is $c$-maximal monotone on $L_{2,\rho}({\mathbb{R}};H)$ by . Thus, in particular $L_{2,\rho}({\mathbb{R}};H)\subseteq{\operatorname{ran}}(\mathcal{T}_{\delta})$ and hence, the assertion follows.

We define the set $$K\coloneqq\{\delta>\|\mathcal{M}'\|+\|\mathcal{N}\|\,;\,\forall g\in L_{2,\rho}({\mathbb{R}};H):\,(\mathcal{T}_{\delta}+\mathcal{B}_{\lambda})^{-1}(g)\in L_{2,\rho}({\mathbb{R}};H)\}.$$

We aim to prove that $K=]\|\mathcal{M}'\|+\|\mathcal{N}\|,\infty[.$ We start with the following observation.

\[lem:uniform\_bound\_K\]Let $\delta\in K.$ Then $$\left|\left(\mathcal{T_{\delta}}+\mathcal{B}_{\lambda}\right)^{-1}(g)\right|_{\rho}\leq\frac{1}{c}|g|_{\rho}$$ for each $g\in L_{2,\rho}({\mathbb{R}};H).$

We set $v\coloneqq\left(\mathcal{T_{\delta}}+\mathcal{B}_{\lambda}\right)^{-1}(g)\in L_{2,\rho}({\mathbb{R}};H).$ Then $\left(\partial_{t,\rho}\mathcal{M}+\delta\right)v=\mathcal{T_{\delta}}v=g-B_{\lambda}(v)\in L_{2,\rho}({\mathbb{R}};H)$ by and hence, $v\in{\operatorname{dom}}(\partial_{t,\rho}\mathcal{M}).$ We then estimate by using $$\begin{aligned}
{\operatorname{Re}}\langle g,v\rangle_{\rho} & ={\operatorname{Re}}\langle\mathcal{T}_{\delta}v+B_{\lambda}(v),v\rangle_{\rho}\\
 & \geq{\operatorname{Re}}\langle(\partial_{t,\rho}\mathcal{M}+\delta)v,v\rangle_{\rho}\\
 & \geq c|v|_{\rho}^{2},\end{aligned}$$ where we again have used that $\partial_{t,\rho}\mathcal{M}+\delta$ is $c$-maximal monotone on $L_{2,\rho}({\mathbb{R}};H)$ since $\delta>\|\mathcal{N}\|.$ By applying the Cauchy-Schwarz inequality on the left hand side, we derive the desired inequality.

\[cor:nonempty\_suffices\]If $K\ne\emptyset$ then $K=]\|\mathcal{M}'\|+\|\mathcal{N}\|,\infty[.$

We prove that for $\delta\in K$ it follows that $B(\delta,c)\cap]\|\mathcal{M}'\|+\|\mathcal{N}\|,\infty[\subseteq K.$ This would yield the assertion. So let $\delta\in K$ and choose $\delta'>\|\mathcal{M}'\|+\|\mathcal{N}\|$ with $|\delta-\delta'|<c.$ For $g\in L_{2,\rho}({\mathbb{R}};H)$ we define the sequence $(v_{n})_{n\in\mathbb{N}}$ in $L_{2,\rho}({\mathbb{R}};H)$ recursively by $v_{0}\coloneqq0$ and $$v_{n+1}\coloneqq\left(\mathcal{T}_{\delta}+\mathcal{B}_{\lambda}\right)^{-1}(g-(\delta'-\delta)v_{n})$$ for $n\in{\mathbb{N}}.$ Note that indeed $v_{n+1}\in L_{2,\rho}({\mathbb{R}};H)$ since $\delta\in K.$ Moreover, since $|(\mathcal{T}_{\delta}+\mathcal{B}_{\lambda})^{-1}|_{\mathrm{Lip},H_{\rho}^{-1}({\mathbb{R}};H)}\leq\frac{1}{c}$ , we infer that the mapping $$H_{\rho}^{-1}({\mathbb{R}};H)\ni w\mapsto\left(\mathcal{T}_{\delta}+\mathcal{B}_{\lambda}\right)^{-1}\left(g-(\delta'-\delta)w\right)\in H_{\rho}^{-1}({\mathbb{R}};H)$$ is a strict contraction and thus, it has a unique fixed point $v\in H_{\rho}^{-1}({\mathbb{R}};H).$ Moreover, $v_{n}\to v$ in $H_{\rho}^{-1}({\mathbb{R}};H)$ as $n\to\infty.$ Note, that the fixed point $v$ satisfies $$(\mathcal{T}_{\delta'}+\mathcal{B}_{\lambda})(v)=g$$ and thus, to complete the 