Oscillation in dynamical systems

Nonlinear Analysrs. Theory, Prmted in Chat Britain.

Merhods

& Appkorions,

Vol

20, No.

OSCILLATION

3, pp. 269-283,

0362-546X/93 $6.00+ .OO ct’ 1993 Pergamon Press Ltd

1993

IN DYNAMICAL

SYSTEMS

GEORGE KARAKOSTAS and YUMEI WV Department of Mathematics, University of Ioannina, P.O. Box 1186, 451 10 Ioannina, Greece (Received

27 June 1991; received for publication

Key words and phrases:

26 February

1992)

Dynamical systems, oscillation, delay differential equations.

1. INTRODUCTION WE INTRODUCE the meaning of oscillation in an abstract dynamical system, investigate its topological pathology and present some illustrative examples and applications. Our interest comes from the fact that one of the greatest problems in the qualitative theory of ordinary and delay differential equations is the description of the oscillatory behavior of solutions of a given equation. This is also shown by the existence of a large number of publications among which some recent ones are [l-19]. (A rich bibliography on the subject can be found in [13].) Most of these articles have been devoted to the investigation of the presence or the absence of oscillatory solutions of scalar differential equations, where the definition of an oscillatory function is the following.

Definition A. A real valued function x defined on an interval of the form [to, +a) is called oscillatory if for any t, 2 t, there are numbers t, s 2 t, such that x(t) = 0 and x(s) # 0. Therefore a nonnegative function would be oscillatory, a fact which seems not to be natural. A natural base can be met in the following definition used in [l].

Definition B. A real valued

function

x defined

on [to, +co) is oscillatory if for any t, 2 t, it

holds inf x(t) < 0 < supx(t). f>l, 121, Moreover in studying systems of equations generalizations to this definition were given. Some of them refer to functions with values in a Hilbert space and so they may apply to more general cases. We present them here.

Definition C. A function x: [to, +co) -+ R” is called oscillatory if at least one coordinate (i E (1,2, . . . . n)) is oscillatory according to definition A. Definition oscillatory

C’. A function x: [to, +a) according to definition A.

+ R” is called

oscillatory if all coordinates

xi

of x are

Definition D. Let H be a Hilbert space with inner product (*, * >. A function x: [to, +oo) -+ H is called oscillatory if there is an h E H such that the function t + (x(t), h) is oscillatory according

to definition

A. 269

270

G. KARAKOSTAS and YUMEI WV

Definition function

D’. A function x: [to, +m) + H is called oscillatory if for any nonzero t --t (x(t), h) is oscillatory according to definition A.

h E H the

Definitions C, C’ were used by Bykov [20] and D, D’ by Domshlak [21]. Moreover another definition of nonoscillation was given by Domshlak [21] which is stated in the following definition.

Definition E. Let E E (0, I] and let h E H (h # 0), where H is a Hilbert x: [to, +m) + His called (h, &]-nonoscillatory if for each large t it holds

space.

A function

(x(t), h) 2 el]x(t)]l * Ilhii. One can see that definition Moreover a natural meaning

Definition B’. A function

B gives a meaning which is stronger than that of definition of oscillation could be given by the following definition.

A.

x: [to, +a~) -+ R is called strongly oscillatory if it holds lim inf x(t) < 0 < lim sup x(t).

I’+CC

(In [2] strong

oscillation

refers to definition

r-r+*

B.)

Getting the geometric interpretation of all these definitions one can see that in some of them the oscillation property remains unchanged under a (small) translation of the coordinate system (definitions B’, D’, E). Moreover each author uses any definition he likes and thinks that it is appropriate for the problem under consideration. It is the purpose of the present paper to push further the meaning of oscillation and strong oscillation by setting it in the framework of the general theory of dynamical systems. The price of this abstract setting is double: first of all it applies to any metric space and hence even to a Banach space. Then oscillation for a delay differential equation refers to the profile space of the solutions and not to their values as is usually done. Second, our definitions may refer to oscillation with respect to a (nonempty) set of states and not only to points or lines. The idea is simple: an orbit oscillates with respect to a set G if it meets the set G at an unbounded sequence of times and no tail of it belongs solely in G. Then the following problem arises: given a set G which of its topological properties can be inherited by the set of all points whose orbits oscillate with respect to G. A great part of the present work is devoted to this problem. Finally, the work closes with some examples and applications to ordinary and delay differential equations. Before proceeding to the strict definitions we must give some auxiliary facts from dynamical systems needed in the sequel. (More things about the theory of dynamical systems can be found, e.g., in [22, 231.) Let R be the real line, R, := {t E R: t L 0) and R_ := (t E I?: t s 0). Also let X be a metric space with metric p. We shall refer to t E IR as the time and to x E X as the state variable. A continuous flow on X is a mapping which (i) (ii) (iii)

?r:mxx-x satisfies the properties: 7-cis continuous; n(0, x) = x, for each x E X; and n(t + s, x) = n(t, n(s, x)), for every x E X and t, s in R.

Oscillation

in dynamical

271

systems

A continuous semi-flow on X is a mapping TC:R, x X -+ X satisfying (i)-(iii) above for t, s in R,. For a flow or semi-flow n on X and any x E X the mapping rc( ., x) is called the motion of x. Then the set y+(x) := {n(t, x): t 2 0) is the positive orbit or trajectory of x. If n is a flow then the orbit of x is the set y(x) := (n(t, x): t E RI. A subset A E X is positively invariant with respect to 7c if for every x E A the positive orbit y+(x) of x is included in A. The set A is invariant if for every x E A we have y(x) 5 A, or equivalently for each t the mapping n(t, *): A -+ A is a homeomorphism. Notice that invariance is defined for flows, while positive invariance is even defined for semi-flows. The o-limit set o(x) (with respect to n) is the set of points y E X such that y = lim z(t,, x) for some sequence tk + +a. For a full orbit through x the a-limit set CY(X)is the set of all z E X with z = lim n(t,, x), for a certain sequence t, + --co. It is easy to see that w(x) and o(x) are positively invariant [22, 231. The (first) positive prolongational limit set J+(x) of a point x E X is the set of all y E X such that there is a sequence (x,,) in X and a sequence (t,) in R, with the property that t, -+ +a, x,, + x and rc(t,, x,) + y. The (first) negative prolongational limit set J-(x) of x is defined as above but the sequence (t,) has to converge to --oo. It is clear that for any x E X we have w(x) E J+(x) and CY(X)E J-(x). Notice that J- is defined for flows and not for semi-flows. It is well known that J+(x), J-(x) are closed and invariant sets [22]. Note that if X is a compact metric space then these sets are nonempty and connected, see [22]. 2. OSCILLATION

Let rc be a semi-flow

on the metric

space (X, p) and let G be a (nonempty)

subset of X.

Definition 2.1. A point x E X will be said to be positively oscillating with respect to G if given any t, E I?+ there are t,, t2 2 t, such that

n(t, 3x1 E G

and

n(t,, xl t-t cl G,

where cl G denotes the closure of G. For any E > 0 let B(G, E) be the open ball in X with center the set G and radius G’ denote the set X/G.

E. Also, let

Definition 2.2. A point x E X will be said to be strongly positively oscillating with respect to G, if there is an E > 0 such that for any t, > 0 there are t, , t2 L t, with the property that x(t,, x) e RG, E)

and

rc(t,, x) $ B(G’, E).

We shall denote by (O:(G)) O+(G) the set of all x E X which oscillating with respect to G. If rc is a flow, then in a similar fashion one can define the sets O-(G) Let C := C(R’, IR) be the set of all continuous functions U: R, topology of uniform convergence on compact sets. This topology metric in C would be defined by

are (strongly)

and O,;(G). --) R endowed with the is metrizable. Indeed, a

+m

pc(U, U,) :=

C $min PI=1

1, i

positively

sup (u(t) - u&)1 s oatsn 1

G. KARAKOSTAS and YUMEIWu

212

For any u E C and t 2 0 we define 7r&t, u) := u, where s 2 0.

&(S) := u(t + s),

Then rrc is a semi-flow in C called the shifting semi-flow. (More things about this semi-flow its usefulness can be found in [23-281.) We define the set

and

c+ := (U E c: U(0) > 0). THEOREM 2.1.

A function u E C oscillates in the sense of definition and only if u E Of(Cf) (respectively u E Of(C’)).

Proof. The first fact is clear. To prove the second sequences sk , rk 4 +oo we have lim u(+) but u $ O:(C+). either

This means

5 --E

one assume

B (B’) (see Introduction)

if

that for some E E (0, 1) and

lim u(rk) 2 e,

and

that for any n E N there is a t, such that for all t 2 I,, it holds

Pc(U,,

c+>< in’

(2.1)

or Pc(U,, (C’)‘)

(2.2)

< !. n

Assume that (2.1) holds. Let no > 41~ and let k, be such that -e/2. Then, from (2.1), it follows that there is a u E C+ such that

sk, 2 t,,. and

u(s,J

I

and therefore

This implies

that

Iu(s,J ;

a contradiction. Conversely,

I

- v(O)/ < 2/n, and so -z&,)

5

-u(sko)

+ u(0) =

Iu(s/J - u(O)1<

k,

Similarly we discuss (2.2). assume that u E O:(C+) and either lim sup u(t) 5 0, f- +m

(2.3)

lim inf u(t) 1 0. f--L+-

(2.4)

or

Oscillation

in dynamical

systems

213

Then there is an E > 0 such that for any t, 2 0 there are t, , t2 2 t, with the property 9u), (C’)‘)

P&M,

2 E

that (2.5)

and u), C’)

PC(Q(t2, Assume first that (2.3) holds. Choose a 6, > 0 with 6, < ~18; then,

(2.6)

from (2.3), it follows

that there is a t, 2 0 such that

for all t 2 I,.

u(t) 5 60 Then there is rr 2 tj satisfying

2 E.

(2.7)

(2.5). This means that a 2 E

Pc0+,

(2.8)

for all v E C with v(O) I 0. Thus u(rr) > 0 and so there is a 6, E (0, S,] such that u(t) > 0 for all t E [TV, T, + S,]. Also from the continuity of u at rr there is a 6, E (0, 1) with a2 5 a,/2 such that if lrl - t 1 I 6, then lu(r1) - W)l Now define

the function

u E (C’)’ -6,

+ t +

this implies

fU(T1+ a,),

t E

[O,621

t

6,.

2 u(z,

Clearly

(2.9)

by the type

u(t) :=

Then (2.8) applies

< ; .

+

to this function

t),

>

and so we get

that there is an index n, such that sup

]u(rr +

t>

-

u(t)1

>

;

OSZSll”

and so sup

]u(rr +

t)

+

a*

-

t

-

OStS6,

Taking ; <

into account sup OSfS6,

+

S,)l

>

(2.10)

;.

2

(2.7), (2.9), (2.10) we, finally,

]u(r1 + t) - 4TJl

fU(T*

+

MrJ

-

as1

obtain

+ S2)l

+ k32 + MT1

a contradiction. Similarly we can prove that (2.4) does not hold. The proof

+ S2)l

1 (

is complete.

_ t 4 -)I

G. KARAKOSTASand YUMEI Wu

214

The result proved in the preceding theorem demonstrates the importance of our definitions and the fact that their general formulation is not quite simple. Moreover this result is the key tool for solving problems in the last section. Furthermore we have the following theorem. THEOREM 2.2. Let (X, p) be a metric space, (C, pc) the preceding metric space, TCa semi-flow in X and rcccthe shifting semi-flow in C. Let I/: X + R be a continuous function, let S E IR and define c, := (U E c: U(0) E S) and t 2 0. u,(t) := V(n(t,x)),

If u., E O+(C,), then x E O+(V-l(S)). U, E O:(C,Y) implies that x E O:( V-‘(S)).

Moreover

if

I/ is uniformly

Proof. Let U, E O+(C,) and f, 2 0 be such that +(I,, u,) E Cs. Then V(n(t, , x)) E S and so n(tl, x) E V-‘(S). Also let t2 L 0 be such that

continuous,

u,(t,)

(Q2(0)

for some sequence

-

pn E S. Now define

I

P,(l

u,(t)

:=

Indeed, if not, we have n(t,, x) = limx,, of V we get V(n(t2, x)) E cl S, namely

= lim Pn,

the sequence

of functions

nt) + W4)t2

(;),

tE

t,’

W&)~

Observe (u,),~ E Now there is

E S, namely

(2.11)

nc(t, 9 u,) $ cl cr. We claim that n(t,, x) I$ cl G, where G := V-‘(S). for some sequence (x,) in G. By the continuity u,(tJ E cl S, or (u,),,(O) E cl S. Therefore

then

[OJ] n’

that U, E C, and u,, -+ (u,),~ (with respect to the pc metric). Thus we have nc(t,, x) = cl Cs contrary to (2.11). This proves that x E O+( V-‘(S)). assume that U, E O:(C,Y), but x $ O:(V-l(S)). The latter means that for each n E /PJ a t, 2 0 such that for all t 2 t, it holds either p(n(t, x), G) < i

(2.12)

p(n(t, x), G’) < f >

(2.13)

n’

or

where G := i/-‘(S). such that

Suppose

that (2.12) holds.

Then

for each t 2 t, there is an x(t, n) E G

p(n(t, x), x(t, n)) < ;.

(2.14)

Oscillation

Since U, E O:(C,),

in dynamical

275

systems

there is an E > 0 such that for each t, L 0 there is a t(t,J L to with

Pc(%(t(to), %), C,) 2 e. Fix such a t, 2 0. Let 6 > 0 be a 6 which corresponds to e/4 in the definition let n, E N with l/n, < 6. Then, from (2.14), we get

(2.15) of uniform

continuity

of I/ and

for each t 2 t,. and therefore

Iwt(t”) (0) - w(t(t,), FJ) I < ; * Put p := V(x(t(t,), no)) and define P

v(s) :=

i

the function

( > 1- f

u E C, by the type

+ $u,Mr,)

1

s E P, 41

+ a,),

I

s > 4,

%(t(kJ + 4,

where 6, > 0 is a 6 which corresponds t(t,). Then, by (2.16), we get

5

to ~14 in the definition

20’;;f, lM(to) + s) -=

(2.16)

of continuity

of U, at the point

+ Zlu,(t(t,))

~,W,))l

- pl

I

<2.;+2.+ and so This implies

that k(QV(to),

which contradicts (2.15). Similarly we can discuss

ux), C,) < e,

(2.13) and the proof 3. SOME

of the theorem

TOPOLOGICAL

is complete.

PROPERTIES

We begin this section with the observation that for any G E Xit holds Oj(G) c O’(G). Also, if G is positively (negatively) invariant then O:(G) = O+(G) = @ (O;(G) = O-(G) = 0). In what follows things proved about O+(G) and O:(G) can be proved in the same way for the sets O-(G) and O;(G) without any further mention. A result which is immediately implied by the definitions is the following lemma. LEMMA3.1. If G E Xand x E Xis a point such that y’(x) n Of(G) The same is true for O:(G). This lemma

implies

that the sets O+(G)

and O:(G)

are positively

# 0, then v’(x) E O+(G).

invariant.

276

G. KARAKOSTAS and YUMEI Wu

When studying oscillation with respect to a set G there may exist points in G which are not traced by orbits of points in O+(G). Actually these points do not affect the oscillation property and subtracting them from G the set O+(G) is not reduced. But for Of(G) this is not true. Indeed we have the following theorem. 3.1. For any G E X it holds (i) Of(G (l O+(G)) = O+(G), and (ii) O:(G n Of(G)) E O:(G).

THEOREM

Proof. (i) If x E O’(G n O+(G)), there is a t 2 0 with n(t, x) E G II O+(G) c O+(G). Hence lemma (3.1) implies that x E O’(G). Now let x E O+(G); then for any to I 0 there are t, , t, L t, such that rr(t,, x) E G and n(t,, x) $ cl G. Since O+(G) is positively invariant, we have n(t,, x) E O+(G) and so rr(t,, x) E G fl O+(G). Also G z G fl O+(G) implies that rc(t,, x) $ cl[G n O+(G)] and therefore x E O’(G n O+(G)). (ii) Let x E O:(G fl O:(G)). Then there is an E > 0 such that for any to L 0 there are t, , t, 2 I, with p(n(t, , x), G fl O:(G)) 1 E and p(n(&, x), [G n Of(G)]‘) L E. Thus n(t2, x) E G fl Of(G) G O:(G) and, by lemma 3.1, we get x E O:(G). Remark 3.1. Equality in (ii) does not always hold, as the following example consider the flow defined by the solutions of the differential equation

i=

-r,

r<

1

0,

r=

1

I r,

r>

1

shows. On the plane

9=1 in polar coordinates. Then for the set G := B((l, 0), i) we observe that O:(G) G f’ O:(G) = ((r, 8): r = 1, -7r/6 < 1.9< n/6) and O:(G fJ O:(G)) = 0.

= ((r, ~9):r = l),

3.2. If G is an open subset of X then we have (i) Of(G) C O+(GC), and (ii) O+(G) is also open provided that G E O+(G).

THEOREM

Proof. (i) Let x E O+(G), rc(t, x) E G and rr(r, x) $ cl G. Then rc(t, x) @ Cc = cl G’ and 7c(r, x) E (cl G)’ = int G’ 5 G’, which imply that x E O+(G’). (ii) Let x E O’(G) and rc(& x) E G. If there is a sequence (x,) in O+(G)’ such that x, -+ x, then, by lemma 3.1, we have n(f, x,) B O+(G). Thus n(f, x,) $ G for all n = 1,2,. . . . But z(t; x,) + rr(t; x) contrary to the openness of G. 3.3. For any G E X it holds (i) O:(G) = O:(GC); (ii) O:(G) is open provided that G c O:(G).

THEOREM

Oscillation

in dynamical

277

systems

Proof. Statement (i) is obvious. To prove statement (ii) we let x E O:(G). E > 0 there is a T L 0 such that p(n(r, x), G’) L E. Thus we have

for some

G G E O;(G).

B By the continuity therefore

Then

of n(r, *) there

is a 6 > 0 such

that

rc(r, B(x, 6)) E B(rc(~, x), c/2)

and

n(r, B(x, 6)) E Of(G). Clearly,

Remark

by lemma

3.1, this implies

that B(x, 6) E Of(G),

which proves

3.2. The example

G E O,:(G)

given in remark 3.1 proves that the result (ii) of theorem 3.3 might not be true.

THEOREM 3.4. Let G E X and assume (a) G E O+(G), G is compact and

without

our theorem. the

condition

that

that either J+([O+(G)Y)

c G’,

(3.1)

C G.

(3.2)

or (b) G’ c O+(G),

G’ is compact

and J+([O+(G)]‘)

(Recall that Jf is the positive

prolongational

limit set.) Then the set O+(G)

is a closed set.

Proof. Assume first that the set O+(G) is not closed. This means that for a point x $ O+(G) there exists a sequence (x,,) in O’(G) such that x, --t x. Since, for each n, we have x,, E O+(G) there are two sequences (tk,J and (rk,J in IR, such that lim tk,, = lim rk,n = +oo k

k

and @k,n

3 Xn)

E

G,

n(rk,n

3 x,)

$

cl G.

(3.3)

Now assume that the condition (a) holds and consider the following two cases: (i) there is an index k such that the sequence (tk,n)n is bounded. Then we can assume that it converges to, say, t;, and so dtk,n

9 Xn)

-+

7’C(fk,X)

E

G

because of closedness of G. The latter relation and our condition give z(ik , x) E Of(G) and by lemma 3.1 we get x E O+(G), a contradiction. (ii) For any k the sequence (tk,n) is not bounded. Thus there exists a sequence (nk) of integers such that tk,nk + +a, as k + +m. By using (3.3) and the fact that G is a compact set we conclude that the set tn(fk,nk

has an accumulation pointy E G. It is obvious set J+(x) of x. Therefore we have J+(x) n G # @

3 x,J:

k E IN)

that y is a point in the positive prolongation and

limit

x E tO+(G)l’,

which contradicts condition (3.1). Thus, under condition (a) the set O+(G) condition (b) holds we work with the sequence (rk.n) to get the same result.

is closed.

If the

278

G. KARAKOSTAS and YUMEIWV

Remark

3.3. The following counterexample shows that the result of the theorem if (3.1) in (a) (or (3.2) in (b)) is not satisfied.

On the cylinder X := ((x, y, z) E R3: x2 + yz = 1) consider of the system of ordinary differential equations x = -y, Here observe

y = x,

the flow defined

may not hold

by the solutions

i = z(1 - 2).

that the set G := ((0, 1, z): + I z 5 1)

is compact,

the set O+(G)

is not

closed

and

contains

= ((x, y, z) E X: 0 < z I

G, but

condition

(3.1)

is not

l] satisfied,

since

for

the

point

p := (0, 1,O) E [O+(G)IC we have

J+(p)

= ((x, y, z) E X: 0 5 z I

l] 3 G.

Also for the set G 2 X, where G’ := ((x, y, z) E X:x we observe

that G’ is compact,

connected.

= ((x, y, z) E X: 0 < z 5 1)

G, but (3.2) is not satisfied, J+([O+(G)]‘)

THEOREM

11,

the set

O+(G) is not closed and contains

2 0, y z 0, + I z I

since

= ((x, y, z) E X: z 5 1) CZG.

3.5. Let G be a set such that G c O+(G). The same fact holds for the set O:(G).

If G is connected

then

O’(G)

is also

Proof. Assume that O+(G) is not connected. Then there exist open sets A, B such that A fl B = @, O+(G) C A U B, A n O+(G) =: A, # 0 and B fl O+(G) =: B, # 0. We claim that the sets A,, B, are positively invariant. Indeed, let x E A, ; since O+(G) is positively invariant, we have rc(t, x) E O+(G) for all t r 0. Assume that n(t,, x) E B, for some t, > 0. Since the function t + rr(t, x) is continuous, the set T := rr([O, tl], x) is connected. But this is not true because of the relations Tc_AUB,

T(7A

2

z-f-u1 2 (x) # 0,

T f? B 1 T fl B, 2 (zr(t, , x)) # 0,

Thus A, is positively invariant. Now, there is an x E A, ; then and so G fI A 2 (rr(i, x)) # 0. G E O’(G) E A U B, G fl A # contradiction. Following the same procedure

AnB=@.

Similarly we show that B, is such. z(t, x) E A, E A for all t 2 0. Also rc(i, x) E G for some t > 0 Similarly we get that G fI B # 0 and therefore observe that 0, G fl B # @ and A fl B = 0. Thus G is not connected, a we can prove the other statement

of the theorem.

Oscillation

in dynamical

4. SOME

APPLICATIONS

219

systems

Here we shall present some applications of the previous theory to ordinary and delay differential equations and we show the way of finding new results. The main tools needed for the applications are the results of theorems 2.1 and 2.2. Then the problem of oscillation in C is postponed to the problem of oscillation in R but in a slightly different sense from that of definition B (or B’) given in the Introduction. In order to discuss the latter problem we use several known results on oscillation for scalar valued functions. In the sequel we will denote by ET the transpose of the matrix E, by (*, -> the known inner product in R” and, if h E R”, by h+(hl) we denote the set of all x E R” such that (x, h) > 0 (=O). Also let I be the identity n x n-matrix. We start with the linear differential system

x = Ax where A is an n x n-matrix

(4.1)

(n > 1).

THEOREM 4.1. Let p, q be real numbers

such that p2 < q and let h be a nonzero

(A3 + 2pA2 + qA)Th = 0.

vector in R” with (4.2)

Then for system (4.1) we have [(A2 + 2pA + qZ)Th]‘\{O) Moreover Proof.

function

E O+(h+).

if p 5 0 then the right set in (4.3) may be replaced Let ~(f, x) be the value of the solution V(x) := (h, x) and let u,(t)

:= Vn(t,

by 0:(/r+).

of (4.1) starting

x)),

(4.3)

from x at t = 0. We define the

t 2 0.

Then I/ is (uniformly) continuous and by theorem 2.2 we have x E O+(h+) provided that 24, E o+(c+). Now, let x0 be a nonzero vector in R” such that <[A2 + 2pA + q]‘h, x0> = 0 and let us define the function f(t)

:= %,(Q

+ 2pkJt)

+ q+$t),

t 2 0.

Then we see that f(t)

= (h, Y’(t; x0) + 2pi(t;x0) + qx(t; x0)) = (h, (A2 + 2pA + q)x(t; x0))

and so

f(0) = 0.

(4.4)

Also f’(t)

= (h, (A3 + 2pA2 + qA)x(t,

x0)> = 0,

t20

and, by (4.4), we get f(t) = 0. This means that uXOsatisfies the equation jt + 2pi + qx = 0 all of whose solutions oscillate in the sense of definition B. Thus, by theorems 2.1 and 2.2 we conclude that x0 E O+(h+). If p 5 0 then x0 oscillates in the sense of definition B’ and therefore x0 E O:(h+). The result (4.3) is proved.

280

G. KARAKOSTASand YUMEI WV

Now consider

the linear

delay differential

equation

x = Ax + Bx(t -

1)

(4.5)

where A, B are n x n-matrices. Let C be the (B)-space of all continuous functions q: [-1, 0] + R” endowed with the sup-norm. Given any a, E C we define n(t, p)(e) to be the solution x,(*, u))(E C) through ~JJat t = 0. This is a semi-flow in C, see, e.g. [27]. THEOREM

4.2. Let h, g be two nonzero /l?lr/t

= 0,

(BT + pZ)(B% Then we have H & O+(G),

vectors

in R” and p > em’ such that

(BT + pZ)A=h

+ AT(Arg

+ ATg) + ATBTg = 0,

+ BTh) = 0

(B’ + pZ)BTg

= 0.

where

H := [u, E C/(0]:
(ATg+ (B’ + pZ)h, ~(1, P)(O)>+ (VT + PZk, ~(0)) = 01

and G := (u, E C: ‘A, v(O)) + (8, rp(-1)) > Proof.

Given

v, E H we define +(t)

0).

the functions

:= (h,x(t;

9)) + (g,x(t

tzo

- 1; P)>,

and f(t)

:= z+(t)

+ pu$D(t - 1).

Then it is easy to see that f(2) = 0 andf(t) = 0 for t > 2. Thus z+ is a nonzero t 1 2 of the scalar delay differential equation j + py(t

- 1) = 0.

solution

for

(4.6)

But such an equation is oscillatory in the sense of definition A because ofp > e-’ (see [12]). We will show that it oscillates in the sense of definition B. Indeed, otherwise we can assume that a solution y of (4.6) is such that y(t) L 0 for t greater than or equal to some to, sup y(s) > 0 for SZf all t and y(tk) = 0, for some sequence tk -+ +a. Then we have j(t) I 0, t 2 t, and so y is nonincreasing. Since y(tk) = 0, we have y(t) = 0 eventually, a contradiction. Therefore UP oscillates in the sense of definition B and so by theorems 2.1 and 2.2 we conclude that v, E O+(G). THEOREM 4.3. Consider the equation (4.5) and assume A = (aij) and B = (bjj) satisfy the following conditions

C

aljajk

=

0,

C

j

b,jbjk

that

+

for some p > e-l

Pb,k

=

0

j

and c aijbjk

f

.i

c ajkbij

for all k = 1, 2, . . . , n. Then we have H c O+(G), H := (u, E C: PrI[An(2, where Pr, denotes

+

Polk

=

0

.i

the 1st coordinate,

where

y?)(O) + (B + pZ)n(l,

and

G := (10 E C: p,(O) > 0).

p)(O)] = 0)

the matrices

Oscillation

in dynamical

Proof. This is as in the proof of the previous defined by

systems

theorem,

281

where now the functions

Z.Q andfare

Up@) := x,(t; V) and f(t) THEOREM 4.4. Consider

the linear

:= L+(t)

+ pucp(t - 1).

system x = ‘4x + (A - p1)x(t

where p > e-’ and det A = 0. If h E R” is a nonzero

- 1)

vector such that ATA

0, then we have

C\{O) = O+(G) where ‘0

GE=

yl~C:

(h, v(s)> ds > 0 .

i Proof.

Given

j

-1

v, E C’\(Ol define the functions ‘f z+(t)

:=

(h, x(s; V)> ti i

t-1

and j-(t) := z.+(t) + pucp(t - 1). Now we proceed

f(l)

as in the proof

of theorem

= (h,-~c[~laCr~dr)

Finally

that it holds

= (h, 41, v)(O)) - (h, x(1; v7E-1)) + P ( h , rr, Cws)

= (“, e’%7(0~+ 11, e-%4

because

4.2 where we have only to observe

- (k v7KN + p(“,

c[r, cd4 ds)

=O

the proof.

a system of the form (4.5) where the matrices

A:= for some numbers where

dr)

+p(h,~~,vWdS)

of A7h = 0. This completes we consider

- NMrl

A, B are defined

by

(;, ;), (11,;) B:=

a, b > 0 with a # b. We observe p = (a - 6)’ ab

that AZ = 0, B2 = 0 and AB + BA = -pI

PO).

G. KARAKOSTASand YUMEI Wu

282 THEOREM

4.5.

Under

the

conditions

above

for

system

(4.5)

we

have

where

H,, E O+(G,)

h E lR2\(O], 0

cp E C([-l,O],

H,, :=

lR2)\(O}:

h, (AB - BA + B)&O)

t

- pB j -1

(

and

(s + 1)&s) d.s

= 0

>

1

0 G,, :=

p E C([-1,

01, IR*):

(

Proof.

Let

v, E

H,,; define

1 -1

the functions

up and

(h, V(S)> ds > 0

. 1

f by

I +(t)

:=

(h, x(s;

i f-l

v)> d.s

and

f(t) By using the properties (nonzero)

solution

of A,

:= z+(t)

B we can see that f(2)

of the delay

differential

write

equation

(4.7)

that its characteristic that

uV E O+(C+) Then

we

completes

can

(z,, and,

z2) oscillates by theorem

assume

the proof

= 0, t > 2. Therefore

up is a

1) = 0.

-

(4.7)

= 22

iz = and observe

= 0 andf(t)

in the form i,

it follows

1).

-

equation

ji + py(t We

+ pucp(t

that

equation in the

-pz,(t

-

A2 + p

sense

2.2, q E O+(G,).

zi 2 0 eventually.

of

1)

e-’ = 0 has no real roots. definition

Assume Then

C.

If zi

By a result

oscillates,

then

that zi does not oscillate

i2 < 0 and

so it cannot

we

of [3] have

but z2 does.

oscillate.

This

of the theorem. REFERENCES

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