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As far as I know, there are two different definitions for pseudomonotone function.

The first definition was introduced by H.Brezis in 1968:

Pseudomonotone function in the sense of Brezis:

Let $V$ be a reflexive Banach space with its dual space $V^\ast$. A mapping $A\colon V\to V^\ast$ is called pseudomonotone iff $A$ is bounded and $\begin{cases} u_n \rightharpoonup u, \\ \limsup\langle Au_n, u_n - u \rangle \le 0\end{cases}$ implies $\liminf \langle Au_n, u_n - v\rangle \ge \langle Au, u - v\rangle$ for all $v\in X$.

The second one refers to Karamardian:

Pseudomonotone in the sense of Karamardian:

Let $V$ be a real Banach space with its dual space $V^\ast$. A mapping $A\colon V\to V^\ast$ is called pseudomonotone iff for all $u, v\in V$, $\langle Av, u - v\rangle \ge 0$ implies $\langle Au, u-v \rangle \ge 0$.

I want to ask for any relation between these two definitions or any article related to this question.

Thank you very much.

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