Correction to: Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

1 Correction to: J Glob Optim https://doi.org/10.1007/s10898-021-01088-x

The results in this paper hold in the setting of a reflexive Banach space endowed with the weak topology, unless otherwise stated.

The assumption iv. in Theorem 2 becomes:

iv. whenever $x,y \in C$, $x_n\in C,$ $x_n \rightharpoonup x$ and $f(x_n, (1-t)x+ty) \ge 0$ for all $t\in [0,1]$ and for all n, then $f(x,y) \ge 0;$

The proof of Proposition 2 becomes:

Proof

Let $x,y \in C$, $x_n\in C,$ $x_n\rightharpoonup x$ and $f(x_n, (1-t)x+ty) \ge 0$ for all $t\in [0,1];$ in particular, $f(x_n,x) \ge 0$ and $f(x_n,y) \ge 0$. Then, $\liminf _{n \rightarrow \infty } f(x_n,x) \ge 0$ and, by B-pseudomonotonicity, $f(x,y) \ge \limsup _{n \rightarrow \infty } f(x_n,y) \ge 0$, for all $y \in C$, that is $f(x,y) \ge 0.$ $\square$

Finally, in Definition 1 convexity of the set C is not essential.