I think monotone operators qualify for question 1:

A mapping $T:X\to X^*$ from a Banach space to its dual is said to be *monotone*, if for all $u,v\in X$ it holds that
$$ \langle Tu - Tv, u-v\rangle\geq 0.$$

This generalizes to relations from $X$ to $X^*$ (i.e. subsets of $X\times X^*$):  $G\subset X\times X^*$ is said to be monotone of for all $(x,u),(y,v)\in G$ it holds that
$$ \langle x-y,u-v\rangle \geq 0.$$

Relations (or multifunction) like this appear, e.g., as subgradients of convex functions on Banach spaces. If $f: X\to ]-\infty,\infty]$ is convex, its subgradient
$$
\partial f(x) = \{x^*\in X^*\ :\  \forall y:\ f(y)\geq f(x) + \langle x^*,y-x\rangle\}
$$
can be viewed either a set-valued mapping $\partial f: X \to 2^{X^*}$ or as a multifunction $\partial f:X\rightrightarrows X^*$ or, by identifying it with its graph $\mathrm{gph}(\partial f) = \{(x^*,x)\in X\times X^*\ : x^*\in\partial f(x)\}$, as subset of $X\times X^*$ (indeed, a monotone one).