Borsuk presentations and the fixed-point index

Retracts, fixed point index and differential equations

Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions. Retractos, ı́ndice de punto fijo y ecuaciones d...

THE FIXED POINT INDEX AND ASYMPTOTIC FIXED POINT THEOREMS FOR ft-SET-CONTRACTIONS

The Szlenk Index and the Fixed Point Property under Renorming

Assume that X is a Banach space such that its Szlenk index Sz X is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R X < 2, where R · is the Garcı́a-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz Y ≤ ω, then, X can be re...

Fixed Point Index in Symmetric Products

Let U be an open subset of a locally compact metric ANR X and let f : U → X be a continuous map. In this paper we study the fixed point index of the map that f induces in the n-symmetric product of X, Fn(X). This index can detect the existence of periodic orbits of period ≤ n of f , and it can be used to obtain the Euler characteristic of the n-symmetric product of a manifold X, χ(Fn(X)). We co...

Fixed Point Index for Composite Type Maps

An index theory is presented for compact absorbing contractive Jc or SJc maps and several new fixed point theorems are given for such maps. AMS(MOS) subject classification: 47H10