IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v143y2009i1d10.1007_s10957-009-9556-x.html
   My bibliography  Save this article

Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces

Author

Listed:
  • J. H. Fan

    (Guangxi Normal University)

  • X. G. Wang

    (Le’An Middle School)

Abstract

By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.

Suggested Citation

  • J. H. Fan & X. G. Wang, 2009. "Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 59-74, October.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:1:d:10.1007_s10957-009-9556-x
    DOI: 10.1007/s10957-009-9556-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-009-9556-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-009-9556-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    2. M. Bianchi & N. Hadjisavvas & S. Schaible, 2004. "Minimal Coercivity Conditions and Exceptional Families of Elements in Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 1-17, July.
    3. Y. B. Zhao & D. Li, 2000. "Strict Feasibility Conditions in Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 107(3), pages 641-664, December.
    4. M. Bianchi & R. Pini, 2005. "Coercivity Conditions for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 79-92, January.
    5. J. Han & Z. H. Huang & S. C. Fang, 2004. "Solvability of Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(3), pages 501-520, September.
    6. M. Bianchi & N. Hadjisavvas & S. Schaible, 2006. "Exceptional Families of Elements for Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 23-31, April.
    7. G. Isac & V. V. Kalashnikov, 2001. "Exceptional Family of Elements, Leray–Schauder Alternative, Pseudomonotone Operators and Complementarity," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 69-83, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ren-you Zhong & Huan-xia Lian & Jiang-hua Fan, 2013. "Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 341-359, November.
    2. César Gutiérrez & Rubén López & Vicente Novo, 2014. "Existence and Boundedness of Solutions in Infinite-Dimensional Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 515-547, August.
    3. Wenjie Mu & Jianghua Fan, 2022. "Existence results for solutions of mixed tensor variational inequalities," Journal of Global Optimization, Springer, vol. 82(2), pages 389-412, February.
    4. Ren-you Zhong & Nan-jing Huang, 2012. "Strict Feasibility for Generalized Mixed Variational Inequality in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 696-709, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. M. Bianchi & N. Hadjisavvas & S. Schaible, 2006. "Exceptional Families of Elements for Variational Inequalities in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 129(1), pages 23-31, April.
    2. Y. Chiang, 2010. "Variational inequalities on weakly compact sets," Journal of Global Optimization, Springer, vol. 46(3), pages 465-473, March.
    3. Ren-you Zhong & Huan-xia Lian & Jiang-hua Fan, 2013. "Exceptional Families of Elements for Optimization Problems in Reflexive Banach Spaces with Applications," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 341-359, November.
    4. I. Konnov, 2014. "On penalty methods for non monotone equilibrium problems," Journal of Global Optimization, Springer, vol. 59(1), pages 131-138, May.
    5. I. Konnov & D. Dyabilkin, 2011. "Nonmonotone equilibrium problems: coercivity conditions and weak regularization," Journal of Global Optimization, Springer, vol. 49(4), pages 575-587, April.
    6. M. Castellani & M. Giuli, 2013. "Refinements of existence results for relaxed quasimonotone equilibrium problems," Journal of Global Optimization, Springer, vol. 57(4), pages 1213-1227, December.
    7. M. Bianchi & R. Pini, 2005. "Coercivity Conditions for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 79-92, January.
    8. Didier Aussel & Parin Chaipunya, 2024. "Variational and Quasi-Variational Inequalities Under Local Reproducibility: Solution Concept and Applications," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1531-1563, November.
    9. G. Isac & S. Z. Németh, 2008. "REFE-Acceptable Mappings: Necessary and Sufficient Condition for the Nonexistence of a Regular Exceptional Family of Elements," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 507-520, June.
    10. Nicuşor Costea & Daniel Alexandru Ion & Cezar Lupu, 2012. "Variational-Like Inequality Problems Involving Set-Valued Maps and Generalized Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 79-99, October.
    11. Massimiliano Giuli, 2013. "Closedness of the Solution Map in Quasivariational Inequalities of Ky Fan Type," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 130-144, July.
    12. Somaye Jafari & Ali Farajzadeh & Sirous Moradi, 2016. "Locally Densely Defined Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 804-817, September.
    13. John Cotrina & Anton Svensson, 2021. "The finite intersection property for equilibrium problems," Journal of Global Optimization, Springer, vol. 79(4), pages 941-957, April.
    14. Ren-you Zhong & Nan-jing Huang, 2012. "Strict Feasibility for Generalized Mixed Variational Inequality in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 696-709, March.
    15. Yiran He, 2017. "Solvability of the Minty Variational Inequality," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 686-692, September.
    16. Ren-you Zhong & Zhen Dou & Jiang-hua Fan, 2015. "Degree Theory and Solution Existence of Set-Valued Vector Variational Inequalities in Reflexive Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 527-549, November.
    17. Truong Duong & Nguyen Tan, 2012. "On the existence of solutions to generalized quasi-equilibrium problems," Journal of Global Optimization, Springer, vol. 52(4), pages 711-728, April.
    18. A. P. Farajzadeh & M. A. Noor, 2010. "On Dual Invex Ky Fan Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 407-413, May.
    19. R. P. Agarwal & M. Balaj & D. O’Regan, 2014. "A Common Fixed Point Theorem with Applications," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 482-490, November.
    20. M. Bianchi & G. Kassay & R. Pini, 2022. "Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization," Journal of Global Optimization, Springer, vol. 82(3), pages 483-498, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:143:y:2009:i:1:d:10.1007_s10957-009-9556-x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.