Abstract
In the present work, we deal with set-valued equilibrium problems, for which we provide sufficient conditions for the existence of a solution. The conditions, that we consider, are imposed not on the whole domain, but rather on a self-segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu–Gale–Nikaïdo-type theorem, with a considerably weakened Walras law in its hypothesis. Furthermore, we consider a noncooperative \(n\)-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fan, K.: A minimax inequality and its application. In: Shisha, O. (ed.) Inequalities, vol. 3, pp. 103–113. Academic Press, New York (1972)
Kristály, A., Varga, C.: Set-valued versions of Ky Fan’s inequality with application to variational inclusion theory. J. Math. Anal. Appl. 282, 8–20 (2003)
Browder, F.E.: The fixed point theory of multi-valued mappings in topological vector spaces. Math. Ann. 177, 283–301 (1968)
László, S., Viorel, A.: Generalized monotone operators on dense sets. arXiv:1310.4453
Luc, D.T.: Existence results for densely pseudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291–308 (2001)
Debreu, G.: Theory of Value. Wiley, New York (1959)
Gale, D.: The law of supply and demand. Math. Scand. 3, 155–169 (1955)
Nikaido, H.: Convex Structures and Economic Theory. Academic Press, San Diego (1968)
Aubin, J.P.: Optima and Equilibria. Springer, Berlin (1993)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Lin, L.J., Yang, M.F., Ansari, Q.H., Kassay, G.: Existence results for Stampacchia and Minty type implicit variational inequalities with multivalued maps. Nonlinear Anal. 61, 1–19 (2005)
László, S.: Multivalued variational inequalities and coincidence point results. J. Math. Anal. Appl. 404, 105–114 (2013)
Lin, K.L., Yang, D.P., Yao, J.C.: Generalized vector variational inequalities. J. Optim. Theory Appl. 92, 117–125 (1997)
Yao, J.C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)
Yao, J.C., Chadli, O.: Pseudomonotone complementary problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Monotonicity, Springer Series Nonconvex Optimization and its Applications, pp. 501–558. Springer, New York (2005)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
Acknowledgments
The authors are grateful to two anonymous referees for their helpful comments and suggestions which led to improvement of the original submitted version of this work. This work was supported by a Grant of the Romanian Ministry of Education, CNCS—UEFISCDI, Project number PN-II-RU-PD-2012-3-0166. This paper is a result of a research made possible by the financial support of the Sectoral Operational Programme for Human Resources Development 2007–2013, co-financed by the European Social Fund, under the project POSDRU/159/1.5/S/132400-“Young successful researchers—professional development in an international and interdisciplinary environment”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dinh The Luc.
Rights and permissions
About this article
Cite this article
László, S., Viorel, A. Densely Defined Equilibrium Problems. J Optim Theory Appl 166, 52–75 (2015). https://doi.org/10.1007/s10957-014-0702-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-014-0702-8
Keywords
- Self-segment-dense set
- Set-valued equilibrium problem
- Debreu–Gale–Nikaïdo-type theorem
- Nash equilibrium