Fredholm solvability

In mathematics, Fredholm solvability encompasses results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, the Fredholm-type properties of the operator involved. The concept is named after Erik Ivar Fredholm.

Let $A$ be a real $n \times n$ -matrix and $b\in \mathbb {R} ^{n}$ a vector.

The Fredholm alternative in $\mathbb {R} ^{n}$ states that the equation $Ax=b$ has a solution if and only if $b^{T}v=0$ for every vector $v\in \mathbb {R} ^{n}$ satisfying $A^{T}v=0$ . This alternative has many applications, for example, in bifurcation theory. It can be generalized to abstract spaces. So, let $E$ and $F$ be Banach spaces and let $T:E\rightarrow F$ be a continuous linear operator. Let $E^{*}$ , respectively $F^{*}$ , denote the topological dual of $E$ , respectively $F$ , and let $T^{*}$ denote the adjoint of $T$ (cf. also Duality; Adjoint operator). Define

(\ker T^{*})^{\perp }=\{y\in F:(y,y^{*})=0{\text{ for every }}y^{*}\in \ker T^{*}\}

An equation $Tx=y$ is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever $y\in (\ker T^{*})^{\perp }$ . A classical result states that $Tx=y$ is normally solvable if and only if $T(E)$ is closed in $F$ .

In non-linear analysis, this latter result is used as definition of normal solvability for non-linear operators.

References

F. Hausdorff, "Zur Theorie der linearen metrischen Räume" Journal für die Reine und Angewandte Mathematik, 167 (1932) pp. 265 [1] [2]
V. A. Kozlov, V.G. Maz'ya, J. Rossmann, "Elliptic boundary value problems in domains with point singularities", Amer. Math. Soc. (1997) [3] [4]
A. T. Prilepko, D.G. Orlovsky, I.A. Vasin, "Methods for solving inverse problems in mathematical physics", M. Dekker (2000) [5][6]
D. G. Orlovskij, "The Fredholm solvability of inverse problems for abstract differential equations" A.N. Tikhonov (ed.) et al. (ed.), Ill-Posed Problems in the Natural Sciences, VSP (1992) [7]