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''for integral equations''
 
''for integral equations''
  
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The homogeneous equation
 
The homogeneous 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/f/f041/f041470/f0414701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi ( x) - \lambda \int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  = 0
 +
$$
  
 
and its transposed equation
 
and its transposed 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/f/f041/f041470/f0414702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\psi ( x) - \lambda \int\limits _ { a } ^ { b }  K ( s, x) \psi ( s)  ds  = 0
 +
$$
  
have, for a fixed value of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f0414703.png" />, either only the trivial solution, or have the same finite number of linearly independent solutions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f0414704.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f0414705.png" />.
+
have, for a fixed value of the parameter $  \lambda $,  
 +
either only the trivial solution, or have the same finite number of linearly independent solutions: $  \phi _ {1} \dots \phi _ {n} $;  
 +
$  \psi _ {1} \dots \psi _ {n} $.
  
 
===Theorem 2.===
 
===Theorem 2.===
 
For a solution of the inhomogeneous equation
 
For a solution of the inhomogeneous 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/f/f041/f041470/f0414706.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\phi ( x) - \lambda \int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  = f ( x)
 +
$$
  
 
to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2):
 
to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2):
  
<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/f/f041/f041470/f0414707.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\int\limits _ { a } ^ { b }  f ( x) \psi _ {j} ( x)  dx  = 0,\  j = 1 \dots n.
 +
$$
  
 
===Theorem 3.===
 
===Theorem 3.===
(the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f0414708.png" />, or the corresponding homogeneous equation (1) has non-trivial solutions.
+
(the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side f $,  
 +
or the corresponding homogeneous equation (1) has non-trivial solutions.
  
 
===Theorem 4.===
 
===Theorem 4.===
 
The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity.
 
The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity.
  
For the Fredholm theorems to hold in the function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f0414709.png" /> it is sufficient that the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147010.png" /> of equation (3) be square-integrable on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147013.png" /> may be infinite). When this condition is violated, (3) may turn out to be a [[Non-Fredholm integral equation|non-Fredholm integral equation]]. When the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147014.png" /> and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1):
+
For the Fredholm theorems to hold in the function space $  L _ {2} [ a, b] $
 +
it is sufficient that the kernel $  K $
 +
of equation (3) be square-integrable on the set $  [ a, b] \times [ a, b] $(
 +
$  a $
 +
and $  b $
 +
may be infinite). When this condition is violated, (3) may turn out to be a [[Non-Fredholm integral equation|non-Fredholm integral equation]]. When the parameter $  \lambda $
 +
and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1):
  
<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/f/f041/f041470/f04147015.png" /></td> </tr></table>
+
$$
 +
\psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b }  \overline{ {K ( s, x) }}\; \psi ( s)  ds  = 0.
 +
$$
  
 
In this case condition (4) is replaced by
 
In this case condition (4) is replaced by
  
<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/f/f041/f041470/f04147016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }  f ( x) \overline{ {\psi _ {j} ( x) }}\; dx  = 0,\ \
 +
j = 1 \dots n.
 +
$$
  
 
These theorems were proved by E.I. Fredholm [[#References|[1]]].
 
These theorems were proved by E.I. Fredholm [[#References|[1]]].
Line 39: Line 73:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.I. Fredholm,  "Sur une classe d'equations fonctionnelles"  ''Acta Math.'' , '''27'''  (1903)  pp. 365–390</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.I. Fredholm,  "Sur une classe d'equations fonctionnelles"  ''Acta Math.'' , '''27'''  (1903)  pp. 365–390</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Instead of the phrases  "transposed equation"  and  "adjoint equation"  one sometimes uses  "adjoint equation of a Fredholm integral equationadjoint equation"  and  "conjugate equation of a Fredholm integral equationconjugate equation"  (cf. [[#References|[a4]]]); in the latter terminology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147017.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041470/f04147018.png" />.
+
Instead of the phrases  "transposed equation"  and  "adjoint equation"  one sometimes uses  "adjoint equation of a Fredholm integral equationadjoint equation"  and  "conjugate equation of a Fredholm integral equationconjugate equation"  (cf. [[#References|[a4]]]); in the latter terminology $  \overline \lambda \; $
 +
is replaced by $  \lambda $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1975)  (Translated from Russian)</TD></TR></table>

Latest revision as of 19:40, 5 June 2020


for integral equations

Theorem 1.

The homogeneous equation

$$ \tag{1 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = 0 $$

and its transposed equation

$$ \tag{2 } \psi ( x) - \lambda \int\limits _ { a } ^ { b } K ( s, x) \psi ( s) ds = 0 $$

have, for a fixed value of the parameter $ \lambda $, either only the trivial solution, or have the same finite number of linearly independent solutions: $ \phi _ {1} \dots \phi _ {n} $; $ \psi _ {1} \dots \psi _ {n} $.

Theorem 2.

For a solution of the inhomogeneous equation

$$ \tag{3 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x) $$

to exist it is necessary and sufficient that its right-hand side be orthogonal to a complete system of linearly independent solutions of the corresponding homogeneous transposed equation (2):

$$ \tag{4 } \int\limits _ { a } ^ { b } f ( x) \psi _ {j} ( x) dx = 0,\ j = 1 \dots n. $$

Theorem 3.

(the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side $ f $, or the corresponding homogeneous equation (1) has non-trivial solutions.

Theorem 4.

The set of characteristic numbers of equation (1) is at most countable, with a single possible limit point at infinity.

For the Fredholm theorems to hold in the function space $ L _ {2} [ a, b] $ it is sufficient that the kernel $ K $ of equation (3) be square-integrable on the set $ [ a, b] \times [ a, b] $( $ a $ and $ b $ may be infinite). When this condition is violated, (3) may turn out to be a non-Fredholm integral equation. When the parameter $ \lambda $ and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1):

$$ \psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b } \overline{ {K ( s, x) }}\; \psi ( s) ds = 0. $$

In this case condition (4) is replaced by

$$ \int\limits _ { a } ^ { b } f ( x) \overline{ {\psi _ {j} ( x) }}\; dx = 0,\ \ j = 1 \dots n. $$

These theorems were proved by E.I. Fredholm [1].

References

[1] E.I. Fredholm, "Sur une classe d'equations fonctionnelles" Acta Math. , 27 (1903) pp. 365–390

Comments

Instead of the phrases "transposed equation" and "adjoint equation" one sometimes uses "adjoint equation of a Fredholm integral equationadjoint equation" and "conjugate equation of a Fredholm integral equationconjugate equation" (cf. [a4]); in the latter terminology $ \overline \lambda \; $ is replaced by $ \lambda $.

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a2] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
[a3] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[a4] P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)
How to Cite This Entry:
Fredholm theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_theorems&oldid=12814
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article