I'm reading a paper by Crandall Liggett ("Generation of semi-groups...") and I have a question about one of the lemmas. I'll included some "background information" as I'm not sure if it's relevant for my question or not.
They say:
A subset $B \subset X \times X$ ($X$ is a Banach space) is called accretive if $(I+\lambda B)^{-1}$ is a function for $\lambda>0$ and: $\lVert (I+\lambda B)^{-1}x- (I+\lambda B)^{-1}y \rVert \leq \lVert x-y \rVert$ for $x,y\in D((I+\lambda B)^{-1})$.
and further:
a set $A \subset X \times X $ and a real number $\omega$ such that $A+\omega I$ is an accretive operator are fixed. If $\lambda$ is real let $J_\lambda$ denote the set$ (I+\lambda A)^{-1}$ and $D_\lambda=D(J_\lambda)$ be its domain.
My question is about the following step in the proof of Lemma 1.2.ii):
Take $[x_1,y_1]\in A$ and $[x,y]\in A$ such that $x_1+\lambda y_1 =x$
For this to be possible one needs to have $x \in D(A)\cap R (I+\lambda A)$, but how does one know that this set is non-empty? Does it follow from one of the assumptions above or something more basic?