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 .

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