Skip to main content
SpringerLink
Log in

Two-Step Iterative Method for Nonconvex Bifunction Variational Inequalities

  • Chapter
  • First Online:
Functional Equations in Mathematical Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 52))

Abstract

In this paper, we suggest and analyze a two-step iterative method for solving nonconvex bifunction variational inequalities. We also discuss the convergence of the iterative method under partially relaxed strongly monotonicity, which is a weaker condition than cocoerciveness.

Mathematics Subject Classification (2000): Primary 49J40; Secondary 90C33

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bounkhel, M., Tadj, L., Hamdi, A.: Iterative schemes to solve nonconvex variational problems. J. Inequal. Pure Appl. Math. 4, 1–14 (2003)

    MathSciNet  Google Scholar 

  2. Clarke, F.H., Ledyaev, Y.S., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer–Verlag, Berlin (1998)

    Google Scholar 

  3. Crespi, G.P., Ginchev, J., Rocca, M.: Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123, 479–496 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crespi, G.P., Ginchev, J., Rocca, M.: Existence of solutions and star-shapedness in Minty variational inequalities. J. Global Optim. 32, 485–494 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Crespi, G.P., Ginchev, J., Rocca, M.: Increasing along rays property for vector functions. J. Nonconvex Anal. 7, 39–50 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Crespi, G.P., Ginchev, J., Rocca, M.: Some remarks on the Minty vector variational principle. J. Math. Anal. Appl. 345, 165–175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fang, Y.P., Hu, R.: Parametric well-posedness for variational inequalities defined by bifunction. Comput. Math. Appl. 53, 1306–1316 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academics Publishers, Dordrecht (2001)

    Google Scholar 

  9. Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P. M.: From Convexity to Nonconvexity. Kluwer Academic Publishers, Dordrecht (2001)

    Book  MATH  Google Scholar 

  10. Glowinski, R., Lions, J.L., Tremolieres, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  11. Lalitha, C.S., Mehra, M.: Vector variational inequalities with cone-pseudomonotone bifunction. Optimization 54, 327–338 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Noor, M.A.: General variational inequalities. Appl. Math. Lett. 1, 119–121 (1988)

    MATH  Google Scholar 

  13. Noor, M.A.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Noor, M.A.: Some new classes of nonconvex functions. Nonlinear Funct. Anal. Appl 11, 165–171 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Noor, M.A.: Some developments in general variational inequalities, Appl. Math. Comput. 152, 199–277 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor, M.A.: Implicit iterative methods for nonconvex variational inequalities. J. Optim. Theory Appl. 143, 619–624 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Noor, M.A.: Projection methods for nonconvex variational inequalities, Optim. Lett. 3, 411–418 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Noor, M.A.: On an implicit method for nonconvex variational inequalities. J. Optim. Theory Appl. 147, 411–417 (2010)

    Google Scholar 

  19. Noor, M.A., Noor, K.I.: Iterative schemes for trifunction variational inequalities, to appear

    Google Scholar 

  20. Noor, M.A., Noor, K.I., Rassias, Th.M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pardalos, P.M., Rassias, Th.M., Khan, A.A.: Nonlinear Analysis and Variational Problems. Springer (2010)

    Google Scholar 

  22. Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352, 5231–5249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Math. Acad. Sci. Paris 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to express my sincere gratitude to Prof. Dr. Themistocles M. Rassias for the kind invitation and Dr. S.M. Junaid Zaidi, Rector, CIIT, Islamabad, Pakistan, for providing excellent research facilities.

This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia under grant N0. KSU.VPP. 108.

Author information

Authors and Affiliations

  1. Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan

    Muhammad Aslam Noor & Khlaida Inayat Noor

  2. Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia

    Muhammad Aslam Noor & Eisa Al-Said

Authors
  1. Muhammad Aslam Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khlaida Inayat Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eisa Al-Said
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Aslam Noor .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, National Technical University of Athens, Iroon Polytechneiou 9, Athens, 15780, Greece

    Themistocles M. Rassias

  2. Institute of Mathematics, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Janusz Brzdek

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Noor, M.A., Noor, K.I., Al-Said, E. (2011). Two-Step Iterative Method for Nonconvex Bifunction Variational Inequalities. In: Rassias, T., Brzdek, J. (eds) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications(), vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0055-4_42

Download citation

Publish with us

Policies and ethics

Search

Navigation