Approximation solvability

A-solvability

Let $X$ and $Y$ be Banach spaces (cf. also Banach space), let $T : X \rightarrow Y$ be a, possibly non-linear, mapping (cf. also Non-linear operator) and let $\Gamma = \{ X _ { n } , P _ { n } ; Y _ { n } , Q _ { n } \}$ be an admissible scheme for $( X , Y )$, which, for simplicity, is assumed to be a complete projection scheme, i.e. $\{ X _ { n } \} \subset X$ and $\{ Y _ { n } \} \subset Y$ are finite-dimensional subspaces with $\operatorname { dim } X _ { n } = \operatorname { dim } Y _ { n }$ for each $n$ and $P _ { n } : Y \rightarrow X_n$ and $Q _ { n } : Y \rightarrow X _ { n }$ are linear projections such that $P _ { n } x \rightarrow x$ and $Q _ { n } y \rightarrow y$ for $x \in X$ and $y \in Y$. Clearly, such schemes exist if both $X$ and $Y$ have a Schauder basis (cf. also Basis; Biorthogonal system). Consider the equation

\begin{equation} \tag{a1} T x = f , \quad x \in X , f \in Y. \end{equation}

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 $x \in X$ of (a1) is somehow established; or

B) approximation solvability of (a1) (with respect to $\Gamma$), in which a solution $x \in X$ of (a1) is obtained as the limit (or at least, a limit point) of solutions $x _ { n } \in X _ { n }$ of finite-dimensional approximate equations:

\begin{equation} \tag{a2} T _ { n } ( x _ { n } ) = Q _ { n } f, \end{equation}

\begin{equation*} x _ { n } \in X _ { n } , Q _ { n } f \in Y _ { n } , T _ { n } = ( Q _ { n } T ) | _{X _ { n }} , \end{equation*}

with $T _ { n } : X _ { n } \rightarrow Y _ { n }$ continuous for each $n$. If $x _ { n }$ and $x$ 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 $X$ and $Y$ are Hilbert spaces (cf. Hilbert space), the projections $P_n$ and $Q _ { n }$ are assumed to be orthogonal (cf. Orthogonal projector). If, for example, $\{ \phi _ { n } \} \subset X$ and $\{ \psi _ { n } \} \subset Y$ are orthogonal bases, then $X _ { n } = \operatorname { span } \{ \phi _ { 1 } , \dots , \phi _ { n } \}$ and $Y _ { n } = \operatorname { span } \{ \psi _ { 1 } , \dots , \psi _ { n } \}$, and $P _ { n } x = \sum _ { i = 1 } ^ { n } ( x , \phi _ { i } ) \phi _ { i }$ and $Q _ { n } y = \sum _ { i = 1 } ^ { n } ( y , \psi _ { i } ) \psi _ { i }$ for $x \in X$, $y \in Y$. In this case, setting , the coefficients $a _ { 1 } ^ { n } , \ldots , a _ { n } ^ { n }$ are determined by (a2), which reduces to the system

\begin{equation*} ( T ( x _ { n } ) , \psi _ { j } ) = ( f , \psi _ { j } ) , j = 1 , \ldots , n. \end{equation*}

A-proper.

In studying the A-solvability of (a1) one may ask: For what type of linear or non-linear mapping $T : X \rightarrow Y$ 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 $T : X \rightarrow Y$ is called A-proper if and only if $T _ { n } : X _ { n } \rightarrow Y _ { n }$ is continuous for each $n$ and such that if $\left\{ x _ { n_j } , : x _ { n_j } \in X _ { n_j } \right\}$ is any bounded sequence satisfying $T _ { n_ j } ( x _ { n_j } ) \rightarrow g$ for some $g \in Y$, then there exist a subsequence $\{ x _ { n_ j } ^ { \prime } \}$ and an $x \in X$ such that $x _ { n_j } ^ { \prime } \rightarrow x$ as $j \rightarrow \infty$ and $T ( x ) = g$, as was first shown in [a2].

It was found (see [a1]) that there are intimate relationships between (unique) A-solvability and A-properness of $T$, shown by the following results:

R1) If $T : X \rightarrow Y$ is a continuous linear mapping, then (a1) is uniquely A-solvable if and only if $T$ 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 $T$ is non-linear and

\begin{equation} \tag{a3} \| T _ { n } ( x ) - T _ { n } ( y ) \| \geq \phi ( \| x - y \| ) \end{equation}

for all $x , y \in X _ { n }$, $n \geq N_0$, where $\phi$ is a continuous function on $\mathbf{R}$ with $\phi ( 0 ) = 0$, $\phi ( t ) > 0$ for $t > 0$ and $\phi ( t ) \rightarrow \infty$ as $t \rightarrow \infty$, then (a1) is uniquely A-solvable for each $f \in Y$ if and only if $T$ is A-proper and one-to-one. If $T$ is continuous, then R2) holds without the condition that $T$ be one-to-one. The result R2) includes various results for strongly monotone or strongly accretive mappings (cf. also Accretive mapping). If $T$ is a continuous linear mapping, then (a3) reduces to

\begin{equation} \tag{a4} \| T _ { n } ( x ) \| \geq c \| x \| \end{equation}

for all $x \in X _ { n }$, $n \geq N_0$, and some $c > 0$. If, in addition, the scheme $\Gamma = \{ X _ { n } , P _ { n } ; Y _ { n } , Q _ { n } \}$ is nested, i.e. $X _ { n } \subset X _ { n + 1} $ and $Y _ { n } \subset Y _ { n + 1 }$ for all $n$, and $Q _ { n } ^ { * } w \rightarrow w$ in $Y ^ { * }$ for each $w \in Y ^ { * }$, then $T$ 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 $f \in Y$. Without this extra condition on $\Gamma$, equation (a1) is uniquely A-solvable if (a1) is solvable for each $f \in Y$, or if either $X$ or $Y$ 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