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Abstract and Applied Analysis
Volume 2013, Article ID 952915, 12 pages
http://dx.doi.org/10.1155/2013/952915
Research Article

A Quasi-Variational Approach for the Dynamic Oligopolistic Market Equilibrium Problem

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 6, 95125 Catania, Italy

Received 18 July 2013; Accepted 19 September 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 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.

Linked References

  1. A. Barbagallo and P. Mauro, “Evolutionary variational formulation for oligopolistic market equilibrium problems with production excesses,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 288–314, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Barbagallo, “Regularity results for evolutionary nonlinear variational and quasi-variational inequalities with applications to dynamic equilibrium problems,” Journal of Global Optimization, vol. 40, no. 1-3, pp. 29–39, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Scrimali, “Quasi-variational inequalities in transportation networks,” Mathematical Models & Methods in Applied Sciences, vol. 14, no. 10, pp. 1541–1560, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Barbagallo and S. Pia, “Weighted quasi-variational inequalities in non-pivot Hilbert spaces and applications,” accepted on Journal of Optimization Theory and Applications.
  5. A. Cournot, “Researches into the mathematical principles of the theory of wealth,” Competition Policy International, vol. 4, no. 1, pp. 283–305, 2008. View at Google Scholar · View at Scopus
  6. J. F. Nash, Jr., “Equilibrium points in n-person games,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 48–49, 1950. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J. Nash, “Non-cooperative games,” Annals of Mathematics. Second Series, vol. 54, pp. 286–295, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Dafermos and A. Nagurney, “Oligopolistic and competitive behavior of spatially separated markets,” Regional Science and Urban Economics, vol. 17, no. 2, pp. 245–254, 1987. View at Google Scholar · View at Scopus
  9. A. Nagurney, “Algorithms for oligopolistic market equilibrium problems,” Regional Science and Urban Economics, vol. 18, no. 3, pp. 425–445, 1988. View at Google Scholar · View at Scopus
  10. A. Nagurney, P. Dupuis, and D. Zhang, “A dynamical systems approach for network oligopolies and variational inequalities,” The Annals of Regional Science, vol. 28, no. 3, pp. 263–283, 1994. View at Publisher · View at Google Scholar · View at Scopus
  11. A. Nagurney, Network Economics: A Variational Inequality Approach, vol. 1 of Advances in Computational Economics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. Barbagallo and M.-G. Cojocaru, “Dynamic equilibrium formulation of the oligopolistic market problem,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 966–976, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. J. Beckmann and J. P. Wallace, “Continuous lags and the stability of market equilibrium,” Economica, vol. 36, pp. 58–68, 1969. View at Google Scholar
  14. A. Barbagallo and A. Maugeri, “Memory term for dynamic oligopolistic market equilibrium problem,” Aplimat—Journal of Applied Mathematics, vol. 3, no. 13, 23 pages, 2010. View at Google Scholar
  15. A. Barbagallo and R. Di Vincenzo, “Lipschitz continuity and duality for dynamic oligopolistic market equilibrium problem with memory term,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 231–247, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Barbagallo and A. Maugeri, “Duality theory for the dynamic oligopolistic market equilibrium problem,” Optimization, vol. 60, no. 1-2, pp. 29–52, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. Daniele, S. Giuffrè, G. Idone, and A. Maugeri, “Infinite dimensional duality and applications,” Mathematische Annalen, vol. 339, no. 1, pp. 221–239, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. P. Daniele and S. Giuffrè, “General infinite dimensional duality and applications to evolutionary network equilibrium problems,” Optimization Letters, vol. 1, no. 3, pp. 227–243, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. P. Daniele, S. Giuffré, and A. Maugeri, “Remarks on general infinite dimensional duality with cone and equality constraints,” Communications in Applied Analysis for Theory and Applications, vol. 13, no. 4, pp. 567–577, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Maugeri and F. Raciti, “Remarks on infinite dimensional duality,” Journal of Global Optimization, vol. 46, no. 4, pp. 581–588, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. B. Donato, “The infinite dimensional Lagrange multiplier rule for convex optimization problems,” Journal of Functional Analysis, vol. 261, no. 8, pp. 2083–2093, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. M. Borwein and A. S. Lewis, “Partially finite convex programming. I. Quasi relative interiors and duality theory,” Mathematical Programming, vol. 57, no. 1, pp. 15–48, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J. Jahn, Introduction to the Theory of Nonlinear Optimization, Springer, Berlin, Germany, 2nd edition, 1996. View at MathSciNet
  24. M. B. Donato, A. Maugeri, M. Milasi, and C. Vitanza, “Duality theory for a dynamic Walrasian pure exchange economy,” Pacific Journal of Optimization, vol. 4, no. 3, pp. 537–547, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. B. Donato and M. Milasi, “Lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 5048–5056, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. Barbagallo and P. Mauro, “Time-dependent variational inequality for an oligopolistic market equilibrium problem with production and demand excesses,” Abstract and Applied Analysis, vol. 2012, Article ID 651975, 35 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. A. Barbagallo and P. Mauro, “On solving dynamic oligopolistic market equilibrium problems in presence of excesses,” Communications in Applied and Industrial Mathematics, vol. 3, no. 1, pp. 1–20, 2012. View at Google Scholar · View at MathSciNet
  28. A. Barbagallo, “Advanced results on variational inequality formulation in oligopolistic market equilirbrium problem,” Filomat, vol. 5, pp. 935–947, 2012. View at Google Scholar
  29. A. Barbagallo and P. Mauro, “Inverse variational inequality approach and applications,” submitted.
  30. A. Barbagallo and P. Mauro, “An inverse problem for the dynamic oligopolistic market equilibrium problem in presence of excesses,” accepted on Procedia—Social and Behavioral Sciences.
  31. O. L. Mangasarian, “Pseudo-convex functions,” Journal of the Society for Industrial and Applied Mathematics, Series A Control, vol. 3, pp. 281–290, 1965. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. N. X. Tan, “Quasi-variational inequality in topological linear locally convex Hausdorff spaces,” Mathematische Nachrichten, vol. 122, pp. 231–245, 1985. View at Google Scholar
  33. K. Kuratowski, Topology. Vol. I, Academic Press, New York, NY, USA, 1966. View at MathSciNet
  34. G. Salinetti and R. J.-B. Wets, “On the convergence of sequences of convex sets in finite dimensions,” SIAM Review, vol. 21, no. 1, pp. 18–33, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. G. Salinetti and R. J.-B. Wets, “Addendum “On the convergence of convex sets infinite dimensions”,” SIAM Review, vol. 22, p. 86, 1980. View at Google Scholar
  36. A. Barbagallo, “Regularity results for time-dependent variational and quasi-variational inequalities and application to the calculation of dynamic traffic network,” Mathematical Models & Methods in Applied Sciences, vol. 17, no. 2, pp. 277–304, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. A. Barbagallo, “Existence and regularity of solutions to nonlinear degenerate evolutionary variational inequalities with applications to dynamic network equilibrium problems,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 1–13, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. A. Barbagallo, “On the regularity of retarded equilibrium in time-dependent traffic equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2406–e2417, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. A. Barbagallo and M.-G. Cojocaru, “Continuity of solutions for parametric variational inequalities in Banach space,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 707–720, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. A. Barbagallo and S. Pia, “Weighted variational inequalities in non-pivot Hilbert spaces with applications,” Computational Optimization and Applications, vol. 48, no. 3, pp. 487–514, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. A. Maugeri, “Convex programming, variational inequalities, and applications to the traffic equilibrium problem,” Applied Mathematics and Optimization, vol. 16, no. 2, pp. 169–185, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet