time horizon, one obtains $\mu(\mathbb{R}_{\geq0})=\frac{1}{2\rho}<\infty$ and hence, this additional condition is superfluous.

The maximal monotonicity of a monotone relation $A$ can be characterised by a range condition on $A$. This is the celebrated Theorem by Minty.

\[thm:Minty\] Let $A\subseteq H\times H$ be monotone. Then the following statements are equivalent:

(i) $A$ is maximal monotone,

(ii) For each $\lambda>0$ and each $z\in H$ there exists $u\in H$ such that $(u,z)\in1+\lambda A,$ i.e. there is $v\in H$ with $(u,v)\in A$ and $u+\lambda v=z.$

(iii) There exists $\lambda>0$ such that for each $z\in H$ there exists $u\in H$ with $(u,z)\in1+\lambda A.$

\[exa:Time\_derivative\]For $\rho>0$ the operator $\partial_{t,\rho}$ is $\rho$-maximal monotone. The monotonicity follows by $$\langle\partial_{t,\rho}\phi,\phi\rangle_{\rho}=\langle\phi,\partial_{t,\rho}^{\ast}\phi\rangle_{\rho}=\langle\phi,-\partial_{t,\rho}\phi\rangle_{\rho}+2\rho|\phi|_{\rho}^{2}\quad(\phi\in{\operatorname{dom}}(\partial_{t,\rho})),$$ which yields $${\operatorname{Re}}\langle\partial_{t,\rho}\phi,\phi\rangle_{\rho}\geq\rho|\phi|_{\rho}^{2}\quad(\phi\in{\operatorname{dom}}(\partial_{t,\rho})).$$ Moreover, for $f\in L_{2,\rho}({\mathbb{R}};H)$ we set $$u\coloneqq\rho\partial_{t,\rho}^{-1}f\in{\operatorname{dom}}(\partial_{t,\rho})$$ and obtain $$u+\frac{1}{\rho}(\partial_{t,\rho}-\rho)u=\frac{1}{\rho}\partial_{t,\rho}u=f,$$ which shows the maximal monotonicity by .

As a consequence of Minty’s theorem and the definition of monotonicity we obtain the following proposition.

Let $A\subseteq H\times H$ be maximal monotone. Then $(1+\lambda A)^{-1}$ is a Lipschitz-continuous mapping defined on the whole Hilbert space $H$ with[^2] $|(1+\lambda A)^{-1}|_{\mathrm{Lip}}\leq1.$ Moreover, for each $\lambda>0$ the so-called *Yosida-approximation* $$A_{\lambda}\coloneqq\frac{1}{\lambda}(1-(1+\lambda A)^{-1}):H\to H$$ is a monotone and Lipschitz-continuous mapping with $|A_{\lambda}|_{\mathrm{Lip}}\leq\frac{1}{\lambda}.$

We conclude this section by some well-known perturbation results for maximal 