Qualitative theory of differential equations in Banach spaces
A branch of functional analysis in which one studies the behaviour on the real axis or on the positive (or negative) semi-axis
(or
) of the solution of the evolution equation in a Banach space. Consider the equations
.
;
.
;
.
, where
is the required function and
the given function with values in a complex Banach space
;
is a linear operator and
is a non-linear operator on
. By the derivative
one means the limit in the norm of
of
as
.
Uniform stability holds for the equation if there exists a constant
such that for any solution
,
, and exponential stability if for any solution
for some
and
.
For equation with a constant bounded operator
, the solution of the Cauchy problem (
given) has the form
. The estimate
holds, where
is a number greater than the real parts of all points of the spectrum of
. Thus, for exponential stability it is necessary and sufficient that the spectrum of
lies in the interior of the left half-plane. In a Hilbert space this holds if and only if there exists a positive-definite form
for which
for every solution of the equation (Lyapunov's theorem). If the spectrum of
is distributed on both sides of the imaginary axis and does not intersect it, then
can be decomposed into a direct sum of subspaces
and
which are invariant with respect to
, and all the solutions are exponentially increasing (decreasing) in
(
) as
. In this case, exponential dichotomy holds for the equation.
If is closed, unbounded and has a dense domain in
, then the Cauchy problem with
is not, in general, well-posed. The existence and properties of the solutions are not determined merely by the distribution of the spectrum of
; the behaviour of its resolvent
must also be specified. Commonly used conditions ensuring the uniform well-posedness of the Cauchy problem on
are provided by the inequalities
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a sufficient condition for their fulfillment being the Hille–Yosida condition
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or the inequality
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When these conditions hold, the solution of the Cauchy problem has the form , where
is a strongly-continuous semi-group of operators for
and
. For uniform (or exponential) stability it suffices that
(respectively,
). If
generates a strongly-continuous semi-group on a Hilbert space, then the sufficiency part of Lyapunov's theorem holds for it, while if it generates a group, then the necessity part holds as well. In a Hilbert space exponential stability is equivalent to
-stability, the property that for all solutions
.
Important for applications is the property of almost-periodicity of the solution or weak almost-periodicity (that is, almost periodicity of the scalar functions
for all
). If all values of an almost-periodic solution lie in a compact set, then the solution is compact. Compactness and weak almost-periodicity imply almost-periodicity. For the equation
, the question of almost-periodicity of the solution is related to the structure of the intersection of the spectrum of
with the imaginary axis. If
is the generating operator of a bounded strongly-continuous semi-group and the above intersection is countable, then for the almost-periodicity of a bounded solution defined on the whole axis it is necessary and sufficient that the limit
exists at each limit point
of the spectrum on the imaginary axis. Furthermore, each uniformly-continuous solution is weakly almost-periodic; it is almost-periodic if it is weakly compact, or if
contains no subspaces isomorphic to the space
of sequences converging to zero, endowed with the max norm. If
is the generating operator of a strongly-continuous semi-group
with the property that the functions
are bounded on
for a set of functionals
which are dense in
, then compactness of a solution implies almost-periodicity.
For equation with a bounded operator
that is continuous in
, the solutions of the Cauchy problem are defined on the whole axis and, by means of the evolution operator
, can be written in the form
. The property of uniform stability is equivalent to the requirement that
; the property of exponential stability is equivalent to
. If
is bounded in integral norm, that is,
, then the general index
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is finite. If , one has exponential stability. For equations with a constant or periodic operator, the formula
holds. It does not hold in the general case. The general index is not altered if
is perturbed by adding a term
for which
as
, or for which the integral
converges at infinity. The size of the general index depends on the behaviour of
at infinity. If the limit
exists and the spectrum of
lies in the interior of the left half-plane, then
. If the operators
,
, form a compact set in the space of bounded operators, if the spectra of all limit operators belong to a half-plane
and if the operator function
has small oscillation, for example, it has the form
for sufficiently small
, or if, for sufficiently large
, it satisfies the Lipschitz condition
with
sufficiently small, then
. In a Hilbert space the condition
is equivalent to the existence of a Hermitian form
such that
and
for every solution
.
If is periodic with period
, i.e. if
, then
. The operator
is called the monodromy operator of equation
. Its spectral radius
is related to the general index by the formula
. The equation has a periodic solution if and only if 1 is an eigen value of
. If the operator
has a logarithm, then the Floquet representation
holds, where
is periodic with period
and
. In particular, the Floquet representation holds if the spectrum of
does not surround the origin; and for this it suffices that
be less than a certain constant, which depends on the geometry of the sphere in
and which is not less than
. For Hilbert space this constant is
. The Floquet representation reduces the question of the behaviour of the solution of the equation to the same question for the equation
with a constant operator
.
Exponential dichotomy holds for equation if for some
the space decomposes into a direct sum of subspaces
and
such that
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when ,
, and
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when ,
. It is supposed here that the subsequences
and
are, in a certain sense, not close to each other. If the general index is finite, this latter requirement holds automatically. For an equation with a periodic
, a necessary and sufficient condition for exponential dichotomy to hold is that the spectrum of the monodromy operator be distributed outside and inside the unit disc without intersecting the unit circle.
For equation with an unbounded operator
whose domain does not depend on
and which satisfies the well-posedness conditions for the Cauchy problem (see above) for every
, the existence of an evolution operator
, defined and strongly continuous in
and
for
, has been proved, under supplementary smoothness conditions. This enables one to carry over to this case a number of the ideas and results described above. However, some difficulties are encountered. E.g., the Floquet representation can be obtained only in a Hilbert space when
, where
is a negative-definite self-adjoint operator and
is a bounded periodic operator satisfying certain extra conditions.
Suppose that is periodic and that the Cauchy problem for equation
is uniformly well-posed. If the intersection of the spectrum of the monodromy operator
with the unit circle is countable, then each bounded uniformly-continuous solution on
is weakly almost-periodic. It is almost-periodic in the case of weak compactness or if
does not contain
. A reflexive space
admits a direct sum decomposition
such that
and
are invariant with respect to
and all the solutions starting in
are almost-periodic, while those starting in
are in a certain sense decreasing: For
,
,
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For the solution of the non-homogeneous equation , the following formula holds:
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For an equation with a bounded operator, this equation is equivalent to the differential equation. In the case of an unbounded operator, this is generally not so, but then this equation is taken as the definition of the (generalized) solution. The fundamental problem for equation is to investigate the properties of the solutions under prescribed properties of the right-hand side. These properties are usually described in terms of the function
belonging to some Banach space of functions on
or
with values in
. If corresponding to each bounded continuous function
there is at least one bounded solution, then the operator
is called weakly regular. If corresponding to each
there is a unique solution
, then
is called regular. For bounded constant
, weak regularity implies regularity. This assertion is no longer true for unbounded
or for bounded periodic
, even in a Hilbert space. If the general index of the equation
is finite, then exponential dichotomy for this equation is equivalent to regularity of
on
. For exponential dichotomy to hold on
it is necessary and sufficient that
be weakly regular on
and that the set of those initial values
to which the bounded solutions of equation
correspond be a complemented subspace of
. If for all solutions of
the inequality
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holds, and if for the solution of the formal adjoint equation the inequality
holds, then the operators
and
are regular. It is not known (1990) whether regularity is preserved on replacing the right-hand side of the first inequality by
. For the regularity of
and
both, a priori estimates are necessary.
If is periodic, then a necessary and sufficient condition for the existence of periodic solutions for each periodic
is that the mapping
be surjective, while for such a solution to be unique, it is necessary and sufficient that the operator
be invertible.
The verification of regularity under known conditions can be reduced to the verification of the regularity of operators with constant coefficients. In the case when is strongly oscillating (for example
with
large), under the assumption that the mean
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exists uniformly with respect to , the operator
is regular if and only if the operator
is regular.
For almost-periodic solutions the specifics of an infinite-dimensional space are already encountered in a generalization of the well-known Bohl–Bohr theorem on the almost-periodicity of a bounded integral of an almost-periodic function, that is, on the almost-periodicity of a solution of the simplest differential equation . If the indefinite integral of an almost-periodic function with values in
is bounded in
and
does not contain
, then the function is almost-periodic. The Bohl–Bohr theorem fails in
. If
is periodic,
is almost-periodic and the intersection of the spectrum of the monodromy operator
with the unit circle is countable, then the same conclusions on the almost-periodicity of solutions stated earlier for the homogeneous equation hold for this case. If
is almost-periodic (as a function with values in the space of bounded operators on
), then in order that for each almost-periodic function
there exists a unique almost-periodic solution, it is necessary and sufficient that
be regular. If equation
has a (weakly) compact solution for
, and the non-trivial (weakly) compact solutions of the homogeneous equation
have the property that
, then equation
has a (weakly) almost-periodic solution.
In the qualitative study of a non-linear equation , it is usually supposed in advance that conditions hold which ensure that solutions exist on
or on
; this imposes an essential constraint on the form of non-linearity of
. A solution
on
is said to be uniformly stable if there exists for each
a
such that for every other solution
the inequality
,
, implies that
is defined for
and that
. A solution is called asymptotically stable if it is uniformly stable and if for some
the inequality
implies that
. If one makes the substitution
in the equation, then after linearization (if this is possible) it takes the form
, where
, and the non-linearity has the property that
. Thus the problem of uniform (asymptotic) stability of the solution
reduces to that of the uniform (or asymptotic) stability of the zero solution of
. In this connection, the equation
is called the linearization equation (or variational equation) of the original equation with respect to the solution
.
If the non-linear part is sufficiently small, then the properties of the solutions are determined by those of the linearized equation. If the general index of the linearized equation is negative and the inequality
,
,
, holds, then for sufficiently small
the zero solution of
is asymptotically stable. If
and the spectrum of
does not intersect the imaginary axis and has a point in the right half-plane, then for sufficiently small
the zero solution is unstable. If the inequality
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holds, then the requirement that there be no points of the spectrum on the imaginary axis can be deleted.
If exponential dichotomy holds on for the linearized equation, then for every
there exist
and
depending on
and
such that the inequalities
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and
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imply the existence, in some neighbourhood of the origin, of two manifolds and
, intersecting at the single point
, such that the solution
starting at
is bounded on the entire axis and
; solutions starting on
(
) approach
exponentially as
(
) and move away from it as
goes to infinity in the opposite direction. Furthermore,
(
) is homeomorphic to the neighbourhood of the space
(
) involved in the definition of exponential dichotomy. If under the above conditions
and
is almost-periodic in
for each
with
, then the solution
is almost-periodic.
Suppose that the autonomous equation has an
-periodic solution
; then the linearized equation
has a single periodic solution
and, consequently, its monodromy operator
has 1 as eigen value. If this eigen value is simple and the remaining spectrum of
lies in the interior of the unit disc and does not surround the origin, then there exist numbers
,
and
such that
for all other solutions of the original equation. This property is called asymptotic orbital stability of the periodic solution. Studies have also been made of other stable invariant manifolds for non-linear equations.
Equations of type with an unbounded operator
correspond in applications to quasi-linear equations of parabolic or hyperbolic type. A theory of "properly" non-linear equations of a specific type has been developed. Thus, if
is a continuous everywhere-defined dissipative operator, then the Cauchy problem
,
, is uniquely solvable on the semi-axis
for any
. The definition of a dissipative operator carries over in a natural fashion to the case of a many-valued operator; for such operators one considers a differential inclusion
rather than a differential equation. If on a reflexive space
, the operator
is closed and dissipative and if the values of
cover the entire space
, then the Cauchy problem is uniquely solvable on
for each
in the domain of
. There are many variants of the latter statement. The solution is given by the formula
, where
is a semi-group of non-linear operators,
,
. For equations with dissipative operators there are also existence theorems for periodic and almost-periodic solutions.
References
[1] | Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian) MR0352638 |
[2] | H.H. Schaefer, "Linear differential equations and function spaces" , Acad. Press (1966) |
[3] | K. Yosida, "Functional analysis" , Springer (1968) MR0239384 Zbl 0830.46001 Zbl 0435.46002 Zbl 0365.46001 Zbl 0286.46002 Zbl 0217.16001 Zbl 0152.32102 Zbl 0126.11504 |
[4] | S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701 |
[5] | V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian) MR0390843 Zbl 0328.47035 |
[6] | A.G. Baskakov, "On the spectral analysis of isometric representations of locally compact Abelian groups" Trudy Mat. Fak. Vorozhen. Inst. , 10 (1973) pp. 96–101 (In Russian) MR0466403 |
[7] | V.V. Zhikov, "Some admissibility and dichotomy questions. The averaging principle" Math. USSR Izv. , 10 (1976) pp. 1307–1332 Izv. Akad. Nauk SSSR Ser. Mat. , 40 : 6 (1976) pp. 1380–1408 |
[8] | V.V. Zhikov, B.M. Levitan, "Favard theory" Russian Math. Surveys , 32 : 2 (1977) pp. 129–180 Uspekhi Mat. Nauk , 32 : 2 (1977) pp. 123–171 MR470405 Zbl 0235.34095 Zbl 0224.34036 Zbl 0198.48103 |
[9] | A.I. Miloslavskii, "Floquet theory for parabolic equations" Funct. Anal. Appl. , 10 : 2 (1976) pp. 151–153 Funktsional. Anal. i Prilozhen. , 10 : 2 (1976) pp. 80–81 MR0473390 |
[10] | V.M. Tyurin, "Functional analysis" , 1 , Ul'yanovsk (1973) (In Russian) |
[11] | B.M. Levitan, V.V. Zhikov, "Almost-periodic functions and differential equations" , Cambridge Univ. Press (1982) (Translated from Russian) MR0690064 Zbl 0499.43005 |
Comments
For a somewhat different approach to the qualitative theory of abstract differential equations see [a1], [a2].
References
[a1] | J.K. Hale, "Asymptotic behavior of dissipative systems" , Amer. Math. Soc. (1988) MR0941371 Zbl 0642.58013 |
[a2] | J.K. Hale, L.T. Magalhes, W.M. Oliva, "An introduction to infinite dimensional dynamical systems" , Springer (1984) MR0725501 Zbl 0533.58001 |
Qualitative theory of differential equations in Banach spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Qualitative_theory_of_differential_equations_in_Banach_spaces&oldid=48368