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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|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|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|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|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|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|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(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  $  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.==
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|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|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|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

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

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|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|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|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|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====
<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>