- Research Article
- Open access
- Published:
Iterative algorithm for solving mixed quasi-variational-like inequalities with skew-symmetric terms in Banach spaces
Journal of Inequalities and Applications volume 2006, Article number: 82695 (2006)
Abstract
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.
References
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:1017531323904
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.
Dien NH: Some remarks on variational-like and quasivariational-like inequalities. Bulletin of the Australian Mathematical Society 1992,46(2):335–342. 10.1017/S0004972700011941
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.2004.05.013
Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961,142(3):305–310. 10.1007/BF01353421
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.
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)90123-2
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.2000.6988
Karamardian S: The nonlinear complementarity problem with applications. II. Journal of Optimization Theory and Applications 1969,4(3):167–181. 10.1007/BF00930577
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/S0004972700018931
Noor MA: Variational-like inequalities. Optimization 1994,30(4):323–330. 10.1080/02331939408843995
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.
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.
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/s10255-004-0185-8
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.
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-z
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/S1052623494250415
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ceng, LC., Ansari, Q.H. & Yao, JC. Iterative algorithm for solving mixed quasi-variational-like inequalities with skew-symmetric terms in Banach spaces. J Inequal Appl 2006, 82695 (2006). https://doi.org/10.1155/JIA/2006/82695
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/JIA/2006/82695