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