P.K. Lin gave the first example of a non-reflexive Banach space (X; II II) with the fixed point property (FPP) for nonexpansive mappings and showed this fact for (`1; II II1) with the equivalent norm k k given by kxk = sup k2N 8k 1 + 8k X1 n=k jxnj; for all x = (xn)n2N 2 `1 : We wonder (c0; k k1) analogue of P.K. Lin’s work and we give positive answer if functions are ane nonexpansive. In our work, for x = (k)k 2 c0, we define jjjxjjj := lim p!1 sup k2N k 0BBBBBB@ X1 j=k j p 2j 1CCCCCCA 1p where k "k 3; k is strictly increasing with k > 2; 8k 2 N; then we prove that (c0; jjjjjj) has the fixed point property for ane jjjjjj-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 0BBBB@ 1 n Xn m=1 xm + x 1CCCCA = lim sup n 0BBBB@ 1 n Xn m=1 xm 1CCCCA + (x) for every weakly null sequence (xn)n and for all x 2 c0, then for every > 0, c0 with the norm k k = ()+jjjjjj has the FPP for ane k k -nonexpansive self-mappings.