Skip to main content
SpringerLink
Log in

Convergence of Iterative Schemes for Multivalued Quasi-Variational Inclusions

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a significant extension of previously known results in this field.

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. Ames, W. F.: Numerical Methods for Partial Differential Equations, Third Edition, Academic Press, New York, 1992.

    Google Scholar 

  2. Attouch, H., Riahi, H. and Moudafi, A.: Quantitative stability analysis for maximal monotone operators and semi-groups of contractions, J. Nonlinear Anal., Theo. Meth. & Appli. 21 (1993), 697–723.

    Google Scholar 

  3. Baiocchi C. and Capelo, A.: Variational and Quasi-Variational Inequalities, J. Wiley and Sons, New York, London, 1984.

    Google Scholar 

  4. Brézis, H.: Opérateurs Maximaux Monotone et Semigroups de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973.

    Google Scholar 

  5. Chang, S. S., Kim, J. K. and Kim, K. H.: On the existence and iterative approximation problems of solutions for set-valued variational inclusions in banach space. To appear in J. Math. Anal. Appli.

  6. Cottle, R.W., Giannessi, F. and Lions, J.L.: Variational Inequalities and Complementarity Problems: Theory and Applications, J. Wiley and Sons, New York, 1980.

    Google Scholar 

  7. Giannessi F. and Maugeri, A.: Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995.

    Google Scholar 

  8. Glowinski, R., Lions, J.L. and Trémolières, R.: Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.

    Google Scholar 

  9. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems, Springer, Berlin, 1984.

    Google Scholar 

  10. Glowinski, R. and Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Publication Co. Philadelphia, 1989.

    Google Scholar 

  11. He, B.: A class of projection and contraction methods for monotone variational inequalities, Appl. Math. Optim. 35 (1997), 69–76.

    Google Scholar 

  12. Martinet, B.: Regularization d'inequations variationelles par approximations successives, Rev. Francaise d'Auto. et Inform. Rech. Oper. 4 (1972), 154–159.

    Google Scholar 

  13. Moudafi, A. and Aslam Noor, M.: Sensitivity analysis for variational inclusions by the Wiener-Hopf equations techniques, J. Appl. Math. Stochastic Anal. 12 (1999), 223–232.

    Google Scholar 

  14. Moudafi, A. and Aslam Noor, M.: New convergence results for iterative methods for set-valued mixed variational inequalities, J. Math. Ineq. Appl. 3 (2000), 223–232.

    Google Scholar 

  15. Nadler, S. B.: Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–488.

    Google Scholar 

  16. Aslam Noor, M.: Algorithms for general monotone mixed variational inequalities, J. Math. Anal. Appl. 229 (1999), 295–303.

    Google Scholar 

  17. Aslam Noor, M.: Projection-splitting algorithms for monotone variational inequalities, Computer Math. Appl. 39 (2000), 73–79.

    Google Scholar 

  18. Aslam Noor, M.: General monotone mixed variational inequalities, J. Natural Geometry 17 (2000), 59–76.

    Google Scholar 

  19. Passty, G. B.: Ergodic convergence to a zero of the sum of maximal monotone operators, J. Math. Anal. Appl. 7 (1978), 591–597.

    Google Scholar 

  20. Robinson, S.M.: Generalized equations and their solutions, Math. Program. Studies 10 (1979), 128–141.

    Google Scholar 

  21. Robinson, S. M.: Normal maps induced by linear transformations, Math. Oper. Research 17 (1992), 691–714.

    Google Scholar 

  22. Rockafellar, R.T.: Monotone operator and the proximal point algorithm, SIAM J. Control. Opt. 14(5) (1976), 877–898.

    Google Scholar 

  23. Shi, P.: Equivalence of variational inequalities with Wiener-Hopf equations, Proc. Amer. Math. Soc. 111 (1991), 339–346.

    Google Scholar 

  24. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38 (2000), 431–446.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département Scientifique Interfacultaire, Univeristé Antilles Guyane, GRIMAAG, F-97200, Schoelcher, Martinique, France

    Abdellatif Moudafi

  2. Mathematics, Etisalat College of Engineering, P.O. Box 980, Sharjah, United Arab Emirates

    Muhammad Aslam Noor

Authors
  1. Abdellatif Moudafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Aslam Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moudafi, A., Noor, M.A. Convergence of Iterative Schemes for Multivalued Quasi-Variational Inclusions. Positivity 8, 75–84 (2004). https://doi.org/10.1023/B:POST.0000023199.34934.9f

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:POST.0000023199.34934.9f

Search

Navigation