Fredholm theorems

for integral equations

Theorem 1.

The homogeneous equation

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

and its transposed equation

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

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

Theorem 2.

For a solution of the inhomogeneous equation

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

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

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

Theorem 3.

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

Theorem 4.

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

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

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

In this case condition (4) is replaced by

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

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

References

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

Comments

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

References

[a1]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a2]	K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
[a3]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[a4]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)