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Fredholm solvability

From Wikipedia, the free encyclopedia

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.

References

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  • 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]