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,

$$ \tag{1 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x),\ \ x \in [ a, b], $$

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

$$ \tag{2 } \psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b } \overline{ {K ( s, x) }}\; \psi ( s) ds = g ( x),\ \ x \in [ a, b], $$

have unique solutions $ \phi , \psi $, for any given functions $ f $ and $ g $, or the corresponding homogeneous equations

$$ \tag{1'} \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = 0, $$

$$ \tag{2'} \psi ( x) - \overline \lambda \; \int\limits _ { a } ^ { b } \overline{ {K ( s, x) }}\; \psi ( s) ds = 0 $$

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

$$ \int\limits _ { a } ^ { b } f ( x) \overline{ {\psi _ {k} ( x) }}\; dx = 0,\ \ k = 1 \dots n, $$

where $ \psi _ {1} \dots \psi _ {n} $ is a complete system of linearly independent solutions of (2'}). Here the general solution of (1) has the form

$$ \phi ( x) = \ \phi _ {*} ( x) + \sum _ {k = 1 } ^ { n } c _ {k} \phi _ {k} ( x), $$

where $ \phi _ {*} $ is some solution of (1), $ \phi _ {1} \dots \phi _ {n} $ is a complete system of linearly independent solutions of (1'}), and the $ c _ {k} $ are arbitrary constants. Similar statements also hold for equation (2).

Let $ T $ be a continuous linear operator mapping a Banach space $ E $ into itself; let $ E ^ {*} $ and $ T ^ {*} $ be the corresponding dual space and dual operator. Consider the equations:

$$ \tag{3 } T ( x) = y,\ \ x, y \in E, $$

$$ \tag{4 } T ^ {*} ( g) = f,\ g, f \in E ^ {*} , $$

$$ \tag{3'} T ( x) = 0,\ x \in E, $$

$$ \tag{4'} T ^ {*} ( g) = 0,\ g \in E ^ {*} . $$

The Fredholm alternative for $ T $ 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 (3'}) and (4'}) have the same finite number of linearly independent solutions $ x _ {1} \dots x _ {n} $ and $ g _ {1} \dots g _ {n} $, respectively; in this case, for equation (3), or (4) respectively, to have a solution, it is necessary and sufficient that $ g _ {k} ( y) = 0 $, $ k = 1 \dots n $, or $ f ( x _ {k} ) = 0 $, $ k = 1 \dots n $, respectively; here the general solution of (3) is given by

$$ x = x ^ {*} + \sum _ {k = 1 } ^ { n } c _ {k} x _ {k} , $$

and the general solution of (4) by

$$ g = g ^ {*} + \sum _ {k = 1 } ^ { n } c _ {k} g _ {k} , $$

where $ x ^ {*} $( respectively, $ g ^ {*} $) is some solution of (3) ((4)), and $ c _ {1} \dots c _ {n} $ are arbitrary constants.

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

1) $ T $ can be represented in the form

$$ T = W + V, $$

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

2) $ T $ can be represented in the form

$$ T = W _ {1} + V _ {1} , $$

where $ W _ {1} $ is an operator with a two-sided continuous inverse and $ V _ {1} $ 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 (1'}) with a continuous kernel $ K $. Then either equation (1) has a continuous solution $ \phi $ for any right-hand side $ f $ or the homogeneous equation (1'}) has a non-trivial solution. In abstract form the alternative may be stated as follows. For a Fredholm operator $ T $ of index zero (cf. Index of an operator) acting on a Banach space the following holds true: Either $ T $ is invertible or $ T $ 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