satisfies $$(\partial_{t,\rho}\mathcal{M}+\delta+\mathcal{B}_{\lambda})(\partial_{t,\rho}u)=\partial_{t,\rho}f.$$ Thus, we are led to consider the differential equation $$(\partial_{t,\rho}\mathcal{M}+\delta+\mathcal{B}_{\lambda})(v)=g.$$ Since $\mathcal{B}_{\lambda}$ is Lipschitz-continuous and maximal monotone on $H_{\rho}^{-1}(\mathbb{R};H)$, the latter equation is well-posed in $H_{\rho}^{-1}({\mathbb{R}};H)$, if we can show that $\partial_{t,\rho}\mathcal{M}+\delta$ is $c$-maximal monotone on $H_{\rho}^{-1}({\mathbb{R}};H)$ in some sense. This will be shown in the next proposition.

\[prop:T\_delta\]Let $\delta>\|\mathcal{M}'\|+\|\mathcal{N}\|.$ Then the mapping $$\partial_{t,\rho}\mathcal{M}+\delta:L_{2,\rho}({\mathbb{R}};H)\subseteq H_{\rho}^{-1}({\mathbb{R}};H)\to H_{\rho}^{-1}({\mathbb{R}};H)$$ is closable and its closure, denoted by $\mathcal{T}_{\delta}$, is $(c+\delta-\|\mathcal{M}'+\mathcal{N}\|)$-maximal monotone.

To prove the closability, let $(v_{n})_{n\in\mathbb{N}}$ be a sequence in $L_{2,\rho}({\mathbb{R}};H)$ such that $v_{n}\to0$ and $\left(\partial_{t,\rho}\mathcal{M}+\delta\right)v_{n}\to y$ for some $y\in H_{\rho}^{-1}({\mathbb{R}};H),$ where both convergences are with respect to the topology in $H_{\rho}^{-1}({\mathbb{R}};H).$ Since clearly $\delta v_{n}\to0$ in $H_{\rho}^{-1}({\mathbb{R}};H),$ we infer that $\partial_{t,\rho}\mathcal{M}v_{n}\to y$ in $H_{\rho}^{-1}({\mathbb{R}};H).$ Moreover, noting that $\partial_{t,\rho}^{-1}v_{n}\to0$ in $L_{2,\rho}({\mathbb{R}};H)$ we derive $$\mathcal{M}v_{n}=\mathcal{M}\partial_{t,\rho}\partial_{t,\rho}^{-1}v_{n}=\partial_{t,\rho}\mathcal{M}\partial_{t,\rho}^{-1}v_{n}-\mathcal{M}'\partial_{t,\rho}^{-1}v_{n}\to0$$ in $H_{\rho}^{-1}({\mathbb{R}};H)$ and hence, $\partial_{t,\rho}\mathcal{M}v_{n}\to0$ in $H_{\rho}^{-2}({\mathbb{R}};H).$ This gives $y=0$ and thus, the operator is closable.

Let now $v\in L_{2,\rho}({\mathbb{R}};H).$ Then we estimate $$\begin{aligned}
{\operatorname{Re}}\left\langle \left(\partial_{t,\rho}\mathcal{M}+\delta\right)v,v\right\rangle _{\rho,-1} & ={\operatorname{Re}}\langle\mathcal{M}v+\partial_{t,\rho}^{-1}\delta v,\partial_{t,\rho}^{-1}v\rangle_{\rho}\\
 & ={\operatorname{Re}}\langle\mathcal{M}\partial_{t,\rho}\partial_{t,\rho}^{-1}v+\partial_{t,\rho}^{-1}\delta v,\partial_{t,\rho}^{-1}v\rangle_{\rho}\\
 & ={\operatorname{Re}}\langle\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta\right)\partial_{t,\rho}^{-1}v,\partial_{t,\rho}^{-1}v\rangle_{\rho}\\
 & \geq(c+\delta-\|\mathcal{M}'+\mathcal{N}\|)|\partial_{t,\rho}^{-1}v|_{\rho}^{2}\\
 & =(c+\delta-\|\mathcal{M}'+\mathcal{N}\|)|v|_{\rho,-1}^{2},\end{aligned}$$ where we have used . Since $L_{2,\rho}({\mathbb{R}};H)$ is a core for $\mathcal{T_{\delta}}$, the $(c+\delta-\|\mathcal{M}'+\mathcal{N}\|)$-monotonicity follows.

For proving the maximal monotonicity, it suffices to show that ${\operatorname{ran}}(\mathcal{T}_{\delta})$ is dense in $H_{\rho}^{-1}({\mathbb{R}};H)$. Note that, since $\delta>\|\mathcal{N}\|$, we have that $\partial_{t,\rho}\mathcal{M}+\delta$ 