Nonlinear Analysis 64 (2006) 2631 – 2642 www.elsevier.com/locate/na Hemiequilibrium-like problems Muhammad Aslam Noor ∗ , Khalida Inayat Noor Mathematics Department, COMSATS Institute of InformationTechnology, Islamabad, Pakistan Received 17 July 2005; accepted 12 September 2005 Abstract In this paper, we introduce and consider a new class of equilibrium problems, known as hemiequilibrium- like problems. This new class includes hemiequilibrium, equilibrium-like problems and several classes of variational inequalities as special cases. A number of iterative methods for solving hemiequilibrium- like problems are suggested and analyzed by using the auxiliary principle technique. We also study the convergence analysis of these iterative methods under some mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.  2005 Elsevier Ltd. All rights reserved. MSC: 49J40; 90C30 Keywords: Hemiequilibrium problems; Hemivariational inequalities; Variational-like inequalities; Auxiliary principle; Proximal methods; Convergence 1. Introduction Equilibrium problems theory has had a great impact and influence in the development of several branches of pure, applied and engineering sciences. It has been shown that equilibrium problems include variational inequalities, complementarity problems, fixed-point, optimization problems and game theory as special cases. Hence collectively equilibrium problems cover a vast range of applications; see Refs. [1,4–6,28,11,20,13,25,21]. It is worth mentioning that almost all the results obtained for the equilibrium problems have been obtained in the setting of convexity. Recently, the concept of convexity has been generalized in several directions.A significant generalization of convex functions is that of invex functions introduced and studied by Hanson [8]. Hanson’s initial result inspired a great deal of subsequent work which has greatly extended the role and applications ∗ Corresponding author. E-mail addresses: noormaslam@hotmail.com (M.A. Noor), khalidanoor@hotmail.com (K.I. Noor). 0362-546X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.09.008