Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \alpha\gt0 \, ,$$we want to prove that there exist a unique $x^*$ such that $F(x^*)^T(x-x^*)\ge 0$ for all $x\in K$.
The uniqueness proof is obvious. Assuming there are two answers we have $(x^*-x')^T(F(x')-F(x^*))\ge\alpha ||x^*-x'||^2$ Which is a contradiction.
But how to prove that there exists answer?
