u$ as $\lambda\to0+.$

With the help of the latter proposition, one can prove the following perturbation result.

\[cor:bounded\_pert\] Let $A,B\subseteq H\times H$ be two maximal monotone relations. Moreover, assume that $A$ is *bounded*, i.e. for each $U\subseteq H$ bounded, the set $$\left\{ v\in H\,;\,\exists u\in U:(u,v)\in A\right\}$$ is bounded. If ${\operatorname{dom}}(A)\cap{\operatorname{dom}}(B)\ne\emptyset$ (i.e. $\exists u,v,w\in H:\,(u,v)\in A$ and $(u,w)\in B$) then $A+B$ is maximal monotone.

The main result
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Throughout, let $\rho>0.$ We begin with stating the hypotheses of the operators and relations involved, which we assume to be valid throughout this section.

\[hyp:assumptions\] Let $\mathcal{M},\mathcal{N},\mathcal{M}'\in L(L_{2,\rho}({\mathbb{R}};H))$ be such that:

(a) $\mathcal{M}\partial_{t,\rho}\subseteq\partial_{t,\rho}\mathcal{M}-\mathcal{M}',$

(b) There exists $c>0$ such that $${\operatorname{Re}}\langle(\partial_{t,\rho}\mathcal{M}+\mathcal{N})\varphi,\varphi\rangle_{\rho}\geq c|\varphi|_{\rho}^{2}$$ for each $\varphi\in C_{c}^{\infty}({\mathbb{R}};H).$

Moreover, let $\mathcal{A}\subseteq L_{2,\rho}({\mathbb{R}};H)\times L_{2,\rho}({\mathbb{R}};H)$ be maximal monotone such that $$\forall u,v\in L_{2,\rho}({\mathbb{R}};H):\,\left((u,v)\in\mathcal{A}\Rightarrow\forall h\geq0:(\tau_{h}u,\tau_{h}v)\in\mathcal{A}\right).$$

It is noteworthy that the perspective on differential inclusions presented here, is that the relation $\mathcal{A}$ is independent of time in the sense that it commutes with time translation, while the time-dependence enter the problem via coefficients, which are incorporated in the operators $\mathcal{M}$ and $\mathcal{N}.$ This is the standard case in problems arising in mathematical physics.

Our main theorem reads as follows.

\[thm:main\] The relation $\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}$ is $c$-maximal monotone. In particular, the 