Let C be a nonempty closed bounded convex subset of a Banach space X whose characteristic of noncompact convexity is less than 1 and T a continuous 1 - χ -contractive SL map (which is not necessarily nonexpansive) from C to K C ( X ) satisfying an inwardness condition, where K C ( X ) is the family of all nonempty compact convex subsets of X . It is proved that T 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).