Fixed point free maps of a closed ball with small measures of noncompactness

We show that in all infinite-dimensional normed spaces it is possible to construct a fixed point free continuous map of the unit ball whose measure of noncompactness is bounded by 2. Moreover, for a large class of spaces (containing separable spaces, Hilbert spaces, and $\ell_\infty\big(S\big)$) even the best possible bound 1 is attained for certain measures of noncompactness

Topological Analysis: From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions

This monograph is an introduction to some special aspects of topology, functional analysis, and analysis for the advanced reader. It also wants to develop a degree theory for function triples which unifies and extends most known degree theories. The book aims to be self-contained and many chapters could even serve as a basis of a course on the covered topics. Only knowledge in basic calculus and of linear algebra is assumed

Merging of degree and index theory

The topological approaches to find solutions of a coincidence equation f1(x)=f2(x) can roughly be divided into degree and index theories. We describe how these methods can be combined. We are led to a concept of an extended degree theory for function triples which turns out to be natural in many respects. In particular, this approach is useful to find solutions of inclusion problems F(x)∈Φ(x). As a side result, we obtain a necessary condition for a compact AR to be a topological group

A general degree for function triples

Consider a fixed class of maps $F$ for which there is a degree theory for the coincidence problem $F(x)=\varphi(x)$ with compact $\varphi$. It is proved that under very natural assumptions this degree extends to a degree for function triples which in particular provides a degree for coincidence inclusions $F(x)\in\Phi(x)$

Fixed point theorems and fixed point index for countably condensing maps

It is proved that there exists a fixed point index theory for operators which are condensing on the countable subsets of the space only. Even weaker compactness assumptions on countable subsets suffice, e.g. conditions with respect to classes of measures of noncompactness, or if measures of noncompactness of countable noncompact sets are not preserved (not necessarily decreased). As an application, we prove a generalization of the Fredholm alternative