Open Access

Projection iterative approximations for a new class of general random implicit quasi-variational inequalities

Journal of Inequalities and Applications20062006:81261

DOI: 10.1155/JIA/2006/81261

Received: 9 November 2005

Accepted: 21 January 2006

Published: 31 May 2006


We introduce a class of projection-contraction methods for solving a class of general random implicit quasi-variational inequalities with random multivalued mappings in Hilbert spaces, construct some random iterative algorithms, and give some existence theorems of random solutions for this class of general random implicit quasi-variational inequalities. We also discuss the convergence and stability of a new perturbed Ishikawa iterative algorithm for solving a class of generalized random nonlinear implicit quasi-variational inequalities involving random single-valued mappings. The results presented in this paper improve and extend the earlier and recent results.


Authors’ Affiliations

Department of Mathematics, Sichuan University of Science and Engineering


  1. Agarwal RP, Huang N-J, Cho YJ: Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings. Journal of Inequalities and Applications 2002,7(6):807–828. 10.1155/S1025583402000425MathSciNetMATHGoogle Scholar
  2. Bharucha-Reid AT: Random Integral Equations. Academic Press, New York; 1972:xiii+267.MATHGoogle Scholar
  3. Chang SS: Fixed Point. Theory and Applications. Chongqing, Chongqing; 1984.Google Scholar
  4. Chang SS: Variational Inequality and Complementarity Problem Theory with Applications. Shanghai Scientific and Technological Literature, Shanghai; 1991.Google Scholar
  5. Chang SS, Huang N-J: Generalized random multivalued quasi-complementarity problems. Indian Journal of Mathematics 1993,35(3):305–320.MathSciNetMATHGoogle Scholar
  6. Chang SS, Huang N-J: Random generalized set-valued quasi-complementarity problems. Acta Mathematicae Applicatae Sinica 1993, 16: 396–405.MATHGoogle Scholar
  7. Chang SS, Zhu YG: Problems concerning a class of random variational inequalities and random quasivariational inequalities. Journal of Mathematical Research and Exposition 1989,9(3):385–393.MathSciNetGoogle Scholar
  8. Cho YJ, Huang N-J, Kang SM: Random generalized set-valued strongly nonlinear implicit quasi-variational inequalities. Journal of Inequalities and Applications 2000,5(5):515–531. 10.1155/S1025583400000308MathSciNetMATHGoogle Scholar
  9. Cho YJ, Shim SH, Huang N-J, Kang SM: Generalized strongly nonlinear implicit quasi-variational inequalities for fuzzy mappings. In Set Valued Mappings with Applications in Nonlinear Analysis, Ser. Math. Anal. Appl.. Volume 4. Taylor & Francis, London; 2002:63–77.Google Scholar
  10. Ding XP: Generalized quasi-variational-like inclusions with nonconvex functionals. Applied Mathematics and Computation 2001,122(3):267–282. 10.1016/S0096-3003(00)00027-8MathSciNetView ArticleMATHGoogle Scholar
  11. Ganguly A, Wadhwa K: On random variational inequalities. Journal of Mathematical Analysis and Applications 1997,206(1):315–321. 10.1006/jmaa.1997.5194MathSciNetView ArticleMATHGoogle Scholar
  12. Himmelberg CJ: Measurable relations. Fundamenta Mathematicae 1975, 87: 53–72.MathSciNetMATHGoogle Scholar
  13. Huang N-J: On the generalized implicit quasivariational inequalities. Journal of Mathematical Analysis and Applications 1997,216(1):197–210. 10.1006/jmaa.1997.5671MathSciNetView ArticleMATHGoogle Scholar
  14. Huang N-J: Random generalized nonlinear variational inclusions for random fuzzy mappings. Fuzzy Sets and Systems 1999,105(3):437–444. 10.1016/S0165-0114(97)00222-4MathSciNetView ArticleMATHGoogle Scholar
  15. Huang N-J, Cho YJ: Random completely generalized set-valued implicit quasi-variational inequalities. Positivity 1999,3(3):201–213. 10.1023/A:1009784323320MathSciNetView ArticleMATHGoogle Scholar
  16. Huang N-J, Long X, Cho YJ: Random completely generalized nonlinear variational inclusions with non-compact valued random mappings. Bulletin of the Korean Mathematical Society 1997,34(4):603–615.MathSciNetGoogle Scholar
  17. Lan H-Y, Huang N-J, Cho YJ: A new method for nonlinear variational inequalities with multi-valued mappings. Archives of Inequalities and Applications 2004,2(1):73–84.MathSciNetMATHGoogle Scholar
  18. Lan H-Y, Kim JK, Huang N-J: On the generalized nonlinear quasi-variational inclusions involving non-monotone set-valued mappings. Nonlinear Functional Analysis and Applications 2004,9(3):451–465.MathSciNetMATHGoogle Scholar
  19. Noor MA, Elsanousi SA: Iterative algorithms for random variational inequalities. Panamerican Mathematical Journal 1993,3(1):39–50.MathSciNetGoogle Scholar
  20. Siddiqi AH, Ansari QH: Strongly nonlinear quasivariational inequalities. Journal of Mathematical Analysis and Applications 1990,149(2):444–450. 10.1016/0022-247X(90)90054-JMathSciNetView ArticleMATHGoogle Scholar
  21. Toskos CP, Padgett WJ: Random Integral Equation with Applications in Life Sciences and Engineering. Academic Press, New York; 1974.Google Scholar
  22. Verma RU: A class of projection-contraction methods applied to monotone variational inequalities. Applied Mathematics Letters 2000,13(8):55–62. 10.1016/S0893-9659(00)00096-3MathSciNetView ArticleMATHGoogle Scholar
  23. Weng XL: 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
  24. Yuan X-Z, Luo X, Li G: Random approximations and fixed point theorems. Journal of Approximation Theory 1996,84(2):172–187. 10.1006/jath.1996.0014MathSciNetView ArticleMATHGoogle Scholar
  25. Zhou HY, Cho YJ, Kang SM: Iterative approximations for solutions of nonlinear equations involving non-self-mappings. Journal of Inequalities and Applications 2001,6(6):577–597. 10.1155/S1025583401000352MathSciNetMATHGoogle Scholar


© Lan 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.