also [@Ber16] and references therein.

Weak convergence theorems for enriched nonexpansive mappings in Hilbert spaces
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A key tool in proving the main result in the previous section has been the fact that the averaged map of a nonexpansive mapping $T$ is asymptotically regular, a property which has been used in conjunction with the demicompactness of $T$.

In the absence of the demicompactness property of the operator $T$, the asymptotic regularity alone does not imply in general the strong convergence of the Krasnoselskij sequence $\{x_n\}_{n=0}^{\infty}$ but the weak convergence can be still assured, as shown by the next theorems which extend Theorems 7 and 8 in Browder and Petryshyn [@BroP67] from nonexpansive mappings to enriched nonexpansive mappings.

\[th2\] Let $C$ be a bounded closed convex subset of a Hilbert space $H$ and $T:C\rightarrow C$ be an enriched nonexpansive operator with $Fix\,(T)=\{p\}$. 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 $$\label{eq4a}
x_{n+1}=(1-\lambda) x_n+\lambda Tx_n,\,n\geq 0,$$ converges weakly to $p$.

We use similar arguments to those in the proof of Theorem \[th1\] to show that $T_\mu x=(1-\mu) x+\mu T x$, which map $C$ into $C$ is nonexpansive. Note also that, in view 