Regularization of Brézis pseudomonotone variational inequalities

Abstract

In this paper we prove the existence of solutions of regularized set-valued variational inequalities involving Brézis pseudomonotone operators in reflexive and locally uniformly convex Banach spaces. By taking advantage of this result, we approximate a general set-valued variational inequality with suitable regularized set-valued variational inequalities, and we show that their solutions weakly converge to a solution of the original one. Furthermore, by strengthening the coercivity conditions, we can prove the strong convergence of the approximate solutions.

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Acknowledgment

We would like to express our gratitude to the referees for their careful reading which improved the presentation of the paper. The research of the second author was supported by a Grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project PN-III-P4-ID-PCE-2016-0190, within PNCDI III.

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Correspondence to M. Bianchi.

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Bianchi, M., Kassay, G. & Pini, R. Regularization of Brézis pseudomonotone variational inequalities. Set-Valued Var. Anal 29, 175–190 (2021). https://doi.org/10.1007/s11228-020-00543-3

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Keywords

  • Set-valued variational inequality
  • B-pseudomonotonicity
  • Approximate solutions
  • Equilibrium problem
  • Navier-Stokes operator

Mathematics Subject Classification (2010)

  • 49J53
  • 49J40
  • 47H05
  • 47J20