thus $u_{\lambda}\in L_{2,\rho}({\mathbb{R}};H)$ is defined. Our first goal is to prove the following theorem.

\[thm:regular\_solution\]Let $\delta>\|\mathcal{M}'\|+\|\mathcal{N}\|$ and $f\in H_{\rho}^{1}({\mathbb{R}};H).$ For $\lambda>0$ set $$u_{\lambda}\coloneqq\left(\partial_{t,\rho}\mathcal{M}-\mathcal{M}'+\delta+\mathcal{A}_{\lambda}\right)^{-1}(f).$$ Then $u_{\lambda}\in H_{\rho}^{1}({\mathbb{R}};H)$ and $$|u_{\lambda}|_{\rho,1}\leq\frac{1}{c}|f|_{\rho,1}.$$

In order to prove this theorem, we define the following mapping $$\mathcal{B}_{\lambda}\coloneqq\partial_{t,\rho}\mathcal{A}_{\lambda}\partial_{t,\rho}^{-1}:H_{\rho}^{-1}({\mathbb{R}};H)\to H_{\rho}^{-1}(\mathbb{R};H).$$ Then, by unitary equivalence, $\mathcal{B}_{\lambda}$ is maximal monotone and Lipschitz-continuous on $H_{\rho}^{-1}({\mathbb{R}};H)$ with $|\mathcal{B}_{\lambda}|_{\mathrm{Lip}}\leq\frac{1}{\lambda}.$ However, we can also interpret $\mathcal{B}_{\lambda}$ as a mapping on $L_{2,\rho}({\mathbb{R}};H)$ as the following proposition shows.

\[prop:properties\_B\]Let $\lambda>0$ and define $$\mathcal{B}_{\lambda}\coloneqq\partial_{t,\rho}\mathcal{A}_{\lambda}\partial_{t,\rho}^{-1}:H_{\rho}^{-1}({\mathbb{R}};H)\to H_{\rho}^{-1}({\mathbb{R}};H).$$ Then $\mathcal{B}_{\lambda}[L_{2,\rho}({\mathbb{R}};H)]\subseteq L_{2,\rho}({\mathbb{R}};H)$ and for $u\in L_{2,\rho}({\mathbb{R}};H)$ we have that $$\begin{aligned}
{\operatorname{Re}}\langle\mathcal{B}_{\lambda}(u),u\rangle_{\rho} & \geq0,\\
|\mathcal{B}_{\lambda}(u)|_{\rho} & \leq\frac{1}{\lambda}|u|_{\rho}.\end{aligned}$$

Let $u\in L_{2,\rho}({\mathbb{R}};H).$ In order to prove $\mathcal{B}_{\lambda}(u)\in L_{2,\rho}({\mathbb{R}};H)$ we need to show that $\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)\in H_{\rho}^{1}({\mathbb{R}};H).$ For doing so, we use the notation from and estimate for $h>0$ $$\begin{aligned}
|D_{h}\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)|_{\rho} & =\frac{1}{h}\left|\tau_{h}\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)-\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)\right|_{\rho}\\
 & =\frac{1}{h}\left|\mathcal{A}_{\lambda}(\tau_{h}\partial_{t,\rho}^{-1}u)-\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)\right|_{\rho}\\
 & \leq\frac{1}{\lambda}|D_{h}\partial_{t,\rho}^{-1}u|_{\rho}.\end{aligned}$$ As $\partial_{t,\rho}^{-1}u\in H_{\rho}^{1}({\mathbb{R}};H)$ the latter estimate shows $\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)\in H_{\rho}^{1}({\mathbb{R}};H)$ according to . Furthermore, letting $h$ tend to 0, we also infer that $$|\mathcal{B}_{\lambda}(u)|_{\rho}=\lim_{h\to0+}|D_{h}\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)|_{\rho}\leq\frac{1}{\lambda}\lim_{h\to0+}|D_{h}\partial_{t,\rho}^{-1}u|_{\rho}=\frac{1}{\lambda}|u|_{\rho}.$$ Moreover, again by using , we estimate $$\begin{aligned}
{\operatorname{Re}}\langle\mathcal{B}_{\lambda}(u),u\rangle_{\rho} & ={\operatorname{Re}}\langle\partial_{t,\rho}\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u),\partial_{t,\rho}\partial_{t,\rho}^{-1}u\rangle_{\rho}\\
 & =\lim_{h\to0+}\frac{1}{h^{2}}{\operatorname{Re}}\langle\tau_{h}\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u)-\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u),\tau_{h}\partial_{t,\rho}^{-1}u-\partial_{t,\rho}^{-1}u\rangle_{\rho}\\
 & =\lim_{h\to0+}\frac{1}{h^{2}}{\operatorname{Re}}\langle\mathcal{A}_{\lambda}(\tau_{h}\partial_{t,\rho}^{-1}u)-\mathcal{A}_{\lambda}(\partial_{t,\rho}^{-1}u),\tau_{h}\partial_{t,\rho}^{-1}u-\partial_{t,\rho}^{-1}u\rangle_{\rho}\\
 & \geq0,\end{aligned}$$ where we have used the monotonicity of $\mathcal{A}_{\lambda}.$

The reason for considering the mapping $\mathcal{B}_{\lambda}$ is the following. Assume that $u_{\lambda}$ is given by . Then, at least formally, $\partial_{t,\rho}u$ 