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In this paper we prove that for every unbounded convex closed set C in c 0 there exists a nonexpansive mapping T : C → C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory.doi:10.1090/s0002-9939-2011-10938-9 fatcat:sqml3pdow5h6vlyy5ci7bxi3ya