Theorem 3.3 in [@Ber07], by considering enriched nonexpansive mappings instead of nonexpansive mappings.

2\) The assumption $Fix\,(T)=\{p\}$ in Theorem \[th2\] may be removed to obtain the following more general result.

\[th3\] Let $C$ be a bounded closed convex subset of a Hilbert space $H$ and $T:C\rightarrow C$ be an enriched nonexpansive operator. Then, for any given $x_0\in C$ and any fixed number $\lambda$, $0<\lambda<1$, the Krasnoselskij iteration $\{x_n\}_{n=0}^{\infty}$ given by converges weakly to a fixed point of $T$.

By the arguments in the proof of Theorem \[th1\], we have $Fix\,(T)=Fix\,(T_\mu)\neq \emptyset$, where, as usually, $T_\mu x=(1-\mu) x+\mu T x$. Moreover, by Theorem \[th1\], $Fix\,(T_\mu)$ is convex. Since $T_\mu$ is nonexpansive, for any $p\in Fix\,(T_\mu)$ and for each $n$ we have $$\|x_{n+1}-p\|\leq \|x_{n}-p\|,\,n\geq 0,$$ which shows that the function $$g(p)=\lim_{n\rightarrow \infty} \|x_{n}-p\|,\,p\in Fix\,(T_\mu),$$ is well defined and is a lower semicontinuous convex function on $Fix\,(T_\mu)$.

The rest of the proof is similar to that of Theorem 8 in Browder and Petryshyn [@BroP67] and therefore is omitted.

Conclusions and further study
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In this paper we introduced and studied the class of [*enriched nonexpansive mappings*]{} in the setting of a Hilbert space $H$. We have shown that any enriched nonexpansive mapping defined 