Namespaces
Variants
Actions

Difference between revisions of "Fredholm solvability"

From Encyclopedia of Mathematics
Jump to: navigation, search
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
How to Cite This Entry:
Fredholm solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_solvability&oldid=24446
This article was adapted from an original article by G. IsacThemistocles M. Rassias (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article