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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.

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1 Answer 1

You can find an example here: Modified projection method for strongly pseudomonotone variational inequalities PD Khanh, PT Vuong, Journal of Global Optimization 58 (2), 341-350

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