$(x,x^*),(y,y^*)\in G(T),\, x\neq y,$ the following implication holds $$\<x^*,y-x\> \ge 0\Longrightarrow  \<y^*,y-x\>> 0;$$

4.  *quasimonotone* if for every $(x,x^*),(y,y^*)\in G(T)$ the following implication holds $$\<x^*,y-x\> > 0 \Longrightarrow  \<y^*,y-x\>\ge 0;$$

5.  *strictly quasimonotone*, if $T$ is quasimonotone, and for all $x,y\in D(T),\,x\neq y$ there exist $z\in(x,y)$ and $z^*\in T(z)$ such that$\<z^*,y-x\> \neq 0.$

The relation among these concepts is shown bellow.

  -------------------------------------------------------- -------------------------------
                             $$                               $T \mbox{ is monotone} $
                             $$                                     $\Downarrow $
   $ T \mbox{ is strictly pseudomonotone}\Longrightarrow$   $T \mbox{ is pseudomonotone}$
                        $\Downarrow$                                $\Downarrow$
   $T \mbox{  is strictly quasimonotone}\Longrightarrow$    $T \mbox{ is quasimonotone}.$
  -------------------------------------------------------- -------------------------------

The concepts of local monotonicity can be defined as follows.

We say that the operator $T$ is locally monotone, (respectively, locally pseudomonotone, locally strict pseudomonotone, locally quasimonotone, locally strict quasimonotone), on its domain $D(T)$, if every $x\in D(T)$ admits an open neighborhood $U_x\subseteq X$ such that the restriction of the operator $T$ on $U_x\cap D(T),\, T\big|_{U_x\cap D(T)}$ is monotone, (respectively, pseudomonotone, strictly pseudomonotone, quasimonotone, strictly quasimonotone).

According to [@KaPiLa1], respectively [@La], local monotonicity, respectively local generalized monotonicity of a real valued function of one real variable in general implies its global counterpart. More precisely the following hold.

\[p21\] Let $I\subseteq\mathbb{R}$ be an open interval and 