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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 an $\textbf{an affine mapping mapping}$ as in the above example.

I am grateful to all your comments and helping.
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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

$$ < \langle F(y)-F(x), y-x\ra\geq y-x\rangle\geq \gamma\|y-x\|^2, \quad \forall x,y\in K. $$+

  • $F$ is strongly pseudomonotone on $K$ if there exists $\gamma>0$ such that $$ \left\geq langle F(x), y-x\rangle\geq 0 \Longrightarrow \left\geq langle F(y), y-x\rangle\geq \gamma\|y-x\|^2 $$ 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 an affine mapping as in the above example.

I am grateful to all your comments and helping.
show/hide this revision's text 1

Two Concepts of Monotonicity

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

$$ < F(y)-F(x), y-x\ra\geq \gamma\|y-x\|^2, \quad \forall x,y\in K. $$ + $F$ is strongly pseudomonotone on $K$ if there exists $\gamma>0$ such that $$ \left\geq 0 \Longrightarrow \left\geq \gamma\|y-x\|^2 $$ 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 $F$ is strongly pseudomonotone but not strongly monotone on $K$. It is interesting to find an affine mapping as in the above example.

I am grateful to all your comments and helping.