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 |
Fredholm alternative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_alternative&oldid=46976