Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $B$ be a map from $V$ to $H$. By Hilbert Triples: Standing Notation and Background §triple the space $V$ is a linear subspace of $H$, so differences of its elements lie in $H$ and the pairing in clause 1 is defined; and for $x\in D(A)$ one has $x\in V$ and $Ax\in H$ by Hilbert Triples: Standing Notation and Background §operator, so the pairing in clause 2 is defined.

1. (Monotonicity)¶ The map $B$ is monotone if

2. ($A$-monotonicity)¶ The map $B$ is $A$-monotone if

3. (Boundedness on $V$-bounded sets)¶ The map $B$ is bounded on $V$-bounded sets if for every positive $R\in\mathbb{R}$ there is $\beta\in\mathbb{R}$ such that every $x\in V$ with $|x|_{V}\le R$ satisfies $|B(x)|_{H}\le\beta$.

4. (Monotone nonlinearity)¶ The map $B$ is a monotone nonlinearity for $(H,V,A)$ if it is monotone, is $A$-monotone, and is bounded on $V$-bounded sets.