Renorming c0 and Closed, Bounded, Convex Sets with Fixed Point Property for Affine Nonexpansive Mappings

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, Hernandez-Linares worked on l(1) and in his Ph.D. thesis, supervisored under Maria Japon, 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.