Variations on some finite-dimensional fixed-point theorems (original) (raw)

Variations on the Brouwer Fixed Point Theorem: A Survey

Mathematics

This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in R n . The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem.

Fixed-point-like theorems on subspaces

Fixed Point Theory and Applications, 2003

We prove a fixed-point-like theorem for multivalued mappings defined on the finite Cartesian product of Grassmannian manifolds and convex sets. Our result generalizes two different kinds of theorems: the fixed-point-like theorem by and the fixed-point theorem by (which generalizes Kakutani's theorem (1941)).

Review Article - A study of some fixed point theorems for various types of maps

International Journal of Mathematics Trends and Technology, 2016

S. The theory has several rather well-defined (yet overlapping) branches. The purely topological theory as well as those topics which lie on the borderline of topology and functional analysis (e.g. those related to Leray-Schauder theory) have their roots in the celebrated theorem of L. E. J. Brouwer. This paper presents a review of the available literature on fixed point theorems for various types of maps.

A Borsuk-Ulam Type Generalization of the Leray-Schauder Fixed Point Theorem

2009

A generalization of the classical Leray-Schauder fixed point theorem, based on the in finite- dimensional Borsuk-Ulam type antipode construction, is proposed. Two completely different proofs based on the projection operator approach and on a weak version of the well known Krein-Milman theorem are presented.

Some fixed point theorems in topological vector spaces

Demonstratio Mathematica, 1996

1. A fixed point theorem for nonself mappings Let A be a subset of a sequentially complete Hausdorff locally convex topological vector space E (over the field J?) with calibration I \ By the terminology of R.T. Moore [6], a calibration T for E means a collection of continuous seminorms p on E which induce the topology of E. Let / , g be nonself mappings from A into E. Let ap, bp, cp, dp and ep be nonnegative real numbers such that ap + bp + cp + dp + ep < 1 and for any x, y in A, and p e r

Generalizations of the Poincaré Birkhoff fixed point theorem

Bulletin of the Australian Mathematical Society, 1977

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

Borsuk-Ulam theorem, fixed point index and chain approximations for maps with multiplicity

Pacific Journal of Mathematics, 1992

In this article we consider m-acyclic maps with respect to a field F and prove the existence of chain approximation for such maps. Furthermore we provide a unified approach to the Borsuk-Ulam theorem and the Bourgin-Yang generalization. Finally we prove the existence of ^-systems for certain m-acyclic maps and define a fixed point index. I. Preliminaries.

A generic view on the theorems of Brouwer and Schauder

Mathematische Zeitschrift, 1993

Several generic fixed point theorems have been established by De Blasi [5, 6], Myjak [9], De Blasi and Myjak [7], Myjak and Sampalmieri [101, Butler [4], Dominguez Benavides [8], Vidossich [121, to quote just a few. However it seems that the well-known and important theorem of Schauder [11] and its finitedimensional version, Brouwer's theorem [31, have not yet been generically investigated. We now fill this gap. For a survey of generic results on convex bodies in Euclidean spaces, see [15]. A large set of fixed points Let E be a Banach space and consider a compact convex set K c E with more than one point. By the Schauder theorem, every continuous function f: K ~ K has at least one fixed point. We shall show here that, in the sense of Baire categories, most continuous functions have uncountably many fixed points. More precisely, the set of all their fixed points is homeomorphic to the Cantor set. This sharply contrasts with the generic finiteness of the fixed point set found by De Blasi [6] in another space of functions and with the behaviour of the non-expansive mappings which have, generically, a single fixed point ([12], see also [9, p. 29]). Of course, the space ~g(K) of all continuous functions f: K-~ K, equipped with the metric of uniform convergence, is complete, hence a Baire space. "Most" means "all, except those in a first category set" (so, a property is generic if it is shared by most elements). Forf~ Cg(K), let Ff be the set of all fixed points off For x ~ E, B(x, r) denotes the open ball of centre x and radius r. The proof of Theorem A below will make use of the following recent result.

Fixed Point and Non-Retract Theorems – Classical Circular Tours

Taiwanese Journal of Mathematics, 2001

We show that the Brouwer fixed point theorem is equivalent to a number of results closely related to the Euclidean spaces or n-simplexes or n-balls. Among them are the Sperner lemma, the KKM theorem, some intersection theorems, various fixed point theorems, an intermediate value theorem, various non-retract theorems, the non-contractibility of spheres, and others.