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Pseudomonotonicity and Economic Equilibrium Problem in Reflexive Banach Space

Zdzisaw Naniewicz

Faculty of Mathematics and Natural Sciences, College of Sciences, Cardinal Stefan Wyszyski University, Dewajtis 5, 01-815 Warsaw, Poland
naniewicz.z{at}acn.waw.pl

The motivation for this paper is the Walrasian general equilibrium model of economy, as formulated by Arrow and Debreu [Arrow, K., G. Debreu. 1954. Existence of an equilibrium for a competitive economy. Econometrica 22 264–290]. The problem considered takes the form of a system of variational inequalities on a reflexive Banach space as the infinite dimensional commodity space. The conditions sufficient for the existence of solutions are provided by means of the theory of pseudomonotone multivalued mapping due to Browder and Hess [Browder, F. E., P. Hess. 1972. Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11 251–294], and the Fenchel duality theory combined with the Galerkin method. The analysis is carried out without any lattice considerations and the commodity space is not required to have interior points. The substantial difference of the presented approach in comparison with currently applied methods is that the preferences are not bound by any variant of the -properness assumption and the consumption sets are not required to have a cone structure. This paper affords new existence results for both the finite and infinite dimensional setting.

Key Words: general equilibrium theory; variational inequalities; pseudo-monotonicity; duality; Galerkin method
History: Received: October 24, 2005; revision received: March 11, 2006;