Difference between revisions of "Fredholm solvability"
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
m (grammatical structure) |
||
Line 11: | Line 11: | ||
In non-linear analysis, this latter result is used as definition of normal solvability for non-linear operators. | 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. | + | The phrase "Fredholm solvability" refers to results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, the Fredholm-type properties of the operator involved. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Hausdorff, "Zur Theorie der linearen metrischen Räume" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 265 {{MR|}} {{ZBL|0003.33104}} {{ZBL|58.1113.05}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.A. Kozlov, V.G. Maz'ya, J. Rossmann, "Elliptic boundary value problems in domains with point singularities" , Amer. Math. Soc. (1997) {{MR|1469972}} {{ZBL|0947.35004}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.T. Prilepko, D.G. Orlovsky, I.A. Vasin, "Methods for solving inverse problems in mathematical physics" , M. Dekker (2000) {{MR|1748236}} {{ZBL|0947.35173}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|0789.35178}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Hausdorff, "Zur Theorie der linearen metrischen Räume" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 265 {{MR|}} {{ZBL|0003.33104}} {{ZBL|58.1113.05}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.A. Kozlov, V.G. Maz'ya, J. Rossmann, "Elliptic boundary value problems in domains with point singularities" , Amer. Math. Soc. (1997) {{MR|1469972}} {{ZBL|0947.35004}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.T. Prilepko, D.G. Orlovsky, I.A. Vasin, "Methods for solving inverse problems in mathematical physics" , M. Dekker (2000) {{MR|1748236}} {{ZBL|0947.35173}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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) {{MR|}} {{ZBL|0789.35178}} </TD></TR></table> |
Revision as of 17:12, 30 October 2016
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
![]() |
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, the 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