Open Access

Generalized Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings

Journal of Inequalities and Applications20082007:020457

Received: 31 July 2007

Accepted: 3 December 2007

Published: 23 January 2008


We consider the solvability of generalized variational inequalities involving multivalued relaxed monotone operators and single-valued nonexpansive mappings in the framework of Hilbert spaces. We also study the convergence criteria of iterative methods under some mild conditions. Our results improve and extend the recent ones announced by many others.


Authors’ Affiliations

Department of Mathematics and the RINS, Gyeongsang National University, Chinju, South Korea
Department of Mathematics, Shijiazhuang University, Shijiazhuang, China


  1. Kinderlehrer D, Starnpacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York, NY, USA; 1980.MATHGoogle Scholar
  2. Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus de l'Académie des Sciences. Paris 1964, 258: 4413–4416.MathSciNetMATHGoogle Scholar
  3. Naniewicz Z, Panagiotopoulos PD: Mathematical Theory of Hemivariational Inequalities and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 188. Marcel Dekker, New York, NY, USA; 1995:xviii+267.Google Scholar
  4. Weng X: Fixed point iteration for local strictly pseudo-contractive mapping. Proceedings of the American Mathematical Society 1991,113(3):727–731. 10.1090/S0002-9939-1991-1086345-8MathSciNetView ArticleMATHGoogle Scholar
  5. Verma RU: Generalized variational inequalities involving multivalued relaxed monotone operators. Applied Mathematics Letters 1997,10(4):107–109. 10.1016/S0893-9659(97)00068-2MathSciNetView ArticleMATHGoogle Scholar


© X. Qin and M. Shang 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.