I nternat. J. Math. Math.
Vol. 6 No. 2 (1983) 243-270
243
DISTRIBUTIONAL AND ENTIRE SOLUTIONS
OF ORDINARY DIFFERENTIAL AND FUNCTIONAL
DIFFERENTIAL EQUATIONS
S.M. SHAH
Department of Mathematics
University of Kentucky
Lexlngton, Kentucky 40502
and
JOSEPH WIENER
Department of Mathematics
Pan American University
Edlnburg, Texas 78539
(Received February 25, 1983)
ABSTRACT.
A brief survey of recent results on distributional and entire solutlosof
ordinary differential equations (ODE) and functional differential equations (FDE) is
given.
Emphasis is made on lnear equations with polynomial coefficients.
Some work
on generallzed-functlon solutions of integral equations is also mentioned.
KEY WORDS AND PHRASES.
Ordinary Differential Equations, Functional Differential
Equations, Integral Equations, Distributional Solutions, Entire Solutions.
19B0 MATHEMATICS SUBJECT CLASSIFICATION CODES.
$4A$0, $4A$5, $4K25, $4K, $0D05,
46FI0, 45E99.
I.
INTRODUCTION AND PRELIMINARIES.
This paper may be considered as a continuation of [I] which contains, in partf-
cular, a survey of recent results on entire solutions of ODE with polynomial
coefffclents.
Integral transformations establish close links between entire and
generalized functions [2].
Therefore, a unified approach may be used fn the study
of both dlstrfb-utional and entire solutions to some classes of linear ODE and,
especially, FDE with linear transformations of the argument [3].
It fs well known
[4] that normal linear homogeneous systems of ODE with fnffnltely dffferentlable
coefficients have no generalized-function solutions other than the classical
S.M. SHAH and J. WIENER
244
In contrast
solutions.
to this case, for equations with singularities in the
coefficients, new solutions in generalized functions may appear as well as some
classical solutions may disappear.
solutions of ODE are discussed.
In Section 2 results on distributional and entire
In Section 3 we study analogous problems for FDE.
Research in this direction, still developed insufficiently, discovers new aspects
and properties in the theory of ODE and FDE.
dissimilarities between the behavior
In fact, there are some striking
of ODE and FDE which deserve further investi-
gat ion.
I.
Distributional solutions to linear homogeneous FDE may be originated either
by singularities of their coefficients or by deviations of argument.
In [5] it has
been proved that the system
x’(t)
Ax(t) + tBx(%t),
-I
% < I
has a solution in the class of distributions
an impossible phenomenon for ODE
without singularities.
2.
In [6] it was shown that a first-order algebraic ODE has no entire
transcendental solutions of order less than
may possess such solutions of zero order
3.
1/2,
whereas even linear first-order FDE
[3], [7].
It is well known [8] that the solution of the initial-value problem for a
normal linear ODE wlth entire coefficients is an entire function.
Let in the linear
FDE
w’Cz)
the functions
I%(z)
Izl
if even
w(0)
a(z), b(z), %(z) Be regular
< i for
< i[9].
a(z)wClCz)) + BCz),
Izl
< i.
w
0
in the disk
Izl
<
I,
and %(0)
O,
Then there is a unique solution of the problem regular in
In general, this solution cannot be extended beyond the circle
a(z), b(z), and %(z)
w’(z)
are entire functions.
i,
Thus, the solution of the eqa_ tiOn
a(z)w(z2),
where a(z) is an entire function wlth positive coefficients, has the circle
as the natural boundary
2.
Izl
[i0], [II].
DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ODE
The number m is called the order of the distribution
Izl
I
EOUATIONS
ORDINARY AND FUNCTIONAL DIFFERENTIAL
m
x
r.
x
()
x ’ o,
(t),
1-0
where
real.
(k)
(2.1)
measure, and the variable t is
denotes the kth derivative of the Dirac
Finite order solutions of linear ODE have been studied mainly for equations
with regular singular points [12
(Wiener [16]).
THEOREM 2.1.
n
Y. a
i--O
i(t)x
(n-i)
(t)
If the equation
(2.2)
0
(re+n- i
C
ai(t)_
with coefficients
In [16] for the first time an existence
16].
(2.1) to any linear ODE was established.
criterion of solutions
in a neighborhood of t
(I)
a
(2)
m satisfies the relation -(m
3)
there exists a nontrlvial solution
0(0)
0,
m+n
E
j--
0 has a solution of order
0, then:
m concentrated on t
mln (j
0Xk+j -n
O,
al(O)
(x0,
(-l)J-laj-i)(0)(k + J
i=o
m
0, 1,
(Wiener [16]).
THEOREM 2.2.
+ n)a 0’(0) +
m)
x
of the system
,n)
(k
t
245
i)!
0
+ n).
Eq. (2.2) has an m order solution with support
0, if the following hypotheses are satisfied:
(i)
For some natural N(0 < N < m
a
(il)
(0)
O,
0,
i
+ n),
rain(N, n);
m is the smallest nonnegative integer root of the relation
M
):.
I--0
(N+l-i)
(O)(m + n
(-I) N+l-i aj
i)!
0, M
mln(N + i, n),
where N denotes the greatest number for which (i) holds;
there exists a nonzero solution of system (3) in Th. 2.1.
From these theorems it follows that if the equation
n
E
t=0
ttatCt)x (i) (t)
with coefficients
ai(t)
(2.3)
0
m
E C
and
an(0)
@
0 has a solution (2.1) of order m, then
S.M. SHAH and J. WIENER
246
n
r.
i=O
(-1)ia i(O) (m
+ i)
(2.4)
O.
Conversely, if m is the smallest nonnegative integer root of (2.4), there exists an
0 [16].
m order solution of (2.3) concentrated on t
This proposition constitutes
the basis for the study of finite order solutions to equations with regular singular
The stated results can be used also in the search of polynomial and rational
points.
solutions to linear ODE with polynomial coefficients.
Thus, we formulate
THEOREM 2.3.
The equation
n
Z
i=O
(air
bi)x(n-i)(t)
+
with constant coefficients
all poles s
i
al,
0
b
and a
i
i, b 0
0
0 has a finiteorder solution if
of the function
n
n
Z
R(s)
(.i s
i=0
i)ai)sn-i-I/ i=0I aisn-i
(n-
are real distinct and all residues r
i
res R(s) are nonnegative integral.
S--S
i
This solution is given by the formula
n
-I
C
x
(d /dr
i-i
sl)ri(t),
C
const
md its order is
n
m=
Er
I--I i.
0 there exists also a solution
If a
n
x- C
H
r
(d!dt
t--I
s:t)
I t
-1.
Polynomial and rational solutions of ODE have been studied extensively [17
25].
In [17] the author deals with the equation
r.
i-O
where a
S
alx
ai{t)
(.t)
0
have mth order derivatives in
[a, b]. Let aio
a /a
i
0
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
(provided a 0 # 0) and
(provided
(a+l,j_ I + ai,j_l) / (al,j_ I + a0,j_l)
aij
al,j_ I + a0,j_ I @
0).
(Sapkarev [17]).
THEOREM 2.4.
247
Eq. (2.5) has a polynomial solution of degree
m if and only if
a{,m_ I + a0,m_ I
(apkarev
THEOREM 2.5.
m
+ i,
+
m
0.
[17]).
Eq. (2.5) has polynomial solutions of degrees m,
i if and only if
n
ai+l,m-1 +
a
0, I,
0 for i
i,m-1
n
i.
Necessary and sufficient conditions for the existence of a maximal number of
polynomial solutions to algebraic differential equations are given in [18] and [19].
Existence of polynomial solutions of an equation of Linard type is studied in [20].
The equation
w’
a0(z)
considered in [21].
Thus, in [22]
n
7.
+
al(z)w
+
n
a (z)w with n polynomials as solutions is
n
In a number of papers additions to Kamke’s treatise are made.
proved that for the generalization
it is
x
+
(-l)i-itix(i)
/ i! +
f(x(n))
i=l
of Clairaut’s equation the following are two solutions:
E
X
i=l
CI,
where
citi /
Cn
I
y +
i=l
where
AI,
+
f(Cn)
are arbitrary
n-I
x
i!
An_ I
constants, and
Ait i,
are arbitrary constants and
f,(y(n))
(_t)n/n!.
In [23] and
[24] rational solutions of Palnlev’s third and fifth equations are studied.
(Gromak, [23]).
THEOREM 2.6.
zww"
where
zw’
0, y
solution has
82/4
2
0 or
ww
3
+ w + 8w + YZw +
87 # 0,
poles and
There exists a rational solution of the equation
4
0, if 8
-+
z,
2k, k
0, +-I
+-2
...; this rational
1 zeros.
The work [25] concerns the study of properties of solutions to the complex
equation (I) Pf
g, where
248
S.M. SHAH and J. WIENER
m
P(z, 8 /z)
k
E
Pk(Z)
k;O
i
z k,
/
1
z
and g is a given holomorphlc or rational function.
( /
x
i / y)
Various conditions guaranteeing
that the solutions of (i) are polynomial or rational functions of a certain type are
In the last part, differential equations of ’Euler type are considered.
obtained.
(Nova[25]).
THEROEM 2.7.
Let
be a simply connected open set in
If u is a regular singular point of P and every solution of Pf
with g e
Ru(),
with a pole at
is rational in
u,
g in 0(
and u E
.
{}),
then P is normal.
Significant contributions to the study of asymptotic properties of the analytic
solutions of algebraic ordinary and partial differential equations are made in [6].
The main properties are the growth of an entire solution, the order of a meromorphic
solution and its exceptional values.
In a certain sense, thls book completes the
In the second chapter of [6], the author studies the
fundamental monograph [26].
algebraic DE
P(z, w, w’)
0.
It is reduced to the form
P0(z
where
Qi(z,
w,
--) -=
r.
QI(z
---)
n) are polynomials In z and n.
solution of (2.6) and let
Substituting w
(2.7)
0
W
Let w(z) be an entire transcendental
be a point on the circle
Izl
r such that
in (2.7) and, dividing its terms by
w(z), z
wn()
gives, with
regard to Maclntyre’s formula [27]
f’ ()/f()
rM’ (r)/M(r)
K(r),
the equation
n
Q0({,
E
K(r))
Qi({, K(r))ji({).
From here it follows that
Q0 (’
The polynomial
Z(r))
Q0(,
o(.i).
(2.8)
K) is called the principal polynomial of Eq. (2.7), and (2.8)
is called the determining equation.
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
THEOREM 2.8.
(Strelitz[6]).
The order and type of an entire transcendental
solution of (2.6) are equal, respectively, to the positive order
of one of the solutions of the determining equation (2.8).
rllm K(r) /r
ojpj,
249
rllm In
M(r) /r
Oj
> 0 and type
Furthermore,
j.
The following proposition shows that not all of the numbers
indicated in Th. 2.8
01
may be the orders of the entire solutions of first order algebraic DE.
THEOREM 2.9.
(Strelltz[6]).
tal solutions of order O <
i
Algebraic DE(2.6) cannot have entire transcenden-
In general,
i
cannot be replaced by a larger number:
there are equations of the form (2.6) that have entire transcendental solutions of
order
I
EXAMPLE 2.1.
2
w
(Strelltz[6]).
+ 4zw’ 2
The equation
i
cos / of order 0
has an entire transcendental solution w
i
The following result is of interest in this connection.
Let R(z, w) be a rational function of z and w.
1
is a
A meromorphlc solution of the equation w’
R(z, w) which is of" order <
THEOREM 2.10.
(Wittlch [26]).
rational function.
In the second chapter of [6] it is also proved that the order of any meromorphlc
solution of a first order algebraic DE is finite.
The orders of the transcendental
entire solutions of second order linear DE with polynomial coefficients have been
[29], [30].
investigated in [28],
degree p and q, respectively.
Set
Suppose that P(z) and O(z) are polynomials of
i
gO
+ max(p,
q).
Let p
_> q +
I.
Then all
transcendental solutions of the equation
(2.9)
w" + P(z)w’ + Q(z)w-- 0
are of the order I
order i +
Here
go
i
q
go"
+ p
go"
If p
I
_< q,
all transcendental solutions are of the
Deviation from this pattern can occur only if
i
q
< p
_< q.
I + p, and there are always solutions of this order; under certain
circumstances, however, a lower order q
THEOREM 2.11.
(Hille [30]).
p
+ I may
also be present.
If in (2.9) either P or Q is an entire
250
S .M. SHAH and J. WIENER
transcendental function while the other is a polynomial, then every transcendental
solution of (2.9) is an entire function of Inflnlteorder.
This is not necessarily
true, however, if both P and Q are entire.
(Wittich [30]).
THEOREM 2.12.
In (2.9) suppose that P and Q are entire
functions and suppose that the equation has a fundamental system
w
I
and w
2
are entire functions of order
01
and
02,
respectively.
Wl(Z), w2(z),
where
Then P and Q are
po lynomials.
Th. 2.12 may be regarded as a converse of Th. 2.11.
(Frei [31]).
THEOREM 2.13.
w
(n)
n
+
the coefficients
7.
i=l
Pi(Z)W
(n-i)
pi(z)(i
transcendental function.
Suppose that in the equation
0
k) are polynomials, and
i, 2,
Pk+l(Z)
is an entire
Under these conditions the equation can have no more than
k linear independent entire transcendental solutions of finite order, whereas all
other solutions of the fundamental system are of infinite order.
The results by Frei,
Pschl, and
Wlttich on the growth of solutions of linear
DE are generalized in the third chapter of [6].
The main tool is the Wlman-Vallron
method, but the case when this method fails is also studied. Nonlinear algebraic
(n)
0 are investigated, too. A necessary conDE of the form P(z, w, w’
w
dition for some complex number a to be a defect value of a meromorphic solution of
finite order is
0)
P(z, a, 0,
0.
We already know that first order algebraic
DE have no entire transcendental solutions of zero order.
In [32] it is shown that
there are algebraic DE of third order that have entire transcendental solutions of
zero order.
THEOREM 2.14.
(Zimogliad [33]).
P(z, w, w’, w")
A second order algebraic differential equation
0
(P is a polynomial of all its variables) cannot have entire transcendental solutions
of zero order.
THEOREM 2.15.
(Shah [34]).
linear homogeneous equation
Let f(z) be an entire solution of an nth order
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
P0 (z)w(n)
+
Pn(Z)W(Z)
+
0
where P. (0 < j < n) are all polynomials.
3
deg
PO
deg
> max deg
Pn
Pj,
(1
251
deg P.
Write max
_< j _< n
d and suppose that
3
0<j<n
1). Thenf(.) is of order 1 and
exponential type T where
and a
I/n
lan / a0
T
lira P (z) /z
d
p
0, n. For cases when the condition on the degree of
z-o P
is not satisfied, see ([34, Th. 1.6]).
P
The Bessel function of integer order n,
z
2w,,
ODE
+ zw’ + (z
z2w
2
2
n )w
+ (z 2- 2Nz
L(L + l))w
M(r,
Jn
are all entire functions.
max
FL(D
z) satisfies the
0 (N a real constant, L a nonnegative integer).
{If k(z) J,
I <k
log M(r, F
r
Consider now vector-vM.ued functions F:
llFCz) [l
satisfies the ODE
0, and the Coulomb wave function
For these functions we have log
fk(l <_ k <_ m)
Jn(Z),
P
1/ I;m.
L)
as r +
.
Suppose that the components
Write
<m}, MCr, F)
!! F(z) lJ-
max
IzJ=r
A vector-valued entire function F is said to be of bounded index
DEFINITION.
(BI) if there exists an integer N such that
max
II F (i) (z) I[
i!
O< i<N
for all z e
I and k
THEOREM 2.16.
function of BIN.
llF(z)
where A
max
0<k<N
>
!1 F(k)(z) Ii
k!
O, i,
The least such integer N is called the index of F.
(Roy and Shah [35]).
Let F:
i
/
m he
a vector-valued entire
Then
II
< A exp((N + i)
11 F(k)(0) !1
(N + 1)
k
Izl)
The result is sharp.
The function F may be of BI but the components
fk may not
theorem, it is shon that if F satisfies an ODE then F and each
be of BI.
fk are
of BI.
R denote the class of all rational functions r(z) bounded at infinity and
(1 < i < m) denote an m
m matrix with entries in R.
Write
In the next
Let
Qi(z)
252
S.M. SHAH and J. WIENER
(a
Qi(z)
pq,i
(z)),
lira
Z-Oo
lapq
]Apq l]
(z)
i
and
sup
(IApq,i],
<_
p, q
(Roy and Shah [35])
THEOREM 2.17.
whose components
1
fl’
f
m
Let F:
+
are all entire functions
be a ve-tor-valued function
Suppose that F satisfies the
ODE
Ln(W,
z, Q)
--w (n) (z) + Ql(Z)- (n-l) (z) +
where g(z) is a vector-valued entire function of BI.
+
Qn(Z)W(Z)
Then each
If the entries of
(Roy and Shah [35]).
THEOREM 2.18.
fl’
fm
are
R then F may not be of BI.
are not in
Qi
satisfies an ODE
fk
of this form (with possibly different n and coefficients), and F,
all of BI.
g(z)
Let w(z)
0 be a vector-valued entire
function satisfying the ODE
L (w, z
n
Q) 0
Then we have:
(i)
lim sup
log
r
ro
where the numbers A
i
(ii)
M(r w)
iffil
are defined above
If the elements of
such that
m
FIn+P
n
< max { i, m
Qi(l <_
(n + p) (n +’p
then the index N, of
F(z),
i <
m) are constant, and p
i)
Cp+
(.n+p
is less than or equal to n
+ p
I.
_>
0 is any integer
<_
i,
The bound on N is
best possible.
Next we compare these growth results with the corresponding ones for solutions
of algebraic difference equations.
THEOREM 2.19.
(Shah [36]).
Let P(t, u, v) be a polynomial with real coeffi-
cients.
Let u(t) be a real continuous solution of a frst order algebraic difference
equation
P(t, u(t), u(t + i))
0 for t
_> t o
Then there exists a positive number
A which depends only on the polynomial P such that
[u(t
lira inf
e (At)
t
2
O.
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
If u(t) is monotonic for t
im
t-+=o
e2(x)
Here
_> to,
,,[u(t)
253
then
0.
e2 (At)
denotes exp(exp x).
The function e (At) cannot be replaced by a function of slower rate of growth,
2
in general.
(Shah [36]).
THEOREM 2.20.
tends to
+
as t
/+=o
Let (t) be an arbitrary increasing function which
_>
real solution u(t) which is continuous for t
point of a sequence {t
n
P(t, u(t), u(t + i))
There exists an equation
such that t
n
-+oo
t
O
0 with a
and which exceeds (t) at each
as n /.
For further results see [37], [38], [39] and [40].
Let f(z) be an entire transcendental function.
J(xr e
where
(r,
f)
max
I =rg(xre
The
(, x)
index is defined as
f)
f) is the central index of the Taylor expansion
I
f(z)
f(i)()(z
)i 1 !.
The author of [41] evaluates the
(e, x)
indices of entire transcendental solutions
of linear ODE with polynomial coefficients.
On the basis of these results some
theorems concerning the distribution of values of these solutions are proved.
2.21.
order 0 and type
c (c
const).
L
llm
Let w(z) be an entire transcendental solution of
of an ordinary linear differential equation with polynomials as
Let n(r, w
coefficients.
w
(Knab [41]).
c) be the counting function of the zeros of the function
Then
SUPr_=n(r
w
p
c) /r < Up.
In [42] the author considers the equation
where
p0(z)w" + pl(z)w’ + p2(z)w
p0(z) # 0, pl(z) and p2(z) are
0,
(2.10)
entire and have real Taylor coefficients about
ny real point.
THEOREM 2.22.
(Lopusans’kll [42]).
Oscillatory real solutions of (2.10) have
254
S.M. SHAH and J. WIENER
only real zeros.
H
(Lopusans kli [42]).
THEOREM 2.23.
w(z) /(z) maps the upper half-plane conformally onto the
only if the function (z)
unit disk, where
solutions of
w(z)
Wl(Z)
(2.10), and
Solutions of (2.10) are oscillatory if and
+
iw2(z)
wj (z)(j
and
I, 2)
are two indenpendent real
their Wronskian is positive on the real axis.
The following characterization of the class HB(Hermlte-Biehler) of entire
functions having all their zeros within the upper half-plane is given in
(Lopusans’kii [42]).
THEOREM 2.24.
An entire function F(z) is of class HB if
and only if on the real axis it is a complex solution of an oscillatory equation of
the form (2.10).
The ODE
[43], where
w(n) (z)
+
p0(z),
Pn-2(z)w
Pn-2(z)
(n-2) (z)
+
+
%nw(z)
p0(z)w(z)
is studied in
ran_ 2,
are polynomials of degrees m
0
respecti-
It is proved that the fundamental system of
vely, and % is a complex parameter.
solutions of the equation, determined by the identity matrix as initial conditions
0, satlsifles the
at z
)I
lw+/-(z,
<
estimates
]I Iz lOexp=l’Xzi,
for all sufficiently large values of
max
(m.
I%1
i +n) /(n
and
Iz I.
The value of 0 is defined by
i),
u<l<n-Z
and c is some positive constant.
Asymptotic properties of the solutions of linear ODE with entire coefficients
are studied in
w
(n)
[44].
Consider the equation
+ an_lW (n-l) +
where all the coefficients a
+
0, I,
ai(z)(i
i
(2.11)
0,
a0w
n
Let f(z) be a meromorphic function in the z-plane.
Z+(If(z)
The function
8(a, f)
8(a, f)
T(r, f)
al -I)
if a
@
L(r
a
f)
i) are entire functions.
Denote:
L(r, a, f)
maXlzl=rZn+If(z)
if a
oo.
is defined as
lira
infr_oL(r
a, f)/T(r, f),
is the usual Nevanlinna characteristic function of f.
solution w(z) of (2.11) a standard solution if
8(a, f)
The authors call
0 for all complex a
# 0,
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
(Boiko and Petrenko [44]).
TIEOREM 2.25.
255
Each fundamental system of solutions
of Eq. (2.11) contains at least one standard solution.
In [45] the author considers the first Palnlev equation w"
i/ (z
solutions are meromorphic of the form w
i
(z
z0)
3
+
,,
z0)2
I / (z
(z), where
,,
z0)
2
6w
(z 0 /10)(z
2
z
+
z whose
0)
2
Yn--O n+2 (z zo)n"
(z)
She represents w as a quotient of two entire functions:
w--
(u’2
I Jw
exp (-
where u
uu") /u
dz
2
(2.12)
dz), and then obtalns
recursion relations for the coeffl-
clents of the power series expansions of the numerator and denominator.
In conclusion, we note that in some recent works [46-50] entire solutions to
DE of infinite order are discussed as well as properties of differential operators
In [46] the author studies the existence of a
in spaces of entire functions.
solution to the equation
7n= 0
anw(n)(z)
f(z) whose growth equals that of the right-
hand side, in the case when f(z) belongs to the class
such that
Ig(x +
iy)
type without multiple roots.
z e
.
0 (i
of exponential
be the operator of convolution with
(), where
Each entire solution w(z) of the equation
k), if and only if
i,
z
k-I
+
+
w(z),
I (1)(z)
>
cle
Wl(Z
I(z) +
where
for all
%
with some constants c I, c > 0.
2
the operator
studies
the
author
In [48]
The operator L
i=
My
is representable in the form w(z)
L w
P
7
Let
Let (z) be an entire function on
(Napalkov [47]).
THEOREM 2.26.
Mw i
g(z)
The following result is proved in [47].
-functlon.
MkW=0
of entire functions
< cexp [(x) + (y)], for any x, y; here the functions (x),
(y) satisfy Hider conditions.
is a
B,
(I)
7 p
i=0
P
7
aikz
k=0
k
p > 0.
is said to be
P
applicable to the set H of entire functions at the point z 0 if the series
0pi(Zo)W(1)(z 0)
(2)
i(z)w(1)(z), pi(z)
converges for any function w from H;
applicable to H in the domain
finite point
Iz
<
if L
P
is
applicable to H at
256
S.M. SHAH and J. WIENER
(3)
strongly regularly applicable to H inside the domain
w e H and R <
I
Iz
<
if, for any
oo,
M(Pi R)M(w(i),R)
<
,
i=0
sup {
where M(f, R)
[fez)I:
Iz I<
R).
Let Q be a bounded simply connected region.
In this remarkable paper the
author gives necessary and sufficient conditions for L
P
to be applicable to the set
R(Q) of exponential functions whose Borel transforms are regular on C(Q).
that if the operator L
sup
where
a--sup {
O<k<p
Izl:
lim
n
z e
R(Q) at p + i distinct points then
is applicable to
P
nla-k
< i/a
Q}. Conversely,
applicable to R(Q) inside
Iz
He proves
(2.12)
if (2.12) holds, then L
P
is strongly regularly
< =o, and maps R(Q) into itself.
In [49] the authors investigate the solvability of a class of functional
equations, containing as a particular case differential equations of finite and of
infinite order with constant coefficients, in the Banach space with weight of entire
functions
B(x,y
{w(z)
g
Aoo llw
I
lw(z lexp(_(x,
sup
y)) <
o.
z=x+iye
Here #(x,y) is a locally bounded function in R 2 with a certain growth for
The author [50] treats an equation Lw
the
pi(z)
are polynomials, deg
Pi
ni’
of all entire functions satisfying lim
lim
f with L
Z
i>0Pi(z)di/dz i,
sup(n i / i) < I, in
SUPr_
(Znlw(reiS)
is a trigonometrically 0-convex function, 0 > 0.
a space
/ro) < g(8).
Izl
where
[0, g(8)]
Here g(8)
It is proved that L is a Noetherian
operator, its index is found and the space of solutions of the corresponding homogeneous equation is investigated.
3.
DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF FDE
Finite order distributional solutions (2.1) of linear FDE have been studied in
[15] and [51].
THEOREM 3.1.
to the system
(Wiener [15]).
The criterion for the existence of solutions (2.1)
257
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
n
tx’(t)
i=0
Ai(t)x(%i t)
with matrices A (t)
roots
TM
C
0 and constants
in a neighborhood of t
%i
# 0
is that some
of the equation
n
I%iI-1%
I
det
i=0
be nonpositive integers.
+ ( + I)E)
Ai(0
(3.1)
0
If m is the smallest of their absolute values there exists
a solution of order m.
From here it follows that the system
n
tx’(t)
A(t)x(t) +
A
i=l
i
(t)x(%it)
0, if
has a solution of order m with support t
Ai(0)
0(i
_>
I) and m +
smallest modulus of the negative integer eigenvalues of the matrix A(0).
is the
This and
similar results were used in [15] to investigate finite order solutions of some im-
portant equations of mathematical physics.
For equations with more general argument
delays we have
THEOR 3.2.
i
i=0
i
The system
A (t)x( (t))
tx’(t)
in which A
15] ).
(Wiener
m
(t) c C
i
(t)
hypotheses are satisfied:
i
g
C
(i)
has a solution (2.1) or order m, if the following
the real zeros
and form a finite or countable set; (2)
(3)
A
tij
(k)(tij)
i(t)
of the functions
0(k
m), for
0
m is the smallest modulus of the nonDositive integer roots of Eq.
are simple
tij
# 0;
(3.1) with
’(0).
In [52] it was shown that, under certain conditions, the system
x’(t)
Z A.(t)x(%.t)
n=0
has a solution
x(t)
Z x n tn)(t)
(3.2)
n=0
in the generalized-function space
(S)’
conjugate to the space
S
of testing functions
S.M. SHAH and J. WIENER
258
(t) that satisfy
the restriction [2]
’’l(n)(t)l
n
ac n
<
n
> i.
To ensure the convergence of series (3.2), it is sufficient to require that for
n/
the vectors x satisfy the inequalities
n
I xn II
bdn n-n0,
<
0 >
since
Z < x
n
n=O
<
for
< 0.
(n)(t), (t)>ll I Z (-l)n (n) (0)xn [1
n=O
l
n=O
l(n)(o)l 11 xn I]
(cdn-O) n
< ab Z
(So)’
<
n=O
If series (3.2) converges, its sum
functional in
<
represents
0 [53].
with the support t
the general form of a linear
Solutions in
(So)’
ODE with polynomial coefficients were studied in [54], [55], [56].
of some linear
The particular
importance of the system
m
l
I
i=0 j=0
(Aij
+
tB..)x(J)(%it
tx(%t)
13
which was considered in [15] is that depending on the coefficients it combines either
equations with a singular or regular point at t
solution of the form (3.2).
tx’ (t)
0 and in both cases there exists a
The equation
Ax(t) + tBx(%t)
(3.3)
provides an interesting example of a system that may have two essentially different
solutions in
(SOB)
0.
concentrated on t
If the matrix A assumes negative integer
eigenvalues, (3.3) has a finite order solution (2.1).
an infinite order solution (3.2), if A
@ -nE
At the same time there exists
for all n >
I.
In [3], [16], [57], and
[58] the foregoing conclusions were extended to comprehensive systems of any order
with countable sets of variable argument deviations.
The basic ideas in the method of
proof are applied to investigate entire solutions of linear FDE.
THEOREM 3.3.
(Cooke and Wiener [3]).
Let the system
m
l
i
I A. (t)x (j) (%ij (t))
lj
j=O
0
(3.4)
259
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
with a finite number of argument deviations, in which x is an r-vector and
r
Aij
are
matrices, satisfy the following hypotheses.
r
(i)
The coefficients A..(t) are polynomials in t of degree not exceeding p:
P
Y. A
.k
k= 0 i
Aij(t
(ii)
%ij(O)
The real-valued functions
At
p
%.. (t)
p > i.
in a neighborhood of the origin,
E C
0 and
o<
(iii)
tk A00(t)
Ioo
< t,
al >_,
+ >_t,
a
(0).
The matrix A is nonsingular and
c
IO0 I-p-1 ]l AII
i>l
Iol-P-Xll Aio
Then in the space of generalized functions
(S)’
p
I!
> O.
with arbitrary
>
there exists
0.
a solution x(t) supported on t
In [3] it is also proved that system (3.4) with a countable set of argument
deviations has a solution (3.2) if, in addition to the conditions of Th. 3.3, there
exists a neighborhood of the origin in which each function
t
Y.
0 and the series
i=
A.
max
j,k
-I
i
A
i
I Aij k II
%ij (t)
has the only zero
converges, where
i
inf
j
i
ij
+
j > I.
The choice of the coefficients in (3.4) enables us to consider both equations with a
singular or regular point and to show that distributional solutions of FDE may be
originated by deviations of the argument.
The authors of [3] also investigate the
system
tPx ’(t)
m
E
EoAij(t)x(j)(%ij(t))
(3.5)
i=0 ]=
the particular cases of which
tPx ’(t)
A(t)x(t)
and
tPx ’(t)
I A i (t)x(%it)
i=O
have been studied in [56] and [57], respectively.
THEOREM 3.4.
(Cooke and Wiener [3]).
Suppose that system (3.5), in which x is
S.M. SHAH and J. WIENER
260
an r-vector and
(i)
The A..(t) are polynomials in t of degree not exceeding p
p+j -2
Z
k=0
Ai-’3(t)
(ii)
%ij
C1
g
(ili)
Aijktk
__>
p
+
j
2:
2.
There exists a neighborhood of the origin in which the real-valued functions
The series
i=O
a.l
inf
j
.
-i
Z
[aijl’
A
A. converges,
i
_>
lio
0 and
have the only zero t
ij (0).
ij
r -matrices, satisfies the following conditions.
are r
Aij
I, inf
lij
> i, for i
_>
0, j
_>
i,
where
Aij k
j,k
Then there is a solution of (3.5) in
(So)’
>
with some
O.
supported on t
The deep study of narrow classes of FDE, and even individual FDE, continues to
remain one of the main problems.
First of all, such equations can have some special,
In addition, we can work out on them in the first
for example applied, interest.
instance methods of studying properties that are similar to properties of equations
without deviation of the argument and are essentially new for equations with devia-
tlon, and then try to extend these methods, and the results obtained, to a broader
In a number of papers [60-66] various authors have continued the study
class of FDE.
(originated in [59]) of the solutions, especially their asymptotic behavior as t
or t
/
,
of the equation
x’ (t)
ax(%t) + bx(t), which
0
arises in certain technical
problems, and also of systems and some more eneral equations of similar form.
These
works concern principally real solutions.
The author of [’67] attacks complicated equations with elegant analytical tools.
He investigates analytic solutions of the FDE
r
s
Z
Z a
j
j--O k=O
kW
(j)
(zqk)
with constant coefficients
0
ajk.
Mellin or Laplace integrals.
0 < q < I,
Its formal solutions are obtained in the form of
The functions occuring in the
difference equations of the form Z
n
v--0
P (qt)G(t + v)
0
inte.rands satisfy
(P(y)
linear
polynomials).
Properties of solutions of such difference equations, in particular the location oF
261
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
singularities and the asymptotic behavior for absolutely large values of t, are
Conditions are derived for formal solutions of Mellin integral type to be
studied.
actual solutions and these are shown to be often expressible as power or Laurent
series.
Solutions of Laplace integral type are shown to be representable as
Finally, questions as to when the llne
Dirichlet series under certain conditions.
of convergence of the Dirichlet series is the natural boundary of the function re-
presented are discussed.
The author asserts that the methods used can be extended
to the case when the coefficients
ajk
are polynomials in z, and to some more general
equations.
In [68] the growth of entire solutions of the FDE
m
I
k--0
akDkw(%m-kz)
0,
d/dz
D
is estimated by means of a suitably constructed comparison function.
9urthermore, an
expllcie representation of all entire solutions is given which in certain cases
leads to conclusions concerning loations and multiplicity of the zeros of particular solutions.
Finally, the growth of the maximum and minimum modulus of the
solutions is compared which implies an estimate of the number of zeros.
m
Lw(z)
I
k=O
akDkw(%m-kz
where a are complex numbers,
k
The FDE
(3.6)
f(z),
is a fixed parameter, 0 <
< i, and the unknown w
and the right member f are entire functions, is considered in [69].
Introducing a
generating function
G(z)
I G zn
n
n=O
G
n
%n(n-l)/2/n!
the author shows that the general solution of (3.6) for f
w(z)
with
(t)
2z---
0 is given by
G(tz)(t) dt,
q(t)/A(t), where
m
A(t)
l
k=0
ak%k(2m_k_ I)/2 k
q is a polynomial of degree <_ m
t
and
F
is a contour enclosing all the zeros of
A.
S.. SHAH and J. WIENER
262
A similar integral representation is given for a solution of (3.6) with f # 0 in
terms of the generalized Borel transform
(t)
f
7.
n=O
n
n+l
/ Gn t
when
Y f z
n=O n
f(z)
In [70] the author discusses the system w’(z)
Aw(lz), 0 < I < i, where A is a corn-
First, the form of all entire solutions is given.
plex constant matrix.
Subsequently,
for z # 0 a special system of particular analytic solutions is constructedhymeans of
which all other solutions may be represented.
all solutions are investigated.
of
The asymptotic properties as z
Furthermore, it is shown that given a specific
asymptotic behavior, there is one and only one solution which possesses that asymptotic behavior.
Given the equation
n
m
w" (z) + E ak (z)w’(IkZ) + 7. b j (z)w(jz)
k=l
j=l
where
ak(z), bj(z)
are in the set 0 <
finite order < 0.
where 0 < I <
0,
are entire functions of finite order < 0 and the constants
zl
I k,
j
< I, the author [71] shows that any solution w(z) is also of
As a special case he discusses the equation w"(z) + p(z)w(Iz)
O,
and p(z) is an entire function of finite order taking real values on
the real axis, and derives an estimate on the type of a solution w(z).
The author of [72] studies properties of solutions to equations of the form
m
7.
w’(Iz)
k=O
where
ak(z)
ak(z)wk(z),
(3.7)
are entire functions such that T(r, a
complex number,
Ill
o, where T(r, w)
k)
o(T(r, w)) as r
is the Nevanlinna function.
,
and I is a
He establishes
the following
THEOREM 3.5.
m= i, then
asr
.
(Mohon’ko [72]).
Let w(z) be an entire solution of (3.7).
w(z) is of zero order, and if m >
The problem
then
In T(r, w)
If
(Zn m/ In o) Zn
r
ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS
n
I a.w
1
n
(i)(z)
exp(ez) Y b.w
1
i=O
(i)
(0)
w
i,
0
i
n- i,
e and I are complex numbers, has been studied with various
bi,
ai,
(i)(%z),
i=O
w
in which
It is proved in [75] that, if
assumptions concerning parameters [73-77].
I #
and
anl
and
I C II
<
263
>
b
I, the solution of the
III
< i,
I,
#
matrix problem
AW(z) + exp (z)[BW(lz) + CW’(lz)], W(0) --W
W’(z)
If
its solution is an entire function.
n
III
[76].
is an entire function of exponential type
0
These results were extended to lin-
ear FDE with polynomial coefficients and countable sets of argument delays in [7],
[3] and [58].
The method of proof employs the ideas developed in the theory of
distributional solutions.
THEOREM 3.6.
W
(p)(z)
W
(ii)
p
Y.
i=0 j=0
Y.
(j)(0)
QiJ
lij
0 <
Wj,
0
j
(3.8)
p
matrices, satisfies the following conditions:
r
(z) are polynomials of degree not exceeding m;
are complex numbers such that
ql <
llijl
the series I
(iii)
Suppose the system
Qij(z)W(J)(lijz)’
and W are r
in which O
(i)
(Wiener[58]).
coefficients of
Qij(z),
< i, (j
o(i)
i), 0 <
p
converges, where
E
and
0
I Qip(O) II
Q(i)
llipl
I Qijk II
q2 <
max
j,k
<
q3
and
< I;
Qijk
are the
< i.
i=O
Then the problem has a unique holomorphic solution, which is an entire function
of order not exceeding m
THEOREM 3.7.
3.6 the parameters
0 <
ql <
--llijl --< q4
THEOREM 3.8.
exists a polynomial
k
E
i=O
+
p.
(Cooke and Wiener [3]).
lij(0 _< j _< p-
If, in addition to the hypotheses of Th.
I) are separated from unity:
< i, the solution of (3.8) is an entire function of zero order.
(Cooke and Wiener [3]).
Q(z) of degree p
m
I A ij (z)W (j) (ij z)
j=O
Q(z)
Under the assumptions of Th. 3.3 there
such that the system
264
S.M. SHAH and J. WIENER
with positive constants
..
has a solution W(z) regular at z
and W(z
-I
is an
entire function of zero order.
(Wiener [51 ]).
THEOREM 3.9.
W’(z)
W(z)
lim
#ith r
x r
ai)
i=OZAi(z)W(z
W
Rez/-oo
The problem
+ Z B i(z)W’(z
i=O
bi )’
0
A, B, W has a unique holomorphic solution which is an entire
matrices
function if
(1)
(ii)
i
a i, b
0
(iii)
m
7.
A (z)
k=l
i
_< Rea.l
(i)
Y. B ekz
ik
k=O
Bi(z)
are complex numbers such that
<
MI
the series I A
B
m
AikekZ
max
k
< oo, 0 < M
(i)
2
and Z B
I Bik II,
_< Rebi _< M3<
(i)
Y.
i=O
and
e
-Rebo1
I B i(0)
oo;
converge where A
I[
e
_Rebi
(1)
max
k
I Aik II,
< I.
The authors [78] propose a method for finding polynomial solutions of the
linear neutral FDE
x’(t)
bx(t) +
n
7.
i=l
a.x’(t-
ri),
where b, ao and r. > 0 are given constants.
1
Meromorphic solutions of a class of
linear differential-difference equations with constant coefficients are investigated
in
[79].
Numerous examples of FDE admitting entire solutions may be found in [40]
and [80].
In conclusion, we mention papers [81] and [82], where singular integral equations have been studied in spaces of generalized functions.
However, it should be
noted that, perhaps, the first work of this kind was [83].
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Gel’fand, I.M. and Shilov, G.E.
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