A Large Class of Non-weakly Compact Subsets in a Renorming of c_0 with FPP

摘要

In 1979, Goebel and Kuczumow showed that very large class of non-weak* compact, closed, bounded and convex subsets of ?~1 has the fixed point property (FPP) for nonexpansive mappings. Later, in 2008, Lin proved that ?~1 can be renormed to have FPP for nonexpansive mappings, co-analogue of Lin's result is still open. However, renorming Co, we prove that Goebel and Kuczumow analogy can be proved under affinity condition. That is, we prove that there exist a renorming of C_0 and a very large class of non-weakly compact, closed, bounded and convex subsets of C_0 with FPP for affine nonexpansive mappings whereas Dowling, Lennard and Turett proved in 2004 that weak compactness is equivalent to FPP for nonexpansive mappings when C_0 is considered with its usual norm instead of any equivalent norm.