Fixed point theorems and fixed point index for countably condensing maps

Abstract

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.