Skip to content


  • Research Article
  • Open Access

Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings

Journal of Inequalities and Applications20072007:064947

  • Received: 1 July 2007
  • Accepted: 3 October 2007
  • Published:


We show that the general variational inequalities are equivalent to the general Wiener-Hopf equations and use this alterative equivalence to suggest and analyze a new iterative method for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality involving multivalued relaxed monotone operators. Our results improve and extend recent ones announced by many others.


  • Variational Inequality
  • Iterative Method
  • Monotone Mapping
  • Nonexpansive Mapping
  • Monotone Operator


Authors’ Affiliations

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China
Department of Mathematics, Shijiazhuang University, Shijiazhuang, 050035, China
Department of Mathematics, Gyeongsang National University, Chinju, 660-701, Korea


  1. Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus de l'Académie des Sciences 1964, 258: 4413–4416.MathSciNetMATHGoogle Scholar
  2. Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560MathSciNetView ArticleMATHGoogle Scholar
  3. Noor MA: General variational inequalities. Applied Mathematics Letters 1988,1(2):119–122. 10.1016/0893-9659(88)90054-7MathSciNetView ArticleMATHGoogle Scholar
  4. Noor MA: Wiener-Hopf equations and variational inequalities. Journal of Optimization Theory and Applications 1993,79(1):197–206. 10.1007/BF00941894MathSciNetView ArticleMATHGoogle Scholar
  5. Noor MA: Projection-proximal methods for general variational inequalities. Journal of Mathematical Analysis and Applications 2006,318(1):53–62. 10.1016/j.jmaa.2005.05.024MathSciNetView ArticleMATHGoogle Scholar
  6. 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
  7. Naniewicz Z, Panagiotopoulos PD: Mathematical Theory of Hemivariational Inequalities and Applications. Volume 188. Marcel Dekker, New York, NY, USA; 1995:xviii+267.Google Scholar
  8. Shi P: Equivalence of variational inequalities with Wiener-Hopf equations. Proceedings of the American Mathematical Society 1991,111(2):339–346. 10.1090/S0002-9939-1991-1037224-3MathSciNetView ArticleMATHGoogle Scholar
  9. Noor MA: Sensitivity analysis for quasi-variational inequalities. Journal of Optimization Theory and Applications 1997,95(2):399–407. 10.1023/A:1022691322968MathSciNetView ArticleMATHGoogle Scholar
  10. Speck F-O: General Wiener-Hopf Factorization Methods, Research Notes in Mathematics. Volume 119. Pitman, Boston, Mass, USA; 1985:vi+157.Google Scholar
  11. Reich S: Constructive techniques for accretive and monotone operators. In Proceedings of the 3rd International Conference on Applied Nonlinear Analysis. Academic Press, New York, NY, USA; 1979:335–345.Google Scholar
  12. Noor MA: General variational inequalities and nonexpansive mappings. Journal of Mathematical Analysis and Applications 2007,331(2):810–822. 10.1016/j.jmaa.2006.09.039MathSciNetView ArticleMATHGoogle Scholar


© Yongfu Su et al. 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.