for integral equations
The homogeneous equation
and its transposed equation
have, for a fixed value of the parameter , either only the trivial solution, or have the same finite number of linearly independent solutions: ; .
For a solution of the inhomogeneous equation
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):
(the Fredholm alternative). Either the inhomogeneous equation (3) has a solution, whatever its right-hand side , or the corresponding homogeneous equation (1) has non-trivial solutions.
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 it is sufficient that the kernel of equation (3) be square-integrable on the set ( and may be infinite). When this condition is violated, (3) may turn out to be a non-Fredholm integral equation. When the parameter and the functions involved in (3) take complex values, then instead of the transposed equation (2) one often considers the adjoint equation to (1):
In this case condition (4) is replaced by
These theorems were proved by E.I. Fredholm .
|||E.I. Fredholm, "Sur une classe d'equations fonctionnelles" Acta Math. , 27 (1903) pp. 365–390|
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 is replaced by .
|[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)|
Fredholm theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_theorems&oldid=12814