limit case of Picard-Banach contractions, i.e., of mappings $T:K\rightarrow K$ which satisfy, in the case of normed linear spaces, a contraction condition of the form $$\|Tx-Ty\|\leq c\cdot \|x-y\|, \forall x,y\in K,$$ where $c\in [0,1)$ is the contraction coefficient.

Banach contraction mapping principle, see for example [@Ber07], assures that any contraction $T$ on a Banach space has a unique fixed point which is the limit of the sequence of its iterates $\{T^n x_0\}$, for any $x_0\in K$. A similar assertion for nonexpansive mappings is not more true.

Indeed, if $K$ is a closed nonempty subset of a Banach space $X$ and $T:K\rightarrow K$ is nonexpansive, it is known that $T$ may not have a fixed point or it may have many fixed points, and third, it may may happen that, even if $T$ has a unique fixed point, the Picard iteration $\{x_n=T^n x_0\}$ may fail to converge to such a fixed point. One of the simplest examples of such a map is $Tx=1-x$ on $[0,1]$ with the usual norm, which gives, for $x_0=1$ say, $x_{2n}=1$ and $x_{2n+1}=0$. Also, rotation about the origin of the unit disk in the plane is another example of nonexpansive mapping having a unique fixed point while 