Abstract
In this work, we present some new fixed point theorems for nonexpansive maps, 1-set contractions, and demi-closed nonexpansive perturbations of nonexpansive maps defined on closed, convex, not necessarily bounded subsets of Banach spaces; the stress will be made on the so-called Furi-Pera boundary condition. The proofs use the Kuratowski measure of noncompactness and rely on a recent compactness result for the approximate fixed point set. To illustrate the results obtained, applications to a fixed point theorem in a Banach algebra and to an integral equation are provided.