# Fredholm theorems

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for integral equations

## Contents

### 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 .

How to Cite This Entry:
Fredholm theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_theorems&oldid=12814
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article