Abstract
In this paper, existence results on variational inequalities and generalized variational inequalities for some nonmonotone operators over closed convex subsets of a real reflexive Banach space are proved. In particular, some surjectivity results and applications to complementarity and generalized complementarity problems are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Browder, F. E.,Existence Theorems for Nonlinear Partial Differential Equations, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, Rhode Island, Vol. 16, pp. 1–60, 1970.
Browder, F. E.,Pseudo-Monotone Operators and the Direct Method of the Calculus of Variations, Archive for Rational Mechanic and Analysis, Vol. 38, pp. 268–277, 1970.
Browder, F. E.,Fixed-Point Theory and Nonlinear Problems, Bulletin of the American Mathematical Society, Vol. 9, pp. 1–39, 1983.
Browder, F. E.,On the Unification of the Calculus of Variations and the Theory of Monotone Nonlinear Operators in Banach Spaces, Proceedings of the National Academy of Sciences, USA, Vol. 56, pp. 419–425, 1966.
Cottle, R. W., andYao, J. C.,Pseudomonotone Complementarity Problems in Hilbert Space, Journal of Optimization Theory and Applications, Vol. 75, pp. 281–295, 1992.
Théra, M.,A Note on the Hartman-Stampacchia Theorem, Proceedings of the International Conference on Nonlinear Analysis and Applications, Edited by V. Lakshmikantham, Marcel Dekker, New York, New York, pp. 573–577, 1987.
Browder, F. E., andHess, P.,Nonlinear Mappings of Monotone Type in Banach Spaces, Journal of Functional Analysis, Vol. 11, pp. 251–294, 1972.
Kravvaritis, D.,Nonlinear Equations and Inequalities in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 67, pp. 205–214, 1979.
Mosco, U.,A Remark on a Theorem of F. E. Browder, Journal of Mathematical Analysis and Applications, Vol. 20, pp. 90–93, 1967.
Stampacchia, G.,Variational Inequalities, Theory and Applications of Monotone Operators, Edited by A. Ghizzetti, Edizioni Oderisi, Gubbio, Italy, pp. 101–192, 1969.
Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.
Harker, P. T., andPang, J. S.,Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48B, pp. 161–220, 1990.
Bazaraa, M. S., Goode, J. J., andNashed, M. Z.,A Nonlinear Complementarity Problem in Mathematical Programming in Banach Space, Proceedings of the American Mathematical Society, Vol. 35, pp. 165–170, 1972.
Saigal, R.,Extension of the Generalized Complementarity Problem, Mathematics of Operations Research, Vol. 1, pp. 260–266, 1976.
Luna, G.,A Remark on the Nonlinear Complementarity Problem, Proceedings of the American Mathematical Society, Vol. 48, pp. 132–134, 1975.
Author information
Authors and Affiliations
Additional information
Communicated by F. Giannessi
This work was partially supported by the National Science Council of the Republic of China under Contracts NSC 81-0208-M-007-34 and NSC 82-0208-M-110-023.
Rights and permissions
About this article
Cite this article
Guo, J.S., Yao, J.C. Variational inequalities with nonmonotone operators. J Optim Theory Appl 80, 63–74 (1994). https://doi.org/10.1007/BF02196593
Issue Date:
DOI: https://doi.org/10.1007/BF02196593