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 × n-matrix and a vector.

The Fredholm alternative in states that the equation has a solution if and only if for every vector satisfying . This alternative has many applications, for example, in bifurcation theory. It can be generalized to abstract spaces. So, let and be Banach spaces and let be a continuous linear operator. Let , respectively , denote the topological dual of , respectively , and let denote the adjoint of (cf. also Duality; Adjoint operator). Define

An equation is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever . A classical result states that is normally solvable if and only if is closed in .

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