there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}_{n=0}^{\infty}$ which converges strongly in $C$. Denote $$\lim_{k\rightarrow \infty} x_{n_k}=q.$$ Then, by the continuity of $T_\mu$ it follows that $$T_\mu x_{n_k} \rightarrow T_\mu q,\, \textnormal{ as } k\rightarrow \infty.$$ Therefore, $\{x_{n_k} -T_\mu x_{n_k} \}$ converges strongly to $0$ and simultaneously, $\{x_{n_k} -T_\mu x_{n_k} \}$ converges strongly to $q-T_\mu q$, which proves that $q=T_\mu q$, i.e., $$q\in Fix\,(T_\mu)=Fix\,(T).$$

The convergence of the entire sequence $\{x_n\}_{n=0}^{\infty}$ to $q$ now follows from the inequality $$\|x_{n+1}-q\|\leq \|x_{n}-q\|, \,n\geq 0,$$ which is a direct consequence of the nonexpansivity of $U_\lambda$ (which, in turn, is a consequence of the nonexpansivity of $T_\mu$).

Hence, for any $x_0\in C$, the Krasnoselskij iteration $\{x_n\}_{n=0}^{\infty}$, given by $$x_{n+1}=U_\lambda x_n=(1-\lambda) x_n+\lambda T_\mu x_n=(1-\lambda) x_n+\lambda [(1-\mu)x_n+\mu T x_n]$$ $$=(1-\lambda) x_n+\lambda [(1-\mu)x_n+\mu T x_n]$$ $$=(1-\lambda \mu) x_n+\lambda \mu Tx_n$$ converges strongly to $q\in Fix\,(T)$ as $n\rightarrow \infty$.

To get exactly the formula for the Krasnoselskij iteration $\{x_n\}_{n=0}^{\infty}$ given above, just simply denote $\lambda:=\lambda \mu\in (0,1)$.

\[rem1\] Theorem \[th1\] is an extension of Lemma 3 of Petryshyn [@Pet] and of its global variant (Theorem 6) in Browder and Petryshyn [@BroP67], by considering instead of nonexpansive mappings the larger class of enriched nonexpansive mappings.

\[rem2\] The class of demicompact 