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Abstract and Applied Analysis
Volume 2012, Article ID 651975, 35 pages
http://dx.doi.org/10.1155/2012/651975
Research Article

Time-Dependent Variational Inequality for an Oligopolistic Market Equilibrium Problem with Production and Demand Excesses

1Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Cintia 80126 Naples, Italy
2Department of Mathematics and Computer Science, University of Catania, Viale A. Doria, 95125 Catania, Italy

Received 2 February 2012; Accepted 20 March 2012

Academic Editor: Kanishka Perera

Copyright © 2012 Annamaria Barbagallo and Paolo Mauro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper is concerned with the variational formulation of the oligopolistic market equilibrium problem in presence of both production and demand excesses. In particular, we generalize a previous model in which the authors, instead, considered only the problem with production excesses, by allowing also the presence of demand excesses. First we examine the equilibrium conditions in terms of the well-known dynamic Cournot-Nash principle. Next, the equilibrium conditions will be expressed in terms of Lagrange multipliers by means of the infinite dimensional duality theory. Then, we show the equivalence between the two conditions that are both expressed by an appropriate evolutionary variational inequality. Moreover, thanks to the variational formulation, some existence and regularity results for equilibrium solutions are proved. At last, a numerical example, which illustrates the features of the problem, is provided.