New families of nonreflexive Banach spaces with the fixed point property

Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets

“…Lin's norm and the ν p (·) norm, it is not difficult to check that, in the case where ν p (·) is equivalent to the · 1 -norm, ν p (·) + λ||| · ||| L is a renorming in 1 which is also near-infinity concentrated for every λ > 0. Therefore we can also deduce (see also [2,Section 4]):…”

“…Nevertheless, there exist some equivalent norms on 1 which do not satisify this condition but they are still sequentially separating [2, Example 3.2]. Furthermore, there exist Banach spaces with sequentially separating norms that are not isomorphic to 1 , although the existence of such a norm implies that the Banach space X is "similar" to 1 , in the sense that it has the Schur property, and so is hereditarily 1 [2,Corollary 7.4]. Recall that a Banach space X is hereditarily 1 if each infinite dimensional closed subspace of X contains a further subspace isomorphic to 1 .…”

“…P. K. Lin's result opened new avenues of research in the fixed point theory of nonexpansive mappings, since he settled negatively the long-standing open question: "Does the fixed point property imply reflexivity?" Since then, many other articles have appeared obtaining sufficient conditions that imply the FPP for equivalent norms on 1 (see for instance [2,6,7,8,10,11,12,15]).…”

About Some Families of Nonexpansive Mappings with respect to Renorming

“…Recently, many works have appeared to be looking for new examples of nonreflexive Banach spaces enjoying the FPP or trying to find some structure on families of equivalent norms with the FPP. In the first sense the works should be mentioned [4][5][6][7]. In the second way the works are remarkable [8,9].…”