Open Access

System of Generalized Implicit Vector Quasivariational Inequalities

Journal of Inequalities and Applications20082007:036845

DOI: 10.1155/2007/36845

Received: 14 February 2007

Accepted: 5 October 2007

Published: 29 January 2008

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]

Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Chongqing Normal University
(2)
College of Economics and Management, Beijing University of Chemical Technology

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/BF00940320MathSciNetView ArticleMATHGoogle 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-PMathSciNetView ArticleMATHGoogle 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/S1446788700033139MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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/BF02191741MathSciNetView ArticleMATHGoogle Scholar
  8. Yang XQ: Vector complementarity and minimal element problems. Journal of Optimization Theory and Applications 1993,77(3):483–495. 10.1007/BF00940446MathSciNetView ArticleMATHGoogle 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-TMathSciNetView ArticleMATHGoogle Scholar
  10. Yang XQ: Generalized convex functions and vector variational inequalities. Journal of Optimization Theory and Applications 1993,79(3):563–580. 10.1007/BF00940559MathSciNetView ArticleMATHGoogle Scholar
  11. Yu SJ, Yao JC: On vector variational inequalities. Journal of Optimization Theory and Applications 1996,89(3):749–769. 10.1007/BF02275358MathSciNetView ArticleMATHGoogle 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-ZMathSciNetView ArticleMATHGoogle 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)00099-2MathSciNetView ArticleMATHGoogle 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:1022640130410MathSciNetView ArticleMATHGoogle 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.1997.5192MathSciNetView ArticleMATHGoogle 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/S0004972700021882MathSciNetView ArticleMATHGoogle 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:1020844423345MathSciNetView ArticleMATHGoogle 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/s001860050024MathSciNetView ArticleMATHGoogle 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/BF02190001MathSciNetView ArticleMATHGoogle 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.1996.0401MathSciNetView ArticleMATHGoogle Scholar
  21. Ansari QH: A note on generalized vector variational-like inequalities. Optimization 1997,41(3):197–205. 10.1080/02331939708844335MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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_8View ArticleGoogle 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)00076-0MathSciNetView ArticleMATHGoogle 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)00199-7MathSciNetView ArticleMATHGoogle 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:1022472103162MathSciNetView ArticleMATHGoogle 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/BF02591749MathSciNetView ArticleMATHGoogle 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/BF00940305MathSciNetView ArticleMATHGoogle 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/S0004972700033116MathSciNetView ArticleMATHGoogle Scholar
  31. Ansari QH, Yao JC: Systems of generalized variational inequalities and their applications. Applicable Analysisl 2000,76(3–4):203–217. 10.1080/00036810008840877MathSciNetView ArticleMATHGoogle 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:1026495115191MathSciNetView ArticleMATHGoogle 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)00202-4MathSciNetView ArticleMATHGoogle 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/S0004972700037904MathSciNetView ArticleMATHGoogle Scholar
  35. Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics. John Wiley & Sons, New York, NY, USA; 1984:xi+518.MATHGoogle 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.1993.1025MathSciNetView ArticleMATHGoogle 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/S0002-9939-1975-0380530-7MathSciNetView ArticleMATHGoogle 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.38.2.121MathSciNetView ArticleMATHGoogle Scholar
  39. Kelley JL, Namioka I: Linear Topological Spaces. Springer, New York, NY, USA; 1963:xv+256.View ArticleGoogle Scholar
  40. Michael E: A note on paracompact spaces. Proceedings of the American Mathematical Society 1953,4(5):831–838. 10.1090/S0002-9939-1953-0056905-8MathSciNetView ArticleMATHGoogle Scholar

Copyright

© J.-W. Peng and X.-P. Zheng 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.