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''A-solvability''
 
''A-solvability''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302702.png" /> be Banach spaces (cf. also [[Banach space|Banach space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302703.png" /> be a, possibly non-linear, mapping (cf. also [[Non-linear operator|Non-linear operator]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302704.png" /> be an admissible scheme for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302705.png" />, which, for simplicity, is assumed to be a complete projection scheme, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302707.png" /> are finite-dimensional subspaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302708.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a1302709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027011.png" /> are linear projections such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027015.png" />. Clearly, such schemes exist if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027017.png" /> have a Schauder basis (cf. also [[Basis|Basis]]; [[Biorthogonal system|Biorthogonal system]]). Consider the equation
+
Let $X$ and $Y$ be Banach spaces (cf. also [[Banach space|Banach space]]), let $T : X \rightarrow Y$ be a, possibly non-linear, mapping (cf. also [[Non-linear operator|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|Basis]]; [[Biorthogonal system|Biorthogonal system]]). Consider the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} T x  = f , \quad x \in X , f \in Y. \end{equation}
  
 
One of the basic problems in [[Functional analysis|functional analysis]] is to  "solve"  (a1). Here,  "solvability"  of (a1) can be understood in (at least) two manners:
 
One of the basic problems in [[Functional analysis|functional analysis]] is to  "solve"  (a1). Here,  "solvability"  of (a1) can be understood in (at least) two manners:
  
A) solvability in which a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027019.png" /> of (a1) is somehow established; or
+
A) solvability in which a solution $x \in X$ of (a1) is somehow established; or
  
B) approximation solvability of (a1) (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027020.png" />), in which a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027021.png" /> of (a1) is obtained as the limit (or at least, a limit point) of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027022.png" /> of finite-dimensional approximate equations:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} T _ { n } ( x _ { n } ) = Q _ { n } f, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027024.png" /></td> </tr></table>
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\begin{equation*} x _ { n } \in X _ { n } , Q _ { n } f \in Y _ { n } , T _ { n } = ( Q _ { n } T ) | _{X _ { n }} ,  \end{equation*}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027025.png" /> continuous for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027028.png" /> are unique, then (a1) is said to be uniquely A-solvable.
+
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027030.png" /> are Hilbert spaces (cf. [[Hilbert space|Hilbert space]]), the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027032.png" /> are assumed to be orthogonal (cf. [[Orthogonal projector|Orthogonal projector]]). If, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027034.png" /> are orthogonal bases, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027040.png" />. In this case, setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027041.png" />, the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027042.png" /> are determined by (a2), which reduces to the system
+
If $X$ and $Y$ are Hilbert spaces (cf. [[Hilbert space|Hilbert space]]), the projections $P_n$ and $Q _ { n }$ are assumed to be orthogonal (cf. [[Orthogonal projector|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027041.png"/>, the coefficients $a _ { 1 } ^ { n } , \ldots , a _ { n } ^ { n }$ are determined by (a2), which reduces to the system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027043.png" /></td> </tr></table>
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\begin{equation*} ( T ( x _ { n } ) , \psi _ { j } ) = ( f , \psi _ { j } ) , j = 1 , \ldots , n. \end{equation*}
  
 
==A-proper.==
 
==A-proper.==
In studying the A-solvability of (a1) one may ask: For what type of linear or non-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027044.png" /> 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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027045.png" /> is called A-proper if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027046.png" /> is continuous for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027047.png" /> and such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027048.png" /> is any bounded sequence satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027049.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027050.png" />, then there exist a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027051.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027053.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027055.png" />, as was first shown in [[#References|[a2]]].
+
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 [[#References|[a2]]].
  
It was found (see [[#References|[a1]]]) that there are intimate relationships between (unique) A-solvability and A-properness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027056.png" />, shown by the following results:
+
It was found (see [[#References|[a1]]]) that there are intimate relationships between (unique) A-solvability and A-properness of $T$, shown by the following results:
  
R1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027057.png" /> is a continuous linear mapping, then (a1) is uniquely A-solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027058.png" /> 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|Galerkin method]]).
+
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|Galerkin method]]).
  
R2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027059.png" /> is non-linear and
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R2) If $T$ is non-linear and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} \| T _ { n } ( x ) - T _ { n } ( y ) \| \geq \phi ( \| x - y \| ) \end{equation}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027063.png" /> is a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027064.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027066.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027068.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027069.png" />, then (a1) is uniquely A-solvable for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027070.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027071.png" /> is A-proper and one-to-one. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027072.png" /> is continuous, then R2) holds without the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027073.png" /> be one-to-one. The result R2) includes various results for strongly monotone or strongly accretive mappings (cf. also [[Accretive mapping|Accretive mapping]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027074.png" /> is a continuous linear mapping, then (a3) reduces to
+
for all $x , y \in X _ { n }$, $n \geq N_0$, where $\phi$ is a continuous function on $\mathbf{R}$ with $\phi ( 0 ) = 0$, $\phi ( t ) &gt; 0$ for $t &gt; 0$ 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|Accretive mapping]]). If $T$ is a continuous linear mapping, then (a3) reduces to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027075.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \tag{a4} \| T _ { n } ( x ) \| \geq c \| x \| \end{equation}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027077.png" />, and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027078.png" />. If, in addition, the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027079.png" /> is nested, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027081.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027082.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027084.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027085.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027086.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027087.png" />. Without this extra condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027088.png" />, equation (a1) is uniquely A-solvable if (a1) is solvable for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027089.png" />, or if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027090.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130270/a13027091.png" /> is reflexive (cf. also [[Reflexive space|Reflexive space]]).
+
for all $x \in X _ { n }$, $n \geq N_0$, and some $c &gt; 0$. 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|Reflexive space]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.V. Petryshyn,  "Approximation-solvability of nonlinear functional and differential equations" , ''Monographs'' , '''171''' , M. Dekker  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.V. Petryshyn,  "On projectional-solvability and Fredholm alternative for equations involving linear A-proper operators"  ''Arch. Rat. Anal.'' , '''30'''  (1968)  pp. 270–284</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  W.V. Petryshyn,  "Approximation-solvability of nonlinear functional and differential equations" , ''Monographs'' , '''171''' , M. Dekker  (1993)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  W.V. Petryshyn,  "On projectional-solvability and Fredholm alternative for equations involving linear A-proper operators"  ''Arch. Rat. Anal.'' , '''30'''  (1968)  pp. 270–284</td></tr></table>

Revision as of 16:56, 1 July 2020

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
How to Cite This Entry:
Approximation solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_solvability&oldid=50144
This article was adapted from an original article by W.V. Petryshyn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article