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 (i) 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 (iii), so the pairing in clause 2 is defined.

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

$$0\le\langle B(x)-B(y),x-y\rangle_{H}\qquad\text{for all }x,y\in V .$$

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

$$0\le\langle B(x),Ax\rangle_{H}\qquad\text{for every }x\in D(A).$$

3. (Boundedness on $V$-bounded sets)(iii) 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)(iv) 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.