1 Introduction

The weak convergence of the extragradient method for solving pseudo-monotone variational inequalities in infinite dimensional Hilbert spaces was studied recently in [1]. The convergence analysis requires the sequential weak continuity of the associated operator. The illustrated example given in [1, Section 4] unfortunately does not satisfy this assumption. We provide in this note new examples satisfying all assumptions required.

2 Updating of Reference 1

Let \(H=\ell _2\), the real Hilbert space, whose elements are the square-summable sequences of real numbers. Let \(\beta >0\) and

$$\begin{aligned} F_\beta (u):=(\beta -\Vert u\Vert )\,u, \quad \forall u \in H. \end{aligned}$$
(1)

It is proved in [1] that the operator \(F_{\beta }\) is pseudo-monotone and Lipschitz continuous. However, the sequential weak continuity of \(F_{\beta }\) was not checked. Indeed, this assumption is not satisfied: let \(\left\{ e_n \right\} \) be the standard basic of H, i.e., \(e_n= (0, \ldots , 1 , 0, \ldots , 0)\) with 1 at the n-th position. Then \(e_1+e_n\) converges weakly to \(e_1\) but \(F_\beta (e_1+e_n) = (\beta -\sqrt{2}) (e_1+e_n)\) converges weakly to \((\beta -\sqrt{2})e_1\) and \((\beta -\sqrt{2})e_1 \not = (\beta -1)e_1 = F_\beta (e_1)\).

A correct example can be found in [2, Example 2.1], where the operator F is pseudo-monotone, Lipschitz continuous and sequentially weakly continuous.

3 Conclusions

We give a corrigendum and new example to illustrate the main results obtained in [1]. The author thanks Radu Boţ for fruiful discussion in finding new examples and Heinz Bauschke for his awesome lectures at the University of Vienna in Dec, 2018, where the author learnt that the projection operator is not weakly continuous [3].