Internat. J. Math. & Math. Sci.
VOL. 17 NO.
(1994) 113-118
113
ON A CLASS OF THIRD ORDER NEUTRAL DELAY DIFFERENTIAL
EQUATIONS WITH PIECEWlSE CONSTANT ARGUMENT
GARYFALOS PAPASCHINOPOULOS
Democritus University of Thrace
School of Engineering
67100 >anthi, Greece
(Received October 26, 1990 and in revised form May 15, 1991)
ABSTRACT. In this paper we study existence, uniqueness and asymptotic stability
the solutions of a class of third order neutral delay differential equations
of
with
piecewise constant argument.
KEY WORDS AND PHRASES. Neutral delay dfferential equations with piecewise constant
argument, difference equation, existence and uniqueness of solutions,
asymptotic
stability.
1980 AMS SUBJECT CLASSIFICATION CODE. 34K05, 34K15.
I. INTRODUCTION
Recently there has been an increase in interest in the study of
equations with piecewise constant argument. See Aftabizadeh, Wiener
Cooke and Wiener
[2], [3],
Huang
[.4],
Lamas, Partheniadis and
differential
Xu
[i],
[5],eartheni-
and
Schinas
In this paper we study the third order neutral delay differential
equations
with piecewise constant argument of the form
d
3
dt
where
te[0,),
3
(y(t) + py(t
I))
l)
--qy([t
p,q are real constants and
E.]
func-
denotes the greatest-integer
tion. It is worthwile to study equations of the form (I) because they include
both
constant and piecewise constant delays.
A function
y;[-l,oo)/R
is a solution of
(I) if the following conditions
are
satisfied:
(1)
(ii)
y is continuous on the set
d
[-I,),
2
dt
d3
2
(y(t) + py(t
I))
+ py(t
I))
(iii) -----(y(t)
3
g(t)
exists on
exists on
[0,o)
[0,)
and
g
is continuous on
except possibly at the points
[0,oo),
t=n
dt
n
6{0,I,...}
(iv)
y
where one-sided third derivatives exist,
satisfies (I) on each interval
In Proposition
Yo: [-I’0]-R
[n,n+l), ne{0,1
of this paper we prove that
continuous on
[-I,0]
and for every
}.
for every
al,a2eR
initial
function
there exists a unique solu-
G. PAPASCHINOPOULOS
114
Yo(t), te[-l,O],
y(t) of (I) such that y(t)
tion
prove also that equation (i) is
y(1)
asymptotically
not
y(2)
and
a
a
2.
(see Proposition
stable
We
2
below).
and the second order differential
We note that similar results for the first
[6].
equations of the form (I) are included in
2. MAIN RESULTS.
We prove now our main results.
PROPOSITION I. Equation (I) has a unique solution y(t) such that
y(t)
Yo(t)
t6E-I,0]
for
p # 0
a_l
t
-I
for
p
0
a
t
0
for
p
0
o
t=
a
a
(2)
t--2
2
Yo:[-I’O+R is a continuous function on [-I,0],
constants and
ao Yo (0)’ a-I Yo (-I) if p # O.
where
are
a_l,a o, al,a 2
real
The solution y(t) is defined by
(_p)n+l yo(
y(t)
l)
p)
2)_ + 0(4
2
+
where
a
2
(.6
ak+l +
2-0) ak+2 +
2p
+ I) +
0(2p-I-2- )
+
a
k
satisfies the difference equation
n
an+4 +
+k--O
(0
(_p)n-k
(P-
3)an+3
(q6
+
3p +
3)an+ 2
(I)
PROOF. Consider a solution y(t) of
t[-l, )
n-l,O
there exists a
y(n)
Then from (I) for
t >
d
O,
(2-3 + 3p
+
l)an+l
satisf.ies
which
such that
(6
+
n < t < n+l.
p)a
(2).
n
For
O.
(4)
each
We set
n(-I 0,...}.
an,
t[n,n+l), n{O,l
it holds
3
dt
3
(y(t) + py(t
(6)
-qan_ I.
I))
If
d
2
(y(t)
2
Bn-dt
/
py(t
at
t
n
to
t[n,n+l),
then by integrating (6) from
d
(7)
ne{O,l ...}
I))
n,
ne(O,l
we take
2
dt
2
(y(t) + py(t
Bn
I))
n)an_ I.
q(t
(8)
Moreover if
c
d
n= (y(t) + PY (t
I))
at
t
n
ne{0,
(9)
NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH PEICEWISE CONSTANT ARGUMENT
by integrating (8) from
(y(t) + py(t
n
I))
to
c
te[n,n+l), ne{0,1
t,
+ (t
n
Finally by integrating (I0) from
n)B
n
n
to
we have
n,2 a
(t
(10)
n-I
te[n,n+1), n@{0,1
t,
115
(5)
and using
we obtain
y(t) + py(t
an+ Pan-1+
1)
[6,p.96]
Applying now Lemma 3
(_p)n+l yo(0-
y(t)
where
t
n
Since
+ 8,
+ py(t
Bn
-
a
n
02
eCk+
n6(O,l,...}.
I,
[-1,o0
)
+ (p
t+n+l
n)3 an_
(11)
k- 6 83 ak-l)
(12)
6
(t
by taking the limit as
}
an-1 +
c
tn+l in (11)
n"
+
n
d2
g(t)
dt
i))
Obviously the functlon
h(t)
if we take the limit as
n- qan-l"
c
By eliminating
b
bn+l, Cn+
(13) by setting
n+l
I)) is continuous on [0,oo).
(9) we obtain for n{0,1
Then
(15)
an-I
an+
Using (13) and (17) for
(2-
l)a +
n
(R6
P)an_ I.
from (14), (16) and the
instead of
2)
%+2 +
+ (P
an+
If we eliminate
(p
J3n + R2
-n+l
n
(14)
from (13), (14), (15) we can easily get
c
n+l
Cn+
-+
art+2 +
(y(t) + py(t
in (I0) and using
t/n+l
(y(t) +
2
n{0,1
is continuous, using (7) we have for
d
+
n,
relatlo
which is derived
Obviously if
a_l,
ao,
p
al,
6
a
n6{0,1
}
)
2
(2p
I) a + (p +
n
5-6
an_ I.
(17)
we can easily take
3
q
)
12
an (3_2 + )4 an_ I.
2
,
there exists a unique solution
and
if
p
from
then it holds
(6- 2p +
an+ 1 +
(16)
(18)
Therefore from (12), (17) and (18) we can easily show that y(t) satlsfles (3).
Now from relatlons (14) and (17) it is easy to prove that a satlsfles the
n
ference equatlon (4).
values
and
(13)
(8) and since the functlon
in
Bn+l
n
n)2
(_p)n-k (ak+ Pak-I +
k=0
n6{0,1
If we take the llmlt as
-2
(II) we get
to
y(t) is continuous in
an+ 1 + pan
+ (t
n
I) +
0 < 8 <
using (5) we get for
n)Cn
(t
p
q
6
a
n
,
of (4) with
(resp. p
there exists a unique solution of (4) with initial values a
a
a
(resp.
0
2
Therefore we proved that if y(t) is a solution of (i) which satisfies
then is defined by (3) and (4) and is uniquely determined by
if
a2).
Yo’ al’ a2
dlf-
initial
q
6
a
(2)
p # 0
G. PAPASCHINOPOULOS
116
ao,
a_l
al, a 2 if p- 0.
Conversely we can easily show that the function
and
is a solution of
y(t)
(I) which satisfies (2). Thus the proof
defined by (3) and (4)
the proposition is
of
completed.
Using (3) and the
[6,pl10-111]
the Corollary
4
be the solution of (I) which satisfies (2). Then
it
same
argument
as in
proof
the
of
we can easily prove the following corollary.
COROLLARY I. Let
y(t)
holds
an+ 2 +
where
n
t
+ 8,
0
<_ 8 <_ I, ne{0,1
}.
In Proposition 2 of this paper we prove that (I) is not asymptotically stable.
We need the following lemma.
LEMMA I. Consider the difference equation
arr+4 + Nlan+3 + N2an+2 + N3arrbl + N4 an
n{-l,0,1,...}
where
and
i
Ni,
1,2,3,4
(19)
0
are real constants.
(19)
Then
is
asymptotically stable if and only if the following conditions are satisfied:
A2
0,
A3
0,
A5
0,
A
A
where
AI
+
3
6
NI + N2 + N3 + N4’ A2 4 + 2N 2N 3
2N2 + 6N4’ 4-4-2N + 2t/3 4N4’ A5
4N 4,
(21)
I +
+
N2 N3 N4"
PROOF. It is known that (19) is asymptotically stable if and only if
root
v
of the characteristic equation
v
satisfies
every
Ivl
< I.
4
+I v3 + /2v2 + 3v + 4
Then it is clear that
transformation
v=
z+l
z_
(22)
0
I # 0.
Therefore using
the
Mobius
NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
to
117
(22) we can take the equation
E2
3
3
where the constants
.,
2
i
1
4
%5
1,2,3,4,5
(23)
are defined in (21).
Ivl
It is easy to show that every root v of (22) satisfies
<
if and only
Rez < 0, where Rez is the real part of z and z is the root of (23). Using
if
Routh-Hurwitz criteria
[7,p.158]
we have
Rez < 0
if and only if all the
condi-
(20) are satisfied. This completes the proof of the lemma.
tions
PROPOSITION
2. Equation (i) is not asymptotically stable.
PROOF. Suppose that (I) is asymptotically stable. Then the difference equation
(4) is also asymptotically stable. We apply Lemma
i
2
q
and
2
-2 < 0 which contradicts the second
1asymptotically
stable.
Then
to (4). It is easy to show that
-2q.
condition of (20). Therefore
(I)
is not
ACKNOWLEDGMENT. I would like to thank the referee for his valuable
suggestions
which led to the presentation of this paper.
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I.
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3.
advanced
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type, Proc. Amer. Math. Soc. 99 (1987), 726-732.
HUANG, Y.K. Oscillations and asymptotic stability of solutions of first order
4.
Oscillatory and periodic solutions
of delay differential equations with piecewlse constant argument,
Proc.
Amer. Math. Soc. 99 (1987), 673-679.
Retarded differential equations with piecewise constant delays, J.Math. Anal. Appl. 99 (1984), 265-297.
neutral differential equations with piecewise constant argument,
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5.
LADAS, G., PARTHENIADIS, E.C. and SCHINAS, J.
6.
PARTHENIADIS, E.C.
7.
COPPEL, W.A. Stability n.d_.asymp_toticbehavior of differential
Heath. Boston, 1965.
J.
Math.
second
Existence theorems for
order differential equations with piecewise constant argument, Differential
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