- Source: Fredholm solvability
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
b
∈
R
n
{\displaystyle b\in \mathbb {R} ^{n}}
a vector.
The Fredholm alternative in
R
n
{\displaystyle \mathbb {R} ^{n}}
states that the equation
A
x
=
b
{\displaystyle Ax=b}
has a solution if and only if
b
T
v
=
0
{\displaystyle b^{T}v=0}
for every vector
v
∈
R
n
{\displaystyle v\in \mathbb {R} ^{n}}
satisfying
A
T
v
=
0
{\displaystyle A^{T}v=0}
. This alternative has many applications, for example, in bifurcation theory. It can be generalized to abstract spaces. So, let
E
{\displaystyle E}
and
F
{\displaystyle F}
be Banach spaces and let
T
:
E
→
F
{\displaystyle T:E\rightarrow F}
be a continuous linear operator. Let
E
∗
{\displaystyle E^{*}}
, respectively
F
∗
{\displaystyle F^{*}}
, denote the topological dual of
E
{\displaystyle E}
, respectively
F
{\displaystyle F}
, and let
T
∗
{\displaystyle T^{*}}
denote the adjoint of
T
{\displaystyle T}
(cf. also Duality; Adjoint operator). Define
(
ker
T
∗
)
⊥
=
{
y
∈
F
:
(
y
,
y
∗
)
=
0
for every
y
∗
∈
ker
T
∗
}
{\displaystyle (\ker T^{*})^{\perp }=\{y\in F:(y,y^{*})=0{\text{ for every }}y^{*}\in \ker T^{*}\}}
An equation
T
x
=
y
{\displaystyle Tx=y}
is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever
y
∈
(
ker
T
∗
)
⊥
{\displaystyle y\in (\ker T^{*})^{\perp }}
. A classical result states that
T
x
=
y
{\displaystyle Tx=y}
is normally solvable if and only if
T
(
E
)
{\displaystyle T(E)}
is closed in
F
{\displaystyle F}
.
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]
Kata Kunci Pencarian:
- Fredholm solvability
- Erik Ivar Fredholm
- Fredholm alternative
- Fredholm theory
- Fredholm integral equation
- Resolvent formalism
- Liouville–Neumann series
- Neumann–Poincaré operator
- Integral transform
- Integral equation
