Английская Википедия:Fredholm solvability

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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 Шаблон:Math be a real Шаблон:Math-matrix and <math>b\in\mathbb R^n</math> a vector.

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

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

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

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]