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.
Similar content being viewed by others
References
Alber, Y., Butnariu, D., Ryazantseva, I.: Regularization of monotone variational inequalities with Mosco approximations of the constraint sets. Set-Valued Anal. 13, 265–290 (2005)
Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Ben Aadi, S., Chadli, O., Koukkous, A.: Evolution hemivariational inequalities for non-stationary Navier-Stokes equations: existence of periodic solutions by an equilibrium problem approach. Minimax Theory Appl. 3(1), 107–130 (2018)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)
Brézis, H.: Équations et inéquations non-linéaires dans les espaces vectoriel en dualité. Ann. Inst. Fourier 18, 115–176 (1968)
Brézis, H., Nirenberg, L., Stampacchia, G.: A remark on Ky Fan’s minimax principle. Boll. Un. Mat. Ital. 6(4), 293–300 (1972)
Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Nat. Acad. Sci. U.S.A. 56, 1080–1086 (1966)
Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972)
Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86, 433–438 (1999)
Fan, K.: A minimax inequality and applications, Inequalities III Shisha ed., pp 103–113. Academic press, Cambridge (1972)
Giannessi, F., Maugeri, A. (eds.): Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York (1995)
Hu, S.H., Papageorgiou, N.S.: Handbook of multivalued analysis, vol. 1. Kluwer, Norwell (1997)
Inoan, D., Kolumbán, J.: On pseudomonotone set-valued mappings. Nonlinear Anal. 68, 47–53 (2008)
Karamardian, S., Schaible, S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)
Kassay, G., Miholca, M.: Existence results for variational inequalities with surjectivity consequences related to generalized monotone operators. J. Optim. Theory Appl. 159, 721–740 (2013)
Kien, B.T., Lee, G.M.: An existence theorem for generalized variational inequalities with discontinuous and pseudomonotone operators. Nonlinear Anal. 74, 1495–1500 (2011)
Kien, B.T., Wong, M.M., Wong, N.C., Yao, J.C.: Solution existence of variational inequalities with pseudomonotone operators in the sense of Brézis. J. Optim. Theory Appl. 140, 249–263 (2009)
Nagurney, A.: Network Economics: a Variational Inequality Approach. Kluwer Academic Publishers, Dordrecht (1993)
Sion, M.: On general minimax theorems, Pac. J. Math. 8, 171–176 (1958)
Tikhonov, A.N.: Regularization of incorrectly posed problems, Soviet. Math. Dokl. 4, 1035–1038 (1963)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, II/B. Nonlinear Monotone Operators. Springer, Berlin (1990)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-020-00543-3
Keywords
- Set-valued variational inequality
- B-pseudomonotonicity
- Approximate solutions
- Equilibrium problem
- Navier-Stokes operator