# Relation between two types of pseudomonotone function

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.