what follows we study the local pseudomonotonicity, respectively the local strict pseudomonotonicity of set-valued operators on self segment-dense sets. We need the following result which proof is straightforward.

\[p33\] Let $F:I\subseteq{\mathbb{R}}\rightrightarrows {\mathbb{R}}$ a set-valued function.

-   If $F(x)> 0$, for every $x\in I$, (respectively $F(x)<0$ for every $x\in I$), then $F$ is strictly pseudomonotone on $I$.

-   If $F(x)=\{0\}$ for some $x\in I$, and $F(z)< 0$ for every $z<x,\,z\in I,$ respectively $F(y)> 0$ for every $x< y,\,y\in I,$ then $F$ is strictly pseudomonotone on $I$.

-   If $I$ is closed from left, $x$ is the left endpoint of $I$ such that $F(x)\ge 0$ and $F(y)>0$ for all $y>x,\, y\in I$, then $F$ is strictly pseudomonotone on $I$.

-   If $I$ is closed from right, $x$ is the right endpoint of $I$ such that $F(x)\le 0$ and $F(y)<0$ for all $y<x,\, y\in I$, then $F$ is strictly pseudomonotone on $I$.

The following result ensures the pseudomonotonicity of a locally pseudomonotone set-valued function on a dense subset.

\[t33\] Let $I\subseteq {\mathbb{R}}$ be an interval and consider the lower semicontinuous set-valued function $F:I\rightrightarrows {\mathbb{R}}.$ Let $J\subseteq I$ dense in $I$ and assume that $F$ is locally [pseudomonotone]{} on $J.$ Further, assume that $0\not\in 