Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 405-410
405
OPERATIONAL CALCULUS AND DIFFERENTIAL EQUATIONS
WITH INFINITELY SMOOTH COEFFICIENTS
DENNIS NEMZER
Department of Mathematics
California State University, Stanlslaus
801 West Monte Vista Avenue
Turlock, CA
95380, U.S.A.
(Received May 2, 1989)
A subring MF of the field of Mikusiski operators is constructed as a
countable union space. Some topological properties of MF are investigated. Then, the
product of an infinitely differentiable function and an element of MF is given and is
ABSTRACT.
used to investigate operational equations with infinitely smooth coefficients.
KEY WORDS AND PHRASES.
Mikusiski operator, infinitely differentiable, operational
equation.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
Primary 44A40, Secondary 46F99, 34A30.
INTRODUCTION.
theory
The
of
generalized
functions
has
been
used
successfully
in
solving
problems in classical analysis as well as in simplifying the theory of differential
equations (Mikusifski
[I],
Zemanian
[2,3]).
In the field of Mikusiski operators, the
inability to define a suitable product of a function and an operator, see Stankovlc
[4], has had
a limiting effect.
For example, the theory and applications of ordinary
differential equations have been restricted to differential equations having constant
In this note we will confine ourselves to the field M of Mikusldski
We will study a subring MF of Mikusifski operators in which the
of an infinitely smooth function and an operator in MF can be suitably
coefficients.
operators
product
[I].
In Section 2 the construction of MF and a convergence in MF are studied.
Then in Section 3, it is shown that an operational equation with infinitely smooth
coefficients has only the classical solution.
Some examples are then given to show
defined.
that this is not the case if singularities are introduced.
2.
THE CONSTRUCTION OF MF AND CONVERGENCE.
The ring of continuous complex-valued functions on
[0,(R)), denoted by C, with
t
addition
and
convolution
((f*g)(t)
quotient field of C is denoted by M
I
f(t-u)g(u)du) has
an is
no
zero
divisors.
The
called the field of Mikusiskl operators.
A typical element x of M, called an operator, is written as x
f/g, where f,g c C.
D. NEMZER
406
<
for t ; 0 and 0 for t
is the function defined as
The integral operator
0.
The
t
f(u)du.
for all f E C, (4*f)(t)
integral operator has the property that,
The
0
inverse of 4, denoted by s, is called the differentiation operator.
"4 (k terms), while 0
will denote 4*4*4*
(k)
will denote the k th derivative
of 0"
k
0,1,2,..., let M
For k
{x
4k*xEC}.
M:
E
By endowing MK with the topology
induced by the countable family of seminorms
(k,m(X) sup{l(4k*x)(t)l:
0 4 t 4 m} for
x in Mk if and only if for each m
Clearly, x
n
Mk
1,2,...
m
>
4k*xn 4k*x
0,
a
is
Frchet
uniformly on
space.
[0,m].
k
Let MF be the countable union space U M
That is, x is an element of MF if x
k=0
k
in MF is said to converge to an
is an element of M for some k. Also, a sequence {x
k
element x in MF if for some some k, x ,x E M n=l,2 ...and {xn} converges to x in the
n}
Since each Mk is complete, MF is sequentially complete.
detailed discussion of countable union spaces see [2].
topology of M k.
MF,
Even though M is considerably larger than
needed
operators
for
For,
applicatlons.
For a more
MF contains many of the important
identifying the locally integrable
by
4"f/4, the collection of locally integrable functions can
be identified with a subrlng of MF. Moreover, MF contains all rational expressions in
function f wlth the operator
s.
We state without proof two lemmas.
LEMMA 2.1. Let
(1)
0n(t)dt=l
(supp
>
for all n and (il) supp
{0n}
Let
each x
S
subset
E
0nO_ [0,En],
of
0n
where
n
is not zero).
0
Then for f
E
C,and for
converges uniformly to f on [0,m].
be a sequence of functions such that on each interval
Then, for f
0 uniformly.
A
{0n’f}
0, the sequence
LEMMA 2.2.
0n
be a sequence of positive functions such that:
is the closure of the set on which
0n
each m
{0n}
a
C and each interval [0,m],
countable
X
space
union
is
0n*f
said
to
[0,m]
0*f
uniformly.
be
dense
if
for
X there is some sequence {xn} in S that converges to x.
THEOREM 2.3.
PROOF.
Let x
C is dense in MF.
k
E M for some k.
Let
{0n}
be a sequence of k-tlmes continuously
dlfferentlable positive functions satisfying the following three properties:
(1)
(iii)
I 0n(t)dt
For
each
for all n; (ll) supp
n,
continuous function f,
k
Thus
], 4
[(4k*x ),On(k)
0n(J)(0)
{On*f}
4K*xeC
0n
c_ [0,1/n], for n=l,2,...,
0, j=0,1,2,...,k.
Then,
by
Lemma
2.1,
for
any
converges uniformly to f on compact subsets of [0,).
for all n and
[(4k*x)*On (k) 1, 4k (4k,x),On 4k*x
407
OPERATIONAL CALCULUS AND DIFFERENTIAL EQUATIONS
uniformly on compact subsets of [0,).
Thus, C is dense in MF.
THEOREM 2.4.
The mapping M
F
M
{k*x*n(k)
That is
converges to x in MF.
x’y, where y e MF,
by x
F given
is sequentially
cont inuou s.
Let {x
PROOF.
be a sequence in MF that converges to x.
k*xn,k*xeC for n
converges
2
uniformly
Hence,
*x
*x
’t
...,
o
[0,m]
and on each interval
k*x.
M
Let y
C for n
F.
Thus
for some k,
the sequence
Then,
{k*x
some
for
i,
by Lemma 2.2, the sequence
1,2,... and
nt *y
e C.
{k
k+i*x*y.
converges uniformly on each interval [0,m] to
That is, {Xn*Y} converges to
x*y in
This establishes the theorem.
The collection of infinitely smooth complex-valued functions on (-,)will be
MF.
We now define the product of an infinitely smooth function and an
denoted by C
operator in M F-
DEFINITION 2.5.
For
given by
C
k
(J)k
[ (-1)J
.x
jffi0
(1)
REMARKS 2.6.
#
k
and x, denoted by
the product of
(J) (k,x)
k-J
C" and
If
(ii)
and x e M
(where
f/o
C, then .f is ordinary multiplication.
f
M
1.xfx for x
m,
f/n
[ (-llJ
n+k
j
>
n+k
F
j =o
F.
g/m
t
)
0 for t
0.
PROOF.
and x
@,#eC"
Suppose that
k
)n
nffiO
k
n
l 7.
n=O
(-i )n
J=O
()
n
(_1) j
[
j =o
k
(.#).x
[
n=0
(-I )n
()
MF,
and xc_M
k.
Thus, F(t)
0 for
(@.#).x= #.(#.x).
Then,
(J)
k-n
(kn) J O
k-J
(n_j)
0,1,2,...,n+k-l.
,(n)
k,x
k-n-J
(2.1)
k-n
The last equality follows by the transformation
Now,
To see that it
Now, let
After differentiating F n+k times we obtain
> 0 and F(J)(0+) 0 for J
#.(f/n) .(k,f/n+k).
Therefore,
THEOREM 2.7. For ,#eC
n-m,g.
if and only if f
J ,[, (J) (k,f)]_
where k is a nonnegatlve integer.
Fn+k)(t)’"
is
f)
It is not clear that multiplication is a well-deflned operation.
Is,notlce that if n
.x,
(k*x)
(kn) (,)(n)
k-n
ufJ+n and
vffin.
(2.2)
D. NEHZER
08
Hence, the
By Lelbnlz’s formula, we see that (2.1) and (2.2) are equivalent.
theorem is proved.
.
EXAMPLES 2.8.
n
n
@.s
(-l)J
()
]--0
0
(where s
(i i
t
m
s
@ (j)(0)s
n-j
for n=O
2 ,oeo
4/ is the identity operator).
6, 6
n
n!
n-m
n
(n-m) s
Example (1) follows by substituting the formula
(-I)m
(1)
REMARKS. 2.9.
@(n)
s n,
nl @(k)(0) sn-(k+l)
+
(see [I])
k=0
.s n
into the definition for
(ll)
Example (ii) follows by substituting @(t)
THEOREM
2.10.
The
mapping
MF
t
MF
into
m
(1).
into Example
@.x, where @eC
by x
given
is
sequentially continuous.
,.PROOF"
k,
to
Now,
that {x
Suppose
*x n’ *xEC
k
*x.
n
.
k
k*(@.x n
uniformly to
converges
x
to
MF.
in
{} J*[@(J)(k*xn)
(-1) j
]=0
C
k and on each interval
J*((J)(k*x)).
Hence,
{.Xn}
That
{*x n}
and on each interval [0 m]
2
Moreover, for each 0 < j
n}
converges uniformly
{J*((J)(k*x n ))}
.x
converges to
some
1,2,.
for n
[0,m],
for
is,
in MF.
converges
This
establishes the theorem.
If we consider C
.x
as a subspace of
f
(t)dtffil. Now,
for t )0, 0
Thus,
{n
(*n)(t)
then the mapping from C
To see this, let
is not sequentially continuous.
positive function such that: (i)
(iii)
MF,
(0)--I,
(il) @(t)ffi0 for
for
nffil,2,...,let
t
nt
f n(U)du
0
converges to 0 in MF.
f
n(t)
@(u)du
into MF given by
be any infinitely smooth
Itl
)
(nt).
I, and
Then,
for
each
n
and
{n.6}
does
I/n.
0
But, for each n,
n.6=n(0)6ffi6.
Hence,
not converge to 0 in MF-
In the theory of distributions, the countable family of semtnorms
Yn,m(,)-sup{I,(n)(t)l:
topology
for
C’.
In
Itlm},
the
n=O,1,2,...,
following
m
theorem,
1,2
we
is
assume
used
to
generate
that C
is
given
a
this
409
OPERATIONAL CALCULUS AND DIFFERENTIAL EQUATIONS
The proofs of the next theorem and corollary are straightforward and thus
topology.
omitted.
THEOREM
2.11.
The
from C
mapping
MF
into
given
by @
@.x
is
sequentially
continuous.
I
Let ,(t)
COROLLARY 2.12.
n=O
MF, .x
Then for x
a t
n
n
for
Itl
<
(R).
an(tn.x).
n--O
3.
DIFFERENTIAL EQUATIONS AND OPERATIONAL EQUATIONS.
to
a
It is known that the only solutions, within the framework of distribution theory,
homogeneous
linear
differential
ordinary
with
equation
infinitely
smooth
But, when the coefficients have singularities,
coefficients are the classical ones.
there may be other distributional solutions (see [5] Littlejohn and Kanwal, and Wiener
[6]).
In this section we will show that in MF the situation is similar.
For
smpllclty, we will only consider second order differential and operational
However, what follows is also true for nth order equations.
equations.
When solving the initial value problem
y’(0)
2
s *y
B2
+
ly’
+ 2y
O, y(0)
B
and
+ a2Y
This follows by the formula given in
(B2 + a1I)6 + s"
By a slmilar argument, the differential equation y" + ly’ + 2y
O,
(s*y)
1
Remark 2.9(i).
where
y" +
using operational methods, the corresponding operational equation Is
C
i, 2
corresponds to the operational equation
s 2,x
+
where y(0)
B
and y’(0)
equations
of
the
constants and
(B2+Bll(O))6
2.x
Thus,
82"
2
form s *x +
we
will
l.(S*X)
+
+
second
consider
2.x
+
a
C
i, 2
THEOREM 3.1.
+
l.(S*X)
Let a
the operatlonal equation
and a
2
be constants and
I,2
s2*x + l.(S*X) + 2.x al +
C
a2s
a2s
=.
order
operational
where a
and a are
2
The only solutlon to
is the classlcal solutlon
to the corresponding dlfferential equation.
k
Let x e M
PROOF.
s 2,x
k+1
(-I) J (k+1
I
+
a
+
such that
@
(j)
(k*x)
k+1-J
k
+
I (-1)J
k
2(J) (k*x)
a2s.
By convolvlng both sides with
k*x
k
0
k+2
we obtain
k+l
+
J=0Z (-1)J (k+lj) J+l,[1(J (k*x)
(3.1)
410
D. NEMZER
Now,
on
each
that
notice
term,
except
the
first
k*x
k*x C(0,).
(0,).
Hence,
we obtain
is also differentiable on
k*x,
in
(0,).
(3.1)
is
differentiable
By an inductive argument,
After differentiating (3.1) k + 2 times we obtain
(k*x)(J)(o+)
0
for j
(3.2)
0,1,2,...,k-I,
(k,x)(k)(O+) a2’ and (k*x)(k+l)(o +) al l(O)a2"
(,k*x)(k+2) + l(k*x) (k+l) + 2(k*x) (k) O.
Moreover,
Now let
(k*x)(k).
g
(3.3)
Then, g is the unique solution to the initial value problem g" +
g(0)
xffig,
and g’(0)
a2’
g(0)
The
2’
a
g’(0)
following
l(0)a2.
al l(0)a2"
examples
By (3.2) and (3.3),
I g’ + 2 g
k*x k,g.
0,
That is,
This establishes the theorem.
demonstrate
that
singularities are allowed in the coefficients.
the
situation
is
different
when
By using the formula given in Example
2.8(ii), it is easy to verify that x is a solution to the corresponding operational
equation.
EXAMPLES 3.2.
(1)
(ii)
x
x
t.(s2*x) + (2-t).(s*x) x
solution to t.(s2*x) + (3-t).(s*x)
6 is a solution to
s is a
0.
x
0.
REFERENCE S
I.
MIKUSINSKI, J., Operational Calculus, Pergamon Press, Oxford, 1959.
2.
ZEMANIAN, A.H.,
Generalized
Integral
Transformatlons,
Dover Publicatlons,
New
York, 1987.
3.
ZEMANIAN, A.H., Distribution Theory and Transform Analysis, Dover Publications,
New York, 1987.
4.
STANKOVIC,
B., On the Impossibility of Extending the Product of
Functions to the Field of Mikusiski Operators, Univ. Novi. Sad.
Continuous
12 (1982),
1-6.
5.
LITTLE JOHN, L.L. and KAN-WAL, R.P., Distributional Solutions of the
Hypergeometric Differential Equation, J. Math. Anal. Appl. 122 (1987), 325345.
6.
WIENER,
J., Generalized Function Solutions of Differential and
Differential Equations, J. Math. Anal. Appl. 88 (1982), 170-182.
Functional