Abstract
Various existence results for variational inequalities in Banach spaces are derived, extending some recent results by Cottle and Yao. Generalized monotonicity as well as continuity assumptions on the operatorf are weakened and, in some results, the regularity assumptions on the domain off are relaxed significantly. The concept of inner point for subsets of Banach spaces proves to be useful.
Similar content being viewed by others
References
Kinderlehrer, D., andStampacchia, G.,An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, New York, 1980.
Thera, M.,A Note on the Hartman-Stampacchia Theorem, Nonlinear Analysis and Applications, Edited by V. Lakshmikantham Dekker, New York, New York, pp. 573–577, 1987.
Yao, J. C.,Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994.
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.
Hartman, G. J., andStampacchia, G.,On Some Nonlinear Elliptic Differential Functional Equations, Acta Matematica, Vol. 115, pp. 271–310, 1966.
Hadjisavvas, N., andSchaible, S.,On Strong Pseudomonotonicity and (Semi) strict Quasimonotonicity, Journal of Optimization Theory and Applications, Vol. 79, pp. 193–155, 1993.
Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.
Schaible, S.,Generalized Monotonicity: A Survey, Generalized Convexity, Edited by S. Komlosi, T. Rapcsak, and S. Schaible, Springer Verlag, Berlin, Germany, pp. 229–249, 1994.
Karamardian, S.,Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.
Harker, P. T., andPang, J. S.,Equilibrium Modeling with Variational Inequalities: Theory, Computation and Applications (to appear).
Cottle, R. W., andYao, J. C.,Pseudomonotone Complementarity Problems in Hilbert Space, Journal of Optimization Theory and Applications, Vol. 75, pp. 281–295, 1992.
Yao, J. C.,Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994.
Shih, M. H., andTan, K. K.,Browder-Stampacchia Variational Inequalities for Multivalued Monotone Operators, Journal of Mathematical Analysis and Applications, Vol. 134, pp. 431–440, 1988.
Avriel, M., Diewert, W. E., Schaible, S., andZang, I.,Generalized Concavity, Plenum Publishing Corporation, New York, New York, 1988.
Do, C.,Bifurcation Theory for Elastic Plates Subjected to Unilateral Conditions, Journal of Mathematical Analysis and Applications, Vol. 60, pp. 435–448, 1977.
Yosida, K.,Functional Analysis, Springer Verlag, Berlin, Germany, 1980.
Fan, K.,A Generalization of Tychonoff's Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961.
Hassouni, A.,Opératurs Quasimonotones: Application a Certains Problèmes Variationnels, Thèse, Université Paul Sabatier, Toulouse, France, 1993.
Karamardian, S., Schaible, S., andCrouzeix, J. P.,Characterizations of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 26, pp. 399–413, 1993.
Isac, G., andThera, M.,Complementarity Problem and the Existence of a Post-Critical Equilibrium State of a Thin Elastic Plate, Journal of Optimization Theory and Applications, Vol. 58, pp. 241–257, 1988.
Author information
Authors and Affiliations
Additional information
This work was completed while the first author was visiting the Graduate School of Management of the University of California, Riverside. The author wishes to thank the School for its hospitality.
Rights and permissions
About this article
Cite this article
Hadjisavvas, N., Schaible, S. Quasimonotone variational inequalities in Banach spaces. J Optim Theory Appl 90, 95–111 (1996). https://doi.org/10.1007/BF02192248
Issue Date:
DOI: https://doi.org/10.1007/BF02192248