Skip to main content

System of Generalized Implicit Vector Quasivariational Inequalities

Abstract

We will introduce a system of generalized implicit vector quasivariational inequalities (in short, SGIVQVI) which generalizes and unifies the system of generalized implicit variational inequalities, the system of generalized vector quasivariational-like inequalities, the system of generalized vector variational inequalities, the system of variational inequalities, the generalized implicit vector quasivariational inequality, as well as various extensions of the classic variational inequalities in the literature, and we present some existence results of a solution for the SGIVQVI without any monotonicity conditions.

[12345678910111213141516171819202122232425262728293031323334353637383940]

References

  1. Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems. Edited by: Cottle RW, Giannessi F, Lions JL. John Wiley & Sons, New York, NY, USA; 1980:151–186.

    Google Scholar 

  2. Chen GY, Cheng GM: Vector variational inequalities and vector optimization. In Lecture Notes in Economics and Mathematical Systems. Volume 285. Springer, Berlin, Germany; 1987:408–456.

    Google Scholar 

  3. Chen GY: Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem. Journal of Optimization Theory and Applications 1992,74(3):445–456. 10.1007/BF00940320

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen GY, 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 

  5. Chen G-Y, Craven BD: Approximate dual and approximate vector variational inequality for multiobjective optimization. Journal of the Australian Mathematical Society. Series A 1989,47(3):418–423. 10.1017/S1446788700033139

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen G-Y, Craven BD: A vector variational inequality and optimization over an efficient set. Zeitschrift für Operations Research 1990,34(1):1–12.

    MathSciNet  MATH  Google Scholar 

  7. Siddiqi AH, Ansari QH, Khaliq A: On vector variational inequalities. Journal of Optimization Theory and Applications 1995,84(1):171–180. 10.1007/BF02191741

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Yang XQ: Vector variational inequality and its duality. Nonlinear Analysis: Theory, Methods & Applications 1993,21(11):869–877. 10.1016/0362-546X(93)90052-T

    Article  MathSciNet  MATH  Google Scholar 

  10. Yang XQ: Generalized convex functions and vector variational inequalities. Journal of Optimization Theory and Applications 1993,79(3):563–580. 10.1007/BF00940559

    Article  MathSciNet  MATH  Google Scholar 

  11. Yu SJ, Yao JC: On vector variational inequalities. Journal of Optimization Theory and Applications 1996,89(3):749–769. 10.1007/BF02275358

    Article  MathSciNet  MATH  Google Scholar 

  12. Lee GM, Kim DS, Lee BS, Cho SJ: Generalized vector variational inequality and fuzzy extension. Applied Mathematics Letters 1993,6(6):47–51. 10.1016/0893-9659(93)90077-Z

    Article  MathSciNet  MATH  Google Scholar 

  13. Lee GM, Kim DS, Lee BS: Generalized vector variational inequality. Applied Mathematics Letters 1996,9(1):39–42. 10.1016/0893-9659(95)00099-2

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin KL, Yang D-P, Yao JC: Generalized vector variational inequalities. Journal of Optimization Theory and Applications 1997,92(1):117–125. 10.1023/A:1022640130410

    Article  MathSciNet  MATH  Google Scholar 

  15. Konnov IV, Yao JC: 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 

  16. Daniilidis A, Hadjisavvas N: Existence theorems for vector variational inequalities. Bulletin of the Australian Mathematical Society 1996,54(3):473–481. 10.1017/S0004972700021882

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang XQ, Yao JC: Gap functions and existence of solutions to set-valued vector variational inequalities. Journal of Optimization Theory and Applications 2002,115(2):407–417. 10.1023/A:1020844423345

    Article  MathSciNet  MATH  Google Scholar 

  18. Oettli W, Schläger D: Existence of equilibria for monotone multivalued mappings. Mathematical Methods of Operations Research 1998,48(2):219–228. 10.1007/s001860050024

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen GY, Li SJ: Existence of solutions for a generalized vector quasivariational inequality. Journal of Optimization Theory and Applications 1996,90(2):321–334. 10.1007/BF02190001

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee GM, Lee BS, Chang S-S: On vector quasivariational inequalities. Journal of Mathematical Analysis and Applications 1996,203(3):626–638. 10.1006/jmaa.1996.0401

    Article  MathSciNet  MATH  Google Scholar 

  21. Ansari QH: A note on generalized vector variational-like inequalities. Optimization 1997,41(3):197–205. 10.1080/02331939708844335

    Article  MathSciNet  MATH  Google Scholar 

  22. Ansari QH: Extended generalized vector variational-like inequalities for nonmonotone multivalued maps. Annales des Sciences Mathématiques du Québec 1997,21(1):1–11.

    MathSciNet  MATH  Google Scholar 

  23. Ding XP, Tarafdar E: Generalized vector variational-like inequalities without monotonicity. In Vector Variational Inequalities and Vector Equilibria. Mathematical Theories. Volume 38. Edited by: Giannessi F. Kluwer Academic, Dordrecht, The Netherlands; 2000:113–124. 10.1007/978-1-4613-0299-5_8

    Chapter  Google Scholar 

  24. Ding XP: The generalized vector quasi-variational-like inequalities. Computers & Mathematics with Applications 1999,37(6):57–67. 10.1016/S0898-1221(99)00076-0

    Article  MathSciNet  MATH  Google Scholar 

  25. Ansari QH, Konnov IV, Yao JC: On generalized vector equilibrium problems. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):543–554. 10.1016/S0362-546X(01)00199-7

    Article  MathSciNet  MATH  Google Scholar 

  26. Chiang Y, Chadli O, Yao JC: Existence of solutions to implicit vector variational inequalities. Journal of Optimization Theory and Applications 2003,116(2):251–264. 10.1023/A:1022472103162

    Article  MathSciNet  MATH  Google Scholar 

  27. Pang J-S: Asymmetric variational inequality problems over product sets: applications and iterative methods. Mathematical Programming 1985,31(2):206–219. 10.1007/BF02591749

    Article  MathSciNet  MATH  Google Scholar 

  28. Cohen G, Chaplais F: Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms. Journal of Optimization Theory and Applications 1988,59(3):369–390. 10.1007/BF00940305

    Article  MathSciNet  MATH  Google Scholar 

  29. Bianchi M: Pseudo P-monotone operators and variational inequalities. In Report 6. Istituto di econometria e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, Italy; 1993.

    Google Scholar 

  30. Ansari QH, Yao JC: A fixed point theorem and its applications to a system of variational inequalities. Bulletin of the Australian Mathematical Society 1999,59(3):433–442. 10.1017/S0004972700033116

    Article  MathSciNet  MATH  Google Scholar 

  31. Ansari QH, Yao JC: Systems of generalized variational inequalities and their applications. Applicable Analysisl 2000,76(3–4):203–217. 10.1080/00036810008840877

    Article  MathSciNet  MATH  Google Scholar 

  32. Ansari QH, Schaible S, Yao JC: System of vector equilibrium problems and its applications. Journal of Optimization Theory and Applications 2000,107(3):547–557. 10.1023/A:1026495115191

    Article  MathSciNet  MATH  Google Scholar 

  33. Allevi E, Gnudi A, Konnov IV: Generalized vector variational inequalities over product sets. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):573–582. 10.1016/S0362-546X(01)00202-4

    Article  MathSciNet  MATH  Google Scholar 

  34. Peng JW: System of generalised set-valued quasi-variational-like inequalities. Bulletin of the Australian Mathematical Society 2003,68(3):501–515. 10.1017/S0004972700037904

    Article  MathSciNet  MATH  Google Scholar 

  35. Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.

    MATH  Google Scholar 

  36. Tian GQ, Zhou J: Quasi-variational inequalities without the concavity assumption. Journal of Mathematical Analysis and Applications 1993,172(1):289–299. 10.1006/jmaa.1993.1025

    Article  MathSciNet  MATH  Google Scholar 

  37. Su CH, Sehgal VM: Some fixed point theorems for condensing multifunctions in locally convex spaces. Proceedings of the American Mathematical Society 1975,50(1):150–154. 10.1090/S0002-9939-1975-0380530-7

    Article  MathSciNet  MATH  Google Scholar 

  38. Fan K: Fixed-point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences of the United States of America 1952,38(2):121–126. 10.1073/pnas.38.2.121

    Article  MathSciNet  MATH  Google Scholar 

  39. Kelley JL, Namioka I: Linear Topological Spaces. Springer, New York, NY, USA; 1963:xv+256.

    Book  Google Scholar 

  40. Michael E: A note on paracompact spaces. Proceedings of the American Mathematical Society 1953,4(5):831–838. 10.1090/S0002-9939-1953-0056905-8

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian-Wen Peng.

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

Peng, JW., Zheng, XP. System of Generalized Implicit Vector Quasivariational Inequalities. J Inequal Appl 2007, 036845 (2008). https://doi.org/10.1155/2007/36845

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/36845

Keywords