The abstract of the talk (see below) may be downloaded in one of the following formats:

Let X be an infinite-dimensional Banach space, and let $B$ and $S$ be its
close unit ball and unit sphere, respectively. A~continuous mapping $
R:B\rightarrow S
$
is said to be a retraction provided that $x=Rx$ for all $%
x\in S$. We prove that in some Banach spaces of continuous functions
for every $\varepsilon >0$ there exists a retraction of the close unit ball
onto the unit sphere being a \hbox{$(1+\varepsilon)$}-set contraction.
This result and the properties of the fixed-point index of a $k$-set
contraction with $k<1$ are the main tools in order to obtain a theorem on the
existence of positive
eigenvalues of $k$-set contractions.

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