Abstract. In 2013, Jiménez--Melado and Llorens--Fuster proved that the renorming of $\ell ^2$, $|x|=\max\{\|x\|_2,p(x)\}$, where $p$ is a seminorm on $\ell ^2$ satisfying certain conditions, has the weak fixed point property. In this paper, we generalize this result for a Banach space having normal structure and Schauder basis. From this, we derive that every Banach space having normal structure and Schauder basis has an equivalent renorming that lacks asymptotic normal structure but has the weak fixed point property.

DOI: 10.14232/actasm-017-339-4

AMS Subject Classification (1991): 47H09, 47H10, 46B20

Keyword(s): nonexpansive mappings, weak fixed point property

Received December 30, 2017 and in final form April 28, 2018. (Registered under 89/2017.)