I read from the book "Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I" (P. 156) and paper "Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications" (P. 171) that strongly monotone operator implies coercivity in the following sense:

$\xi$-monotone on $K$ for some $\xi>1$ if there exists a constant $c>0$ such that $$ (F(x)-F(y))^T(x-y) \geq c\|x-y\|^{\xi}, \quad \forall x, y \in K . $$ A strongly monotone $F$ means 2-monotone, which implies the satisfaction of the following coercive condition $$ \exists y \text{ s.t.} \lim _{\substack{x \in K \\\|x\| \rightarrow \infty}} \frac{F(x)^T(x-y)}{\|x\|^{\frac{\xi+1}{2}}}=\infty . $$

I don't quite understand 1. how to related the norm $\| x-y\|$ to $\| x\|$ 2. Does it implicitly assume that set $K$ is unbounded?

Any hints are sincerely appreciated!



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