Quasivariational Inequalities for a Dynamic Competitive Economic Equilibrium Problem
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Research Article Quasivariational Inequalities for a Dynamic Competitive Economic Equilibrium Problem Maria Bernadette Donato, Monica Milasi, and Carmela Vitanza Department of Mathematics, University of Messina, Contrada Papardo, Salita Sperone 31, 98166 Messina, Italy Correspondence should be addressed to Carmela Vitanza, [email protected] Received 3 February 2009; Revised 24 August 2009; Accepted 12 October 2009 Recommended by Siegfried Carl The aim of this paper is to consider a dynamic competitive economic equilibrium problem in terms of maximization of utility functions and of excess demand functions. This equilibrium problem is studied by means of a time-dependent quasivariational inequality which is set in the Lebesgue space L2 0, T , R. This approach allows us to obtain an existence result of timedependent equilibrium solutions. Copyright q 2009 Maria Bernadette Donato et al. 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.
1. Introduction The theory of variational inequality was born in the 1970s, driven by the solution given by G. Fichera to the Signorini problem on the elastic equilibrium of a body under unilateral constraints and by Stampacchia’s work on defining the capacitory potential associated to a nonsymmetric bilinear form. It is possible to attach to this theory a preliminary role in establishing a close relationship between theory and applications in a wide range of problems in mechanics, engineering, mathematical programming, control, and optimization 1–4. In this paper, a dynamic competitive economic equilibrium problem by using a variational formulation is studied. It was Walras 5 who, in 1874, laid the foundations for the study of the general equilibrium theory, providing a succession of models, each taking into account more aspects of a real economy. The rigorous mathematical formulation of the general equilibrium problem, with possibly nonsmooth but convex data, was elaborated by Arrow and Debreu 6 in the 1954. In 1985, Border in 7 elaborated a variational inequality formulation of a Walrasian price equilibrium. By means of the variational formulation, Dafermos in 8 and Zhao in 9 proved some qualitative results for the solutions to the Walrasian problem in the static case. Moreover, Nagurney and Zhao 10 see also Zhao 9, Dafermos and Zhao
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Journal of Inequalities and Applications
11 considered the static Walrasian price equilibrium problem as a network equilibrium problem over an abstract network with very simple structure that consists of a single origin-destination pair of nodes and single links joining the two nodes. Furthermore, the characterization of Walrasian price equilibrium vectors as solutions of a variational inequality induces efficient algorithms for their computation for further details see also Nagurney’s book 12, Chapter 9, and its complete bibliography. In 13 it was proven ho
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