Schauder fixed point theorem

The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if $K$ is a nonempty convex subset of a Hausdorff topological vector space $V$ and $T$ is a continuous mapping of $K$ into itself such that $T(K)$ is contained in a compact subset of $K$ , then $T$ has a fixed point.

A consequence, called Schaefer's fixed point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was discovered earlier by Juliusz Schauder and Jean Leray. The statement is as follows:

Let $T$ be a continuous and compact mapping of a Banach space $X$ into itself, such that the set

\{ x \in X : x = \lambda T x \mbox{ for some } 0 \leq \lambda \leq 1 \}

is bounded. Then $T$ has a fixed point.

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).

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. ISBN 3-540-41160-7.
E. Zeidler, Nonlinear Functional Analysis and its Applications, I - Fixed-Point Theorems

External links

Hazewinkel, Michiel, ed. (2001), "Schauder theorem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
"Schauder fixed point theorem". PlanetMath. with attached proof (for the Banach space case).

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

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Schauder fixed point theorem

History

See also

References

External links