Math. Meth. Oper. Res. (2006) 63: 553–565 DOI 10.1007/s00186-005-0052-2 ORIGINAL ARTICLE E. Allevi · A. Gnudi · I. V. Konnov The proximal point method for nonmonotone variational inequalities Received: 15 July 2004 / Accepted: 15 April 2005 / Published online: 29 June 2006 © Springer-Verlag 2006 Abstract We consider an application of the proximal point method to variational inequality problems subject to box constraints, whose cost mappings possess order monotonicity properties instead of the usual monotonicity ones. Usually, conver- gence results of such methods require the additional boundedness assumption of the solutions set. We suggest another approach to obtaining convergence results for proximal point methods which is based on the assumption that the dual varia- tional inequality is solvable. Then the solutions set may be unbounded. We present classes of economic equilibrium problems which satisfy such assumptions. Keywords Proximal point method · Multivalued variational inequalities · Box-constrained sets · Nonmonotone mappings 1 Introduction Let X be a nonempty convex set in the real n-dimensional space R n , and let G : X → (R n ) be a multivalued mapping [Here and below (A) denotes the family of all nonempty subsets of a set A]. Then one can define the multivalued variational inequality problem (VI for short): Find a point x ∗ ∈ X such that E. Allevi Department of Quantitative Methods, Brescia University, Contrada Santa Chiara, 50, Brescia 25122, Italy A. Gnudi Department of Mathematics, Statistics, Informatics and Applications, Bergamo University,Via dei Caniana, 2, Bergamo 24127, Italy I. V. Konnov (B ) Department of Applied Mathematics, Kazan University, ul. Kremlevskaya,18, Kazan 420008, Russia E-mail: Igor.konnov@ksu.ru