TIME-DEPENDENT VARIATIONAL INEQUALITIES AND

APPLICATIONS TO EQUILIBRIUM PROBLEMS

Fabio Raciti

Department of Mathematics and Computer Science - University of Catania, Italy

                      e-mail fraciti@dmi.unict.it

Laura Scrimali

Department of Mathematics and Computer Science - University of Catania, Italy

                     e-mail scrimali@dmi.unict.it

In the last years several authors have dealt with time-dependent problems in the framework of variational inequalities. Very recently time has been introduced in variational models describing traffic problems as well as equilibrium markets problems. In this note we present two generalizations of time-dependent variational inequalities. The first topic concerns delay effects in economic networks. In economic markets models the importance of delay derives from the fact that, even if the propagation of information through the network could be considered in certain cases instantaneous, producers and consumers usually take their decisions sometimes after they get all the data available. In particular we are interested in spatial price equilibrium problems, where one wants to compute the commodity supply prices, demand prices and trade flows that satisfy a certain equilibrium condition. Our approach extends previous work by A. Nagurney, and successively by P. Daniele and A. Maugeri. The second application concerns equilibrium problems in transportation networks. In this case, variational analysis is exploited to the aim of surveying dynamic and elastic models, when capacity constraints are introduced on all the routes. The elasticity of travel demands requires a formulation in terms of quasi-variational inequalities. Therefore, by applying some recent results by M. De Luca and A. Maugeri, we shall provide a theorem for the existence of solutions. Finally, we shall propose some stability and sensitivity results in order to clarify the behavior of equilibrium solutions, when some changes in data occur.

Presented by Fabio Raciti and Laura Scrimali