#### Reducible functional differential equations

Journal of
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
S.M. SHAH 0 1 2
0 Functional Differential Equation , Argument Deviation, Involu-
1 Department of Mathematics Pan American University Edinburg , Texas 78539 , USA
2 Department of Mathematics University of Kentucky Lexington , Kentucky 40506 , USA
This is the first part of a survey on analytic solutions of functional differential equations (FDE). Some classes of FDE that can be reduced to ordinary differential equations are considered since they often provide an insight into the structure of analytic solutions to equations with more general argument deviations. Reducible FDE also find important applications in the study of stability of differential-difference equations and arise in a number of biological models. tion.
AND PHRASES
I. INTRODUCTION.
1980 MATHEt.TICS SUBJECT CLASSIFICATION CODES.
34K05, 34K20, 34K09.
In [
1-4
] a method has been discovered for the study of a special class of
functional differential equations
differential equations with involutions.
This basi
cally algebraic approach was developed also in a number of other works and culminated
in the monograph [
5
].
Though numerous papers continue to appear in this field [
6-10
],
some aspects of the theory still require further investigation.
In connection with
the DurDoses of our article we mention only such topics as hiher-order equations
with rotation of the argument, equations in partial derivatives with involutions,
influence of the method on the study of systems with deviations of more general
nature, and solutions in spaces of generalized and entire functions.
In [12] we proved that the solution is obtained very simply by a differentiation of
(2.1). As a matter of fact,
x"(t)
x(),
x’(t)
-- x’(
t
t2x"(t) + x(t)
whence,
Cons equ en t ly,
x(t) r[ClCS(--z In t) + C2sin(Z-z in t)].
Substituting x(t) in (2.1), we obtain CI=C2, and finally,
x(t)
C
cos(- In t
).
Obviously, the key to the solution is the fact that the function f(t)
interval (0, ) one-to-one onto itself and that the relation
f(f(t)) t,
i/t maps the
(2.3)
or, equivalently,
t E G, and fn (t) t for n
functions are involutions.
EXAMPLE 2.1. f(t)
EXAMPLE 2.2.
I,
m
i. It is easy to check that the following
c
t on R
(_oo, oo), where c is an arbitrary real.
f(t)
-at for t _> 0,
-t/a for t <_ O,
on R, where a > 0 is arbitrary [
5
].
EXAfP LE 2.3.
t k for 0 < t < i,
on (0, oo), where k is an arbitrary positive integer [
5
].
EXAMPLE 2.4. The function f(z) ez, where E exp(2i/m), is an involution of
order m on the complex plane.
EXAMPLE 2.5. The function [13]
We denote the set of all such functions by I. The graph of each f e I is
symmetric about the line x t in the (t, x) plane. Conversely, if F is the set of
points of the (t, x) plane, symmetric about the line x t and which contains for
each t a single point with abscissa t, then F is a graph of a function from I. One
of the methods for obtaining strong involutions is the following [14]. Assume that a
real function g(t, x) is defined on the set of all ordered pairs of real numbers and
is such that if g(t, x) O, then g(x, t) 0 (in particular, this is fulfilled if g
is symmetric, i.e., g(t, x) g(x, t)). If to each t there corresponds a single real
x f(t) such that g(t, x) 0, then f g I. For example,
g(t, x) t + x c,
then
f(t)
c- t.
If we take
then
S. Mo SHAH AND J. WIENER
g(t, x)
f(t) 3Jc t 3
Every continuous function f I is strictly decreasing [15]. Hence,
t-l_iomo f(t) +oo, tl-i+mo f(t) _o. (2.4)
THEOREM 2.1. A continuous strong involution f(t) has a unique fixed point.
PROOF. The continuous function @(t) f(t) t satisfies relations of the form
(2.4) and, therefore, has a zero which is unique by virtue of its strict monotonicity.
We also consider hyperbolic involutary mappings
f(t) Ytt+- (2 + > 0) (2.5)
which leave two points fixed. We introduce the following definition.
DEFINITION 2.2. A relation of the form
F(t, x(fl(t)) x(fk(t))
fk(t) are involutions, is called a differential equation with
in which fl(t),
involutions [i].
THEOM 2.2([1]). Let the equation
x’(t) F(t, x(t), x(f(t))) (2.6)
satisfy the following hypotheses.
(i) The function f(t) is a continuously differentiable strong involution with
a fixed point tO.
x(n) (fl(t))
x (n) (fk(t)))=O,
PROOF. Eq. (2.8) is obtained by differentiating (2.6).
x"(t) F + xF(t) x (t) + x(fF(t)) x (f(t))f’(t).
Indeed, we have
The fixed point of the involution f(t)
(2.11) is
x()
xO;
a-t is t o
a/2. The initial condition for
where x is an unknown function and where the following conditions are fulfilled:
p" r 2, n and every t G.
THEOREM 2.5 ([]7]). If conditions (1)-(3) are satisfied, then every p-times
differentiable solution of Eq. (2.14) is a component of the solution of a system of
ordinary differential equations with argument t only. This system is obtained from
Fq. (2.14).
To investigate the equation x’(t) f(x(t), x(-t)), the author of [6] denotes
y(t) x(-t) and obtains
M f’ i(t) ddt
Then the solution of the linear ordinary differential equation
kn=O ak(f(t))MLx(t) x(t) kn=ZOak(f(t))M@(t) + @(f(t))
with the initial conditions
x (k) (t O
xk,
k
Mx(t) lt=t0 Xk + Mk*(t)It=to,
0
k
n
1,
0
n
i
is a solution of problem (3.1)-(3,2).
PROOF. By successively differentiating (3.1) n times, we obtain
x(f(t))
Lx(t)
(t)
MOLx(t)
MO(t),
(3.3)
(3.4)
x’(f(t))
x"(f(t))
f’(t) ddt Lx(t)
f’(t) ddt (t)
1
MLx(t)
M(t),
1
f,(t) ddt MLx(t)
I
f’(t) dd- M(t)
M2Lx(t)
M2(t),
x (n) (f(t)) f’ I(t) ddt Mn-ILx(t) f’ i(t) ddt Mn-l(t)
These relations are multiplied by a0(f(t)) al(f(t)
and the results are added together:
n
Z ak(f(t))x (k) (f(t))
k=O
By virtue of f(f(t))
nZ ak(f(t))Mx(t )- nF. ak(f(t))Mk(t).
k=O k--O
t, it follows from (3.1) that
an (f(t))
MnLx(t)
Mn(t).
respectively
n
k=O ak(f(t))x (k) (f(t)) x(t) + (f(t)).
Thus, we obtain Eq. (3,4). In order that the solution of this equation satisfies
problem (3.1)-(3.2), we need to pose the following initial conditions for (3.4):
values of the function x(t) and of its n 1 derivatives at the point t O should
equal Xk, k O, n i, from (3.2), while the values x (n) (tO x (2n-l) (t O
are determined from the relations
the
Mx(t)
0
n
i
by substituting the values t O and xk for t and x (k) (t).
THEOREM 3.2 ([i]), The equation
x
(t)
x(
(3.6)
is integrable in quadratures and has a fundamental system of solutions of the form
ta(In t) sin(b In t), ta(In t) j cos(b In t), (3.7)
a and b are real and is a nonnegative integer.
PROOF. By an n-fold differentiation Eq. (3.6) is reduced to the Euler equation
k=nI2+nl b(kn) tkx (k) (t) x(t). (3.8)
For n i this follows from (2.2). Let us assume that the assertion is true for n
and prove its validity for n + i. In accordance with formula (3.3), we introduce for
Eq. (3.6) the operator
M -t 2 ddt
On the basis of (3.4) and (3.8) we have
Mnx (n) (t) l2n
k=n+l
b(kn) tkx (k) (t),
Then
Mnx (n+l (t)
2n
I
k=n+l
b (n) tkx(k+l) (t).
k
Mn+ix (n+l) (t)
-t 2 d
2n
k=n+lZ bk(n)tkx(k+l)__ (t)
is reduced by an (n+l) -fold differentiation to the Euler equation
Mn+ix(n+l)(t)
x(t).
At the same time we established the recurrence relation
b k(n+l)= -(k-l)"bk(_n)1 bk(_n)2
n + 2 <_ k < 2n + 2,
b(n’n’= 0 b(2nn)+l 0
connecting the coefficients of the Euler equations
k=n2+nlb kn) t__ (k) (t) x(t) and k=2nnF,++22 bk(n+l)tkx(k)(t): x(t),
which correspond to the equations
x (n) (t)
x(
and x (n+l) (t)
x(_t)i.
It is well known [18] that the Euler equation has a fundamental system of solutions
of the form (3.7), where a + bi is a root of the characteristic equation and j is a
nonnegative integer smaller than its multiplicity. The theorem is proved.
EXAMPLE 3.1. The investigation of the nonhomogeneous equation [I]
x’(t)
x( ) + 9(t), 0 < t < o, 9(t) g C 1 (0,
reduces to the problem
t2x"(t) + x(t)
x(1)
The solution is
x(1)
x0
t29’(t)
O,
x’(1)
-e2 --- A2 > O,
(3.9)
-a2
XXlo_% 2 [(a-%2) t kl + (%l-a)t%2].
x(f(t)) with an involution f(t) has been studied in [19].
Consider the equation [13] with respect to the unknown function x(t):
x’(t) a(t)x(f(t)) + b(t),
(i) The function f maps an open set G onto G.
(2) The function f can be iterated in the following way:
fl(t) f(t) fk(t) f(fk_l(t)) fm (t)
t (t E G)
(3.13)
where m is the least natural number for which the last relation holds.
(3) The functions a(t), b(t) and f(t) are m 1 times differentiable on G, and
x(t) is m times differentiable on the same set.
THEOREM 3.4 (3]). Eq. (3.13), for which conditions (I)-(3) hold, can be
reduced to a linear differential equation of order m.
EXAMPLE 3.3. Consider the equation 1161
x’(t) x(f(t)), f(t) (l-t) -I (3 14)
and G (_o% 0)U(0, I)U(lo + oo). For f we have f3(t) t on G. In this case (3.14)
is reducible to the equation
t2(l-t)2x (t) 2t2(l-t)x"(t) x(t) O.
THEOREM 3.5 ([i]). In the system
x’(t) Ax(t) + Bx(c-t), x(c/2) x0 (3.15)
let A and B be constant commutative r xr matrices, x be an r-dimensional vector,
and B be nonsingular.
Then the solution of the system
x"(t) (A2-B2)x(t)
x(cl2) x0, x’(cl2) =(A+B)x0
is the solution of problem (3.15).
In [7] it has been proved that the equation
has the general solution
while the equation
t-2) + c4[sin(’] In t) + 3+I/ cos( In t)].
It follows from here that, by appropriate choice of c I, c2, c3, and c4, we can
obtain both oscillating and nonoscillating solutions of the above equations. On the
other hand, it is known that, for ordinary second-order equations, all solutions are
either simultaneously oscillating or simultaneously nonoscillating. It has been
also proved in [7] that the system
x’(t) A(t)x(t) + f(t, x(tl-)) 1 <_ t <
II f(t, x())II <-- II x()ll q,
where 6 > 0 and q _> 1 are constants, is stable with respect to the first
approximat ion.
t a-.- [tkx)kt (t)]
kt k x (k) (t)
which proves the assertion.
The functional differential equation
Q’(t) AQ(t) + BQT(T t), < t < (3.17)
where A, B are n x n constant matrices, T _> 0, Q(t) is a differentiable n n matrix
and QT(t) is its transpose, has been studied in [20]. Existence, uniqueness and an
algebraic representation of its solutions are given. This equation, of considerable
interest in its own right, arises naturally in the construction of Liapunov
functionals for retarded differential equations of the form x’(t) Cx(t) + Dx(t-I), where
C, D are constant n n matrices. The role played by the matrix Q(t) is analogous to
the one played by a positive definite matrix in the construction of Liapunov functions
for ordinary differential equations. It is shown that, unlike the infinite
dimensionality of the vector space of solutions of functional differential equations, the
linear vector space of solutions to (3.17) is of dimension n2. Moreover, the authors
give a complete algebraic characterization of these n2 linearly independent solutions
which parallels the one for ordinary differential equations, indicate computationally
simple methods for obtaining the solutions, and allude to the variation of constants
formula for the nonhomogeneous problem.
The initial condition for (3.17) is
where si, and s,j are, respectively, the i th row and the j th colun of S; further,
let there correspond to the n> n matrix S the n2-vector s (Sl, Sn,)T. With
this notation Eqs. (3.19) and (3.20) can be rewritten as
r(t
t)B
-I IA
(t
and
q() [kl,, kn,]T, r(-) [k,Tl, kT,n T
which, with the obvious correspondence and for simplicity of notation, are denoted as
p’(t) Cp(t), p(T/2) PT/2" (3.21)
Here p(t) is an 2n2-vector and C is a 2n2 2n 2 constant matrix. (3.21) is used in
provinR the followin result:
THEOREM 3.7 ([20]). Eq. (3.17) with the initial condition (3.18) has a unique
solution Q(t) for < t < oo.
Examination of the proof makes it clear that knowledge of the solution to (3.21)
each .,
nj,r Zr=s I n. mj, Zj m.=3 2n2"
given by
where the generalized eigenmatrix pair (Lji ,r’ Mj ,ri)associated with the eigenvalue
satisfies the equations
T
L.2,r0 M.3,r0 0. But this result demonstrates that if %.3 is a solution of (3.22),
-%. will also be a solution; moreover, %. and -%. have the same geometric
multiplici3 3 3
(-I) q+i the n2 linearly independent solutions of (3 19) given by
Zj ,qr(t)
W.3 qr(t)
exp(j(t )T)
q (t
i=l
Tq-i
1
(q- i):
exp(-j (t
T q (t
)) Y
i=l
(q
)T q-i
i): (-I) q+i
r L i
j r
i
Mj ,r
+
Mj r
Lj r
satisfy the condition
T
Zj, Wj,r
But this is precisely condition (3.20)" it therefore follows that the expressions
Zj,rq(t)
q (t
l
i= I
)T q-i
(q- i)!
[exp(%j(t )T)Lj,ri +
(-1) q+i exp(-Ej(t -zT) )Mj,riT
(3"25)
are n 2 linearly independent solutions of (3.17).
THEOREM 3.8 ([20]). Eq. (3.17) has n 2 linearly independent solutions given by
Eq. (3.25), where the generalized eigenmatrix pairs (Lj, ri Mj,ri satisfy Eq. (3.24)
for one of the elements of the pair (j, -j), each of which is a solution of Eq.
(3.22).
Eq. (3.17) has been used in [
22
] for the construction of Liapunov functionals and
also encountered in a somewhat different form in [
23
].
Some problems of mathematical physics lead to the study of initial and boundary
value problems for equations in partial derivatives with deviating arguments. Since
research in this direction is developed poorly, the investigation of equations with
involutions is of certain interest. They can be reduced to equations without
argument deviations and, on the other hand, their study discovers essential differences
that may appear between the behavior of solutions to functional differential
equations and the corresponding equations without argument deviations.
The solution of the mixed problem with homogeneous boundary conditions and
initial values at the fixed point t o of the involution f(t) for the equations
ut(t, x) auXX (t x) + buXX (f(t) x)
(3.26)
and
Cn T’n(t O -%n (a+b)Cn
satisfies Eq. (3.29). The following theorems illustrate striking dissimilarities
between equations of the form (3.26) and (3.27) and the corresponding equations
without argument deviations.
THEOREM 3.9. The solution of the problem
ut(t x) auxx (t, x) + huxx (c-t, x), (3.31)
u(t, 0) u(t, ) 0, u(cl2, x) (x)
is unbounded as t +, if a b # 0. If Ibl < lal, b # 0, expansion (3,28) diverges
for all t # c/2.
PROOF. By separating the variables, we obtain
2 2
Tn,(t)=_o_zn (aTn (t) + bTn (c-t)), Tn (c/2) Cn (3.32)
The initial conditions for equations (3.31) and (3.32) are posed at the fixed point
of the involution f(t) c t. In this case, Eq. (3.30) takes the form
4 4
T("t)n n4 ,b 2 a 2) Tn (t)
Tn (c/2)
The completion of the proof is a result of simple computations. Depending on the
relations between the coefficients a and b, the following possibilities may occur:
2n 2 2 a2 (t-c)
(i) Tn (t) Cn (cos 2
a+b
b2_a2
s in
(x), ut(O x)
Tn (0) An T’n(0) Bn
by successive differentiation of which we obtain
Tn(3)(t) 22a2n 2 T(t) + T262-n2 T’n(-t)
From Eq. (3.33) we find
and also
T (4) (t)
n
T"n(-t)
r2 a 2n 2 T"n(t)
2
7r 22b2n2 T"n (-t)
2a2n2
2
Tn(_t)
22b2n2 Tn (t)
Tn(4)(t) + 22a22n2 Tn"(t) +
with the initial conditions
Thus, Eq. (3.34) is reduced to the fourth-order ordinary differential equation
4 (a44-b4)n4 Tn (t) 0
(3) Finally, the inequality a 2 < b 2 leads to the result
n 2b_a2
Bn sinh n b2-a 2 t.
Of some interest is the equation
ut(t, x)
(t +_____B
Auxxyt
x)
with the hyperbolic involution
having two fixed points
which is a generalization of Eq. (2.1). Differentiation changes (3.37) to the form
(yt )2T"n(t) + A2A244n4 Tn(t) 0.
(3.38)
o. For
the functions Tn (t) are oscillatory.
4. EQUATIONS WITH ROTATION OF THE ARGUMENT
An equation that contains, along with the unknown function x(t) and its
derivatives, the value x(-t) and, possibly, the derivatives of x at the point -t, is called
a differential equation with reflection. An equation in which as well as the unknown
function x(t) and its derivatives, the values x(1t-a I) X(mt-am and the
corresponding values of the derivatives appear, where gl’ m are mth roots of
unity and al’ m are complex numbers, is called a differential equation with
rotation. For m 2 this last definition includes the previous one. Linear first-order
equations with constant coefficients and with reflection have been examined in detail
in [
5
]. There is also an indication (p. 169) that "the problem is much more
difficult in the case of a differential equation with reflection of order greater than
one". Meanwhile, general results for systems of any order with rotation appeared in
[
3
], [
4
], [
9
], and [
24
].
Consider the scalar equation
k=En0 akx(k) (t) k=nE0bkX (k) (ct) + lp(t), m__ 1 (4.1)
AIAoX (BiBoX)(e2t) + AI@ + (Bo)(et),
and act on this relation by A2. From
AoX
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
.
(B0x)(Et) +
Finally, this process leads to the ordinary differential equation
(AO(m-l) B 0(m-l))x mj-=ZIO
I,
and I is the identity operator. Thus, (4.1) is reduced to the ODE (4.2) of order
mn. ’I make the initil onditions for (4.2) agree with the riRinal probl,m, it
necessary to attach to ’onditions (4.1) the additional relati,,s
(A0(j)
gk(j+l) B0(j))x(k)(t)] t=0
Y. g ikAJ-i) B i-l)(*k) (t) It__0
i=O
(j
0
m-o 2; k
0
n- i).
System (4.3) has a unique solution for x (k)(O)(n < k _< mn- I), iff
anj # (eibn)J
(0 < i < m- i, i < j < m- i)
These considerations enable us to formulate
THEOREM 4 1 ([
9
]) If @EC (m-l)n and inequalities (4.4) are fulfilled, the
solution of ordinary differential equation (4.2) with initial conditions (4.1)-(4.3)
satisfies problem (4.1).
THEOREM 4.2 ([
9
]). If g # i, the substitution
y x exp(at/l e)
exp(ct)(By)(et) +
(4.2)
(4 3)
(4 4)
(4.5)
with operators A and B defined by (4.1) to
Px (Qx)(et) + exp(-c,t/l
where P and Q are linear differential operators of order n with constant coefficients
and E is the identity matrix.
THEOREM 4.3. ([
3
]). If e is a root of unity (e # I), Icll < 1, and the matrix A
is commuting with B and C, then problem (4.6) is reducible to an ordinary linear
system with constant coefficients.
The following particular case of Eq. (4.1) has been investigated in [27].
THEOREM 4.4. ([27]). Suppose we are given a differential equation with
reflection of order n with constant coefficients
kn=7.O [ak-x(k) (t) + bkX (k) (-t)] y(t). (4.8)
bj_kb k
O, I,
n and j
k + I
k + n,
n
(c) the polynomial 7. %2jtj has simple roots uq only, where
j--O
J
k=ZO Cjk for 0 < j < n,
k=j-n j k for n < j 5_ 2n,
n
x(t)
Then every solution of Eq. (4.8) is of the form
Cjk
(-l)n+j-k(an2-bn2)(aj_ ka k-bj_kb k).
(4.10)
(4.11)
where the Ck are arbitrary constants and (t) is a solution of the equation
THEOREM 4.5 ([
9
]). Suppose that the coefficients of the equation
n
Y. ak(t)x (k) (t)
k=O
x(et) + (t), x (k) (0)
xk, k
O,
n
1 (4.9)
belong to C (m-l)n em
i, a (0) # 0 and
ej
n
l E-Jka,_(eJt)dk/dt k,
k=O
0,
m- 1.
Then the solution of the linear ordinary differential equation
L (m-l) x(t)
0
m-i
x(t) + Z (Lk(m-l) )(ek-lt) + (em-lt)
k=l
(Lk(m-l)
Lm_ILm_2
L
k 0
k < m
with the initial conditions
satisfies problem (4,9),
x (k) (0)
Xk(k
0
n
I)
n(m-1)
1
kx (k) (0) + (k) (0) ,k=O
PROOF. Applying the operator LI to (4.9) and taking into account that
we get
(LoX)(et)
x(e2t) + (et)
LIeOX(t) x(2t) + Ll(t) + (t)
and act on this equation by L2 to obtain
It is easy to verify the relations
In particular,
(tAdldt + B) m X(t) X(t). (4.13)
This is Euler’s equation with matrix coefficients. Since its order is higher than
that of (4.12) we substitute the general solution of (4.13) in (4.12) and equate the
coefficients of the like terms in the corresponding logarithmic sums to find the
additional unknown constants.
EXAMPLE 4.2. We connect with the equation [
9
]
tx’(t) 2x(t) x(et), e3 1 (4.14)
with constant coefficients A and B, det A # 0 and em
and has a solution
X(t) e(t)tA-IB
where the matrix P(t) is a finite linear combination of exponential functions.
PROOF. The transition from (4.15) to an ordinary equation is realized by means
of the operators
(4.16)
1 is Integrable in closed form
(4.17)
L. 0
in consequence of which we obtain the relation
(Ad/dt t-IB) m X(t) em(m-l)/2X(t).
e-J(Ad/dt t-IB), j
m
Xk(t) exp(ktA-l)tA-IB, k
I,
mo
Their linear combination represents the general solution of (4.15).
EXAMPLE 4.3. In accordance with (4.17) to the equation [
9
]
tx’(t) 3x(t) + tx(-t) (4.18)
there correspond two ordinary relations
x’(t) (3t-I + i)x(t), x’(t) (3t-I i)x(t).
We substitute into (4.18) the linear combination of their solutions
x(t) t3(Clexp(it) + C2exp(-it))
and find C 2 IC I. A solution of (4.18) is
x(t) Ct3(slnt + cost).
Biological models often lead to systems of delay or functional differential
equations (FDE) and to questions concerning the stability of equilbrium solutions of
such equations. The monographs [28] and [29] discuss a number of examples of such
models which describe phenomena from population dynamics, ecology, and physiology.
The work [29] is mainly devoted to the analysis of models leading to reducible FDE.
A necessary and sufficient condition for the reducibility of a FDE to a system of
ordinary differential equations is given by the author of [
30
]. His method is
frequently used to study FDE arising in biological models. We omit these topics and
refer to a recent paper [
31
]. For the study of analytic solutions to FDE, which will
be the main topic in the next part of our paper, we also mention survey [
32
].
I0. WIENER, J. and AFTABIZADEH, A.R. Boundary value problems for differential
equations with reflection of the argument, Internat. J. Math. & Math. Sci. (to
appear)
Ii. SILBERSTEIN, L. Solution of the equation f’(x)
(1940), 185-186.
f(I/x), Philos. Maga.zine 30
12. WIENER, J. On Silberstein’s functional equation, Uen. Zap. Ryazan. Pedagog.
Inst. 41 (1966), 5-8.
13.
14. SHISHA, O. and MEHR, C.B. On involutions, J. Nat. Bur. Stand. 71B (1967), 19-2(%
15. BOGDANOV, Yu.S. On the functional equation xn
t, DAN BSSR 5 (1961), 235-237.
16.
17.
LUI, R. On a class of functional differential equations, Publ. Electrotechn.
Fac. Univ. Belgra_de Ser. Math. Phys. 461-497 (1974), 31-32.
LUI, R. Functional differential equations whose arguments form a finite
group, Publ. Electrotechn. Fac. Univ.. Be.lgrade. Ser. Math. Phys. 498-541
(1975), 133-135.
18. CODDINGTON, E.A. and LEVINSON, N. Theory of ordinary differential equations,
McGraw-Hill, New York, 1955.
19. KULLER, R.G. On the differential equation f’
Mag. 42 (1969), 195-200.
I, Math.
20. CASTELAN, W.G. and INFANTE, E.F. On a functional equation arising in the
stability theory of difference-differential equations, _Q.rt. Appl. Math. 35
(1977), 311-319.
27. MABIC-KULMA, B. On an equation with reflection of order n, S_tudia .Math. 35
(1970), 69-76.
28. CUSHING, J.M. Integrodifferential equations and delay models in population
dynamics, in "Lecture Notes in Biomathematics, No. 20", Springer-Verlag,
Berlin, 1977.
McDONALD, N. Time lags in biological models, in "Lecture Notes in
Biomathematics, No. 27", Springer-Verlag, Berlin, 1978.
ns Research
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0 for k 2 . WIENER , J. Differential equations in partial derivatives with involutions , Differencial'nye Uravnenija 7 ( 1970 ), 1320 - 1322 .
3, WIENER , J. Differential equations with periodic transformations of the argument , Izv. Vys. Uebn. Zaved. Radiofizika 3 _ ( 1973 ), 481 - 484 .
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5. PRZEWORSKA-ROLEWICZ , D. Equations with transformed argument. An algebraic approach , Panstwowe Wydawnictwo Naukowe, Warszawa, 1973 .
ARKOVSKII , A.N. Functional-differential equations with a finite group of argument transformations, Asotic behavior of solutions of functionaldifferential equati_ons Akad. Nauk Ukrain . SSR, Inst.Mat., Kiev ( 1978 ), 118 - 142 .
KURDANOV , Kh.Yu. The influence of an argument deviation on the behavior of solutions of differential equations , Differencial'nye Uravnenija 15 ( 1979 ), 944 .
KISIELEWICZ , M. (Editor) Functional differential systems and related topics , Proceedings of the First International Conference held at Blaej_ewko, _May 19 _-. 26 1979 . Highe,r College of Engineering, Institute of Mathematics and Physics, Zie]ona Gora (Poland) , 1980 .
9. COOKE , K. and WIENER , J. Distributional and analytic solutions of functional differential equations , J. Math. Anal. Appl . 98 ( 1984 ), 111 - 129 .
LUI , R. On a functional differential equation , Publ. Electrotechn. Fac. Univ. .Belgrad .e,.. Ser. Ma.th. Phys . 338 - 352 ( 1971 ), 55 - 56 .
21. BELLMAN , R. Introduction to matrix analysis, McGraw- Hill , New York, 1960 .
22. REPIN , I.M. Quadratic Liapunov functionals for systems with delays , Prikl. Matem. Mekh . 29 ( 1965 ), 564 - 566 .
23. DATKO , R. An algorithm for computing Liapunov functionals for some differential difference equations , in Ordinary differential equations , 1971 NRL-MRC Conference , Academic Press ( 1972 ), 387 - 398 .
24. WIENER , J. Periodic mappings in the study of functional differential equations , Differencial'nye Uravn.e.nija 3 , Ryazan ( 1974 ) 34 - 45 .
25. BRUWIER , L. Sur l'application du calcul cymbolique a la iresolution d'equations fonctionnelles , Bull. Soc. R. Sci. Liege 17 ( 1948 ), 220 - 245 .
26. VALEEV , K.G. On solutions of some functional equations, Isis.led. po Integro-diff . Uravn. v Kirgizii 5 ( 1968 ), 85 - 89 .
30. FARGUE , D.M. Rducibilit des systmes hrditaires a des systmes dynamiques , C. R. Acad . Sci. Paris Ser. B 277 ( 1973 ), 471 - 473 .
31. BUSENBERG , S. and TRAVIS , C. On the use of reducible-functional differential equations in biological models , J. Math. Anal. Appl . 89 ( 1982 ), 46 - 66 .
32. SHAH , S.M. and WIENER , J. Distributional and entire solutions of ordinary differential and functional differential equations, Internat . J. Math. & Math. Sci. 6 ( 2 ), ( 1983 ), 243 - 270 .
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