Fixed Point Property Under Renormings in Non-Reflexive Banach Spaces : Some techniques and examples

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.