$\{x_n=T^n x_0\} (x_0\neq 0)$ does not converge.

These aspects made the study of nonexpansive mappings one the major and most active research areas of nonlinear analysis since the mid-1960’s. We mention the early contributions in this respect, due to Mann [@Mann] in 1953 and to Krasnoselskij [@Kra55] in 1955, respectively, who considered instead of Picard iteration (which does not converge, in general, for nonexpansive mappings) an explicit averaged iteration of the form $x_{n+1}=f(x_n,T x_n), n\geq 0$. We state in the following the classical result due to Krasnoselskij [@Kra55].

\[tK\] Let $X$ be a uniformly convex Banach space and $C$ a closed subset of $X$. If $T:C \rightarrow  C$ is nonexpansive and $\overline{T(C)}$ is compact, then the mapping defined by $
T_{\frac{1}{2}} x = \frac{1}{2}x + \frac{1}{2} Tx
$ has the property that its sequence of iterates always converges to a point of $T$. Since $T$ and $T_{\frac{1}{2}}$ have the same fixed points, the limit of a convergent sequence given by $
x_{n+1}= \frac{1}{2} x_n+\frac{1}{2}T x_n, n\geq 0
$ is necessarily a fixed point of $T$.

The following two problems were important in the study of nonexpansive mappings and were approached, amongst others, by [@Sch], [@Bro65]-[@Bro65c], [@Ede66], [@Hal], [@Dot] etc. :

