The fixed point property for some generalized nonexpansive mappings and renormings

In 2008, T. Suzuki [33] defined one of most relevant extensions of the notion of nonexpansivity. This notion was extended in [15] defining the, so called, CλCλ-mappings. In the last years, some papers have appeared trying to extend the most important results about existence of fixed points for nonexpansive mappings to this wider class of mappings (see, for instance, [2], [5], [9], [10] and [15]). In this paper we continue this project proving that Banach spaces with extended unconditional bases satisfy the fixed point property for CλCλ-mappings if the unconditional constant of the basis is small enough. We also begin a new project on renorming with the fixed point property for CλCλ-mappings, proving that any separable Banach space can be renormed to satisfy the fixed point property for CλCλ-mappings in a stable sense, that is, the space with the new norm satisfies the fixed point property which, in addition, is inherited by those isomorphic spaces which are close to it in the sense of the Banach–Mazur distance.