Regularization of Brézis pseudomonotone variational inequalities in topological vector spaces

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In this paper, the definition of Brézis pseudomonotone (B-pseudomonotone) is extended to topological vector spaces. Some propositions, by using Banach–Alaoglu theorem and the notations of radially continuity and locally bounded, in order a set-valud mapping to be B-pseudomonotone are provided. Three Kinds of variational inequalities problems, by investigating the relationship between their solution sets, are introduced. An existence theorem for a solution of a variational inequality, by applying KKM theory and Berge's Theorem, is given. Also, an existence result for a solution of strong variational inequality by using Sion's Lemma is stated. Finally, the strong variational inequality problem, by using the solution sets of regularized variational inequalities problems depend on a sequence of set-valued mappings is approximated. The results of this article extend the main results given in Bianchi et al. (Regularization of Brézis pseudomonotone variational inequalities. Set-Valued Var. Anal. 2021;29:175–190), from reflexive Banach spaces to Hausdorff topological vector spaces.