# Linear differential equation in a Banach space

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 $f ( t)$.

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