I nter. J Mah. & Mh S i.
(1982) 729-736
Vol. 5 No.
729
DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF
LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
JOSEPH WIENER
Department of Hathemat
Pan American University
Edinburg, Texas 78539
(Received February 26, 1982)
ABSTRACT.
A unified approach to the study of generallzed-functlon and entire solu-
tions to linear functional differential equations with polynomial coefficients is
suggested.
KEY WORDS AND PHRASES.
Functional
Differential Equations,
Distributional Solutions,
Entire So lutions.
Mathematics Subject
I.
Classification Codes.
34K$5, $4K99, 46F]0, ZODOS.
INTRODUCTION AND PRELIMINARIES.
An interesting survey of recent results on entire solutions of ordinary differential equations with polynomial coefficients is given in [I].
In the present paper
we continue the study of distributional solutions to linear functional differential
equations (FDE) in accordance with the concepts outlined in [2] and [3].
There are
profound and close links between spaces of generalized and entire functions [4].
Therefore, the basic ideas in the method of proof may also be applied to the study
of entire solutions to linear FDE, especially with linear transformations of the
argument.
We investigate such linear homogeneous FDE with infinitely smooth coeffl-
clents that have solutions in the class of singular functlonals which is impossible
for analogous ordinary differential equations.
Solutions of linear FDE with
countable sets of variable argument deviations are considered in the generalizedfunction space
($80)’
conjugate to the space
satisfy the restriction [4]
I(n) ct)I
< acn
nnB
8 > 1.
$80 of testing functions (t)
that
730
J. WIENER
(n)
In the sequel,
denotes the nth derivative of the Dirac
measure and <’f, $ >
is the value of the functional f applied to the testing function
variable t.
of the real
The norm of a matrix is defined to be
and E is the identity matrix.
In [2], it has been proved that under certain con-
dltions the system
Y.
Y.
+/-=o j=o
+
(Aij
tBtj)x(J)(;jt)
tx(%t)
has a solution
r.
x(t)
x
n=0
in
(S0)’
n
(n)(t)
B >
with arbitrary
(1.1)
To ensure the convergence of series (I.i), it is
i.
sufficient to require that for n
II x,n II
<
bdn n-nO,
/
the vectors x satisfy the inequalities
n
> I
(1.2)
since
n
n=0
<
r.
n=O
for
8 < P.
n
n=O
l C"> <o>1 II x II
(cdnB-P) n
< ab
<
n=O
If series (i.i) converges, its sum represents the general form of a
linear functional in
(S0)’
with the support t
0 [5].
Some recent developments in
astrophysics posed new problems about the existence of distributional solutions also
to certain integral equations [2].
2.
EXISTENCE OF DISTRIBUTIONAL SOLUTIONS.
We look for solutions of the form (I.i) to the system
m
Y.
Z A (t) x (j)
lj
i=o j=o
LEMMA.
%’(t o
If t
o
+/-j (t))
is a fixed point of the function %(t) E C
O, then in some neighborhood of
0
(n)((t)
(2 .I)
0
t
o
(n)(t
t
tO,
o /
n
(%’(tO)) 1%’
I (-=
and
LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
PROOF.
Since
%(t)
t
%(t 0)
t
o
(t
t
o (t), (t 0)
o
There exists a neighborhood D
0,
of t
O
O
to,
%(t 0)
%(t0)
such that
t0--+ t o
0, t @
(t)
for
0 and,
0.
to,
8(n) c%(t)
But for t
%’(to).
at all points of which
o
assuming the opposite we find a sequence
hence, %’(t O
In D for t @
731
t
o
t
o)
8
(n)
( (t) (t
to))
O.
to,
(n)
(X(t)
(n) :X’ (t (t
o)
t
(2.3)
o)).
Thus, (2.3)holds for all t e D O and it remains to observe that
8
8
/ cn Icl
(n)(ct)
(n)(t)
Aij (t)
Let (2.1), in which x(t) is an r-vector and
THEOREM 2.1.
are r x r
matrices, satisfy the following hypotheses.
(1)
0 <
(ii)
XO0 (to/I
< 1,
The coefficients
Iij (to/I
kf i0Y" Ai]k(t
The series
max
J,k
(iv)
t
I Ai
E
II Aij k ![,
e C
I have a
> 1,
i
+ J
cow,non
fixed point t and
o
Z 1.
are polynomials of degree not exceeding p:
Aij(t)
P
Ai](t)
(iii)
%ij (t)
The real-valued functions
k
o
A00(t)
to)P,
A(t
p
_>
I.
converges where
;k
i
inf
j
Iklj(t o
i
+
j > 1.
The matrix A is nonsingular ad
-p-1
Xo(t O)
c
II A I!
Then, in some neighborhood of
to,
-i>lr
X’io(tO) l-p-1 II qop 11
there exists a solution x(t)
>
o.
g
(S)’ with
arbitrary 13 > 1:
x(t)
I:
n=O
PROOF.
x
n
8(n)(t to).
k
By virtue of (2.2) and the formula t
6(n)(t)
for n > k, and 0 for n < k, we obtain the equations
(-l)kn!8(n-k)(t) / (n-k)!,
732
J. WIENER
Z (-I) k
i,j,k
Xn(n+j-k)(t
t
o /
(n + j
k)!
ltjl
n+j
o,
+/-j
’j (to)
ij
for the unknowns x
Z
+ j)!
Y. (n
n+j >_k
Aij k
n
(-l)k(n
of the solution x(t).
+ k)’
lij
-I
-n-k
Hence,
ij
Aijk Xn+k-j
0, n > 0
iJ !-1 aij
Aijk Xn+k-j
/ (n +,p)! +
which can be written as
-n-k
(-l)P-k(n+k)
Y.
k-j<p
+
Imi0]-I i0-n-p Ai0p)Xn+p
Z
i>0
Since
0
0(k < p), the first sum does not include terms with
A00k
to (iv), the coefficients of
]{
Z
:0
Xn+p
00"
According
are nonsingular matrices and
-i
lei0 I-i %0-n-p A) I’ !
IIl[
[oo n
Consequent ly,
m+p-i
IIn+pll!q+ k:0’ llX+k_II,
where V is some positive constant.
(2.4)
0<q<
Using the notation
(2.5)
0<i<n
we conclude from
(2.4) that
II Xn+p II
< H(m + p)qn+
For large n, there is (m
and
Mn+p Mn+p_ I.
M
n
=M
N
+
PMn+p_I
p)qn+p
< I.
Hence,
II Xn+p
<
Mn+p_ I
Thus, starting with some N,
(2.6)
n >N
The application of (2.6) to (2.4) successively yields:
II XN+p+k II
]I
< H (m+p)
qN+PMN,
LINEAR
FbRqCTIONAL
DIFFERENTIAL EQUATIONS
733
II XN+p+(m+p)+k II <_ p2 (m + p)2qN+pqN+p+(m+p)
11 XN+p+2 (m+p)+kll
<
3 (re+p) 3qN+qqN+p+(m+p) qN+p+2 (m+P)MN
(0 < k < m + p
I).
The conj ecture
I! XN+p+n(m+p)+k It < n+l(m+p)n+lqn(N+p)+n(n+l) (re+p)/2
(2.7)
may readily be ascertained by induction for all n and the mentioned values of k, and
proves the theorem since the condition 0 < q < I makes it more restrictive than
(1.2).
3.
EXISTENCE OF ENTIRE SOLUTIONS.
We apply the method of the previous section to prove the existence of entire
solutions of linear FDE with polynomial coefficients and to evaluate their order of
growth.
THEOREM 3.1.
X
Suppose the system
p
(e)(t)
Z
i=0 j
in which
(i)
Qij and
Qij (t)
(ii)
%ij
X are (r x )
X
(j)(0) =Xj
0,
j
p
i
matrices, satisfies the following conditions:
are polynomials of degree not exceeding m;
are complex numbers such that
0 <
(iii)
Y.=0Qij(t)X (j)(%ijt),
<-- l%ij]
ql
the series I
Q(i)
< l, (j=0,
p
converges, where
are the coefficients of
Qij (t),
Q(i)
and
i=0Z
l),
max
j,k
0 <
q2
<-- flip] < q3
1[ Qijk ![
II Qip(O) II
and
< l;
Qijk
< lo
Then the problem has a unique holomorphich solution, which is an entire
function or order not exceeding m
PROOF.
+
p.
The expansions
m
Qij(t)
Z Q ij ktk X(t)= Z
n=0
k=-0
imply that
X
(p)(t)
Z Xn+p t n(n+p)’ /n’
n--0
n
(3.1)
734
J. WIENER
X
(j)(%ijt)
X
Qlj(t)
Z
n--O
(j)
%j Xn+jtn(n + J)!
(ijt)=
Z
t
k=0
l
n!(n + p
y.
Z
lij QijkXn+j_k (n
+
k)! / (n
j
relations for the matrices X
p-I m n!(n
+ j k)’
Z
Z Z
i=0 J=0 k=0 (n+p)’(n-k)’
n.ip Qip0 Xn+p
Z
i=0
n-k
Z
n--0
and yield the following recursion
(E-
m
n
/ n!
k)!
%n.-k
ip Qipk Xn+p-k
(n+p)!(n-k)!
iffi0 k=l
n
k)!
n
n-k
ij Qijk Xn+J-k
_>
(3.2)
0
Hypotheses (ii) and (iii) ensure the existence of the inverse matrices
-I
(E- Y.
Qipo for all n:
i=O
n_ip
Qip0 )-1
in_
Z
(E
ip
i=0
ll(v- -r. inip Qip0
i:0
-i
Z
k=O
II
iZ n Qipo
i:0Z II Qip0 II )-1
=0
!
k
ip
(i-
Therefore, formulas (3.2) determine the coefficients X uniquely and, since
n
n’(n + j
n!(n + p
k)’ / (n + p)’(n
k)! / (n + p)!(n
! (n +
k)! < i,
k)!
p)-i
O<_J
<p-i
we obtain, by virtue of (iii),
I1 xn+p II
p-I m
<
+
n
n
For large n, there Is q3
II
x
II <_
p
<--
c(n +
11 Xn+j_k ]]
J=0 k=0
(n + p) -1 and
p)-I
+ bq n3
m
I
lI Xn+p_k ]l
m+p-i
E
k=0
llXn+km
Here a, b, c are some positive constants.
Wi:h the notation
(3.3)
(2.5),
(3.3) that
IIXn+p I[ <
c(m +
P)Mn+p_ l
/ (n + p).
Starting with some N,
c(m + p) / (n + p) < 1,
llXn+p II <_ Mn+p_ 1 Hn+p Mn+p_ I
and it remains to apply (2.6) successively to (3.3):
]IXN+p+k II <_.
c(m +
p)
/ (N + p),
it follows from
LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
[l+Ft(m+p)+k II < c2(m + p)2
II .,X+p+2(m+p)+kll
3
c (m
<_
+ p) 3
(0 < k < m + p
735
/ (N + p)(N + Cm + p) + p)
/ (N+p)(N+ (re+p) +p)(N+2(m+p) +p)
I).
Now it can be proved easily that, for all n,
n
II +p+n(m+p)+k II <_ cn+l(m + p)n+N
/
--]
(N + i(m + p) + p).
i--O
Thus,
II +p+n(m+p)+k I
<_ cn+l(m + P)
/ n!
and the solution X(t) is an entire function whose order of growth does not exceed
m+p.
If, in addition to the hypotheses of Theorem 3.1, the parameters
THEOREM 3.2.
%iJ (0 < J
<p
0 <
i) are separated from unity:
ql
<-- l%ij
< q4 < i, the solution
of (3.1) is an entire function of zero order.
PROOF.
Under the conditions of Theorem 2.1, the system
m
Z
(t)X (j) (ijt)
7. A
with real constants
0 <
l=ool
0
aiJ
l= jl
<
>_
+ J
_>
has a distributional solution (i.I), the coefficients Xn of which satisfy inequalities (1.2) and are determined with the exactness to arbitrary
apply to (3.4) the Laplace transformation assuming
same
notation for
x(P) (sloo)
(3.1).
A-I
s/00
7.
p
7.
(_l)p-k
Xn
+j>o k=0
t, and
satisfy (2.7).
hence, the parameters
results of [6].
positive and retaining the
O0/eij
oi-I Aijk(SJx(,loij))
%ij
%tJ
(k)
(n)(t)
is s
n
These estimates use only the moduli of
may be complex.
0
reduce this equation to the form
This proves the theorem, since the transform of
coefficients
We
X(t) and its transform:
p+l
+ *oo
The substitutions
ij
Xp_ I.
X0,
and the
lJ
Theorems 3.1 and 3.2 generalize the
J. WIENER
736
The problem
THEOREMS 3.3.
7 A
i=O
F’(z)
F(z
i(z)
F(z)
lira
+
i)
7 B
t=0
F’ (z
i(z)
(3.5)
bi),
F0
Rez+with (r
a
matrices A, B, F has a unique holomorphlc solution which is an
r)
entire function if:
(1)
(il)
Ai(z
al,
0
(iii)
m
7 A
k=l
b
kz
_< Reai _< M I
<
the series 7 A
(1)
,
PROOF
k=0
B
ik
kz
e
0 <
M2 _< Rebi < M3
<
(i) and 7. B (i) e -Rebi
max
k
II B i(0) II e-Rebi
7
i=0
m
7
Bi(z)
are complex numbers such that
i
converge where A
and
ik
e
II Aik II B(i)
max
k
II B
< i.
The substitutions t
e
z
e -ai
8i
e -bl
and F(z)
reduce (3.5) to (3.1) of the .first order with the initial condition X(0)
X(t)
F0
REFERENCES
I.
Fricke, G.H.; Roy, Ranjan and Shah, S. M. Bounded index entire solutions of
ordinary differential equations and summabillty methods, Internat. J.
Math. & Math. Scl. 4(3), (1981), 417-434.
2.
Wiener, Joseph. Generalized-functlon solutions of linear systems. J. Differ.
Equat. 38(2), (1980), 301-315.
3.
Wiener, Joseph.
4.
Gel’fand, I.M. and Shilov, G.E. Generalized functions, Vol. 2, Academic Press,
New York, 1968.
5.
Mityagin, B.S. On the infinitely differentiable function with the given values
of the derivatives at a point, Dokl. Akad. Nauk SSSR 138(2), (1961),
Generallzed-function solutions of differential and f,lnctional
differential equations, J. Math. Anal. and Appl. 88(i), (1982), 170-182.
289-292.
6.
Wiener, Joseph. On the entire solutions of a linear functional differential
system, Differ. Uravn. 13(3), (1977), 551-555.