Oscillation theorems for second-order functional differential equations

equations

Citation for published version (APA):

Brands, J. J. A. M. (1977). Oscillation theorems for second-order functional differential equations. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7712). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977

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,

TECHNOLOGICAL UNIVERSITY EINDHOVEN

Department of Mathematics

Memorandum 1977-12 august 1977

Oscillation theorems for second-order functional differential eqhations

by J.J.A.M. Brands Technological University Department of Mathematics PO BOX 513, Eindhoven The Netherlands

J.J.A.M. Brands

Technological University, Eindhoven, The Netherlands

ABSTRACT

This paper presents some comparison theorems on the oscillatory behaviour of solutions of second-order functional differential equations.

Here we state one of the main results in a simplified form: Let q, t₁,t₂

be nonnegative continuous functions on (O,C(» such that t 1 - t

_z

is a

bounded function on [l,~) and t - tl (t) ~ 00 if t ~~. Then

yet)

+ q(t)y(t tl(t»

=

0 is oscillatory if and only if

yet)

+ q(t)y(t t

2(t» = 0 is oscillatory.

1. INTRODUCTION

Throughout C(A) is the set C+ and C

t are

to ₀

this paper we use the following notations and abbreviations:

of all continuous mappings of A into :R, C =C«to'oo»,

to

the sets of all nonnegative and all nonpositive functions in C

t ' respectively. Constant functions are simply denoted by their value.

By Y Se denote the set consisting of all functions y E Co with the property

that jyj is positive and concave (and hence monotonically non-decreasing)

on (0,"") and

I

y( t)

I

= o( t), (t ~ ""). We recall that a func tio n y E

Co

is called concave if for all a E [O,lJ and all positive numberst

l, t2

we have y(at} + (I - a)t

Z) ~ ay(tl) + (I - a)y(tZ)' By a deZay we mean

a function T €

C~

with the property that t - t(t)

~ ~

if t

~ ~.

REMARK. With obvious modifications our considerations will remain valid if

OSCILLATION OF DELAY-DIFFERENTIAL EQUATIONS

- 2

-The goal of this paper is to present comparison theorems about the oscillatory behaviour of solutions of second-order functional differential equations. In order to give a clear idea of our method we postpone the general case to section 7 and first treat equations

yet) + q(t)y(t - T(t» • 0 ,

+

where q € Co and T is a delay. We shall refer to this equation by the symbol D

q,T

By a

soLution

of D we mean a continuous function y on some interval q,T

(to'oo) where to ~ 0, with the property that there is a number t l, tl ~ to' such that t - T(t) > to if t > tl and such that y is twice continuously differentiable for t > t

l, satisfying

D

q,T for t equation D is called o8cilZato~y if every solution of D

q,T q,T

> t

l • An has

ar-bitrarily large (positive) zeros. A solution y of D is called nont~iviaZ

q,T

if for every r > 0 there is a number t, t > r, such that yet) ~ o. A solution y of D is called a

positive (negative)

solution if there is

q,T

a number to' such that yet) > 0

«

0) for t > to.

The oscillatory behaviour of special cases of second-order functional differential equations has been already the subject of several papers

([lJ, [3], [5J, [6J, [8J, [9J). The main results of the present paper are of a different nature and seem to be new.

2. MAIN RESULTS

In section 4 we give a proof of the following +

THEOREM I.

Let

ql,q2

be functions in

CO'

and Let

T

1,T2

be delays.

Suppose that for

eve~y

positive

y € Y

there are nonnegative constants

M1,M2,

such that

4

for aZl

t

sufficientZy Zarge

4

the following two

yet) - M2 > 0 ,

Then we have: if

D

is oscillatory then

D

is oscillatory.

ql,t₁ q2,T₂

Several interesting consequences of theorem I are treated in

section 5. In section 6 we shall show that a condition of the form

DO

f

eut2q(t)dt < "", with u < 2,

never guanantees that all nontrivial solutions of

y(

t) + q( t)y( t - 1) .. 0

(with q € C~), are nonoscillatory. This result shows that theorem 6

of [6J is incorrect.

Finally, in section 7, we generalize the results of sections 3,4,5

to larger classes of second-order functional differential equations.

3. PRELIMINARIES

In the sequel we need the following lemmas

+

LEMMA 1.

Let

q € Co and

let

T

be a delay. Suppose that there is a positi7)e

funotion

y €

C~o'

whioh is differentiabZe lor

t > t I

(where

to

~

0 and t 1

is

suoh that

t - T(t) > to

for

t >

tl),satisfying

co

yet)

~

f

q(8)y(S - T(s»ds,

t

Then

D

has a positive

soZution.v~

whiah moreover satisfies

q,T

vet) !> yet), _(t > to) and vet) ~ 0, (t ~ 00) •

We have 4 -Vn+1(t) = yet), =

J

t ""

""

q(s)v (s - t(s»ds _n

v₂(t)

=

f

q(s)y(s - T(s»ds

~

yet) .. vI(t) ,

t

nOlI 1,2, ••• ) .

It follows that v

2(t) ~ vIet) for t > tl, whence

It ""

f

q(s)v₂(s - T(s»ds < "",

t

easily follows by induction that:

vn+,(t) ~ v (t) J ,( t > _t J ' n == n 0 :s _vn+1(t) :s vn(t)

,

(t > _{t 1 '} 0 ~ _vn+1(t) ~

v

(t)

,

(t > t l , n 1,2, ••• ) n =: 1,2, ••• ) n .. 2,3,. •• )

We conclude that vn tends to a positive limit function v and that vn

tends to a limit function w. Clearly vet) ~ y(t) , (t > to) and

vet) ~ y(t_l) , (t ~ t

I). It is now a routine matter to prove that

v -

w,

and (by Lebesque's theorem on dominated convergence) that

""

vet)

=

J

q(s)v(s - T(s»ds ,

t

Hence v'is a positive solution of D Obviously vet) ~ 0 (t ~ (0) •

REMARK. Clearly lemma 1 holds for a larger class of L's,namely

LEMMA 2.

Let

to

be a nonnegative number. Let

u

be a positive

and

concave

function in

C

t

suah that

u(t) - oCt), (t + =).

Then there is a number

t

l, tl > to'

2nd a positive funation

y E Y

suoh that

yet)

=

u(t), (t 2 t1),

-I -1

and moreover

3

if

t3 2 t2 2 t

l,

then

t2 u(t 2) 2 t3 u(t3).

The proof is elementary. The lemma is easily visualized by drawing a diagram.

4. PROOF OF THEOREM 1

We assume that D is oscillatory and D is not. Then there

ql,L₁ q2,LZ

exists a solution u of D which has no large positive zeros. Without qZ,LZ

loss of generality we may assume that u is a positive solution. We select numbers t

l, t2, such that u(t) > 0, (t > t1) and t - L2(t) > tl, (t > t2). Clearly u(t) is nonnegative and monotonically non-increasing for t > t

2. Integrating D from t to x where t

_z

~ t < x, we get

qz

,r

Z

x

u( t) - u(x) =

I

q2 (s)u(s - L 2(S) )ds •

t

Letting x +~, it follows that

and

J

Q2(s)u(s - L2(s»ds <

~

, t co u(t) 2

J

Q2(s)u(s - t _2(s»ds, t

By lemma 1 it follows that we may suppose, without loss of generality,

that u(t) +

a

(t + co), Hence, from lemma 2 and the conditions of theorem 1,

we infer that there are nonnegative numbers M

6

-with t3 ~ t2 + M

I, such that, for all t > t 3, we have:

Put yet)

=

u(t - M

I) - M2• Then

ql(t)y(t-'I(t» =ql(t)(u(t-'l(t) -M I) -M2) sq2(t)u(t-'2(t», (t > t_3)·

Hence

t t

By application of lemma 1 it follows that D has a positive solution; q I" I

this contradicts the assumption.

5. APPLICATIONS OF THEOREM I

An immediate consequence of theorem 1 is

THEOREM 2. Let q E

C~.

Let '1' '2 be deZays such that '1 - '2 is bounded

on some intel'vaZ [to'co). Then D is osciUatory if and on~y if D is

q"1 Q"2

osci~~atol'Y·

PROOF. Take MI

₌

_max

I,

1 - L

21 ,

M2

=

0 and apply theorem 1. t~to

+

EXAMPLE 1. Let q E Co and let L be a delay. If D is oscillatory then

'y'(t) ( - 1 q,L

+ 1 + aCt - L(t» )q(t)y(t - L(t»

=

0 is oscillatory for all real a.

PROOF. Let y L Y be a pos1t1ve function. If a ~ 0 then trivially

large. If a < 0 then, for all t sufficiently large,

q(t)(y(t - T(t» -

bO»::;;

(I + aCt - t(t»-l)q(t)y(t -

ret»~

• +

EXAMPLE 2. Let q € CO' Let t be a delay which is bounded on some interval

[ta,oo). Then we have: D

q,T is oscillatory if and only if D q,O is oscillatory. PROOF. Apply theorem 2.

REMARK 1. If in example 2 the condition "t is bounded" is replaced by the condition " t

-I

r (t) -+

0,

(t -+ co)" then we get a false statement as is shown by the following example:

Let

and

-2 t(t) • t(1 + 2 log t)(1 + log t) •

Then D a is oscillatory (see [4, p 20J), but D _{q, q,r} has the solution yet)

= t!.

+

EXAMPLE 3. Let q E

Co

and let T be a delay. Then we have: If

yet)

+ (I - t-1T(t»q(t)y(t)

=

a is oscillatory then

yet)

+ q(t)y(t - T(t»

=

0 is oscillatory.

PROOF. Apply lemma 2 and theorem I.

REMARK 2. Observing the proof of theorem 1 we see that the condition in theorem 1 can be weakened as follows: For every positive function y € Y

with the property that 00

I

Q2(t)y(t - T₂(t»dt < 00 ,

there are nonnegative constants MI , M2 such that, for all t sufficiently large, the following two inequalities hold:

yet) - M2 >

a ,

co 00

J

ql(s)(y(s - rl(s) - M

1) - M2)ds ::;;

J

q2(s)y(s - L2(s»ds •

8

-We shall not pay attention to this generalization of theorem 1, except for one example.

+

EXAHPLE 4. Let q E Co and let a be a positive number. Then we have:

yet) + q(t)y(t) • 0 is oscillatory if and only if

.. -1

y(t) + (a ft+o q(s)ds)y(t) • 0 is oscillatory.

t

PROOF. Let y E Y be a positive function. Then, changing the order of integration, we easily derive that

.., co s+a

J

q(s)y(s - a)ds ~

J

N-l(J

~

t+a t s

for all sufficiently large t.

6. A COUNTEREXAMPLE

q(cr)dcr)y(s)ds

co

~

f

q(s)y(s)ds •

t

It is well-known (for instance see [4, p 4]) that all nontrivial solutions of

yet) + q(t)y(t)

=

0 ,

where q E C;, are nonoscillatory if (1) is not oscillatory. We shall

provide an example showing that such a statement is not true for all equations

yet) + q(t)y(t - 1) • 0 •

Besides, we remark that such a situation is also possible for non-linear equations without delay; for example yet) +

!

t-3y3(t) = 0

(1 )

(2)

has both nontrivial oscillatory and nonoscillatory solutions ([7, p 33, 34J). Moreover, in the case of equation (1) one can give conditions, which

tq(t)dt < QOII is such a condition. Our example will show also that

there are no conditions of the form

f

at 2

e q(t)dt < QO with a < 2,

which guarantee that all nontrivial solutions of (2) are nonoscillatory. First we give some heuristics. Since (2), and hence (1), (example of section 5) has to be not oscillatory, we try a function q(t) which tends rather fast to zero i f t -+ 00, This implies that the "characteristic

2 -z

equation" of (2) z + qe ... 0 has solutions z with 1m z ~ 'IT. Moreover,

it can be proved that large s'uccessive zeros of a nontrivial oscillatory solution of (2) are less than 1 apart, in the case that (2) is not oscillatory. Therefore it is reasonable to try a solution of the form yet) ...

r(t)ei~(t)

with

~(t)

> 'IT.

We are now able to start a precise calculation. All asymptotic statements will be with respect to t -+ QO. Substituting in (2)

yet) ... exp(- ~(t) + i'IT(t - Set»~), ~ =

n

and using the abbreviation

~f = f(t - I) - f(t), we get

.

n

= -

('IT 2 (I - S) • 2 + 'lTacot 'IT~e) - 2'IT(1 _8)cot • _{'IT~e • n + n} 2 sin 'lT68

e-~~q

... 2'IT(1 -

e

)n + 'ITS.

Several choices for

e

are possible. We shall calculate the case

-) 2

Set) ... t • Setting p ...

n -

2t + 2t in (3), we find

-2. 2

t p

=

aCt) + 8(t)p + y(t)p ,

h a -)

were a, ~, y have asymptotic series in terms of powers of t ,the first

~ew terms being given by

aCt) ~ _{- 4 +} 4t- 1 + •••

,

S(t) ~ _{2 -} 2t- 1 _{+ ••.}

,

y( t)

=

t -2 (3) (4)

- 10

-Hence (5) is a so-called unstable Riccati equation, which has exactly one bounded solution p. This solution has an asymptotic series

which formally satisfies equation (5). For a detailed treatment of this result on Riccati equations see [2, p 184 - 186J. By means of

sub-. 2 -2

stitution we find rO .. 2, r 1 .. O. It follows that wet) '" 2t - 2t + 2 + OCt ), hence

Applying this to (4) we find

4 -2t2 + 4t - 11/3 -1

q(t) .. 4t e (I + OCt

» ,

which corresponds with nonoscillatory equations (I) and (2). Neverthe-less we have found a nontrivial oscillating solution of (2), and, moreover,

< 00 if a _< 2.

-n

REMARK. We can get even sharper results. If we choose S (t) .. t , where n is a 2 -1 n+2

positive integer, then we find similarly wet) ~ 2(n + 2n) t ,

( -I n+ I n . ( -t

q t)

=

exp [- 2n t + OCt )J; and ~f we choose

e

t)

=

e then we get 1JJ(t)

~

2et and q(t) .. exp [- (1 - e-l)et(1 + 0(1»].

7.

GENERALIZATIONS

First we introduce same more notation. We shall use Church's lambda notation: A Af(x) means a function with domain A and with function

x€

values given by the expression f (x). For example; if f .. A R x2 then

2 x€

f(x) .. x for all real x, The letter 6 always denotes a variable which runs through all reals of [O,lJ, Therefore we simply write

Ae

instead

We shall also use some abbreviations. C is the Banach space

+

-C([O,I]) with the sup-norm. C and C are the sets of all non-negative and all nonpositive functions in C, respectively.

G

to is the set of all functions g € C«to'~) x C) satisfying, for all t > to' the conditions:

( i) g(t,O) == 0 (H) _{if q> I ,(jl2} E:

c ,

+ such that f+l J + g(t,Ql2)'

-

Ql 2 € C then g(t,Qll) 2:

-(iii) _{if ipl,Ql2} € C

,

such that Ql₁ - Ql₂ € C then g(t,'P_j ) s g(t,({lZ)'

We write g E G if we do not want to specify to'

Let t1,t

_Z

be nonnegative numbers. If g E

G

t and if L is a delay 2

such that t - L(t) > tl for t > t

2, then f(t,y) • g(t,A e yet - T(t)e» defines a function a domain including on (t

2,oo) x C . _tl The set of all functions

£

with

(t2'~) x C , such that the restriction of f to

t}

the form (*) is denoted by F t ' If we do not want t}, 2

to specify t

1,t2 then we write f € F.

We remark that for a given f E F the representation in the form At,yg(t,A_e yet - T(t)e» is not unique.

+

EXAMPLE. Let q € CO' Let f(t,y) • q(t)y(t - I),

Then f(t,y) • g(t,A_e yet - T(t)e» with g(t,q»

=

q(t)q>(l), T(t)

=

but also with g(t,Ql)

=

q(t)ip(!), .(t)

=

2.

12

-The class of equations we want to consider are of the form yet) + f(t,y) - 0 ,

where f L F. We shall refer to this equation by the symbol D

f or D g,! . By a solution of D

f we mean a continuous function y on some interval

(to'~)' with to ~ 0 , having the property that there is a number t_I,

tl ~ to' such that f € F_t t and such that y is twice continuously

0' 1

differentiable on (tl'~)' satisfying D

f for t > tl• The definitions of "D

f is oscillatory", "nontrivial solution", "positive and negative solutions" are the same as in the introduction. The results are analogous to previous results.

+

LEMMA 1'.

Let

to ~ O.

Let

f € F t

and "let

Y E C

t

be differentiable

0' t} 0

for

t > t l ,

satisfying

~

yet)

~

f

f(s,y)ds , t

Then

D

f

has a positive solution

v,

which moreover satisfies

v ( t) !:> Y ( t) , (t > to) and

v (

t) + 0 (t + co) •

The proof is analogous to the proof of lemma 1.

THEOREM I'.

Let

f 1 ,f 2 € F

simultaneous.ly. Suppose that for every

y € Y

there are non:negative constan,ts

M} ,M

2,

such that

l

for aU

t

SUfficientLy

Zarge

l

the foLLowing two inequaUties hold:

ly(t)1 - M2 > 0 ,

Ifj(t,Aty(t - M₁) - M2 sgn y)1 s; If₂(t,y)1 •

(sgn y

equaLs

+1

or

-1

accopding to whether

y

is a positive

or

a negative

function). Then we have: If

D

f

is osciZlatory then

Df

is oscillatory.

1 2

The proof is analogous to the proof of theorem 1, but one remark may be useful. The phrase "without loss of generality we may assume

that u is a positive solution" must be changed in "We assume u to be a positive solution. If u is a negative solution then the

same with f

l,f2 instead of f1,f2• (f is defined by f(t,y) Further we assume that £1

=

A

t,ygl(t,A

e

yet - 't(t)e» £2

=

A

t ,y8 Z(t,y(t - "2(t)6» where gl and g2 in G and "1

delays. I I ,

proof is the

III - f ( t, -y».

and

and "2 are

THEOREM 2'.

Let

g € G

and let

1

1,12

be

de~ays,

suoh that

"1 - 12

is

bounded on some interval

[to'~)'

Then

0

is osoiZlatory if and only

. g, "1

if

0

is osoiZlatory.

g,1

2

Ex&~LE I. Let f E F. Let f E F be defined for all real ~ by

-I (l. f~(t,y) ... f(t,A

t(! + ~t )y(t». Then we have: If Of is oscillatory then Of is oscillatory for all real ~.

~

EXAMPLE 2. Let (l. be a positive number. Let q € c~. Let 1 be a delay.

Then we have: If yet) + q(t)(1 +

~(I

+ Iy(t - t(t)I)-I)y(t - t(t) ... 0 is oscillatory then yet) + q(t)y(t - 1(t» ... 0 is oscillatory.

PROOF. Let y € Y. Without loss of generality we may assume that y is

a positive function. We define a constant M as follows: If yet) + L, (t + ~)

-I

then M

=

L(l + a) ; if yet) +

=,

(t + 00) then M

=

~. Then we have, for all t sufficiently large, that q(t)(1

+~(l

+y(t-t<'t» -M)-I)(y(t-t(t) -M)

~ q{t)y(t - 1(t» and yet) - M > O. Then we apply theorem I'.

EXAMPLE 3. Let ql,q2 be functions

inc~.

Let 1

1,12 be delays. Suppose that 12 is bounded on some interval [t

o

'=)' Then we have

is oscillatory if and only if yet) + ql(t)y(t-'t(t» +q2(t)y(t) ... 0 is oscillatory. The latter equation is oscillatory if

.. -I

14

-PROOF. Apply theorem 1 I and lemma 2.

Finally we present a sufficient condition for D to be oscillatory

gtT in the case that T is unbounded.

THEOREM 4.

Let

g E G.

Let

T

be a

de~ay ~hich

is differentiabZe on some

interval

(tl'~)

and

f(t) < 1

for

t > t

1•

Then

D g,T

is osciZZatory if

D

*

is oscillatory. Here

g*

is defined by

g ,0

*

-)

g (t,<p) '" (l-Hh(t») g(h(t),rp), (t > tl ' rp E C) ,

where

h E C t

is defined by

h(t) - T(h(t» = t. I

PROOF, We suppose that D

*

°

is oscillatory and D is not. As in the

g , g,T

proof of theorem 1 I we may assume that there exists a positive solution

u of Dg,T' Then there is a number t

_z

with t2 > t) such that u(t-,(t» >0

if t > t2 and

00

u(t)

~

f

g(s,Aeu(s - T(s)8»ds ,

t

Let t3 be such that t - T(t) > t2 if t > t3' Then, since u(t) ~ 0

if t > t

2, we have

00

u(t)

~

I

g(s,AeU(. - T(s»)ds ,

t Put s - h(o). Then

s

u(t) ~

f

g(h(o), Aeu(o»(1 - f(h(o») do -1

t-T(t) co

~

f

g(h(o),Aeu(o»(l t -I - Hh(o») do,

Application of lemma I' completes the proof.

EXAMPLE 4. Let q E

C~.

Then we have: If yet) +2q(2t)y3(t) 0 is

oscillatory then yet) +

g(t)y3(~t)

... 0 is oscillatory.

REMARK. Applying lemma 2 and theorem I ' we get the result:

- 1 3 . 3

If yet) +

8'

q(t)y (t) ... 0 lS oscillatory then yet) +q(t)y Ot) '" 0

is oscillatory. This statement is weaker than the statement of example 3, which can be seen as follows: If we put t ... 2s, y(2s) ... w(s), then the equation yet) +

~

q(t)y3(t) ... 0 is transformed into the equation

wI! (s ) + ~ q ( 28 )",,3 ( s) 0: O.

ACKNOWLEDGEMENTS

I would like to thank N.G. de Bruijn and M.L.J. Hautus for their help and valuable suggestions.

REFERENCES

1. Bradley, J.S., Oscillation theorems for a second-order delay equation. J. Di££. Eq. 8 (1970), 397-403.

2. Bruijn, N.G. de, Asymptotic Methods in Analysis. North-Holland Publishing Co. Amsterdam. First Ed. 1958, Third Ed. 1970.

3. Burton, T. and Grimmer, R., Oscillation, continuation and uniqueness of solutions of retarded differential equations. Trans. Amer. Math. Soc. 179 (1973),193-209.

4. Coppel, W.A., Disconjugacy. Lecture Notes in Mathematics 220. Springer-Verlag, Berlin etc. 1971.

5. Gollwitzer, H.E., On nonlinear oscillation for a second-order delay equation. J. Math. Anal. Appl. 26 (1969), 385-389.

6. Kung, G.C.T., Oscillation and nonoscillation of differential equations with a time lag. SIAM. J. Appl. Math. 21 (1971), 207-213.

16

-7. Moore, R.A. and Nehari, Z., Nonoscillation theorems for a class of nonlinear differential equations. Trans. Amer. Math. Soc. 93 (1959),

30-52.

S.

Staikos, V.A. and Petsoulas, A.G., Some oscillation criteria for

second-order nonlinear delay-differential equations. J. Math. Anal.

Appl. 30 (1970), 695-701.

9. Waltman, P., A note on an oscillation criterion for an equation with

17 -GREEK LETTERS a alpha

a

beta y gamma A capital delta ql phi If psi A lambda

•

n eta p rho t tau

e

theta 0' sigma MATHEMATICAL SYMBOLS

(the usual notation for "is element of")