Fredholm theorems

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for integral equations

Theorem 1.

The homogeneous equation

 (1)

and its transposed equation

 (2)

have, for a fixed value of the parameter , either only the trivial solution, or have the same finite number of linearly independent solutions: ; .

Theorem 2.

For a solution of the inhomogeneous equation

 (3)

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):

 (4)

Theorem 3.

(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.

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 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 [1].

References

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