Open Access

Iterative algorithm for solving mixed quasi-variational-like inequalities with skew-symmetric terms in Banach spaces

Journal of Inequalities and Applications20062006:82695

DOI: 10.1155/JIA/2006/82695

Received: 1 April 2006

Accepted: 28 May 2006

Published: 17 August 2006


We develop an iterative algorithm for computing the approximate solutions of mixed quasi-variational-like inequality problems with skew-symmetric terms in the setting of reflexive Banach spaces. We use Fan-KKM lemma and concept of -cocoercivity of a composition mapping to prove the existence and convergence of approximate solutions to the exact solution of mixed quasi-variational-like inequalities with skew-symmetric terms. Furthermore, we derive the posteriori error estimates for approximate solutions under quite mild conditions.


Authors’ Affiliations

Department of Mathematics, Shanghai Normal University
Department of Mathematics, Aligarh Muslim University
Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals
Department of Applied Mathematics, National Sun Yat-sen University


  1. Ansari QH, Yao J-C: Iterative schemes for solving mixed variational-like inequalities. Journal of Optimization Theory and Applications 2001,108(3):527–541. 10.1023/A:1017531323904MathSciNetView ArticleMATHGoogle Scholar
  2. Antipin AS: Iterative gradient prediction-type methods for computing fixed points of extremal mapping. In Parametric Optimization and Related Topics, IV (Enschede, 1995), Approx. Optim.. Volume 9. Edited by: Guddat J, Jonden HTh, Nizicka F, Still G, Twitt F. Lang, Frankfurt am Main; 1997:11–24.Google Scholar
  3. Dien NH: Some remarks on variational-like and quasivariational-like inequalities. Bulletin of the Australian Mathematical Society 1992,46(2):335–342. 10.1017/S0004972700011941MathSciNetView ArticleMATHGoogle Scholar
  4. Ding XP, Yao J-C: Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces. Computers & Mathematics with Applications 2005,49(5–6):857–869. 10.1016/j.camwa.2004.05.013MathSciNetView ArticleMATHGoogle Scholar
  5. Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961,142(3):305–310. 10.1007/BF01353421MathSciNetView ArticleMATHGoogle Scholar
  6. Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam; 1981:xxix+776.Google Scholar
  7. Hanson MA: On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications 1981,80(2):545–550. 10.1016/0022-247X(81)90123-2MathSciNetView ArticleMATHGoogle Scholar
  8. Huang N-J, Deng C-X: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Mathematical Analysis and Applications 2001,256(2):345–359. 10.1006/jmaa.2000.6988MathSciNetView ArticleMATHGoogle Scholar
  9. Karamardian S: The nonlinear complementarity problem with applications. II. Journal of Optimization Theory and Applications 1969,4(3):167–181. 10.1007/BF00930577MathSciNetView ArticleMATHGoogle Scholar
  10. Lee C-H, Ansari QH, Yao J-C: A perturbed algorithm for strongly nonlinear variational-like inclusions. Bulletin of the Australian Mathematical Society 2000,62(3):417–426. 10.1017/S0004972700018931MathSciNetView ArticleMATHGoogle Scholar
  11. Noor MA: Variational-like inequalities. Optimization 1994,30(4):323–330. 10.1080/02331939408843995MathSciNetView ArticleMATHGoogle Scholar
  12. Siddiqi AH, Ansari QH, Ahmad R: Some remarks on variational-like inequalities. In Mathematics and Its Applications in Industry and Business. Edited by: Siddiqi AH, Ahmad K. Narosa, New Delhi; 2000:101–108.Google Scholar
  13. Tseng P: Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming. Mathematical Programming, Series B 1990,48(2):249–263.View ArticleMathSciNetMATHGoogle Scholar
  14. Zeng L-C: Iterative algorithm for finding approximate solutions of a class of mixed variational-like inequalities. Acta Mathematicae Applicatae Sinica. English Series 2004,20(3):477–486. 10.1007/s10255-004-0185-8MathSciNetView ArticleMATHGoogle Scholar
  15. Zeng LC: Iterative approximation of solutions to generalized set-valued strongly nonlinear mixed variational-like inequalities. Acta Mathematica Sinica (Chinese Series) 2005,48(5):879–888.MathSciNetMATHGoogle Scholar
  16. Zeng LC, Schaible S, Yao J-C: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-zMathSciNetView ArticleMATHGoogle Scholar
  17. Zhu DL, Marcotte P: Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM Journal on Optimization 1996,6(3):714–726. 10.1137/S1052623494250415MathSciNetView ArticleMATHGoogle Scholar


© Lu-Chuan Ceng et al. 2006

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.