Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces

A Note on “Existence Results for Noncoercive Mixed Variational Inequalities in Finite Dimensional Spaces”

“…In the proof of [1,Corollary 3.1], we say "Since assumption (T h) holds immediately for T (x) = Ax + a". This is not correct, as shown in the example given in the paper itself [1, page 127].…”

“…This is not correct, as shown in the example given in the paper itself [1, page 127]. So, given a matrix A ∈ R n×n , a vector a ∈ R n , and a closed and convex set K ⊂ R n , we consider the following assumption: (Ah): The pair (A, h) has the MVIP on K , with MVIP as defined in [1,Definition 3.1].…”

“…Analogously, since the proof of [1,Theorem 3.2] is based on [1, Corollary 3.1], we rewrite this theorem as follows:…”

“…Finally, in the remainder of the paper, whenever [1,Theorem 3.2] is used, assumption (Ah) should be added.…”

Solving Mixed Variational Inequalities Beyond Convexity

“…The first motivation behind this study comes from [19,20], where existence results for mixed variational inequalities involving quasiconvex functions were provided, hence the question of numerically solving such problems arose naturally. As far as we know in the literature one can find only iterative methods for solving mixed variational inequalities without convexity assumptions where the involved functions are taken continuous in works like [32][33][34], however these algorithms are either implicitly formulated or merely conceptual schemes, no implementations of them being known to us.…”