Renorming co and Closed, Bounded, Convex Sets with Fixed Point Property for Affine Nonexpansive Mappings

摘要

In 2008, P.K. Lin provided the first example of a nonreflexive space that can be renormed to have fixed point property for nonexpansive mappings. This space was the Banach space of absolutely summable sequences l~1 and researchers aim to generalize this to c_0, Banach space of null sequences. Before P.K. Lin's intriguing result, in 1979, Goebel and Kuczumow showed that there is a large class of non-weak* compact closed, bounded, convex subsets of l~1 with fixed point property for nonexpansive mappings. Then, P.K. Lin inspired by Goebel and Kuczumow's ideas to give his result. Similarly to P.K. Lin's study, Hernández-Linares worked on L~1 and in his Ph.D. thesis, supervisored under Maria Japón, showed that L~1 can be renormed to have fixed point property for affine nonexpansive mappings. Then, related questions for c_0 have been considered by researchers. Recently. Nezir constructed several equivalent norms on c_0 and showed that there are non-weakly compact closed, bounded, convex subsets of c_0 with fixed point property for affine nonexpansive mappings. In this study, we construct a family of equivalent norms containing those developed by Nezir as well and show that there exists a large class of non-weakly compact closed, bounded, convex subsets of c_0 with fixed point property for affine nonexpansive mappings.