Non-Fredholm integral equation
For example, the Fourier integral equation
has the solution
where is an arbitrary positive constant; an infinite set of linearly independent solutions corresponds to the eigen value of (1), that is, for equation (1) Fredholm's theorem that a homogeneous equation has finitely many linearly independent solutions, does not hold.
In the case of the Lalesco–Picard integral equation
every is an eigen value, namely, to every positive number correspond the two independent solutions
Consequently, for equation (2) Fredholm's theorem, that the set of eigen values of an equation is at most countable, does not hold.
The theory has been worked out in detail for two classes of non-Fredholm integral equations: equations in which the unknown function stands under the sign of an improper integral in the sense of the principal value (singular integral equations, cf. Singular integral equation); and equations in which the unknown function stands under the sign of an integral convolution transformation (cf. Integral equation of convolution type). For such equations, generally speaking, the Fredholm alternative is violated, as well as the fact that the numbers of linearly independent solutions of the homogeneous equation and its adjoint are equal.
|||I.I. Privalov, "Integral equations" , Moscow-Leningrad (1937) (In Russian)|
|||I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian)|
|[a1]||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)|
Non-Fredholm integral equation. B.V. Khvedelidze (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Non-Fredholm_integral_equation&oldid=13328