a\) what additional conditions on the structure of the ambient space $X$ and / or on the properties of $T$ must be added to assure that a nonexpansive mapping has at least one fixed point ?

b\) how one can locate and approximate such a fixed point ?

Apart from the theoretical aspects mentioned above, nonexpansive mappings are extremely important from the point of view of their applications because, see [@Bru]:

"1) Nonexpansive maps are intimately connected with the monotonicity methods developed since the early 1960Õs and constitute one of the first classes of nonlinear mappings for which fixed point theorems were obtained by using the fine geometric properties of the underlying Banach spaces instead of compactness properties.

2\) Nonexpansive mappings appear in applications as the transition operators for initial value problems of differential inclusions of the form $0 \in du +T(t)u$, where the operators $\{T(t)\}$ are, in general, set-valued and are accretive or dissipative and minimally continuous."

A simple search in MathScinet shows the impressive number of 3606 indexed papers that bear the term “nonexpansive” in their title and 5826 papers that hold the term “nonexpansive” anywhere, as by November 30, 2018.

Having in view the significant interest of 