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Generalized vector quasi-variational-like inequalities
Journal of Inequalities and Applications volume 2006, Article number: 59387 (2006)
Abstract
Using maximal element theorem, we prove some existence theorems for the two types of generalized vector quasi-variational-like inequalities with non-monotonicity and non-compactness.
References
Ansari QH: A note on generalized vector variational-like inequalities. Optimization 1997,41(3):197–205. 10.1080/02331939708844335
Ansari QH: Extended generalized vector variational-like inequalities for nonmonotone multivalued maps. Annales des Sciences Mathématiques du Québec 1997,21(1):1–11.
Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, New York; 1984:xi+518.
Bianchi M, Hadjisavvas N, Schaible S: Vector equilibrium problems with generalized monotone bifunctions. Journal of Optimization Theory and Applications 1997,92(3):527–542. 10.1023/A:1022603406244
Chadli O, Chiang Y, Huang S: Topological pseudomonotonicity and vector equilibrium problems. Journal of Mathematical Analysis and Applications 2002,270(2):435–450. 10.1016/S0022-247X(02)00079-3
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
Chen GY, Cheng GM: Vector Variational Inequalities and Vector Optimization, Lecture Notes in Economics and Mathematical Systems. Volume 285. Springer, Berlin; 1987.
Chen GY, Craven BD: Approximate dual and approximate vector variational inequality for multiobjective optimization. Australian Mathematical Society. Journal. Series A. 1989,47(3):418–423. 10.1017/S1446788700033139
Chen GY, Craven BD: A vector variational inequality and optimization over an efficient set. Zeitschrift für Operations Research 1990,34(1):1–12.
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
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
Daniilidis A, Hadjisavvas N: Existence theorems for vector variational inequalities. Bulletin of the Australian Mathematical Society 1996,54(3):473–481. 10.1017/S0004972700021882
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
Ding XP, Tarafdar E: Generalized vector variational-like inequalities with-pseudomonotone set-valued mappings. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optim. Appl.. Volume 38. Edited by: Giannessi F. Kluwer Academic, Dordrecht; 2000:125–140. 10.1007/978-1-4613-0299-5_9
Ding XP, Tarafdar E: Generalized vector variational-like inequalities without monotonicity. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optim. Appl.. Volume 38. Edited by: Giannessi F. Kluwer Academic, Dordrecht; 2000:113–124. 10.1007/978-1-4613-0299-5_8
Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems (Proc. Internat. School, Erice, 1978). Edited by: Cottle RW, Giannessi F, Lions J-L. John Wiley & Sons, Chichester; 1980:151–186.
Hadjisavvas N, Schaible S: From scalar to vector equilibrium problems in the quasimonotone case. Journal of Optimization Theory and Applications 1998,96(2):297–309. 10.1023/A:1022666014055
Kim WK: Existence of maximal element and equilibrium for a nonparacompact-person game. Proceedings of the American Mathematical Society 1992,116(3):797–807.
Konnov IV, Yao JC: On the generalized vector variational inequality problem. Journal of Optimization Theory and Applications 1997,206(1):42–58.
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
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
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
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
Lin LJ, Yu ZT, Kassay G: Existence of equilibria for multivalued mappings and its application to vectorial equilibria. Journal of Optimization Theory and Applications 2002,114(1):189–208. 10.1023/A:1015420322818
Oettli W: A remark on vector-valued equilibria and generalized monotonicity. Acta Mathematica Vietnamica 1997,22(1):213–221.
Oettli W, Schläger D: Existence of equilibria for monotone multivalued mappings. Mathematical Methods of Operations Research 1998,48(2):219–228. 10.1007/s001860050024
Peng JW: Equilibrium problems for-spaces. Mathematica Applicata (Wuhan) 1999,12(3):81–87.
Qun L: Generalized vector variational-like inequalities. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optim. Appl.. Volume 38. Edited by: Giannessi F. Kluwer Academic, Dordrecht; 2000:353–369.
Schaefer HH: Topological vector spaces, Graduate Texts in Mathematics. Volume 3. Springer, New York; 1971:xi+294.
Siddiqi AH, Ansari QH, Khaliq A: On vector variational inequalities. Journal of Optimization Theory and Applications 1995,84(1):171–180. 10.1007/BF02191741
Su CH, Sehgal VM: Some fixed point theorems for condensing multifunctions in locally convex spaces. Proceedings of the American Mathematical Society 1975, 50: 150–154. 10.1090/S0002-9939-1975-0380530-7
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
Yang XQ: Generalized convex functions and vector variational inequalities. Journal of Optimization Theory and Applications 1993,79(3):563–580. 10.1007/BF00940559
Yang XQ: Vector complementarity and minimal element problems. Journal of Optimization Theory and Applications 1993,77(3):483–495. 10.1007/BF00940446
Yang XQ: Vector variational inequality and its duality. Nonlinear Analysis 1993,21(11):869–877. 10.1016/0362-546X(93)90052-T
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
Yu SJ, Yao JC: On vector variational inequalities. Journal of Optimization Theory and Applications 1996,89(3):749–769. 10.1007/BF02275358
Zhou J, Chen G: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. Journal of Mathematical Analysis and Applications 1988,132(1):213–225. 10.1016/0022-247X(88)90054-6
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Peng, JW., Yang, XM. Generalized vector quasi-variational-like inequalities. J Inequal Appl 2006, 59387 (2006). https://doi.org/10.1155/JIA/2006/59387
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DOI: https://doi.org/10.1155/JIA/2006/59387