has the property that whenever $\{u_{n}\}$ is a bounded sequence in $H$ and $\{Tu_{n}-u_{n}\}$ is strongly convergent, then there exists a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ which is strongly convergent.

[@BroP67] \[def2\] Let $H$ be a Hilbert space and $C$ a closed convex subset of $H$. A mapping $T:C\rightarrow C$ is called *asymptotically regular* (on $C$) if, for each $x\in C$, $$\|T^{n+1}x-T^n x\|\rightarrow 0  \textnormal{ as } n\rightarrow \infty.$$

The following Lemma, which is adapted after Corollary to Theorem 5 in [@BroP67] will be used in the proof of the main result of this section.

\[lem1\] Let $H$ be a Hilbert space and $C$ a closed convex subset of $H$. If the mapping $U:C\rightarrow C$ is nonexpansive and $Fix\,(U)\neq \emptyset$ then, for any given $\lambda\in (0,1)$, the mapping $U_\lambda=I+(1-\lambda) U$ maps $C$ into $C$, has the same fixed points as $U$ and is asymptotically regular.

Now we can state and prove the main result of this section.

\[th1\] Let $C$ be a bounded closed convex subset of a Hilbert space $H$ and $T:C\rightarrow C$ be a $b$-enriched nonexpansive and demicompact mapping. 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 