Fredholm alternative

A statement of an alternative that follows from the Fredholm theorems. In the case of a linear Fredholm integral equation of the second kind,

(1)

the Fredholm alternative states that either equation (1) and its conjugate equation

(2)

have unique solutions , for any given functions and , or the corresponding homogeneous equations

(1prm)

(2prm)

have non-zero solutions, where the number of linearly independent solutions is finite and is the same for both equations.

In the second case equation (1) has a solution if and only if

where is a complete system of linearly independent solutions of (2prm). Here the general solution of (1) has the form

where is some solution of (1), is a complete system of linearly independent solutions of (1prm), and the are arbitrary constants. Similar statements also hold for equation (2).

Let be a continuous linear operator mapping a Banach space into itself; let and be the corresponding dual space and dual operator. Consider the equations:

(3)

(4)

(3prm)

(4prm)

The Fredholm alternative for means the following: 1) either the equations (3) and (4) have solutions, for arbitrary right-hand sides, and then their solutions are unique; or 2) the homogeneous equations (3prm) and (4prm) have the same finite number of linearly independent solutions and , respectively; in this case, for equation (3), or (4) respectively, to have a solution, it is necessary and sufficient that , , or , , respectively; here the general solution of (3) is given by

and the general solution of (4) by

where (respectively, ) is some solution of (3) ((4)), and are arbitrary constants.

Each of the following two conditions is necessary and sufficient for the Fredholm alternative to hold for the operator .

1) can be represented in the form

where is an operator with a two-sided continuous inverse and is a compact operator.

2) can be represented in the form

where is an operator with a two-sided continuous inverse and is a finite-dimensional operator.

References

[1]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) pp. Chapt. 1 (Translated from Russian)
[2]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[3]	L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)

Comments

The precise form of the Fredholm alternative is as follows: Consider the equations (1) and (1prm) with a continuous kernel . Then either equation (1) has a continuous solution for any right-hand side or the homogeneous equation (1prm) has a non-trivial solution. In abstract form the alternative may be stated as follows. For a Fredholm operator of index zero (cf. Index of an operator) acting on a Banach space the following holds true: Either is invertible or has a non-trivial kernel (cf. Kernel of a linear operator; Kernel of an integral operator).

References

[a1]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977)
[a2]	A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) pp. Chapt. 5