Distributional and entire solutions of ordinary differential and functional differential equations
I nternat. J. Math. Math.
Vol.
S.M. SHAH 0 1
Equations, Integral Equations, Distributional Solutions, Entire Solutions.
0 Department of Mathematics Pan American University Edlnburg , Texas 78539 , USA
1 Department of Mathematics University of Kentucky Lexlngton , Kentucky 40502 , USA
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 generallzedfunctlon solutions of integral equations is also mentioned.
AND PHRASES; Ordinary Differential Equations; Functional Differential

DISTRIBUTIONAL AND ENTIRE SOLUTIONS
OF ORDINARY DIFFERENTIAL AND FUNCTIONAL
DIFFERENTIAL EQUATIONS
I. INTRODUCTION AND PRELIMINARIES.
This paper may be considered as a continuation of [I] which contains, in
partfcular, 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 dlstrfbutional 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 generalizedfunction solutions other than the classical
solutions. In contrast 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. In Section 2 results on distributional and entire
solutions of ODE are discussed. 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. In fact, there are some striking
dissimilarities between the behavior of ODE and FDE which deserve further
investigat 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 firstorder algebraic ODE has no entire
transcendental solutions of order less than 1/2, whereas even linear firstorder FDE
may possess such solutions of zero order [
3
], [
7
].
3. It is well known [
8
] that the solution of the initialvalue problem for a
normal linear ODE wlth entire coefficients is an entire function. Let in the linear
FDE
w’Cz) a(z)wClCz)) + BCz), w(0) w0
the functions a(z), b(z), %(z) Be regular in the disk Izl < I, and %(0) O,
I%(z) < i for Izl < i. Then there is a unique solution of the problem regular in
Izl < i[
9
]. In general, this solution cannot be extended beyond the circle Izl i,
if even a(z), b(z), and %(z) are entire functions. Thus, the solution of the eqa_tiOn
w’(z) a(z)w(z2),
where a(z) is an entire function wlth positive coefficients, has the circle Izl I
as the natural boundary [i0], [II].
2. DISTRIBUTIONAL AND ENTIRE SOLUTIONS OF ODE
The number m is called the order of the distribution
ORDINARY AND FUNCTIONAL DIFFERENTIAL EOUATIONS
0 has a solution of order
x 1mr. 0x () (t), x ’ o,
where (k) denotes the kth derivative of the Dirac measure, and the variable t is
real. Finite order solutions of linear ODE have been studied mainly for equations
with regular singular points [12 16]. In [
16
] for the first time an existence
criterion of solutions (2.1) to any linear ODE was established.
THEOREM 2.1. (Wiener [
16
]). If the equation
nY. ai(t)x (ni) (t) 0
iO
with coefficients ai(t)_ C (re+ni in a neighborhood of t
m concentrated on t 0, then:
(I) a0(0) 0,
(2) m satisfies the relation (m + n)a 0’(0) + al(O) O,
3) there exists a nontrlvial solution (x0, xm) of the system
mj+En0Xk+j mnlni=(jo ,n)(l)Jlaji)(0)(k + J i)! 0
(k 0, 1, m + n).
THEOREM 2.2. (Wiener [
16
]). Eq. (2.2) has an m order solution with support
t 0, if the following hypotheses are satisfied:
(i) For some natural N(0 < N < m + n),
a (0) O, i 0, rain(N, n);
(2.1)
(2.2)
rn. (1)iai(O) (m + i) O. (2.4)
i=O
Conversely, if m is the smallest nonnegative integer root of (2.4), there exists an
m order solution of (2.3) concentrated on t 0 [
16
]. This proposition constitutes
the basis for the study of finite order solutions to equations with regular singular
points. The stated results can be used also in the search of polynomial and rational
solutions to linear ODE with polynomial coefficients.
Thus, we formulate
THEOREM 2.3. The equation
Zn (air + bi)x(ni)(t) 0
i=O
with constant coefficients al, bi and a0 i, b0 0 has a finiteorder solution if
all poles si of the function
R(s) inZ=0 (.is (n i)ai)sniI/ inI=0aisni
are real distinct and all residues r i
res R(s) are nonnegative integral.
SS i
This solution is given by the formula
n
x C I (d /dr sl)ri(t), C
ii
const
md its order is
n
m= IEIr i.
If an
0 there exists also a solution
s:t) r I t1.
x C H (d!dt
tI
Polynomial and rational solutions of ODE have been studied extensively [17
25]. In [
17
] the author deals with the equation
ir. Oalx
(.t)
0
where aS
ai{t) have mth order derivatives in [a, b]. Let aio
ai / a0
THEOREM 2.4. (Sapkarev [
17
]). Eq. (2.5) has a polynomial solution of degree
m if and only if
P(z, 8 /z)
m
kE;OPk(Z)
k i zk, / z
1 ( / x
i / y)
and g is a given holomorphlc or rational function. Various conditions guaranteeing
that the solutions of (i) are polynomial or rational functions of a certain type are
obtainTeHdE.ROEMIn2.t7h.e la(sNtovpaa[r2t5,]).diffLeetrentibaelaeqsuiamtpiloynscoonfne’cEtuelder opteynpeseatreicnonsiadnedreud.E .
If u is a regular singular point of P and every solution of Pf g in 0( {}),
with g e Ru(), is rational in with a pole at u, 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
fundamental monograph [
26
]. In the second chapter of [
6
], the author studies the
algebraic DE
P(z, w, w’) 0.
It is reduced to the form
P0(z w, ) = r. QI(z W)
0
(2.7)
where Qi(z, n) are polynomials In z and n. Let w(z) be an entire transcendental
solution of (2.6) and let be a point on the circle Izl r such that
Substituting w w(z), z in (2.7) and, dividing its terms by wn() gives, with
regard to Maclntyre’s formula [27]
f’ ()/f() rM’ (r)/M(r) K(r),
the equation n
Q0({, K(r)) E Qi({, K(r))ji({).
From here it follows that
Q0 (’ Z(r)) o(.i). (2.8)
The polynomial Q0(, K) is called the principal polynomial of Eq. (2.7), and (2.8)
is called the determining equation.
THEOREM 2.8. (Strelitz[
6
]). The order and type of an entire transcendental
solution of (2.6) are equal, respectively, to the positive order Oj > 0 and type
of one of the solutions of the determining equation (2.8). Furthermore,
rllm K(r) /r
ojpj, rllm In M(r) /r
The following proposition shows that not all of the numbers 01 indicated in Th. 2.8
may be the orders of the entire solutions of first order algebraic DE.
THEOREM 2.9. (Strelltz[
6
]). Algebraic DE(2.6) cannot have entire
transcendental solutions of order O < i 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. (Strelltz[
6
]). The equation
w2 + 4zw’ 2 i
has an entire transcendental solution w cos / of order 0 i
The following result is of interest in this connection.
THEOREM 2.10. (Wittlch [
26
]). Let R(z, w) be a rational function of z and w.
A meromorphlc solution of the equation w’ R(z, w) which is of" order < 1 is a
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
investigated in [28], [
29
], [
30
]. Suppose that P(z) and O(z) are polynomials of
degree p and q, respectively. Set gO i + max(p, q). Let p _> q + I. Then all
transcendental solutions of the equation
w" + P(z)w’ + Q(z)w 0 (2.9)
are of the order I + p go" If p _< qI, all transcendental solutions are of the
order i + qi go" Deviation from this pattern can occur only if qi < p _< q.
Here go I + p, and there are always solutions of this order; under certain
circumstances, however, a lower order q p + I may also be present.
THEOREM 2.11. (Hille [
30
]). If in (2.9) either P or Q is an entire
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.
THEOREM 2.12. (Wittich [
30
]). In (2.9) suppose that P and Q are entire
functions and suppose that the equation has a fundamental system Wl(Z), w2(z), where
wI and w2 are entire functions of order 01 and 02, respectively. Then P and Q are
po lynomials.
Th. 2.12 may be regarded as a converse of Th. 2.11.
THEOREM 2.13. (Frei [
31
]). Suppose that in the equation
w (n) + 7n. Pi(Z)W (ni) 0
i=l
the coefficients pi(z)(i i, 2, k) are polynomials, and Pk+l(Z) is an entire
transcendental function. 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 WlmanVallron
method, but the case when this method fails is also studied. Nonlinear algebraic
DE of the form P(z, w, w’ w (n) 0 are investigated, too. A necessary
condition for some complex number a to be a defect value of a meromorphic solution of
finite order is P(z, a, 0, 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
]). A second order algebraic differential equation
P(z, w, w’, w") 0
(P is a polynomial of all its variables) cannot have entire transcendental solutions
of zero order.
THEOREM 2.15. (Shah [
34
]). Let f(z) be an entire solution of an nth order
linear homogeneous equation
P0 (z)w(n) +
+ Pn(Z)W(Z)
0
and aP zliroa PP (z) /zd p 0, n. For cases when the condition on the degree of P
is not satisfied, see ([34, Th. 1.6]).
The Bessel function of integer order n, Jn(Z), satisfies the ODE
z2w,, + zw’ + (z 2 n2 )w 0, and the Coulomb wave function FL(D z) satisfies the
ODE z2w + (z2 2Nz L(L + l))w 0 (N a real constant, L a nonnegative integer).
For these functions we have log M(r, Jn r log M(r, FL) as r + .
Consider now vectorvM.ued functions F: 1/ I;m. Suppose that the components
fk(l <_ k <_ m) are all entire functions. Write
llFCz) [l max {Ifk(z) J, I < k <m}, MCr, F)
max !! F(z)
lJIzJ=r
DEFINITION. A vectorvalued entire function F is said to be of bounded index
(BI) if there exists an integer N such that
II F (i) (z) I[ > !1 F(k)(z) Ii
i! k!
max
O< i<N
for all z e I and k O, i, The least such integer N is called the index of F.
THEOREM 2.16. (Roy and Shah [
35
]). Let F: i / m he a vectorvalued entire
function of BIN. Then
llF(z) II < A exp((N + i) Izl)
where A 0m<akx<N 11 F((Nk)+(01)) k!1 The result is sharp.
The function F may be of BI but the components fk may not be of BI. In the next
theorem, it is shon that if F satisfies an ODE then F and each fk are of BI. Let
R denote the class of all rational functions r(z) bounded at infinity and Qi(z)
(1 < i < m) denote an m m matrix with entries in R. Write
and
Qi(z)
(apq,i (z)), lira lapq i (z)
ZOo
]Apq l]
sup (IApq,i], 1 <_ p, q
THEOREM 2.17. (Roy and Shah [
35
]) Let F: + be a vetorvalued function
whose components fl’ fm are all entire functions Suppose that F satisfies the
ODE
Ln(W, z, Q) w (n) (z) + Ql(Z) (nl) (z) + + Qn(Z)W(Z) g(z)
where g(z) is a vectorvalued entire function of BI. Then each fk satisfies an ODE
of this form (with possibly different n and coefficients), and F, fl’ fm are
all of BI. If the entries of Qi are not in R then F may not be of BI.
THEOREM 2.18. (Roy and Shah [
35
]). Let w(z) 0 be a vectorvalued entire
function satisfying the ODE
Ln (w, z Q) 0
Then w(ei) havlei:m sup log M(r w) < max { i, m n
ro r iffil
where the numbers Ai are defined above
(ii) If the elements of Qi(l <_ i < m) are constant, and p _> 0 is any integer
such that
m FIn+P (n + p) (n +’p i) (.n+p Cp+ <_ i,
then the index N, of F(z), is less than or equal to n + p 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
coefficients. 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 e2 (At)
t
If u(t) is monotonic for t _> to, then
max I =rg(xre f)
f(z)
I f(i)()(z
where (r, f) is the central index of the Taylor expansion
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. (Knab [
41
]). Let w(z) be an entire transcendental solution of
order 0 and type of an ordinary linear differential equation with polynomials as
coefficients. Let n(r, w c) be the counting function of the zeros of the function
w c (c const). Then
L llm SUPr_=n(r w c) /rp < Up.
In [
42
] the author considers the equation
p0(z)w" + pl(z)w’ + p2(z)w 0, (2.10)
where p0(z) # 0, pl(z) and p2(z) are entire and have real Taylor coefficients about
ny real point.
THEOREM 2.22. (Lopusans’kll [
42
]). Oscillatory real solutions of (2.10) have
only real zeros.
THEOREM 2.23. (LopusHans kli [
42
]). Solutions of (2.10) are oscillatory if and
only if the function (z) w(z) /(z) maps the upper halfplane conformally onto the
unit disk, where w(z) Wl(Z) + iw2(z) and wj (z)(j I, 2) are two indenpendent real
solutions of (2.10), and their Wronskian is positive on the real axis.
The following characterization of the class HB(HermlteBiehler) of entire
functions having all their zeros within the upper halfplane is given in
THEOREM 2.24. (Lopusans’kii [
42
]). 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 w(n) (z) + Pn2(z)w (n2) (z) + + p0(z)w(z) %nw(z) is studied in
[
43
], where p0(z), Pn2(z) are polynomials of degrees m0 ran_2,
respectively, and % is a complex parameter. It is proved that the fundamental system of
solutions of the equation, determined by the identity matrix as initial conditions
at z 0, satlsifles the estimates
lw+/(z, )I < ]I Iz lOexp=l’Xzi,
for all sufficiently large values of I%1 and Iz I. The value of 0 is defined by
u<l<mnaZx (m. i +n) /(n i),
and c is some positive constant.
Asymptotic properties of the solutions of linear ODE with entire coefficients
are studied in [
44
]. Consider the equation
w (n) + an_lW (nl) +
TIEOREM 2.25. (Boiko and Petrenko [
44
]). 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" 6w2 + z whose
solutions are meromorphic of the form w i/ (z z0) 2 (z 0 /10)(z z 0) 2
i (z z0) 3 + I / (z z0)2 ,, (z), where ,, (z) YnO n+2 (z zo)n"
She represents w as a quotient of two entire functions:
w (u’2 uu") /u2 (2.12)
where u exp ( I dz Jw dz), and then obtalns recursion relations for the
coefflclents of the power series expansions of the numerator and denominator.
In conclusion, we note that in some recent works [
4650
] entire solutions to
DE of infinite order are discussed as well as properties of differential operators
in spaces of entire functions. In [
46
] the author studies the existence of a
solution to the equation 7n= 0 anw(n)(z) f(z) whose growth equals that of the
righthand side, in the case when f(z) belongs to the class B, of entire functions g(z)
such that Ig(x + iy) < cexp [(x) + (y)], for any x, y; here the functions (x),
(y) satisfy Hider conditions. Let (z) be an entire function on of exponential
type without multiple roots. Let My be the operator of convolution with (), where
is a functlon. The following result is proved in [
47
].
THEOREM 2.26. (Napalkov [
47
]). Each entire solution w(z) of the equation
MkW=0 is representable in the form w(z) zkIWl(Z + + w(z), where
Mwi 0 (i i, k), if and only if I(z) + I (1)(z) > cle for all
z e % with some constants c I, c 2 > 0.
In [
48
] the author studies the operator
k7=P0aikz k p > 0.
LPw i7=0p i(z)w(1)(z), pi(z)
. (I) applicable to the set H of entire functions at the point z 0 if the series
The operator LP is said to be
7 i= 0pi(Zo)W(1)(z0) converges for any function w from H;
(2) applicable to H in the domain Iz < if LP is applicable to H at
finite point
(3) strongly regularly applicable to H inside the domain Iz <
w e H and R < oo,
if, for any
where asup { Izl: z e Q}. Conversely, if (2.12) holds, then LP is strongly regularly
applicable to R(Q) inside Iz < =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
{w(z) g Aoo llw II
B(x,y z=xs+uipye
Here #(x,y) is a locally bounded function in R2 with a certain growth for Izl
The author [
50
] treats an equation Lw f with L Z i>0Pi(z)di/dz i, where
the pi(z) are polynomials, deg Pi ni’ lim sup(n i / i) < I, in a space [0, g(8)]
of all entire functions satisfying lim SUPr_ (Znlw(reiS) /ro) < g(8). Here g(8)
is a trigonometrically 0convex function, 0 > 0. 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.
lw(z lexp(_(x, y)) < o.
tx’(t) in=0Ai(t)x(%it)
with matrices A (t) CTM in a neighborhood of t
roots of the equation
det inI=0 I%iI1% Ai(0 + ( + I)E) 0
be nonpositive integers. If m is the smallest of their absolute values there exists
a solution of order m.
From here it follows that the system
tx’(t) A(t)x(t) + in=l Ai (t)x(%it)
has a solution of order m with support t 0, if Ai(0) 0(i _> I) and m + is the
smallest modulus of the negative integer eigenvalues of the matrix A(0). This and
similar results were used in [
15
] to investigate finite order solutions of some
important equations of mathematical physics. For equations with more general argument
delays we have
THEOR 3.2. (Wiener 15] ). The system
0 and constants %i # 0 is that some
(3.1)
tx’(t)
i=0Ai (t)x( i (t))
in which Ai (t) c Cm i (t) g C has a solution (2.1) or order m, if the following
hypotheses are satisfied: (i) the real zeros tij of the functions i(t) are simple
and form a finite or countable set; (2) A (k)(tij) 0(k 0 m), for tij # 0;
(3) m is the smallest modulus of the nonDositive integer roots of Eq. (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)
n=Z0xntn)(t)
in the generalizedfunction space (S)’ conjugate to the space S of testing functions
(3.2)
(t) that satisfy the restriction [2]
’’l(n)(t)l < acnnn
> i.
To ensure the convergence of series (3.2), it is sufficient to require that for n/
the vectors xn satisfy the inequalities
since
II xn II < bdn nn0, 0 >
Z < xn(n)(t), (t)>ll II Z (l)n (n) (0)xn [1 <
n=O n=O
< l l(n)(o)l 11 xn I] < ab Z (cdnO) n <
n=O n=O
for < 0. If series (3.2) converges, its sum represents the general form of a linear
functional in (So)’ with the support t 0 [
53
]. Solutions in (So)’ of some linear
ODE with polynomial coefficients were studied in [
54
], [
55
], [
56
]. The particular
importance of the system
il=0 jm=I0 (Aij + tB1.3.)x(J)(%it tx(%t)
which was considered in [
15
] is that depending on the coefficients it combines either
equations with a singular or regular point at t 0 and in both cases there exists a
solution of the form (3.2). The equation
tx’ (t) Ax(t) + tBx(%t) (3.3)
provides an interesting example of a system that may have two essentially different
solutions in (SOB) concentrated on t 0. If the matrix A assumes negative integer
eigenvalues, (3.3) has a finite order solution (2.1). At the same time there exists
an infinite order solution (3.2), if A @ nE 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
li jm=IOA.lj (t)x (j) (%ij (t)) 0 (3.4)
with a finite number of argument deviations, in which x is an rvector and Aij are
r r matrices, satisfy the following hypotheses.
(i) The coefficients A..(t) are polynomials in t of degree not exceeding p:
Aij(t k=PY. 0Ai.ktk A00(t) Atp p > i.
(ii) The realvalued functions %.. (t) E C in a neighborhood of the origin,
0 and
o< Ioo < t, al >_, + >_t, a (0).
(iii) The matrix A is nonsingular and
c
IO0 Ip1 ]l AII
IolPXll Aiop 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
tPx ’(t) i=E0 ]m=EoAij(t)x(j)(%ij(t))
the particular cases of which
tPx ’(t) A(t)x(t)
and
tPx ’(t)
iI=OAi (t)x(%it)
(3.5)
(ili) The series Z
i=O
.i A. converges, where
with constant coefficients ajk. Its formal solutions are obtained in the form of
Mellin or Laplace integrals. The functions occuring in the inte.rands satisfy linear
difference equations of the form Zvn0 P (qt)G(t + v) 0 (P(y) polynomials).
Properties of solutions of such difference equations, in particular the location oF
singularities and the asymptotic behavior for absolutely large values of t, are
studied. Conditions are derived for formal solutions of Mellin integral type to be
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
Dirichlet series under certain conditions. Finally, questions as to when the llne
of convergence of the Dirichlet series is the natural boundary of the function
represented 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
kI0akDkw(%mkz) 0, D d/dz
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. The FDE
Lw(z) km=IOakDkw(%mkz f(z),
where ak are complex numbers, 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
(3.6)
G(z)
n=IOGnzn Gn
%n(nl)/2/n!
the author shows that the general solution of (3.6) for f
0 is given by
w(z)
2z
G(tz)(t) dt,
with (t) q(t)/A(t), where
m
A(t) k=l0ak%k(2m_k_ I)/2 tk
q is a polynomial of degree <_ m
and F is a contour enclosing all the zeros of A.
S.. SHAH and J. WIENER
A similar integral representation is given for a solution of (3.6) with f # 0 in
terms of the generalized Borel transform
when
(t)
n=7.O f n / Gntn+l
f(z) n=YO fnz
In [
70
] the author discusses the system w’(z) Aw(lz), 0 < I < i, where A is a
cornplex constant matrix. First, the form of all entire solutions is given. Subsequently,
for z # 0 a special system of particular analytic solutions is constructedhymeans of
which all other solutions may be represented. The asymptotic properties as z of
all solutions are investigated. 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
m n
w" (z) + k=El ak (z)w’(IkZ) + j=7.l bj (z)w(jz) 0,
i=nIOa.1w (i)(z) exp(ez) in=YOb.1w (i)(%z),
w (i) (0) wi, i 0 n i,
in which ai, bi, e and I are complex numbers, has been studied with various
assumptions concerning parameters [
7377
]. It is proved in [
75
] that, if III I,
I # and anl > bn its solution is an entire function. If III < i, #
and II C II < I, the solution of the matrix problem
W’(z) AW(z) + exp (z)[BW(lz) + CW’(lz)], W(0) W0
is an entire function of exponential type [
76
]. These results were extended to
linear 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. (Wiener[
58
]). Suppose the system
W (p)(z) Y. Yp. Qij(z)W(J)(lijz)’
i=0 j=0
W (j)(0) Wj, j 0 p
in which O and W are r r matrices, satisfies the following conditions:
(i) QiJ (z) are polynomials of degree not exceeding m;
(ii) lij are complex numbers such that
(3.8)
0 < ql < llijl < i, (j 0 p i), 0 < q2 < llipl < q3 < I;
(iii) the series I o(i) converges, where Q(i) max II Qijk II and Qijk are the
j,k
coefficients of Qij(z), and E II Qip(O) II < i.
i=O
Then the problem has a unique holomorphic solution, which is an entire function
of order not exceeding m + p.
THEOREM 3.7. (Cooke and Wiener [
3
]). If, in addition to the hypotheses of Th.
3.6 the parameters lij(0 _< j _< p I) are separated from unity:
0 < ql < llijl < q4 < i, the solution of (3.8) is an entire function of zero order.
THEOREM 3.8. (Cooke and Wiener [
3
]). Under the assumptions of Th. 3.3 there
exists a polynomial Q(z) of degree p such that the system
ik=EO jm=IOAij (z)W (j) (ij z) Q(z)
linear neutral FDE
matrices A, B, W has a unique holomorphic solution which is an entire
(1) Ai (z)
m
7. AikekZ Bi(z)
k=l
m
k=Y.OBikekz
(ii) ai, bi are complex numbers such that
0 _< Rea.l < MI < oo, 0 < M2 _< Rebi _< M3< oo;
(iii) the series I A (i) and Z B (i) eRebo1 converge where A (1)
max II Bik II, and Y. II Bi(0) I[ e
k i=O
_Rebi
< I.
The authors [
78
] propose a method for finding polynomial solutions of the
max II Aik II,
k
..
with positive constants
has a solution W(z) regular at z
entire function of zero order.
THEOREM 3.9. (Wiener [
51
]). The problem
W’(z)
where b, ao and r. > 0 are given constants. Meromorphic solutions of a class of
1
linear differentialdifference 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
].
2. Gel’fand, I.M. and Shilov, G.E. Generalized Functions, Vol. 2, Academic Press,
New York, 1968.
4. Gel’fand, I.M. and Shilov, G.E. Generalized Functions, Vol. I, Academic Press,
New York, 1968.
S.M. SHAB and J. WIENER
27. Macintyre, A.J. On Bloch’s theorem. Math. Z. 44(1939), 536540.
28. Wittich, Hans.
gleichungen.
Ganze transzendente Lsungen algebraischer
Differential
Gtt Nachrichten, (1952), 277288.
40. Bellman, Richard; Cooke, Kenneth L. DifferentlalDifference Equations, Acadic
Press, New York, 1963.
66. Pandolfi, L. Some observations on the asymptotic behavior of the solutions of
the equation (t)=A(t)x(%t)+B(t)x(t), %>0. J. Math. Anal. Appl. 67 (2), (1979),
483489.
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