Nonlinear Analysis 42 (2000) 71 – 83 www.elsevier.nl/locate/na Resolvent equations for set-valued mixed variational inequalities Muhammad Aslam Noor a;∗ , Themistocles M. Rassias b a Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5 b Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece Received 13 November 1997; accepted 2 May 1998 Keywords: Variational inequalities; Resolvent equations; Iterative algorithms; Convergence 1. Introduction In recent years, variational inequality theory has appeared as an elegant and fascinat- ing branch of applicable mathematics. Variational inequalities arise in various models for a large number of mathematical, physical, regional, and other problems. Variational inequalities have been extended and generalized in many directions using novel and innovative techniques. For the recent state of the art, one may see [1–27] and the references therein. One of the most dicult and important problem in variational in- equality theory is the development of an ecient and implementable algorithm. There is a substantial number of numerical methods including the projection technique and its variant forms, auxiliary principle technique, Newton and descent framework. The applicability of the projection method is limited due to the fact that it is not easy to nd the projection except in very special cases. Secondly, the projection method cannot be used to suggest iterative algorithms for variational inequalities of type (2.1) due to the presence of the nonlinear term ’. In this paper, we use the concept of the resolvent operator to suggest a number of iterative methods for solving the set-valued mixed variational inequalities. We prove * Corresponding author. E-mail addresses: noor@mscs.dal.ca (M.A. Noor), trassias@math.nuta.gr (T.M. Rassias) 0362-546X/00/$ - see front matter c  2000 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(98)00332-0