and set $$u_{\lambda}\coloneqq\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}_{\lambda}\right)^{-1}(f).$$ Moreover, set $$v\coloneqq\left(\mathcal{T}_{\delta}+\mathcal{B}_{\lambda}\right)^{-1}(\partial_{t,\rho}f).$$ We note that $v\in L_{2,\rho}({\mathbb{R}};H)$ by and that $$|v|_{\rho}\leq\frac{1}{c}|\partial_{t,\rho}f|_{\rho}=\frac{1}{c}|f|_{\rho,1}$$ by . We compute $$\begin{aligned}
\partial_{t,\rho}\mathcal{M}\partial_{t,\rho}^{-1}v-\mathcal{M}'\partial_{t,\rho}^{-1}v+\delta\partial_{t,\rho}^{-1}v & =\mathcal{M}v+\delta\partial_{t,\rho}^{-1}v\\
 & =\partial_{t,\rho}^{-1}\mathcal{T}_{\delta}v\\
 & =\partial_{t,\rho}^{-1}(\partial_{t,\rho}f-\mathcal{B}_{\lambda}(v))\\
 & =f-\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}v)\end{aligned}$$ and hence, $$\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}_{\lambda}\right)(\partial_{t,\rho}^{-1}v)=f,$$ which shows $u_{\lambda}=\partial_{t,\rho}^{-1}v\in H_{\rho}^{1}({\mathbb{R}};H).$ Moreover, $$|u_{\lambda}|_{\rho,1}=|v|_{\rho}\leq\frac{1}{c}|f|_{\rho,1}.\tag*{\qedhere}$$

\[cor:maxmonaux\]Let $\delta>\|\mathcal{M}'\|+\|\mathcal{N}\|$. Then $\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}$ is $c$-maximal monotone and hence, $$\mathcal{S}_{\rho,\mathrm{aux}}\coloneqq\left(\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}\right)^{-1}:L_{2,\rho}({\mathbb{R}};H)\to L_{2,\rho}({\mathbb{R}};H)$$ is Lipschitz-continuous. Moreover, for $f\in H_{\rho}^{1}({\mathbb{R}};H)$ we have that $\mathcal{S}_{\rho,\mathrm{aux}}(f)\in H_{\rho}^{1}({\mathbb{R}};H)\cap{\operatorname{dom}}\left(\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}\right)^{-1}\right).$

We already know that $\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}$ is $c$-monotone by . For proving the maximal monotonicity it suffices to show that there is a dense subset $D\subseteq L_{2,\rho}({\mathbb{R}};H)$ such that for each $f\in D$ there exists $u\in L_{2,\rho}({\mathbb{R}};H)$ with $$(u,f)\in\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}.$$ We show that the latter is true for $D=H_{\rho}^{1}({\mathbb{R}};H).$ So, let $f\in H_{\rho}^{1}({\mathbb{R}};H).$ For showing that a solution $u\in L_{2,\rho}({\mathbb{R}};H)$ exists, it suffices to prove that $\sup_{\lambda>0}|\mathcal{A}_{\lambda}(u_{\lambda})|_{\rho}<\infty$ for $$u_{\lambda}\coloneqq\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}_{\lambda}\right)^{-1}(f)$$ by . According to we know that $u_{\lambda}\in H_{\rho}^{1}({\mathbb{R}};H)$ and that $$|u_{\lambda}|_{\rho,1}\leq\frac{1}{c}|f|_{\rho,1}$$ for each $\lambda>0.$ Thus, we can estimate $$\begin{aligned}
\left|\mathcal{A}_{\lambda}(u_{\lambda})\right|_{\rho} & =\left|f-\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'\right)u_{\lambda}-\delta u_{\lambda}\right|_{\rho}\\
 & =\left|f-\mathcal{M}\partial_{t,\rho}u_{\lambda}-\delta u_{\lambda}\right|_{\rho}\\
 & \leq|f|_{\rho}+\|\mathcal{M}\||u_{\lambda}|_{\rho,1}+\frac{\delta}{\rho}|u_{\lambda}|_{\rho,1}\\
 & \leq|f|_{\rho}+\frac{1}{c}\left(\|\mathcal{M}\|+\frac{\delta}{\rho}\right)|f|_{\rho,1}.\end{aligned}$$ Summarising, we have shown the $c$-maximal monotonicity of $\overline{\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}}$. Moreover, $u_{\lambda}\to u$ as $\lambda\to0+$ in $L_{2,\rho}({\mathbb{R}};H)$ and $$(u,f)\in\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}$$ by , i.e. $u=\mathcal{S}_{\rho,\mathrm{aux}}(f)\in{\operatorname{dom}}\left(\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}\right)^{-1}\right)$. Moreover, since $(u_{\lambda})_{\lambda>0}$ is bounded in $H_{\rho}^{1}({\mathbb{R}};H),$ we infer that also $u=\mathcal{S}_{\rho,\mathrm{aux}}(f)\in H_{\rho}^{1}({\mathbb{R}};H)$ by .

With this result at hand, 