for some $x^*\in T(x)$ then $\<y^*,y-x\>\ge0$ for all $y^*\in T(y).$

Let now $x,y\in D(T)$ arbitrary and assume that $\<x^*,y-x\>>0.$ We show that $\<y^*,y-x\>\ge0,$ for all $y^*\in T(y).$ Since $T$ is lower semicontinuous, according to Remark \[r21\], for every sequence $({x_n})\subseteq D(T)$ such that $x_n\longrightarrow x$ there exists a sequence of elements $x^*_n \in T(x_n)$ such that $x^*_{n}\longrightarrow x^*,$ respectively for every sequence $({y_n})\subseteq D(T)$ such that $y_n\longrightarrow y$ and for every $y^*\in T(y)$ there exists a sequence of elements $y^*_n \in T(y_n)$ such that $y^*_{n}\longrightarrow y^*.$ Let us fix $y^*\in T(y).$ Since $D$ is dense in $D(T)$, one can consider $(x_n),(y_n)\subseteq D.$ Obviously $\<x_n^*,y_n-x_n\>\longrightarrow \<x^*,y-x\>>0,$ hence $\<x_n^*,y_n-x_n\>>0$ if $n$ is big enough. According to the first part of the proof $\<y_n^*,y_n-x_n\>\ge0.$ By taking the limit $n\longrightarrow\infty$ one obtains $\<y^*,y-x\>\ge 0.$

According to the next corollary the previous result improves Theorem 3.6 from [@La].

Let $T:D(T){\longrightarrow}X^*$ be a continuous 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: $$\forall x,y\in D(T),\, z\in[x,y]\cap(D(T)\setminus D) \mbox{ one has }\<T(z),y-x\>\neq0.$$ Then $T$ is strictly quasimonotone.

Locally pseudomonotone and locally strictly pseudomonotone operators
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