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An equation of the form
 
An equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
A _ {0} ( t) \dot{u}  = A _ {1} ( t) u + g ( t) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591603.png" />, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591604.png" />, are linear operators in a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591606.png" /> is a given function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591607.png" /> an unknown function, both with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591608.png" />; the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l0591609.png" /> is understood to be the limit of the difference quotient with respect to the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916010.png" />.
+
where $  A _ {0} ( t) $
 +
and $  A _ {1} ( t) $,  
 +
for every $  t $,  
 +
are linear operators in a [[Banach space|Banach space]] $  E $,  
 +
$  g ( t) $
 +
is a given function and $  u ( t) $
 +
an unknown function, both with values in $  E $;  
 +
the derivative $  \dot{u} $
 +
is understood to be the limit of the difference quotient with respect to the norm of $  E $.
  
 
==1. Linear differential equations with a bounded operator.==
 
==1. Linear differential equations with a bounded operator.==
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916012.png" />, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916013.png" />, are bounded operators acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916015.png" /> has a bounded inverse for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916016.png" />, then (1) can be solved for the derivative and takes the form
+
Suppose that $  A _ {0} ( t) $
 +
and $  A _ {1} ( t) $,  
 +
for every $  t $,  
 +
are bounded operators acting in $  E $.  
 +
If $  A _ {0} ( t) $
 +
has a bounded inverse for every $  t $,  
 +
then (1) can be solved for the derivative and takes the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{u}  = A ( t) u + f ( t) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916018.png" /> is a bounded operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916021.png" /> are functions with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916022.png" />. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916024.png" /> are continuous (or, more generally, are measurable and integrable on every finite interval), then the solution of the [[Cauchy problem|Cauchy problem]]
+
where $  A ( t) $
 +
is a bounded operator in $  E $,  
 +
and $  f ( t) $
 +
and $  u ( t) $
 +
are functions with values in $  E $.  
 +
If the functions $  A ( t) $
 +
and $  f ( t) $
 +
are continuous (or, more generally, are measurable and integrable on every finite interval), then the solution of the [[Cauchy problem|Cauchy problem]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\dot{u}  = A ( t) u ,\  u ( s)  = u _ {0} ,
 +
$$
  
exists for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916026.png" /> and is given by the formula
+
exists for any $  u _ {0} \in E $
 +
and is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916027.png" /></td> </tr></table>
+
$$
 +
u ( t)  = U ( t , s ) u _ {0} ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
U ( t , s ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916029.png" /></td> </tr></table>
+
$$
 +
= \
 +
I + \int\limits _ { s } ^ { t }  A ( t _ {1} )  dt _ {1} + \sum
 +
_ { n= } 2 ^  \infty  \int\limits _ { s } ^ { t }  \int\limits _ { s } ^ { {t _ n} } \dots \int\limits _ { s } ^ { {t } _ {2} } A ( t _ {n} ) \dots A ( t _ {1} )  d t _ {1} \dots d t _ {n}  $$
  
is the [[Evolution operator|evolution operator]] of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916030.png" />. The solution of the Cauchy problem for equation (2) is determined by the formula
+
is the [[Evolution operator|evolution operator]] of the equation $  \dot{u} = A ( t) u $.  
 +
The solution of the Cauchy problem for equation (2) is determined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916031.png" /></td> </tr></table>
+
$$
 +
u ( t)  = U ( t , s ) u _ {0} + \int\limits _ { s } ^ { t }  U ( t , \tau ) f (
 +
\tau )  d \tau .
 +
$$
  
 
From (4) one obtains the estimate
 
From (4) one obtains the estimate
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\| U ( t , s ) \|  \leq    \mathop{\rm exp}
 +
\left \{ \int\limits _ { s } ^ { t }  \| A ( \tau ) \|  d \tau \right \} ;
 +
$$
  
 
a refinement of it is:
 
a refinement of it is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5prm)</td></tr></table>
+
$$ \tag{5'}
 +
\| U ( t , s ) \|  \leq    \mathop{\rm exp} \left \{ \int\limits _ { s } ^ { t }  r _ {A} ( \tau ) d \tau \right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916034.png" /> is the [[Spectral radius|spectral radius]] of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916035.png" />. The evolution operator has the properties
+
where $  r _ {A} ( \tau ) $
 +
is the [[Spectral radius|spectral radius]] of the operator $  A ( \tau ) $.  
 +
The evolution operator has the properties
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916036.png" /></td> </tr></table>
+
$$
 +
U ( s , s )  = I ,\  U ( t , \tau ) U ( \tau , s )  = U ( t , s),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916037.png" /></td> </tr></table>
+
$$
 +
U ( t , \tau )  = [ U ( \tau , t ) ]  ^ {-} 1 .
 +
$$
  
In the study of (2) the main attention has been focused on the behaviour of its solutions at infinity, in dependence on the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916039.png" />. An important characteristic of the equation is the general (or singular) exponent
+
In the study of (2) the main attention has been focused on the behaviour of its solutions at infinity, in dependence on the behaviour of $  A ( t) $
 +
and $  f ( t) $.  
 +
An important characteristic of the equation is the general (or singular) exponent
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916040.png" /></td> </tr></table>
+
$$
 +
\kappa  = \overline{\lim\limits}\; _ {\tau , s \rightarrow \infty } 
 +
\frac{1} \tau
 +
  \mathop{\rm ln}
 +
\| U ( \tau + s , s ) \| .
 +
$$
  
 
Equations with periodic and almost-periodic coefficients have been studied in detail (see [[Qualitative theory of differential equations in Banach spaces|Qualitative theory of differential equations in Banach spaces]]).
 
Equations with periodic and almost-periodic coefficients have been studied in detail (see [[Qualitative theory of differential equations in Banach spaces|Qualitative theory of differential equations in Banach spaces]]).
  
Equation (2) can also be considered in the complex plane. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916042.png" /> are holomorphic in a simply-connected domain containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916043.png" />, then the formulas (3), (4), (5), (5prm) remain valid if the integrals are understood to be integrals over a rectifiable arc joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916045.png" />.
+
Equation (2) can also be considered in the complex plane. If the functions $  A ( t) $
 +
and $  f ( t) $
 +
are holomorphic in a simply-connected domain containing the point $  s $,  
 +
then the formulas (3), (4), (5), (5'}) remain valid if the integrals are understood to be integrals over a rectifiable arc joining $  s $
 +
and $  t $.
  
A number of other questions arises in the case when the original linear equation is not solvable for the derivative. If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916046.png" /> is boundedly invertible everywhere except at one point, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916047.png" />, then in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916048.png" /> the equation reduces to the form
+
A number of other questions arises in the case when the original linear equation is not solvable for the derivative. If the operator $  A _ {0} ( t) $
 +
is boundedly invertible everywhere except at one point, say $  t = 0 $,  
 +
then in the space $  E $
 +
the equation reduces to the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
a ( t) \dot{u}  = A ( t) u + f ( t) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916050.png" /> is a scalar function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916051.png" />. Here the main attention is focused on the study of the behaviour of the solutions in a neighbourhood of the origin, and the analytic and non-analytic cases are distinguished.
+
where $  a ( t) $
 +
is a scalar function and $  a ( 0) = 0 $.  
 +
Here the main attention is focused on the study of the behaviour of the solutions in a neighbourhood of the origin, and the analytic and non-analytic cases are distinguished.
  
 
===The analytic case.===
 
===The analytic case.===
 
For the simplest equation
 
For the simplest equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916052.png" /></td> </tr></table>
+
$$
 +
t \dot{u}  = A u
 +
$$
  
with a constant operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916053.png" />, the evolution operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916054.png" /> has the form
+
with a constant operator $  A $,  
 +
the evolution operator $  U ( t) = U ( t , 0 ) $
 +
has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916055.png" /></td> </tr></table>
+
$$
 +
U ( t)  = e ^ {A  \mathop{\rm ln}  t } ,
 +
$$
  
and the solutions are not single-valued: as one goes round the origin in the positive direction they are multiplied by the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916056.png" />.
+
and the solutions are not single-valued: as one goes round the origin in the positive direction they are multiplied by the operator $  e ^ {2 \pi i A } $.
  
 
Consider an equation with a regular singularity
 
Consider an equation with a regular singularity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
t \dot{u}  = \left ( \sum _ { k= } 0 ^  \infty  A  ^ {(} k) t  ^ {k} \right ) u ,
 +
$$
  
where the series on the right-hand side converges in a neighbourhood of the origin. If one looks for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916058.png" /> in the form of a series
+
where the series on the right-hand side converges in a neighbourhood of the origin. If one looks for the operator $  U ( t) $
 +
in the form of a series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916059.png" /></td> </tr></table>
+
$$
 +
U ( t)  = \left ( \sum _ { k= } 0 ^  \infty  U  ^ {(} k) t  ^ {k} \right ) e ^ {A  ^ {(} 0)  \mathop{\rm ln}  t } ,
 +
$$
  
then for the determination of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916060.png" /> one obtains the system of equations
+
then for the determination of the coefficients $  U  ^ {(} k) $
 +
one obtains the system of equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916061.png" /></td> </tr></table>
+
$$
 +
A  ^ {(} 0) U  ^ {(} 0) - U  ^ {(} 0) A  ^ {(} 0)  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916062.png" /></td> </tr></table>
+
$$
 +
( A  ^ {(} 0) - kI ) U  ^ {(} k) - U  ^ {(} k) A  ^ {(} 0)  = - \sum _ { j= } 1 ^ { k }  A  ^ {(} j) u ^ {( k- j ) } ,\  k = 1 , 2 , .  . . .
 +
$$
  
For this system to be solvable, that is, for (7) to be formally solvable, it is sufficient that the spectra of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916064.png" /> do not intersect (cf. [[Spectrum of an operator|Spectrum of an operator]]), or, equivalently, that there are no points differing by an integer in the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916065.png" />. Under this condition the series
+
For this system to be solvable, that is, for (7) to be formally solvable, it is sufficient that the spectra of the operators $  A  ^ {(} 0) $
 +
and $  A  ^ {(} 0) - kI $
 +
do not intersect (cf. [[Spectrum of an operator|Spectrum of an operator]]), or, equivalently, that there are no points differing by an integer in the spectrum of $  A  ^ {(} 0) $.  
 +
Under this condition the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916066.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 0 ^  \infty  U ^ {( k) } {t  ^ {k} }
 +
$$
  
converges in the same neighbourhood of zero as the series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916067.png" />. Now, if there are finitely many integers representable as differences of points of the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916068.png" />, and each of them is an isolated point of the spectrum of the transformer
+
converges in the same neighbourhood of zero as the series for $  A ( t) $.  
 +
Now, if there are finitely many integers representable as differences of points of the spectrum of $  A  ^ {(} 0) $,  
 +
and each of them is an isolated point of the spectrum of the transformer
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916069.png" /></td> </tr></table>
+
$$
 +
\mathfrak A X  = A  ^ {(} 0) X - X A  ^ {(} 0) ,
 +
$$
  
 
then there is a solution of the form
 
then there is a solution of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916070.png" /></td> </tr></table>
+
$$
 +
U ( t)  = \left ( I + \sum _ { k= } 1 ^  \infty  U _ {k} (  \mathop{\rm ln}  t ) t
 +
^ {k} \right ) e ^ {A  ^ {(} 0)  \mathop{\rm ln}  t } ,\  0 < | t | < \rho ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916071.png" /> are entire functions of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916072.png" />, satisfying for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916073.png" /> the condition
+
where the $  U _ {k} $
 +
are entire functions of the argument $  \mathop{\rm ln}  t $,  
 +
satisfying for every $  \epsilon > 0 $
 +
the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916074.png" /></td> </tr></table>
+
$$
 +
\| U _ {k} (  \mathop{\rm ln}  t ) \|  \leq  C _  \epsilon  e ^ {\epsilon
 +
|  \mathop{\rm ln}  t | } .
 +
$$
  
If the integer points of the spectrum of the transformer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916075.png" /> are poles of its resolvent, then the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916076.png" /> are polynomials.
+
If the integer points of the spectrum of the transformer $  \mathfrak A $
 +
are poles of its resolvent, then the functions $  U _ {k} $
 +
are polynomials.
  
 
In the case of an irregular singularity, the differential equation
 
In the case of an irregular singularity, the differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916077.png" /></td> </tr></table>
+
$$
 +
t  ^ {m} \dot{u}  = \left ( \sum _ { k= } 0 ^ { m- }  1 A  ^ {(} k) t  ^ {k} \right ) u
 +
$$
  
has been considered in a Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916078.png" /> (for example, in the algebra of bounded operators on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916079.png" />). Under certain restrictions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916080.png" /> it reduces by means of Laplace integrals to an equation with a regular singularity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916081.png" /> in the algebra of matrices with entries from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916082.png" />.
+
has been considered in a Banach algebra $  \mathfrak B $(
 +
for example, in the algebra of bounded operators on a Banach space $  E $).  
 +
Under certain restrictions on $  A  ^ {(} 0) $
 +
it reduces by means of Laplace integrals to an equation with a regular singularity $  ( m = 1 ) $
 +
in the algebra of matrices with entries from $  \mathfrak B $.
  
 
===The non-analytic case.===
 
===The non-analytic case.===
 
Suppose that in the equation
 
Suppose that in the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916083.png" /></td> </tr></table>
+
$$
 +
t  ^ {n} \dot{u}  = A ( t) u + f ( t) ,\  0 \leq  t \leq  T ,
 +
$$
  
the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916085.png" /> are infinitely differentiable. In the finite-dimensional case a complete result has been obtained: If the equation has a formal solution in the form of a power series, then it has a solution that is infinitely differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916086.png" /> for which the formal series is the Taylor series at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916087.png" />. In the infinite-dimensional case there is only a number of sufficient conditions for the existence of infinitely-differentiable solutions.
+
the functions $  A ( t) $
 +
and $  f ( t) $
 +
are infinitely differentiable. In the finite-dimensional case a complete result has been obtained: If the equation has a formal solution in the form of a power series, then it has a solution that is infinitely differentiable on $  [ 0 , T ] $
 +
for which the formal series is the Taylor series at the point $  t = 0 $.  
 +
In the infinite-dimensional case there is only a number of sufficient conditions for the existence of infinitely-differentiable solutions.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916088.png" />. If the spectrum of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916089.png" /> does not intersect the imaginary axis, then there is a family of infinitely-differentiable solutions that depends on an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916090.png" /> belonging to the invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916091.png" /> corresponding to the part of the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916092.png" /> lying in the left half-plane. Any solution that is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916093.png" /> appears in this family. If the whole spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916094.png" /> lies in the left half-plane, then there is only one infinitely-differentiable solution.
+
Suppose that $  m > 1 $.  
 +
If the spectrum of the operator $  A ( 0) $
 +
does not intersect the imaginary axis, then there is a family of infinitely-differentiable solutions that depends on an arbitrary element $  g  ^ {-} $
 +
belonging to the invariant subspace of $  A ( 0) $
 +
corresponding to the part of the spectrum of $  A ( 0) $
 +
lying in the left half-plane. Any solution that is continuous on $  [ 0 , T] $
 +
appears in this family. If the whole spectrum of $  A ( 0) $
 +
lies in the left half-plane, then there is only one infinitely-differentiable solution.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916095.png" />. If there are no negative integers in the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916096.png" />, then there is a unique infinitely-differentiable solution. Under similar assumptions about the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916097.png" />, equations of the form (6) have been considered in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l05916099.png" /> have finite smoothness, and the solutions have the same smoothness.
+
Suppose that $  m = 1 $.  
 +
If there are no negative integers in the spectrum of $  A ( 0) $,  
 +
then there is a unique infinitely-differentiable solution. Under similar assumptions about the operator $  A ( 0) $,  
 +
equations of the form (6) have been considered in which $  a ( t) $
 +
and $  f ( t) $
 +
have finite smoothness, and the solutions have the same smoothness.
  
A rather different picture emerges when the differential equation is unsolvable for the derivative for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160100.png" />, for example when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160101.png" /> is a constant non-invertible operator. Suppose that in the equation
+
A rather different picture emerges when the differential equation is unsolvable for the derivative for all $  t $,  
 +
for example when $  A $
 +
is a constant non-invertible operator. Suppose that in the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
A \dot{u}  = B u
 +
$$
  
the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160104.png" /> are bounded in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160106.png" /> is a non-invertible [[Fredholm-operator(2)|Fredholm operator]]. Suppose that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160107.png" /> is continuously invertible for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160108.png" />. Then there are decompositions into direct sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160110.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160112.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160113.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160115.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160116.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160117.png" /> is invertible on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160118.png" /> and maps onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160119.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160120.png" /> is finite-dimensional. All solutions of (8) lie in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160121.png" /> and have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160122.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160123.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160124.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160126.png" />. For an inhomogeneous equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160127.png" />, a solution exists only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160128.png" /> has a certain smoothness and under certain compatibility conditions for the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160129.png" /> and its derivatives with the initial data. The number of derivatives that certain components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160130.png" /> must have and the number of compatibility conditions are equal to the maximal length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160131.png" />-adjoint chains of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160132.png" />. If these conditions are satisfied, the solution of the Cauchy problem is unique.
+
the operators $  A $
 +
and $  B $
 +
are bounded in the space $  E $
 +
and $  A $
 +
is a non-invertible [[Fredholm-operator(2)|Fredholm operator]]. Suppose that the operator $  A + \epsilon B $
 +
is continuously invertible for sufficiently small $  \epsilon $.  
 +
Then there are decompositions into direct sums $  E= N  ^ {(} 1) + M  ^ {(} 1) $
 +
and $  E = N  ^ {(} 2) + M  ^ {(} 2) $
 +
such that $  A $
 +
and $  B $
 +
map $  N  ^ {(} 1) $
 +
into $  N  ^ {(} 2) $
 +
and $  M  ^ {(} 1) $
 +
into $  M  ^ {(} 2) $.  
 +
The operator $  A $
 +
is invertible on $  M  ^ {(} 1) $
 +
and maps onto $  M  ^ {(} 2) $.  
 +
The subspace $  N  ^ {(} 1) $
 +
is finite-dimensional. All solutions of (8) lie in the subspace $  M  ^ {(} 1) $
 +
and have the form $  \mathop{\rm exp} ( \widetilde{A}  {}  ^ {-} 1 Bt ) u _ {0} $,  
 +
where $  \widetilde{A}  $
 +
is the restriction of $  A $
 +
to $  M  ^ {(} 1) $
 +
and $  u _ {0} \in M  ^ {(} 1) $.  
 +
For an inhomogeneous equation $  A \dot{u} = Bu + f ( t) $,  
 +
a solution exists only if $  f ( t) $
 +
has a certain smoothness and under certain compatibility conditions for the values of $  f( t) $
 +
and its derivatives with the initial data. The number of derivatives that certain components of $  f ( t) $
 +
must have and the number of compatibility conditions are equal to the maximal length of $  B $-
 +
adjoint chains of the operator $  A $.  
 +
If these conditions are satisfied, the solution of the Cauchy problem is unique.
  
If the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160133.png" /> is non-invertible for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160134.png" />, then all solutions of (8) lie in a subspace that has, generally speaking, infinite deficiency (cf. also [[Deficiency subspace|Deficiency subspace]]). The solution of the Cauchy problem for it is not unique. For the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160135.png" /> in the inhomogeneous equation infinitely many differentiability conditions and compatibility conditions are required.
+
If the operator $  A + \epsilon B $
 +
is non-invertible for all $  \epsilon $,  
 +
then all solutions of (8) lie in a subspace that has, generally speaking, infinite deficiency (cf. also [[Deficiency subspace|Deficiency subspace]]). The solution of the Cauchy problem for it is not unique. For the function $  f( t) $
 +
in the inhomogeneous equation infinitely many differentiability conditions and compatibility conditions are required.
  
 
==2. Linear differential equations with an unbounded operator.==
 
==2. Linear differential equations with an unbounded operator.==
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160136.png" /> is invertible for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160137.png" />, so that (1) can be solved for the derivative and takes the form
+
Suppose that $  A _ {0} ( t) $
 +
is invertible for every $  t $,  
 +
so that (1) can be solved for the derivative and takes the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160138.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
$$ \tag{9 }
 +
\dot{u}  = A ( t) u + f ( t) ,
 +
$$
  
and suppose that here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160139.png" /> is an unbounded operator in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160140.png" />, with dense domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160141.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160142.png" /> and with non-empty [[Resolvent set|resolvent set]], and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160143.png" /> is a given function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160144.png" /> an unknown function, both with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160145.png" />.
+
and suppose that here $  A ( t) $
 +
is an unbounded operator in a space $  E $,  
 +
with dense domain of definition $  D ( A ( t) ) $
 +
in $  E $
 +
and with non-empty [[Resolvent set|resolvent set]], and suppose that $  f ( t) $
 +
is a given function and $  u ( t) $
 +
an unknown function, both with values in $  E $.
  
Even for the simplest equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160146.png" /> with an unbounded operator, solutions of the Cauchy problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160147.png" /> need not exist, they may be non-unique, and they may be non-extendable to the whole semi-axis, so the main investigations are devoted to the questions of existence and uniqueness of the solutions. A solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160148.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160149.png" /> is understood to be a function that takes values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160150.png" />, is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160151.png" /> and satisfies the equation. Sometimes this definition is too rigid and one introduces the concept of a weak solution as a function that has the same properties on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160152.png" /> and is only continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160153.png" />.
+
Even for the simplest equation $  \dot{u} = Au $
 +
with an unbounded operator, solutions of the Cauchy problem $  u ( 0) = u _ {0} $
 +
need not exist, they may be non-unique, and they may be non-extendable to the whole semi-axis, so the main investigations are devoted to the questions of existence and uniqueness of the solutions. A solution of the equation $  \dot{u} = Au $
 +
on the interval $  [ 0, T ] $
 +
is understood to be a function that takes values in $  D ( A) $,  
 +
is differentiable on $  [ 0, T ] $
 +
and satisfies the equation. Sometimes this definition is too rigid and one introduces the concept of a weak solution as a function that has the same properties on $  ( 0 , T ] $
 +
and is only continuous at 0 $.
  
Suppose that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160154.png" /> has a resolvent
+
Suppose that the operator $  A $
 +
has a resolvent
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160155.png" /></td> </tr></table>
+
$$
 +
R ( \lambda , A )  = ( A - \lambda I )  ^ {-} 1
 +
$$
  
for all sufficiently large positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160156.png" /> and that
+
for all sufficiently large positive $  \lambda $
 +
and that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160157.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {\lambda \rightarrow \infty }  \lambda  ^ {-} 1  \mathop{\rm ln}  \| R
 +
( \lambda , A ) \|  = < T .
 +
$$
  
 
Then the weak solution of the problem
 
Then the weak solution of the problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160158.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>
+
$$ \tag{10 }
 +
\dot{u}  = Au ,\  u ( 0)  = u _ {0}  $$
  
is unique on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160159.png" /> and can be branched for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160160.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160161.png" />, then the solution is unique on the whole semi-axis. This assertion is precise as regards the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160162.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160163.png" />.
+
is unique on $  [ 0 , T - h ] $
 +
and can be branched for $  t = T - h $.  
 +
If $  h = 0 $,  
 +
then the solution is unique on the whole semi-axis. This assertion is precise as regards the behaviour of $  R ( \lambda , A ) $
 +
as $  \lambda \rightarrow \infty $.
  
If for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160164.png" /> there is a unique solution of the problem (10) that is continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160165.png" />, then this solution can be extended to the whole semi-axis and can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160166.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160167.png" /> is a [[Strongly-continuous semi-group|strongly-continuous semi-group]] of bounded operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160169.png" />, for which the estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160170.png" /> holds. For the equation to have this property it is necessary and sufficient that
+
If for every $  u _ {0} \in D ( A) $
 +
there is a unique solution of the problem (10) that is continuously differentiable on $  [ 0 , T ] $,
 +
then this solution can be extended to the whole semi-axis and can be represented in the form $  u ( t) = U ( t) u _ {0} $,  
 +
where $  U ( t) $
 +
is a [[Strongly-continuous semi-group|strongly-continuous semi-group]] of bounded operators on $  [ 0 , \infty ) $,
 +
$  U ( 0) = I $,  
 +
for which the estimate $  \| U ( t) \| \leq  M e ^ {\omega t } $
 +
holds. For the equation to have this property it is necessary and sufficient that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160171.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>
+
$$ \tag{11 }
 +
\| ( \lambda - \omega )  ^ {m} R  ^ {m} ( \lambda , A ) \|  \leq  M
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160172.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160173.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160174.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160175.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160176.png" />. These conditions are difficult to verify. They are satisfied if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160177.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160178.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160179.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160180.png" /> is a [[Contraction semi-group|contraction semi-group]]. This is so if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160181.png" /> is a maximal [[Dissipative operator|dissipative operator]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160182.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160183.png" /> is not differentiable (in any case for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160184.png" />); it is often called the generalized solution of (10). Solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160185.png" /> can be constructed as the limit, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160186.png" />, of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160187.png" /> with bounded operators, under the same initial conditions. For this it is sufficient that the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160188.png" /> commute, converge strongly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160189.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160190.png" /> and that
+
for all $  \lambda > \omega $
 +
and $  m = 1 , 2 \dots $
 +
where $  M $
 +
does not depend on $  \lambda $
 +
and $  m $.  
 +
These conditions are difficult to verify. They are satisfied if $  \| ( \lambda - \omega ) R ( \lambda , A ) \| \leq  1 $,  
 +
and then $  \| U ( t) \| \leq  e ^ {\omega t } $.  
 +
If $  \omega = 0 $,  
 +
then $  U ( t) $
 +
is a [[Contraction semi-group|contraction semi-group]]. This is so if and only if $  A $
 +
is a maximal [[Dissipative operator|dissipative operator]]. If $  u _ {0} \notin D ( A) $,  
 +
then the function $  U ( t) u _ {0} $
 +
is not differentiable (in any case for $  t = 0 $);  
 +
it is often called the generalized solution of (10). Solutions of the equation $  \dot{u} = Au $
 +
can be constructed as the limit, as $  n \rightarrow \infty $,  
 +
of solutions of the equation $  \dot{u} = A _ {n} u $
 +
with bounded operators, under the same initial conditions. For this it is sufficient that the operators $  A _ {n} $
 +
commute, converge strongly to $  A $
 +
on $  D ( A) $
 +
and that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160191.png" /></td> </tr></table>
+
$$
 +
\| e ^ {t A _ {n} } \|  \leq  M e ^ {\omega t } .
 +
$$
  
If the conditions (11) are satisfied, then the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160192.png" /> (Yosida operators) have these properties.
+
If the conditions (11) are satisfied, then the operators $  A _ {n} = - nI - n  ^ {2} R ( \lambda , A ) $(
 +
Yosida operators) have these properties.
  
Another method for constructing solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160193.png" /> is based on Laplace transformation. If the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160194.png" /> is defined on some contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160195.png" />, then the function
+
Another method for constructing solutions of the equation $  \dot{u} = A u $
 +
is based on Laplace transformation. If the resolvent of $  A $
 +
is defined on some contour $  \Gamma $,  
 +
then the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160196.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>
+
$$ \tag{12 }
 +
u ( t)  = -  
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma  e ^ {\lambda t } R (
 +
\lambda , A ) u _ {0}  d \lambda
 +
$$
  
 
formally satisfies the equation
 
formally satisfies the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160197.png" /></td> </tr></table>
+
$$
 +
\dot{u}  = A u +
 +
\frac{1}{2 \pi i }
 +
\int\limits _  \Gamma  e ^ {\lambda t } \
 +
d \lambda u _ {0} .
 +
$$
  
If the convergence of the integrals, the validity of differentiation under the integral sign and the vanishing of the last integral are ensured, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160198.png" /> satisfies the equation. The difficulty lies in the fact that the norm of the resolvent cannot decrease faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160199.png" /> at infinity. However, on some elements it does decrease faster. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160200.png" /> is defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160201.png" /> and if
+
If the convergence of the integrals, the validity of differentiation under the integral sign and the vanishing of the last integral are ensured, then $  u ( t) $
 +
satisfies the equation. The difficulty lies in the fact that the norm of the resolvent cannot decrease faster than $  | \lambda |  ^ {-} 1 $
 +
at infinity. However, on some elements it does decrease faster. For example, if $  R ( \lambda , A ) $
 +
is defined for $  \mathop{\rm Re}  \lambda \geq  \alpha $
 +
and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160202.png" /></td> </tr></table>
+
$$
 +
\| R ( \lambda , A ) \|  \leq  M  | \lambda |  ^ {k} ,\  k \geq  - 1 ,
 +
$$
  
for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160203.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160204.png" /> formula (12) gives a solution for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160205.png" />. In a "less good" case, when the previous inequality is satisfied only in the domain
+
for sufficiently large $  | \lambda | $,  
 +
then for $  \Gamma = ( - i \infty , i \infty ) $
 +
formula (12) gives a solution for any $  u _ {0} \in D ( A ^ {[ k ] + 3 } ) $.  
 +
In a "less good" case, when the previous inequality is satisfied only in the domain
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160206.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Re}  \lambda  \geq  \alpha  |  \mathop{\rm Im}  \lambda |  ^ {a} ,\  0
 +
< a < 1
 +
$$
  
(weakly hyperbolic equations), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160207.png" /> is the boundary of this domain, one obtains a solution only for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160208.png" /> belonging to the intersection of the domains of definition of all powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160209.png" />, with definite behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160210.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160211.png" />.
+
(weakly hyperbolic equations), and $  \Gamma $
 +
is the boundary of this domain, one obtains a solution only for an $  u _ {0} $
 +
belonging to the intersection of the domains of definition of all powers of $  A $,  
 +
with definite behaviour of $  \| A  ^ {n} u _ {0} \| $
 +
as $  n \rightarrow \infty $.
  
Significantly weaker solutions are obtained in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160212.png" /> goes into the left half-plane, and one can use the decrease of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160213.png" /> on it. As a rule, the solutions have increased smoothness for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160214.png" />. If the resolvent is bounded on the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160215.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160216.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160217.png" /> is a smooth non-decreasing concave function that increases like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160218.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160219.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160220.png" /> the function (12) is differentiable and satisfies the equation, beginning with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160221.png" />; as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160222.png" /> increases further, its smoothness increases. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160223.png" /> increases like a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160224.png" /> with exponent less than one, then the function (12) is infinitely differentiable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160225.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160226.png" /> increases like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160227.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160228.png" /> belongs to a [[Quasi-analytic class|quasi-analytic class]] of functions; if it increases like a linear function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160229.png" /> is analytic. In all these cases it satisfies the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160230.png" />.
+
Significantly weaker solutions are obtained in the case when $  \Gamma $
 +
goes into the left half-plane, and one can use the decrease of the function $  | e ^ {\lambda t } | $
 +
on it. As a rule, the solutions have increased smoothness for $  t > 0 $.  
 +
If the resolvent is bounded on the contour $  \Gamma $:  
 +
$  \mathop{\rm Re}  \lambda = - \psi ( |  \mathop{\rm Im}  \lambda | ) $,  
 +
where $  \psi ( \tau ) $
 +
is a smooth non-decreasing concave function that increases like $  \mathop{\rm ln}  \tau $
 +
at $  \infty $,  
 +
then for any $  u _ {0} \in E $
 +
the function (12) is differentiable and satisfies the equation, beginning with some $  t _ {0} $;  
 +
as $  t $
 +
increases further, its smoothness increases. If $  \psi ( \tau ) $
 +
increases like a power of $  \tau $
 +
with exponent less than one, then the function (12) is infinitely differentiable for $  t > 0 $;  
 +
if $  \psi ( \tau ) $
 +
increases like $  \tau / \mathop{\rm ln}  \tau $,  
 +
then $  u ( t) $
 +
belongs to a [[Quasi-analytic class|quasi-analytic class]] of functions; if it increases like a linear function, then $  u ( t) $
 +
is analytic. In all these cases it satisfies the equation $  \dot{u} = A u $.
  
The existence of the resolvent on contours that go into the left half-plane may be obtained, by using series expansion, from the corresponding estimates on vertical lines. If for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160231.png" />,
+
The existence of the resolvent on contours that go into the left half-plane may be obtained, by using series expansion, from the corresponding estimates on vertical lines. If for $  \mathop{\rm Re}  \lambda \geq  \gamma $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160232.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>
+
$$ \tag{13 }
 +
\| R ( \lambda , A ) \|  \leq  M ( 1 + |  \mathop{\rm Im}  \lambda | ) ^
 +
{- \beta } ,\  0 < \beta < 1 ,
 +
$$
  
then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160233.png" /> there is a solution of problem (10). All these solutions are infinitely differentiable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160234.png" />. They can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160235.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160236.png" /> is an infinitely-differentiable semi-group for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160237.png" /> having, generally speaking, a singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160238.png" />. For its derivatives one has the estimates
+
then for every $  u _ {0} \in D ( A) $
 +
there is a solution of problem (10). All these solutions are infinitely differentiable for $  t > 0 $.  
 +
They can be represented in the form $  u ( t) = U ( t) u _ {0} $,  
 +
where $  U ( t) $
 +
is an infinitely-differentiable semi-group for $  t > 0 $
 +
having, generally speaking, a singularity at $  t = 0 $.  
 +
For its derivatives one has the estimates
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160239.png" /></td> </tr></table>
+
$$
 +
\| U ( k) ( t) \|  \leq  M _ {k} t ^ {1 - ( k+ 1 ) / \beta } e ^ {
 +
\omega t } .
 +
$$
  
If the estimate (13) is satisfied for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160240.png" />, then all generalized solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160241.png" /> are analytic in some sector containing the positive semi-axis.
+
If the estimate (13) is satisfied for $  \beta = 1 $,  
 +
then all generalized solutions of the equation $  \dot{u} = Au $
 +
are analytic in some sector containing the positive semi-axis.
  
The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160242.png" /> is called an abstract parabolic equation if there is a unique weak solution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160243.png" /> satisfying the initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160244.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160245.png" />. If
+
The equation $  \dot{u} = Au $
 +
is called an abstract parabolic equation if there is a unique weak solution on $  [ 0 , \infty ] $
 +
satisfying the initial condition $  u ( 0) = u _ {0} $
 +
for any $  u _ {0} \in E $.  
 +
If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160246.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>
+
$$ \tag{14 }
 +
\| R ( \lambda , A ) \|  \leq  M  | \lambda - \omega |  ^ {-} 1 \ \
 +
\textrm{ for }  \mathop{\rm Re}  \lambda > \omega ,
 +
$$
  
 
then the equation is an abstract parabolic equation. All its generalized solutions are analytic in some sector containing the positive semi-axis, and
 
then the equation is an abstract parabolic equation. All its generalized solutions are analytic in some sector containing the positive semi-axis, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160247.png" /></td> </tr></table>
+
$$
 +
\| \dot{u} ( t) \|  \leq  t  ^ {-} 1 C e ^ {\omega t }  \| u _ {0} \| ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160248.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160249.png" />. Conversely, if the equation has the listed properties, then (14) is satisfied for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160250.png" />.
+
where $  C $
 +
does not depend on $  u _ {0} $.  
 +
Conversely, if the equation has the listed properties, then (14) is satisfied for the operator $  A $.
  
If problem (10) has a unique weak solution for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160251.png" /> for which the derivative is integrable on every finite interval, then these solutions can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160252.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160253.png" /> is a strongly-continuous semi-group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160254.png" />, and every weak solution of the inhomogeneous equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160255.png" /> with initial condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160256.png" /> can be represented in the form
+
If problem (10) has a unique weak solution for any $  u _ {0} \in D ( A) $
 +
for which the derivative is integrable on every finite interval, then these solutions can be represented in the form $  u ( t) = U ( t) u _ {0} $,  
 +
where $  U ( t) $
 +
is a strongly-continuous semi-group on $  ( 0 , \infty ) $,
 +
and every weak solution of the inhomogeneous equation $  \dot{v} = Av + f ( t) $
 +
with initial condition $  v ( 0) = 0 $
 +
can be represented in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160257.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>
+
$$ \tag{15 }
 +
v ( t)  = \int\limits _ { 0 } ^ { t }  U ( t- s ) f ( s) ds .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160258.png" /> is defined for any continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160259.png" />, hence it is called a generalized solution of the inhomogeneous equation. To ensure that it is differentiable, one imposes smoothness conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160260.png" />, and the "worse" the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160261.png" />, the "higher" these should be. Thus, under the previous conditions, (15) is a weak solution of the inhomogeneous equation if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160262.png" /> is twice continuously differentiable; if (11) is satisfied, then (15) is a solution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160263.png" /> is continuously differentiable; if (13) is satisfied with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160264.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160265.png" /> is a weak solution if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160266.png" /> satisfies a Hölder condition with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160267.png" />. Instead of smoothness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160268.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160269.png" /> one can require that the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160270.png" /> belong to the domain of definition of the corresponding power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160271.png" />.
+
The function $  v ( t) $
 +
is defined for any continuous $  f ( t) $,  
 +
hence it is called a generalized solution of the inhomogeneous equation. To ensure that it is differentiable, one imposes smoothness conditions on $  f ( t) $,  
 +
and the "worse" the semi-group $  U ( t) $,  
 +
the "higher" these should be. Thus, under the previous conditions, (15) is a weak solution of the inhomogeneous equation if $  f ( t) $
 +
is twice continuously differentiable; if (11) is satisfied, then (15) is a solution if $  f ( t) $
 +
is continuously differentiable; if (13) is satisfied with $  \beta > 2/3 $,  
 +
then $  v ( t) $
 +
is a weak solution if $  f ( t) $
 +
satisfies a Hölder condition with exponent $  \gamma > 2 ( 1 - 1/ \beta ) $.  
 +
Instead of smoothness of $  f ( t) $
 +
with respect to $  t $
 +
one can require that the values of $  f ( t) $
 +
belong to the domain of definition of the corresponding power of $  A $.
  
 
For an equation with variable operator
 
For an equation with variable operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160272.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>
+
$$ \tag{16 }
 +
\dot{u}  = A ( t) u ,\ \
 +
0 \leq  t \leq  T ,
 +
$$
 +
 
 +
there are some fundamental existence and uniqueness theorems about solutions (weak solutions) of the Cauchy problem  $  u ( s) = u _ {0} $
 +
on the interval  $  s \leq  t \leq  T $.  
 +
If the domain of definition of  $  A ( t) $
 +
does not depend on  $  t $,
 +
 
 +
$$
 +
D ( A ( t) )  \equiv  D ( A),
 +
$$
  
there are some fundamental existence and uniqueness theorems about solutions (weak solutions) of the Cauchy problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160273.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160274.png" />. If the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160275.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160276.png" />,
+
if the operator  $  A ( t) $
 +
is strongly continuous with respect to  $  t $
 +
on $  D ( A) $
 +
and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160277.png" /></td> </tr></table>
+
$$
 +
\| \lambda R ( \lambda , A ( t) ) \|  \leq  1
 +
$$
  
if the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160278.png" /> is strongly continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160279.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160280.png" /> and if
+
for  $  \lambda > 0 $,
 +
then the solution of the Cauchy problem is unique. Moreover, if  $  A ( t) $
 +
is strongly continuously differentiable on $  D ( A) $,
 +
then for every  $  u _ {0} \in D ( A) $
 +
a solution exists and can be represented in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160281.png" /></td> </tr></table>
+
$$
 +
u ( t)  = U ( t , s ) u _ {0} ,
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160282.png" />, then the solution of the Cauchy problem is unique. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160283.png" /> is strongly continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160284.png" />, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160285.png" /> a solution exists and can be represented in the form
+
where  $  U ( t , s ) $
 +
is an [[Evolution operator|evolution operator]] with the following properties:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160286.png" /></td> </tr></table>
+
1)  $  U ( t , s ) $
 +
is strongly continuous in the triangle  $  T _  \Delta  $:  
 +
$  0 \leq  s \leq  t \leq  T $;
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160287.png" /> is an [[Evolution operator|evolution operator]] with the following properties:
+
2)  $  U ( t , s ) = U ( t , \tau ) U ( \tau , s ) $,
 +
0 \leq  s \leq  \tau \leq  t \leq  T $,
 +
$  U ( s , s ) = I $;
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160288.png" /> is strongly continuous in the triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160289.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160290.png" />;
+
3) $  U ( t , s ) $
 +
maps  $  D ( A) $
 +
into itself and the operator
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160291.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160292.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160293.png" />;
+
$$
 +
A ( t) U ( t , s ) A  ^ {-} 1 ( s)
 +
$$
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160294.png" /> maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160295.png" /> into itself and the operator
+
is bounded and strongly continuous in  $  T _  \Delta  $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160296.png" /></td> </tr></table>
+
4) on  $  D ( A) $
 +
the operator  $  U ( t , s ) $
 +
is strongly differentiable with respect to  $  t $
 +
and  $  s $
 +
and
  
is bounded and strongly continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160297.png" />;
+
$$
  
4) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160298.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160299.png" /> is strongly differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160300.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160301.png" /> and
+
\frac{\partial  U }{\partial  t }
 +
  =  A ( t) U ,\ \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160302.png" /></td> </tr></table>
+
\frac{\partial  U }{\partial  s }
 +
  = - U A ( s) .
 +
$$
  
The construction of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160303.png" /> is carried out by approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160304.png" /> by bounded operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160305.png" /> and replacing the latter by piecewise-constant operators.
+
The construction of the operator $  U ( t , s ) $
 +
is carried out by approximating $  A ( t) $
 +
by bounded operators $  A _ {n} ( t) $
 +
and replacing the latter by piecewise-constant operators.
  
In many important problems the previous conditions on the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160306.png" /> are not satisfied. Suppose that for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160307.png" /> there are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160309.png" /> such that
+
In many important problems the previous conditions on the operator $  A ( t) $
 +
are not satisfied. Suppose that for the operator $  A ( t) $
 +
there are constants $  M $
 +
and $  \omega $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160310.png" /></td> </tr></table>
+
$$
 +
\| R ( \lambda , A ( t _ {k} ) ) \dots R ( \lambda , A ( t _ {1} ) ) \|
 +
\leq  M ( \lambda - \omega )  ^ {-} k
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160311.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160312.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160313.png" />. Suppose that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160314.png" /> there is densely imbedded a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160315.png" /> contained in all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160316.png" /> and having the following properties: a) the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160317.png" /> acts boundedly from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160318.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160319.png" /> and is continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160320.png" /> in the norm as a bounded operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160321.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160322.png" />; and b) there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160323.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160324.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160325.png" /> such that
+
for all $  \lambda > \omega $,  
 +
0 \leq  t _ {1} \leq  \dots \leq  t _ {k} \leq  T $,
 +
$  k = 1 , 2, . . . $.  
 +
Suppose that in $  E $
 +
there is densely imbedded a Banach space $  F $
 +
contained in all the $  D ( A ( t) ) $
 +
and having the following properties: a) the operator $  A ( t) $
 +
acts boundedly from $  F $
 +
to $  E $
 +
and is continuous with respect to $  t $
 +
in the norm as a bounded operator from $  F $
 +
to $  E $;  
 +
and b) there is an isomorphism $  S $
 +
of $  F $
 +
onto $  E $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160326.png" /></td> </tr></table>
+
$$
 +
S A ( t) S  ^ {-} 1  = A ( t) + B ( t) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160327.png" /> is an operator function that is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160328.png" /> and strongly measurable, and for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160329.png" /> is integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160330.png" />. Then there is an evolution operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160331.png" /> having the properties: 1); 2); 3') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160332.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160333.png" /> is strongly continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160334.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160335.png" />; and 4') on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160336.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160337.png" /> is strongly differentiable in the sense of the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160338.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160339.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160340.png" />. This assertion makes it possible to obtain existence theorems for the fundamental quasi-linear equations of mathematical physics of hyperbolic type.
+
where $  B ( t) $
 +
is an operator function that is bounded in $  E $
 +
and strongly measurable, and for which $  \| B ( t) \| $
 +
is integrable on $  [ 0 , T ] $.  
 +
Then there is an evolution operator $  U ( t , s ) $
 +
having the properties: 1); 2); 3') $  U ( t , s ) F \subset  F $
 +
and $  U ( t , s ) $
 +
is strongly continuous in $  F $
 +
on $  T _  \Delta  $;  
 +
and 4') on $  F $
 +
the operator $  U ( t , s ) $
 +
is strongly differentiable in the sense of the norm of $  E $
 +
and  $  \partial  U / \partial  t = A ( t) U $,
 +
$  \partial  U / \partial  s = - U A ( s) $.  
 +
This assertion makes it possible to obtain existence theorems for the fundamental quasi-linear equations of mathematical physics of hyperbolic type.
  
The method of frozen coefficients is used in the theory of parabolic equations. Suppose that, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160341.png" />, to the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160342.png" /> corresponds an operator semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160343.png" />. The unknown evolution operator formally satisfies the integral equations
+
The method of frozen coefficients is used in the theory of parabolic equations. Suppose that, for every $  t _ {0} \in [ 0 , T ] $,
 +
to the equation $  \dot{u} = A ( t _ {0} ) u $
 +
corresponds an operator semi-group $  U _ {A ( t _ {0}  ) } ( t) $.  
 +
The unknown evolution operator formally satisfies the integral equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160344.png" /></td> </tr></table>
+
$$
 +
U ( t , s )  = U _ {A ( t) }  ( t - s ) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160345.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _ { s } ^ { t }  U _ {A ( t) }  ( t - s ) [
 +
A ( \tau ) - A ( t) ] U ( \tau , s )  d \tau ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160346.png" /></td> </tr></table>
+
$$
 +
U ( t , s )  = U _ {A ( s) }  ( t - s ) +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160347.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _ { s } ^ { t }  U ( t , \tau ) [ A ( \tau ) -
 +
A ( s) ] U _ {A ( s) }  ( \tau - s )  d \tau .
 +
$$
  
When the kernels of these equations have weak singularities, one can prove that the equation has solutions and also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160348.png" /> is an evolution operator. The following statement has the most applications: If
+
When the kernels of these equations have weak singularities, one can prove that the equation has solutions and also that $  U ( t , s ) $
 +
is an evolution operator. The following statement has the most applications: If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160349.png" /></td> </tr></table>
+
$$
 +
D ( A ( t) )  \equiv  D ( A) ,\ \
 +
\| R ( \lambda , A ( t) ) \|  < \
 +
M ( 1 + | \lambda | )  ^ {-} 1
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160350.png" /> and
+
for $  \mathop{\rm Re}  \lambda \geq  0 $
 +
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160351.png" /></td> </tr></table>
+
$$
 +
\| [ A ( t) - A ( s) ] A  ^ {-} 1 ( 0) \|  \leq  C  | t - s |  ^  \rho
 +
$$
  
(a Hölder condition), then there is an evolution operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160352.png" /> that gives a weak solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160353.png" /> of the Cauchy problem for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160354.png" />. Uniqueness of the solution holds under the single condition that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160355.png" /> is continuous (in a Hilbert space). An existence theorem similar to the one given above holds for the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160356.png" /> with a condition of type (13) and for a certain relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160357.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160358.png" />.
+
(a Hölder condition), then there is an evolution operator $  U ( t , s ) $
 +
that gives a weak solution $  U ( t , s ) u _ {0} $
 +
of the Cauchy problem for every $  u _ {0} \in E $.  
 +
Uniqueness of the solution holds under the single condition that the operator $  A ( t) A  ^ {-} 1 ( 0) $
 +
is continuous (in a Hilbert space). An existence theorem similar to the one given above holds for the operator $  A ( t) $
 +
with a condition of type (13) and for a certain relation between $  \beta $
 +
and $  \rho $.
  
The assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160359.png" /> is constant does not make it possible in applications to consider boundary value problems with boundary conditions depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160360.png" />. Suppose that
+
The assumption that $  D ( A ( t) ) $
 +
is constant does not make it possible in applications to consider boundary value problems with boundary conditions depending on $  t $.  
 +
Suppose that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160361.png" /></td> </tr></table>
+
$$
 +
\| R ( \lambda , A ( t) ) \|  \leq  M
 +
( 1 + | \lambda | )  ^ {-} 1 ,\ \
 +
\mathop{\rm Re}  \lambda > 0 ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160362.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\frac{d A  ^ {-} 1 ( t) }{dt}
 +
-
 +
\frac{d A  ^ {-} 1
 +
( s) }{ds}
 +
\right \|  \leq  K  | t - s |  ^  \alpha  ,\  0 < \alpha < 1 ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160363.png" /></td> </tr></table>
+
$$
 +
\left \|
 +
\frac \partial {\partial  t }
 +
R ( \lambda , A ( t) )
 +
\right \|  \leq  N  | \lambda | ^ {\rho - 1 } ,\  0 \leq  \rho \leq  1 ,
 +
$$
  
in the sector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160364.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160365.png" />; then there is an evolution operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160366.png" />. Here it is not assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160367.png" /> is constant. There is a version of the last statement adapted to the consideration of parabolic problems in non-cylindrical domains, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160368.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160369.png" /> lies in some subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160370.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160371.png" />.
+
in the sector $  |  \mathop{\rm arg}  \lambda | \leq  \pi - \phi $,  
 +
$  \phi < \pi / 2 $;  
 +
then there is an evolution operator $  U ( t , s ) $.  
 +
Here it is not assumed that $  D ( A ( t) ) $
 +
is constant. There is a version of the last statement adapted to the consideration of parabolic problems in non-cylindrical domains, in which $  D ( A ( t) ) $
 +
for every $  t $
 +
lies in some subspace $  E ( t) $
 +
of $  E $.
  
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160372.png" /> for equation (16) formally satisfies the integral equation
+
The operator $  U ( t , s ) $
 +
for equation (16) formally satisfies the integral equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160373.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>
+
$$ \tag{17 }
 +
U ( t , s )  = I +
 +
\int\limits _ { s } ^ { t }  A ( \tau ) U ( \tau , s ) d \tau .
 +
$$
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160374.png" /> is unbounded, this equation cannot be solved by the method of successive approximation (cf. [[Sequential approximation, method of|Sequential approximation, method of]]). Suppose that there is a family of Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160375.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160376.png" />, having the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160377.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160378.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160379.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160380.png" /> is bounded as an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160381.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160382.png" />:
+
Since $  A ( t) $
 +
is unbounded, this equation cannot be solved by the method of successive approximation (cf. [[Sequential approximation, method of|Sequential approximation, method of]]). Suppose that there is a family of Banach spaces $  E _  \alpha  $,  
 +
0 \leq  \alpha \leq  1 $,  
 +
having the property that $  E _  \beta  \subset  E _  \alpha  $
 +
and $  \| x \| _  \alpha  \leq  \| x \| _  \beta  $
 +
for $  \alpha < \beta $.  
 +
Suppose that $  A ( t) $
 +
is bounded as an operator from $  E _  \beta  $
 +
to $  E _  \alpha  $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160383.png" /></td> </tr></table>
+
$$
 +
\| A ( t) \| _ {E _  \beta  \rightarrow E _  \alpha  }
 +
\leq  C ( \beta - \alpha )  ^ {-} 1 ,
 +
$$
  
and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160384.png" /> is continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160385.png" /> in the norm of the space of bounded operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160386.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160387.png" />. Then in this space the method of successive approximation for equation (17) will converge for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160388.png" />. In this way one can locally construct an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160389.png" /> as a bounded operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160390.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160391.png" />. In applications this approach gives theorems of Cauchy–Kovalevskaya type (cf. [[Cauchy–Kovalevskaya theorem|Cauchy–Kovalevskaya theorem]]).
+
and that $  A ( t) $
 +
is continuous with respect to $  t $
 +
in the norm of the space of bounded operators from $  E _  \beta  $
 +
to $  E _  \alpha  $.  
 +
Then in this space the method of successive approximation for equation (17) will converge for $  | t - s | \leq  ( \beta - \alpha ) ( Ce )  ^ {-} 1 $.  
 +
In this way one can locally construct an operator $  U ( t , s ) $
 +
as a bounded operator from $  E _  \beta  $
 +
to $  E _  \alpha  $.  
 +
In applications this approach gives theorems of Cauchy–Kovalevskaya type (cf. [[Cauchy–Kovalevskaya theorem|Cauchy–Kovalevskaya theorem]]).
  
For the inhomogeneous equation (9) with known evolution operator, for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160392.png" /> the solution of the Cauchy problem is formally written in the form
+
For the inhomogeneous equation (9) with known evolution operator, for the equation $  \dot{u} = A ( t) u $
 +
the solution of the Cauchy problem is formally written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160393.png" /></td> </tr></table>
+
$$
 +
u ( t)  = U ( t , s ) u _ {0} +
 +
\int\limits _ { s } ^ { t }  U ( t , \tau ) f ( \tau )  d \tau .
 +
$$
  
This formula can be justified in various cases under certain smoothness conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059160/l059160394.png" />.
+
This formula can be justified in various cases under certain smoothness conditions on $  f ( t) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.G. Krein, "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Functional analysis'' , Math. Reference Library , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.P. Glushko, "Degenerate linear differential equations" , Voronezh (1972) (In Russian) {{MR|}} {{ZBL|0265.34011}} {{ZBL|0252.34072}} {{ZBL|0241.34008}} {{ZBL|0235.34011}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian) {{MR|0352638}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S.P. Zubova, K.I. Chernyshovas, "A linear differential equation with a Fredholm operator acting on the derivative" , ''Differential Equations and their Applications'' , '''14''' , Vilnius (1976) pp. 21–29 (In Russian) (English abstract) {{MR|0470716}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.N. Kuznetsov, "Differentiable solutions to degenerate systems of ordinary equations" ''Funct. Anal. Appl.'' , '''6''' : 2 (1972) pp. 119–127 ''Funktional. Anal. i Prilozhen.'' , '''6''' : 2 (1972) pp. 41–51 {{MR|}} {{ZBL|0259.34005}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S.G. Krein, G.I. Laptev, "An abstract scheme for the consideration of parabolic problems in noncylindrical regions" ''Differential Eq.'' , '''5''' (1969) pp. 1073–1081 ''Differentsial. Uravn.'' , '''5''' : 8 (1969) pp. 1458–1469 {{MR|}} {{ZBL|0254.35064}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> Yu.I. Lyubich, "The classical and local Laplace transformation in an abstract Cauchy problem" ''Russian Math. Surveys'' , '''21''' : 3 (1966) pp. 1–52 ''Uspekhi Mat. Nauk'' , '''21''' : 3 (1966) pp. 3–51 {{MR|}} {{ZBL|0173.12002}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> L.V. Ovsyannikov, "A singular operator in a scale of Banach spaces" ''Soviet Math. Dokl.'' , '''6''' (1965) pp. 1025–1028 ''Dokl. Akad. Nauk SSSR'' , '''163''' : 4 (1965) pp. 819–822 {{MR|}} {{ZBL|0144.39003}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> P.E. Sobolevskii, "Equations of parabolic type in a Banach space" ''Trudy Moskov. Mat. Obshch.'' , '''10''' (1961) pp. 297–350 (In Russian) {{MR|0141900}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> R. Beals, "Laplace transform methods for evolution equations" H.G. Garnir (ed.) , ''Boundary value problems for linear evolution equations: partial differential equations. Proc. NATO Adv. Study Inst. Liège, 1976'' , Reidel (1977) pp. 1–26 {{MR|0492648}} {{ZBL|0374.35039}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A. Friedman, "Uniqueness of solutions of ordinary differential inequalities in Hilbert space" ''Arch. Rat. Mech. Anal.'' , '''17''' : 5 (1964) pp. 353–357 {{MR|0171181}} {{ZBL|0143.16701}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> T. Kato, "Linear evolution equations of "hyperbolic" type II" ''J. Math. Assoc. Japan'' , '''25''' : 4 (1973) pp. 648–666 {{MR|0326483}} {{ZBL|0262.34048}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> F. Trèves, "Basic linear partial differential equations" , Acad. Press (1975) {{MR|0447753}} {{ZBL|0305.35001}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> J. Miller, "Solution in Banach algebras of differential equations with irregular singular point" ''Acta Math.'' , '''110''' : 3–4 (1963) pp. 209–231 {{MR|0153939}} {{ZBL|0122.35303}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.G. Krein, "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Functional analysis'' , Math. Reference Library , Moscow (1972) (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.P. Glushko, "Degenerate linear differential equations" , Voronezh (1972) (In Russian) {{MR|}} {{ZBL|0265.34011}} {{ZBL|0252.34072}} {{ZBL|0241.34008}} {{ZBL|0235.34011}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian) {{MR|0352638}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S.P. Zubova, K.I. Chernyshovas, "A linear differential equation with a Fredholm operator acting on the derivative" , ''Differential Equations and their Applications'' , '''14''' , Vilnius (1976) pp. 21–29 (In Russian) (English abstract) {{MR|0470716}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.N. Kuznetsov, "Differentiable solutions to degenerate systems of ordinary equations" ''Funct. Anal. Appl.'' , '''6''' : 2 (1972) pp. 119–127 ''Funktional. Anal. i Prilozhen.'' , '''6''' : 2 (1972) pp. 41–51 {{MR|}} {{ZBL|0259.34005}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S.G. Krein, G.I. Laptev, "An abstract scheme for the consideration of parabolic problems in noncylindrical regions" ''Differential Eq.'' , '''5''' (1969) pp. 1073–1081 ''Differentsial. Uravn.'' , '''5''' : 8 (1969) pp. 1458–1469 {{MR|}} {{ZBL|0254.35064}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> Yu.I. Lyubich, "The classical and local Laplace transformation in an abstract Cauchy problem" ''Russian Math. Surveys'' , '''21''' : 3 (1966) pp. 1–52 ''Uspekhi Mat. Nauk'' , '''21''' : 3 (1966) pp. 3–51 {{MR|}} {{ZBL|0173.12002}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> L.V. Ovsyannikov, "A singular operator in a scale of Banach spaces" ''Soviet Math. Dokl.'' , '''6''' (1965) pp. 1025–1028 ''Dokl. Akad. Nauk SSSR'' , '''163''' : 4 (1965) pp. 819–822 {{MR|}} {{ZBL|0144.39003}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> P.E. Sobolevskii, "Equations of parabolic type in a Banach space" ''Trudy Moskov. Mat. Obshch.'' , '''10''' (1961) pp. 297–350 (In Russian) {{MR|0141900}} {{ZBL|}} </TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> R. Beals, "Laplace transform methods for evolution equations" H.G. Garnir (ed.) , ''Boundary value problems for linear evolution equations: partial differential equations. Proc. NATO Adv. Study Inst. Liège, 1976'' , Reidel (1977) pp. 1–26 {{MR|0492648}} {{ZBL|0374.35039}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A. Friedman, "Uniqueness of solutions of ordinary differential inequalities in Hilbert space" ''Arch. Rat. Mech. Anal.'' , '''17''' : 5 (1964) pp. 353–357 {{MR|0171181}} {{ZBL|0143.16701}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> T. Kato, "Linear evolution equations of "hyperbolic" type II" ''J. Math. Assoc. Japan'' , '''25''' : 4 (1973) pp. 648–666 {{MR|0326483}} {{ZBL|0262.34048}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> F. Trèves, "Basic linear partial differential equations" , Acad. Press (1975) {{MR|0447753}} {{ZBL|0305.35001}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> J. Miller, "Solution in Banach algebras of differential equations with irregular singular point" ''Acta Math.'' , '''110''' : 3–4 (1963) pp. 209–231 {{MR|0153939}} {{ZBL|0122.35303}} </TD></TR></table>

Revision as of 22:17, 5 June 2020


An equation of the form

$$ \tag{1 } A _ {0} ( t) \dot{u} = A _ {1} ( t) u + g ( t) , $$

where $ A _ {0} ( t) $ and $ A _ {1} ( t) $, for every $ t $, are linear operators in a Banach space $ E $, $ g ( t) $ is a given function and $ u ( t) $ an unknown function, both with values in $ E $; the derivative $ \dot{u} $ is understood to be the limit of the difference quotient with respect to the norm of $ E $.

1. Linear differential equations with a bounded operator.

Suppose that $ A _ {0} ( t) $ and $ A _ {1} ( t) $, for every $ t $, are bounded operators acting in $ E $. If $ A _ {0} ( t) $ has a bounded inverse for every $ t $, then (1) can be solved for the derivative and takes the form

$$ \tag{2 } \dot{u} = A ( t) u + f ( t) , $$

where $ A ( t) $ is a bounded operator in $ E $, and $ f ( t) $ and $ u ( t) $ are functions with values in $ E $. If the functions $ A ( t) $ and $ f ( t) $ are continuous (or, more generally, are measurable and integrable on every finite interval), then the solution of the Cauchy problem

$$ \tag{3 } \dot{u} = A ( t) u ,\ u ( s) = u _ {0} , $$

exists for any $ u _ {0} \in E $ and is given by the formula

$$ u ( t) = U ( t , s ) u _ {0} , $$

where

$$ \tag{4 } U ( t , s ) = $$

$$ = \ I + \int\limits _ { s } ^ { t } A ( t _ {1} ) dt _ {1} + \sum _ { n= } 2 ^ \infty \int\limits _ { s } ^ { t } \int\limits _ { s } ^ { {t _ n} } \dots \int\limits _ { s } ^ { {t } _ {2} } A ( t _ {n} ) \dots A ( t _ {1} ) d t _ {1} \dots d t _ {n} $$

is the evolution operator of the equation $ \dot{u} = A ( t) u $. The solution of the Cauchy problem for equation (2) is determined by the formula

$$ u ( t) = U ( t , s ) u _ {0} + \int\limits _ { s } ^ { t } U ( t , \tau ) f ( \tau ) d \tau . $$

From (4) one obtains the estimate

$$ \tag{5 } \| U ( t , s ) \| \leq \mathop{\rm exp} \left \{ \int\limits _ { s } ^ { t } \| A ( \tau ) \| d \tau \right \} ; $$

a refinement of it is:

$$ \tag{5'} \| U ( t , s ) \| \leq \mathop{\rm exp} \left \{ \int\limits _ { s } ^ { t } r _ {A} ( \tau ) d \tau \right \} , $$

where $ r _ {A} ( \tau ) $ is the spectral radius of the operator $ A ( \tau ) $. The evolution operator has the properties

$$ U ( s , s ) = I ,\ U ( t , \tau ) U ( \tau , s ) = U ( t , s), $$

$$ U ( t , \tau ) = [ U ( \tau , t ) ] ^ {-} 1 . $$

In the study of (2) the main attention has been focused on the behaviour of its solutions at infinity, in dependence on the behaviour of $ A ( t) $ and $ f ( t) $. An important characteristic of the equation is the general (or singular) exponent

$$ \kappa = \overline{\lim\limits}\; _ {\tau , s \rightarrow \infty } \frac{1} \tau \mathop{\rm ln} \| U ( \tau + s , s ) \| . $$

Equations with periodic and almost-periodic coefficients have been studied in detail (see Qualitative theory of differential equations in Banach spaces).

Equation (2) can also be considered in the complex plane. If the functions $ A ( t) $ and $ f ( t) $ are holomorphic in a simply-connected domain containing the point $ s $, then the formulas (3), (4), (5), (5'}) remain valid if the integrals are understood to be integrals over a rectifiable arc joining $ s $ and $ t $.

A number of other questions arises in the case when the original linear equation is not solvable for the derivative. If the operator $ A _ {0} ( t) $ is boundedly invertible everywhere except at one point, say $ t = 0 $, then in the space $ E $ the equation reduces to the form

$$ \tag{6 } a ( t) \dot{u} = A ( t) u + f ( t) , $$

where $ a ( t) $ is a scalar function and $ a ( 0) = 0 $. Here the main attention is focused on the study of the behaviour of the solutions in a neighbourhood of the origin, and the analytic and non-analytic cases are distinguished.

The analytic case.

For the simplest equation

$$ t \dot{u} = A u $$

with a constant operator $ A $, the evolution operator $ U ( t) = U ( t , 0 ) $ has the form

$$ U ( t) = e ^ {A \mathop{\rm ln} t } , $$

and the solutions are not single-valued: as one goes round the origin in the positive direction they are multiplied by the operator $ e ^ {2 \pi i A } $.

Consider an equation with a regular singularity

$$ \tag{7 } t \dot{u} = \left ( \sum _ { k= } 0 ^ \infty A ^ {(} k) t ^ {k} \right ) u , $$

where the series on the right-hand side converges in a neighbourhood of the origin. If one looks for the operator $ U ( t) $ in the form of a series

$$ U ( t) = \left ( \sum _ { k= } 0 ^ \infty U ^ {(} k) t ^ {k} \right ) e ^ {A ^ {(} 0) \mathop{\rm ln} t } , $$

then for the determination of the coefficients $ U ^ {(} k) $ one obtains the system of equations

$$ A ^ {(} 0) U ^ {(} 0) - U ^ {(} 0) A ^ {(} 0) = 0 , $$

$$ ( A ^ {(} 0) - kI ) U ^ {(} k) - U ^ {(} k) A ^ {(} 0) = - \sum _ { j= } 1 ^ { k } A ^ {(} j) u ^ {( k- j ) } ,\ k = 1 , 2 , . . . . $$

For this system to be solvable, that is, for (7) to be formally solvable, it is sufficient that the spectra of the operators $ A ^ {(} 0) $ and $ A ^ {(} 0) - kI $ do not intersect (cf. Spectrum of an operator), or, equivalently, that there are no points differing by an integer in the spectrum of $ A ^ {(} 0) $. Under this condition the series

$$ \sum _ { k= } 0 ^ \infty U ^ {( k) } {t ^ {k} } $$

converges in the same neighbourhood of zero as the series for $ A ( t) $. Now, if there are finitely many integers representable as differences of points of the spectrum of $ A ^ {(} 0) $, and each of them is an isolated point of the spectrum of the transformer

$$ \mathfrak A X = A ^ {(} 0) X - X A ^ {(} 0) , $$

then there is a solution of the form

$$ U ( t) = \left ( I + \sum _ { k= } 1 ^ \infty U _ {k} ( \mathop{\rm ln} t ) t ^ {k} \right ) e ^ {A ^ {(} 0) \mathop{\rm ln} t } ,\ 0 < | t | < \rho , $$

where the $ U _ {k} $ are entire functions of the argument $ \mathop{\rm ln} t $, satisfying for every $ \epsilon > 0 $ the condition

$$ \| U _ {k} ( \mathop{\rm ln} t ) \| \leq C _ \epsilon e ^ {\epsilon | \mathop{\rm ln} t | } . $$

If the integer points of the spectrum of the transformer $ \mathfrak A $ are poles of its resolvent, then the functions $ U _ {k} $ are polynomials.

In the case of an irregular singularity, the differential equation

$$ t ^ {m} \dot{u} = \left ( \sum _ { k= } 0 ^ { m- } 1 A ^ {(} k) t ^ {k} \right ) u $$

has been considered in a Banach algebra $ \mathfrak B $( for example, in the algebra of bounded operators on a Banach space $ E $). Under certain restrictions on $ A ^ {(} 0) $ it reduces by means of Laplace integrals to an equation with a regular singularity $ ( m = 1 ) $ in the algebra of matrices with entries from $ \mathfrak B $.

The non-analytic case.

Suppose that in the equation

$$ t ^ {n} \dot{u} = A ( t) u + f ( t) ,\ 0 \leq t \leq T , $$

the functions $ A ( t) $ and $ f ( t) $ are infinitely differentiable. In the finite-dimensional case a complete result has been obtained: If the equation has a formal solution in the form of a power series, then it has a solution that is infinitely differentiable on $ [ 0 , T ] $ for which the formal series is the Taylor series at the point $ t = 0 $. In the infinite-dimensional case there is only a number of sufficient conditions for the existence of infinitely-differentiable solutions.

Suppose that $ m > 1 $. If the spectrum of the operator $ A ( 0) $ does not intersect the imaginary axis, then there is a family of infinitely-differentiable solutions that depends on an arbitrary element $ g ^ {-} $ belonging to the invariant subspace of $ A ( 0) $ corresponding to the part of the spectrum of $ A ( 0) $ lying in the left half-plane. Any solution that is continuous on $ [ 0 , T] $ appears in this family. If the whole spectrum of $ A ( 0) $ lies in the left half-plane, then there is only one infinitely-differentiable solution.

Suppose that $ m = 1 $. If there are no negative integers in the spectrum of $ A ( 0) $, then there is a unique infinitely-differentiable solution. Under similar assumptions about the operator $ A ( 0) $, equations of the form (6) have been considered in which $ a ( t) $ and $ f ( t) $ have finite smoothness, and the solutions have the same smoothness.

A rather different picture emerges when the differential equation is unsolvable for the derivative for all $ t $, for example when $ A $ is a constant non-invertible operator. Suppose that in the equation

$$ \tag{8 } A \dot{u} = B u $$

the operators $ A $ and $ B $ are bounded in the space $ E $ and $ A $ is a non-invertible Fredholm operator. Suppose that the operator $ A + \epsilon B $ is continuously invertible for sufficiently small $ \epsilon $. Then there are decompositions into direct sums $ E= N ^ {(} 1) + M ^ {(} 1) $ and $ E = N ^ {(} 2) + M ^ {(} 2) $ such that $ A $ and $ B $ map $ N ^ {(} 1) $ into $ N ^ {(} 2) $ and $ M ^ {(} 1) $ into $ M ^ {(} 2) $. The operator $ A $ is invertible on $ M ^ {(} 1) $ and maps onto $ M ^ {(} 2) $. The subspace $ N ^ {(} 1) $ is finite-dimensional. All solutions of (8) lie in the subspace $ M ^ {(} 1) $ and have the form $ \mathop{\rm exp} ( \widetilde{A} {} ^ {-} 1 Bt ) u _ {0} $, where $ \widetilde{A} $ is the restriction of $ A $ to $ M ^ {(} 1) $ and $ u _ {0} \in M ^ {(} 1) $. For an inhomogeneous equation $ A \dot{u} = Bu + f ( t) $, a solution exists only if $ f ( t) $ has a certain smoothness and under certain compatibility conditions for the values of $ f( t) $ and its derivatives with the initial data. The number of derivatives that certain components of $ f ( t) $ must have and the number of compatibility conditions are equal to the maximal length of $ B $- adjoint chains of the operator $ A $. If these conditions are satisfied, the solution of the Cauchy problem is unique.

If the operator $ A + \epsilon B $ is non-invertible for all $ \epsilon $, then all solutions of (8) lie in a subspace that has, generally speaking, infinite deficiency (cf. also Deficiency subspace). The solution of the Cauchy problem for it is not unique. For the function $ f( t) $ in the inhomogeneous equation infinitely many differentiability conditions and compatibility conditions are required.

2. Linear differential equations with an unbounded operator.

Suppose that $ A _ {0} ( t) $ is invertible for every $ t $, so that (1) can be solved for the derivative and takes the form

$$ \tag{9 } \dot{u} = A ( t) u + f ( t) , $$

and suppose that here $ A ( t) $ is an unbounded operator in a space $ E $, with dense domain of definition $ D ( A ( t) ) $ in $ E $ and with non-empty resolvent set, and suppose that $ f ( t) $ is a given function and $ u ( t) $ an unknown function, both with values in $ E $.

Even for the simplest equation $ \dot{u} = Au $ with an unbounded operator, solutions of the Cauchy problem $ u ( 0) = u _ {0} $ need not exist, they may be non-unique, and they may be non-extendable to the whole semi-axis, so the main investigations are devoted to the questions of existence and uniqueness of the solutions. A solution of the equation $ \dot{u} = Au $ on the interval $ [ 0, T ] $ is understood to be a function that takes values in $ D ( A) $, is differentiable on $ [ 0, T ] $ and satisfies the equation. Sometimes this definition is too rigid and one introduces the concept of a weak solution as a function that has the same properties on $ ( 0 , T ] $ and is only continuous at $ 0 $.

Suppose that the operator $ A $ has a resolvent

$$ R ( \lambda , A ) = ( A - \lambda I ) ^ {-} 1 $$

for all sufficiently large positive $ \lambda $ and that

$$ \overline{\lim\limits}\; _ {\lambda \rightarrow \infty } \lambda ^ {-} 1 \mathop{\rm ln} \| R ( \lambda , A ) \| = h < T . $$

Then the weak solution of the problem

$$ \tag{10 } \dot{u} = Au ,\ u ( 0) = u _ {0} $$

is unique on $ [ 0 , T - h ] $ and can be branched for $ t = T - h $. If $ h = 0 $, then the solution is unique on the whole semi-axis. This assertion is precise as regards the behaviour of $ R ( \lambda , A ) $ as $ \lambda \rightarrow \infty $.

If for every $ u _ {0} \in D ( A) $ there is a unique solution of the problem (10) that is continuously differentiable on $ [ 0 , T ] $, then this solution can be extended to the whole semi-axis and can be represented in the form $ u ( t) = U ( t) u _ {0} $, where $ U ( t) $ is a strongly-continuous semi-group of bounded operators on $ [ 0 , \infty ) $, $ U ( 0) = I $, for which the estimate $ \| U ( t) \| \leq M e ^ {\omega t } $ holds. For the equation to have this property it is necessary and sufficient that

$$ \tag{11 } \| ( \lambda - \omega ) ^ {m} R ^ {m} ( \lambda , A ) \| \leq M $$

for all $ \lambda > \omega $ and $ m = 1 , 2 \dots $ where $ M $ does not depend on $ \lambda $ and $ m $. These conditions are difficult to verify. They are satisfied if $ \| ( \lambda - \omega ) R ( \lambda , A ) \| \leq 1 $, and then $ \| U ( t) \| \leq e ^ {\omega t } $. If $ \omega = 0 $, then $ U ( t) $ is a contraction semi-group. This is so if and only if $ A $ is a maximal dissipative operator. If $ u _ {0} \notin D ( A) $, then the function $ U ( t) u _ {0} $ is not differentiable (in any case for $ t = 0 $); it is often called the generalized solution of (10). Solutions of the equation $ \dot{u} = Au $ can be constructed as the limit, as $ n \rightarrow \infty $, of solutions of the equation $ \dot{u} = A _ {n} u $ with bounded operators, under the same initial conditions. For this it is sufficient that the operators $ A _ {n} $ commute, converge strongly to $ A $ on $ D ( A) $ and that

$$ \| e ^ {t A _ {n} } \| \leq M e ^ {\omega t } . $$

If the conditions (11) are satisfied, then the operators $ A _ {n} = - nI - n ^ {2} R ( \lambda , A ) $( Yosida operators) have these properties.

Another method for constructing solutions of the equation $ \dot{u} = A u $ is based on Laplace transformation. If the resolvent of $ A $ is defined on some contour $ \Gamma $, then the function

$$ \tag{12 } u ( t) = - \frac{1}{2 \pi i } \int\limits _ \Gamma e ^ {\lambda t } R ( \lambda , A ) u _ {0} d \lambda $$

formally satisfies the equation

$$ \dot{u} = A u + \frac{1}{2 \pi i } \int\limits _ \Gamma e ^ {\lambda t } \ d \lambda u _ {0} . $$

If the convergence of the integrals, the validity of differentiation under the integral sign and the vanishing of the last integral are ensured, then $ u ( t) $ satisfies the equation. The difficulty lies in the fact that the norm of the resolvent cannot decrease faster than $ | \lambda | ^ {-} 1 $ at infinity. However, on some elements it does decrease faster. For example, if $ R ( \lambda , A ) $ is defined for $ \mathop{\rm Re} \lambda \geq \alpha $ and if

$$ \| R ( \lambda , A ) \| \leq M | \lambda | ^ {k} ,\ k \geq - 1 , $$

for sufficiently large $ | \lambda | $, then for $ \Gamma = ( - i \infty , i \infty ) $ formula (12) gives a solution for any $ u _ {0} \in D ( A ^ {[ k ] + 3 } ) $. In a "less good" case, when the previous inequality is satisfied only in the domain

$$ \mathop{\rm Re} \lambda \geq \alpha | \mathop{\rm Im} \lambda | ^ {a} ,\ 0 < a < 1 $$

(weakly hyperbolic equations), and $ \Gamma $ is the boundary of this domain, one obtains a solution only for an $ u _ {0} $ belonging to the intersection of the domains of definition of all powers of $ A $, with definite behaviour of $ \| A ^ {n} u _ {0} \| $ as $ n \rightarrow \infty $.

Significantly weaker solutions are obtained in the case when $ \Gamma $ goes into the left half-plane, and one can use the decrease of the function $ | e ^ {\lambda t } | $ on it. As a rule, the solutions have increased smoothness for $ t > 0 $. If the resolvent is bounded on the contour $ \Gamma $: $ \mathop{\rm Re} \lambda = - \psi ( | \mathop{\rm Im} \lambda | ) $, where $ \psi ( \tau ) $ is a smooth non-decreasing concave function that increases like $ \mathop{\rm ln} \tau $ at $ \infty $, then for any $ u _ {0} \in E $ the function (12) is differentiable and satisfies the equation, beginning with some $ t _ {0} $; as $ t $ increases further, its smoothness increases. If $ \psi ( \tau ) $ increases like a power of $ \tau $ with exponent less than one, then the function (12) is infinitely differentiable for $ t > 0 $; if $ \psi ( \tau ) $ increases like $ \tau / \mathop{\rm ln} \tau $, then $ u ( t) $ belongs to a quasi-analytic class of functions; if it increases like a linear function, then $ u ( t) $ is analytic. In all these cases it satisfies the equation $ \dot{u} = A u $.

The existence of the resolvent on contours that go into the left half-plane may be obtained, by using series expansion, from the corresponding estimates on vertical lines. If for $ \mathop{\rm Re} \lambda \geq \gamma $,

$$ \tag{13 } \| R ( \lambda , A ) \| \leq M ( 1 + | \mathop{\rm Im} \lambda | ) ^ {- \beta } ,\ 0 < \beta < 1 , $$

then for every $ u _ {0} \in D ( A) $ there is a solution of problem (10). All these solutions are infinitely differentiable for $ t > 0 $. They can be represented in the form $ u ( t) = U ( t) u _ {0} $, where $ U ( t) $ is an infinitely-differentiable semi-group for $ t > 0 $ having, generally speaking, a singularity at $ t = 0 $. For its derivatives one has the estimates

$$ \| U ( k) ( t) \| \leq M _ {k} t ^ {1 - ( k+ 1 ) / \beta } e ^ { \omega t } . $$

If the estimate (13) is satisfied for $ \beta = 1 $, then all generalized solutions of the equation $ \dot{u} = Au $ are analytic in some sector containing the positive semi-axis.

The equation $ \dot{u} = Au $ is called an abstract parabolic equation if there is a unique weak solution on $ [ 0 , \infty ] $ satisfying the initial condition $ u ( 0) = u _ {0} $ for any $ u _ {0} \in E $. If

$$ \tag{14 } \| R ( \lambda , A ) \| \leq M | \lambda - \omega | ^ {-} 1 \ \ \textrm{ for } \mathop{\rm Re} \lambda > \omega , $$

then the equation is an abstract parabolic equation. All its generalized solutions are analytic in some sector containing the positive semi-axis, and

$$ \| \dot{u} ( t) \| \leq t ^ {-} 1 C e ^ {\omega t } \| u _ {0} \| , $$

where $ C $ does not depend on $ u _ {0} $. Conversely, if the equation has the listed properties, then (14) is satisfied for the operator $ A $.

If problem (10) has a unique weak solution for any $ u _ {0} \in D ( A) $ for which the derivative is integrable on every finite interval, then these solutions can be represented in the form $ u ( t) = U ( t) u _ {0} $, where $ U ( t) $ is a strongly-continuous semi-group on $ ( 0 , \infty ) $, and every weak solution of the inhomogeneous equation $ \dot{v} = Av + f ( t) $ with initial condition $ v ( 0) = 0 $ can be represented in the form

$$ \tag{15 } v ( t) = \int\limits _ { 0 } ^ { t } U ( t- s ) f ( s) ds . $$

The function $ v ( t) $ is defined for any continuous $ f ( t) $, hence it is called a generalized solution of the inhomogeneous equation. To ensure that it is differentiable, one imposes smoothness conditions on $ f ( t) $, and the "worse" the semi-group $ U ( t) $, the "higher" these should be. Thus, under the previous conditions, (15) is a weak solution of the inhomogeneous equation if $ f ( t) $ is twice continuously differentiable; if (11) is satisfied, then (15) is a solution if $ f ( t) $ is continuously differentiable; if (13) is satisfied with $ \beta > 2/3 $, then $ v ( t) $ is a weak solution if $ f ( t) $ satisfies a Hölder condition with exponent $ \gamma > 2 ( 1 - 1/ \beta ) $. Instead of smoothness of $ f ( t) $ with respect to $ t $ one can require that the values of $ f ( t) $ belong to the domain of definition of the corresponding power of $ A $.

For an equation with variable operator

$$ \tag{16 } \dot{u} = A ( t) u ,\ \ 0 \leq t \leq T , $$

there are some fundamental existence and uniqueness theorems about solutions (weak solutions) of the Cauchy problem $ u ( s) = u _ {0} $ on the interval $ s \leq t \leq T $. If the domain of definition of $ A ( t) $ does not depend on $ t $,

$$ D ( A ( t) ) \equiv D ( A), $$

if the operator $ A ( t) $ is strongly continuous with respect to $ t $ on $ D ( A) $ and if

$$ \| \lambda R ( \lambda , A ( t) ) \| \leq 1 $$

for $ \lambda > 0 $, then the solution of the Cauchy problem is unique. Moreover, if $ A ( t) $ is strongly continuously differentiable on $ D ( A) $, then for every $ u _ {0} \in D ( A) $ a solution exists and can be represented in the form

$$ u ( t) = U ( t , s ) u _ {0} , $$

where $ U ( t , s ) $ is an evolution operator with the following properties:

1) $ U ( t , s ) $ is strongly continuous in the triangle $ T _ \Delta $: $ 0 \leq s \leq t \leq T $;

2) $ U ( t , s ) = U ( t , \tau ) U ( \tau , s ) $, $ 0 \leq s \leq \tau \leq t \leq T $, $ U ( s , s ) = I $;

3) $ U ( t , s ) $ maps $ D ( A) $ into itself and the operator

$$ A ( t) U ( t , s ) A ^ {-} 1 ( s) $$

is bounded and strongly continuous in $ T _ \Delta $;

4) on $ D ( A) $ the operator $ U ( t , s ) $ is strongly differentiable with respect to $ t $ and $ s $ and

$$ \frac{\partial U }{\partial t } = A ( t) U ,\ \ \frac{\partial U }{\partial s } = - U A ( s) . $$

The construction of the operator $ U ( t , s ) $ is carried out by approximating $ A ( t) $ by bounded operators $ A _ {n} ( t) $ and replacing the latter by piecewise-constant operators.

In many important problems the previous conditions on the operator $ A ( t) $ are not satisfied. Suppose that for the operator $ A ( t) $ there are constants $ M $ and $ \omega $ such that

$$ \| R ( \lambda , A ( t _ {k} ) ) \dots R ( \lambda , A ( t _ {1} ) ) \| \leq M ( \lambda - \omega ) ^ {-} k $$

for all $ \lambda > \omega $, $ 0 \leq t _ {1} \leq \dots \leq t _ {k} \leq T $, $ k = 1 , 2, . . . $. Suppose that in $ E $ there is densely imbedded a Banach space $ F $ contained in all the $ D ( A ( t) ) $ and having the following properties: a) the operator $ A ( t) $ acts boundedly from $ F $ to $ E $ and is continuous with respect to $ t $ in the norm as a bounded operator from $ F $ to $ E $; and b) there is an isomorphism $ S $ of $ F $ onto $ E $ such that

$$ S A ( t) S ^ {-} 1 = A ( t) + B ( t) , $$

where $ B ( t) $ is an operator function that is bounded in $ E $ and strongly measurable, and for which $ \| B ( t) \| $ is integrable on $ [ 0 , T ] $. Then there is an evolution operator $ U ( t , s ) $ having the properties: 1); 2); 3') $ U ( t , s ) F \subset F $ and $ U ( t , s ) $ is strongly continuous in $ F $ on $ T _ \Delta $; and 4') on $ F $ the operator $ U ( t , s ) $ is strongly differentiable in the sense of the norm of $ E $ and $ \partial U / \partial t = A ( t) U $, $ \partial U / \partial s = - U A ( s) $. This assertion makes it possible to obtain existence theorems for the fundamental quasi-linear equations of mathematical physics of hyperbolic type.

The method of frozen coefficients is used in the theory of parabolic equations. Suppose that, for every $ t _ {0} \in [ 0 , T ] $, to the equation $ \dot{u} = A ( t _ {0} ) u $ corresponds an operator semi-group $ U _ {A ( t _ {0} ) } ( t) $. The unknown evolution operator formally satisfies the integral equations

$$ U ( t , s ) = U _ {A ( t) } ( t - s ) + $$

$$ + \int\limits _ { s } ^ { t } U _ {A ( t) } ( t - s ) [ A ( \tau ) - A ( t) ] U ( \tau , s ) d \tau , $$

$$ U ( t , s ) = U _ {A ( s) } ( t - s ) + $$

$$ + \int\limits _ { s } ^ { t } U ( t , \tau ) [ A ( \tau ) - A ( s) ] U _ {A ( s) } ( \tau - s ) d \tau . $$

When the kernels of these equations have weak singularities, one can prove that the equation has solutions and also that $ U ( t , s ) $ is an evolution operator. The following statement has the most applications: If

$$ D ( A ( t) ) \equiv D ( A) ,\ \ \| R ( \lambda , A ( t) ) \| < \ M ( 1 + | \lambda | ) ^ {-} 1 $$

for $ \mathop{\rm Re} \lambda \geq 0 $ and

$$ \| [ A ( t) - A ( s) ] A ^ {-} 1 ( 0) \| \leq C | t - s | ^ \rho $$

(a Hölder condition), then there is an evolution operator $ U ( t , s ) $ that gives a weak solution $ U ( t , s ) u _ {0} $ of the Cauchy problem for every $ u _ {0} \in E $. Uniqueness of the solution holds under the single condition that the operator $ A ( t) A ^ {-} 1 ( 0) $ is continuous (in a Hilbert space). An existence theorem similar to the one given above holds for the operator $ A ( t) $ with a condition of type (13) and for a certain relation between $ \beta $ and $ \rho $.

The assumption that $ D ( A ( t) ) $ is constant does not make it possible in applications to consider boundary value problems with boundary conditions depending on $ t $. Suppose that

$$ \| R ( \lambda , A ( t) ) \| \leq M ( 1 + | \lambda | ) ^ {-} 1 ,\ \ \mathop{\rm Re} \lambda > 0 ; $$

$$ \left \| \frac{d A ^ {-} 1 ( t) }{dt} - \frac{d A ^ {-} 1 ( s) }{ds} \right \| \leq K | t - s | ^ \alpha ,\ 0 < \alpha < 1 ; $$

$$ \left \| \frac \partial {\partial t } R ( \lambda , A ( t) ) \right \| \leq N | \lambda | ^ {\rho - 1 } ,\ 0 \leq \rho \leq 1 , $$

in the sector $ | \mathop{\rm arg} \lambda | \leq \pi - \phi $, $ \phi < \pi / 2 $; then there is an evolution operator $ U ( t , s ) $. Here it is not assumed that $ D ( A ( t) ) $ is constant. There is a version of the last statement adapted to the consideration of parabolic problems in non-cylindrical domains, in which $ D ( A ( t) ) $ for every $ t $ lies in some subspace $ E ( t) $ of $ E $.

The operator $ U ( t , s ) $ for equation (16) formally satisfies the integral equation

$$ \tag{17 } U ( t , s ) = I + \int\limits _ { s } ^ { t } A ( \tau ) U ( \tau , s ) d \tau . $$

Since $ A ( t) $ is unbounded, this equation cannot be solved by the method of successive approximation (cf. Sequential approximation, method of). Suppose that there is a family of Banach spaces $ E _ \alpha $, $ 0 \leq \alpha \leq 1 $, having the property that $ E _ \beta \subset E _ \alpha $ and $ \| x \| _ \alpha \leq \| x \| _ \beta $ for $ \alpha < \beta $. Suppose that $ A ( t) $ is bounded as an operator from $ E _ \beta $ to $ E _ \alpha $:

$$ \| A ( t) \| _ {E _ \beta \rightarrow E _ \alpha } \leq C ( \beta - \alpha ) ^ {-} 1 , $$

and that $ A ( t) $ is continuous with respect to $ t $ in the norm of the space of bounded operators from $ E _ \beta $ to $ E _ \alpha $. Then in this space the method of successive approximation for equation (17) will converge for $ | t - s | \leq ( \beta - \alpha ) ( Ce ) ^ {-} 1 $. In this way one can locally construct an operator $ U ( t , s ) $ as a bounded operator from $ E _ \beta $ to $ E _ \alpha $. In applications this approach gives theorems of Cauchy–Kovalevskaya type (cf. Cauchy–Kovalevskaya theorem).

For the inhomogeneous equation (9) with known evolution operator, for the equation $ \dot{u} = A ( t) u $ the solution of the Cauchy problem is formally written in the form

$$ u ( t) = U ( t , s ) u _ {0} + \int\limits _ { s } ^ { t } U ( t , \tau ) f ( \tau ) d \tau . $$

This formula can be justified in various cases under certain smoothness conditions on $ f ( t) $.

References

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How to Cite This Entry:
Linear differential equation in a Banach space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_differential_equation_in_a_Banach_space&oldid=47651
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article