Internat. J. Math. & Math. Sci.
VOL. 14 NO.
(1991) 111-126
III
ON APPROXIMATION OF THE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
BY USING PIECEWlSE CONSTANT ARGUMENTS
ISTVAN
GYRi
Department of Mathematics, The University of Rhode Island
Kingston, Rhode Island 02881 USA
and
Computing Centre of A. Szent-Gyrgyi Medical University
6720 Szeged, Pecsi u. 4/a., Hungary.
H
(Received June 12, 1989 and in revised form June 30, 1989)
By using the Gronwall
ABSTRACT
Bellman inequality we prove some
limit
relations
between the solutions of delay differential equations with continuous arguments and the
of
solutions
some
arguments(EPCA).
related
EPCA are
delay differential equations with piecewise constant
strongly related to some discrete difference equations
numerical analysis, therefore the results can be used to compute numerical
arising in
of delay differential
solutions
equations of neutral
equations.
type by applying a
We also consider the delay differential
generalization of the Gronwall
Bellman
inequality.
KEY
WORDS
AND
PHRASES.
Differential
equations
of
retarded
differential equations with piecewlse constant arguments,
neutral
and
numerical
types,
approximation of
solutions, difference equations.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
34K, 39A, 65L.
INTRODUCTION.
Recently, Cooke and Wiener [4] introduced a new class of delay differential
These, so called delay differential equations with piecewise constant
equations.
arguments (EPCA) consolidate several properties of the continuous dynamical systems
generated by delay differential equations ([9]) and of the discrete dynamical systems
generated
by
equations
with
equations
(see, e.g.
difference equations ([i0]). In fact, a wide class of delay differential
piecewise
equations of some
Our aim in this
solutions
constant
arguments
can
be
rearranged
to
some
difference
[I], [2], [8]) which remind us of the numerical approximating
delay dfferential equations.
paper is to establish some approximating results for the
of delay differential equations via the
solutions
differential equations with piecewise constant arguments.
Consider the
delay differential equation
of
some
related delay
I.
112
GY’RI
m
(t) + Po(t)x(t) +
m
i
for i
where
Pi
R+
R
Let
[,]
denote
0
E Pi(t)x(t-Ti)
i-i
Set
max
lim
integer function, N
greatest
k e N, (k
For a fixed
(i.I)
0
0,I
are positive real numbers and for i
Ti
are continuous functions.
the
>
t
I),
--
i
integers.
the set of nonnegative
and define three delay differential equations
h
set
m,
with piecewise constant arguments as follows
m
6(t) + Po(t)u(t) +
t
-[
Pi(t)u([
i-i
and
+(t) + Po(t)v([
t
t
]h) + X
t
]h)w([
t
m
]h) +
pi([
t
i-i
>
0
]]h) -O,
t
]]h)
-[
Pi(t)v([
i-I
and
(t) + po([
i
t
]h)w([
0
-
-[
and
some
EPCA
special
with
arguments
(1.2) h
>
]]h)
(1.3) h
0
O,t
)
O.
(1.4) h
Wiener, Cooke and Shah in [5]
by
Note here, the theory of EPCA was initiated
[ii]
t
t
and
]h have been introduced in [6].
In the second section of this paper we show that the solutions of Uh(t), Vh(t)
of equations (1.2) h, (1.3) h and (1.4) h, respectively, approximate the
and Wh(t)
0
solutions x(t) of Eq. (i.i) uniformly on any compact interval of [0,), as h
an
LI
is
if
(I.i)
on
uniform
are
[0,),
these
approximations
We also prove that
perturbation of (t)
0, that is for all i
are based on the well-known Gronwall
In the
third
section
we
0,I
m,
t<
Pi(t)
Our
proofs
Bellman lemma.
of
some
extend
results
these
delay
to
by using a generalization of the Gronwall
differential equations of neutral type
Belman lemma for the nonnegatlve solutions of the following inequality
y(t)
where
)
0
f(t)
f(t) + by(t-) +
t
afoY(S-u)ds
t ) 0
is a given nondecreaslng continuous function and
a
> O, b
0,
0
and
are given constants.
2. DELAY DIFFERENTIAL EQUATIONS.
Consider the delay differential equation
m
(t) + Po(t)x(t) + X Pi(t)x(t-i)
0
t
->
0
(2.1)
i-I
where we state the following hypotheses
(H I
for i
1,2
m,
i are
positive real numbers and
max
lim
i
(H 2) for i
0,i
m, the functions Pi
R are continuous.
The initial conditions associated with (2.1) are of the following type
e c
x(s)
s
(s), -T
0
C([-,0],R)
R+
(2.2)
APPROXIMATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
(HI)
It is known ([3]) that under hypotheses
(H2)
and
113
the initial value problem
(2.1) and (2.2) has a unique solution on [oT,) which is continuous on [-T,) and
The solution of (2.1) and (2.2) is denoted by
[0,)
continuously differentiable on
x()(t).
Let
> I be
k
a fixed integer and set
h
By using
h
we define the
m
i
t
+ E Pi(t)u([
i=l
and
(t) + P0(t)v([
t
m
]h) +
Pi(t)v([
i-i
and
t
(t) + p0([
]h)w([
m
t
--
following three delay differential equations with plecewise
constant arguments"
6(t) + P0(t)u(t)
(2.3)
]h) +
Y
pi([
i-I
]]h)
T
t
t
0
i
>
t
-
]]h) -0
T
t
]h)w([
(2.4) h
0
t
i
> 0
]]h)
(2.5) h
(2.6) h
0, t>0.
With these equations we associate the following initial conditions
u(jh)
#(jh)
for
j
ok
0
v(jh)
#(jh)
for
j
-k
0
w(jh)
#(jh)
for
j
-k
0
and
(2.7) h
(2.8) h
and
(2.9) h
e C
respectively, where
By solution of (2.4) h and (2.5) h we mean a function Uh(#)(t) which is defined
0} U (0,) and satisfies the following properties:
(-k
set
on the
(a) Uh()(t) is continuous on [0,)
t
) with the
[0
(b) the derivative h()(t) exists at each point
possible exception of the points
Uh()(t)
u(t)
(2.8)h
and (2.6) h
First we
Eq.(2.1)
[nh, (n+l)h)
finite one-sided
each
on
Vh()(t)
and
Wh()(t)
interval
of the initial value problems
(2.9) h, respectively, are analogous and they are omitted.
Assume
that
hypotheses (H I) and (H 2)
hold
and
C
PROOF.
t
satisfies
Then each one of the initial value problems (2.4) h
(2.9) h has a unique solution.
and (2.6) h
e
(2.8)h
where
prove the following existence and uniqueness result.
LEMMA 2.1.
and
N),
n e N
for
The definitions of the solutions
(2.5)h
nh, (n
exist
derivatives
(c) the function
[nh,(n+l)h]
t
-
Consider the initial value problem (2.4) h
one has
t
Ti
]]h
(n
ki)h
(2.7) h.
k >_ I)
h
(2.7) h,
For all
(2.5) h
n
>
0 and
GY’O’RI
I.
114
ri
ki
where
i
i
Thus for
t
E
n
[nh,(n+l)h)
(2.4)h
is equivalent to the following equation
m
6(t) + P0(t)u(t) +
E Pi(t)u((n-ki)h)
i-I
0
By using the initial condition (2.7) h we find that u(t)
(2.7)h on [0,h) if and only if u(t) is a solution of
is a solution of
(2.4)h
and
m
6(t) + P0(t)u(t) +
0
Pi(t)(-kih)
0
< h
t
i-I
and
(0)
u(0)
But
initial
this
value
problem
[O,h)
and
exactly
has
which
solution
one
is
and
(h)
are defined, as
easily show, by the method of steps, that
u(t)
exists and is unique on
differentiable
on
u(h)
By using an argument similar
(2.8) h and (2.6) h
(2.5)h
The proof is complete.
[0,)
solution on
C
Let
be
defined
where C I denotes the set
of
C -{@
by
continuously
Then one can
[0,(R))
to that given above one can prove that each one of
(2.9) h has one and only one
value problems
the, initial
h
t
CI
E
m
(0-)
+
P0(0)(0)
+ X
0
Pi(0)(-ri)
i-i
differentiable
continuously
of
maps
[-r
0]
into
R.
We now are ready to prove our convergence theorem.
THEOREM 2.1.
(HI)
Assume that hypotheses
and
(H2)
C
hold and
is given.
Then the following statements are valid:
x()(t), Uh()(t), Vh()(t) and Wh()(t
value problems (2.1)- (2.2), (2.4)h- (2.7)h (2.5)h- (2.8)h and
respectively, satisfy the following relations for all T > 0
llm max ]x(#)(t)
lira max x()(t)
Vh(#)(t)
Uh()(t)
(a)
lim max [x()(t)
h0 0.<t.<T
CI 0
If
(b)
(2.6)h- (2.9)h
h0 0,<t.<T
h0 0,<t,<T
M
of the initial
the solutions
then
0
Wh()(t)
T > 0
all
for
(2.10)
there
exist
constants L-
L(T,) and
such that
M(T,#)
x(#)(t)
Uh(#)(t)
< Lh
0
-<
t
<- T,
h
> 0
(2.11)
and
Vh()(t) -<
x()(t)
(c)
on any
If for all
compact
N- N(T,#)
i
Mh
0,I
subinterval of [0,)
-<
0
n
< T,
t
h > 0
the functions
C
and
(2.12)
Pi(t)
then
are Lipschitz-contlnuous
there
exists
a
constant
such that
x()(t)
Wh(#)(t)
< Nh
0
-<
t
-< T
h > 0
REMARK 2.1. It is known from Lemma 2.1 in [7] that
and dense set in C
(2.13)
C
is
a
nonempty
APPROXIMATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
PROOF OF THEOREM 2.1.
-Uh()(t)
Consider the solutions
(a)
Then from (2.1) and
value problems (2.1)
(2.4)h
(2.2) and (2.4) h
x()(t)
(2.7)h
and
Uh(t
respectively.
we find
m
(t)
x(t)
-- --- - --
of initial
115
6h(t) --Po(t)(x(t)
Uh(t))
Pi(t)(x(t-Ti)-u([
Ti
t
])h))
i-i
for all
Thus
> 0
t
the
r(t) <-
-Ix(t)
(t)
function
Uh(t)l
m
t
;0(
(s) +
Po(S)
Y-
s
([
and for all t
>
0
lh)
]]h)ds + f(t)
i-I
where
;
f(t)-
IPi(S)
(s) +
s
[Pi(S)[Ix(s-i)-u([
m
t
[0 d PO (s)
(0) -0
satisfies
Pi(S)nX(S-,i)
i
On the other hand,
s
x([
]]h
lh)
for all
s
t )0
ds
s ) 0
and by using the initial conditions one has
for all
Z
s
([
K"
i
I
i
n
]]h)-
and for all
s
tK-
Therefore
we
T
s
I(( K"
s
)
[-]h
]]h)
s
Uh(
T
i
K"
]]h)I-o,
such that
0
0
that the function
find
i
y(t)
max (s)
satisfies
the inequality
0st
m
;vi_Ell Pi(S) lY(s)ds + f(t)
y(t)
where we used that
f(t)
inequality ([3]) we find
y(t)
t
0
is a monotone increasing function.
f(t)
e
t
)
By Gronwall
0
By using the definition of the greatest integer function, we find for all
and for all
s ) 0
-h
Set
(x’t,h)
s
T
s
]]h
i
h
Bellman
(2.14)
i
1,2
(2.15)
max
Then from (2.15) it follows that
[x(s-Ti)-x(
for all
0
s
t
s
and for all
m
iO
]]h)
i
o(x’t,h)
i
m
Also,
(2.17)
m
I. G Y’O’RI
116
and clearly (2.14) yields
t
0
y(t) -<
t(t)
Since
e
m
E
ds
Pi(S)
i=O
t
-Ix(t)-Uh(t)
e
Pi(S)
(x’t,h), t > 0
ds
x()(t)-Uh()(t)
t
x()(t)-Uh()(t)
m
E
0 i=l
y(t), we obtain
m
0 i-0 Pi(S)
ds
Y
Pi(S)
(x’t,h)
ds
(2.18)
i
for all
>
h-
0
and
t
>
0
But the solution
(2.2) is continuous on
x()(t) of (2.1)
(2.18) it can be easily seen that for all fixed T > 0
[-T,)
and hence from
m
x()(t)-Uh()(t) -<
max
as
m
Oi_lIPi(S) s (x’T,h)
T
i 0
e
O,
(2.19)
0
h,
By using an argument similar to that given above, we have
m
max
lx((t)-Vh()(t)
;
e
i 0
0tT
as
Pi(S)
s (x;T,h) 0,
(2.20)
h 0
It remains to consider the solutions
the equations (2.1) and (2.6) h, respectively.
t
(t)-(t)-p0([
t
]h)(x([
x(t)
x()(t)
and
w(t)
Wh(#)(t
of
In that case
t
]h)-w([
]h)) +
m
Ti
i
t
t
t
pi([]h)(x([
-[-]]h))+(Po(t)-Po([]h))x([]h
-[-]]h)-w([
i-I
m
t
i
t
(Pi(t)-Pi([]h))x([-[-]]h)
i-i
m
Z
i-i
for all
Set
(t)
+
t
P0(t)(x(t)-x([]h))
+
+
t
Pi(t)(x(t-i)-x([ -[--]]h))
t ) 0
x(t)-w(t)
t
>
Then
0
(0)
0
and by integrating we obtain
m
(t)
i
s
s
s
; (I PO([]h) l([]h)
+ Zlpi([lh) l([ -[-llh)ds+g(t),
i-1
tO,
(2.21)
where
v
m
i
s
s
t
s
z lpi(s)-pi([lh)llx([K
g(t)-fod
Po(s)-Po([lh)l([lh)+
-[--llh) lds
m
S
T
i
+
t>0
117
APPROXIMATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
But for all
I
i
and for all
m
-"r
s
>
the inequality
0
i
[- [-]l
yields
([
Set
(s)
y(t) -max
0<s<t
for
i
s
([
i
s
g-
t
11
h )- 0
0
Then
]]h) < y(s)
<
i
i
<
and
m
s
> 0
Moreover, g(t) is a monotone increasing function and hence (2.21) yields
t
0
y(t) <
-
Y
Ipi([
l)y(s)ds
]h
+ g(t)
t
>
>
0
0
i-0
By Gronwall-Bellman lemma (see, e.g. [3]) this yields
m
t
y(t) < e
s
;0 i-0[PiC[
]h)
ds
for all
g(t)
t
(2.22)
On the other hand,
m
for all
O, where
t
m
I_ Pi(S) s e(x’t,h)
Ix<s>
e(x’t,h)
By using the definitions of
T > 0
(2.23)
is defined in (2.16).
r(t)
and
y(t), from (2.22) and (2.23) we obtain for all
m
max
xC)(t)-w(#)(t)l -<
i 0
e
as max I*(,)1
(; i_Zlpi(s)-pi([lh)l
0
T<s<T
0<t<T
m
+
;zlPi(s)lds
iO-
o
(x’T,h))
as
h 0
(2.24)
The proof of statement (a) is complete.
(b)
x()(t)
all T > 0
Assume that
e
C
Then
it
can
of (2.1)
be
easily
(2.2) is continuously dtfferenttable
there exists a constant K
K(T) such that
v(x’T,h)
K h
h
on
seen that the
[-T,)
and
0
solution
clearly for
(2.25)
Set
m
L
Ke
i 0
m
i 1
and
m
M
i0
Ke
m
t 0
Then
(2.11)
and
if we assume that
(2.12)
Pi(t)’s
follow from
(2.19) and (2.20)
respectively
are Lipschltz-contlnuous on any compact interval
Moreover,
[0,T], then
I. cYS’i
118
there exists a constant
m
for all
Set
m2
ml(T
]h)
-< ml(s-[
s
IPi(s)-Pi(
X
i-O
such that
mI
s
]h) -<
mlh
[O,T]
s
x
max
0-<s-<T i-0
Pi (s)l
and
m
NThen
(2.24)
and
em2T(Tml + ;0 i-0X IPi(S)l ds)
(2.25)
yield inequality
(2.13)
the proof
and
of
the
theorem is
complete.
In the next result we give a condition which guarantees that the first two
approximations are uniform on the half line [0,m)
Assume that hypotheses (H I) and
THEOREM 2.2.
(H2)
are satisfied and for all i
m
0,i
; Pi<t)l
C
Let
Vh()(t)
<
dt
(2.26)
be a given initial function.
of equations (2.1),
(2.4) h and
Then the solutions
(2.5)h
x()(t),
Uh()(t
and
respectively, satisfy the following
relations
sup
Ix(#)(t)
Uh()(t)
0
as
h
0
(2.27)
Ix()(t)
Vh()(t)
0
as
h
0
(2.28)
t,O
and
sup
PROOF.
By using (2.26)
(2.2) is bounded on
it can be easily seen that the solution of (2.1)
[0,), and hence from (2.1) we find
m
;;i
Thus for all
h
0
a(h)
sup
x(t)l <
> 0
h6t6T
T)h
and
Pi(t)l dt
as
h
0
6 h
,(;h)- max {] (t2)-(tl)
and
using the definition (2.16) of (x;T,h) and (2.29) we obtain
t2-tl
Set
tI
t2
T
0
(x’T,h)
htT
for all h > 0
b(h)
In that case, by using (2.26), (2.19) and (2.20) yield
m
max ((#;h)
max
OtT
and
x(@) (t) -Uh(#) (t)
a(h)}
e
i 0
m
Inui-IX Pi(S) ds b(h),
[-,,0]
By
APPROXIMATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
119
m
x()(t)-Vh()(t) -<
max
e C
#
respectively. Since
0i_OE IPi(S)
a(h)
and
and clearly (2.27)
0
h
m
i0
e
0
h
as
T
ds b(h)
>
0
we find that
0
and (2.28) are satisfied.
b(h)
0
as
The proof of the theorem is
complete.
following result shows
The
that
(2.4)h
equations
f(n+l)
af(n)
Assume
THEOREM 2.3.
1
Set
is an integer.
h-
T
Then
(a)
-
of (2.4) h
t
-a(n)e
for all
nh
-<
t
< (n+l)h and
the
first
and (H 2) are satisfied and
(2.7)h
> 0
k
(2.30)
m
is given by
ds a(n-ki)
nhPi <s);eOP0(Sl)dSl
i-i
n
(2.6) h are
s
-;Po(S)ds i;0Po(s)ds m
Uh()(t
(HI)
i- 1
for all
Uh(#) (t)
the solution
paper
f(n)
that the hypotheses
ri
ki
and
(2.5) h and
In this
strongly related to some discrete difference equations.
forward difference of a function f(n) is denoted by
t
where
{a(n)}
is a
(2.31)
sequence which satisfies the
difference equation
a(n+l) -a(n)e
; Po(s)ds
m
Z
e
i-I
a(n)
(b)
-<
nh
;PO(Sl)dSlds
r(n+llh(s)
nh i e
#(nh)
Vh(#)(t
t
(l-[nhPO(U)du)b(n)
t
< (n+l)h and
n
>
a(n-kl)-O
n>O
(2.8) h is given by
of (2.5) h
m
t
+ X
i-I
[nhPi(u)du b(n-k i)
0
(2.32)
0
-k
n
the solution
Vh()(t)
for all
+l)hpo (s)ds +
where
{b(n)
(2.33)
is a
sequence which satisfies the
difference equation
b(n) +
+l)pho(u)du b(n)
;
m
+ X
i-I
b(n)
(c)
the solution
(n+l)hp i (s)ds
nh
(nh)
of
O, n > 0
(2.34)
0
-k
n
Wh()(t
b(n-ki)
(2.6) h
(2.9) h is given by
m
Wh()(t)
-c(n)
(Po(nh)c(n) +
Pi(nh)c(n-ki))(t-nh)
(2.35)
i-i
for all
nh
<
t
< (n+l)h and
difference equation
n
>- O,
where
{c(n)}
is a sequence which satisfies the
GY0"RI
I.
120
m
Ae(n) +
+ h
hP0(nh)c(n)
i-I
c(n)
We prove (a) only.
PROOF.
be omitted.
d
(u(t)e
for all
0P0(s)ds
m
-[
Pi(t)u([
i-I
t ) 0
m
i-i
t
i-I
hPi(s)e
where we use the notation
a(n)
]]h)e
Then we have
0P0(Sl)dSl
t e
(2.37)
[nh,(n+l)h),
s
t
SnhPi(s)e
oPO(Sl)dSlu([
0P0(Sl)dSl ds
Ti
s
a(n-ki)
for
n
0,I
{a(n)}
(2.32).
satisfies
]]h)ds
>-
0
(2.38)
{a(n)} defined by
by continuity of the
But,
The proof of the
(n+l)h, (2.31) yields (2.32).
solution, as t
(2.7)h.
satisfies (2.31) with the sequence
-Uh(#)(t
We now show that
(2.38).
Ti
n- -k
for
u(nh)
u(t)
(2.4)h
s
m
a(n)e
This means that
- -of
Uh($)(t)
it follows, for
Po(S)ds
Po(S)ds
(2.36)
The proofs of (b) and (c) are similar and they will
t
I
Po(s)ds-u(nh)e
0]
t
By integrating, from (2.37)
u(t)e
>
n
0
u(t)
Consider the solution
t
d-
n- -k
#(nh)
0
Pi(nh)c(n-ki)
theorem
is
REMARK 2.2. Consider the nonlinear delay differential equation
(t) + f(t,x(t),x(t-;l)
x(t-,m)) -0 t ) 0
-
with the initial condition
where
and
m
R+xR
f
max
lin
T
x(t) -#(t)
R is a continuous
0
and
function
i
for
----- -----f
provided that
is
Lipschitz
a
(2.39)
e C
#
and
(2.40)
all
i
i
m
Then one can generalize Theorem 2.1 for the initial
(2.40)
(2.39)
t
complete.
continuous
value
function
i > 0
problem
and
the
approximating equations are given as follows"
6(t) + f(t,u(t), u([
and
(t) + f(t,v([
t
]h),v([
and
(t) +
where
i
t
h-
and
k
> I
I
t
t
f([]h,w([]h),w([
]]h)
t
]]h)
v([
[-]]h)
w([
TI
Tm
t
u([
is a given integer.
]]h)) -0
t
m
t
m
(2.41)
]]h)) -0
(2.42)
]]h) -0
(2.43)
Note that the equations (2.42) and (2.43)
can be rearranged to some difference equations while (2.41) cannot
3. NEUTRAL CASE.
Consider the delay differential equation of neutral type
APPROXIMATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
d
[x(t) + qx(t-r)] + px(t-y)
dwhere
p, q e R, p
0
#
F, r e (0,)
and
0
and
121
(3.1)
0
t
(3.2)
max {F,r}
r
The initial conditions associated with (3.1) are of the following type
c C([-r,0],R)
0
s
-r
x(s)
(s)
(3.3)
Then it is known ([3]) that the initial value problem (3.1) and (3.3) has one and only
The unique solution of (3.1) and (3.2) is denoted by x().
one solution on [-,)
Let
k
r
and
T
h
be a given integer and
i
)
Set
#h"
(3.4)
In this section we approximate the solutions of (3.1) with the solutions of the equation
d
d--
t
qu(t-mhh)) + pu([
(u(t) +
h]h)
0
Note that Eq. (3.4) has two delayed arguments
is
the
while
continuous
second
one
is
for
t ) 0
t-mhh
piecewlse
(3.5)
t
and
h
]h
The first one
Therefore
constant.
the
initial
condition associated with (3.4) is of "mixed" type
u(s)
h(S)
-mhh
where for all
-mh
i
h(S)
(ih)
A function
defined on the
0
and
u(jh)
-I
and
s
s
(i+l)h-s
0
"h
(3.6)
+ #((i+l)h) s-lh
h
h
u(t)
Uh(#)(t
[-mhh,0 U {-h
set
#(Jh), j
(ih,(i+l)h]
(3.7)
is a solution of (3.5)
0}
U
(0,m)
and
and
(3.6) if
satisfies
the
u(t) is
following
properties:
(a)
u(t)
(b)
the
is continuous on
derivative
[0,-)
u(t) +
of
qu(t-mhh
exists
with the possible exception of the points
each
at
nh
t
(n
point t
N),
[0, -)
where finite
one-sided derivatives exist.
(c)
the function u(t) satisfies Eq. (3.5) on each interval [nh,(n+l)h) for nz0
and the initial condition (3.6) is satisfied.
With initial value problem (3.5) and (3.6) we associate the delay difference
equation of neutral type
A(a(n) + qa(n-mh)
+ pha(n-h)
0
for
-k
0
n
)
0
(3.8)
and the initial condition
a(n)
where
A
#(nh)
for
n
(3.9)
denotes the first forward
difference, i.e., for a function f(n)
f(n)
f(n)
f(n+l)
Note here, the initial value problem (3.8) and (3.9) has a unique solution which is
denoted by ah(#)(n
We now are ready to state and to prove the following basic existence and
uniqueness result for the initial value problem (3.5) and (3.6).
LEMMA 3.1.
inteEer.
Let
#
Assume
C
be
that (3.2)
Eiven.
is satisfied and
h
T
v where
k
> I
is an
Then the initial value problem (3.5) and (3.6) has a
I. GYORI
122
unique solution
u(#)(t)
Moreover,
u()(t)
is given
n+l)h-t
u()(t)
a(#)(n)
+ a()(n+l) t-nh
h
h
for all t e [nh,(n+l)h) and for all n
0
where
(3.8) and (3.9).
PROOF.
d
d--
(3.10)
a(#)(n)
It can be easily seen that a function
n e N
and only if for all
and
is the unique solution of
u(t)
is a solution of (3.5) if
[nh,(n+l)h)
t
(u(t) + qu(t-mhh)) +
by
pu((n-h)h
0
(3.11)
qu((n-mh)h
phu((n-h)h
(3.12)
or equivalently
-qu(t-mhh) +
u(t)
u(nh) +
By the method of steps it can be easily seen that Eq. (3.12) has one and only one
solution satisfying (3.6), and clearly the initial value problem (3.5) and (3.6) has a
unique solution u(t)
By continuity of the solution, as t
(n+l)h,
Uh()(t)
(3.12) yields
qu((n-mh)h)) + phu((n-h)h
(u((n+l)h) +
for
(nh)
u(nh)
and
0
-k
0
for
n
0
This means that
a(n)
u(nh) where a(n)
is’the unique solution of (3.8) and (3.9). But from (3.11) it follows that the function
is a piecewise linear function on 0
t <
u(t) + qu(t-mhh
On the other hand,
s
0
u(t) is a piecewise linear function on -mhh
Therefore u(t) is a piecewise
linear function on [0,) and clearly (3.10) is satisfied. The proof is complete.
We now are in a position to state the following convergence theorem.
THEOREM 3.1.
(3.3) and (3.5)
h-
T
0
x()(t)
Uh()(t)
and
C
is a fixed initial
of the initial value problems
(3.6), respectively, satisfy the following limit relation
Uh(#)(t)l
x(#)(t)
sup
as
Assume that (3.2) is satisfied and
Then the solutions
function.
(3.1)
n
(3.13)
+
k
or equivalently as
T > 0
for all
0
The proof of the above theorem uses a generalization of the Gronwall-Bellman
lemma for a continuous solution
y
f(t) + by(t-) + a
y(t)
[-a,)
[0,m)
t
;0Y(S-u)ds
t
of the inequality
0
(3.14)
provided that
(i)
a
> 0, b
(ii) f
LEMMA 3.2.
y
[-u,m)
[0,m)
0,
[0,m)
>
[0,)
0
> 0 are
and
constants and
a
max {,).
is a continuous and nondecreaslng function.
Assume that (i) and (li) are satisfied and the continuous function
satisfies (3.14).
Then there exists a unique constant
c
> 0
such
that
cbe
-c
+ ae "c
(3.15)
c
and
y(t) -< d(t)e ct
where
for
t
> 0
(3.16)
APPROXIMATION OF SOI,UTIONS OF DELAY DIFFERENTIAL EQUATIONS
d(t)
f(t)
max
e-CSy(s)|
max
_a<s<O
1-be’C
Moreover,
holds.
(i)
d(t)
t
>
(3.17)
0
It is clear that equation (3.15) has a unique solution
PROOF.
that
for
be
i
f(t)+
max
l-be
-c
Let
e’CSy(s)
+ }
max
-c
> 0 be given
> 0
for
123
c
> 0
(3.18)
-o,
t
provided
and set
-asO
where
f(t)
t
0
f(0)
-o
t
f(t)
< 0
Then it can be easily seen that
d(t)e ct
y(t) <
t e
for all
and from (3.14) it follows
[-o,0]
((t)+) +
y(t) <
(3.19)
y(s-)ds
by(t-) + a
We now show that (3.19) is valid for all
t
> 0
(3.20)
0
t
Otherwise, there exists
tI
> 0 such
that
y(t) <
dt(t)e ct
-o
t
< tI
d(t I) e ctl
and y(t I)
(3.21)
But in that case (3.20) yields
Y(tl) <
f(t)
Since
((tl)+)
d(tl-)e c(tl’) +
+
is monotone nondecreasing we have that
Y(tl) < (tl)+
a
C
e
+
dz(tl) e
-c d(tl)
f(tl)+
f(t)
and
d(t)
are
Hence by applying (3.15) we obtain
monotone nondecreasing.
(tl)+
id(s-)eC(S’)ds
a
(1-be
+
CeCt I
+
a
C
e -c
d(tl)(e ctl
a
d(t l)eCtl (e -e +-C
C)d(tl)
+
e -c
I)
i) +
d(tl)e ctl
ct
d(tl)e I
But from (3.18) it follows
and hence
(l-be’C)d(tl)
(t I) +
y(t I) <
d(tl)eCtl
This contradicts (3.21), and clearly (3.19) is satisfied for all
0 .From (3.18) it
t
can be easily seen that
d(t)
lim d(t)
0+
t
therefore (3.19) implies (3.16), as
REMARK 3.1.
c- a
If
f(t)
cO
0+
0
The proof is complete.
[0,)
is constant on
is the unique positive solution of (3.15) and
Lemma 3.2 reduces to the well known
REMARK 3.2.
Gronwll-Bellman
d(t)
and
co
b
0
and
for all
t
0
then
> 0.
Thus
inequality ([3]).
One can easily generalize Lemma 3.2 for the case of several delays
I. GYORI
124
in the following form:
[-a,)
y
Assume that
y(t) -< f(t) +
[0,)
is a continuous function such that
n
t
0
biY(t-i) +
i-1
(a0Y(S)
>
+ I ajy(s-j))ds
j-1
t
aj >
-< J
0
provided that
(i)’
bi
> O, i > 0,
-<
(i
-< ),
i
and
0
(0
n)
and
n
0, (ljn), are constants
j
j-0
[0,)
[0,)
(ii)’ f
Then the equation
c
-
aj >
i-I
for
max
max
max
i’
u]
lj n"
li
n
bie ci
d(t)- max {(I
and
is a continuous and nondecreasing function.
+ a0 +
c
aje
j -t
has exactly one positive root c and
d(t)e ct
y(t)
where
0
0
t
bie’Ci)’If(t),
e’CSy(s)}
max
-os0
0
t
We now are in a position to prove Theorem 3.1.
PROOF OF THEOREM 3.1.
Uh()(t)
Consider the solutions
of the initial value problems (3.1)
x()(t)
x(t)
(3.3) and (3.5)
and
Uh(t
(3.6), respectively.
Set
[
Yh (t)
x(0)
Since
-Uh(0)
x(t)
Uh(t)
t
0
>
0
-T-<t<0
Yh(t)
we find that
from (3.1) and (3.5) it follows
d
d--
(x(t)
Uh(t +
quh (t.mhh)) + px(t- r)
qx(t-r)
pu([
and clearly
t(Yh(t)
x(t-mhh))) + pyh([
+ qyh(t-,h) + q(x(t-r)
x([
p(x(t-’)
t
lh]h))
-0
0
t
Thus, from (3.3) and (3.6), by integration we find
Yh(t) + qyh(t-mhh) +
+ p
x(t-,h)) +
q(x(t-r)
(x(s-r)
S
x([
lhlh))ds
p
t
’oYh([
-0
Set
fh (t)
q]
max
0,<s,<t
Ix(s-r) -x(s-mhh)J +
p
;
--
is continuous on
x(s.F) -x([
t
s
t
[-T,).
t
lh]h)
+
lYh(t’mhh)l
+
I
S yh([
-0
lh]h) +
h]h)ds
> 0
lh]h)l ds
t
In that case
Yh(t)l "< fh (t)
On the other hand,
h]h)ds
t
>
0
->
0
APPROXIMATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS
Let
Zh(t)
s
yh(
Yh(S)l
max
-Tst
0
h]h)
fh (t)
Zh(t)
s
wherever
4 Zh(t-mhh)
+
Yh(t)
Since
0
h]
+ P
0
-mhh
and
0
t
125
we find
t
;oZh(S)ds
t ) 0
(3.22)
By virtue of Lemma 3.2, we have
dh(t)e cht
Zh(t)
where
0
t
is the unique positive solution of
ch
and
dh(t)
max
el 4 e’Cmhh + pl c
((I-I 4 e -Chmhh )’Ifh(t)
(I-I 4
(3.23)
max
e
-asO
e’chmhh)-Ifh(t)
ChSzh(s)}
t ) 0
Thus
Yh(t)l
Zh(t)
(i-I
e
-Chmhh )’Ifh(t
Now, by using the continuity of
for
OtT
O, as
fh(t)
t
>
0
(3.24)
and an argument similar to that given in the
x(t)
proof of Theorem 2.1 one can see that for all
max
all
T > 0
0
h
(3.25)
On the other hand,
mhh
r
]h
r
as
h
therefore from (3.23) it can be seen that
0
ch
co
as
O+
h
where
co
is the
unique positive solution of
4e "cr
+ a- c
Thus (3.24) and (3.25) yield
max
OtT
for all
T > 0
Yh(t)l
x(#)(t)
max
OtT
Uh(#)(t)l
0
as
h
0
The proof of the theorem is complete.
REMARK 3.3. Applying Remark 3.2 one can easily generalize Theorem 3.1 for the
case of several delays.
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i.
2.
3.
4.
5.
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A.R. Aftabizadeh and J.Wiener, Oscillatory and periodic solutions for
systems of two first order linear differential equations with piecewise
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R.Bellman and K.L.Cooke, Differential-Difference Equations, Academic
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K.L.Cooke and J.Wiener, Neutral differential equations with piecewise
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K.L.Cooke and J.Wiener, Retarded differential equations with plecewise
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6.
I. GYORI
K.L.Cooke and J.Wiener, Stability regions for linear equations with
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7. I. Gyori, Two approximation techniques for functional differential
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