Journal of the
Korean Mathematical Society JKMS

Using an index theory for countably condensing maps, we show the existence of eigenvalues for countably $k$-set contractive maps and countably condensing maps in an infinite dimensional Banach space $X$, under certain condition that depends on the quantitative characteristic, that is, the infimum of all $k\ge 1$ for which there is a countably $k$-set-contractive retraction of the closed unit ball of $X$ onto its boundary.

Keywords: eigenvalues, countably condensing maps, fixed point index

MSC numbers: 47J10, 47H09, 47H11