Internat. J. Math. & Math. Sci.
(1985) 1-27
Vol. 8 No.
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
S.M. SHAH
Department of Mathematics
University of Kentucky
Lexington, Kentucky 40506
and
JOSEPH WIENER
Department of Mathematics
Pan American University
Edinburg, Texas 78539
(Received December 5, 1984)
ABSTRACT.
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.
KEY WORDS AND PHRASES.
Functional
Differential Equation, Argument Deviation, Involu-
tion.
1980 MATHEt.TICS SUBJECT CLASSIFICATION CODES.
I.
34K05, 34K20, 34K09.
INTRODUCTION.
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
some aspects of the theory still require further investigation.
[6-10],
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.
S. M. SHAH AND J. WIENER
2
2.
DIFFERENTIAL EQUATIONS WITH INVOLUTIONS.
In studying equations with a deviating argument, not only the general properties
are of interest, but also the selection and analysis of the individual classes of
such equations which admit of simple methods of investigation.
In this section we
consider a special type of functional differential equations that can be transformed
into ordinary differential equations and thus provide an abundant source of relations
with analytic solutions.
Silberstein
[ii] studied the equation
x(),
x’(t)
0 < t <
(2.1)
--
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’(
t
whence,
t2x"(t)
Cons equ en t ly,
+ x(t)
r[ClCS(--z
x(t)
C2sin(Z-z
CI=C2,
cos(-
C
(2.2)
0.
In t) +
Substituting x(t) in (2.1), we obtain
x(t)
x(t),
t
In t
in
t)].
and finally,
).
Obviously, the key to the solution is the fact that the function f(t)
interval
(0, )
i/t maps the
one-to-one onto itself and that the relation
f(f(t))
(2.3)
t,
or, equivalently,
f(t)
is satisfied for each t
A function f(t)
(2.3),
7
f
-I
(t)
(0, oo).
t that maps a set G onto itself and satlsles on G condition
is called an involution.
coincides with its own inverse.
fl(t)
f(t)
In other words, an involution is a mapping which
Let
fn+l(t)
denote the iterations of a function f:G
f(f (t))
n
G.
n
1
2
A function f:G
involution of order m if there exists an integer m
_>
G is said to be an
2 such that f (t)
m
t for each
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
t E G, and f (t)
n
I,
t for n
It is easy to check that the following
i.
m
3
functions are involutions.
EXAMPLE 2.1.
f(t)
(_oo, oo), where
t on R
c
c is an arbitrary real.
EXAMPLE 2.2.
_> 0,
-at for t
f(t)
-t/a for
on
<_ O,
R, where a > 0 is arbitrary [5].
EXAfP LE 2.3.
on
t
(0, oo), where k
EXAMPLE 2.4.
k
t
for 0 < t < i,
is an arbitrary positive integer
ez, where
The function f(z)
[5].
exp(2i/m), is an involution of
E
order m on the complex plane.
EXAMPLE 2.5.
The function
[13]
t,
t e
f(t)
(0, I)U(I, 2)U.. U(m-2, m-l),
t+l, t
[t-(m-l),
(m-l, m)
t
(_o, O)U(O, I)U... U(m-l, m)U(m,
is an involution of order m on G
DEFINITION 2.1.
(_oo, O)U(m, +oo),
A real function f(t)
t of a real variable t, defined on the
whole axis and satisfying relation (2.3) for all t, is called a strong involution
[].
We denote the set of all such functions by I.
t in the
metric about the line x
points of the
(t, x) plane.
(t, x) plane, symmetric about
The graph of each f e I is sym-
Conversely, if F is the set of
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)
is symmetric, i.e.,
x
O, then g(x, t)
g(t, x)
f(t) such that g(t, x)
g(x, t)).
g
+
x
t
then
f(t)
If to each t there corresponds a single real
0, then f
g(t, x)
c- t.
0 (in particular, this is fulfilled if g
I.
For example,
c,
S. Mo SHAH AND J. WIENER
4
If we take
g(t, x)
t
3
3
+ x
c,
then
3Jc
f(t)
lim f(t)
t-+o
+oo,
t-_oo
Hence,
(2.4)
_o.
A continuous strong involution f(t) has a unique fixed point.
THEOREM 2.1.
PROOF.
3
I is strictly decreasing [15].
Every continuous function f
lim f(t)
t
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
t+
f(t)
which leave two points fixed.
in which
involutions
(2.5)
> 0)
We introduce the following definition.
x(fk(t))
x(fl(t))
fk(t)
fl(t),
+
A relation of the form
DEFINITION 2.2.
F(t,
(2
Yt-
x(n) (fl(t))
x
(n)
(fk(t)))=O,
are involutions, is called a differential equation with
[i].
THEOM 2.2([1]).
Let the equation
F(t, x(t), x(f(t)))
x’(t)
(2.6)
satisfy the following hypotheses.
(i)
The function f(t) is a continuously differentiable strong involution with
a fixed point t
(ii)
O.
The function F is defined and is continuously differentiable in the whole
space of its arguments.
(iii)
The given equation is uniquely solvable with respect to x(f(t)):
G(t, x(t), x’(t)).
x(f(t))
(2.7)
Then the solution of the ordinary differential equation
x"(t)
F + F x’(t) +
F
x(t)
x(f(t)) f’(t)F(f(t), x(f(t)), x(t))
(2.8)
(where x(f(t)) is given by expression (2.7)) with the initial conditions
x(t
is a solution of
0)
x0,
x’(t
O)
F(to, Xo, x O)
(2.8’)
Eq. (2.6) with the initial condition
x(t
O)
x
O.
(2.9)
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
PROOF.
Eq. (2.8)
x"(t)
F + F x (t) +
F
x(t)
x(f(t)) x (f(t))f’(t).
is obtained by differentiating
But from (2.6) and the relation f(f(t))
x’(f(t))
(2.6).
5
Indeed, we have
it follows that
F(f(t), x(f(t)), x(t)).
The second of the initial conditions (2.8’)is a compatibility condition and is found
from Eq. (2.6), with regard to (2.9) and f(t
0)
t
O
It is especially clear to see
the role of involutions in equations which do not contain
and x(t) explicitly.
In
this case,
F(x(t)).
x’(f(t))
THEOREM 2.3 ([i]).
Assume that in the equation
x’(t)
F(x(f(t)))
(2.10)
the function f(t) is a continuously differentiable strong involution with a fixed
point t
o
and the function F is defined, continuously differentiable, and strictly
monotonic on
(_oo, o).
Then the solution of the ordinary differential equation
x"(t)
F’(x(f(t)))F(x(t))f’(t),
F-l(x’(t))
x(f(t))
with the initial conditions
x(t
is a solution of
COROLLARY.
form
O)
Xo,
x’(t
o
F(Xo)
Eq. (2.10) with the initial condition
X(to)
x
O.
Theorems 2.2 and 2.3 remain valid if f(t) is an involution of the
(2.5), while the
equations are considered on one of the intervals
(_oo, a/y)
or
(I, ).
REMARK.
Let t
O
be the fixed point of an involution f(t).
(2.10) are retarded equations, whereas for t < t
EXAMPLE 2.6.
O
For t >
to,
(2.6) and
they are of advanced type.
By differentiating the equation [1]
I
x(a-t)
x’(t)
(2.11)
and taking into account that
I
x(t)
x’(a-t)
we obtain the ordinary differential equation
d2x
dt
2
i
x(t)
dx 2
()
(2.12)
S.M. SHAH AND J. WIENER
6
The fixed point of the involution f(t)
a-t is t
a/2.
o
The initial condition for
(2.11) is
x()
x
O;
the corresponding conditions for (2.12) are
x()
Eq. (2.12)
1
x’()
x o,
x0
quadratures:
is integrable in
a
x(t)
x
t-
exp
0
2
0
x
This is the solution of the original equation
The topic of the paper
[16]
(2.11).
is the equation
x(n)(t))
F(t, x(t), x’(t)
(2.13)
x(f(t)),
where x is an unknown function.
THEOREM 2.4
([16]).
Let the following conditions be satisfied:
(I) The function f maps the open set G into G, G being a subset of the set R of
real numbers.
(2)
The function f has
fl(t)
iterations such that
f(t)
fk(t)
f(fk_l(t))
f (t)
t
for each t e G, where m is the smallest natural number for which the last expression
holds.
(3)
The function f has derivatives up to, and inclusive of, the order mn
for each t g G, f’(t) # 0 for each t
(4)
The function
(5)
G.
F(t, Ul, u2,
arguments for each t g G and
R (r
Ur
n
Un+ I)
I,
is mn
n
n times differentiable of its
+ I) and Fu
n+l
The unknown function x has derivatives up to, and inclusive of, the order
mn on G.
In this case there exists an ordinary differential equation of order mn such
that each solution of Eq. (2.13) is simultaneously a solution of this differential
equat ion.
Let us consider the functional differential equation [17]
(k
(k)
F(fl(t), x(fl(t))
x
l)
(fl(t))
X(fn(t))
x
n
(f (t)))=0
(2 14)
where x is an unknown function and where the following conditions are fulfilled:
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
(I)
The functions
fl’
superposition of functions,
f
n
fl(t)
form a finite group of order n with respect to
t, and map the open set G into G, G being the
largest open set wherein all expressions appearing in this paper are defined.
(2)
The functions x and f (r
max(k
where p
(3)
p"
kn)
I
n) have derivatives up to the order p,
i
r
so that f’(t)
r
# 0 for every
For the function F at least one relation
r
2,
n and every t
THEOREM 2.5
([]7]).
I
G and r
t
F
x (s) (f)
r
# 0
n.
O,
is valid for s
G.
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
x(-t) and obtains
y(t)
y’(t) =-x’(-t) =-f(x(-t), x(t)) =-f(y, x).
Hence, the solutions of the original equation correspond
to the solutions of the
system of ordinary differential equations
d__x
dt
y(O).
with the condition x(O)
dy
y)
f(x
dt
_f(y
x)
From the qualitative analysis of the solutions of
the associated system he derives qualitative information about the solutions of the
equation with transformed argument.
The linear case is discussed in some detail.
Several examples of more general equations are also considered.
Boundary-value problems for differential equations with reflection of the argument are studied in
3.
[I0].
LINEAR EQUATIONS
In this section we study equations of the form
n
Y.
Lx(t)
k=O
ak(t)x(k)(t)
x(f(t)) + (t)
(3.1)
with an involution f(t).
THEOREM 3.1 ([i]).
Suppose that the initial conditions
(k)
(t
x
O)
are posed for
x
k
Eq. (3.1) in which the coefficients
ak(t)
the function
strong (or hyperbolic (2.5)) involution f(t) with fixed point t
cn(_m,
oo) (or
cn(/y,
oo)).
If n > I, then f’(t)
(3.2)
n- 1
k= 0
# O.
o
(t), and the
belong to the class
We introduce the operator
S.M. SHAH AND J. WIENER
8
d
i
M
(3.3)
f’ (t) dt
Then the solution of the linear ordinary differential equation
n
n
ak(f(t))MLx(t)
k=O
Z
x(t)
k=O
ak(f(t))M@(t)
(3.4)
+ @(f(t))
with the initial conditions
x
Mx(t)
lt=t 0
x(f(t))
0
k
n
i
(3.1)-(3,2).
MOLx(t)
(t)
Lx(t)
d
f’(t) dt Lx(t)
x"(f(t))
d
1
MLx(t)
f,(t) dt
f’(t) dt (t)
n-ILx(t)
i
d
f’ (t) dt
M(t),
MLx(t)
d
I
M(t)
f’(t) d-
d
I
M
f’ (t) dt
These relations are multiplied by
times, we obtain
MO(t),
d
1
x’(f(t))
(f(t))
1,
n
0
By successively differentiating (3.1) n
PROOF.
(n)
k
x k,
O
Mk*(t)It=to,
+
Xk
is a solution of problem
x
(k) (t
M2Lx(t)
Mn-l(t)
M2(t),
Mn(t).
MnLx(t)
a0(f(t)) al(f(t)
a
n
(f(t))
respectively
and the results are added together:
n
Z
k=O
ak(f(t))x
By virtue of f(f(t))
(k)
n
(f(t))
n
ak(f(t))Mx(t )- k--OF. ak(f(t))Mk(t).
k=O
Z
t, it follows from (3.1) that
n
k=O
ak(f(t))x
Thus, we obtain Eq. (3,4).
(k)
(f(t))
x(t) + (f(t)).
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
equal
Xk,
k
O,
n
1 derivatives at the point t should
O
i, from (3.2), while the values x
(n)
(t
x
O
are determined from the relations
Mx(t)
x
(k)(f(t))
the
+
Mk@(t),
by substituting the values t and x for t and x
O
k
(k)
k
(t).
0
n
i
(2n-l) (t
O
9
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
THEOREM 3.2 ([i]),
The equation
(t)
x
is
x(
(3.6)
integrable in quadratures and has a fundamental system of solutions of the form
ta(In
t)
a and b are real and
PROOF.
sin(b In t), ta(In t) j cos(b In t),
is a nonnegative integer.
By an n-fold differentiation Eq. (3.6) is reduced to the Euler equation
2n
b(kn) tkx (k) (t)
I
k=n+l
For n
(3.7)
i this follows from (2.2).
x(t).
(3.8)
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 d
dt
On the basis of (3.4) and (3.8) we have
2n
l
k=n+l
Mnx (n) (t)
Mnx (n+l
2n
I
(t)
b
k=n+l
b(kn) tkx (k) (t),
(n) tkx(k+l)
k
(t).
Then
Mn+ix (n+l) (t)
-t
2 d
2n
k=n+lZ bk(n)tkx(k+l)__
2n
(n) k+l (k+l)
l KD
t
(t)
x
k
k=n+l
(t)
2n
(n) k+2 (k+2)
l
b
(t)
x
t
k
k=n+l
Consequently, the equation
x
(n+l)
x()
(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
(n)
b (n+l)= -(k-l)"bk_
k
1
b(n’n’=
0
b(n)
2n+l
(n)
bk_ 2
n
+ 2 <_ k < 2n + 2,
0
connecting the coefficients of the Euler equations
2n
b
k=n+l
k
n) t__ (k)
2n+2
(t)
x(t) and
F,
k=n+2
b
(n+l)tkx(k)(t):
k
x(t),
S.M. SHAH AND J. WIENER
i0
which correspond to the equations
(n)
x
(t)
x(
and
x
(n+l)
_i
x().
(t)
t
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.
EXAMPLE 3.1.
The theorem is proved.
The investigation of the nonhomogeneous equation
x’(t)
)
+ 9(t),
x(1)
x
x(
0 < t < o,
9(t)
g
[I]
1
C (0,
0
reduces to the problem
t2x"(t)
+ x(t)
x(1)
O,
t29’(t)
x’(1)
x
+ 9(1).
0
The solution is
x(t)
Xo/- cos(
it
2
f
In t) +
-3
u
/2
i
THEOREM 3.3 ([i]).
(-
/ In _t_)
u
sin(---
+ 9(1))
[u2
(u)
r sin(
()
In t) +
du.
The solution of the equation
tx(),
x’(t)
x(1)
x
0 < t <
(3.9)
0
is representable in closed form.
PROOF.
After differentiation Eq. (3.9) is reduced to the form
t2x"(t)
tx’(t) +
2x(t)
0
with the initial conditions
x(1)
Putting t
Xo,
x’(1)
ax
0.
exp u, we obtain
x"(u)
(l+8)x’(u) +
e2x(u)
The roots of the characteristic equation
2
are
(I + S) +
kl,2
Consider various cases:
(1)
(I+B) 2
4
-e
2
-
2
A 2 > O,
2
0
et2.
O,
II
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
kl
Xo
Xl_% 2 [(a-%2) t
x(t)
(1+6)
(2)
-a
2
O,
x(t)
(i+6)
(3)
Xot
2
_A
I,
$
2
(1+6)
/211
1+6
+ (a
t].
__)In
< 0,
1+6
c
Xot(l+6)/2[cos(Aln"
x(t)
For a
(%l-a)t%2].
+
t) +
2
A
sin(& In
t)]
0 the latter formula yields the solution of the Silberstein equation
(2.1).
EXAMPLE 3.2.
at
The equation
r
x(t) +
btr+l
(t)
x
ct
s+l
I
(t > o)
+ at s x ’(),
x(
(3.10)
where a, b, c, d, r, s are real constants and x is an unknown function, was explored
in
17].
Let b
2
d
2
at
# O.
r
x()
By putting
x(t) + bt
r+l
y(t), Eq. (3.10) becomes
x’(t)
ct
s+l
y(t)
dt
s+2
y’(t).
(3.11)
I is
If
substituted for t in (3.10), we get
at
-r
y(t)
bt
-r+l
y’(t)
-s-I
ct
x(t) + dt
-s
x’(t).
(3.12)
From (3.11) and (3.12) Euler’s equation is obtained:
where A
If b
t2x"(t) + (r-s)tx’(t) + (Bs-Br+A2-m2+B)x(t)
(bc-ad)/(b2-d2), B (cd-ab)/(b2-d2).
d and a
c, Eq. (3.10) is equivalent to the system of equations
ax(t) + btx’(t)
If b
d and a
0,
u()
u(t),
tr-s-lu(t).
c, (3.10) reduces to the functional equation
x(tl_)
In the case of b
t
-d and a
ax(t) + btx’(t)
In the case of b
r-s-I
-c
Eq. (3.10) reduces to the system
u()
u(t)
-d and a
#
x()
The equation x’(t)
x(t).
-c
-t
r-s-I
u(t).
(3.10) reduces to the functional equation
-t
r-s-I
x(t).
x(f(t)) with an involution f(t) has been studied in [19].
S. M. SHAH AND J. WIENER
12
[13]
Consider the equation
with respect to the unknown function x(t):
a(t)x(f(t)) + b(t),
x’(t)
(3.13)
(i)
The function f maps an open set G onto G.
(2)
The function f can be iterated in the following way:
fl(t)
fk(t)
f(t)
f(fk_l(t))
f (t)
t
m
(t E G)
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
x’(t)
(_o% 0)U(0, I)U(lo + oo).
and G
1161
(l-t) -I
x(f(t)), f(t)
For f we have
f3(t)
t on G.
(3 14)
In this case (3.14)
is reducible to the equation
t2(l-t)2x
THEOREM 3.5 ([i]).
2t2(l-t)x"(t)
(t)
x(t)
O.
In the system
Ax(t) + Bx(c-t), x(c/2)
x’(t)
let A and B be constant commutative r xr
x
(3.15)
0
matrices, x be an r-dimensional vector,
and B be nonsingular.
Then the solution of the system
(A2-B2)x(t)
x"(t)
x(cl2)
is the solution of
In
[7]
x
0,
x’(cl2) =(A+B)x0
problem (3.15).
it has been
proved that the equation
t2x"(t)
f
x()
O,
0 < t <
has the general solution
x(t)
c
l(r t
2
+
t-2)
c2[sin(r
+
In t)
cos( In
t)],
while the equation
t2x"(t)
+ I
x(--It
0
has the general solution
x(t)
c3(t
2
t-2)
+
c4[sin(’]
In t) +
3+I/ cos( In t)].
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
It follows from here that, by appropriate choice of c
I,
c2, c3,
13
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.
also proved in
[7]
that the system
x’(t)
x(tl-))
II x()ll q,
A(t)x(t) + f(t,
x())II <--
II f(t,
where 6 > 0 and q
It has been
_> 1 are constants,
<_
1
t <
is stable with respect to the first approxima-
t ion.
For the equation
n
Z
k=0
aktkx(k) (t) x(),
0 < t <
(3.16)
we prove the following result.
THEOREM 3.6. Eq.(3.16) is reducible by the substitution t
e
s
to a linear ordi-
nary differential equation with constant coefficients and has a fundamental system of
solutions of the form (3.7).
PROOF.
Put
e
s
and x(e s)
y(s), then tx’(t)
tkx(k)(t)
y’(s).
Assume that
Ly(s),
where L is a linear differential operator with constant coefficients.
From the
relation
tk+l
x
(k+l)
a-.- [tkxk (t)]
)t
(t)
t
(t)
L[y’(s)
kt
k
x
(k)
(t)
we obtain
t
k+l
x
(k+l)
ky(s)],
which proves the assertion.
The functional differential equation
Q’(t)
where A, B are n x n
and
QT(t)
AQ(t) + BQ T(T
constant matrices, T
_> 0, Q(t)
is its transpose, has been studied in
(3.17)
< t <
t),
is a differentiable n
[20].
algebraic representation of its solutions are given.
n matrix
Existence, uniqueness and an
This equation, of considerable
interest in its own right, arises naturally in the construction of Liapunov functio-
nals for retarded differential equations of the form x’(t)
C, D are constant n
n matrices.
Cx(t) + Dx(t-I), where
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
14
S.M. SHAH AND J. WIENER
It is shown that, unlike the infinite dimen-
for ordinary differential equations.
sionality of the vector space of solutions of functional differential equations, the
linear vector space of solutions to (3.17) is of dimension n
give a complete algebraic characterization of these n
2
2.
Moreover, the authors
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
Q(-)
where K is an arbitrary n
(3.18)
K,
Eq. (3.17)
n matrix.
Q’(t)
AQ(t) + BR(t),
R’(t)
-Q(t)B
is intimately related to the system
(3.19)
T,
T
R(t)A
with the initial conditions
Q(
K,
T
R(-)
K
T
(3.20)
2
For any two n n matrices P, S, let the n x n
(or direct) product
[21]
si,
and
s,j
matrix
PS
denote the Kronecker
and introduce the notation for the n x n matrix
S
where
2
(sij)
Sl*
(n,)
Sn*
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
Sn,)T.
(Sl,
s
With
this notation Eqs. (3.19) and (3.20) can be rewritten as
r(t
t)B
-I
IA
(t
and
q()
T
kn,]T, r(-)
[kl,,
[k,l,
T
k,n
T
which, with the obvious correspondence and for simplicity of notation, are denoted as
p’(t)
Here p(t) is an
2n2-vector
Cp(t),
and C is a
p(T/2)
2n2
2n
2
(3.21)
PT/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)
15
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
immediately yields the sol’ution of (3.17)-(3.18).
But (3.21) is a standard initial-
value problem in ordinary differential equations; the structure of the solutions of
2
Furthermore, since the 2n
such problems is well known.
2n
2
matrix C has a very
special structure, it is possible to recover the structure of the solutions of Eq.
(3.17).
I’
Let
%p’
2n2’
p
be the distinct eigenvalues of the matrix C, that
is, solutions of the determinantal equation
det[%I
.,
each
nj,r Zr=s I
I,
n.
C]
O,
(3.22)
p, with algebraic multiplicity m. and geometric multiplicities
mj, Zj
m.=3 2n2"
Then 2n
2
linearly independent solutions of (3.21) are
given by
j
q (t)
j
r
n.,
i,
where q
exp(% (t-
(t
q
T
))
Z
T q-i
-)
ej
(q-i)’
i=l
i
(3.23)
r
and the 2n 2 linearly independent eigenvectors and generalized
eigenvectors are given by
i
3,r
C]e.
[% I
j
i-I
-e.
e
3,r
0
j,s
O.
A change of notation, and a return from the vector to the matrix form, shows that 2n
linearly independent solutions of (3.19) are given by
r
Yj
q
T
q(t)J
exp(%j(t )
r
(t-
(q-i)
i=l
Mj
r
i
where the generalized eigenmatrix pair (L
j
,r’
i
r
Mj ,ri)associated
with the eigenvalue
satisfies the equations
A)L
(%. I
3
L
i.
3,
BM
r
i.
i-I
L.
3, r
3, r
(3.24)
i
B
j,r
T
i- 1
j,r
i
(% I + AT)
+ Mj,r
j
M
The structure of these equations is a most particular one; indeed, if they are multi-
plied by -I, transposed, and written in reverse order, they yield
(-
j
A)M. i
I
I
3,r
0
L.
T
2,r
M.
0
B
T
I
j,r
+L
i
j,r
(_%
i- 1
3,r
T
M.
I+
AT
L
i_I
T
j,r
T
0.
But this result demonstrates that if %. is a solution of (3.22),
3
3,r
-%. will also be
3
i
BL.3,r
.T
.T
.T
M.
T
3,r
a solution;
moreover, %. and -%. have the same geometric multiplici3
3
S.M. SHAH AND J. WIENER
16
Hence, the distinct eigenvalues always
ties and the same algebraic multiplicity.
3’
i
i
%. are (L., r,
((-i)
i+l
Mj,r),
.T
1
(-i)
Mj,r
responding to
%.
(-I) q+i
2
the n
and if the generalized eigenmatrix pairs corresponding to
%j),
appear in pairs (%.
i+l
the generalized eigenmatrix pairs corresponding to
.T
I
Lj ,r
).
-%j
will be
These remarks imply that if the solution (3.23) cor-
is added to the solution
(3.23) corresponding to -%. multiplied by
linearly independent solutions of (3 19) given by
Zj ,rq(t)
T
exp(j(t ))
W. q(t)
r
r
Tq-i
(t
q
1
L
(q- i):
i=l
exp(-j
T
) q-i
T
(t
q
Y
)) i=l
(q
+
i
Mj ,r
3
(t
i
j r
Mj
r
Lj
r
(-I) q+i
i):
satisfy the condition
T
Zj,
Wj,r
But this is precisely condition (3.20)"
Z
q(t)
j,r
q
l
i= I
(t
it therefore follows that the expressions
T q-i
)
[exp(%j(t ))Lj,r
(-1) q+i
are n
2
i
T
(q- i)!
+
exp(-Ej(t
T
-)
z
i
T
)Mj,r
(3"25)
linearly independent solutions of (3.17).
THEOREM 3.8
([20]).
2
Eq. (3.17) has n linearly independent solutions given by
Eq. (3.25), where the generalized eigenmatrix pairs
for one of the elements of the pair
(j, -j),
i
(Lj, r
Mj,ri
satisfy Eq. (3.24)
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 argu-
ment deviations and, on the other hand, their study discovers essential differences
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
17
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
u
t(t,
x)
of the involution f(t) for the equations
o
au
XX
x) + bu
(t
XX
(f(t)
x)
(3.26)
and
utt(t
can be found
a2u
x)
XX
b2u
x) +
(t
(f(t)
XX
(3.27)
x)
by the method of separation of the variables.
Thus, for (3.26) the
functions T (t) in the expansion
n
u(t, x)
T (t)X (x)
l
n
n=l
(3.28)
n
are determined from the relation
T’(t)
-% aT (t)
n
n
% bT (f(t))
n
n
T (t
n
n
O)
C
n
(3.29)
Its investigation is carried out by means of Theorem 3.1, according to which the
solution of the equation
%2(a2-b2)f’(t)T
(t)
n
n
T"(t) =-% a(l+f’(t))T’(t)
n
n
n
(3.30)
with the initial conditions
T (t
n
satisfies Eq. (3.29).
C
O
T’(t
n
n
-% (a+b)C
O
n
n
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)
for all t
xx
(t, x) + hu xx (c-t, x),
u(t, )
u(t, 0)
is unbounded as t +,
au
b
if a
# 0.
0, u(cl2, x)
If
Ibl
<
lal,
(3.31)
(x)
b
# 0,
expansion (3,28) diverges
# c/2.
PROOF.
By separating the variables,
we obtain
2 2
T,(t)=__n
n
oz
(aT (t) + bT (c-t)), T (c/2)
n
n
n
C
(3.32)
n
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
n4
T"(t)n
T (c/2)
n
C
,b
2
n’ T’(c/2)
n
a
2) Tn (t)
2 2
n
-2
(a + b)C
n
18
S. M. SHAH AND J. WIENER
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:
2 a2 (t-)c
2 2
n
C (cos
T (t)
(i)
2
n
n
2 2
n
a+b
s in
b2_a2
2 2
n
(2)
T (t)
C (i
(3)
T (t)
=Cn [(I
n
/a 2
I
n
u
b
2 2
2
n
exp(----
a_-----)
(i +
IEOREM 3.10.
c
--)), (lal
(a+b)(t-
n
/aa
2_
)exp(-
The solution of the equation
tt
a2u xx (t,
(t, x)
x) +
u(t, )
u(t, O)
is unbounded as t /,
if a
2
O, u(O, x)
2.
b
22
n
In the case a
(lal
<
Ibl);
<
lt).
+
c
/a2_b 2 (t-))],
(Ib
x)
(x),
),
Ibl);
c
/a2_b 2 (t -))
b2u xx (-t,
satisfying the boundary and initial conditions
2_a2 (t- c_),
2
(3.33)
ut(O
x)
(x),
2 < 2
b expansion (3.28) diverges
for all t # 0.
PROOF.
Separation of the variables gives for the functions T (t) the relation
n
222
a n
T(t)
2
T (0)
n
T (t)
2b2n 2
2
n
A
n
T’(0)
n
B
Tn(-t)
n
by successive differentiation of which we obtain
2 2 2
a n
T
+
T’(-t)
n
(3)(t)
T
(4)
n
(t)
2 T(t)
2 2 2
r a n
2
T"(t)
n
262n2
T7r
2b2n 2
2
n
T" (-t)
n
From Eq. (3.33) we find
222
T"(-t)
n
a n
2
Tn(_t)
2b22
n
(t)
2 Tn
and also
222
a n
2
Tn(-t)
a
2
7
T"(t) +
n
a ,n,
z242
b22 Tn(t)"
(3.34)
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
19
Thus, Eq. (3.34) is reduced to the fourth-order ordinary differential equation
T(4)(t)
n
+
22a2n2
2
4
T"(t) +
n
(a4-b4)n 4
4
T (t)
0
n
with the initial conditions
An, T’(O)n
rn(O)
T
(3)
n
2(a2+b 2) n 2
An,
2
Bn, T"(O)n
2(a2-b2)n2
(0)
_2
n
It remains to consider various cases that may arise depending on the characteristic
roots.
(I)
2 < 2
a
For b
T (t)
A cos
n
n
(2)
If a
2
b
2
then
T (t)
A
n
n
(3)
+
t
na t
cos
+ Bn t.
2
2
Finally, the inequality a < b leads to the result
T (t)
n
n/a2+b 2
A.n cos
t
+
B
n
2
2_a
b
n
sinh
n
b 2-a 2
t.
Of some interest is the equation
u
t(t,
x)
Au
(t +_____B
x)
xxyt
(3.35)
with the hyperbolic involution
having two fixed points
t
_+A
0
y
1
_e-A
y
The search of a solution in the region
(e/y, o)[0, 1]
(or(-oo, a/y)x[O, 1])
satis-
fying the conditions
u(t, 0)
u(t, g)
0,
u(t0,
x)
(x) (or
u(t!,
x)
O(x))
(3.36)
leads to the relation
T’(t)
n
Az2n2
2
+ 8
)
Tn (y-_
at
(3.37)
S.M. SHAH AND J. WIENER
20
Eq. (2.1).
which is a generalization of
)2T"(t)
n
(yt
Iyt- al
The substitution
Differentiation changes (3.37) to the form
A2A2 4n4 Tn(t)
4
+
(3.38)
0.
permits integration of (3.38) in closed form.
exps
Omitting the calculations, we formulate a qualitative result.
The solution of problem (3.35)-(3.36) is unbounded as t
THEOREM 3.11.
o.
For
the functions T (t) are oscillatory.
n
4.
EQUATIONS WITH ROTATION OF THE ARGUMENT
An equation that contains, along with the unknown function x(t) and its deriva-
tives, the value x(-t) and, possibly, the derivatives of x at the point -t, is called
An equation in which as well as the unknown
a differential equation with reflection.
function x(t) and its derivatives, the values
x(1t-aI)
X(mt-am
m are mth
responding values of the derivatives appear, where gl’
m are complex numbers,
ty and
al’
tion.
For m
and the corroots of uni-
is called a differential equation with rota-
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 diffi-
cult 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
n
akx(k) (t)
E
k=0
x
(k)
with complex constants
(0)
bk,
ak,
variable coefficients.
n
E
k=0
bkX
Xk,
(k)
k
(ct) + lp(t),
0
n
j
and apply A to
I
E
e, then the method
k=0
k
ake-Jk
dt
(j
0
te
k
n
Bj
m-
given equation
Z
k=0
i),
(4.1)
is extended to some systems with
Turning to (4.1) and assuming that
d
1
n- 1
introduce the operators
A
m__
k
bkJk dtk
d
is smooth enough, we
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
AoX (B0x)(Et)
21
.
+
S inc e
(BIBoX)(E2t) + (B0@)(gt)
AI[(BoX)(Et)]
we obtain
AIAoX (BiBoX)(e2t) + AI@ + (Bo)(et),
and act on this relation by A2.
From
A2[(BIBoX)(2t)] (B2BIBoX)(E3t) + (BIBo@)(F2t),
A2[(Bo)(Et)]
(AiB0)(et)
it follows that
A2AIAo x (B2BIBoX)(e
3
+
t
A2AI + (AIBo)(Et)
+
(BIBo@)(E2t).
Finally, this process leads to the ordinary differential equation
(AO(m-l)
B
m-I
(m-l))x
0
Z
(A
j=O
(m_l_J)B(J_l)
) (e j t)
0
(4.2)
I
where
A (j)= A A
AO)=
B
-1)
and I is the identity operator.
mn.
A
j j-1
i
B (j)
i’
i
B B
j
j-l’’"
B
i’
i <
I,
Thus, (4.1) is reduced to the ODE (4.2) of order
’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) B (j))x(k)(t)]
0
(j
0
Y.
t=0
m-o 2; k
System (4.3) has a unique solution for x
a
j
n
#
g
i=O
(eib)J
n
ikAJ-i) B i-l)(k)
* (t) It__ 0
0
(4 3)
n- i).
(k)(O)(n < k _< mn-
(0 < i < m- i, i < j < m- i)
I), iff
(4 4)
These considerations enable us to formulate
THEOREM 4 1
tion of
([9])
If
@EC
(m-l)n
and inequalities (4.4) are fulfilled, the solu-
ordinary differential equation (4.2) with initial conditions (4.1)-(4.3)
satisfies problem (4.1).
THEOREM 4.2
([9]).
If
y
transfor.s
g
# i, the substitution
x exp(at/l
e)
the equation
Ay
exp(ct)(By)(et) +
(4.5)
S. M. SHAH AND J. WIENER
22
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
Pk’ qk
and
an’ qn
Pn
hn.
m
Under assumptions (4.4) and
COROLLARY.
I, (4.5)
is reducible to a linear
ordinary differential equation with constant coefficients.
REMARK.
lanl # Ibnl.
Conditions (4.4) hold if, in particular,
Theorems 4.1 and
4.2 sharpen the corresponding results of [25] and [26] established for homogeneous
equations (4.1) and (4.5) by operational methods under the restriction
EXAMPLE 4.1.
x expt reduces the equation
The substitution y
y’(t)
(5y(-t) + 2y’(-t)) exp2t, y(O)
lanl
>
[9]
YO
to the form
x’(t) + x(t)
7x(-t) + 2x’(-t), x(O)
Therefore (4.2) gives for x(t) the ODE
x(O)
x’(O)
Y0’
-6y 0.
x"
16x
The unknown solution
y(t)
Yo(5
YO"
0 with the initial conditions
s
exp(-3t)
exp5t)/4.
The analysis of the matrix equation
AX(t) +
X’(t)
exp(at)[BX(et) + CX’(et)],
x(0)
with constant
(4.6)
E
(complex) coefficients was carried
out in
[3].
The norm of a matrix is
defined to be
lcll
max
.leij !,
(4.7)
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
THEOREM 4.4.
([27]).
[27].
Suppose we are given a differential equation with reflec-
tion of order n with constant coefficients
n
7.
k=O
We suppose that
(a)
a
2
n
b
2
n
(k)
[a-x(k)
(t) + bkX
(-t)]
k
y(t).
(4.8)
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
(b)
aj_ka k
(c)
the polynomial
J
O, I,
0 for k
bj_kb k
n
7.
%2jt
j--O
j
n and j
has simple roots u
q
k
23
+ I
k
+ n,
only, where
for 0 < j < n,
k=ZO Cjk
(-l)n+j-k(an2-bn2)(aj_ ka k-bj_kb k).
Cjk
n
jk
k=j-n
for n < j 5_ 2n,
Then every solution of Eq. (4.8) is of the form
(-l)n(a n2-
x(t)
b
n
2
m=O
n
Z Ck
k=O
q=l
k
d
(a
Uq
(t)
are arbitrary constants and
n
(t)
b
k/^
n
Z
where the C
[(-l)mam
Z
n
e
k
m
t
q
-
(-t)] +
b
k
e
q
t
is a solution of the equation
2
(d--- Uq)(t)
y(t).
q=l
([9]).
THEOREM 4.5
Suppose that the coefficients of the equation
n
Y.
k=O
belong to C
(m-l)n
em
(t)
x(et) + (t), x
(k)
(0)
x
k
k,
O,
n
1 (4.9)
i, a (0) # 0 and
n
ej
(k)
ak(t)x
l
k=O
E-Jka,_(eJt)dk/dt k,
0,
(4.10)
m- 1.
Then the solution of the linear ordinary differential equation
L
(m-l)
0
m-i
Z (L (m-l)
x(t) +
x(t)
k=l
(L
(m-l)
k
L
Lm_ILm_2
k
)(ek-lt)
k
+
k < m
0
with the initial conditions
x
(k)
(0)
Xk(k
0
n
1),
Lox(k) (t) It=o
n(m-1)
(em-lt)
_
(4.11)
I)
k (k)
x
(0)
+ (k) (0) ,k=O
1
satisfies problem (4,9),
PROOF.
Applying the operator L I to (4.9) and taking into account that
S. M. SHAH AND J. WIENER
24
(LoX)(et) x(e2t)
+ (et)
we get
x(2t)
LIeOX(t)
+
+ (t)
Ll(t)
and act on this equation by L to obtain
2
x(e3t)
L2LILoX(t)
+
L2LI(t)
+
(L2)(et)
+
(2t).
It is easy to verify the relations
(L.x)(Jt)
3
x(e j+l t) +
(eJt),
0
m-
I.
In particular,
(L
m_l
Thus, the use of the operator
THEOREM 4.6
([9]).
x)
(em-lt
Lm_ I
x(t) +
(em-lt).
at the conclusive stage yields
(4.11).
The system
tAX’(t) + BX(t)
(4.12)
X(Et)
with constant matrices A and B is integrable in the closed form if e
PROOF.
m
I, det A
0.
For (4.12) the operators L. defined by formula (4.10) are
3
tAd/dt + B.
L.
3
Hence, on the basis of the previous theorem, (4.12) is reducible to the ordinary
system
(tAdldt + B) m X(t)
X(t).
This is Euler’s equation with matrix coefficients.
(4.13)
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.
We connect with the equation [9]
EXAMPLE 4.2.
tx’(t)
2x(t)
(td/dt
2)
x(et),
e3
(4.14)
1
the relation
3
x(t)
x(t).
The substitution of its general solution
x(t)
into
C1 t3
(4.14) gives C
+
2
C
t
3
3/2
(C2sin(23- lnt) + C3cos(
A solution of (4.14) is
0.
x= Ct
THEOREM 4.7 ([9]).
lnt))
3
The system
tAX’(t)
BX(t) + tX(et)
(4.15)
25
REDUCIBLE FUNCTIONAL DIFFERENTIAL EQUATIONS
with constant coefficients A and
B, det A # 0 and e
m
1 is Integrable in closed form
and has a solution
e(t)t
X(t)
A-IB
(4.16)
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
t-IB),
e-J(Ad/dt
L.
0
j
m-
in consequence of which we obtain the relation
t-IB) m
(Ad/dt
Since e re(m-l)/2=
+_I,
X(t)
e
m
+
(ekE
k=l
gk
(4.17)
it takes the form
[Ad/dt
where
m(m-l)/2X(t).
are the m-order roots of i or -I.
AX’(t)
(ekE
+
t-IB)]
X(t)
0
The solutions of the equations
t-IB)x(t)
are matrices
exp(ktA-l)t A-IB,
Xk(t)
I,
k
mo
Their linear combination represents the general solution of (4.15).
In accordance with (4.17)
EXAMPLE 4.3.
to the equation
[9]
(4.18)
3x(t) + tx(-t)
tx’(t)
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)
and find C
2
IC I.
x(t)
t3(Clexp(it)
+
C2exp(-it))
A solution of (4.18) is
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 fre-
S. M. SHAH AND J. WIENER
26
quently used
to
study FDE arising in biological models.
refer to a recent paper
[31].
We omit these topics and
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].
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