Semi-group of non-linear operators
A one-parameter family of operators ,
, defined and acting on a closed subset
of a Banach space
, with the following properties:
1) for
,
;
2) for any
;
3) for any , the function
(with values in
) is continuous with respect to
on
.
A semi-group is of type
if
![]() |
A semi-group of type is called a contraction semi-group.
As in the case of semi-groups of linear operators (cf. Semi-group of operators), one introduces the concept of the generating operator (or infinitesimal generator) of the semi-group
:
![]() |
for those elements for which the limit exists. If
is a contraction semi-group,
is a dissipative operator. Recall that an operator
on a Banach space
is dissipative if
for
,
. A dissipative operator may be multi-valued, in which case
in the definition stands for any of its values at
. A dissipative operator is said to be
-dissipative if
for
. If
is of type
, then
is dissipative.
The fundamental theorem on the generation of semi-groups: If is a dissipative operator and
contains
for sufficiently small
, then there exists a semi-group
of type
on
such that
![]() |
where and the convergence is uniform on any finite
-interval. (The existence of
can also be proved if one replaces the condition
by the weaker condition
![]() |
where is the distance between sets.)
For any operator one has a corresponding Cauchy problem
![]() | (*) |
If the problem (*) has a strong solution, i.e. if there exists a function which is continuous on
, absolutely continuous on any compact subset of
, takes values in
for almost all
, has a strong derivative for almost all
, and satisfies the relation (*), then
. Any function
is a unique integral solution of the problem (*).
Under the assumptions of the fundamental theorem, if is a reflexive space and
is closed (cf. Closed operator), then the function
yields a strong solution of the Cauchy problem (*) for
, with
almost everywhere, where
is the set of elements of minimal norm in
. In that case the generating operator
of the semi-group
is densely defined:
. If, moreover,
and
are uniformly convex, then the operator
is single-valued and for all
there exists a right derivative
; this function is continuous from the right on
, and continuous at all points with the possible exception of a countable set; in this case
and
.
If is reflexive (or
, where
is separable) and
is a single-valued operator and has the property that
in
and
in the weak topology
(respectively, in
) imply
, then
,
, and
is a weakly (weak-
) continuously-differentiable solution of the problem (*). In the non-reflexive case, examples are known where the assumptions of the fundamental theorem hold with
and the functions
do not even have weak derivatives on
at any
,
.
Let be a continuous operator, defined on all of
, such that
is dissipative. Then
for
,
, and for any
the problem (*) has a unique continuously-differentiable solution on
, given by
. If
is continuous on its closed domain
, then it will be the generating operator of a semi-group of type
on
if only and only if
is dissipative and
for
.
In a Hilbert space , a contraction semi-group on a set
may be extended to a contraction semi-group on a closed convex subset
of
. Moreover, the generating operator
of the extended semi-group is defined on a set dense in
. There exists a unique
-dissipative operator such that
and
. If
is
-dissipative, then
is convex and there exists a unique contraction semi-group
on
such that
.
Let be a convex semi-continuous functional defined on a real Hilbert space
and let
be its subdifferential; then the operator
(for all
such that
is non-empty) is dissipative. The semi-group
possesses properties similar to those of a linear analytic semi-group. In particular,
(
) for any
, and
is a strong solution of the Cauchy problem (*), with
![]() |
for all ,
. If
attains its minimum, then
converges weakly to some minimum point as
.
Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems. Let ,
,
be Banach spaces; let
,
be operators defined and single-valued on
,
, respectively, satisfying the assumptions of the fundamental theorem for the same type
; let
be linear operators,
. Then convergence of the resolvents (cf. Resolvent) (
,
)
![]() |
for implies convergence of the semi-groups
![]() |
uniformly on any finite closed interval.
The multiplicative formulas developed by S. Lie in the finite-dimensional linear case can be generalized to the non-linear case. If ,
and
are single-valued
-dissipative operators on a Hilbert space and the closed convex set
is invariant under
and
, then, for any
,
![]() | (**) |
This formula is also valid in an arbitrary Banach space for any
, provided
is a densely-defined
-dissipative linear operator and
is a continuous dissipative operator defined on all of
. In both cases
![]() |
![]() |
Examples of non-linear differential operators satisfying the conditions of the fundamental theorem on the generation of semi-groups are given below. In each case only the space and the boundary conditions are indicated, while
is not described. In all examples,
is a bounded domain in
with smooth boundary;
are multi-valued maximal monotone mappings
,
,
; and
is a continuous strictly-increasing function,
.
Example 1.
,
,
,
on
.
Example 2.
,
,
on
.
Example 3.
,
,
on
.
Example 4.
or
,
,
on
.
Example 5.
,
, where
with values in
,
.
Example 6.
,
, where
is continuous.
References
[1] | V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian) |
[2] | H. Brézis, "Opérateurs maximaux monotones et semigroups de contractions dans les espaces de Hilbert" , North-Holland (1973) |
[3] | H. Brézis, A. Pazy, "Convergence and approximation of semigroups of nonlinear operators in Banach spaces" J. Funct. Anal. , 9 : 1 (1972) pp. 63–74 |
[4] | M.G. Crandall, T.M. Liggett, "Generation of semi-groups of nonlinear transformations on general Banach spaces" Amer. J. Math. , 93 : 2 (1971) pp. 265–298 |
[5] | Y. Kobayashi, "Difference approximation of Gauchy problems for quasi-dissipative operators and generation of nonlinear semigroups" J. Math. Soc. Japan , 27 : 4 (1975) pp. 640–665 |
[6] | Y. Konishi, "On the uniform convergence of a finite difference scheme for a nonlinear heat equation" Proc. Japan. Acad. , 48 : 2 (1972) pp. 62–66 |
[7] | R.H. Martin, "Differential equations on closed subsets of a Banach space" Trans. Amer. Math. Soc. , 179 (1973) pp. 399–414 |
[8] | G.F. Webb, "Continuous nonlinear perturbations of linear accretive operators in Banach spaces" J. Funct. Anal. , 10 : 2 (1972) pp. 191–203 |
[9] | M.I. [M.I. Khazan] Hazan, "Nonlinear evolution equations in locally convex spaces" Soviet Math. Dokl. , 14 : 5 (1973) pp. 1608–1614 Dokl. Akad. Nauk SSSR , 212 : 6 (1973) pp. 1309–1312 |
[10] | M.I. [M.I. Khazan] Hazan, "Differentiability of nonlinear semigroups and the classical solvability of nonlinear boundary value problems for the equation ![]() |
Comments
See also Semi-group of operators; One-parameter semi-group.
The formula (**) above, especially in the form
![]() |
which holds, e.g., when are self-adjoint operators on a separable Hilbert space so that
, defined on
, is self-adjoint, is known as the Trotter product formula, [a5], [a4].
References
[a1] | Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , CWI Monographs , 5 , North-Holland (1987) |
[a2] | A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) |
[a3] | R.H. Martin, "Nonlinear operators and differential equations in Banach spaces" , Wiley (1976) |
[a4] | B. Simon, "Functional integration and quantum physics" , Acad. Press (1979) pp. 4–6 |
[a5] | H. Trotter, "On the product of semigroups of operators" Proc. Amer. Math. Soc. , 10 (1959) pp. 545–551 |
Semi-group of non-linear operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-group_of_non-linear_operators&oldid=48659