Evolutionary variational inequality with long-term memory and applications to economic networks

The aim of this paper is to present the relation between an evolutionary variational inequality with long-term memory and Lagrange multipliers. More precisely, we study the oligopolistic market equilibrium problem in which the profit function depends also on previous events of the market by means of a long-term memory which takes into account the previous states of the equilibrium. Moreover, thanks to the variational formulation, we are able to show existence and regularity results for equilibrium solutions. Then, we apply the infinite dimensional duality theory through which we obtain the existence of Lagrange multipliers which are great utility in order to understand the behaviour of the market. Finally, an example is provided, which allows to analyse the influence of the long-term memory on the equilibrium solution.