Skip to main content

Advertisement

SpringerLink
Log in

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle technique. The convergence of three-step iterative sequences generated by the algorithm is also proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cubiotti P. Existence of solutions for lower semi-continuous quasi equilibrium problems[J]. Comput Math Appl, 1995, 30(12):11–12.

    Article  MATH  MathSciNet  Google Scholar 

  2. Noor M A. Auxiliary principle for generalized mixed variational-like inequalities[J]. J Math Anal Appl, 1997, 215(1):78–85.

    Article  Google Scholar 

  3. Noor M A. Some recent advances in variational inequalities I[J]. New Zealand J Math, 1997, 26(2):53–80.

    MATH  MathSciNet  Google Scholar 

  4. Noor M A. Generalized variational-like inequalities[J]. Math Comput Modelling, 1998, 27(3):93–101.

    Article  MathSciNet  Google Scholar 

  5. Panagiotopoulos P D, Stavroulakis G E. New types of variational principles based on the notion of quasi-differentiability[J]. Acta Mech, 1992, 94(3/4):171–194.

    Article  MATH  MathSciNet  Google Scholar 

  6. Panagiotopoulos P D. Inequality problems in mechanics and applications[M]. Boston: Birkhäuser, 1985.

    Google Scholar 

  7. Parida J, Sen A. A variational-like inequality for multi-functions with applications[J]. J Math Anal Appl, 1987, 124(1):73–81.

    Article  MATH  MathSciNet  Google Scholar 

  8. Tian G. Generalized quasi variational-like inequality problem[J]. Math Oper Res, 1993, 18(3):752–764.

    Article  MATH  MathSciNet  Google Scholar 

  9. Yao J C. The generalized quasi variational inequality problem with applications[J]. J Math Anal Appl, 1991, 158(1):139–160.

    Article  MATH  MathSciNet  Google Scholar 

  10. Yao J C. Existence of generalized variational inequalities[J]. Oper Res Lett, 1994, 15(1):35–40.

    Article  MATH  MathSciNet  Google Scholar 

  11. Huang N J. On the generalized implicit quasivariational inequalities[J]. J Math Anal Appl, 1997, 216(1):197–210.

    Article  MATH  MathSciNet  Google Scholar 

  12. Huang N J. Mann and Ishikwa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions[J]. Comput Math Appl, 1998, 35(1):1–7.

    Article  MATH  MathSciNet  Google Scholar 

  13. Glowinski R, Lions J L, Tremolieres R. Numerical analysis of variational inequalities[M]. North-Holland, Amsterdam, 1981.

    Google Scholar 

  14. Chang S S, Xiang S W. On the existence of solutions for a class of quasi-bilinear variational inequalities[J]. J Systems Sci Math Sci, 1996, 16(3):136–140.

    MathSciNet  Google Scholar 

  15. Ding Xieping. On the generalized mixed variational-like inequalities[J]. J Sichuan Normal Univ, 2003, 22(5):494–503 (in Chinese).

    Google Scholar 

  16. Ding Xieping. Predictor-corrector iterative algorithms for solving generalized mixed quasi-variational-like inequalities[J]. J Comput Appl Math, 2005, 182(1):1–12.

    Article  MATH  MathSciNet  Google Scholar 

  17. Siddiqi A H, Ansari Q H. Strongly nonlinear qusi-variational inequalities[J]. J Math Anal Appl, 1990, 149(2):444–450.

    Article  MATH  MathSciNet  Google Scholar 

  18. Noor M A. Splitting methods for pseudomonotone general mixed variational inequalities[J]. J Global Optim, 2000, 18(1):75–89.

    Article  MATH  MathSciNet  Google Scholar 

  19. Tseng P. A modified forward-backward splitting method for maximal monotone mappings[J]. SIAM J Control Optim, 2000, 38(2):431–466.

    Article  MATH  MathSciNet  Google Scholar 

  20. Xu H K. Iterative algorithms for nonlinear operators[J]. Journal of London Mathematical Society, 2002, 66(2):240–256.

    Article  MATH  Google Scholar 

  21. Huang Nanjing, Deng Chuanxian. Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities[J]. J Math Anal Appl, 2001, 256(2):345–359.

    Article  MATH  MathSciNet  Google Scholar 

  22. Glowinski R, Le Tallec P. Augmented Lagrange and operator splitting methods in nonlinear mechanics[M]. SIAM, Philadelphia, 1989.

    Google Scholar 

  23. Huang N J, Liu Y P, Tang Y, Bai M R. On the generalized set-valued strongly nonlinear implicit variational inequalities[J]. Comput Math Appl, 1998, 37(10):1–7.

    Article  MathSciNet  Google Scholar 

  24. Yao J C. Abstract variational inequality problems and a basic theorem of complementarity[J]. Comput Math Appl, 1993, 25(1):73–79.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, P. R. China

    Xu Hai-li  (徐海丽) & Guo Xing-ming  (郭兴明) (Professor, Doctor)

Authors
  1. Xu Hai-li  (徐海丽)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guo Xing-ming  (郭兴明)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo Xing-ming  (郭兴明).

Additional information

Contributed by GUO Xing-ming

Project supported by the National Natural Science Foundation of China (No.10472061)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, Hl., Guo, Xm. Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. Appl Math Mech 28, 721–729 (2007). https://doi.org/10.1007/s10483-007-0602-x

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-007-0602-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation