I am looking for a general definition of monotone condition for a function $G: \mathbb{R}^m \to \mathbb{R}^m$, and since I did not find a unique definition of monotone condition for multivariable function (see Monotonicity of function of two variables), I was wondering if the definition of monotone operator could be a valid one.
$G: U \subset\mathbb{R}^m \to \mathbb{R}^m$ monotone if $$\langle G(u_1)-G(u_2),u_1-u_2\rangle \geq 0,$$ for all $u_1,u_2 \in U$.