<s>
In	O
mathematics	O
,	O
an	O
abstract	B-Algorithm
differential	I-Algorithm
equation	I-Algorithm
is	O
a	O
differential	O
equation	O
in	O
which	O
the	O
unknown	O
function	O
and	O
its	O
derivatives	O
take	O
values	O
in	O
some	O
generic	O
abstract	O
space	O
(	O
a	O
Hilbert	O
space	O
,	O
a	O
Banach	O
space	O
,	O
etc	O
.	O
)	O
.	O
</s>
<s>
Adding	O
boundary	O
conditions	O
can	O
often	O
be	O
translated	O
in	O
terms	O
of	O
considering	O
solutions	O
in	O
some	O
convenient	O
function	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
where	O
the	O
unknown	O
function	O
belongs	O
to	O
some	O
function	B-Algorithm
space	I-Algorithm
,	O
and	O
is	O
an	O
operator	O
(	O
usually	O
a	O
linear	B-Architecture
operator	I-Architecture
)	O
acting	O
on	O
this	O
space	O
.	O
</s>
<s>
Very	O
often	O
,	O
the	O
study	O
of	O
other	O
abstract	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
amounts	O
(	O
by	O
e.g.	O
</s>
<s>
The	O
theory	O
of	O
abstract	B-Algorithm
differential	I-Algorithm
equations	I-Algorithm
has	O
been	O
founded	O
by	O
Einar	O
Hille	O
in	O
several	O
papers	O
and	O
in	O
his	O
book	O
Functional	O
Analysis	O
and	O
Semi-Groups	O
.	O
</s>
<s>
Let	O
and	O
be	O
two	O
linear	B-Architecture
operators	I-Architecture
,	O
with	O
domains	O
and	O
,	O
acting	O
in	O
a	O
Banach	O
space	O
.	O
</s>
<s>
According	O
to	O
the	O
definition	O
of	O
well-posed	B-Algorithm
problem	I-Algorithm
by	O
Hadamard	O
,	O
the	O
Cauchy	O
problem	O
is	O
said	O
to	O
be	O
well	O
posed	O
(	O
or	O
correct	O
)	O
on	O
if	O
:	O
</s>
<s>
Additionally	O
,	O
if	O
is	O
dense	O
in	O
,	O
the	O
operator	O
can	O
be	O
extended	O
to	O
a	O
bounded	O
linear	B-Architecture
operator	I-Architecture
defined	O
on	O
the	O
entire	O
space	O
.	O
</s>
<s>
where	O
the	O
unknown	O
is	O
a	O
function	O
,	O
is	O
given	O
and	O
,	O
for	O
each	O
,	O
is	O
a	O
given	O
,	O
closed	O
,	O
linear	B-Architecture
operator	I-Architecture
in	O
with	O
domain	O
,	O
independent	O
of	O
and	O
dense	O
in	O
,	O
is	O
called	O
time-dependent	O
Cauchy	O
problem	O
.	O
</s>
<s>
An	O
operator	O
valued	O
function	O
with	O
values	O
in	O
(	O
the	O
space	O
of	O
all	O
bounded	O
linear	B-Architecture
operators	I-Architecture
from	O
to	O
)	O
,	O
defined	O
and	O
strongly	O
continuous	O
jointly	O
in	O
for	O
,	O
is	O
called	O
a	O
fundamental	O
solution	O
of	O
the	O
time-dependent	O
problem	O
if	O
:	O
</s>
<s>
the	O
partial	O
derivative	O
exists	O
in	O
the	O
strong	B-Algorithm
topology	I-Algorithm
of	O
,	O
belongs	O
to	O
for	O
,	O
and	O
is	O
strongly	O
continuous	O
in	O
for	O
;	O
</s>
<s>
where	O
is	O
a	O
nonlinear	B-Architecture
operator	I-Architecture
with	O
domain	O
,	O
is	O
called	O
nonlinear	O
Cauchy	O
problem	O
.	O
</s>
