# 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)

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).