Abstract
We show that many equilibrium problems fulfill the common laws expressed by a set of conditions and that the equilibrium solution is obtained as a solution to a Variational Inequality. In particular we study the traffic equilibrium problem in the continuum case and we solve the problem to express this problem by means of a Variational Inequality.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Auslender, Noncoercive Optimization Problems, Mathematics of Operation Research (1996).
C. Baiocchi, G. Buttazzo, F. Gastaldi, F. Tomarelli, General existence theorems for unilateral problems in continuum mechanics, Arch. Rational Mech. Anal. 100 (1988), 149–189.
C. Baiocchi, A. Capelo, Variational and quasivariational inequalities: applications to free boundary problems, J. Wiley and Sons, Chichester (1984).
V. Barbu, Optimal control of Variational Inequalities, Research Notes in Mathematics 100 (1984).
H. Brezis, Problème Unilatéraux, J. Math. Pures et Appl. 51 (1972), 1–168.
S. Dafermos, Continuum Modelling of Transportation Networks, Transportation Res. 14 B (1980), 295–301.
P. Daniele, Lagrangean Function for Dynamic Variational Inequalities, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 58 (1999), 101–119.
P. Daniele-A. Maugeri, Vector Variational Inequalities and a Continuum Modelling of traffic Equilibrium Problem, In Vector Variational Inequalities and Vector Equilibria, F. Giannessi Ed., Kluwer Academic Publishers (2000), 97–111.
G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Sez. I (8) 7 (1964), 71–140.
G. Fichera, Boundary value problems in elasticity with unilateral constraints, Handbuch der Physik, IV a/2, Springer-Verlag, Berlin Heidelberg New York (1972), 347–389.
G. Fichera, Problemi unilaterali nella statica dei sistemi continui, problemi attuali di meccanica teorica e applicata, Atti del Convegno Internazionale a ricordo di Modesto Ponetti, Torino (1977), 171–178.
J. Gwinner, On continuum Modelling of Large Dense Networks in Urban Road Traffic, In Mathematics in Transport Planning and Control (J.D. Griffiths ed.), IMA Conference, Cardiff (1988).
J.L. Lions, G. Stampacchia, Variational Inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519.
P.L. Lions, Mathematical Topics in Fluid Mechanics, Clarendon Press., Oxford, 1996.
A. Maugeri. New Classes of Variational Inequalities and Applications to Equilibrium Problems, Methods of Operation Research 53 (1985), 129–131.
A. Maugeri, New classes of Variational Inequalities and Applications to Equilibrium Problems, Rendiconti Accademia Nazionale delle Scienze detta dei XL 11 (1987), 277–284.
A. Nagurney, Network Economics — A Variational Inequality Approach, Kluwer Academic Publishers, 1993.
B.D. Reddy, F. Tomarelli, The obstacle problem for an elastoplastic body, Appl. Math. Optim. 21 (1990), 89–110.
A. Signorini, Questioni di elasticità non linearizzata e semilinearizzata, Rend. Mat. 18 (1959), 95–139.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Maugeri, A. (2001). Equilibrium Problems and Variational Inequalities. In: Giannessi, F., Maugeri, A., Pardalos, P.M. (eds) Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Nonconvex Optimization and Its Applications, vol 58. Springer, Boston, MA. https://doi.org/10.1007/0-306-48026-3_12
Download citation
DOI: https://doi.org/10.1007/0-306-48026-3_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-0161-1
Online ISBN: 978-0-306-48026-3
eBook Packages: Springer Book Archive