P.K. Lin gave the first example of a non-reflexive Banach space (X,||?||)
with the fixed point property (FPP) for nonexpansive mappings and showed
this fact for (l1,||?||1) with the equivalent norm ||?|| given by ||x|| = sup
k?N 8k/1+8k ?1,n=k |xn|, for all x = (xn)n?N ? l1. We wonder (c0, ||?||1)
analogue of P.K. Lin?s work and we give positive answer if functions are
affine nonexpansive. In our work, for x = (?k)k ? c0, we define |||x||| := lim
p?? sup ?k?N ?k (?1,j=k |?j|p/2j)1/p where ?k ?k 3, k is
strictly increasing with ?k > 2, ?k ? N, then we prove that (c0,|||?|||) has
the fixed point property for affine |||?|||-nonexpansive self-mappings. Next,
we generalize this result and show that if ?(?) is an equivalent norm to the
usual norm on c0 such that lim sup n ?(1/n ?n,m=1 xm + x) = lim
sup n ?(1/n ?n,m=1 xm) + ?(x) for every weakly null sequence
(xn)n and for all x ? c0, then for every ? > 0, c0 with the norm ||?||? =
?(?)+?|||?||| has the FPP for affine ||?||?-nonexpansive self-mappings.