ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

k-Set contractive retractions and k-set contractions in spaces of continuous functions

Giulio Trombetta Dipartimento di Matematica Arcavacata di Rende Università della Calabria, Italy

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 → S is said to be a retraction provided that x = Rx for all x ∈ S. We prove that in some Banach spaces of continuous functions for every ε > 0 there exists a retraction of the close unit ball onto the unit sphere being a (1 + ε)-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.