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

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 $\lambda$.