Abstract
In this paper, some existence results for a nonlinear complementarity problem involving a pseudo-monotone mapping over an arbitrary closed convex cone in a real Hilbert space are established. In particular, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space. Applications to a class of nonlinear complementarity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Karamardian, S.,Generalized Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 8, pp. 161–168, 1971.
Allen, G.,Variational Inequalities, Complementarity Problems, and Duality Theorems, Journal of Mathematical Analysis and Applications, Vol. 58, pp. 1–10, 1977.
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.
Borwein, J. M.,Alternative Theorems for General Complementarity Problems, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, New York, New York, Vol. 259, pp. 194–203, 1985.
Dash, A. T., andNanda, S.,A Complementarity Problem in Mathematical Programming in Banach Space, Journal of Mathematical Analysis and Applications, Vol. 98, pp. 328–331, 1984.
Gowda, M. S., andSeidman, T. I.,Generalized Linear Complementarity Problems, Mathematical Programming, Vol. 46, pp. 329–340, 1990.
Habetler, G. J., andPrice, A. L.,Existence Theory for Generalized Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 7, pp. 223–239, 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. 48, pp. 161–220, 1990.
Isac, G.,On Some Generalization of Karamardian's Theorem on the Complementarity Problem, Bollettino UMI, Vol. 7, 2-B, pp. 323–332, 1988.
Isac, G.,Nonlinear Complementarity Problem and Galerkin Method, Journal of Mathematical Analysis and Applications, Vol. 108, pp. 563–574, 1985.
Isac, G.,Problèmes de Complémentarité [en Dimension Infinie], Minicourse, Publication du Département de Mathématique, Université de Limoges, Limoges, France, 1985.
Isac, G., andThéra M.,Complementarity Problem and the Existence of the Post-Critical Equilibrium State of a Thin Elastic Plate, Journal of Optimization Theory and Applications, Vol. 58, pp. 241–257, 1988.
Karamardian, S.,Complementarity Problems over Cones with Monotone and Pseudo-Monotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.
Moré, J. J.,Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.
Nanda, S., andNanda, S.,A Nonlinear Complementarity Problem in Mathematical Programming in Hilbert Space, Bulletin of the Australian Mathematical Society, Vol. 20, pp. 233–236, 1979.
Barbu, V., andPrecupanu, T.,Convexity and Optimization in Banach Spaces, Sijthoff and Noordhoff, Bucharest, Romania, 1978.
Hartman, G. J., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.
Kinderlehrer, D., andStampacchia, G.,An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, New York, 1980.
Théra, M.,A Note on the Hartman-Stampacchia Theorem, Nonlinear Analysis and Applications, Edited by V. Lakshmikantham, Marcel Dekker, New York, New York, pp. 573–577, 1987.
Holmes, R. B.,Geometric Functional Analysis and Its Applications, Springer-Verlag, New York, New York, 1975.
Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.
Karamardian, S., andSchaible, S.,First-Order Characterizations of Generalized Monotone Maps, Manuscript, Graduate School of Management, University of California, Riverside, California, 1989.
Bourbaki, N.,Topological Vector Spaces, Chapters 1–5, Translated by H. G. Eggleston and S. Madan, Springer-Verlag, Berlin, Germany, 1987.
Fan, K.,A Generalization of the Alaoglu-Bourbaki Theorem and Its Applications, Mathematische Zeitschrift, Vol. 88, pp. 48–60, 1965.
Do, C.,Bifurcation Theory for Elastic Plates Subjected to Unilateral Conditions, Journal of Mathematical Analysis and Applications, Vol. 60, pp. 435–448, 1977.
Author information
Authors and Affiliations
Additional information
Communicated by S. Karamardian
This research was partially supported by the National Science Foundation Grant DMS-89-13089, Department of Energy Grant DE-FG03-87-ER-25028, and Office of Naval Research Grant N00014-89-J-1659. The authors would like to express their sincere thanks to Professor S. Schaible, School of Administration, University of California, Riverside, for his helpful suggestions and comments. They also thank the referees for their comments and suggestions that improved this paper substantially.
Rights and permissions
About this article
Cite this article
Cottle, R.W., Yao, J.C. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl 75, 281–295 (1992). https://doi.org/10.1007/BF00941468
Issue Date:
DOI: https://doi.org/10.1007/BF00941468