Browder fixed-point theorem

The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for uniformly convex Banach spaces. It asserts that if $K$ is a nonempty convex closed bounded set in uniformly convex Banach space and $f$ is a mapping of $K$ into itself such that $\|f(x)-f(y)\|\leq \|x-y\|$ (i.e. $f$ is non-expansive), then $f$ has a fixed point.

History

Following the publication in 1965 of two independent versions of the theorem by Felix Browder and by William Kirk, a new proof of Michael Edelstein showed that, in a uniformly convex Banach space, every iterative sequence $f^{n}x_{0}$ of a non-expansive map $f$ has a unique asymptotic center, which is a fixed point of $f$ . (An asymptotic center of a sequence $(x_{k})_{k\in \mathbb {N} }$ , if it exists, is a limit of the Chebyshev centers $c_{n}$ for truncated sequences $(x_{k})_{k\geq n}$ .) A stronger property than asymptotic center is Delta-limit of T.C. Lim, which in the uniformly convex space coincides with the weak limit if the space has the Opial property.

References

F. E. Browder, Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. U.S.A. 54 (1965) 1041–1044
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965) 1004–1006.
M. Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206-208.

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert) LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex 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
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–Mazur Banach–Saks Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman 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
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

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Browder fixed-point theorem

History

See also

References