operators contains, among many classes of operators (see [@Pet]), the compact operators and, in particular, the completely continuous and strongly continuous operators.

Hence, from Theorem \[th1\] (and also from Corollary \[cor1\]) one obtains the pioneering result of Krasnoselskij from 1955 ([@Kra55], [@Kra56]) $$\frac{1}{2}(x_n+Tx_n)\rightarrow q\in Fix\,(T) \,(\textnormal{ as } n\rightarrow \infty$$ which was subsequently extended by Schaefer in 1957 ([@Sch]) to the general Krasnoselskij scheme $$(1-\lambda)x_n+\lambda Tx_n\rightarrow q\in Fix\,(T), \,(\textnormal{ as } n\rightarrow \infty,\, 0<\lambda <1),$$ a result established in the general setting of a uniformly Banach space.

The above results for compact operators have been extended to strictly convex Banach spaces by Edelstein in [@Ede66].

(Theorem 6, [@BroP67]) \[cor1\] Let $C$ be a bounded closed convex subset of a Hilbert space $H$ and $T:C\rightarrow C$ be a nonexpansive and demicompact operator. Then the set $Fix\,(T)$ of fixed points of $T$ is a nonempty convex set and there exists $\lambda\in \left(0,1\right)$ such that, for any given $x_0\in C$, the Krasnoselskij iteration $\{x_n\}_{n=0}^{\infty}$ given by $$x_{n+1}=(1-\lambda) x_n+\lambda Tx_n,\,n\geq 0,$$ converges strongly to a fixed point of $T$.

Any nonexpansive mapping is a $0$-enriched nonexpansive mapping. Hence, Corollary \[cor1\] follows from Theorem \[th1\] for $b=0$ , that is, for $\mu=1$.

Remind that a 