Two Concepts of Monotonicity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T23:26:38Z http://mathoverflow.net/feeds/question/103292 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103292/two-concepts-of-monotonicity Two Concepts of Monotonicity Jane 2012-07-27T10:20:51Z 2012-07-27T10:32:51Z <p>Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that</p> <ul> <li>$F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that</li> </ul> <p>$$\langle F(y)-F(x), y-x\rangle\geq \gamma\|y-x\|^2, \quad \forall x,y\in K.$$</p> <ul> <li>$F$ 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 $x,y\in K$.</li> </ul> <p>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 $K$.</p> <p>$\textbf{Question:}$ 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.</p> <p>I am grateful to all your comments and helping.</p>