Fredholm theorems

for integral equations

Theorem 1.[edit]

The homogeneous equation

$$\begin{array}{}\text{(1)}& \varphi (x)-\lambda \underset{a}{\overset{b}{\int }}K(x,s)\varphi (s)ds=0\end{array}$$

and its transposed equation

$$\begin{array}{}\text{(2)}& \psi (x)-\lambda \underset{a}{\overset{b}{\int }}K(s,x)\psi (s)ds=0\end{array}$$

have, for a fixed value of the parameter $\lambda$, either only the trivial solution, or have the same finite number of linearly independent solutions: ${\varphi }_1\dots {\varphi }_n$; ${\psi }_1\dots {\psi }_n$.

Theorem 2.[edit]

For a solution of the inhomogeneous equation

$$\begin{array}{}\text{(3)}& \varphi (x)-\lambda \underset{a}{\overset{b}{\int }}K(x,s)\varphi (s)ds=f(x)\end{array}$$

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):

$$\begin{array}{}\text{(4)}& \underset{a}{\overset{b}{\int }}f(x){\psi }_j(x)dx=0,j=1\dots n.\end{array}$$

Theorem 3.[edit]

(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.[edit]

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):

$$\psi (x)-\bar{\lambda }\underset{a}{\overset{b}{\int }}\overline{K(s,x)}\psi (s)ds=0.$$

In this case condition (4) is replaced by

$$\underset{a}{\overset{b}{\int }}f(x)\overline{{\psi }_j(x)}dx=0,j=1\dots n.$$

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

References[edit]

Comments[edit]

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 $\bar{\lambda }$ is replaced by $\lambda$.

References[edit]