I am currently trying to understand this paper, but I am stuck about how to visualise the following part. We are looking at a Cauchy problem for the non-linear evolution operator $u'(t)= (A+B)u(t)$ for $t\geq 0$, and the derivation of mild solutions.
Here is the extract that I am currently having difficulty with:
In order to impose the local continuity for the nonlinear operator $B$ from $D$ into $X$, we employ a vector-valued functional $\phi=(\phi_i)^n_{i=1}$ on $X$ to $\bar{\mathbb{R}}^n_+$ such that $D \subset D(\phi):= \{x \in X; \phi_i (x) < \infty $ for all $i = 1,2,...,n\}$, and the order '$\leq$' in $\mathbb{R}^n$ defined in the way that $\alpha = (\alpha_i)^n_{i=1} \leq \beta = (\beta_i)_{i=1}^n $ if an only if $\alpha_i \leq \beta_i$ for all $i = 1,2,...,n$.
($\phi$) For each $\alpha \in \mathbb{R}_+^n$, the level set $D_{\alpha}:= \{ v \in D; \phi(v) \leq \alpha \}$ is closed in X.
(B) For each $\alpha \in \mathbb{R}_+^n$, the operator $B$ is continuous on $D_\alpha$ in X.
My issue is understanding precisely what the vector-valued functional is and what role it plays.
I would appreciate any help!
I think it might be best to think of $\phi$ as providing a 'norm' for $x$, in the sense that it associates an $n$-tuple of positive real numbers to $x$, and that each $\phi_i(x)$ is finite for $x \in D$. The functionals $\phi_i$ do not need to be (semi)norms themselves, since they are not specified to be absolutely homogeneous or subadditive (see e.g. here). Apparently, only the fact that they map $x$ onto the positive reals, is sufficient for them to be useful.
Then, you can think of $D_\alpha$ as being the pre-image of the closed $n$-block (or hyperrectangle) $R_\alpha \subset \mathbb{R}_+^n$, given by $R_\alpha = \left\{ y \in \mathbb{R}_+^n\,|\, y \leq \alpha\right\}$, using the order '$\leq$' as described in your question.
Now, remember the topological definition of continuity of a function between two topological spaces using the preimage of a closed set (see e.g. here), and I think you'll see why it is useful to introduce these functionals $\phi_i$.