JAMC J Appl Math Comput (2011) 35: 595–605 DOI 10.1007/s12190-010-0380-0 Bifunction hemivariational inequalities Muhammad Aslam Noor · Khalida Inayat Noor · Zhenyu Huang Received: 19 October 2009 / Published online: 27 January 2010 © Korean Society for Computational and Applied Mathematics 2010 Abstract In this paper, we introduce and consider a new class of variational inequali- ties, which is called the bifunction hemivariational inequality. This new class includes several classes of variational inequalities as special cases. A number of iterative meth- ods for solving bifunction hemivariational inequalities 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. Keywords Bifunction hemivariational inequalities · Variational-like inequalities · Auxiliary principle · Proximal methods · Convergence 1 Introduction Variational inequality theory, which can be viewed as an important and significant extension of the variational principles, has had a great impact and influence in the The research of Prof. Zhengyu Huang is supported by National Natural Science Foundation of China (NSFC Grant No. 10871092). M.A. Noor ( ) · K.I. Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan e-mail: noormaslam@hotmail.com K.I. Noor e-mail: khalidanoor@hotmail.com M.A. Noor Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia Z. Huang Mathematics Department, Nanjing University, Nanjing 210093, China e-mail: zhenyu@nju.edu.cn