” F]. In; (tezn‘Tj (t))]
(5‘7) 2k -1 q *-
s -fj<t)[v y(tm YT]. 1(t)[yTj 1<T1+Tn2 th1(t62n-Tj>)].

... * _ . o .
By (iii), Fj[u1(t€2n Tj(t))] is positive for t > T1 + T

and does not tend to zero as t a a because of (i). Thus

1

* .
Fj[u1(t62n-Tj(t))] 2 kj,l > 0 if t > T2. Moreover, by

Lemma 2.2 there is a constant kj 2 > O and a T3 2 T2 for

there is a kj 1 > O and a T* 2 T + T for which
I

47

which y.r (t) 2 kj'2y(t) . Hence, for t > T3,

j,l

I -
(5.8) wk(t) S kj,lkj,2[y'r.

1(T1+T)]2qy(t)fD2ky(t)1'1fj(t).
3:

By Lemma 2.1, t2kD2ky(t) s 22k(2k+1)! y(t) for t 2 2T1.

For T* = max (2T1.T3), we have

2k

I -
(5.9) wk 5 klt fj(t)
_ 2q -2k ,-1
where k1 — kj,lkj,2[y7j 1(T1 + T)] 2 (2k+l). .
Integrating (5.9) from T* to a,
0
lim wk(t)-wk(T*) s - K1 S t2kf.(t)dt = - .
t-u- T* 3

Noting that o s lim wk(t), it follows that wk(T*) = ., which
11-00
is absurd since Dn‘lzo(T*) > O and D2ky(T*) # 0.

Now suppose n is even and k 2 n/2 or n is odd and

k 2 (n+l)/2. Define

'wk(t) = [DZk-nzo(t)]-1Dn_lz (t).

0

Equation (5.4) becomes:

2k-n 2k-n+g

wk' (t) = [D o

20 (t) 1"113’2o (t) - [92k'"z° (t)f215“1.2o(t) D (t) .

2ky<t>

zo(t): (5.6) remains unchanged. By Lemma

Equations (5.5), (5.7) and (5.8) remain valid with D

replaced by D2k-n

2.1, we have for t 2 T*

n-l
t2kD2k “2°(t) s 22k(2k-n)£ M n [(2k-n)Mm-1

J=1

+ j]y(t).

*
Thus, for t 2 T

, _ 2k
wk (t) 5 k2 t fj(t).

48

where

-1
_ 2q -2k _ -1 -1n _ -1 .
k2 _. kj'lkj'2[yT. (T1+T) 2 (2k n): M .11 [(2k n)Mm + 1]

3,1 1:1

For the case that n is odd and k = (n-l)/2, we de-

fine:

-1

wk(t) = [02ky(t)]’1 D“ zo(t).

The only change in the proof is that by Lemma 2.1,

n-l
t2kD2ky(t) s 22k M n (Mm 1 + i).

i=1
The rest of the arguments proceed as before and the theorem
is proved.
The following results are obtained easily upon consid-

ering more carefully the proof of Theorem 5.1.

Qorollagy 5.1. Under the hypotheses of Theorem 5.1, equation

(5.1) has no solutions of type BS (3 = k,...,n-l).

gorollagy 5.2. Suppose, in addition to (i), (ii), (iii) and

(5.2), there is some integer k = O,l,...,2n-2 for which
Q
Stkfj(t)dt = + s.

k 1
P;L_

Then (5.1) has no solutions of type BS (3 = ],...,n-l).

Corollagy 5.3. Suppose, in addition to (i), (ii) and (5.2),
there is some integer k = l,...,2n-l and some j = 1,...,N

for which Fj(u) # 0 if xk % O and

S” fj(t)dt = + m .

Then (5.1) has no solutions of type 83' s = [g],...,n-l.

-1

49

Denote by I1 the set of indices i (l S i s N) for
which Fi(u) is nondecreasing with respect to x. for each

j (l s j 3 2n). Let I2 k denote the set of indices i
(1 s i s N) for which xk1 Fi k(xk) is nonincreasing with
respect to xk, where Fi,k(xk) is the function obtained
from Fi(u) by setting xj = O for all j # k. Finally'let

Il,k = I1 0 12,k' In terms of these notions we may give a

different type of nonoscillation criterion.

Theorem 522. Let (5.1) satisfy, in addition to (i), (ii) and

(iii), the following conditions:

(iv) Il k # ¢ for some k = l,...,2n-l:

(v) there is a nonnegative function @(t) such that for
all c 2 l,
(D
S [@(t) Z c‘1t1"’2“ti(t)r'i R(tzn-k) - Pk (t) (¢'(t))2)dt = + .. ,
1611 k '
where
P;1(t) = 4Nk(t - T)2“-k-1 @(t) and
_ n-k-l _
22n-kln.'M1'I (an1+j),1sksn
-1 j=1
Nk. — _
22n-k (2n-k):, k 2 n + l, k=k+l.

Then (5.1) has no solutions of type Bn-l'

Proof: Suppose y(t) is a solution of type Bn-l' If lSksn,

let

50
w(t) = - @(t)n“‘1zo<t)/Dk‘1y<t-T)

Using previous notation, z(t) < O for t > T1 + T. A simple

computation shows that
w'<t) = - @(t)D“zo(t)/Dk‘1y<t-T) — @‘(t)D“‘1zo(t)/D1"1y(t)
+ Q(t)Dn-lzo(t)Dky(t-T)/[Dk.1y(t-T)]2 .

We note that since Dnzo(t) < O,

k 2n-k-l n-l 2n-k-l n-l
D y(t T) 2 Nk(t T) D zo(t-T) 2 Nk(t-T) D zo(t),
where
_ _‘ n-k-l _
Nk1= 221.1 k (n-l)! Mrlfl (an 1 + j)
j=1

substituting for Dnzo(t) from equation (5.1) and using (i),

(ii) and (iv), one finds

(5.10) w'<t)2¢>(t) 23 fi(t)rnk'1y(t-T)]'1Fi (Dk'1y(t—T))+ a(t).

. ,k
léIl'k
‘where

a(t) = Nk(t-T)2“'k’1s'1(t) (z(t))2 + 6’1(t)4>"(t)z(t).

Completing the square as suggested by the last two terms, we
have

(5.11)
a(t) = uké'1<t)(t—T)2“‘k’1r22(t)+4¢(t>é'<t)Pk(t>z<t)]

Nké‘1(t)<t-T)2“'k’1i[z(n+2: (tmxtwkuz) 12-4¢2(t)<§’(t»’g3<t)1

4uk ¢<t)(t-T)2“‘k‘1pfi(t>(¢'<t))2

N

Pk(t)<¢‘<t))2 .

51

Moreover, since Dnzo(t) < 0, there is a constant c1 2 l

and a t1 2 T1 + T for which

(5.12) Dk-1y(t) s clth-k , t 2 t1 .

We then have the chain of inequalities

[ k’]. 1"]. “ltk-an

D Y(t-T)]'1Fi'k(D y(t—T)) 2 c1 t2n-k

i,k (C1 )

(5.13)

-l k-2n 2n-k
2 c1 t Fi'k(t ).

‘where the first inequality follows because of (5.12) and the

fact that i E I and the second because i 6 I1 and

2,k

c 2 l.

l
Combining (5.10), (5.11) and (5.13), we have

2n-k

(5.14) w’ (t) 2 @(t) 23 o'1'1t1"1’“fi (t)Fi'k(t -Pk(t)(§’(t))2.

1611,k
Integrating this from t1 to t and using (v), it follows
that w(t) is positive for sufficiently large t, which is
a contradiction since ‘w(t) < O for t > T1 + T.

Now suppose k 2 n + 1, then k - l 2 n, and we let
w(t) = - é(t)Dn‘lzo(t)/Dk_n~lzo(t-T).

Since y is of type Bn-l' ‘we have ‘w(t) < 0 for t:>T14-T.

n-l

k-n-l
D 20(t-T)

n I 11"]. k-
w’(t)==- @(t)D zo(t)/ zo(t-T)-§ (t)D zo(t)D

1

+ §(t)Dn- zo(t)Dk-nqgt‘qubk-n-lzo(t‘T)12 -

Substituting for Dnzo(t) from (5.1) EDd using (i), (ii) and

(iv), we obtain

52

w’(t):=§(t) z) fi(t)[nk‘"‘1zo(t-T)]‘1F. nk‘“‘1zo(t-T))

(
. i,k
lEIl'k
+ Q (t) ,

2n—k-l

where a(t) 2 -[4N;(t-T) §(t)]_1({>1(t))2 as in (5.11)

with N; = [22”.k (Zn-k) !]-l(5 . 12) then becomes
k-n-l * 2n-k ‘ *
D zo(t) 5 c1 t , t 2 t1 ,

*
where c1 2 and t: 2 T1 + T. The inequalities of (5.13) now

become
k-n-l _ -l k-n-l _ * -l k-2n * 2n—k
[D 20(t T)] Fi,k(D zot T)) 2 (c1) t Fi,k(c1t )
* -l k-2n 2n-k

Thus (5.14) remains valid with c1 replaced by c; and Nk

*

replaced by Ni. An integration from t1 to t results in

the same contradiction as before.

gemegk: By requiring, instead of (ii), that Fi(cu)==cFi(u),
we may assume in the proof of Theorem 5.2 that c1 = l (or
c; = l) and take c = l/t to obtain as a trivial corollary
integral criteria independent of the parameter c and thus

easier to apply fOr a specific verification.

Corollary 5.4. Let (5.1) satisfy, in addition to (i), (iii)

and (iv), the following conditions

(ii)’ Fi(cu) = cFi(u) , i E Il,k7 and

(v) there is a nonnegative function §(t) for which

53

2n-k-1

SIMt) 2 fi<t)Fi,k(1H4Nk‘t‘T) Mt) 1‘1(¢’<t))2)dt=+n

ieILk
Then (5.1) has no solutions of type Bn-l‘
If we denote by 12+k the set of indices i (l s i s N)
. -1 . . . .
for which xk Fi,k(xk) is nondecreasing with respect to xk

+

+ .
for l s k s 2n-l and let Il,k = I1 n 12,k' we may modify

the estimates of Theorem 5.2 and state the following result.

Theorem 5.3. Let (5.1) satisfy, in addition to (i), (ii)

and (iii), the following conditions:

(iv) Il+k # ¢ for some k = l,...,2n-l :

(v) there is a nonnegative function §(t) such that for
all c 2 l and for all d > 0 ,

-1tk-2n 2n-k-1

3 (Ht) 2 + c )-Pk(t) (¢’<t)>2ldt=+s.

1611,k

Then (5.1) has no solutions of type Bn-l .

fi(t)Fi,k(dt

Proof: Suppose y(t) is a solution of (5.1) of type Bn-l'

If 1 s k s n, let
‘w(t) = - ¢(t)n“‘1zo(t)/nk'§gt-T).
As in Theorem 5.2, (5.10) and (5.11) imply that
w’ (t) 2 e (t). 23 + :51 (t) [Dk'1y (t-T) 1‘1Fi k(131°"1y(1:-'r)) -
leIl,k '

Pk (t) @‘1 (t) (c: ’ (t) ) 2.

where P;1(t) = 4Nk(t-T)2n-k_1§. As before, there is a

c2 2 max (N1.....Nn)=ll and a t2 2 T1 + T such that

54

n-2
D 20(t) S c2(t T), t 2 t2

Since i E Il+k' 'we have the following chain of inequalities

-l 2n-k- n-
FiLk[N (t-T) 1D tow-T)]

 

 

k-l -l k-l
[D y(t-T)] F- (D y(t-T))z .. - _ _.
i,k N f(t-T)2n k an 220(t-T)
F. [N-1(t-'l)2n-k-1Dn-zz (t -'r)]
i k 0 2
(5.16) 2 —_22 _1 1‘2n-k'
c2N (t-T)
= c-1(t-T)k-2nF. [d(t-T)2n-k-1]a
i,k
where c = c N“1 2 l and d = Nlen-zz (t -T).
2 o 2
For n + l s k S 2n-l, let
‘w(t) = -¢(t)Dn-lzo(t)/Dk‘n-lzo(t-T)
Since (5.16) holds with DF-1y(t-T) replaced by k"n-lzow-T),

the result now follows upon integration from t2 to t as

in Theorem 5.2.

Remark: Since the criteria in Theorem 5.3 depends on two pa-
rameters c and d, it seems difficult to apply. Moreover,
the discrepancy in the power of t in the Fi,k term gives
rise to a weaker test for oscillation. Application of Theorem

5.3 to the equation

(5.17) [r(t)y”(t)]” + p(t)yT(t) = 0

shows that a proper choice of §(t) results in criteria
which agrees to a large extent with previous results. Here
2n = 4, f1(t) = p(t) and F1(u) = x1. Conditions (i), (ii)

and (iii) are clearly valid; xllFl 1(x1) = 1 'which is

55

. . . . 3-6
triVially nondecreasing. Letting 9(t) = t . 0 < 6 S 2.

(v) becomes

a _ _ 2
S[Ct2 6P(t) —fii%]dt=+u.

For 6 > 0, this is equivalent to
G 2-6
(5.18) S t p(t)dt = + ..

Thus, if (5.18) holds for any 0 < 6 S 2, there are no solu-
tions of (5.17) of type B1.
Theorem 5.4. Let (5.1) satisfy, in addition to (i), (ii)
and (iii) the following conditions:

(iv) I1+k # ¢ for some k = l,...,2n-2:

(v) there is a nonnegative function 6(t) which for all

c > 0 satisfies

Sum Z '1tFk+1"2“ . k(ot2“"k‘1) - Pk (t) (2’ (t))2]dt = + ..
ie11,1: '

Then (5.1) has no solutions of type Bn-l'

Proof: It is sufficient to note that

 

-k-l -
[k-l ”-1 k- » F1 k[N1--(t -T)2“D n“ 2 zo(t 24)]
D y(t-T F2 (D 1y(t- T 2-—#__
l k+l-2n 2n-k-l
(t-T) Fi'k[co(t T) ]
-l tk+l- -2n 2n-k-l
2C Fi,k(Ct )0
k+l-2n k+l-2n -l n-2

where c = 2 co = 2 N D zo(t2-T) > O .

56

Remark: Theorem 5.4 corrects the inadequacy of Theorem 5.3.

If we consider the equation (5.17) again and apply Theorem

5.4 with @(t) = t2, (v) becomes

 

Q

S [t2p(t) - 1 2 Jdt = + a .
which is equivalent to

'2
(5.19) S t p(t)dt = + a .

Hence (5.19) implies the nonexistence of solutions of (5.17)
of type Bn-l This criterion was already established in

section two for the simpler equation (1.1).

LIST OF REFERENCES

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

LIST OF REFERENCES

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403.

F. Burkowski, Oscillation theorems for a second order
nonlinear functional differential equation, J. Math.
Anal. Appl. 33 (1971), 258-262.

L. E. El'sgol'ts, Introduction to the theory of differen-
tial equations with deviating arguments, Holden-Day, San
Francisco, 1966.

H. E. Gollwitzer, 0n nonlinear oscillations for a sec-
ond order delay equation, J. Math. Anal. Appl. 26
(1969), 285-289.

I. T. Kiguradze, Oscillation properties of solutions of
certain ordinary differential equations (Russian), Dokl.
Akad. Nauk SSR 144 (1962), 33-36; translated as Soviet
Math. Dokl. 3 (1962), 649-652.

G. Ladas, Oscillation and asymptotic behavior of solu-
tions of differential equations with retarded arguments,
J. Differential Equations 10 (1971), 281-290.

W. Leighton and Z. Nehari, 0n the oscillation of solu-
tions of self adjoint linear differential equations of
fourth order, Trans. Amer. Math. Soc. 89 (1958), 325-
377.

S. B. Norkin, Differential equations of the second order
with retarded argument, Some prOblems of the theory of
vibrations of systems with retardation, Translations of
Mathematical Monographs, Amer. Math. Soc., Vol. 31, 1972,
285 pp.

V. A. Staikos and A. G. Petsoulas, Some oscillation cri-

teria for second order nonlinear delay differential equ-
ations, J. Math. Anal. Appl. 30 (1970), 695-701.

57

[10]

[11]

[12]

58

James S. W. Wong, Second order oscillation with re-
tarded arguments, Proc. Conference on Ordinary Differ-
ential Equations, June 14-23, 1971, edited by Leonard
Weiss, Academic Press, New YOrk, 1972, 581-596. '

P. K. Wong, On a class of nonlinear fourth order dif-
ferential equations, Annali di Matematica pura ed
applicata 81 (1969» 331-346.

T. Yoshizawa, Oscillatory property of solutions of sec-
ond order differential equations, Tohoku Math. J._22
(1970), 619-634.

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