Fredholm equation

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)