I'm currently reading Craig-Wayne's Newton's Method and Periodic Solutions of Nonlinear Wave Equations. My background includes undergraduate functional analysis and PDE, and there are some notations which I would like to clarify.
On page 8, $(2.8)$ introduces the equation $W(U) + V(\Omega)U = 0$, and down the page, it explains that $W : \mathcal{H}_\sigma \to \mathcal{H}_{\sigma - \gamma}$ is a real analytic map for every $0 < \gamma \leq \sigma < \bar{\sigma}$, but did not define $W(U)$ explicitly, and no other information about $W(U)$ is provided prior to page 9.
- I would like to know if $W(U)$ is a notation for some commonly defined map?
- $V(\Omega)$ is explicitly defined in $(2.9)$. Am I right to say that $V(\Omega)U$ is the notation for $V(\Omega)(U)$, i.e. $V(\Omega)$ with the input $U$?
It also then says that $W(0) = 0$ and $D_UW(0) = 0$. On the next page, they introduced the linearised operator $H(U) = V(\Omega) + DW(U)$.
- I would also like to clarify the meaning of $D_UW$ and $DW(U)$.
Any help is appreciated.