Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that
- $F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that
$$ \langle F(y)-F(x), y-x\rangle\geq \gamma\|y-x\|^2, \quad \forall x,y\in K. $$
- $F$ is strongly pseudomonotone on $K$ if there exists $\gamma>0$ such that $$ \langle F(x), y-x\rangle\geq 0 \Longrightarrow \langle F(y), y-x\rangle\geq \gamma\|y-x\|^2 $$ for all $x,y\in K$.
It is easily to verify that strongly monotone implies strongly pseudomonotone. The converse is not true in general. For example, in one-dimensional case $$ F(x)=(2-x), \quad K=[0,1], $$ the mapping $F$ is strongly pseudomonotone but not strongly monotone on $K$.
$\textbf{Question:}$ Can we find a mapping $F: K\rightarrow \mathbb{R}^n (n\geq 2)$ such that $\text{int}K\ne \emptyset$ ($K$ has a nonempty interior) and $F$ is strongly pseudomonotone but not strongly monotone on $K$. It is interesting to find $\textbf{an affine mapping}$ as in the above example.
I am grateful to all your comments and helping.
