Fixed points of multimaps which are not necessarily nonexpansive

Let [inline-graphic not available: see fulltext] be a nonempty closed bounded convex subset of a Banach space [inline-graphic not available: see fulltext] whose characteristic of noncompact convexity is less than [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] a continuous [inline-graphic not available: see fulltext]- [inline-graphic not available: see fulltext]-contractive [inline-graphic not available: see fulltext] map (which is not necessarily nonexpansive) from [inline-graphic not available: see fulltext] to [inline-graphic not available: see fulltext] satisfying an inwardness condition, where [inline-graphic not available: see fulltext] is the family of all nonempty compact convex subsets of [inline-graphic not available: see fulltext]. It is proved that [inline-graphic not available: see fulltext] has a fixed point. Some fixed points results for noncontinuous maps are also derived as applications. Our result contains, as a special case, a recent result of Benavides and Ramírez (2004).