Consider a subset C of a Banach space (X, .). Let T be a mapping
from a set C to itself, it is said that a point x in C is a fixed
point for T if Tx=x. This mapping is a nonexpansive mapping if Tx -
Ty x - y for all x and y belonging to C. It is said that a Banach
space X has the fixed point property (FPP) if every nonexpansive
mapping defined from a closed convex bounded subset into itself has
a fixed point. For a long time, it was conjectured that all Banach
spaces with the FPP had to be reflexive. In 2008, it was given an
unexpected answer to this conjecture: it was found the first known
nonreflexive Banach space with the FPP. On the other hand, in 2009,
it was proved that every reflexive Banach space can be renormed to
have the FPP. This leads us to the following question: Which type
of nonreflexive Banach spaces can be renormed to have the FPP? So,
the main object of this book is to study new families of
nonreflexive Banach spaces which can be renormed to have the FPP
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