We give conditions under which a self-mapping on a bounded closed convex subset, containing zero, of a reflexive Banach space is never nonexpansive, i.e., there is no renorming with respect to which the mapping is nonexpansive. This provides with a unifying procedure to prove that the classical mappings of Kakutani, Nirenberg and P. K. Lin are not uniformly Lipschitzian in any bounded closed convex subset of \(\ell _2\) which they leave invariant. In an analogous way, the well known fixed point free mappings of Goebel-Kirk-Thele and Baillon, although they are uniformly Lipschitzian, are also seen to be never nonexpansive in any subset of \(\ell _2\) where they are self-mappings.
Fixed point Nonexpansive mapping Classical fixed point free mappings
Mathematics Subject Classification
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The first author has been supported by MINECO and FEDER Project MTM2014-57838-C2-2-P. The second author has been supported by Grant MTM2012-34847-C02-02.
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