The Schauder fixed-point theorem is an extension of the
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simples ...
to
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s, which may be of infinite dimension. It asserts that if
is a nonempty
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
closed subset of a
Hausdorff topological vector space
and
is a continuous mapping of
into itself such that
is contained in a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
subset of
, then
has a
fixed point.
A consequence, called Schaefer's fixed-point theorem, is particularly useful for proving existence of solutions to
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
.
Schaefer's theorem is in fact a special case of the far reaching
Leray–Schauder theorem which was proved earlier by
Juliusz Schauder
Juliusz Paweł Schauder (; 21 September 1899, Lwów, Austria-Hungary – September 1943, Lwów, Occupied Poland) was a Polish mathematician of Jewish origin, known for his work in functional analysis, partial differential equations and m ...
and
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
Life and career
He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
.
The statement is as follows:
Let
be a continuous and compact mapping of a Banach space
into itself, such that the set
:
is bounded. Then
has a fixed point. (A ''compact mapping'' in this context is one for which the image of every bounded set is
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sin ...
.)
History
The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the
Scottish book. In 1934,
Tychonoff proved the theorem for the case when ''K'' is a compact convex subset of a
locally convex space. This version is known as the Schauder–Tychonoff fixed-point theorem. B. V. Singbal proved the theorem for the more general case where ''K'' may be non-compact; the proof can be found in the appendix of Bonsall's book (see references).
See also
*
Fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...
s
*
Banach fixed-point theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
*
Kakutani fixed-point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex set, convex, compact set, compact subset of a Euclidean sp ...
References
* J. Schauder, ''Der Fixpunktsatz in Funktionalräumen'', Studia Math. 2 (1930), 171–180
* A. Tychonoff, ''Ein Fixpunktsatz'', Mathematische Annalen 111 (1935), 767–776
* F. F. Bonsall, ''Lectures on some fixed point theorems of functional analysis'', Bombay 1962
* D. Gilbarg,
N. Trudinger, ''Elliptic Partial Differential Equations of Second Order''. .
* E. Zeidler, ''Nonlinear Functional Analysis and its Applications, ''I'' - Fixed-Point Theorems''
External links
*
*
* .
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Fixed-point theorems
Theorems in functional analysis
Topological vector spaces