x>1.
\end{array}
\right.$$ Then $F$ is lower semicontinuous and locally [quasimonotone]{} on ${\mathbb{R}}$, but is not globally [quasimonotone]{}.

It is easy to check that $F$ is locally [quasimonotone]{} and lower semicontinuous on ${\mathbb{R}}.$ On the other hand, for $x=-2,\,y=2$ and $x^{\ast }=1\in F(x)$ we have $\<x^{\ast },y-x\>=4>0,$ but for $y^{\ast }=-1\in F(y)$ we have $\<y^{\ast },y-x\>=-4<0,$ which shows that $F$ is not [quasimonotone]{}.

Next we define the local generalized monotonicity of an operator on a subset of its domain.

\[d2.1\] Let $T:X\rightrightarrows X^*$ be a set-valued operator and let $D\subseteq D(T).$ We say that $T$ is locally quasimonotone, (respectively, locally monotone, locally pseudomonotone, locally strictly pseudomonotone, locally strictly quasimonotone), on $D$, if every $x\in D$ admits an open neighborhood $U_x$, such that the restriction $T|_{U_x\cap D}$ is [quasimonotone]{}, (respectively, [monotone]{}, [pseudomonotone]{}, [strictly pseudomonotone]{}, [strictly quasimonotone]{}).

\[r2.2\]

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Some global generalized monotonicity results for single-valued operators
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Having Definition \[d2.1\] in mind, an important question is related to which properties must the subset $D\subseteq D(T)$ have such that the local generalized monotonicity on $D$ implies the global monotonicity of the same kind. The following results concerning single-valued operators, established in [@KaPiLa1], respectively [@La], will serve as a starting point for 