that $z>y.$ According to the previous part of the proof there exist an open neighborhood $U_y$ of $y,$ respectively $U_z$ of $z$ such that $F(u)\ge 0$ for all $u\in{U_y\cap I,}$ respectively $F(v)\le0$ for all $v\in{U_z\cap I}.$

Let $u\in U_y\cap U_x\cap J,$ respectively $v\in U_z\cap U_x\cap J,$ such that $u<v,$ and with the property that there exist $u^*\in F(u),\, u^*>0$, respectively $v^*\in F(v),\,v^*<0.$ Such $u^*$ and $v^*$ must exist since $F$ is strictly quasimonotone on $U_x\cap J.$ Then, $\<u^*,v-u\>>0$ and $\<v^*,v-u\><0,$ contradiction.

We have shown, that $F$ is locally strictly quasimonotone on $I$. According to Proposition \[p31\], $F$ is globally strictly quasimonotone on $I$ and the proof is completely done.

In what follows we extend Theorem \[t31\] for lower semicontinuous operators that are locally strictly quasimonotone on a self segment-dense subset of their convex domain.

\[t32\] Let $X$ be a real Banach space and let $X^*$ be its topological dual. Let $T:X\rightrightarrows X^*$ be a lower semicontinuous operator with convex domain $D(T).$ Let $D\subseteq D(T)$ self segment-dense in $D(T)$ and assume that $T$ is locally strictly quasimonotone on $D.$ Assume further, that the following condition holds: $$\label{e1}
\forall x,y\in D(T),\, z\in[x,y]\cap(D(T)\setminus D)\mbox{ and }z^*\in T(z) \mbox{ one has }\<z^*,y-x\>\neq0.$$ Then $T$ 