Skip to main content
Journal of Inequalities and Applications
Download PDF
  • Research Article
  • Open access
  • Published:

On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces

Journal of Inequalities and Applications volume 2007, Article number: 094092 (2007) Cite this article

Abstract

The purpose of this paper is to study the solvability for a class of generalized strong vector variational-like inequalities in reflexive Banach spaces. Firstly, utilizing Brouwer's fixed point theorem, we prove the solvability for this class of generalized strong vector variational-like inequalities without monotonicity assumption under some quite mild conditions. Secondly, we introduce the new concept of pseudomonotonicity for vector multifunctions, and prove the solvability for this class of generalized strong vector variational-like inequalities for pseudomonotone vector multifunctions by using Fan's lemma and Nadler's theorem. Our results give an affirmative answer to an open problem proposed by Chen and Hou in 2000, and also extend and improve the corresponding results of Fang and Huang (2006).

[1234567891011121314151617]

References

  1. Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems (Proceedings of an International School, Erice, 1978). Edited by: Cottle RW, Giannessi F, Lions JL. John Wiley & Sons, Chichester, UK; 1980:151–186.

    Google Scholar 

  2. Chen G-Y, Hou S-H: Existence of solutions for vector variational inequalities. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and Its Applications. Volume 38. Edited by: Giannessi F. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:73–86. 10.1007/978-1-4613-0299-5_5

    Chapter  Google Scholar 

  3. Chen G-Y, Yang XQ: The vector complementary problem and its equivalences with the weak minimal element in ordered spaces. Journal of Mathematical Analysis and Applications 1990,153(1):136–158. 10.1016/0022-247X(90)90270-P

    Article  MathSciNet  MATH  Google Scholar 

  4. Giannessi F (Ed): Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and Its Applications. Volume 38. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xiv+523.

    Google Scholar 

  5. Huang NJ, Li J, Thompson HB: Implicit vector equilibrium problems with applications. Mathematical and Computer Modelling 2003,37(12–13):1343–1356. 10.1016/S0895-7177(03)90045-8

    Article  MathSciNet  MATH  Google Scholar 

  6. Fang Y-P, Huang N-J: Strong vector variational inequalities in Banach spaces. Applied Mathematics Letters 2006,19(4):362–368. 10.1016/j.aml.2005.06.008

    Article  MathSciNet  MATH  Google Scholar 

  7. Yang XQ: Vector complementarity and minimal element problems. Journal of Optimization Theory and Applications 1993,77(3):483–495. 10.1007/BF00940446

    Article  MathSciNet  MATH  Google Scholar 

  8. Yang XQ: Vector variational inequality and vector pseudolinear optimization. Journal of Optimization Theory and Applications 1997,95(3):729–734. 10.1023/A:1022694427027

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang XQ, Goh CJ: On vector variational inequalities: application to vector equilibria. Journal of Optimization Theory and Applications 1997,95(2):431–443. 10.1023/A:1022647607947

    Article  MathSciNet  MATH  Google Scholar 

  10. Konnov IV, Yao J-C: On the generalized vector variational inequality problem. Journal of Mathematical Analysis and Applications 1997,206(1):42–58. 10.1006/jmaa.1997.5192

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang N-J, Fang Y-P: On vector variational inequalities in reflexive Banach spaces. Journal of Global Optimization 2005,32(4):495–505. 10.1007/s10898-003-2686-z

    Article  MathSciNet  MATH  Google Scholar 

  12. Zeng L-C, Yao J-C: Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. Journal of Global Optimization 2006,36(4):483–497. 10.1007/s10898-005-5509-6

    Article  MathSciNet  MATH  Google Scholar 

  13. Fang YP, Huang NJ: On the strong vector variational inequalities. In Research Report. Department of Mathematics, Sichuan University, Sichuan, China; 2002.

    Google Scholar 

  14. Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.

    Article  MathSciNet  MATH  Google Scholar 

  15. Brouwer LEJ: Beweis der invarianz des-dimensionalen gebiets. Mathematische Annalen 1911,71(3):305–313. 10.1007/BF01456846

    Article  MathSciNet  Google Scholar 

  16. Browder FE: A new generation of the Schauder fixed point theorem. Mathematische Annalen 1967,174(4):285–290. 10.1007/BF01364275

    Article  MathSciNet  MATH  Google Scholar 

  17. Fan K: A generalization of Tychonoff's fixed point theorem. Mathematische Annalen 1961,142(3):305–310. 10.1007/BF01353421

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Lu-Chuan Ceng

  2. General Education Center, China Medical University, Taichung, 404, Taiwan

    Yen-Cherng Lin

  3. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    Jen-Chih Yao

Authors
  1. Lu-Chuan Ceng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yen-Cherng Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jen-Chih Yao.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Ceng, LC., Lin, YC. & Yao, JC. On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces. J Inequal Appl 2007, 094092 (2007). https://doi.org/10.1155/2007/94092

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/94092

Keywords