that the definition of lower semicontinuity is equivalent to the following (see [@AF] Def. 1.4.2).

\[r21\] $T$ is lower semicontinuous at $x\in D(T)$, if and only if, for every sequence $({x_n})\subseteq D(T)$ such that $x_n\longrightarrow x$ and for every $x^*\in T(x)$ there exists a sequence of elements $x^*_n \in T(x_n)
$ such that $x^*_{n}\longrightarrow x^*.$

Obviously, when $T$ is single-valued then upper semicontinuity and also lower semicontinuity become the usual notion of continuity.

\[d3.1\] Let $T:X\longrightarrow X^*$ be a single valued operator. We say that $T$ is hemicontinuous at $x\in X$, if for all $(t_n)_{n\in{{\mathbb N}}}\subset {{\mathbb R}},\,t_n{\longrightarrow} 0,\, {n\longrightarrow\infty}$ and $y\in X,$ we have $T(x+t_n y)\rightharpoonup^* T(x),\, n\longrightarrow \infty,$ where $"\rightharpoonup^*"$ denotes the convergence with respect to the $weak^*$ topology of $X^*$.

Local generalized monotonicities
--------------------------------

In what follows we recall the definitions of several monotonicity concepts (see, for instance [@HKS; @L-S; @PQ]). The operator $T:X\rightrightarrows X^*$ is called:

1.  * monotone* if for all $(x,x^*),(y,y^*)\in G(T)$ one has the following $$\left\langle x^*-y^*,x-y\right\> \geq0;$$

2.  * pseudomonotone* if for all $(x,x^*),(y,y^*)\in G(T),$ the following implication holds $$\<x^*,y-x\> \ge 0\Longrightarrow  \<y^*,y-x\>\ge 0,$$ or equivalently, for all $(x,x^*),(y,y^*)\in G(T),$ one has $$\<x^*,y-x\> > 0\Longrightarrow  \<y^*,y-x\>>0;$$

3.  *strictly pseudomonotone* if for all 