Approximation solvability
A-solvability
Let and
be Banach spaces (cf. also Banach space), let
be a, possibly non-linear, mapping (cf. also Non-linear operator) and let
be an admissible scheme for
, which, for simplicity, is assumed to be a complete projection scheme, i.e.
and
are finite-dimensional subspaces with
for each
and
and
are linear projections such that
and
for
and
. Clearly, such schemes exist if both
and
have a Schauder basis (cf. also Basis; Biorthogonal system). Consider the equation
![]() | (a1) |
One of the basic problems in functional analysis is to "solve" (a1). Here, "solvability" of (a1) can be understood in (at least) two manners:
A) solvability in which a solution of (a1) is somehow established; or
B) approximation solvability of (a1) (with respect to ), in which a solution
of (a1) is obtained as the limit (or at least, a limit point) of solutions
of finite-dimensional approximate equations:
![]() | (a2) |
![]() |
with continuous for each
. If
and
are unique, then (a1) is said to be uniquely A-solvable.
Although the concepts A) and B) are distinct in their purpose, they are not independent. In fact, sometimes knowledge of A) is essential for B) to take place.
If and
are Hilbert spaces (cf. Hilbert space), the projections
and
are assumed to be orthogonal (cf. Orthogonal projector). If, for example,
and
are orthogonal bases, then
and
, and
and
for
,
. In this case, setting
, the coefficients
are determined by (a2), which reduces to the system
![]() |
A-proper.
In studying the A-solvability of (a1) one may ask: For what type of linear or non-linear mapping is it possible to show that (a1) is uniquely A-solvable? It turns out that the notion of an A-proper mapping is essential in answering this question.
A mapping is called A-proper if and only if
is continuous for each
and such that if
is any bounded sequence satisfying
for some
, then there exist a subsequence
and an
such that
as
and
, as was first shown in [a2].
It was found (see [a1]) that there are intimate relationships between (unique) A-solvability and A-properness of , shown by the following results:
R1) If is a continuous linear mapping, then (a1) is uniquely A-solvable if and only if
is A-proper and one-to-one. This is the best possible result, which includes as a special case all earlier results for the Galerkin or Petrov–Galerkin method (cf. also Galerkin method).
R2) If is non-linear and
![]() | (a3) |
for all ,
, where
is a continuous function on
with
,
for
and
as
, then (a1) is uniquely A-solvable for each
if and only if
is A-proper and one-to-one. If
is continuous, then R2) holds without the condition that
be one-to-one. The result R2) includes various results for strongly monotone or strongly accretive mappings (cf. also Accretive mapping). If
is a continuous linear mapping, then (a3) reduces to
![]() | (a4) |
for all ,
, and some
. If, in addition, the scheme
is nested, i.e.
and
for all
, and
in
for each
, then
is A-proper and one-to-one if and only if (a4) holds. In particular, by R1), equation (a1) is uniquely A-solvable for each
. Without this extra condition on
, equation (a1) is uniquely A-solvable if (a1) is solvable for each
, or if either
or
is reflexive (cf. also Reflexive space).
References
[a1] | W.V. Petryshyn, "Approximation-solvability of nonlinear functional and differential equations" , Monographs , 171 , M. Dekker (1993) |
[a2] | W.V. Petryshyn, "On projectional-solvability and Fredholm alternative for equations involving linear A-proper operators" Arch. Rat. Anal. , 30 (1968) pp. 270–284 |
Approximation solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_solvability&oldid=50144