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Asymptotic constancy for systems of delay differential equations
Nonlinear
Analysis.
Theory,
Methods
Vol. 30. No. 7, pp. 4595S4606, 1991 Proc. 2nd World Congress of Nonlinear Analysfs 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X197 $17.00 + 0.00
& Applications,
Pergamon
PII: SO362-546X(W)OO316-7
ASYMPTOTIC
CONSTANCY DIFFERENTIAL KOUICHI
Department
of Mathematical
Key word and phrases: of the space C.
Delay
Sciences,
differential
FOR SYSTEMS EQUATIONS
OF DELAY
MUKAKAMI Tokushima
equation,
IJniversity,
asymptotic
Tokushima.
constancy,
periodic
770, JAPAN solut.ion,
decomposition
1. INTRODUCTION
We consider
the asymptotic
behavior
x’(t) = A(z(t
of the solution
- rI) - x(t - rz)),
of 0 < r1 < r2.
A E R”““,
(1.1)
For scalar delay differential equations, several results can be found in the literature. there are few results for the higher dimensional case such as (1.1). In case of the scalar equation x’(t) the following
theorem
= a(x(t - ~-1) - x(t - Y-Z)),
is known
a E R,
However,
0 5 r1 < r2,
(1.2)
[l]
A. Consider the solution of (1.2). (i) If a < 0 and r1 = 0, all solutions of (1.2) tend to constant
THEOREM
(ii) If 0 < a -< (iii)
1j a >
’ all solutions 7-z - 7’1’
as t + 03.
limits
as t + K’.
of (1.2) which are unbounded
as t --+ 03
of (1.2) tend to constant
1 there are solutions r2 - r, ’
limits
The purpose of this paper is to extend the above result to the system (1.1). First we consider the two-dimensional case of (1.1). By the transformation an appropriate matrix P, we can write (1.1) as y’(t) = P-‘AP(y(t
iz)
,
where al, u2, a and 0 (0 < (Q] 5 x/2) case of (1.1).
(ii) A = (E
with
- rl) - y(t - rz)).
Thus we assume that A of (1.1) is either of the following (i) A = (2
x(t) = Py(t)
a) ,
are real numbers.
three matrices (iii) A = a (?,”
in Jordan cc:)
After that, we consider
Form. ,
(1.3)
the n-dimensional
2. PRELIMINARIES
Let C = C([-Q,O], R”). For the solution r(t) of (l.l), let LC~(T) = x(t + 7) for r E [-ra,O]. Then we have zt E C. Define the solution operator T(t) such as zt = T(t)4 for 4 E C. Since the family {T(t) : t 2 0} is the strongly continuous semigroup of linear operators on C, there exists the infinitesimal generator a of T(t) such that (1.1) can be written as d -xt dt
. = Aq.
4595
Second World Congress of Nonlinear Analysts
4596 The spectrum
o(a)
of a is given by roots of the characteristic det A(x)
= 0,
A(x)
For any X E U(A), there exists the generalized space C can be decomposed as
= XI - Ae+lX eigenspace
C=
equation + Ae-T2X.
PA of a associated
(2.2) with
X such that the
Px@Qx.
Let A = {X E o(A) 1 Fk X 2 0) and @ be a basis for the generalized eigenspace P,, of a associated with A. Let C’ = C([O, ~1, RIXn), and define the bilinear form {+,, 4) for any $ E C’, $I E C as
The formal adjoint for the generalized
operator a* is defined by the relation ($,, &) = (a*$, 4). We let 9 be a basis eigenspace of a* associated with A. Then for any 4 E C, we have
where
Moreover,
any solution
of (2.1) with the initial
function
4 E C at t = 0 satisfies that (2.4)
where B is defined by a+ = 9B. To compute a basis for the generalized LEMMA
2.1. N(A
where y = col( yl,
- XI)”
coincides
eigenspace,
with functions
Pz PI
‘. ..’
Pk Pk-1 ,
- XI)k
where ,0 = (&,
lemma is useful.
4 of the form
. , yk) satisfies Aky = 0 with P* 0
Also, N(a*
the following
coincides
, &)
with functions
satisfies ,ijAk = 0.
A(J)(x) P Jfl = ~ j!
$ of the form
’
j = 0,1,2 , . . .
Second World Congress of Nonlinear Analysts
4597
3. RESULTS
3.1.
Diagonal
Matrix.
We consider
the case that A is given by ,
We have the following LEMMA
al,a2 E R.
(3.1)
result for the roots of (2.2).
3.1. Assume
that A is given by (3.1). Let w = K 7-l +7-2
w < Uj < & (j = 1,2), then X = 0 is a double root of (2.2), and the sin(r, - rr)w remaining roots of (2.2) have negative real parts.
(i) If -
(j = 1,2), then X = 0 and X = f2iw
(4 v aj = -sin(r2w TI)w the remaining
roots of (2.2) have negative real parts.
If aj = ~ ’
(iii)
7-P -
are doable roots of (2.2), and
(j = 1,2), then X = 0 is a quadmple
root of (2.2), and the remaining
roots
Tl
of (2.2) have negative real parts. Proof.
Let pi(X)
= X - alemT1’ + alePp2’ det A(x)
=
and p2(X) = X - a2e-“lX
we have
1 XI - AeC”’ + AeeTZXI x - ale-nx + ale-(‘2x 0 0 X - u2eerLX + u2e-‘2x
= =
Pl(NP2(0
Thus it suffices to consider roots of pi(X) pi(X), we only consider pr (X) = 0. First we can see that pi(O) = 0, p’,(O) = is a simple root of pr(X) = 0 for al # 1/(r2 Next we consider pure imaginary roots pi(X) = 0. From pi(x) = pi(X), X = -iy is Re p,(iy) Im p,(iy) That
+ azemTzl. Then
= 0 and ps(X)
= 0. Since p2(X)
be much the same as
1 + ai(rr - 7-s) and p?(O) = -ar(rf - ~22). Hence X = 0 - ~1) and a double root for ai =: 1/(r2 - ri). of pi(X) = 0. Suppose that X = iy, y # 0 is a root of also a root of pi(X) = 0. From p,(iy) = 0, we have
= -ai
cosriy
+ a1 cosrzy
= 0,
= y + al sin riy - al sin rzy = 0.
is 2al sin CT1 + 4Y 2
sin (d
= o 2
(3.2)
and y-2alcos
(7 +rz)Y sin (7-Z -b)Y =o, (3.3) 2 2 If ai = 0, then X = 0 is the only root of p,(X) = 0. Thus we assume that ai # 0. From (3.2), we have sin b-1 + 7.2)Y = 0 or sin (" _Zr2’Y = 0. In the latter case, (3.3) implies that y = 0. This 2 contradicts that y # 0. Therefore we obtain that
(7-l + T2)Y = kn 2
Also (3.3) implies
or
Y= -
Tl
2kn
=2kw
(k=il,f2,...).
+rz
that (-l)kal
sin(r2 - rl)kw
= kw
(k = &1,&2,.‘.).
(3.5)
4598
of Nonlinear
Second World Congress
Suppose that -
sm(r,
w - rl)w
Analysts
1 From T2 - 7-l
we have Ial sin(rz - rl)kwl
w sin(rz - rl)w
<
I sin(rz - rl)kwl
< (IcwJ
(k = fl,
*2,
)
This means that the relation (3.5) does not hold. Hence pi(X) = 0 has no pure imaginary except for 0. We know that X = 0 is the only root of pi(X) = 0 for al = 0. From this, together
fi
da we conclude
aI # ~
1
?-2 -
1 x=0
T-1 ’
= 0 with Re X 2 0.
w then (3.4) and (3.5) imply that X = f2iw sin(r* - ri)w ’
are simple roots of pi(X) Ial sin(rz - ri)kwl
Thus pi(X)
for
that X = 0 is the only root of pi(X)
If ai = -
X = f2iw
=o
= 0 has no pure imaginary
We have the following
I sin(rs - ri)kwl
(k = f2, f3,.
< lbwl
behavior
of solution
THEOREM 3.1. Assume that A is given by (3.1). Let ~(4) 7r functiondEC. Letw=-.
be th e solution
+ bo
as
t--+00,
where b. = (I + (r-1 - rz)A)-l(d(O)
Uj
= - sin(r2:
. . ).
of (1.1).
+rz
44)
(ii) If
= 0. Since
root except for 0 and f2iw.
result for the asymptotic
Tl
are roots of pi(X)
= 0. Also we have
w . sin(ra - ri)w
5
root with
- As”
T1)w (j = L2)7 then 44)
-+ 60 + Wl
as
t-m,
-rz
d(t)&)
of (1.1) tith
the initial
Second
World
Congress
of Nonlinear
4599
Analysts
where cos2wt
bo =
(I+
sin2wt
0
(r-1 -r&4)-‘(@)
- A/-“4(t)& -p2
0
OaT(t (4(O) +J-1‘1
J
QT(t + rZ)A@(t)dt). -7-2 Proof (i) Let A = (0). Using Lemma 2.1, we will give a basis of the generalized eigenspace associated with A. From Lemma 3.1 (i), we know that X = 0 is the double root of (2.2). So we compute a basis of N(XI - a)’ at X = 0. From (2.2), we have
Thus ,y = coZ( (3,
(i))
+ q)Ac#+)dt
Pl = P2 =
A(0) = 0, A’(0) = I + (q - Q)A,
A2 =
(;
2)
and Y = -1((f)
= (;
-
‘+(rl,,-‘dA)~ satisfy A27 = 0. Therefore we have bases for
T(:)I
IV(XI - a)2 at X = 0 such as el=c>eo+(~)teo=(~),
ez=(~)eo+(~)teo=
(3.
So we have a basis for the generalized eigenspace of 2 associated with A as * = (ehe2) =
1 0 o 1 .
( >
Similarly, a basis for the generalized eigenspace of a* associated with A is given by \k = aT = I. F’rom (2.3), we have (‘I’, a) = (a’, a) = I + (q - rz)A. Thus any q5in C can be written as cj=#‘~+dQ~ where d PA = bo =
a+,
(a)-‘{*,(b)
= bo
(I + (~1 - ~z)A)-l(4(0)
- A/-”
Since A+=@B,
B=
(
-72
0 0 o o , >
we have T(t)q@
= @eBtbo = bo.
Therefore we conclude that q(r))
--) T(t)#‘,’
= b.
(t -+ co).
4(r)&).
4600
Second World
Congress
of Nonlinear
Analysts
(ii) Let A = (0, f2iw). From the case (i), we know that the identity matrix is a basis of N(XI-A)2 at X = 0. So we compute a basis of N(XI - a)” at X = f2iw. From (2.2), we have Pl
=
= (; 2) = (; ;);
A2
Then
y = col( (:)
N(XI
- 2)” at X = f2iw
= f2iwl~ 2iwI = 0, = I - A(q - r2) COS(T~ - T~)Wk inI.
A(A2iw) A’(f2iw)
Pz =
j (i))
and Y = 4@)
satisfy A27 = 0. Thus
l ($)
we have bases for
as
Define new bases for N(XI - a)2 at X = ~t2iw such as
el = f(41 +
42),
e2 =
Then a basis for the generalized
;($I
-
eigenspace
i = (I, a),
W) = (
By the same way, a basis for the generalized From (2.3), we have
Therefore
any q!~E C can be written
452),
e3
=
iCO3
of a associated
+
$41,
e4
=
Ai43
-
with A is given by
cos2wt sin 2wt 0 0 0 0 cos2wt sin2wt > . eigenspace of a* associated
with A is given by \ir = CT.
as
cp=c+4P^+c$Q~, where 9 PA =
bo =
q9,~)-‘(!i,
4) = bo + a&,
(I + (~1- r,)A)-‘(4(O)
bl=(i(l(4(O) +sO -PI
- A/-” $(t)dt), -r‘2
s 0
CDT(t+ r,)A4(t)dt
-
aT(t + rz)A$(t)dt) --T‘*
From a@ = @B, we have
[T(t)@](r) = +(0)esct+‘) Therefore
we can conclude
This completes
the proof.
= @(t + T).
that q(4)
41).
-+ T(t)$pA = bo + @‘tb,
(t -+ cc)
Second World Congress of Nonlinear Analysts Remark
tends to -oc as r1 -+ 0. Therefore, w sin(r, - rl)w of (1.1) tend to constant limits as t + co.
1. We can see that -
and pi = 0, all solutions Remark
2. If ai = -
1
7-2 -
or a2 = ~
T-2 -
Tl
Then a basis for the generalized Or (ii
4601
Y ;
f).
1 Tl ’
then the multiplicity
of X = 0 is triple
eigenspace associated with zero is given by
Thus there exist solutions
if a3 < 0 (j = 1,2)
of (1.1) which are unbounded
(ii
or quadruple.
Y p(:
Y f)
as t + CYZ
1 w or a3 > (j = 1,2), then there exists a root of (2.2) sin(r2 - ri)w r2 - 7-1 whose real part is positive. Thus there exist solutions of (1.1) which are unbounded as t --+ KI. Remark
3. If aj < -
3.2. Triangular
Matrix.
We consider
the case that A is given by a E R.
We have the following
results.
LEMMA 3.2. Assume that A is gzven by (3.6). 1 then X = 0 is a double w
(9 Ij(ii)
(iii)
TOOt
of (2.2), and the remaining
W
then X = 0 and X = f2iw are double roots of (2.2), and the sin(rz - rl)w’ remaining roots of (2.2) have negative real parts. 1 Ija=then X = 0 is a quadruple root of (2.2), and the remaining roots of (2.2) ?-2- Tl’ have negative real parts.
If
a=-
Pro0 j. Let p(X) = X - aeClx detA(x)
+ aeemx. Then we have x - ae-Plx + ae-r2x 0
=
-e-‘lX + e-rzX x _ ae-“x + ae-‘2x
Thus the rest of the proof is the same as the proof of Lemma
= P(X)P(X).
3.1.
THEOREM3.2. Assume that A is given y (3.6). Let ~~(4) be the solution junction
q5. 1j -
W
sin(rz - Ti)w
x
T-1 + 7-2’ +
bo
as
of (1.1) with the initial
then
tAcc
where b. = (I + (rl - r2)A)-‘(qS(O)
- A/-”
@(t)dt). -m
Proof
The proof is the same as the proof of Theorem
3.1 (i).
1 Remark 4. If a 5 or a 2 then there exists a basis for the generalized sin(rz - ri)w r2 - r1 ’ eigenspace associated with zero, which is unbounded as t + 00. Thus there exist solutions of (1.1) which are unbounded as t --+ 03. W
Second World
4602 3.3.
Complex
Matrix.
Congress
of Nonlinear
Analysts
the case that A is given by
We consider
a E R,
0 < 101 5 ;.
(3.7)
0 3.3. Suppose that A is given by (3.7). Let w1 = -~ - PI and w2 = ~ 7-l fT2’ ~1 + r2
LEMMA
w2
(i) If -
zz
0, and the
then (2.2) has a double root X = 0 and two simple roots X = k2iwl, w1 sin(r2 - rl)wl ’ and the remaining roots have negative real parts.
(ii) If a = -
(4
Tl)w2 9 then (2.2) has a double root X = 0 and two simple roots X = k2iwz,
If a = sin(r2y
and the remaining Proof.
Let pi(X) detA(x)
roots have negative real parts.
= X - ae”ee-rlX
+ aieewrzx and pz(X)
X - aem”’ -aeerl’sin
= =
= X - ae-‘ee-P1x
cos 0 + aeCpzx cos 0 0 + ae-p2x sin 0
+ a-fee-‘zx.
Then we have
aeCrlA sin 0 - ae-+ sin 0 X - aeepl’ cos 0 + uemp2’ cos 0
(A-ae
= =
7’ cos 0 + aePzx cos 0)’ + (ae-“A sin 0 - aemrzx sin 0)’ - aeerzx sin t9))” (A-ae -3’ cos 0 + ae-p2x cos 0)’ - {i(ae-rlxsinO (A _ ae’ee-‘lX + ae’ee-P2~)(~ _ ae-2ee-“LX + ae-%ee-r2A)
=
Pl(%72(4.
Thus it suffices to consider roots of p1 (X) = 0 and pz(X) = 0. Since p, (1) = p2(X), we only consider pl(X) = 0. Also we assume that 0 > 0. First we can see that X = 0 is always a simple root of pi(X) = 0, because ~~(0) = 0 and p;(O) = 1+ aeiB(rl - T2) # 0. Suppose that X = iy, y # 0 is a root of PI(X) = 0. We remark that X = -iy is a root of pz(X) = 0. From pl(iy) = 0, we have Fte pl(iy)
= -acos(O
- rly)
+ acos(8 - rzy) = 0,
Im pl (iy) = y - a sin(O - rly) + a sin(O - r2y) = 0. That
is 2a sin
T.L$Sy-e>
~0
sin(?$I!y)
(
(3.8)
and y-2acos
(
T
y-0)
If a = 0, then X = 0 is the only root of pl(X) have sin
Tl
+rz
my ( This contradicts
- 0
=Oorsin(vy)
sin(yy)
=O.
(3.9)
= 0. Thus we assume that a # 0. From (3.8), we = 0. In the latter
case, (3.9) implies
that y = 0.
>
that y # 0. Thus we have ~1 +
-y-0=&r7-2
2
or
y=
2(0 + h) Tl
+7-2
(k = 0, +1, f2, ‘. . ).
(3.10)
Second World
Also (3.9) implies
Congress
of Nonlinear
4603
Analysts
that (3.11) w1 sin(T2 - rl)wl
Suppose that -
w2
sm(r2 - r1)w2 ’
From
Bfkrr Wl PI+rz I4 < sin(r2 - T~)w~ 5 sin(z(0 + k7r))
(k = fl,
+2, *3, ‘. . ),
we have (asin(z(B+I;n))I
< IsI
(Ic=O,fl,f2,...).
This means that the relation (3.11) does not hold. Hence pi(X) = 0 has no pure imaginary except for 0. We know that X = 0 is the only root of pi(X) = 0 for a = 0. From this, together ax
aa
~ = 0, we conclude
that X = 0 is the only root of pi(X)
root with
= 0 with Re X 2 0.
X-O Ifa=-
w1
sin(r2 - rl)wl
’
then (3.10) and (3.11) imply that X = -2iwl
p;(-2iwl)
we have that X = -2iwl
for k = 1, f2,f3,
=
1 + arlei(Q+2r14
=
1 + a(rr - r2) cos(rl
#
0,
is a simple root of pi(X)
. . . . Thus pi(X)
In the same way, if a = -
is a root ofp,(X)
= 0. Since
_ ar2ei(Q+2w1) - r2)wl
- i0
= 0. Also we have
= 0 has no pure imaginary
root except for 0 and -2iwr.
w2 then X = 0 and X = 2iw2 are the only and simple roots sin(r2 - rl)w2 ’
ofp,(X)=OwithReXzO. THEOREM bnction
3.3. Suppose that A is given by (3.7). Let ~~(4) be the solution e 4. Let wl = -sign(e)- ~-14 and w2 = -. 1 7-1 + r2
(i) Ij -
w1 sin(r2 - rl)wI
w2
sin(r2 - rl)w2 ’ 44)
then
+ bo
as
t + co,
where b. = (I + (~1 - TZ)A)~~(@(O) (ii) If a = -
w1 sin(r2 - rl)wl
’
then
- A/-”
-72
d(t)dt)
of (1.1) with the initial
Second World
4604
where
a(t)= ( t cos 2wlt sin 2wl
bo = =
bl
Congress
- sin 2wlt cos2w1t ) )
(I + (~1 - r,)A)-‘(4(O) ((l-
tL;r-$wl
Ij a =
B= - A/-”
Analysts
( 2L -p2
-?
) 7
$(t)dt),
)I + f (~1 + r,)B)-’
@‘(t (+(o) +Jo -PI (iii)
of Nonlinear
+ r,)A$(t)dt
-
J
’ @‘(t + r,)A$(t)dt). -YZ
then w2 sin(rz - rI)wZ ’
where cos 2wzt sin 2wzt
5%(t) =
- sin 2wzt cos2wd ) 3 B = ( 2&
bo =
(I + (~1 - e)A)-l(q?~(O) - A/-‘~
bl
((1 -
=
-?z
h
- rz)w2
t4c
-
)I + f(rl
(i) Let A = (0).
d(t)&),
+ ra)B)-’
44
(d(O) + lo GT(t + q)A4(t)dt -‘I Proof.
-2ow2 ) )
- Jo aT(t + rz)Ad(t)dt). -r2
From (2.2), we have that Pl = A(0)
Thus a basis for the generalized
P2 = A’(0)
= 0,
eigenspace
= I + (q - rz)A.
of A associated
@= (el,e2), el= (t), and a basis for the generalized eigenspace of a* associated the proof is the same as the proof of Theorem 3.1 (i).
with A is
e2= (3, with A is 9 = aT = I. Thus the rest of
Fr om the case (i), we know that the identity matrix is a basis for N(XI-A) (ii) Let A = (0, *2iwl}. at X = 0. So we compute a basis for N(XI - a) at X = *2iwl. F’rom (2.2), we have
Pl = A(f2iw,) = +2.&l - Aer2irlwl + ,4,+~~1 = Thus bases for N(XI
-2wleFie
- a) at X = f2iwl &(t)
= (la)
sin 0 are eaiwlt,
4*(t) = (a> e-z’wlt
We define a new basis as el = k (A+
421,
e2 = k ($2 - 41)
SecondWorld
Congress
of Nonlinear
Analysts
4605
Then a basis for the generalized eigenspace of a associated with A is given by
& = (I, CD), a(t) =
cos 2qt
- sin 2wlt
( sin 2wlt
cos2w1t
>s
Also a basisfor the generalizedeigenspaceof a* associatedwith A is given by
9= 0f , f&(t) =a)‘(t). From (2.3), we have I+ (i&C)
(q - rz)A
=
0
.
Thus any I$ in C can be written as where 4 PA =
b(‘i’, +)-‘(+,
4) = b. + @bl
bo =
(I + (7-1 - ~z)A)-~($(o)
bl
((1 - tJ;;Ty$:,
=
- A/-T1 q+(7)&), -p2
)I + ;(Tl + Tz)B)-’ 0
(N-3 + Jo aT(t + rl)Acj(t)dt
-
-rI
J -rz
@(t
+ rz)Ad(t)dt)
Since a@ = @B, we have This meansthat
T(t)4
[T(t)+](T) = G(0)eBct+‘) = @(t + 7). pi\ = bo + CDtbl. Therefore we concludethat
44)
--) bo+ Wl
(t
3 co).
(iii) The proof is the sameas the proof of the case(ii). w2 w1 or a > then there exists a root of (2.2) whose sin(r2 - 7-1)~~ sin(r2 - Irl)w2’ real part is positive. Thus there exist solutionsof (1.1) which are unbounded as t -+ 03.
Renark
5. If a < -
n-dimensional System. Consider the casethat A is an R x n matrix. Let a&ok (k = 1,2,. . , n) be eigenvaluesof A. Then (2.2) can be written as
3.4.
det A(x)
=
fi(X - akeiske-‘lx + akeiBke-QX)= 0. k=l
Thus we have the following result. THEOREM 3.4. Let akeiek(l&l 5 7~/2; k = 1,2, . . , n) be eigenvaluesof A. Suppose that Wlk
sin(T2
-
T1)Wlk
.
WZk
sln(T2- ?-l)@k’
where wlk
=
r - l@kl ___ 7-1 +T2 ’
W2k
=
-. Tl
ok +T2
(k=
1,2,...,n)
Second
4606
Then the solution
~(4)
World
Congress
FIG.
1. Example
of (1.1) vrith the initial G(d)
+
of Nonlinear
Analysts
of Theorem
fiLnction bo
3.4
4 E C([-r2,
01, R”) satisfies that
0 + ml
where b. = (I + (rl - r,)A)-‘(4(O)
Example
1. Consider
#(t)dt). -P2 the case that ~1 = 1, ~2 = 2, and A and 4 is given by A=
(‘[
We have the asymptotic equilibrium the solution of (1.1) with the initial
if
ii),
- A/-”
4(t)=
(a$$).
point as b. = col( 11/2,3,1) by Theorem function 4, tends to b0 as t -+ 00.
3.4. Fig. 1 shows that
REFERENCES 1. K.COOKE 16, 75-101 2. O.DIEKMANN,
Complts, 3. 4. 5. 6.
& J.YORKE, (1973). S.A.VAN
Some
Equations
Modelling
Growth
Processes
and
Gonorrhea
Epidemics
Math.
Biosci.
GILS, S.M.VERDUYN LUNEL & H.-O.WALTHER, Delay Eq-uatrons: ~&nctzonal-, and Nonlinear Analysis. Springer-Verlag (1995). Theory of finctional Diflerential Equatzons, Springer-Verlag (1977). & S.M.VERDUYN LUNEL, Ithtd~tion to finctional Diflerential Equations, Springer-Verlag (1993). Delay Differrntzal Equations With Apphcatronr in Population Dynamzcs, Academic Press (1993).
J.HALE, J.HALE Y.KUANG, K.MURAKAMI, tion, Funkcial.
Asymptotic Constancy Ekvac. 39, 519-540 (1996).
and
Periodic
Solutions
for Linear
Autonomous
Delay
Differential
Equa-
Analysis.
Theory,
Methods
Vol. 30. No. 7, pp. 4595S4606, 1991 Proc. 2nd World Congress of Nonlinear Analysfs 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X197 $17.00 + 0.00
& Applications,
Pergamon
PII: SO362-546X(W)OO316-7
ASYMPTOTIC
CONSTANCY DIFFERENTIAL KOUICHI
Department
of Mathematical
Key word and phrases: of the space C.
Delay
Sciences,
differential
FOR SYSTEMS EQUATIONS
OF DELAY
MUKAKAMI Tokushima
equation,
IJniversity,
asymptotic
Tokushima.
constancy,
periodic
770, JAPAN solut.ion,
decomposition
1. INTRODUCTION
We consider
the asymptotic
behavior
x’(t) = A(z(t
of the solution
- rI) - x(t - rz)),
of 0 < r1 < r2.
A E R”““,
(1.1)
For scalar delay differential equations, several results can be found in the literature. there are few results for the higher dimensional case such as (1.1). In case of the scalar equation x’(t) the following
theorem
= a(x(t - ~-1) - x(t - Y-Z)),
is known
a E R,
However,
0 5 r1 < r2,
(1.2)
[l]
A. Consider the solution of (1.2). (i) If a < 0 and r1 = 0, all solutions of (1.2) tend to constant
THEOREM
(ii) If 0 < a -< (iii)
1j a >
’ all solutions 7-z - 7’1’
as t + 03.
limits
as t + K’.
of (1.2) which are unbounded
as t --+ 03
of (1.2) tend to constant
1 there are solutions r2 - r, ’
limits
The purpose of this paper is to extend the above result to the system (1.1). First we consider the two-dimensional case of (1.1). By the transformation an appropriate matrix P, we can write (1.1) as y’(t) = P-‘AP(y(t
iz)
,
where al, u2, a and 0 (0 < (Q] 5 x/2) case of (1.1).
(ii) A = (E
with
- rl) - y(t - rz)).
Thus we assume that A of (1.1) is either of the following (i) A = (2
x(t) = Py(t)
a) ,
are real numbers.
three matrices (iii) A = a (?,”
in Jordan cc:)
After that, we consider
Form. ,
(1.3)
the n-dimensional
2. PRELIMINARIES
Let C = C([-Q,O], R”). For the solution r(t) of (l.l), let LC~(T) = x(t + 7) for r E [-ra,O]. Then we have zt E C. Define the solution operator T(t) such as zt = T(t)4 for 4 E C. Since the family {T(t) : t 2 0} is the strongly continuous semigroup of linear operators on C, there exists the infinitesimal generator a of T(t) such that (1.1) can be written as d -xt dt
. = Aq.
4595
Second World Congress of Nonlinear Analysts
4596 The spectrum
o(a)
of a is given by roots of the characteristic det A(x)
= 0,
A(x)
For any X E U(A), there exists the generalized space C can be decomposed as
= XI - Ae+lX eigenspace
C=
equation + Ae-T2X.
PA of a associated
(2.2) with
X such that the
Px@Qx.
Let A = {X E o(A) 1 Fk X 2 0) and @ be a basis for the generalized eigenspace P,, of a associated with A. Let C’ = C([O, ~1, RIXn), and define the bilinear form {+,, 4) for any $ E C’, $I E C as
The formal adjoint for the generalized
operator a* is defined by the relation ($,, &) = (a*$, 4). We let 9 be a basis eigenspace of a* associated with A. Then for any 4 E C, we have
where
Moreover,
any solution
of (2.1) with the initial
function
4 E C at t = 0 satisfies that (2.4)
where B is defined by a+ = 9B. To compute a basis for the generalized LEMMA
2.1. N(A
where y = col( yl,
- XI)”
coincides
eigenspace,
with functions
Pz PI
‘. ..’
Pk Pk-1 ,
- XI)k
where ,0 = (&,
lemma is useful.
4 of the form
. , yk) satisfies Aky = 0 with P* 0
Also, N(a*
the following
coincides
, &)
with functions
satisfies ,ijAk = 0.
A(J)(x) P Jfl = ~ j!
$ of the form
’
j = 0,1,2 , . . .
Second World Congress of Nonlinear Analysts
4597
3. RESULTS
3.1.
Diagonal
Matrix.
We consider
the case that A is given by ,
We have the following LEMMA
al,a2 E R.
(3.1)
result for the roots of (2.2).
3.1. Assume
that A is given by (3.1). Let w = K 7-l +7-2
w < Uj < & (j = 1,2), then X = 0 is a double root of (2.2), and the sin(r, - rr)w remaining roots of (2.2) have negative real parts.
(i) If -
(j = 1,2), then X = 0 and X = f2iw
(4 v aj = -sin(r2w TI)w the remaining
roots of (2.2) have negative real parts.
If aj = ~ ’
(iii)
7-P -
are doable roots of (2.2), and
(j = 1,2), then X = 0 is a quadmple
root of (2.2), and the remaining
roots
Tl
of (2.2) have negative real parts. Proof.
Let pi(X)
= X - alemT1’ + alePp2’ det A(x)
=
and p2(X) = X - a2e-“lX
we have
1 XI - AeC”’ + AeeTZXI x - ale-nx + ale-(‘2x 0 0 X - u2eerLX + u2e-‘2x
= =
Pl(NP2(0
Thus it suffices to consider roots of pi(X) pi(X), we only consider pr (X) = 0. First we can see that pi(O) = 0, p’,(O) = is a simple root of pr(X) = 0 for al # 1/(r2 Next we consider pure imaginary roots pi(X) = 0. From pi(x) = pi(X), X = -iy is Re p,(iy) Im p,(iy) That
+ azemTzl. Then
= 0 and ps(X)
= 0. Since p2(X)
be much the same as
1 + ai(rr - 7-s) and p?(O) = -ar(rf - ~22). Hence X = 0 - ~1) and a double root for ai =: 1/(r2 - ri). of pi(X) = 0. Suppose that X = iy, y # 0 is a root of also a root of pi(X) = 0. From p,(iy) = 0, we have
= -ai
cosriy
+ a1 cosrzy
= 0,
= y + al sin riy - al sin rzy = 0.
is 2al sin CT1 + 4Y 2
sin (d
= o 2
(3.2)
and y-2alcos
(7 +rz)Y sin (7-Z -b)Y =o, (3.3) 2 2 If ai = 0, then X = 0 is the only root of p,(X) = 0. Thus we assume that ai # 0. From (3.2), we have sin b-1 + 7.2)Y = 0 or sin (" _Zr2’Y = 0. In the latter case, (3.3) implies that y = 0. This 2 contradicts that y # 0. Therefore we obtain that
(7-l + T2)Y = kn 2
Also (3.3) implies
or
Y= -
Tl
2kn
=2kw
(k=il,f2,...).
+rz
that (-l)kal
sin(r2 - rl)kw
= kw
(k = &1,&2,.‘.).
(3.5)
4598
of Nonlinear
Second World Congress
Suppose that -
sm(r,
w - rl)w
Analysts
1 From T2 - 7-l
we have Ial sin(rz - rl)kwl
w sin(rz - rl)w
<
I sin(rz - rl)kwl
< (IcwJ
(k = fl,
*2,
)
This means that the relation (3.5) does not hold. Hence pi(X) = 0 has no pure imaginary except for 0. We know that X = 0 is the only root of pi(X) = 0 for al = 0. From this, together
fi
da we conclude
aI # ~
1
?-2 -
1 x=0
T-1 ’
= 0 with Re X 2 0.
w then (3.4) and (3.5) imply that X = f2iw sin(r* - ri)w ’
are simple roots of pi(X) Ial sin(rz - ri)kwl
Thus pi(X)
for
that X = 0 is the only root of pi(X)
If ai = -
X = f2iw
=o
= 0 has no pure imaginary
We have the following
I sin(rs - ri)kwl
(k = f2, f3,.
< lbwl
behavior
of solution
THEOREM 3.1. Assume that A is given by (3.1). Let ~(4) 7r functiondEC. Letw=-.
be th e solution
+ bo
as
t--+00,
where b. = (I + (r-1 - rz)A)-l(d(O)
Uj
= - sin(r2:
. . ).
of (1.1).
+rz
44)
(ii) If
= 0. Since
root except for 0 and f2iw.
result for the asymptotic
Tl
are roots of pi(X)
= 0. Also we have
w . sin(ra - ri)w
5
root with
- As”
T1)w (j = L2)7 then 44)
-+ 60 + Wl
as
t-m,
-rz
d(t)&)
of (1.1) tith
the initial
Second
World
Congress
of Nonlinear
4599
Analysts
where cos2wt
bo =
(I+
sin2wt
0
(r-1 -r&4)-‘(@)
- A/-“4(t)& -p2
0
OaT(t (4(O) +J-1‘1
J
QT(t + rZ)A@(t)dt). -7-2 Proof (i) Let A = (0). Using Lemma 2.1, we will give a basis of the generalized eigenspace associated with A. From Lemma 3.1 (i), we know that X = 0 is the double root of (2.2). So we compute a basis of N(XI - a)’ at X = 0. From (2.2), we have
Thus ,y = coZ( (3,
(i))
+ q)Ac#+)dt
Pl = P2 =
A(0) = 0, A’(0) = I + (q - Q)A,
A2 =
(;
2)
and Y = -1((f)
= (;
-
‘+(rl,,-‘dA)~ satisfy A27 = 0. Therefore we have bases for
T(:)I
IV(XI - a)2 at X = 0 such as el=c>eo+(~)teo=(~),
ez=(~)eo+(~)teo=
(3.
So we have a basis for the generalized eigenspace of 2 associated with A as * = (ehe2) =
1 0 o 1 .
( >
Similarly, a basis for the generalized eigenspace of a* associated with A is given by \k = aT = I. F’rom (2.3), we have (‘I’, a) = (a’, a) = I + (q - rz)A. Thus any q5in C can be written as cj=#‘~+dQ~ where d PA = bo =
a+,
(a)-‘{*,(b)
= bo
(I + (~1 - ~z)A)-l(4(0)
- A/-”
Since A+=@B,
B=
(
-72
0 0 o o , >
we have T(t)q@
= @eBtbo = bo.
Therefore we conclude that q(r))
--) T(t)#‘,’
= b.
(t -+ co).
4(r)&).
4600
Second World
Congress
of Nonlinear
Analysts
(ii) Let A = (0, f2iw). From the case (i), we know that the identity matrix is a basis of N(XI-A)2 at X = 0. So we compute a basis of N(XI - a)” at X = f2iw. From (2.2), we have Pl
=
= (; 2) = (; ;);
A2
Then
y = col( (:)
N(XI
- 2)” at X = f2iw
= f2iwl~ 2iwI = 0, = I - A(q - r2) COS(T~ - T~)Wk inI.
A(A2iw) A’(f2iw)
Pz =
j (i))
and Y = 4@)
satisfy A27 = 0. Thus
l ($)
we have bases for
as
Define new bases for N(XI - a)2 at X = ~t2iw such as
el = f(41 +
42),
e2 =
Then a basis for the generalized
;($I
-
eigenspace
i = (I, a),
W) = (
By the same way, a basis for the generalized From (2.3), we have
Therefore
any q!~E C can be written
452),
e3
=
iCO3
of a associated
+
$41,
e4
=
Ai43
-
with A is given by
cos2wt sin 2wt 0 0 0 0 cos2wt sin2wt > . eigenspace of a* associated
with A is given by \ir = CT.
as
cp=c+4P^+c$Q~, where 9 PA =
bo =
q9,~)-‘(!i,
4) = bo + a&,
(I + (~1- r,)A)-‘(4(O)
bl=(i(l(4(O) +sO -PI
- A/-” $(t)dt), -r‘2
s 0
CDT(t+ r,)A4(t)dt
-
aT(t + rz)A$(t)dt) --T‘*
From a@ = @B, we have
[T(t)@](r) = +(0)esct+‘) Therefore
we can conclude
This completes
the proof.
= @(t + T).
that q(4)
41).
-+ T(t)$pA = bo + @‘tb,
(t -+ cc)
Second World Congress of Nonlinear Analysts Remark
tends to -oc as r1 -+ 0. Therefore, w sin(r, - rl)w of (1.1) tend to constant limits as t + co.
1. We can see that -
and pi = 0, all solutions Remark
2. If ai = -
1
7-2 -
or a2 = ~
T-2 -
Tl
Then a basis for the generalized Or (ii
4601
Y ;
f).
1 Tl ’
then the multiplicity
of X = 0 is triple
eigenspace associated with zero is given by
Thus there exist solutions
if a3 < 0 (j = 1,2)
of (1.1) which are unbounded
(ii
or quadruple.
Y p(:
Y f)
as t + CYZ
1 w or a3 > (j = 1,2), then there exists a root of (2.2) sin(r2 - ri)w r2 - 7-1 whose real part is positive. Thus there exist solutions of (1.1) which are unbounded as t --+ KI. Remark
3. If aj < -
3.2. Triangular
Matrix.
We consider
the case that A is given by a E R.
We have the following
results.
LEMMA 3.2. Assume that A is gzven by (3.6). 1 then X = 0 is a double w
(9 Ij(ii)
(iii)
TOOt
of (2.2), and the remaining
W
then X = 0 and X = f2iw are double roots of (2.2), and the sin(rz - rl)w’ remaining roots of (2.2) have negative real parts. 1 Ija=then X = 0 is a quadruple root of (2.2), and the remaining roots of (2.2) ?-2- Tl’ have negative real parts.
If
a=-
Pro0 j. Let p(X) = X - aeClx detA(x)
+ aeemx. Then we have x - ae-Plx + ae-r2x 0
=
-e-‘lX + e-rzX x _ ae-“x + ae-‘2x
Thus the rest of the proof is the same as the proof of Lemma
= P(X)P(X).
3.1.
THEOREM3.2. Assume that A is given y (3.6). Let ~~(4) be the solution junction
q5. 1j -
W
sin(rz - Ti)w
x
T-1 + 7-2’ +
bo
as
of (1.1) with the initial
then
tAcc
where b. = (I + (rl - r2)A)-‘(qS(O)
- A/-”
@(t)dt). -m
Proof
The proof is the same as the proof of Theorem
3.1 (i).
1 Remark 4. If a 5 or a 2 then there exists a basis for the generalized sin(rz - ri)w r2 - r1 ’ eigenspace associated with zero, which is unbounded as t + 00. Thus there exist solutions of (1.1) which are unbounded as t --+ 03. W
Second World
4602 3.3.
Complex
Matrix.
Congress
of Nonlinear
Analysts
the case that A is given by
We consider
a E R,
0 < 101 5 ;.
(3.7)
0 3.3. Suppose that A is given by (3.7). Let w1 = -~ - PI and w2 = ~ 7-l fT2’ ~1 + r2
LEMMA
w2
(i) If -
zz
0, and the
then (2.2) has a double root X = 0 and two simple roots X = k2iwl, w1 sin(r2 - rl)wl ’ and the remaining roots have negative real parts.
(ii) If a = -
(4
Tl)w2 9 then (2.2) has a double root X = 0 and two simple roots X = k2iwz,
If a = sin(r2y
and the remaining Proof.
Let pi(X) detA(x)
roots have negative real parts.
= X - ae”ee-rlX
+ aieewrzx and pz(X)
X - aem”’ -aeerl’sin
= =
= X - ae-‘ee-P1x
cos 0 + aeCpzx cos 0 0 + ae-p2x sin 0
+ a-fee-‘zx.
Then we have
aeCrlA sin 0 - ae-+ sin 0 X - aeepl’ cos 0 + uemp2’ cos 0
(A-ae
= =
7’ cos 0 + aePzx cos 0)’ + (ae-“A sin 0 - aemrzx sin 0)’ - aeerzx sin t9))” (A-ae -3’ cos 0 + ae-p2x cos 0)’ - {i(ae-rlxsinO (A _ ae’ee-‘lX + ae’ee-P2~)(~ _ ae-2ee-“LX + ae-%ee-r2A)
=
Pl(%72(4.
Thus it suffices to consider roots of p1 (X) = 0 and pz(X) = 0. Since p, (1) = p2(X), we only consider pl(X) = 0. Also we assume that 0 > 0. First we can see that X = 0 is always a simple root of pi(X) = 0, because ~~(0) = 0 and p;(O) = 1+ aeiB(rl - T2) # 0. Suppose that X = iy, y # 0 is a root of PI(X) = 0. We remark that X = -iy is a root of pz(X) = 0. From pl(iy) = 0, we have Fte pl(iy)
= -acos(O
- rly)
+ acos(8 - rzy) = 0,
Im pl (iy) = y - a sin(O - rly) + a sin(O - r2y) = 0. That
is 2a sin
T.L$Sy-e>
~0
sin(?$I!y)
(
(3.8)
and y-2acos
(
T
y-0)
If a = 0, then X = 0 is the only root of pl(X) have sin
Tl
+rz
my ( This contradicts
- 0
=Oorsin(vy)
sin(yy)
=O.
(3.9)
= 0. Thus we assume that a # 0. From (3.8), we = 0. In the latter
case, (3.9) implies
that y = 0.
>
that y # 0. Thus we have ~1 +
-y-0=&r7-2
2
or
y=
2(0 + h) Tl
+7-2
(k = 0, +1, f2, ‘. . ).
(3.10)
Second World
Also (3.9) implies
Congress
of Nonlinear
4603
Analysts
that (3.11) w1 sin(T2 - rl)wl
Suppose that -
w2
sm(r2 - r1)w2 ’
From
Bfkrr Wl PI+rz I4 < sin(r2 - T~)w~ 5 sin(z(0 + k7r))
(k = fl,
+2, *3, ‘. . ),
we have (asin(z(B+I;n))I
< IsI
(Ic=O,fl,f2,...).
This means that the relation (3.11) does not hold. Hence pi(X) = 0 has no pure imaginary except for 0. We know that X = 0 is the only root of pi(X) = 0 for a = 0. From this, together ax
aa
~ = 0, we conclude
that X = 0 is the only root of pi(X)
root with
= 0 with Re X 2 0.
X-O Ifa=-
w1
sin(r2 - rl)wl
’
then (3.10) and (3.11) imply that X = -2iwl
p;(-2iwl)
we have that X = -2iwl
for k = 1, f2,f3,
=
1 + arlei(Q+2r14
=
1 + a(rr - r2) cos(rl
#
0,
is a simple root of pi(X)
. . . . Thus pi(X)
In the same way, if a = -
is a root ofp,(X)
= 0. Since
_ ar2ei(Q+2w1) - r2)wl
- i0
= 0. Also we have
= 0 has no pure imaginary
root except for 0 and -2iwr.
w2 then X = 0 and X = 2iw2 are the only and simple roots sin(r2 - rl)w2 ’
ofp,(X)=OwithReXzO. THEOREM bnction
3.3. Suppose that A is given by (3.7). Let ~~(4) be the solution e 4. Let wl = -sign(e)- ~-14 and w2 = -. 1 7-1 + r2
(i) Ij -
w1 sin(r2 - rl)wI
w2
sin(r2 - rl)w2 ’ 44)
then
+ bo
as
t + co,
where b. = (I + (~1 - TZ)A)~~(@(O) (ii) If a = -
w1 sin(r2 - rl)wl
’
then
- A/-”
-72
d(t)dt)
of (1.1) with the initial
Second World
4604
where
a(t)= ( t cos 2wlt sin 2wl
bo = =
bl
Congress
- sin 2wlt cos2w1t ) )
(I + (~1 - r,)A)-‘(4(O) ((l-
tL;r-$wl
Ij a =
B= - A/-”
Analysts
( 2L -p2
-?
) 7
$(t)dt),
)I + f (~1 + r,)B)-’
@‘(t (+(o) +Jo -PI (iii)
of Nonlinear
+ r,)A$(t)dt
-
J
’ @‘(t + r,)A$(t)dt). -YZ
then w2 sin(rz - rI)wZ ’
where cos 2wzt sin 2wzt
5%(t) =
- sin 2wzt cos2wd ) 3 B = ( 2&
bo =
(I + (~1 - e)A)-l(q?~(O) - A/-‘~
bl
((1 -
=
-?z
h
- rz)w2
t4c
-
)I + f(rl
(i) Let A = (0).
d(t)&),
+ ra)B)-’
44
(d(O) + lo GT(t + q)A4(t)dt -‘I Proof.
-2ow2 ) )
- Jo aT(t + rz)Ad(t)dt). -r2
From (2.2), we have that Pl = A(0)
Thus a basis for the generalized
P2 = A’(0)
= 0,
eigenspace
= I + (q - rz)A.
of A associated
@= (el,e2), el= (t), and a basis for the generalized eigenspace of a* associated the proof is the same as the proof of Theorem 3.1 (i).
with A is
e2= (3, with A is 9 = aT = I. Thus the rest of
Fr om the case (i), we know that the identity matrix is a basis for N(XI-A) (ii) Let A = (0, *2iwl}. at X = 0. So we compute a basis for N(XI - a) at X = *2iwl. F’rom (2.2), we have
Pl = A(f2iw,) = +2.&l - Aer2irlwl + ,4,+~~1 = Thus bases for N(XI
-2wleFie
- a) at X = f2iwl &(t)
= (la)
sin 0 are eaiwlt,
4*(t) = (a> e-z’wlt
We define a new basis as el = k (A+
421,
e2 = k ($2 - 41)
SecondWorld
Congress
of Nonlinear
Analysts
4605
Then a basis for the generalized eigenspace of a associated with A is given by
& = (I, CD), a(t) =
cos 2qt
- sin 2wlt
( sin 2wlt
cos2w1t
>s
Also a basisfor the generalizedeigenspaceof a* associatedwith A is given by
9= 0f , f&(t) =a)‘(t). From (2.3), we have I+ (i&C)
(q - rz)A
=
0
.
Thus any I$ in C can be written as where 4 PA =
b(‘i’, +)-‘(+,
4) = b. + @bl
bo =
(I + (7-1 - ~z)A)-~($(o)
bl
((1 - tJ;;Ty$:,
=
- A/-T1 q+(7)&), -p2
)I + ;(Tl + Tz)B)-’ 0
(N-3 + Jo aT(t + rl)Acj(t)dt
-
-rI
J -rz
@(t
+ rz)Ad(t)dt)
Since a@ = @B, we have This meansthat
T(t)4
[T(t)+](T) = G(0)eBct+‘) = @(t + 7). pi\ = bo + CDtbl. Therefore we concludethat
44)
--) bo+ Wl
(t
3 co).
(iii) The proof is the sameas the proof of the case(ii). w2 w1 or a > then there exists a root of (2.2) whose sin(r2 - 7-1)~~ sin(r2 - Irl)w2’ real part is positive. Thus there exist solutionsof (1.1) which are unbounded as t -+ 03.
Renark
5. If a < -
n-dimensional System. Consider the casethat A is an R x n matrix. Let a&ok (k = 1,2,. . , n) be eigenvaluesof A. Then (2.2) can be written as
3.4.
det A(x)
=
fi(X - akeiske-‘lx + akeiBke-QX)= 0. k=l
Thus we have the following result. THEOREM 3.4. Let akeiek(l&l 5 7~/2; k = 1,2, . . , n) be eigenvaluesof A. Suppose that Wlk
sin(T2
-
T1)Wlk
.
WZk
sln(T2- ?-l)@k’
where wlk
=
r - l@kl ___ 7-1 +T2 ’
W2k
=
-. Tl
ok +T2
(k=
1,2,...,n)
Second
4606
Then the solution
~(4)
World
Congress
FIG.
1. Example
of (1.1) vrith the initial G(d)
+
of Nonlinear
Analysts
of Theorem
fiLnction bo
3.4
4 E C([-r2,
01, R”) satisfies that
0 + ml
where b. = (I + (rl - r,)A)-‘(4(O)
Example
1. Consider
#(t)dt). -P2 the case that ~1 = 1, ~2 = 2, and A and 4 is given by A=
(‘[
We have the asymptotic equilibrium the solution of (1.1) with the initial
if
ii),
- A/-”
4(t)=
(a$$).
point as b. = col( 11/2,3,1) by Theorem function 4, tends to b0 as t -+ 00.
3.4. Fig. 1 shows that
REFERENCES 1. K.COOKE 16, 75-101 2. O.DIEKMANN,
Complts, 3. 4. 5. 6.
& J.YORKE, (1973). S.A.VAN
Some
Equations
Modelling
Growth
Processes
and
Gonorrhea
Epidemics
Math.
Biosci.
GILS, S.M.VERDUYN LUNEL & H.-O.WALTHER, Delay Eq-uatrons: ~&nctzonal-, and Nonlinear Analysis. Springer-Verlag (1995). Theory of finctional Diflerential Equatzons, Springer-Verlag (1977). & S.M.VERDUYN LUNEL, Ithtd~tion to finctional Diflerential Equations, Springer-Verlag (1993). Delay Differrntzal Equations With Apphcatronr in Population Dynamzcs, Academic Press (1993).
J.HALE, J.HALE Y.KUANG, K.MURAKAMI, tion, Funkcial.
Asymptotic Constancy Ekvac. 39, 519-540 (1996).
and
Periodic
Solutions
for Linear
Autonomous
Delay
Differential
Equa-