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An integral equation of the form
 
An integral equation of the form
  
<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/f041420/f0414201.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
K ( x, s) \phi ( s) \
 +
ds  = f ( x),\ \
 +
x \in [ a, b],
 +
$$
  
 
— a Fredholm equation of the first kind, or one of the form
 
— a Fredholm equation of the first kind, or one of the form
  
<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/f041420/f0414202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi ( x) - \lambda
 +
\int\limits _ { a } ^ { b }
 +
K ( x, s) \phi ( s) \
 +
ds  = f ( x),\ \
 +
x \in [ a, b],
 +
$$
  
 
— a Fredholm equation of the second kind, if the integral operator
 
— a Fredholm equation of the second kind, if the integral operator
  
<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/f041420/f0414203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
K \phi ( x)  = \
 +
\int\limits _ { a } ^ { b }
 +
K ( x, s) \phi ( s) ds,\ \
 +
x \in [ a, b],
 +
$$
  
is completely continuous in some function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f0414204.png" />. It is assumed that the free term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f0414205.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f0414206.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f0414207.png" />. An important example of a Fredholm equation is one in which the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f0414208.png" /> satisfies the condition
+
is completely continuous in some function space $  E $.  
 +
It is assumed that the free term f $
 +
and the function $  \phi $
 +
belong to $  E $.  
 +
An important example of a Fredholm equation is one in which the kernel $  K $
 +
satisfies the condition
  
<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/f041420/f0414209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
B  ^ {2}  = \
 +
\int\limits _ { a } ^ { b }
 +
\int\limits _ { a } ^ { b }
 +
| K ( x, s) |  ^ {2} \
 +
dx  ds  < \infty ,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142010.png" />.
+
and $  E = L _ {2} ([ a, b]) $.
  
The numerical parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142011.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142014.png" /> can take either real or complex values. For Fredholm equations of the first kind see [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]; [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]] and [[Ill-posed problems|Ill-posed problems]]. Below Fredholm equations of the second kind only are considered.
+
The numerical parameter $  \lambda $
 +
and the functions f $,  
 +
$  \phi $
 +
and $  K $
 +
can take either real or complex values. For Fredholm equations of the first kind see [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]; [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]] and [[Ill-posed problems|Ill-posed problems]]. Below Fredholm equations of the second kind only are considered.
  
 
==The method of successive approximation to solutions of Fredholm equations of the second kind.==
 
==The method of successive approximation to solutions of Fredholm equations of the second kind.==
 
This was the first method that was proposed for solving equation (1). To state this method, suppose that (1) is written in the form
 
This was the first method that was proposed for solving equation (1). To state this method, suppose that (1) is written in the form
  
<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/f041420/f04142015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\phi ( x)  = \
 +
f ( x) + \lambda K \phi ( x),\ \
 +
x \in [ a, b].
 +
$$
  
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142016.png" /> satisfies the condition (3), and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142017.png" />. Let the initial approximation to the desired solution be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142018.png" />; if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142019.png" />-th approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142020.png" /> has been constructed, then
+
Assume that $  K $
 +
satisfies the condition (3), and that $  E = L _ {2} ([ a, b]) $.  
 +
Let the initial approximation to the desired solution be $  \phi _ {0} = f $;  
 +
if the $  ( n - 1) $-
 +
th approximation $  \phi _ {n - 1 }  $
 +
has been constructed, then
  
<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/f041420/f04142021.png" /></td> </tr></table>
+
$$
 +
\phi _ {n}  = \
 +
f + \lambda K \phi _ {n - 1 }  ;
 +
$$
  
 
in this case
 
in this case
  
<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/f041420/f04142022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\phi _ {n}  = \
 +
\sum _ {m = 0 } ^ { n }
 +
\lambda  ^ {m} K _ {m} f,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142023.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142024.png" />-th iterated kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142025.png" />. The function (5) is a partial sum of the series
+
where $  K _ {m} $
 +
denotes the $  m $-
 +
th iterated kernel of $  K $.  
 +
The function (5) is a partial sum of the series
  
<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/f041420/f04142026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
\sum _ {m = 0 } ^  \infty 
 +
\lambda  ^ {m} K _ {m} f,
 +
$$
  
which is called the Neumann (or Liouville–Neumann) series. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142027.png" />, then (6) converges in the quadratic mean to a solution of (1), and this solution is unique (see, for example, [[#References|[5]]]). If there is a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142028.png" /> such that
+
which is called the Neumann (or Liouville–Neumann) series. If $  | \lambda | < B  ^ {-} 1 $,  
 +
then (6) converges in the quadratic mean to a solution of (1), and this solution is unique (see, for example, [[#References|[5]]]). If there is a positive constant $  A $
 +
such that
  
<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/f041420/f04142029.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
| K ( x, s) |  ^ {2} \
 +
ds  \leq  A,\ \
 +
x \in [ a, b],
 +
$$
  
then (6) converges absolutely and uniformly. Generally speaking, (6) diverges if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142030.png" />. Indeed, this is the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142031.png" /> has an eigen value. But if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142032.png" /> has no eigen values (as, for example, in the case of a [[Volterra kernel|Volterra kernel]]), then (6) converges for every value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142033.png" />.
+
then (6) converges absolutely and uniformly. Generally speaking, (6) diverges if $  | \lambda | \geq  B  ^ {-} 1 $.  
 +
Indeed, this is the case if $  K $
 +
has an eigen value. But if $  K $
 +
has no eigen values (as, for example, in the case of a [[Volterra kernel|Volterra kernel]]), then (6) converges for every value of $  \lambda $.
  
 
==Fredholm's method for solving a Fredholm equation of the second kind.==
 
==Fredholm's method for solving a Fredholm equation of the second kind.==
The method of successive approximation enables one to construct solutions of (1), generally speaking, only for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142034.png" />. A method that makes it possible to solve (1) for any value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142035.png" /> was first proposed by E.I. Fredholm (1903). Under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142036.png" /> is continuous on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142037.png" />, and that the free term and the desired solution are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142038.png" />, the following is a brief description of the gist of this method.
+
The method of successive approximation enables one to construct solutions of (1), generally speaking, only for small values of $  \lambda $.  
 +
A method that makes it possible to solve (1) for any value of $  \lambda $
 +
was first proposed by E.I. Fredholm (1903). Under the assumption that $  K $
 +
is continuous on the square $  [ a, b] \times [ a, b] $,  
 +
and that the free term and the desired solution are continuous on $  [ a, b] $,  
 +
the following is a brief description of the gist of this method.
 +
 
 +
Divide the interval  $  [ a, b] $
 +
into  $  n $
 +
equal parts of length  $  h = ( b - a)/n $.
 +
If the integral in (1) would be replaced by a Riemann sum, the exact equation (1) would be replaced by the approximation
 +
 
 +
$$ \tag{7 }
 +
\phi ( x) - \lambda h
 +
\sum _ {j = 1 } ^ { n }
 +
K ( x, s _ {j} )
 +
\phi ( s _ {j} )  =  f ( x),\ \
 +
x \in [ a, b].
 +
$$
  
Divide the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142039.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142040.png" /> equal parts of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142041.png" />. If the integral in (1) would be replaced by a Riemann sum, the exact equation (1) would be replaced by the approximation
+
Set  $  x = s _ {1} \dots s _ {n} $
 +
successively in (7) to determine the approximate values of the unknown function  $  \phi $
 +
at the points  $  s _ {j} $,  
 +
thus obtaining the linear algebraic 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/f/f041/f041420/f04142042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{8 }
 +
\phi _ {i} - \lambda h
 +
\sum _ {j = 0 } ^ { n }
 +
K _ {ij} \phi _ {j}  = f _ {i} ,\ \
 +
i = 1 \dots n,
 +
$$
  
Set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142043.png" /> successively in (7) to determine the approximate values of the unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142044.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142045.png" />, thus obtaining the linear algebraic system
+
where  $  f ( s _ {i} ) = f _ {i} $,
 +
$  \phi ( s _ {i} ) = \phi _ {i} $,
 +
$  K ( s _ {i} , s _ {j} ) = K _ {ij} $.  
 +
Whether the system (8) has a solution or not depends on the value of the determinant
  
<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/f041420/f04142046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$
 +
\Delta ( \lambda ) = \
 +
\left |
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142049.png" />. Whether the system (8) has a solution or not depends on the value of the determinant
+
\begin{array}{ccc}
 +
1 - \lambda hK _ {11} - \lambda hK _ {12}  &\dots  &- \lambda hK _ {1n}  \\
 +
\dots  &\dots  &\dots  \\
 +
- \lambda hK _ {n1} - \lambda hK _ {n2}  &\dots  &1 - \lambda hK _ {nn}  \\
 +
\end{array}
 +
\
 +
\right | ,
 +
$$
  
<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/f041420/f04142050.png" /></td> </tr></table>
+
which is a polynomial in  $  \lambda $.
 +
If  $  \lambda $
 +
is not one of the roots of this polynomial, then (8) has a solution. Solving this system and substituting the resulting values  $  \phi _ {j} = \phi ( s _ {j} ) $
 +
in (7), an approximate solution of (1) is obtained:
  
which is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142052.png" /> is not one of the roots of this polynomial, then (8) has a solution. Solving this system and substituting the resulting values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142053.png" /> in (7), an approximate solution of (1) is obtained:
+
$$ \tag{9 }
 +
\phi ( x) \approx \
 +
f ( x) + \lambda
  
<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/f041420/f04142054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
\frac{Q ( x, s _ {1} \dots s _ {n} ; \lambda ) }{\Delta ( \lambda ) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142056.png" /> are polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142057.png" />. The method presented is one of the possible versions for constructing an approximate solution of the Fredholm equation (1) (see [[#References|[6]]]).
+
where $  Q $
 +
and $  \Delta $
 +
are polynomials in $  \lambda $.  
 +
The method presented is one of the possible versions for constructing an approximate solution of the Fredholm equation (1) (see [[#References|[6]]]).
  
One might expect that in the limit, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142058.png" /> in such a way that the Riemann sum (7) tends to the integral in (1), the limit of the right-hand side of (9) becomes an exact solution of (1). Using formal limit transitions in analogous expressions, Fredholm established a formula that should represent a solution of (1):
+
One might expect that in the limit, as $  n \rightarrow \infty $
 +
in such a way that the Riemann sum (7) tends to the integral in (1), the limit of the right-hand side of (9) becomes an exact solution of (1). Using formal limit transitions in analogous expressions, Fredholm established a formula that should represent a solution of (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/f041420/f04142059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>
+
$$ \tag{10 }
 +
\phi ( x)  = \
 +
f ( x) + \lambda
 +
\int\limits _ { a } ^ { b }
 +
R ( x, s; \lambda )
 +
f ( s) ds,
 +
$$
  
 
where
 
where
  
<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/f041420/f04142060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>
+
$$ \tag{11 }
 +
R ( x, s;  \lambda )  = \
 +
 
 +
\frac{D ( x, s; \lambda ) }{D ( \lambda ) }
 +
,
 +
$$
 +
 
 +
$$ \tag{12 }
 +
D ( \lambda )  = \sum _ {m = 0 } ^  \infty 
 +
\frac{(- 1)  ^ {m} }{m! }
 +
A _ {m} \lambda  ^ {m} ,
 +
$$
 +
 
 +
$$ \tag{13 }
 +
D ( x, s; \lambda )  = \sum _ {m = 0 } ^  \infty 
 +
\frac{(- 1)
 +
^ {m} }{m! }
 +
B _ {m} ( x, s) \lambda  ^ {m} ,
 +
$$
  
<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/f041420/f04142061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(12)</td></tr></table>
+
$$ \tag{14 }
 +
A _ {m}  = \int\limits _ { a } ^ { b }  \dots \int\limits _ { a } ^ { b }  K \left ( \begin{array}{c}
 +
s _ {1} \dots s _ {m} \\
 +
s _ {1} \dots s _ {m}
 +
\end{array}
 +
\right ) ds _ {1} \dots ds _ {m} ,
 +
$$
  
<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/f041420/f04142062.png" /></td> <td valign="top" style="width:5%;text-align:right;">(13)</td></tr></table>
+
$$ \tag{15 }
 +
B _ {m} ( x, s)  = \int\limits _ { a } ^ { b }  \dots \int\limits _ { a } ^ { b }  K \left ( \begin{array}{c}
 +
x, s _ {1} \dots s _ {m} \\
 +
s,\
 +
s _ {1} \dots s _ {m}
 +
\end{array}
 +
\right ) ds _ {1} \dots ds _ {m} ,
 +
$$
  
<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/f041420/f04142063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(14)</td></tr></table>
+
$$
 +
K \left ( \begin{array}{c}
 +
x _ {1} \dots x _ {n} \\
 +
s _ {1} \dots s _ {n}
 +
\end{array}
 +
\right )  = \left |
 +
\begin{array}{ccc}
 +
K ( x _ {1} , s _ {1} )  &\dots  &K ( x _ {1} , s _ {n} )  \\
 +
\dots  &\dots  &\dots  \\
 +
K ( x _ {n} , s _ {1} )  &\dots  &K ( x _ {n} , s _ {n} ) \\
 +
\end{array}
 +
\right | .
 +
$$
  
<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/f041420/f04142064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(15)</td></tr></table>
+
To calculate  $  A _ {m} $
 +
and  $  B _ {m} ( x, s) $,
 +
instead of the formulas (14) and (15) one can make use of the following recurrence relations:
  
<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/f041420/f04142065.png" /></td> </tr></table>
+
$$
 +
A _ {0}  = 1,\ \
 +
B _ {0} ( x, s)  = \
 +
K ( x, s),\ \
 +
A _ {m}  = \int\limits _ { a } ^ { b }
 +
B _ {m - 1 }  ( s, s)  ds,
 +
$$
  
To calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142067.png" />, instead of the formulas (14) and (15) one can make use of the following recurrence relations:
+
$$
 +
B _ {m} ( x, s)  = K ( x, s) A _ {m} - m \int\limits _ { a } ^ { b }  K ( x, t) B _ {m - 1 }  ( t, s) dt,
 +
$$
  
<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/f041420/f04142068.png" /></td> </tr></table>
+
$$
 +
= 1, 2 , .  . . .
 +
$$
  
<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/f041420/f04142069.png" /></td> </tr></table>
+
The series (12) and (13) are called Fredholm series. The function  $  D ( \lambda ) $
 +
is called the Fredholm determinant of  $  K $;  
 +
$  D ( x, s; \lambda ) $
 +
is called the first Fredholm minor for  $  D ( \lambda ) $;  
 +
and the function (11) is called the resolvent (or solving kernel or reciprocal kernel) of  $  K $(
 +
or of equation (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/f041420/f04142070.png" /></td> </tr></table>
+
The justification of the limit transitions mentioned above, which lead to (10), was carried out by D. Hilbert (see [[Integral equation|Integral equation]]). Fredholm, having constructed the series (12) and (13), then proved directly and rigorously that they converge for all finite values of  $  \lambda $
 +
and that (13), moreover, converges uniformly with respect to  $  x $
 +
and  $  s $
 +
on  $  [ a, b] \times [ a, b] $.
 +
The establishment of a connection between  $  D ( \lambda ) $
 +
and  $  D ( x, s; \lambda ) $
 +
enabled him to prove the following proposition: If  $  D ( \lambda ) \neq 0 $,
 +
then equation (1) has one and only one solution, which is expressed by formula (10).
  
The series (12) and (13) are called Fredholm series. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142071.png" /> is called the Fredholm determinant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142072.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142073.png" /> is called the first Fredholm minor for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142074.png" />; and the function (11) is called the resolvent (or solving kernel or reciprocal kernel) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142075.png" /> (or of equation (1)).
+
If follows from this proposition that a value of  $  \lambda $
 +
that is not a root of the Fredholm determinant is a regular value for the homogeneous equation associated with (1):
  
The justification of the limit transitions mentioned above, which lead to (10), was carried out by D. Hilbert (see [[Integral equation|Integral equation]]). Fredholm, having constructed the series (12) and (13), then proved directly and rigorously that they converge for all finite values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142076.png" /> and that (13), moreover, converges uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142078.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142079.png" />. The establishment of a connection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142081.png" /> enabled him to prove the following proposition: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142082.png" />, then equation (1) has one and only one solution, which is expressed by formula (10).
+
$$ \tag{1h }
 +
\phi ( x) - \lambda
 +
\int\limits _ { a } ^ { b }
 +
K ( x, s) \phi ( s)
 +
ds  = 0,\ \
 +
x \in [ a, b],
 +
$$
  
If follows from this proposition that a value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142083.png" /> that is not a root of the Fredholm determinant is a regular value for the homogeneous equation associated with (1):
+
that is, in this case the equation has only the zero solution. If $  \lambda $
 +
is a root of the equation  $  D ( \lambda ) = 0 $,
 +
then  $  \lambda $
 +
is a pole of the resolvent (11) of equation (1h) and an eigen value of this latter equation. In order to construct by the Fredholm method the eigen functions belonging to this eigen value, one introduces the concept of the  $  p $-
 +
th minor of  $  D ( \lambda ) $.
 +
Let
  
<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/f041420/f04142084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1h)</td></tr></table>
+
$$
 +
B _ {0} \left (
 +
\begin{array}{c}
  
that is, in this case the equation has only the zero solution. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142085.png" /> is a root of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142086.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142087.png" /> is a pole of the resolvent (11) of equation (1h) and an eigen value of this latter equation. In order to construct by the Fredholm method the eigen functions belonging to this eigen value, one introduces the concept of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142089.png" />-th minor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142090.png" />. Let
+
x _ {1} \dots x _ {p} \\
  
<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/f041420/f04142091.png" /></td> </tr></table>
+
s _ {1} \dots s _ {p}
 +
\end{array}
  
<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/f041420/f04142092.png" /></td> </tr></table>
+
\right )  = K \left (
 +
\begin{array}{c}
  
<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/f041420/f04142093.png" /></td> </tr></table>
+
x _ {1} \dots x _ {p} \\
  
Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142095.png" />-th minor for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142096.png" /> is the series
+
s _ {1} \dots s _ {p}
 +
\end{array}
  
<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/f041420/f04142097.png" /></td> <td valign="top" style="width:5%;text-align:right;">(16)</td></tr></table>
+
\right ) ,
 +
$$
  
<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/f041420/f04142098.png" /></td> </tr></table>
+
$$
 +
B _ {m} \left ( \begin{array}{c}
 +
x _ {1} \dots x _ {p} \\
 +
s _ {1} \dots s _ {p}
 +
\end{array}
 +
\right )  = \int\limits _ { a } ^ { b }  \dots \int\limits _ { a } ^ { b }  K \left ( \begin{array}{c}
 +
x _ {1} \dots x _ {p} ,\
 +
t _ {1} \dots t _ {m} \\
 +
s _ {1} \dots s _ {p} , t _ {1} \dots t _ {m}
 +
\end{array}
 +
\right ) \times
 +
$$
  
which becomes equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f04142099.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420100.png" />. The series (16) is absolutely convergent for all finite values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420101.png" /> and converges uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420103.png" /> satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420106.png" />. Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420107.png" /> is an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420108.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420110.png" /> since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420111.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420112.png" /> be the multiplicity of the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420113.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420114.png" />. There is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420115.png" /> such that all minors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420116.png" /> of orders less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420117.png" /> are identically equal to zero, while the minor of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420118.png" /> is different from zero. There is some collection of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420120.png" /> such that
+
$$
 +
\times
 +
dt _ {1} \dots dt _ {m} .
 +
$$
  
<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/f041420/f041420121.png" /></td> </tr></table>
+
Then the  $  p $-
 +
th minor for  $  D ( \lambda ) $
 +
is the series
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420122.png" /> is called the rank (or multiplicity) of the eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420123.png" />. The functions
+
$$ \tag{16 }
 +
D \left (
 +
\begin{array}{c}
  
<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/f041420/f041420124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(17)</td></tr></table>
+
x _ {1} \dots x _ {p} \\
 +
 
 +
s _ {1} \dots s _ {p}
 +
\end{array}
 +
;  \lambda
 +
\right ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
\sum _ {m = 0 } ^  \infty 
 +
\frac{(- 1)  ^ {m} }{m! }
 +
 
 +
B _ {m} \left ( \begin{array}{c}
 +
x _ {1} \dots x _ {p} \\
 +
s _ {1} \dots s _ {p}
 +
\end{array}
 +
\right ) \lambda ^ {m + p } ,
 +
$$
 +
 
 +
which becomes equal to  $  D ( x, s; \lambda ) $
 +
for  $  p = 1 $.
 +
The series (16) is absolutely convergent for all finite values of  $  \lambda $
 +
and converges uniformly with respect to  $  x _ {1} \dots x _ {p} $,
 +
$  s _ {1} \dots s _ {p} $
 +
satisfying the conditions  $  a \leq  x _ {k} \leq  b $,
 +
$  a \leq  s _ {k} \leq  b $,
 +
$  k = 1 \dots p $.
 +
Suppose now that  $  \lambda $
 +
is an eigen value of  $  K $;  
 +
$  D ( \lambda _ {0} ) = 0 $,
 +
$  \lambda _ {0} \neq 0 $
 +
since  $  D ( 0) = 1 $.
 +
Let  $  r $
 +
be the multiplicity of the root  $  \lambda _ {0} $
 +
of the equation  $  D ( \lambda ) = 0 $.  
 +
There is a natural number  $  q \leq  r $
 +
such that all minors of  $  D ( \lambda _ {0} ) $
 +
of orders less than  $  q $
 +
are identically equal to zero, while the minor of order  $  q $
 +
is different from zero. There is some collection of values  $  x _ {1}  ^  \prime  \dots x _ {q}  ^  \prime  $,
 +
$  s _ {1}  ^  \prime  \dots s _ {q}  ^  \prime  $
 +
such that
 +
 
 +
$$
 +
D \left (
 +
\begin{array}{c}
 +
 
 +
x _ {1}  ^  \prime  \dots x _ {q}  ^  \prime
 +
\\
 +
 
 +
s _ {1}  ^  \prime  \dots s _ {q}  ^  \prime 
 +
\end{array}
 +
;  \lambda _ {0} \right )  \neq  0.
 +
$$
 +
 
 +
The number  $  q $
 +
is called the rank (or multiplicity) of the eigen value  $  \lambda _ {0} $.  
 +
The functions
 +
 
 +
$$ \tag{17 }
 +
\phi _ {k} ( x)  = \
 +
 
 +
\frac{D \left (
 +
\begin{array}{c}
 +
 
 +
x _ {1}  ^  \prime  \dots x _ {k - 1 }  ^  \prime  ,\
 +
x, x _ {k + 1 }  ^  \prime  \dots x _ {q}  ^  \prime
 +
\\
 +
 
 +
s _ {1}  ^  \prime  \dots s _ {k - 1 }  ^  \prime  ,\
 +
s _ {k}  ^  \prime  , s _ {k + 1 }  ^  \prime  \dots
 +
s _ {q}  ^  \prime 
 +
\end{array}
 +
; \lambda _ {0} \right ) }{D \left (
 +
\begin{array}{c}
 +
 
 +
x _ {1}  ^  \prime  \dots x _ {q}  ^  \prime
 +
\\
 +
 
 +
s _ {1}  ^  \prime  \dots s _ {q}  ^  \prime 
 +
\end{array}
 +
; \lambda _ {0} \right ) }
 +
 
 +
$$
  
 
are linearly independent solutions of (1h).
 
are linearly independent solutions of (1h).
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420125.png" /> has eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420126.png" />. These functions are called a complete system of eigen functions of (1h) (or of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420127.png" />) belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420128.png" /> if any other eigen function belonging to this eigen value is a linear combination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420129.png" />.
+
Suppose that $  \lambda _ {0} $
 +
has eigen functions $  \phi _ {1} \dots \phi _ {q} $.  
 +
These functions are called a complete system of eigen functions of (1h) (or of the kernel $  K $)  
 +
belonging to $  \lambda _ {0} $
 +
if any other eigen function belonging to this eigen value is a linear combination of $  \phi _ {1} \dots \phi _ {q} $.
 +
 
 +
If  $  \lambda _ {0} $
 +
is an eigen value of the homogeneous equation (1h) of multiplicity  $  q $,
 +
then it is also an eigen value of multiplicity  $  q $
 +
for the transposed equation to (1h):
 +
 
 +
$$ \tag{1ht }
 +
\psi ( x) - \lambda _ {0} \int\limits _ { a } ^ { b }
 +
K ( s, x) \psi ( s) \
 +
ds  = 0,
 +
$$
 +
 
 +
where a complete system of eigen functions for (1h) is defined by the formulas (17), and for (1ht) by similar formulas constructed for the transposed kernel  $  K ( s, x) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420130.png" /> is an eigen value of the homogeneous equation (1h) of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420131.png" />, then it is also an eigen value of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420132.png" /> for the transposed equation to (1h):
+
If $  \lambda _ {0} $
 +
is an eigen value of $  K $
 +
of multiplicity $  q $,  
 +
then equation (1) has a solution if and only if the following conditions are satisfied:
  
<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/f041420/f041420133.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1ht)</td></tr></table>
+
$$ \tag{18 }
 +
\int\limits _ { a } ^ { b }
 +
f ( t) \psi _ {k} ( t) \
 +
dt  = 0,\ \
 +
k = 1 \dots q,
 +
$$
  
where a complete system of eigen functions for (1h) is defined by the formulas (17), and for (1ht) by similar formulas constructed for the transposed kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420134.png" />.
+
where $  \psi _ {1} \dots \psi _ {q} $
 +
constitute a complete system of eigen functions of (1ht). If the conditions (18) are satisfied, then all solutions of (1) are determined by the formula
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420135.png" /> is an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420136.png" /> of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420137.png" />, then equation (1) has a solution if and only if the following conditions are satisfied:
+
$$
 +
\phi ( x)  = \
 +
f ( x) + \int\limits _ { a } ^ { b }
 +
H ( x, s) f ( s)  ds +
 +
\sum _ {k = 1 } ^ { q }
 +
c _ {k} \phi _ {k} ( x),
 +
$$
  
<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/f041420/f041420138.png" /></td> <td valign="top" style="width:5%;text-align:right;">(18)</td></tr></table>
+
where  $  c _ {1} \dots c _ {q} $
 +
are arbitrary constants,  $  \{ \phi _ {k} \} $
 +
is a complete system of eigen functions of (1h), and the function  $  H $
 +
is defined by the equation
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420139.png" /> constitute a complete system of eigen functions of (1ht). If the conditions (18) are satisfied, then all solutions of (1) are determined by the formula
+
$$
 +
H ( x, s) = \
  
<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/f041420/f041420140.png" /></td> </tr></table>
+
\frac{D \left ( \begin{array}{c}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420141.png" /> are arbitrary constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420142.png" /> is a complete system of eigen functions of (1h), and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420143.png" /> is defined by the equation
+
x, x _ {1}  ^  \prime  \dots x _ {q}  ^  \prime
 +
\\
  
<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/f041420/f041420144.png" /></td> </tr></table>
+
s, s _ {1}  ^  \prime  \dots s _ {q}  ^  \prime 
 +
\end{array}
 +
; \lambda _ {0} \right ) }{D \left ( \begin{array}{l}
  
A continuous kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420145.png" /> has at most a countable set of eigen values, which can only have the limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420146.png" />.
+
x _ {1}  ^  \prime  \dots x _ {q}  ^  \prime
 +
\\
 +
 
 +
s _ {1}  ^  \prime  \dots s _ {q}  ^  \prime 
 +
\end{array}
 +
;  \lambda _ {0} \right ) }
 +
.
 +
$$
 +
 
 +
A continuous kernel $  K $
 +
has at most a countable set of eigen values, which can only have the limit point $  \lambda = 0 $.
  
 
The propositions stated above for the equation (1) are called the [[Fredholm theorems|Fredholm theorems]]. Fredholm extended these theorems to the case of a system of such equations, and also to the case of one class of kernels with a weak singularity (see [[Integral operator|Integral operator]]).
 
The propositions stated above for the equation (1) are called the [[Fredholm theorems|Fredholm theorems]]. Fredholm extended these theorems to the case of a system of such equations, and also to the case of one class of kernels with a weak singularity (see [[Integral operator|Integral operator]]).
Line 149: Line 515:
 
In the Fredholm theorems one often considers, instead of the transposed equation (1ht), the adjoint equation to (1):
 
In the Fredholm theorems one often considers, instead of the transposed equation (1ht), 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/f041420/f041420147.png" /></td> </tr></table>
+
$$
 +
\psi ( x) - \overline{ {\lambda _ {0} }}\;
 +
\int\limits _ { a } ^ { b }
 +
\overline{ {K ( s, x) }}\;
 +
\psi ( s)  ds  = 0.
 +
$$
  
 
In this case the conditions (18) are replaced by the conditions
 
In this case the conditions (18) are replaced by the conditions
  
<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/f041420/f041420148.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
f ( t) \overline{ {\psi _ {k} ( t) }}\; \
 +
dt  = 0,\ \
 +
k = 1 \dots q.
 +
$$
  
The Fredholm method described above was generalized by T. Carleman [[#References|[9]]] (see also [[#References|[7]]], [[#References|[11]]]) to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420151.png" /> in (1) are assumed to be square-integrable. Under these assumptions the results of Fredholm stated above are valid.
+
The Fredholm method described above was generalized by T. Carleman [[#References|[9]]] (see also [[#References|[7]]], [[#References|[11]]]) to the case when f $,  
 +
$  \phi $
 +
and $  K $
 +
in (1) are assumed to be square-integrable. Under these assumptions the results of Fredholm stated above are valid.
  
 
In addition to the method of successive approximation and the Fredholm method for solving Fredholm equations, E. Schmidt, influenced by research of Hilbert, developed a method based on the construction, independent of the Fredholm theory, of a theory of equations (1) with a real symmetric kernel.
 
In addition to the method of successive approximation and the Fredholm method for solving Fredholm equations, E. Schmidt, influenced by research of Hilbert, developed a method based on the construction, independent of the Fredholm theory, of a theory of equations (1) with a real symmetric kernel.
  
The research of Hilbert and Schmidt prepared the ground for an abstract account of the Fredholm theory. Hilbert turned his attention to the fact that the Fredholm theory basically depends on the property of so-called complete continuity (compactness) of the integral transform with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420152.png" />. Hilbert formulated this property for bilinear forms. F. Riesz (see [[#References|[8]]]) showed that the main results of the Fredholm theory remain valid if the integral operator in (1) is replaced by an arbitrary completely-continuous operator acting on a complete function space. The research of Riesz was supplemented by J. Schauder (see [[#References|[10]]]) by means of introducing the concept of an adjoint operator in a Banach space, which made it possible to give a conclusive abstract formulation of the analogues of the Fredholm theorems in Banach spaces. These theorems are often called the Riesz–Schauder theorems. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420153.png" /> appearing in the statements of these theorems given below is assumed to act on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420154.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420155.png" /> denotes the Banach space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420156.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420157.png" /> the adjoint operator.
+
The research of Hilbert and Schmidt prepared the ground for an abstract account of the Fredholm theory. Hilbert turned his attention to the fact that the Fredholm theory basically depends on the property of so-called complete continuity (compactness) of the integral transform with kernel $  K $.  
 +
Hilbert formulated this property for bilinear forms. F. Riesz (see [[#References|[8]]]) showed that the main results of the Fredholm theory remain valid if the integral operator in (1) is replaced by an arbitrary completely-continuous operator acting on a complete function space. The research of Riesz was supplemented by J. Schauder (see [[#References|[10]]]) by means of introducing the concept of an adjoint operator in a Banach space, which made it possible to give a conclusive abstract formulation of the analogues of the Fredholm theorems in Banach spaces. These theorems are often called the Riesz–Schauder theorems. The operator $  V $
 +
appearing in the statements of these theorems given below is assumed to act on a Banach space $  E $;  
 +
$  E  ^ {*} $
 +
denotes the Banach space dual to $  E $,  
 +
and $  V  ^ {*} $
 +
the adjoint operator.
  
 
===Theorem 1.===
 
===Theorem 1.===
 
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/f041420/f041420158.png" /></td> <td valign="top" style="width:5%;text-align:right;">(19)</td></tr></table>
+
$$ \tag{19 }
 +
\phi - \lambda V \phi  = 0,\ \
 +
\phi \in E,
 +
$$
  
 
and its adjoint equation
 
and its adjoint 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/f041420/f041420159.png" /></td> <td valign="top" style="width:5%;text-align:right;">(20)</td></tr></table>
+
$$ \tag{20 }
 +
\psi - \lambda V  ^ {*} \psi  = 0,\ \
 +
\psi \in E  ^ {*} ,
 +
$$
  
have only the zero solution or the same number of linearly independent solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420161.png" />.
+
have only the zero solution or the same number of linearly independent solutions $  \phi _ {1} \dots \phi _ {q} $,  
 +
$  \psi _ {1} \dots \psi _ {q} $.
  
 
===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/f041420/f041420162.png" /></td> <td valign="top" style="width:5%;text-align:right;">(21)</td></tr></table>
+
$$ \tag{21 }
 +
\phi - \lambda V \phi  = f,\ \
 +
f, \phi \in E,
 +
$$
  
to exist, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420164.png" />; if these conditions are satisfied and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420165.png" /> is any solution of (21), then its general solution has the form
+
to exist, it is necessary and sufficient that $  \phi _ {k} ( f  ) = 0 $,  
 +
$  k = 1 \dots q $;  
 +
if these conditions are satisfied and if $  \phi _ {0} $
 +
is any solution of (21), then its general solution has the form
  
<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/f041420/f041420166.png" /></td> </tr></table>
+
$$
 +
\phi _ {0} +
 +
\sum _ {k = 1 } ^ { q }
 +
c _ {k} \phi _ {k} ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420167.png" /> are arbitrary constants.
+
where the $  c _ {k} $
 +
are arbitrary constants.
  
 
===Theorem 3.===
 
===Theorem 3.===
For each value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420168.png" />, the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420169.png" /> contains at most a finite number of eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420170.png" />, that is, values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420171.png" /> for which the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420172.png" /> has non-zero solutions.
+
For each value of $  r \neq 0 $,  
 +
the disc $  | \lambda | \leq  r $
 +
contains at most a finite number of eigen values of $  V $,  
 +
that is, values of $  \lambda $
 +
for which the equation $  \phi - \lambda V \phi = 0 $
 +
has non-zero solutions.
  
 
These theorems make it possible to prove the Fredholm theorems for an equation (1) in the case of a variety of concrete classes of integral operators (2), for example if the given and desired functions are square-integrable.
 
These theorems make it possible to prove the Fredholm theorems for an equation (1) in the case of a variety of concrete classes of integral operators (2), for example if the given and desired functions are square-integrable.
  
Instead of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420173.png" /> as domain of integration one can consider some bounded or unbounded measurable set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420174.png" /> in a space of any number of dimensions. Instead of the ordinary integral one can take the Lebesgue–Stieltjes integral relative to a non-negative measure.
+
Instead of the interval $  [ a, b] $
 +
as domain of integration one can consider some bounded or unbounded measurable set $  D $
 +
in a space of any number of dimensions. Instead of the ordinary integral one can take the Lebesgue–Stieltjes integral relative to a non-negative measure.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Goursat,  "Cours d'analyse mathématique" , '''3''' , Gauthier-Villars  (1923)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,  "Lectures on the theory of integral equations" , Graylock  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.V. Lovitt,  "Linear integral equations" , Dover, reprint  (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.G. Mikhlin,  "Linear integral equations" , Hindushtan Publ. Comp. , Delhi  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.V. Kantorovich,  V.I. Krylov,  "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.G. Mikhlin,  ''Dokl. Akad. Nauk SSSR'' , '''42''' :  9  (1944)  pp. 387–390</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F. Riesz,  "Ueber lineare Funktionalgleichungen"  ''Acta Math.'' , '''41'''  (1918)  pp. 71–98</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  T. Carleman,  "Zur Theorie der linearen Integralgleichungen"  ''Math. Z.'' , '''9'''  (1921)  pp. 196–217</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J. Schauder,  ''Studia Math.'' , '''2'''  (1930)  pp. 183–196</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  F. Smithies,  "The Fredholm theory of integral equations"  ''Duke Math. J.'' , '''8'''  (1941)  pp. 107–130</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Goursat,  "Cours d'analyse mathématique" , '''3''' , Gauthier-Villars  (1923)  pp. Chapt. 2</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,  "Lectures on the theory of integral equations" , Graylock  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.V. Lovitt,  "Linear integral equations" , Dover, reprint  (1950)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.G. Mikhlin,  "Linear integral equations" , Hindushtan Publ. Comp. , Delhi  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.V. Kantorovich,  V.I. Krylov,  "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.G. Mikhlin,  ''Dokl. Akad. Nauk SSSR'' , '''42''' :  9  (1944)  pp. 387–390</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  F. Riesz,  "Ueber lineare Funktionalgleichungen"  ''Acta Math.'' , '''41'''  (1918)  pp. 71–98</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  T. Carleman,  "Zur Theorie der linearen Integralgleichungen"  ''Math. Z.'' , '''9'''  (1921)  pp. 196–217</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J. Schauder,  ''Studia Math.'' , '''2'''  (1930)  pp. 183–196</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  F. Smithies,  "The Fredholm theory of integral equations"  ''Duke Math. J.'' , '''8'''  (1941)  pp. 107–130</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Line 202: Line 610:
 
For  "m-th iterated kernel"  see [[Iterated kernel|Iterated kernel]].
 
For  "m-th iterated kernel"  see [[Iterated kernel|Iterated kernel]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420175.png" />, then (6) converges uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420176.png" /> on bounded sets. A result about pointwise convergence of (6) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420177.png" /> can be found in [[#References|[a4]]]. Additional references for the topics treated in the article are [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]].
+
If $  | \lambda | < B  ^ {-} 1 $,  
 +
then (6) converges uniformly with respect to f $
 +
on bounded sets. A result about pointwise convergence of (6) for $  | \lambda | \geq  B  ^ {-} 1 $
 +
can be found in [[#References|[a4]]]. Additional references for the topics treated in the article are [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]].
  
 
==Fredholm equations of the first kind.==
 
==Fredholm equations of the first kind.==
 
Such Fredholm equations,
 
Such Fredholm 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/f/f041/f041420/f041420178.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
( K \phi ) ( x)  = \int\limits _ { a } ^ { b }
 +
K ( x , s ) \phi ( s)  d s  = \
 +
f ( x) ,\  x \in [ a , b ] ,
 +
$$
  
are ill-posed (see [[Ill-posed problems|Ill-posed problems]]). The notion of solution one usually uses is the notion of the best approximate solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420179.png" /> defined by
+
are ill-posed (see [[Ill-posed problems|Ill-posed problems]]). The notion of solution one usually uses is the notion of the best approximate solution $  K ^ { + } \phi $
 +
defined 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/f041420/f041420180.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\| K ^ { + } \phi - f \|  \rightarrow  \inf ,\ \
 +
\| \phi \|  \rightarrow  \inf ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420181.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420182.png" />-norm; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420183.png" /> is the  "Moore–Penrose generalized inverseMoore–Penrose generalized inverse"  (see [[#References|[a3]]]). The domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420184.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420185.png" />, which is a dense, but usually proper subset of the image space, since, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420186.png" /> is compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420187.png" /> is closed if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420188.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420189.png" /> has a [[Degenerate kernel|degenerate kernel]]. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420190.png" /> denotes the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420191.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420192.png" /> is non-closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420193.png" /> is unbounded, i.e. the solution of (a2), even if it exists, depends discontinuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041420/f041420194.png" />. For numerical methods for Fredholm equations of the first kind see [[Regularization method|Regularization method]] and [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]].
+
where $  \| \cdot \| $
 +
denotes the $  L _ {2} $-
 +
norm; $  K ^ { + } $
 +
is the  "Moore–Penrose generalized inverseMoore–Penrose generalized inverse"  (see [[#References|[a3]]]). The domain of definition $  D ( K ^ { + } ) $
 +
equals $  R ( K) \oplus R ( K )  ^  \perp  $,  
 +
which is a dense, but usually proper subset of the image space, since, if $  K $
 +
is compact, $  R ( K) $
 +
is closed if and only if $  \mathop{\rm dim}  R ( K) < \infty $,  
 +
i.e. $  K $
 +
has a [[Degenerate kernel|degenerate kernel]]. Here, $  R ( K) $
 +
denotes the range of $  K $.  
 +
If $  R ( K) $
 +
is non-closed, then $  K ^ { + } $
 +
is unbounded, i.e. the solution of (a2), even if it exists, depends discontinuously on f $.  
 +
For numerical methods for Fredholm equations of the first kind see [[Regularization method|Regularization method]] and [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hochstadt,  "Integral equations" , Wiley  (1973)</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">  M.Z. Nashed (ed.) , ''Genealized inverses and applications'' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Suzuki,  "On the convergence of Neumann series in Banach space"  ''Math. Ann.'' , '''220'''  (1976)  pp. 143–146</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Widom,  "Lectures on integral equations" , Amer. Book Comp.  (1969)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a7]</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  (1958)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hochstadt,  "Integral equations" , Wiley  (1973)</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">  M.Z. Nashed (ed.) , ''Genealized inverses and applications'' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Suzuki,  "On the convergence of Neumann series in Banach space"  ''Math. Ann.'' , '''220'''  (1976)  pp. 143–146</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Widom,  "Lectures on integral equations" , Amer. Book Comp.  (1969)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a7]</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  (1958)  (Translated from Russian)</TD></TR></table>

Revision as of 19:40, 5 June 2020


An integral equation of the form

$$ \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) \ ds = f ( x),\ \ x \in [ a, b], $$

— a Fredholm equation of the first kind, or one of the form

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

— a Fredholm equation of the second kind, if the integral operator

$$ \tag{2 } K \phi ( x) = \ \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds,\ \ x \in [ a, b], $$

is completely continuous in some function space $ E $. It is assumed that the free term $ f $ and the function $ \phi $ belong to $ E $. An important example of a Fredholm equation is one in which the kernel $ K $ satisfies the condition

$$ \tag{3 } B ^ {2} = \ \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } | K ( x, s) | ^ {2} \ dx ds < \infty , $$

and $ E = L _ {2} ([ a, b]) $.

The numerical parameter $ \lambda $ and the functions $ f $, $ \phi $ and $ K $ can take either real or complex values. For Fredholm equations of the first kind see Integral equation with symmetric kernel; Fredholm equation, numerical methods and Ill-posed problems. Below Fredholm equations of the second kind only are considered.

The method of successive approximation to solutions of Fredholm equations of the second kind.

This was the first method that was proposed for solving equation (1). To state this method, suppose that (1) is written in the form

$$ \tag{4 } \phi ( x) = \ f ( x) + \lambda K \phi ( x),\ \ x \in [ a, b]. $$

Assume that $ K $ satisfies the condition (3), and that $ E = L _ {2} ([ a, b]) $. Let the initial approximation to the desired solution be $ \phi _ {0} = f $; if the $ ( n - 1) $- th approximation $ \phi _ {n - 1 } $ has been constructed, then

$$ \phi _ {n} = \ f + \lambda K \phi _ {n - 1 } ; $$

in this case

$$ \tag{5 } \phi _ {n} = \ \sum _ {m = 0 } ^ { n } \lambda ^ {m} K _ {m} f, $$

where $ K _ {m} $ denotes the $ m $- th iterated kernel of $ K $. The function (5) is a partial sum of the series

$$ \tag{6 } \sum _ {m = 0 } ^ \infty \lambda ^ {m} K _ {m} f, $$

which is called the Neumann (or Liouville–Neumann) series. If $ | \lambda | < B ^ {-} 1 $, then (6) converges in the quadratic mean to a solution of (1), and this solution is unique (see, for example, [5]). If there is a positive constant $ A $ such that

$$ \int\limits _ { a } ^ { b } | K ( x, s) | ^ {2} \ ds \leq A,\ \ x \in [ a, b], $$

then (6) converges absolutely and uniformly. Generally speaking, (6) diverges if $ | \lambda | \geq B ^ {-} 1 $. Indeed, this is the case if $ K $ has an eigen value. But if $ K $ has no eigen values (as, for example, in the case of a Volterra kernel), then (6) converges for every value of $ \lambda $.

Fredholm's method for solving a Fredholm equation of the second kind.

The method of successive approximation enables one to construct solutions of (1), generally speaking, only for small values of $ \lambda $. A method that makes it possible to solve (1) for any value of $ \lambda $ was first proposed by E.I. Fredholm (1903). Under the assumption that $ K $ is continuous on the square $ [ a, b] \times [ a, b] $, and that the free term and the desired solution are continuous on $ [ a, b] $, the following is a brief description of the gist of this method.

Divide the interval $ [ a, b] $ into $ n $ equal parts of length $ h = ( b - a)/n $. If the integral in (1) would be replaced by a Riemann sum, the exact equation (1) would be replaced by the approximation

$$ \tag{7 } \phi ( x) - \lambda h \sum _ {j = 1 } ^ { n } K ( x, s _ {j} ) \phi ( s _ {j} ) = f ( x),\ \ x \in [ a, b]. $$

Set $ x = s _ {1} \dots s _ {n} $ successively in (7) to determine the approximate values of the unknown function $ \phi $ at the points $ s _ {j} $, thus obtaining the linear algebraic system

$$ \tag{8 } \phi _ {i} - \lambda h \sum _ {j = 0 } ^ { n } K _ {ij} \phi _ {j} = f _ {i} ,\ \ i = 1 \dots n, $$

where $ f ( s _ {i} ) = f _ {i} $, $ \phi ( s _ {i} ) = \phi _ {i} $, $ K ( s _ {i} , s _ {j} ) = K _ {ij} $. Whether the system (8) has a solution or not depends on the value of the determinant

$$ \Delta ( \lambda ) = \ \left | \begin{array}{ccc} 1 - \lambda hK _ {11} - \lambda hK _ {12} &\dots &- \lambda hK _ {1n} \\ \dots &\dots &\dots \\ - \lambda hK _ {n1} - \lambda hK _ {n2} &\dots &1 - \lambda hK _ {nn} \\ \end{array} \ \right | , $$

which is a polynomial in $ \lambda $. If $ \lambda $ is not one of the roots of this polynomial, then (8) has a solution. Solving this system and substituting the resulting values $ \phi _ {j} = \phi ( s _ {j} ) $ in (7), an approximate solution of (1) is obtained:

$$ \tag{9 } \phi ( x) \approx \ f ( x) + \lambda \frac{Q ( x, s _ {1} \dots s _ {n} ; \lambda ) }{\Delta ( \lambda ) } , $$

where $ Q $ and $ \Delta $ are polynomials in $ \lambda $. The method presented is one of the possible versions for constructing an approximate solution of the Fredholm equation (1) (see [6]).

One might expect that in the limit, as $ n \rightarrow \infty $ in such a way that the Riemann sum (7) tends to the integral in (1), the limit of the right-hand side of (9) becomes an exact solution of (1). Using formal limit transitions in analogous expressions, Fredholm established a formula that should represent a solution of (1):

$$ \tag{10 } \phi ( x) = \ f ( x) + \lambda \int\limits _ { a } ^ { b } R ( x, s; \lambda ) f ( s) ds, $$

where

$$ \tag{11 } R ( x, s; \lambda ) = \ \frac{D ( x, s; \lambda ) }{D ( \lambda ) } , $$

$$ \tag{12 } D ( \lambda ) = \sum _ {m = 0 } ^ \infty \frac{(- 1) ^ {m} }{m! } A _ {m} \lambda ^ {m} , $$

$$ \tag{13 } D ( x, s; \lambda ) = \sum _ {m = 0 } ^ \infty \frac{(- 1) ^ {m} }{m! } B _ {m} ( x, s) \lambda ^ {m} , $$

$$ \tag{14 } A _ {m} = \int\limits _ { a } ^ { b } \dots \int\limits _ { a } ^ { b } K \left ( \begin{array}{c} s _ {1} \dots s _ {m} \\ s _ {1} \dots s _ {m} \end{array} \right ) ds _ {1} \dots ds _ {m} , $$

$$ \tag{15 } B _ {m} ( x, s) = \int\limits _ { a } ^ { b } \dots \int\limits _ { a } ^ { b } K \left ( \begin{array}{c} x, s _ {1} \dots s _ {m} \\ s,\ s _ {1} \dots s _ {m} \end{array} \right ) ds _ {1} \dots ds _ {m} , $$

$$ K \left ( \begin{array}{c} x _ {1} \dots x _ {n} \\ s _ {1} \dots s _ {n} \end{array} \right ) = \left | \begin{array}{ccc} K ( x _ {1} , s _ {1} ) &\dots &K ( x _ {1} , s _ {n} ) \\ \dots &\dots &\dots \\ K ( x _ {n} , s _ {1} ) &\dots &K ( x _ {n} , s _ {n} ) \\ \end{array} \right | . $$

To calculate $ A _ {m} $ and $ B _ {m} ( x, s) $, instead of the formulas (14) and (15) one can make use of the following recurrence relations:

$$ A _ {0} = 1,\ \ B _ {0} ( x, s) = \ K ( x, s),\ \ A _ {m} = \int\limits _ { a } ^ { b } B _ {m - 1 } ( s, s) ds, $$

$$ B _ {m} ( x, s) = K ( x, s) A _ {m} - m \int\limits _ { a } ^ { b } K ( x, t) B _ {m - 1 } ( t, s) dt, $$

$$ m = 1, 2 , . . . . $$

The series (12) and (13) are called Fredholm series. The function $ D ( \lambda ) $ is called the Fredholm determinant of $ K $; $ D ( x, s; \lambda ) $ is called the first Fredholm minor for $ D ( \lambda ) $; and the function (11) is called the resolvent (or solving kernel or reciprocal kernel) of $ K $( or of equation (1)).

The justification of the limit transitions mentioned above, which lead to (10), was carried out by D. Hilbert (see Integral equation). Fredholm, having constructed the series (12) and (13), then proved directly and rigorously that they converge for all finite values of $ \lambda $ and that (13), moreover, converges uniformly with respect to $ x $ and $ s $ on $ [ a, b] \times [ a, b] $. The establishment of a connection between $ D ( \lambda ) $ and $ D ( x, s; \lambda ) $ enabled him to prove the following proposition: If $ D ( \lambda ) \neq 0 $, then equation (1) has one and only one solution, which is expressed by formula (10).

If follows from this proposition that a value of $ \lambda $ that is not a root of the Fredholm determinant is a regular value for the homogeneous equation associated with (1):

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

that is, in this case the equation has only the zero solution. If $ \lambda $ is a root of the equation $ D ( \lambda ) = 0 $, then $ \lambda $ is a pole of the resolvent (11) of equation (1h) and an eigen value of this latter equation. In order to construct by the Fredholm method the eigen functions belonging to this eigen value, one introduces the concept of the $ p $- th minor of $ D ( \lambda ) $. Let

$$ B _ {0} \left ( \begin{array}{c} x _ {1} \dots x _ {p} \\ s _ {1} \dots s _ {p} \end{array} \right ) = K \left ( \begin{array}{c} x _ {1} \dots x _ {p} \\ s _ {1} \dots s _ {p} \end{array} \right ) , $$

$$ B _ {m} \left ( \begin{array}{c} x _ {1} \dots x _ {p} \\ s _ {1} \dots s _ {p} \end{array} \right ) = \int\limits _ { a } ^ { b } \dots \int\limits _ { a } ^ { b } K \left ( \begin{array}{c} x _ {1} \dots x _ {p} ,\ t _ {1} \dots t _ {m} \\ s _ {1} \dots s _ {p} , t _ {1} \dots t _ {m} \end{array} \right ) \times $$

$$ \times dt _ {1} \dots dt _ {m} . $$

Then the $ p $- th minor for $ D ( \lambda ) $ is the series

$$ \tag{16 } D \left ( \begin{array}{c} x _ {1} \dots x _ {p} \\ s _ {1} \dots s _ {p} \end{array} ; \lambda \right ) = $$

$$ = \ \sum _ {m = 0 } ^ \infty \frac{(- 1) ^ {m} }{m! } B _ {m} \left ( \begin{array}{c} x _ {1} \dots x _ {p} \\ s _ {1} \dots s _ {p} \end{array} \right ) \lambda ^ {m + p } , $$

which becomes equal to $ D ( x, s; \lambda ) $ for $ p = 1 $. The series (16) is absolutely convergent for all finite values of $ \lambda $ and converges uniformly with respect to $ x _ {1} \dots x _ {p} $, $ s _ {1} \dots s _ {p} $ satisfying the conditions $ a \leq x _ {k} \leq b $, $ a \leq s _ {k} \leq b $, $ k = 1 \dots p $. Suppose now that $ \lambda $ is an eigen value of $ K $; $ D ( \lambda _ {0} ) = 0 $, $ \lambda _ {0} \neq 0 $ since $ D ( 0) = 1 $. Let $ r $ be the multiplicity of the root $ \lambda _ {0} $ of the equation $ D ( \lambda ) = 0 $. There is a natural number $ q \leq r $ such that all minors of $ D ( \lambda _ {0} ) $ of orders less than $ q $ are identically equal to zero, while the minor of order $ q $ is different from zero. There is some collection of values $ x _ {1} ^ \prime \dots x _ {q} ^ \prime $, $ s _ {1} ^ \prime \dots s _ {q} ^ \prime $ such that

$$ D \left ( \begin{array}{c} x _ {1} ^ \prime \dots x _ {q} ^ \prime \\ s _ {1} ^ \prime \dots s _ {q} ^ \prime \end{array} ; \lambda _ {0} \right ) \neq 0. $$

The number $ q $ is called the rank (or multiplicity) of the eigen value $ \lambda _ {0} $. The functions

$$ \tag{17 } \phi _ {k} ( x) = \ \frac{D \left ( \begin{array}{c} x _ {1} ^ \prime \dots x _ {k - 1 } ^ \prime ,\ x, x _ {k + 1 } ^ \prime \dots x _ {q} ^ \prime \\ s _ {1} ^ \prime \dots s _ {k - 1 } ^ \prime ,\ s _ {k} ^ \prime , s _ {k + 1 } ^ \prime \dots s _ {q} ^ \prime \end{array} ; \lambda _ {0} \right ) }{D \left ( \begin{array}{c} x _ {1} ^ \prime \dots x _ {q} ^ \prime \\ s _ {1} ^ \prime \dots s _ {q} ^ \prime \end{array} ; \lambda _ {0} \right ) } $$

are linearly independent solutions of (1h).

Suppose that $ \lambda _ {0} $ has eigen functions $ \phi _ {1} \dots \phi _ {q} $. These functions are called a complete system of eigen functions of (1h) (or of the kernel $ K $) belonging to $ \lambda _ {0} $ if any other eigen function belonging to this eigen value is a linear combination of $ \phi _ {1} \dots \phi _ {q} $.

If $ \lambda _ {0} $ is an eigen value of the homogeneous equation (1h) of multiplicity $ q $, then it is also an eigen value of multiplicity $ q $ for the transposed equation to (1h):

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

where a complete system of eigen functions for (1h) is defined by the formulas (17), and for (1ht) by similar formulas constructed for the transposed kernel $ K ( s, x) $.

If $ \lambda _ {0} $ is an eigen value of $ K $ of multiplicity $ q $, then equation (1) has a solution if and only if the following conditions are satisfied:

$$ \tag{18 } \int\limits _ { a } ^ { b } f ( t) \psi _ {k} ( t) \ dt = 0,\ \ k = 1 \dots q, $$

where $ \psi _ {1} \dots \psi _ {q} $ constitute a complete system of eigen functions of (1ht). If the conditions (18) are satisfied, then all solutions of (1) are determined by the formula

$$ \phi ( x) = \ f ( x) + \int\limits _ { a } ^ { b } H ( x, s) f ( s) ds + \sum _ {k = 1 } ^ { q } c _ {k} \phi _ {k} ( x), $$

where $ c _ {1} \dots c _ {q} $ are arbitrary constants, $ \{ \phi _ {k} \} $ is a complete system of eigen functions of (1h), and the function $ H $ is defined by the equation

$$ H ( x, s) = \ \frac{D \left ( \begin{array}{c} x, x _ {1} ^ \prime \dots x _ {q} ^ \prime \\ s, s _ {1} ^ \prime \dots s _ {q} ^ \prime \end{array} ; \lambda _ {0} \right ) }{D \left ( \begin{array}{l} x _ {1} ^ \prime \dots x _ {q} ^ \prime \\ s _ {1} ^ \prime \dots s _ {q} ^ \prime \end{array} ; \lambda _ {0} \right ) } . $$

A continuous kernel $ K $ has at most a countable set of eigen values, which can only have the limit point $ \lambda = 0 $.

The propositions stated above for the equation (1) are called the Fredholm theorems. Fredholm extended these theorems to the case of a system of such equations, and also to the case of one class of kernels with a weak singularity (see Integral operator).

The Fredholm alternative follows by combining the Fredholm theorems.

In the Fredholm theorems one often considers, instead of the transposed equation (1ht), the adjoint equation to (1):

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

In this case the conditions (18) are replaced by the conditions

$$ \int\limits _ { a } ^ { b } f ( t) \overline{ {\psi _ {k} ( t) }}\; \ dt = 0,\ \ k = 1 \dots q. $$

The Fredholm method described above was generalized by T. Carleman [9] (see also [7], [11]) to the case when $ f $, $ \phi $ and $ K $ in (1) are assumed to be square-integrable. Under these assumptions the results of Fredholm stated above are valid.

In addition to the method of successive approximation and the Fredholm method for solving Fredholm equations, E. Schmidt, influenced by research of Hilbert, developed a method based on the construction, independent of the Fredholm theory, of a theory of equations (1) with a real symmetric kernel.

The research of Hilbert and Schmidt prepared the ground for an abstract account of the Fredholm theory. Hilbert turned his attention to the fact that the Fredholm theory basically depends on the property of so-called complete continuity (compactness) of the integral transform with kernel $ K $. Hilbert formulated this property for bilinear forms. F. Riesz (see [8]) showed that the main results of the Fredholm theory remain valid if the integral operator in (1) is replaced by an arbitrary completely-continuous operator acting on a complete function space. The research of Riesz was supplemented by J. Schauder (see [10]) by means of introducing the concept of an adjoint operator in a Banach space, which made it possible to give a conclusive abstract formulation of the analogues of the Fredholm theorems in Banach spaces. These theorems are often called the Riesz–Schauder theorems. The operator $ V $ appearing in the statements of these theorems given below is assumed to act on a Banach space $ E $; $ E ^ {*} $ denotes the Banach space dual to $ E $, and $ V ^ {*} $ the adjoint operator.

Theorem 1.

The homogeneous equation

$$ \tag{19 } \phi - \lambda V \phi = 0,\ \ \phi \in E, $$

and its adjoint equation

$$ \tag{20 } \psi - \lambda V ^ {*} \psi = 0,\ \ \psi \in E ^ {*} , $$

have only the zero solution or the same number of linearly independent solutions $ \phi _ {1} \dots \phi _ {q} $, $ \psi _ {1} \dots \psi _ {q} $.

Theorem 2.

For a solution of the inhomogeneous equation

$$ \tag{21 } \phi - \lambda V \phi = f,\ \ f, \phi \in E, $$

to exist, it is necessary and sufficient that $ \phi _ {k} ( f ) = 0 $, $ k = 1 \dots q $; if these conditions are satisfied and if $ \phi _ {0} $ is any solution of (21), then its general solution has the form

$$ \phi _ {0} + \sum _ {k = 1 } ^ { q } c _ {k} \phi _ {k} , $$

where the $ c _ {k} $ are arbitrary constants.

Theorem 3.

For each value of $ r \neq 0 $, the disc $ | \lambda | \leq r $ contains at most a finite number of eigen values of $ V $, that is, values of $ \lambda $ for which the equation $ \phi - \lambda V \phi = 0 $ has non-zero solutions.

These theorems make it possible to prove the Fredholm theorems for an equation (1) in the case of a variety of concrete classes of integral operators (2), for example if the given and desired functions are square-integrable.

Instead of the interval $ [ a, b] $ as domain of integration one can consider some bounded or unbounded measurable set $ D $ in a space of any number of dimensions. Instead of the ordinary integral one can take the Lebesgue–Stieltjes integral relative to a non-negative measure.

References

[1] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) pp. Chapt. 1 (Translated from Russian)
[2] E. Goursat, "Cours d'analyse mathématique" , 3 , Gauthier-Villars (1923) pp. Chapt. 2
[3] I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian)
[4] W.V. Lovitt, "Linear integral equations" , Dover, reprint (1950)
[5] S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian)
[6] L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)
[7] S.G. Mikhlin, Dokl. Akad. Nauk SSSR , 42 : 9 (1944) pp. 387–390
[8] F. Riesz, "Ueber lineare Funktionalgleichungen" Acta Math. , 41 (1918) pp. 71–98
[9] T. Carleman, "Zur Theorie der linearen Integralgleichungen" Math. Z. , 9 (1921) pp. 196–217
[10] J. Schauder, Studia Math. , 2 (1930) pp. 183–196
[11] F. Smithies, "The Fredholm theory of integral equations" Duke Math. J. , 8 (1941) pp. 107–130

Comments

See also Noetherian integral equation.

Concerning the terminology transposed/adjoint equation see Fredholm theorems. A completely-continuous operator is nowadays usually called a compact operator.

For "m-th iterated kernel" see Iterated kernel.

If $ | \lambda | < B ^ {-} 1 $, then (6) converges uniformly with respect to $ f $ on bounded sets. A result about pointwise convergence of (6) for $ | \lambda | \geq B ^ {-} 1 $ can be found in [a4]. Additional references for the topics treated in the article are [a1], [a2], [a5].

Fredholm equations of the first kind.

Such Fredholm equations,

$$ \tag{a1 } ( K \phi ) ( x) = \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s = \ f ( x) ,\ x \in [ a , b ] , $$

are ill-posed (see Ill-posed problems). The notion of solution one usually uses is the notion of the best approximate solution $ K ^ { + } \phi $ defined by

$$ \tag{a2 } \| K ^ { + } \phi - f \| \rightarrow \inf ,\ \ \| \phi \| \rightarrow \inf , $$

where $ \| \cdot \| $ denotes the $ L _ {2} $- norm; $ K ^ { + } $ is the "Moore–Penrose generalized inverseMoore–Penrose generalized inverse" (see [a3]). The domain of definition $ D ( K ^ { + } ) $ equals $ R ( K) \oplus R ( K ) ^ \perp $, which is a dense, but usually proper subset of the image space, since, if $ K $ is compact, $ R ( K) $ is closed if and only if $ \mathop{\rm dim} R ( K) < \infty $, i.e. $ K $ has a degenerate kernel. Here, $ R ( K) $ denotes the range of $ K $. If $ R ( K) $ is non-closed, then $ K ^ { + } $ is unbounded, i.e. the solution of (a2), even if it exists, depends discontinuously on $ f $. For numerical methods for Fredholm equations of the first kind see Regularization method and Fredholm equation, numerical methods.

References

[a1] H. Hochstadt, "Integral equations" , Wiley (1973)
[a2] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
[a3] M.Z. Nashed (ed.) , Genealized inverses and applications , Acad. Press (1976)
[a4] N. Suzuki, "On the convergence of Neumann series in Banach space" Math. Ann. , 220 (1976) pp. 143–146
[a5] H. Widom, "Lectures on integral equations" , Amer. Book Comp. (1969)
[a6] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a7] 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 (1958) (Translated from Russian)
How to Cite This Entry:
Fredholm equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_equation&oldid=46977
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article