cannot be weakened. As consequences we obtain some results that improve the results stated in [@La].

Locally quasiomonotone and locally strictly quasimonotone operators
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In this paragraph we study the local quasimonotonicity, respectively the local strict quasimonotonicity of set-valued operators on dense sets. As it was expected some additional conditions are needed in order to assure that local generalized monotonicity on a dense subset of a lower semicontinuous set-valued operator implies its global counterpart. Such conditions will be given and by an example we show that our conditions are essential in obtaining these results.

Let $F:I\subseteq{\mathbb{R}}\rightrightarrows {\mathbb{R}}$ be a set-valued function. Let us introduce the following notations. $$F(x)\ge 0\Leftrightarrow x^*\ge 0,\,\forall x^*\in F(x),\mbox{ and } F(x)>0, \mbox{ if } F(x)\ge0\mbox{ and }0\not\in F(x).$$ Obviously the inequalities $F(x)\le 0$, respectively $F(x)<0$ can be introduced analogously.

\[l31\] 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 quasimonotone on $J.$ Then, for any fixed $x\in I$ one has $F(x)\ge0,$ or $F(x)\le 0.$

Let $x\in I$, and suppose, that there exist $x_1^*,x_2^*\in F(x),\, x_1^*>0,\,x_2^*<0.$ Consider the intervals $V_1=(x_1^*-\epsilon,x_1^*+ \epsilon),$ respectively $V_2=(x_2^*-\epsilon,x_2^*+\epsilon),$ where $\epsilon>0$ such that $x_1^*-\epsilon>0$ 