Linear differential equation in a Banach space
An equation of the form
![]() | (1) |
where and
, for every
, are linear operators in a Banach space
,
is a given function and
an unknown function, both with values in
; the derivative
is understood to be the limit of the difference quotient with respect to the norm of
.
1. Linear differential equations with a bounded operator.
Suppose that and
, for every
, are bounded operators acting in
. If
has a bounded inverse for every
, then (1) can be solved for the derivative and takes the form
![]() | (2) |
where is a bounded operator in
, and
and
are functions with values in
. If the functions
and
are continuous (or, more generally, are measurable and integrable on every finite interval), then the solution of the Cauchy problem
![]() | (3) |
exists for any and is given by the formula
![]() |
where
![]() | (4) |
![]() |
is the evolution operator of the equation . The solution of the Cauchy problem for equation (2) is determined by the formula
![]() |
From (4) one obtains the estimate
![]() | (5) |
a refinement of it is:
![]() | (5prm) |
where is the spectral radius of the operator
. The evolution operator has the properties
![]() |
![]() |
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 and
. An important characteristic of the equation is the general (or singular) exponent
![]() |
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 and
are holomorphic in a simply-connected domain containing the point
, then the formulas (3), (4), (5), (5prm) remain valid if the integrals are understood to be integrals over a rectifiable arc joining
and
.
A number of other questions arises in the case when the original linear equation is not solvable for the derivative. If the operator is boundedly invertible everywhere except at one point, say
, then in the space
the equation reduces to the form
![]() | (6) |
where is a scalar function and
. 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
![]() |
with a constant operator , the evolution operator
has the form
![]() |
and the solutions are not single-valued: as one goes round the origin in the positive direction they are multiplied by the operator .
Consider an equation with a regular singularity
![]() | (7) |
where the series on the right-hand side converges in a neighbourhood of the origin. If one looks for the operator in the form of a series
![]() |
then for the determination of the coefficients one obtains the system of equations
![]() |
![]() |
For this system to be solvable, that is, for (7) to be formally solvable, it is sufficient that the spectra of the operators and
do not intersect (cf. Spectrum of an operator), or, equivalently, that there are no points differing by an integer in the spectrum of
. Under this condition the series
![]() |
converges in the same neighbourhood of zero as the series for . Now, if there are finitely many integers representable as differences of points of the spectrum of
, and each of them is an isolated point of the spectrum of the transformer
![]() |
then there is a solution of the form
![]() |
where the are entire functions of the argument
, satisfying for every
the condition
![]() |
If the integer points of the spectrum of the transformer are poles of its resolvent, then the functions
are polynomials.
In the case of an irregular singularity, the differential equation
![]() |
has been considered in a Banach algebra (for example, in the algebra of bounded operators on a Banach space
). Under certain restrictions on
it reduces by means of Laplace integrals to an equation with a regular singularity
in the algebra of matrices with entries from
.
The non-analytic case.
Suppose that in the equation
![]() |
the functions and
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
for which the formal series is the Taylor series at the point
. In the infinite-dimensional case there is only a number of sufficient conditions for the existence of infinitely-differentiable solutions.
Suppose that . If the spectrum of the operator
does not intersect the imaginary axis, then there is a family of infinitely-differentiable solutions that depends on an arbitrary element
belonging to the invariant subspace of
corresponding to the part of the spectrum of
lying in the left half-plane. Any solution that is continuous on
appears in this family. If the whole spectrum of
lies in the left half-plane, then there is only one infinitely-differentiable solution.
Suppose that . If there are no negative integers in the spectrum of
, then there is a unique infinitely-differentiable solution. Under similar assumptions about the operator
, equations of the form (6) have been considered in which
and
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 , for example when
is a constant non-invertible operator. Suppose that in the equation
![]() | (8) |
the operators and
are bounded in the space
and
is a non-invertible Fredholm operator. Suppose that the operator
is continuously invertible for sufficiently small
. Then there are decompositions into direct sums
and
such that
and
map
into
and
into
. The operator
is invertible on
and maps onto
. The subspace
is finite-dimensional. All solutions of (8) lie in the subspace
and have the form
, where
is the restriction of
to
and
. For an inhomogeneous equation
, a solution exists only if
has a certain smoothness and under certain compatibility conditions for the values of
and its derivatives with the initial data. The number of derivatives that certain components of
must have and the number of compatibility conditions are equal to the maximal length of
-adjoint chains of the operator
. If these conditions are satisfied, the solution of the Cauchy problem is unique.
If the operator is non-invertible for all
, 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
in the inhomogeneous equation infinitely many differentiability conditions and compatibility conditions are required.
2. Linear differential equations with an unbounded operator.
Suppose that is invertible for every
, so that (1) can be solved for the derivative and takes the form
![]() | (9) |
and suppose that here is an unbounded operator in a space
, with dense domain of definition
in
and with non-empty resolvent set, and suppose that
is a given function and
an unknown function, both with values in
.
Even for the simplest equation with an unbounded operator, solutions of the Cauchy problem
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
on the interval
is understood to be a function that takes values in
, is differentiable on
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
and is only continuous at
.
Suppose that the operator has a resolvent
![]() |
for all sufficiently large positive and that
![]() |
Then the weak solution of the problem
![]() | (10) |
is unique on and can be branched for
. If
, then the solution is unique on the whole semi-axis. This assertion is precise as regards the behaviour of
as
.
If for every there is a unique solution of the problem (10) that is continuously differentiable on
, then this solution can be extended to the whole semi-axis and can be represented in the form
, where
is a strongly-continuous semi-group of bounded operators on
,
, for which the estimate
holds. For the equation to have this property it is necessary and sufficient that
![]() | (11) |
for all and
where
does not depend on
and
. These conditions are difficult to verify. They are satisfied if
, and then
. If
, then
is a contraction semi-group. This is so if and only if
is a maximal dissipative operator. If
, then the function
is not differentiable (in any case for
); it is often called the generalized solution of (10). Solutions of the equation
can be constructed as the limit, as
, of solutions of the equation
with bounded operators, under the same initial conditions. For this it is sufficient that the operators
commute, converge strongly to
on
and that
![]() |
If the conditions (11) are satisfied, then the operators (Yosida operators) have these properties.
Another method for constructing solutions of the equation is based on Laplace transformation. If the resolvent of
is defined on some contour
, then the function
![]() | (12) |
formally satisfies the equation
![]() |
If the convergence of the integrals, the validity of differentiation under the integral sign and the vanishing of the last integral are ensured, then satisfies the equation. The difficulty lies in the fact that the norm of the resolvent cannot decrease faster than
at infinity. However, on some elements it does decrease faster. For example, if
is defined for
and if
![]() |
for sufficiently large , then for
formula (12) gives a solution for any
. In a "less good" case, when the previous inequality is satisfied only in the domain
![]() |
(weakly hyperbolic equations), and is the boundary of this domain, one obtains a solution only for an
belonging to the intersection of the domains of definition of all powers of
, with definite behaviour of
as
.
Significantly weaker solutions are obtained in the case when goes into the left half-plane, and one can use the decrease of the function
on it. As a rule, the solutions have increased smoothness for
. If the resolvent is bounded on the contour
:
, where
is a smooth non-decreasing concave function that increases like
at
, then for any
the function (12) is differentiable and satisfies the equation, beginning with some
; as
increases further, its smoothness increases. If
increases like a power of
with exponent less than one, then the function (12) is infinitely differentiable for
; if
increases like
, then
belongs to a quasi-analytic class of functions; if it increases like a linear function, then
is analytic. In all these cases it satisfies the equation
.
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 ,
![]() | (13) |
then for every there is a solution of problem (10). All these solutions are infinitely differentiable for
. They can be represented in the form
, where
is an infinitely-differentiable semi-group for
having, generally speaking, a singularity at
. For its derivatives one has the estimates
![]() |
If the estimate (13) is satisfied for , then all generalized solutions of the equation
are analytic in some sector containing the positive semi-axis.
The equation is called an abstract parabolic equation if there is a unique weak solution on
satisfying the initial condition
for any
. If
![]() | (14) |
then the equation is an abstract parabolic equation. All its generalized solutions are analytic in some sector containing the positive semi-axis, and
![]() |
where does not depend on
. Conversely, if the equation has the listed properties, then (14) is satisfied for the operator
.
If problem (10) has a unique weak solution for any for which the derivative is integrable on every finite interval, then these solutions can be represented in the form
, where
is a strongly-continuous semi-group on
, and every weak solution of the inhomogeneous equation
with initial condition
can be represented in the form
![]() | (15) |
The function is defined for any continuous
, hence it is called a generalized solution of the inhomogeneous equation. To ensure that it is differentiable, one imposes smoothness conditions on
, and the "worse" the semi-group
, the "higher" these should be. Thus, under the previous conditions, (15) is a weak solution of the inhomogeneous equation if
is twice continuously differentiable; if (11) is satisfied, then (15) is a solution if
is continuously differentiable; if (13) is satisfied with
, then
is a weak solution if
satisfies a Hölder condition with exponent
. Instead of smoothness of
with respect to
one can require that the values of
belong to the domain of definition of the corresponding power of
.
For an equation with variable operator
![]() | (16) |
there are some fundamental existence and uniqueness theorems about solutions (weak solutions) of the Cauchy problem on the interval
. If the domain of definition of
does not depend on
,
![]() |
if the operator is strongly continuous with respect to
on
and if
![]() |
for , then the solution of the Cauchy problem is unique. Moreover, if
is strongly continuously differentiable on
, then for every
a solution exists and can be represented in the form
![]() |
where is an evolution operator with the following properties:
1) is strongly continuous in the triangle
:
;
2) ,
,
;
3) maps
into itself and the operator
![]() |
is bounded and strongly continuous in ;
4) on the operator
is strongly differentiable with respect to
and
and
![]() |
The construction of the operator is carried out by approximating
by bounded operators
and replacing the latter by piecewise-constant operators.
In many important problems the previous conditions on the operator are not satisfied. Suppose that for the operator
there are constants
and
such that
![]() |
for all ,
,
. Suppose that in
there is densely imbedded a Banach space
contained in all the
and having the following properties: a) the operator
acts boundedly from
to
and is continuous with respect to
in the norm as a bounded operator from
to
; and b) there is an isomorphism
of
onto
such that
![]() |
where is an operator function that is bounded in
and strongly measurable, and for which
is integrable on
. Then there is an evolution operator
having the properties: 1); 2); 3')
and
is strongly continuous in
on
; and 4') on
the operator
is strongly differentiable in the sense of the norm of
and
,
. 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 , to the equation
corresponds an operator semi-group
. The unknown evolution operator formally satisfies the integral equations
![]() |
![]() |
![]() |
![]() |
When the kernels of these equations have weak singularities, one can prove that the equation has solutions and also that is an evolution operator. The following statement has the most applications: If
![]() |
for and
![]() |
(a Hölder condition), then there is an evolution operator that gives a weak solution
of the Cauchy problem for every
. Uniqueness of the solution holds under the single condition that the operator
is continuous (in a Hilbert space). An existence theorem similar to the one given above holds for the operator
with a condition of type (13) and for a certain relation between
and
.
The assumption that is constant does not make it possible in applications to consider boundary value problems with boundary conditions depending on
. Suppose that
![]() |
![]() |
![]() |
in the sector ,
; then there is an evolution operator
. Here it is not assumed that
is constant. There is a version of the last statement adapted to the consideration of parabolic problems in non-cylindrical domains, in which
for every
lies in some subspace
of
.
The operator for equation (16) formally satisfies the integral equation
![]() | (17) |
Since 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
,
, having the property that
and
for
. Suppose that
is bounded as an operator from
to
:
![]() |
and that is continuous with respect to
in the norm of the space of bounded operators from
to
. Then in this space the method of successive approximation for equation (17) will converge for
. In this way one can locally construct an operator
as a bounded operator from
to
. 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 the solution of the Cauchy problem is formally written in the form
![]() |
This formula can be justified in various cases under certain smoothness conditions on .
References
[1] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 MR0617913 Zbl 0435.46002 |
[2] | S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701 |
[3] | E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501 |
[4] | , Functional analysis , Math. Reference Library , Moscow (1972) (In Russian) |
[5] | V.P. Glushko, "Degenerate linear differential equations" , Voronezh (1972) (In Russian) Zbl 0265.34011 Zbl 0252.34072 Zbl 0241.34008 Zbl 0235.34011 |
[6] | Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian) MR0352638 |
[7] | 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) MR0470716 |
[8] | 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 Zbl 0259.34005 |
[9] | 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 Zbl 0254.35064 |
[10] | 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 Zbl 0173.12002 |
[11] | 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 Zbl 0144.39003 |
[12] | P.E. Sobolevskii, "Equations of parabolic type in a Banach space" Trudy Moskov. Mat. Obshch. , 10 (1961) pp. 297–350 (In Russian) MR0141900 |
[13] | 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 MR0492648 Zbl 0374.35039 |
[14] | A. Friedman, "Uniqueness of solutions of ordinary differential inequalities in Hilbert space" Arch. Rat. Mech. Anal. , 17 : 5 (1964) pp. 353–357 MR0171181 Zbl 0143.16701 |
[15] | T. Kato, "Linear evolution equations of "hyperbolic" type II" J. Math. Assoc. Japan , 25 : 4 (1973) pp. 648–666 MR0326483 Zbl 0262.34048 |
[16] | F. Trèves, "Basic linear partial differential equations" , Acad. Press (1975) MR0447753 Zbl 0305.35001 |
[17] | J. Miller, "Solution in Banach algebras of differential equations with irregular singular point" Acta Math. , 110 : 3–4 (1963) pp. 209–231 MR0153939 Zbl 0122.35303 |
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