Skip to main content

Advertisement

SpringerLink
Log in

Existence of Solutions and an Algorithm for Mixed Variational-Like Inequalities in Banach Spaces

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the class of mixed variational-like inequalities in reflexive Banach spaces. By applying a minimax inequality due to the author, some existence and uniqueness theorems for solutions of mixed variational-like inequalities are proved. Next, by applying the auxiliary problem technique, we suggest an innovative iterative algorithm to compute approximate solutions of the mixed variational-like inequality. Finally, convergence criteria are also discussed.

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. X. P. Ding (1997) ArticleTitleRandom Mixed Variational-Like Inequalities in Topological Vector Spaces Journal of the Sichuan Normal University 20 1–13

    Google Scholar 

  2. X. P. Ding (1998) ArticleTitleAlgorithm of Solutions for Mixed Nonlinear Variational-Like Inequalities in Reflexive Banach Space Applied Mathematics and Mechanics 19 521–529 Occurrence Handle0917.49008

    MATH  Google Scholar 

  3. Q. H. Ansari J. C. Yao (2001) ArticleTitleIterative Schemes for Solving Mixed Variational-Like Inequalities Journal of Optimization Theory and Applications 108 527–541 Occurrence Handle10.1023/A:1017531323904 Occurrence Handle1828671 Occurrence Handle0999.49008

    Article  MathSciNet  MATH  Google Scholar 

  4. M. A. Noor (1995) ArticleTitleNonconvex Functions and Variational Inequalities Journal of Optimization Theory and Applications 87 615–630 Occurrence Handle10.1007/BF02192137 Occurrence Handle0840.90107 Occurrence Handle1364116

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Parida M. Sahoo A. Kumar (1989) ArticleTitleA Variational-Like Inequality Problem Bulletin of the Australian Mathematical Society 39 225–231 Occurrence Handle998015 Occurrence Handle0649.49007

    MathSciNet  MATH  Google Scholar 

  6. N. H. Dien (1992) ArticleTitleSome Remarks on Variational-Like and Quasivariational-Like Inequalities Bulletin of the Australian Mathematical Society 46 335–342 Occurrence Handle0773.90071 Occurrence Handle1183788 Occurrence Handle10.1017/S0004972700011941

    Article  MATH  MathSciNet  Google Scholar 

  7. J. C. Yao (1994) ArticleTitleExistence of Generalized Variational Inequalities Operation Research Letters 15 35–40 Occurrence Handle0874.49012 Occurrence Handle95e:49015

    MATH  MathSciNet  Google Scholar 

  8. X. P. Ding (1997) ArticleTitleGeneral Algorithm of Solutions for Nonlinear Variational Inequalities in Banach Spaces Computers and Mathematics with Applications 34 131–137 Occurrence Handle10.1016/S0898-1221(97)00194-6 Occurrence Handle0888.65080 Occurrence Handle1483163

    Article  MATH  MathSciNet  Google Scholar 

  9. X. P. Ding (1999) ArticleTitleGeneral Algorithm of Random Solutions for Random Nonlinear Variational Inequalities in Banach Spaces Stochastic Analysis and Applications 17 383–394 Occurrence Handle0939.47054 Occurrence Handle1686987

    MATH  MathSciNet  Google Scholar 

  10. P. T. Harker J. S. Pang (1990) ArticleTitleFinite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Appliactions Mathematical Programming 48B 161–221 Occurrence Handle1073707

    MathSciNet  Google Scholar 

  11. G. Cohen (1988) ArticleTitleAuxiliary Problem Principle Extended to Variational Inequalities Journal of Optimization Theory and Applications 59 325–333 Occurrence Handle10.1007/BF00940305 Occurrence Handle0628.90066 Occurrence Handle974037

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Isac (1993) Complementarity Problems Springer Berlin Germany

    Google Scholar 

  13. M. A. Noor (1988) ArticleTitleGeneral Variational Inequalities Applied Mathematics Letters 1 119–122 Occurrence Handle0655.49005 Occurrence Handle953368

    MATH  MathSciNet  Google Scholar 

  14. M. A. Noor (1990) ArticleTitleMixed Variational Inequalities Applied Mathematics Letters 3 73–75 Occurrence Handle0714.49014 Occurrence Handle91a:47090

    MATH  MathSciNet  Google Scholar 

  15. M. A. Noor (1992) ArticleTitleGeneral Algorithm for Variational Inequalities Journal of Optimization Theory and Applications 73 409–413 Occurrence Handle10.1007/BF00940189 Occurrence Handle0794.49009 Occurrence Handle1163268

    Article  MATH  MathSciNet  Google Scholar 

  16. M. A. Noor K. I. Noor T. M. Rassias (1993) ArticleTitleSome Aspects of Variational Inequalities Journal of Computational and Applied Mathematics 47 285–312 Occurrence Handle10.1016/0377-0427(93)90058-J Occurrence Handle1251844 Occurrence Handle0788.65074

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Tseng (1990) ArticleTitleFurther Applications of a Splitting Algorithm to Decomposition in Variational Inequalities and Convex Programming Mathematical Programming 48 249–263 Occurrence Handle10.1007/BF01582258 Occurrence Handle0725.90079 Occurrence Handle1073710

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Karamardian (1969) ArticleTitleThe Nonlinear Complementarity Problem with Applications, Part 2 Journal of Optimization Theory and Applications 4 161–187 Occurrence Handle252036

    MathSciNet  Google Scholar 

  19. R. U. Verma (1997) ArticleTitleOn Generalized Variational Inequalities Involving Relaxed Lipschitz and Relaxed Monotone Operators Journal of Mathematical Analysis and Applications 213 387–392 Occurrence Handle10.1006/jmaa.1997.5556 Occurrence Handle0902.49009 Occurrence Handle1469383

    Article  MATH  MathSciNet  Google Scholar 

  20. M. A. Hanson (1981) ArticleTitleOn Sufficiency of the Kuhn-Tucker Conditions Journal of Mathematical Analysis and Applications 80 545–550 Occurrence Handle10.1016/0022-247X(81)90123-2 Occurrence Handle0463.90080 Occurrence Handle614849

    Article  MATH  MathSciNet  Google Scholar 

  21. X. P. Ding E. Tarafdar (1995) ArticleTitleExistence and Uniqueness of Solutions for a General Nonlinear Variational Inequality Applied Mathematics Letters 8 31–36 Occurrence Handle10.1016/0893-9659(94)00106-M Occurrence Handle1355147 Occurrence Handle0824.49009

    Article  MathSciNet  MATH  Google Scholar 

  22. J. X. Zhou G. Chen (1988) ArticleTitleDiagonal Convexity Conditions for Problems in Convex Analysis and Quasivariational Inequalities Journal of Mathematical Analysis and Applications 132 213–225 Occurrence Handle10.1016/0022-247X(88)90054-6 Occurrence Handle942365 Occurrence Handle0649.49008

    Article  MathSciNet  MATH  Google Scholar 

  23. X. P. Ding K. K. Tan (1992) ArticleTitleA Minimax Inequality with Applications to Existence of Equilibrium Point and Fixed-Point Theorems Colloquium Mathematicum 63 233–247 Occurrence Handle1180636 Occurrence Handle0833.49009

    MathSciNet  MATH  Google Scholar 

  24. D. Pascali S. Sburian (1978) Nonlinear Mappings of Monotone Type Sijthoff and Noordhoff International Publishers Alphen aan den Rijn, Netherlands Occurrence Handle0423.47021

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan, P. R. China

    X. P. Ding

Authors
  1. X. P. Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by H. P. Benson

This research was supported by NSF, Sichman Education Department of China, Projects 2003A081 and SZD0406.

The author expresses his sincere thanks to Professor H.P. Benson and the anonymous referees for careful comments leading to the present version of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, X.P. Existence of Solutions and an Algorithm for Mixed Variational-Like Inequalities in Banach Spaces. J Optim Theory Appl 127, 285–302 (2005). https://doi.org/10.1007/s10957-005-6540-y

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-005-6540-y

Keywords

Search

Navigation