Let $\mathcal H$ be a Hilbert space and $A \in \mathcal L(\mathcal H)$ be accretive, i.e. $\langle -Av + Aw, v-w \rangle \le 0$ for all $v,w \in \mathcal{H}$. I want to show that then $R_{\lambda} := (I + \lambda A)^{-1} \in \mathcal L(\mathcal H)$ exists for every $\lambda \ge 0$ and also that it is a metric map, i.e. $\|R_{\lambda}v - R_{\lambda}w\| \le \|v-w\|$ for all $v,w \in \mathcal{H}$.
My approach: For $\lambda \neq 0$, I can try to find a solution for $(I + \lambda A)v = f$ with the recursion formula
$\frac {v^{n+1} - v^{n}}{\tau} + (I + \lambda A)v^n = f $ for arbitrary $v^0$ and some suitable $\tau > 0$. How to continue?
[Different solutions are also welcomed]
Another word for metric map is nonexpansive operator. To see that $R_{\lambda A}$ is nonexpansive we first define the Cayley operator,
$$C_{\lambda A} := 2R_{\lambda A} - I$$
where $I$ is the identity. We will show that the Cayley operator is nonexpansive and the nonexpansiveness of $R_{\lambda A}$ will follow since it is the average of $C_{\lambda A}$ and the identity (i.e. $R_{\lambda A} = \frac{1}{2}(C_{\lambda A}+I)$)
Let $(x,u)\in gra(R_{\lambda A})$ and $(y,v)\in gra(R_{\lambda A})$, then we have
\begin{equation} |C_{\lambda A}(x) - C_{\lambda A}(y)|_2^2 =|2R_{\lambda A}(x)-x - 2R_{\lambda A}(y)+y|_2^2 = |2(u-v)+y-x|_2^2\\ = 4|u-v|_2^2 - 4\langle u-v, x-y\rangle +|x-y|_2^2 \leq |x-y|_2^2 \end{equation}
Thus $C_{\lambda A}$ is nonexpansive and we are done.