The null space and the range of a convolution operator in a fading memory space
Authors: Staffans Olof J. 
Published: June 2010

We study the convolution equation

(
∗
)

(\ast )

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu asterisk x prime left-parenthesis t right-parenthesis plus v asterisk x left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis left-parenthesis negative normal infinity greater-than t greater-than normal infinity right-parenthesis">

μ

∗

x
′

(
t
)
+
v

∗

x
(
t
)
=
f
(
t
)

(
−
∞
>
t

>
∞
)

\mu \; \ast \;x’(t) + v\; \ast \;x(t) = f(t) \quad ( - \infty > t\, > \infty )

\]
, as well as a perturbed version of

(
∗
)

(\ast )

, namely

(
∗
∗
)

(\ast \ast )

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu asterisk x prime left-parenthesis t right-parenthesis plus v asterisk x left-parenthesis t right-parenthesis equals upper F left-parenthesis x right-parenthesis left-parenthesis t right-parenthesis left-parenthesis negative normal infinity greater-than t greater-than normal infinity right-parenthesis period">

μ

∗

x
′

(
t
)
+
v

∗

x
(
t
)
=

F
(
x
)

(
t
)

(
−
∞
>
t
>
∞
)
.

\mu \;\ast \;x’(t) + v\,\ast \;x(t) = \,F(x)\,(t)\quad ( - \infty > t > \infty ).

\]
Here

x
x

is a

R

n

{{\mathbf {R}}^n}

-valued function on

(
−
∞
,
∞
)
,

x
′

(
t
)
=
d
x
(
t
)

/

d
t

( - \infty ,\infty ),x’(t) = dx(t)/dt

, and

μ
\mu

and

ν
\nu

are matrix-valued measures. If

μ
\mu

and

ν
\nu

are supported on

[
0
,
∞
)

[0,\infty )

, with

μ
\mu

atomic at zero, then

(
∗
)

(\ast )

can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that

μ
\mu

and

ν
\nu

are supported on

[
0
,
∞
)

[0,\infty )

, prescribe an initial condition of the type

x
(
t
)
=
ξ
(
t
)

(
t
⩽
0
)

x(t) = \xi (t)\,(t \leqslant 0)

, and require

(
∗
)

(\ast )

and

(
∗
∗
)

(\ast \ast )

to hold only for

t
⩾
0

t \geqslant 0

. Instead we permit

(
∗
)

(\ast )

and

(
∗
∗
)

(\ast \ast )

to be of "Fredholm" type, i.e.

μ
\mu

and

ν
\nu

need not vanish on

(
−
∞
,
0
)

( - \infty ,0)

, we restrict the growth rate of

x
x

and

f
f

at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of

(
∗
)

(\ast )

and

(
∗
∗
)

(\ast \ast )

on the whole real line, satisfying conditions like
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue x left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-slanted-equals upper C eta left-parenthesis t right-parenthesis left-parenthesis negative normal infinity greater-than t greater-than normal infinity right-parenthesis">

|

x
(
t
)

|

⩽
C
η
(
t
)

(
−
∞
>
t
>
∞
)

|x(t)| \leqslant C\eta (t)\;( - \infty > t > \infty )

, where

C
C

is a constant, depending on

x
x

, and

η
\eta

is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of

(
∗
)

(\ast )

has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.