proof, we need to show that $v\in L_{2,\rho}({\mathbb{R}};H).$ For doing so, it suffices to show that $\sup_{n\in\mathbb{N}}|v_{n}|_{\rho}<\infty$ by . Using , we estimate $$|v_{n+1}|_{\rho}\leq\frac{1}{c}(|g|_{\rho}+|\delta'-\delta||v_{n}|_{\rho})$$ for $n\in{\mathbb{N}}$ and thus, by induction $$|v_{n}|_{\rho}\leq\frac{1}{c}|g|_{\rho}\sum_{j=0}^{n-1}\left(\frac{|\delta'-\delta|}{c}\right)^{j}\quad(n\in{\mathbb{N}}).$$ As $|\delta'-\delta|<c,$ we infer the boundedness of $(v_{n})_{n\in{\mathbb{N}}}$ and hence, the assertion follows.

\[prop:K is all\]$K=]\|\mathcal{M}'\|+\|\mathcal{N}\|,\infty[.$

We choose $\delta>\frac{1}{\lambda}+\|\mathcal{N}\|+\|\mathcal{M}'\|$ and prove that $\delta\in K.$ Then the assertion follows by . Let $g\in L_{2,\rho}({\mathbb{R}};H)$ and define the sequence $(v_{n})_{n\in{\mathbb{N}}}$ in $L_{2,\rho}({\mathbb{R}};H)$ recursively by $v_{0}\coloneqq0$ and $$v_{n+1}\coloneqq\mathcal{T}_{\delta}^{-1}(g-B_{\lambda}(v_{n}))=(\partial_{t,\rho}\mathcal{M}+\delta)^{-1}(g-\mathcal{B}_{\lambda}(v_{n}))\quad(n\in{\mathbb{N}}).$$ Note that $\mathcal{B}_{\lambda}$ leaves $L_{2,\rho}({\mathbb{R}};H)$ invariant by and that $\partial_{t,\rho}\mathcal{M}+\delta$ is $c$-maximal monotone on $L_{2,\rho}({\mathbb{R}};H)$ and thus, $v_{n+1}$ lies indeed in $L_{2,\rho}({\mathbb{R}};H).$ Indeed, $\partial_{t,\rho}\mathcal{M}+\delta$ is even $\left(c+\delta-\|\mathcal{N}\|\right)$-maximal monotone and thus, we estimate $$|v_{n+1}|_{\rho}\leq\frac{1}{c+\delta-\|\mathcal{N}\|}\left(|g|_{\rho}+|\mathcal{B}_{\lambda}(v_{n})|_{\rho}\right)\leq\frac{1}{c+\delta-\|\mathcal{N}\|}\left(|g|_{\rho}+\frac{1}{\lambda}|(v_{n})|_{\rho}\right)\quad(n\in{\mathbb{N}}),$$ where we have used in the second inequality. Thus, by induction we can estimate $$|v_{n}|_{\rho}\leq\frac{1}{c+\delta-\|\mathcal{N}\|}|g|_{\rho}\sum_{j=0}^{n-1}\frac{1}{\lambda^{j}\left(c+\delta-\|\mathcal{N}\|\right)^{j}}\quad(n\in{\mathbb{N}})$$ and thus, by the choice of $\delta,$ the sequence $(v_{n})_{n\in{\mathbb{N}}}$ is bounded in $L_{2,\rho}({\mathbb{R}};H).$ Moreover, since $\mathcal{T}_{\delta}$ is $(c+\delta-\|\mathcal{M}'+\mathcal{N}\|)$-maximal monotone by , we infer by the choice of $\delta$ that $|\mathcal{T}_{\delta}|_{\mathrm{Lip},H_{\rho}^{-1}({\mathbb{R}};H)}<\lambda$ and hence $$H_{\rho}^{-1}({\mathbb{R}};H)\ni w\mapsto\mathcal{T}_{\delta}^{-1}(g-\mathcal{B}_{\lambda}(w))\in H_{\rho}^{-1}({\mathbb{R}};H)$$ is a strict contraction and hence, has a unique fixed point $v\in H_{\rho}^{-1}({\mathbb{R}};H)$ which satisfies $$(\mathcal{T}_{\delta}+\mathcal{B}_{\lambda})(v)=g.$$ Since $v_{n}\to v$ in $H_{\rho}^{-1}({\mathbb{R}};H)$, we infer that $v\in L_{2,\rho}({\mathbb{R}};H)$ by and thus, $\delta\in K$.

We now can come to the

Let $f\in H_{\rho}^{1}({\mathbb{R}};H)$ 