ABUNDANCE OF EQUIVALENT NORMS ON c(0) WITH FIXED POINT PROPERTY FOR AFFINE NONEXPANSIVE MAPPINGS

In 1979, K. Goebel and T. Kuczumow showed that a large class of closed, bounded, convex (c.b.c.), non-weak*-compact subsets K of l(1) has the fixed point property for nonexpansive mappings. Later, in 2008, P.K. Lin proved that l(1) can be renormed to have the fixed point property for nonexpansive mappings. Then, Nezir recently worked on c(0)-analogue of Goebel and Kuczumow's theorem with an equivalent norm and showed that there exists a large class of equivalent norms parallel to.parallel to on c(0) for which there exist non-weakly compact closed, bounded, convex subsets that have the fixed point property for affine parallel to.parallel to-nonexpansive mappings. In fact, he sees that his examples are closed, convex hulls of some asymptotically isometric (ai) c(0)-summing basic sequences whereas Lennard and Nezir in 2011 showed that the closed, convex hull of any ai c(0)-summing basic sequence fails the fixed point property for affine parallel to.parallel to(infinity)-nonexpansive mappings. In this work, we show that equivalent norms with fixed point property for affine nonexpansive mappings are somewhat abundant. Firstly, we construct many types of equivalent norms and even show some norms are exactly the same as the natural norm while it is not clear to see that in the beginning, and then we show with our new type of equivalent norms c(0) do not contain any asymptotically isometric copy of c(0). Next, we see that Nezir's equivalent norms are not the only ones with fixed point property for affine nonexpansive mappings on his sets.