### 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.

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.

#### Keywords

Fixed point theorem; countably condensing operator; fixed point index; degree theory; measure of noncompactness; sequential measure of noncompactness; multivalued mapping; fundamental set; ultimately compact operator; Fredholm operator; positive operator

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