If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$.
Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces with its dual space $V^*$, via:
The operator $A\colon V \to V^*$ is called monotone if $$ \langle A(u) - A(v), u-v \rangle \geq 0,$$ for all $u$, $v \in V$.
So my question is: Does monotonicity of $A$ imply, that the Frechet derivative of $A$ is positive, i.e. $\langle A'(w)v,v\rangle \geq 0$ for any $w \in V$ and for all $v\in V$.
Any idea or reference is appreciated.