Fredholm solvability
Let be a real
-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, e.g. in bifurcation theory. It can be generalized to abstract spaces. So, let and
be Banach spaces (cf. Banach space) 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
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An equation is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever
(cf. also Normal solvability). 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.
The phrase "Fredholm solvability" refers to results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, Fredholm-type properties of the operator involved.
References
[a1] | F. Hausdorff, "Zur Theorie der linearen metrischen Räume" J. Reine Angew. Math. , 167 (1932) pp. 265 Zbl 0003.33104 Zbl 58.1113.05 |
[a2] | V.A. Kozlov, V.G. Maz'ya, J. Rossmann, "Elliptic boundary value problems in domains with point singularities" , Amer. Math. Soc. (1997) MR1469972 Zbl 0947.35004 |
[a3] | A.T. Prilepko, D.G. Orlovsky, I.A. Vasin, "Methods for solving inverse problems in mathematical physics" , M. Dekker (2000) MR1748236 Zbl 0947.35173 |
[a4] | 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) Zbl 0789.35178 |
Fredholm solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_solvability&oldid=24446